Selected Papers of K C Chou (20th Century Physics) (World Scientific Series 20th Century Physics)

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SE~ECTED PHPERS OF H. C. CHOU

World Scientific Series in 20th Century Physics Published

Vol. 22 A Quest for Symmetry -

Selected Works of Bunji Sakita

edited by K. Kikkawa, M. Virasoro and S. R. Wadia

Vol. 23 Selected Papers of Kun Huang (with Commentary) edited by B. -F. Zhu

Vol. 24 Subnuclear Physics - The First 50 Years: Highlights from Erice to ELN by A. Zichichi edited by O. Barnabei, P. Pupillo and F. Roversi Monaco Vol. 25 The Creation of Quantum Chromodynamics and the Effective Energy by V. N. Gribov, G. 't Hooft, G. Veneziano and V. F. Weisskopf edited by L. N. Upatov

Vol. 26 A Quantum Legacy - Seminal Papers of Julian Schwinger edited by K. A. Milton Vol. 27 Selected Papers of Richard Feynman (with Commentary) edited by L. M. Brown

Vol. 28 The Legacy of Leon Van Hove edited by A. Giovannini Vol. 29 Selected Works of Emil Wolf (with Commentary) edited by E. Wolf

Vol. 30 Selected Papers of J. Robert Schrieffer -

In Celebration of His 70th Birthday

edited by N. E. Bonesteel and L. P. Gor'kov

Vol. 31 From the Preshower to the New Technologies for Supercolliders of Antonino Zichichi edited by B. H. Wiik, A. Wagner and H. Wenninger

In Honour

Vol. 32 In Conclusion - A Collection of Summary Talks in High Energy Physics edited by J. D. Bjorken Vol. 33 Formation and Evolution of Black Holes in the Galaxy - Selected Papers with Commentary edited by H. A. Bethe, G. E. Brown and C.-H. Lee Vol. 35 A Career in Theoretical Physics, 2nd Edition by P. W. Anderson

Vol. 36 Selected Papers (1945-1980) with Commentary by Chen Ning Yang

Vol. 37 Adventures in Theoretical Physics - Selected Papers with Commentaries by Stephen L. Adler

Vol. 38 Matter Particled - Patterns, Structure and Dynamics - Selected Research Papers of Yuval Ne'eman edited by R. Ruffini and Y. Verbin

Vol. 39 Searching for the Superworld - A Volume in Honour of Antonino Zichichi on the Occasion of the Sixth Centenary Celebrations of the University of Turin, Italy edited by S. Ferrara and R. M. Mossbauer

Vol. 40 Murray Gell-Mann -

Selected papers

edited by H. Fritzsch

Vol. 41 Many-Body Theory of Molecules, Clusters, and Condensed Phases edited by N. H. March and G. G. N. Angilella

For information on Vols. 1-21, please visit http://www.worldscibooks.comlseries/wsscp_series.shtml

World Scientific Series in 20th Century Physics

Vol. 42

SE~ECTEO PRPERS OF

H. C. CHOU editor

Yue-Liang Wu Chinese Academy ofSciences, China

'~world Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING. SHANGHAI· HONG KONG· TAIPEI· CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

The editors and the publisher would like to thank the following publishers for their permission to reproduce the articles found in this volume American Institute of Physics American Physical Society Birhauser Verlag Chinese Physical Society Editorial office of "Progress of Theoretical Physics" Editorial office of "Journal of Experimental and Theoretical Physics" Elsevier Science in China Press Springer-Verlag

SELECTED PAPERS OF K C CHOU World Scientific Series in 20th Century Physics - Vol. 42 Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4280-37-2 ISBN-lO 981-4280-37-2

Printed in Singapore by World Scientific Printers

v

Foreword

Kuang-Chao Chou (Guang-Zhao Zhou) is a world-renowned theoretical physicist. Before I met him in the early nineteen seventies, I have already known his many significant research contributions for more than a decade from the late nineteen fifties. His published papers have won uniformly high praises by the international scientific community and his articles are always written with depth and elegance. Kuang-Chao was the originator of the helicity amplitude analysis. That concept was introduced by him in 1957, followed by a full mathematical analysis of this very effective tool to his work in high energy physics research. In 1959, he perfected an important theorem in T and P violations, which states that under CPT invariance, while T reflection in not conserved, the decay branching ratios of particle and antiparticle to different states can be different even though their total decay widths are the same. In addition to his work on resonances, Kuang-Chao also pioneered the use of nuclear absorption to detect the weak magnetism in weak interactions, as well as a large body of theoretical analysis on the important photo-nuclear reactions through the use of dispersion theory. In 1960, Kuang-Chao analytically demonstrated the partial conservation of axial current (PCAC), which has been regarded as one of the cornerstones in particle physics. For this fundamental contribution, he was recognized internationally as one of the founding fathers of PCAC. In the nineteen eighties after the Cultural Revolution, under the organization and guidance by Kuang-Chao, the younger generation of Chinese scientists were able to make many significant research works on grand unification theory, CP violation, non-linear sigma model effective Lagrangian theory, spontaneous symmetry breaking in super-symmetry, topological aspects of quantum field theory and its relation to anomaly. Many of these results (for example in topological aspects of quantum field theory and its relation to anomaly) have already attracted recognition from the international as well as Chinese physics communities. In statistical physics and condensed matter physics, the research group directed by Kuang-Chao systemized the Green's function formalism in non-equilibrium statistical mechanics. Their method was applied to the studies of laser, plasma, critical dynamics, random quenching, etc.

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In addition to the remarkable achievement made by Kuang-Chao in many fields of theoretical physics, he has been one of the most important scientists in the development of science and technology in China and to the promotion of the international science exchange and cooperation. Kuang-Chao Chou served as the director of Institute of Theoretical Physics at the Chinese Academy of Sciences, the Dean of the Science School of Tsinghua University, the President of Chinese Academy of Sciences, the Chairman of the China Association for Science and Technology, the Executive Vice-President of China Commission for Promoting International Science and Technology and the vice Chairman of China Association for Peace and Disarmament. He is an Academician of the Chinese Academy of Sciences and has also been elected as Foreign Associate of the US National Academy of Sciences, Fellow of the Third World Academy of Science, and Foreign Member of USSR Academy of Sciences, and the European Academy of Arts, Sciences and Humanities.

T

1>.

vii

Introduction

Zhou Guang-Zhao's scientific career is divided into several periods:

I. Early period (around 1955-1956). He was working in China. II. DUBNA period (1957-1960). Zhou was prolific in this period, publishing many papers in the JETP. I was in the USA at the time and had studied several of his papers, especially his work on PCAC. In the USA he was famous as the most brilliant young theorist in Dubna. III. National defence period (1961-1979). Zhou returned to China and ceased publication in scientific journals. But I had seen in a museum display in Qin-Hai Province the cover page of the design report of the first Chinese atomic bomb. It bears the names of Zhou and Deng Jia-Xian. Rumors say Zhou had made crucial contributions to that design, using analytic methods without the use of computers. IV. Later period (1979-around 1987). Zhou was at CERN, later at the VPI in the USA for several years, then back to China. In this period Zhou resumed his interest in academic research and was active in several frontier area of theoretical physics, often with young collaborators. V. Recent period (1987). Zhou is busy with Academy of Science matters after the late 1980s. But he still keeps up his interest in physics research.

I had first met Zhou at a dinner in Beijing hosted by Premier Zhou En-Lai in the early 1970s, almost forty years ago. During these forty years we have become close friends. I am deeply impressed by how he had transformed himself in the 1980s from a research physicist into an influential and highly respected policy maker and administrator. His success is partly rooted in his genes, of course, but I believe, also very much in his principled Chinese cultural background. Zhou is a first rate physicist: broad, powerful and very quick in grasping new ideas. His style of doing physics reminds me of that of Landau, Salam, and of Teller. But in personal relationship Zhou is a perfect Confucian gentleman, without

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the aggressive edges that characterize so many famous US, European and Russian theoretical physicists.

C. N. Yang

ix

Introduction

Professor Kuang-Chao Chou (Guang-Zhao Zhou) was born in Changsha, Hunan Province in 1929. Upon graduated from the Department of Physics at Tsinghua University in 1951, he enrolled as a postgraduate student in theoretical physics at Peking University where he became as a faculty member in 1954. He then immediately carried out several research works on nuclear physics and particle physics published in Chinese journals [lc,2c,3c]. In the period 1957-1960, Prof. Kuang-chao Chou worked as a researcher at the Joint Institute for Nuclear Research (JINR), Dubna in the Soviet Union, where he published more than 30 papers and made significant contributions in several important subjects concerning symmetries. His first paper at JINR was the issue on a symmetry property of the new Gell-Mann theory [1]. He then paid special attention to the spins and parities of particles [2,3,6], which enabled him to establish the relativistic theory of reactions for the polarized particles [7,8] and especially for the massless polarized particles [13], where he introduced, for the first time, the concept of helicity amplitude and its corresponding mathematical description, which has been shown to be very powerful in analyzing high energy scattering amplitude. He studied at the first time the particle-antiparticle asymmetry (CP violation) in the hyperon decays [10,14,23]' where he proved an important theorem in CP violation, i.e., when CPT is invariant but T reflection is not conserved, the decay branching ratios of particle and anti-particle to different states may be different although total decay widths are the same. He also noticed in the early time the possible symmetry properties for the 7l'-K system [22]. Its group symmetry was realized late on as the SU(3) symmetry by Cell-Mann in 1961. The renowned important paper written by Kuang-Chao Chou was on the pseudovector current and lepton decays of baryons and mesons [26], which has been an initial paper for proving the theorem of partial conservation of axial current (PCAC) in a simple and concise way, and it has been reputed to be an outstanding contribution in the research of hadron physics. He also conducted some interesting works concerning charge symmetry properties [11,18], and a series research works on dispersion relation, photo-nuclear reaction and scattering processes of mesons and hyperons [16,20,23-25,27,28,30-34,9c,10c,l1c]. He

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also calculated at the first time the mass difference between neutron and proton [4c]. With collaborators, he also studied the capture of muon in nucleus [5c,6c,7c,8c]. It was a very productive period for Prof. Kuang-chao Chao, his works immediately attracted special attentions worldwide, for instance, his most significant and fundamental work on PCAC [26] motivated the renowned paper on "Dynamical model of elementary particles based on analogy with superconductivity" by Y. Nambu and G. Jona-Lasinio (Phys. Rev. Vol. 122, No.1 and Vol. 124, No.1, 1961), for which Y. Nambu won the Nobel prize in 2008. Due to those remarkable achievements, Prof. Kuang-chao Chou was highly praised by the international scientific community and became a world-renowned theoretical physicist. The period 1960's was a golden time for particle physics. It was also the most creative period for Prof. Kuang-chao Chou carrying out the important initial works in the frontiers of particle physics. Nevertheless, it was in this important time, Prof. Kuang-chao Chou made an alternative and unusual choice that he decided to give up his research interest and the advanced working and living conditions at JINR when he learnt that the project of atomic Bomb in China met some hard problems in its first stage of design and all the relevant experts from Soviet Union had to leave China. In 1961, Prof. Kuang-chao Chou returned back to China and took part in the theoretical study and design of the atomic Bomb, which was known to be the most mysterious project. He then analyzed carefully all formalisms used in the computations and checked in detail all the numerical calculations. Eventually, he solved the hard problems caused in the design of atomic Bomb and made a decisive and right judgment based on the first physical principle. It was this breakthrough, made by Prof. Kuang-chao Chou, which played a crucial rule for the success of the first atomic Bomb (1964) in China. During the period 1961-1978, Prof. Kuang-chao Chou made important contributions in the development of atomic energy, applied physics and computational mathematics, which included the high temperature and high density physics, explosion mechanics, mechanics of radiative fluid, neutron physics, plasma physics and computational mechanics. In 1978, Institute of Theoretical Physics (ITP) was approved by the vice Premier Deng Xiao-ping to be established at the Chinese Academy of Sciences (CAS), Prof. Kuang-chao Chou joined the Institute of Theoretical Physics. In the period 19791987, he was able to work again on the frontiers of theoretical physics. In the early 1980s, he worked as a guest researcher and professor in the Virginia Polytechnic Institute, USA and the European Organization for Nuclear Research. Prof. Kuang-chao Chou made a number of important contributions in multitude research areas of theoretical physics. In the field of high energy particle physics, under his leadership, some significant achievements have been yielded at several interesting subjects, which included the gauge field theories [3541,43,45,49,52,12c,13c,14c,15c], nonlinear sigma models in symmetric coset and

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curved spaces [46,47,50,51,56,64,68]' U(l) anomaly and chiral dynamical theories [42,44,48,53,57,73]. The most important observations by Prof. Kuang-chao Chou and his collaborators were the gauge invariance and anomaly-free condition [58,16c,17c] of the Wess-Zumino-Witten effective action and the topological origin of gauge anomalies [60]. In the paper [59], it was firstly pointed out that Wittens effective Lagrangian for chiral field missed some important terms. Thus the topological properties of gauge fields were extensively explored in several papers [62,63,67,18c]. Late on, a simplified derivation of Chern-Simons cochain was realized [71] and it has been adopted to obtain the general Chern-Simons characteristic classes [72,19c,20c]. In particular, several interesting physical applications have been investigated, which included the derivation of the anomalous term in Virasoro algebra [69], the effective action of sigma model anomalous with external gauge fields [70], the possible origin of B-vacuum etc. At the same time, Prof. Kuang-chao Chou also paid special attention to the phenomenology of weak interactions. In particular, an interesting observation was made on the correlation between the top quark mass and direct CP violation in kaon decays [65]. As a consequence, a heavy top quark with mass much larger than W-boson mass was predicted and emphasized, and a fourth generation quark was also motivated [66] at that time. In addition to the gauge field theories and particle physics, Prof. Kuang-chao Chou (Guang-Zhao Zhou) has also made important contributions in the fields of statistical physics and condensed matter physics. With collaborators, the closed time path Green's functions has systematically been developed [79,80,21c,22c] and applied to the critical dynamics [81,23c,24c] and statistical physics [82,83,84,85,27c,28c]. A nice review article was published in Phys. Rep. 118, 1 (1985) which has widely been cited internationally. It has been shown that the method of closed time path Green's functions allows one to treat, in a unified way, both equilibrium and nonequilibrium systems. The method has also been applied to the random system [86,87,90] and the disorder electron system [88] as well as the Bohr-Sommerfield quantization for the fractional quantum Hall effect system [93]. The other interesting works conducted the spontaneous symmetry breaking [92] and time reversal invariance [95,25c,26c] in the nonequilibrium system as well as the influence functional [94]. In 1980s, Prof. Kuang-chao Chou has led, as the director of ITP-CAS, several key research directions at ITP, which concern: quantum field theory and particle physics, nuclear physics and plasma physics, statistical physics and condensed matter physics, gravity and astrophysics. For his outstanding contributions to theoretical physics, applied physics, atomic energy and basic research sciences in various aspects, and also for his great capability of leadership, in 1987, Prof. Kuang-chao Chou was pointed by the government to serve as the president of Chinese Academy of Sciences (CAS). Once again, he has to give up his own research interest and to

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make alternative contributions for leading the development of sciences and technology at the CAS. In 1996, Prof. Kuang-chao Chou was considering to return back to the scientific research. He presented in 1995 the opening address for the 17th International Lepton-Photon Conference which was regarded as the most influenced International Conference on particle physics held in Beijing. After that conference, he was actually thinking about the most hot topics in particle physics, which involved CP violation, origin of mass, neutrino physics, supersymmetric theories, low energy phenomena of grand unification theories [74-77] and a general discussion on CP and CPT violating observable in meson decays[78]. It has been shown that the 18 unknown quantities in the standard model of particle physics can well be predicted by a minimal set of 5 parameters in a class of supersymmetric grand unification model [74], about ten relations among CP violating-phase, masses and mixing angles of quarks and leptons have been obtained, which remain consistent with the experimental data. In 1997, Prof. Kuang-chao Chou would wish to realize his desire of doing theoretical research. Nevertheless, it was unexpected that he was elected to be the vice Chairman of Standing Committee of the National People's Congress of China and pointed again by the government to serve as the Chairman of the China Association for Sciences and Technology. Until 2006, he was honored as the honorary Chairman of the Chinese Association of Sciences and Technology. Since 2006, he has been invited to serve as the member of International Advisory Committee and the honorary Chairman of the Scientific Council for the Institute of Theoretical Physics at the Chinese Academy of Sciences. In this volume, the most papers written by Prof. Kuang-chao Chou (GuangZhao Zhou) are printed. All the collected papers have been arranged into four parts according to the subjects of research areas and the languages of publishing journals. Part I (in English) and Part III (in Chinese) are the papers on Field Theories, Particle Physics and Nuclear Physics, Part II (in English) and Part IV (in Chinese) are the papers on Statistical Physics and Condensed Matter Physics. From the published papers, it is seen how Prof. Kuang-chao Chou caught up the frontiers of theoretical physics in various periods and carried out the creative research works with initial ideas and motivations, as well as how he has worked in different key research directions of theoretical physics and made significant contributions to various interesting research subjects and interdisciplinary areas. Due to the outstanding research works and significant contributions, Prof. Kuang-chao Chou has earned numerous national and international awards which include: National Prize first order in Natural Science, Prize first order from Qiushi Foundation of HongKong, Gian Carlo Wick Commemorative gold medal from the World Federation of Scientists, the Chinese Meritorious Service Medals for Nuclear Scientists and Satellite Pioneers. He was elected in 1980 to be the Academician of

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Chinese Academy of Sciences. He has also been elected as Foreign Associate of the US National Academy of Sciences, Fellow of the Third World Academy of Science, and Foreign Member of USSR Academy of Sciences, Czechoslovak Academy of Sciences, Bulgarian Academy of Sciences, Romania Academy of Sciences, Mongolian Academy of Sciences, the European Academy of Arts, Sciences and Humanities, Membre fondateur Academie Francophone d'Ingenieurs. He has been honored to be the honorary PhD of several universities which include the City College of New York in USA, McGill University in Canada, Chinese University of Hong Kong. He also served as the Executive Vice-President of China Commission for Promoting International Science and Technology, the vice Chairman and honorary Chairman of China Association for Peace and Disarmament. He was conferred Commendatore dellOrdine Ai Merito Della Republica Italiana in 1993. The editor would like to thank Dr. Shan-Gui Zhou and Dr. Ling-Mei Cheng for their invaluable assistance in preparing this collection. The assistance of Ms. Lian-Zi Wei and Ms. Ji Zhang was also very helpful for completing and publishing this volume. I would like to thank Prof. K.K. Phua for providing many useful suggestions in publishing this volume. Yue-Liang Wu

K avli Institute for Theoretical Physics China Key Laboratory of Frontiers in Theoretical Physics Institute of Theoretical Physics, Chinese Academy of Sciences

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Contents

Foreword (T. D. Lee)

v

Introduction (C. N. Yang)

vii

Introduction (Y. L. Wu)

ix

Part I l.

2.

3.

4.

5.

6.

7.

8.

9.

Field theory, particle physics and nuclear physics

Concerning a symmetry property of the new Gell-Mann theory, Zh. Eksp. Teor. Fiz. 33 (1957) 1058. Soviet Physics JETP 6 (1958) 815. Spins and parities of the hyperfragment H\ and of the K meson, (with M. I. Shirokov), Zh. Eksp. Teor. Fiz. 33 (1957) 1072. Soviet Physics JETP 6 (1958) 828. Spins and parities of the AH4 hyperfragment and K-meson, (with M. I. Shirokov), Nuc!. Phys. 6 (1958) 10. On non-uniqueness of nucleon-nucleon scattering phase shifts, (with L. G. Zastavenko and R. M. Ryndin), Nuc!. Phys. 6 (1958) 669. Phase indeterminacies in nucleon-nucleon scattering, (with L. G. Zastavenko and R. M. Ryndin), Zh. Eksp. Teor. Fiz. 34 (1958) 526. Soviet Physics JETP 7 (1958) 363. On the determination of the relative parities of elementary particles, Zh. Eksp. Teor. Fiz. 34 (1958) 1027. Soviet Physics JETP 34 (1958) 710. The relativistic theory of reactions involving polarized particles, (with M. I. Shirokov), Zh. Eksp. Teor. Fiz. 34 (1958) 1230. Soviet Physics JETP 34 (1958) 85l. Selection rules in reactions involving polarized particles, Zh. Eksp. Teor. Fiz. 35 (1958) 783. Soviet Physics JETP 35 (1959) 543. The Shapiro integral transformation, (with L. G. Zastavenko), Zh. Eksp. Teor. Fiz. 35 (1958) 1417. Soviet Physics JETP 35 (1959) 990.

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10. A note on the decay of the ~:-hyperon and its antiparticle, Nuc!. Phys. 9 (1958/59) 652. 11. The universal Fenni interaction and the capture of muons in hydrogen, (with V. Maevskii), Zh. Eksp. Tear. Fiz. 35 (1958) 1581. Soviet Physics JETP 35 (1959) 1106.

12. Charge symmetry properties and representations of the extended Lorentz group in the theory of elementary particles, (with V. I. Ogievetskii), Zh. Eksp. Tear. Fiz. 36 (1959) 264. Soviet Physics JETP 36 (1959) 179. 13. Reactions involving polarized particles of zero rest mass, Zh. Eksp. Tear. Fiz. 36 (1959) 909. Soviet Physics JETP 36 (1959) 642. 14. Some symmetry properties in processes of antihyperon production with annihilation of antinucleons, Zh. Eksp. Tear. Fiz. 36 (1959) 938. Soviet Physics JETP 36 (1959) 663. 15. On the problem of investigating the interaction between 1t mesons and hyperons (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 37 (1959) 283. Soviet Physics JETP 37 (1960) 199. 16. Electromagnetic mass of the K meson, (with V. I. Ogievetskii), Zh. Eksp. Tear. Fiz. 37 (1959) 866. Soviet Physics JETP 37 (1960) 616. 17. Dispersion relations for the scattering of y quanta by nucleons, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 37 (1959) 1714. Soviet Physics JETP 37 (1960) 1213. 18. Charge symmetry properties and representations of the extended Lorentz group in the theory of elementary particles, (with V. I. Ogievetsky), Nuc!. Phys. 10 (1959) 235. 19. Integral transformations of the I. S. Shapiro type for particles of zero mass, (with L. G. Zastavenko), Zh. Eksp. Tear. Fiz. 38 (1960) 134. Soviet Physics JETP 11 (1960) 97. 20. Scattering of gamma-ray quanta by nucleons near the threshold for meson production, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 38 (1960) 201. Soviet Physics JETP 11 (1960) 147.

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21. On the production of an electron-positron pair by a neutrino in the field ofa nucleus, (with A. M. Badalyan), Zh. Eksp. Tear. Fiz. 38 (1960) 664. Soviet Physics JETP 11 (1960) 477. 22. Possible symmetry properties for the n-K system, Zh. Eksp. Tear. Fiz. 38 (1960) 1015. Soviet Physics JETP 11 (1960) 730. 23. On the decay of ~ hyperons, Zh. Eksp. Tear. Fiz. 38 (1960) 1342. Soviet Physics JETP 11 (1960) 966. 24. Dispersion relations and analysis of the energy dependence of cross sections near thresholds of new reactions, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 39 (1960) 112. Soviet Physics JETP 12 (1961) 82. 25. Inelastic final-state interactions and near-threshold singularities, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 39 (1960) 364. Soviet Physics JETP 12 (1961) 258. 26. On the pseudovector current and lepton decays of baryons and mesons, Zh. Eksp. Tear. Fiz. 39 (1960) 703. Soviet Physics JETP 12 (1961) 492. 27. Elastic scattering of gamma quanta by nuclei, (with L. I. Lapidus) Zh. Eksp. Tear. Fiz. 39 (1960) 1056. Soviet Physics JETP 12 (1961) 735. 28. The elastic scattering of y rays by deuterons below the pion-production threshold, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 39 (1960) 1286. Soviet Physics JETP 12 (1961) 898. 29. On the pion-pion resonance in the p-state, (with Ho Tso-Hsiu), Zh. Eksp. Tear. Fiz. 39 (1960) 1485. Soviet Physics JETP 12 (1961) 1032. 30. On the role of the single-meson pole diagram in scattering of gamma quanta by protons, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 41 (1961) 294. Soviet Physics JETP 14 (1962) 210. 31. Low-energy limit of the yN-scattering amplitude and crossing symmetry, (with L. I. Lapidus), Zh. Eksp. Tear. Fiz. 41 (1961) 491. Soviet Physics JETP 14 (1962) 352.

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32. On the K +N~A(~)+y process, (with L. I. Lapidus), Zh. Eksp. Teor. Fiz. 41 (1961) 1310. Soviet Physics JETP 14 (1962) 932. 33. Scattering of photons by nucleons, (with L. I. Lapidus), Zh. Eksp. Teor. Fiz. 41 (1961) 1546. Soviet Physics JETP 14 (1962) 1102. 34. A suggested experiment to determine the spin ofYl * and the parities of A-~, A-Y1 *, and ~-Y1 *, (with Su Zhao-bin and Gao Chong-shou), Scientia Sinica 12 (1963) 451. 35. The pure gauge fields on a coset space, (with Tu Tung-sheng and Yean Tu-Nan), Scientia Sinica 22 (1979) 37. 36. Soliton-soliton scattering and the semi-classical approximation for the problem of scattering in three-dimensional space, (with Dai Yuan-ben), Scientia Sinica 22 (1979) 281. 37. The non-topological soliton with a non-Abelian internal symmetry, (with Zhu Zhong-yuan, Dai Yuan-ben and Wu Yong-shi), Scientia Sinica 23 (1980) 40. 38. On the vacuum of the pure gauge fields on coset, Scientia Sinica 23 (1980) 431. 39. A model of electro-weak interaction in SU(3) X U(1) gauge theory, (with Gao Chong-shou), Scientia Sinica 23 (1980) 566. 40. Electro-weak theory in SU(3), (with Gao Chong-shou), Chinese Science Bulletin 25 (1980) 21. 41. Disorder parameter and duality, (with Xian Ding-chang), Chinese Science Bulletin 25 (1980) 635. 42. Axial U(l) anomaly and chiral symmetry-breaking in QCD, AlP. Conf. Proc. 72 (1981) 621. 43. Possible SU(4)cxSU(3)fXU(l) model, (with Chong-shou Gao), Phys. Rev. D 23 (1981) 2690. 44. On the quantization and the renormalization of the pure-gauge fields on the coset-space, (with Ruan Tu-nan), J Univ. Sci. Tech. China 11 (1981) 15. 45. The U(l) anomalous Ward identities and chiral dynamics, Chinese Science Bulletin 27 (1982) 147. 46. New non-linear cr model on symmetric spaces, (with Song Xing-chang), Commun. Theor. Phys. 1 (1982) 69. 47. The Hn sigma-models and self-dual SU(n) Yang-Mills fields in four-dimensional Euclidean space, (with Song Xing-chang), Commun. Theor. Phys. 1 (1982) 185.

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249 255 262

xix

48. On the dynamical problems and the Fermion spectra in the Rishon model, (with Dai Yuan-ben), Commun. Theor. Phys. 1 (1982) 725. 49. Wilson loop integral and string wave functional, (with Li Xiao-yuan), Scientia Sinica, Ser. A 25 (1982) 264. 50. Backlund transformation, local and nonlocal conservation laws for nonlinear cr models on symmetric coset spaces, (with Song Xing-chang), Scientia Sinica, Ser. A 25 (1982) 716. 51. Local conservation laws for various nonlinear cr models, (with Song Xing-chang), Scientia Sinica, Ser. A 25 (1982) 825. 52. Composite gauge bosons in a nonabelian theory, (with Swee-Ping Chi a and Charles B. Chiu), Phys. Lett. 109 B (1982) 457. 53. Massless baryons and anomalies in chiral (QCD)z, (with D. Amati and S. Yankielowicz), Phys. Lett. 110 B (1982) 309. 54. On the determination of effective potentials in supersymmetric theories, (with D. Amati), Phys. Lett. 114 B (1982) 129. 55. Koba-Nie1sen-Olesen scaling and production mechanism in high energy collisions, (with Liu Lian-sou and Men Ta-chong), Phys. Rev. D 28 (1983) 1080. 56. Nonlinear cr model on multidimensional curved space with certain cylindrical symmetry, (with Song Xing-chang), Commun. Theor. Phys. 2 (1983) 971. 57. Kac-Moody algebra for two dimensional principal chiral models, (with Song Xing-chang), Commun. Theor. Phys. 2 (1983) 1391. 58. On the gauge invariance and anomaly-free condition of the Wess-Zumino-Witten effective action, (with Guo Han-ying, Wu Ke and Song Xing-chang), Phys. Lett. 134B (1984) 67. 59. On Witten's effective Lagrangian for chiral field, (with Guo Han-ying, Wu Ke and Song Xing-chang), Commun. Theor. Phys. 3 (1984) 73. 60. The topological origins of Gauge anomalies, (with Guo Han-ying, Wu Ke and Song Xing-chang), Commun. Theor. Phys. 3 (1984) 125.

281

287 295

302

311

316

320

323

329

341

349

352

360

xx 61. On the dynamical symmetry breaking for the N = I pure supersymmetric 365 Yang-Mills model, (with Dai Yuan-ben and Chang Chao-hsi), Commun. Theor. Phys. 3 (1984) 221. 62. The unified scheme of the effective action and chiral anomalies in any even 374 dimensions, (with Han-ying Guo, Xiao-yuan Li, Ke Wu and Xing-chang Song), Commun. Theor. Phys. 3 (1984) 491. 382 63. Symmetric and asymmetric anomalies and effective Lagrangian, (Han-ying Guo, Ke Wu and Xing-chang Song), Commun. Theor. Phys. 3 (1984) 593. 64. On the two dimensional non-linear cr model with Wess-Zumino term (I) 393 classical theory, (with Dai Yuan-ben), Commun. Theor. Phys. 3 (1984) 767. 65. CP violation from the standard model, 397 Proceedings of Europhysics Topical Conference on Flavor Mixing in Weak Interactions, (Ettore Majorana International Science Series: Physical Sciences, Vol. 20), 609. 424 66. Top quark mass and the fourth generation of quark, (with Wu Yue-liang and Xie Yan-bo), Chinese Physics Letters 1 (1984) 47. 67. Anomalies of arbitrary gauge group and the its reduction group, Einstein and 426 Lorentz anomalies, (with Han-ying Guo and Ke Wu), Commun. Theor. Phys. 4 (1985) 91. 68. On the two dimensional non-linear cr model with Wess-Zumino term (II) 437 quantum theory, (with Dai Yuan-ben), Commun. Theor. Phys. 4 (1985) 123. 69. Derivation of the anomalous term in Virasoro algebra by topological method, 442 (with Wu Yue-liang and Xie Yan-bo), Commun. Theor. Phys. 5 (1986) 359. 70. Effective action of sigma model anomalous with external gauge fields, 448 (with Yue-liang Wu and Yan-bo Xie), Modern Phys. Lett. A 1 (1986) 23. 71. A simplified derivation of Chern-Simons co chain and a possible origin of 453 9-vacuum term, (with Wu Yue-liang and Xie Yan-bo), Commun. Theor. Phys. 7 (1987) 27. 72. The general Chern-Simons characteristic classes and their physical 464 applications, (with Wu Yue-liang and Xie Yan-bo), Commun. Theor. Phys. 8 (1987) 341. 73. Signature for chiral-symmetry breaking at high temperatures, 491 (with Lay-nam Chang and Ngee-pong Chang), Phys. Rev. D 43 (1991) 596.

xxi

74. CP violation, fermion masses and mixings in a predictive SUSY S0(10)x~(48)xU(1) model with small tanf3, (with Y. L. Wu), Phys. Rev. D 53 (1996) R3492. 75. Low energy phenomena in a model with symmetry group SUSY S0(10)x~(48)xU(1), (with Wu Vue-liang), Science in China, Ser. A, 39 (1996) 65. 76. A solution to the puzzles of CP violation, neutrino oscillation, fermion masses and mixings in an SUSY GUT model with small tanj3, (with Vue-liang Wu), Nuc!. Phys. B (Proc. Supp!.) 52A (1997) 159. 77. A possible unification model for all basic forces, (with Wu Vue-liang), Science in China, Ser. A, 41 (1998) 324. 78. Searching for rephase-invariant CP- and CPT-violating observables in meson decays, (with W. F. Palmer, E. A. Paschos and Y. L. Wu), Euro. Phys. J C 16 (2000) 279.

Part II

Statistical physics and condensed matter physics

79. Renormalization of the closed time path Green's functions in nonequilibrium statistical field theory, (with Su Zhao-bing), Chin. Phys. 1 (1981) 635. 80. Dyson equation and Ward-Takahashi identities of the closed time path Green's function, (with Su Zhao-bing), Chin. Phys. 1 (1981) 645. 81. Closed time path Green's functions and critical dynamics, (with Zhao-bin Su, Bai-lin Hao and Lu Yu), Phys. Rev. B 22 (1980) 3385. 82. On theory of the statistical generating functional for the order parameter (1) general formalism, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Commun. Theor. Phys. 1 (1982) 295. 83. On theory of the statistical generating functional for the order parameter (II) density matrix and the field-theoretical structure of the generating functional, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Commun. Theor. Phys. 1 (1982) 307. 84. On theory of the statistical generating functional for the order parameter (III) effective action formalism for the order parameter, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Commun. Theor. Phys. 1 (1982) 389.

497

501

514

518 524

533

535

545

559

582

594

606

xxii

85. On an approximate form of the coupled equations of the order parameter with the weak electromagnetic field for the ideal superconductor, (with Su Zhao-bin), Commun. Theor. Phys. 1 (1982) 669. 86. On a dynamic theory of quenched random system, (with Su Zhao-bin and Yu Lu), Commun. Theor. Phys. 2 (1983) 1181. 87. A dynamical theory of the infinite range random Ising model, (with Su Zhao-bin and Yu Lu), Commun. Theor. Phys. 2 (1983) 1191. 88. A dynamical theory of random quenched system and its application to infinite-ranged Ising model, (with Su Zhao-bin and Yu Lu), Proceedings of the First Asia-Pacific Physics Conference, (World Scientific Pub. Co.), 474. 89. Symmetry and Ward identities for disordered electron system, (with Lin Jian-cheng and Shen Yu), Commun. Theor. Phys. 3 (1984) 139. 90. Does Parisi's solution of the Sherrington-Kirkpatrick model locate on the absolute maximum of the free energy? (with Shen Yu), Commun. Theor. Phys. 3 (1984) 263. 91. Equilibrium and nonequilibrium formalisms made unified, (with Zhao-bin Su, Bai-lin Hao and Lu Yu), Phys. Rep. 118 (1985) 1. 92. Spontaneous symmetry breaking and Nambu-Goldstone mode in a non-equilibrium dissipative system, (with Zhao-bin Su), Prog. Thea. Phys. Supp. 86 (1986) 34. 93. The canonical description and Bohr-Sommerfeld quantization for the fractional quantum Hall effect system, (with Zhao-bin Su, Han-bin Pang and Yan-bo Xie), Phys. Lett A 123 (1987) 249. 94. Influence functional and closed-time-path Green's function, (with Zhao-bin Su, Liao-yuan Chen and Xiao-tong Yu), Phys. Rev. B37 (1988) 9810. 95. Time reversal invariance and its application to nonequilibrium stationary states, (with Z. B. Su), Thirty Years Since Parity Nonconservation: A Symposium for T. D. Lee, (Birkhauser Verlag, 1988) 117.

621

632

641

652

667

677

682

812

821

826

829

xxiii

Part III

1.

~~, *-\LT~J:I~~~J:I Field theory, particle physics and nuclear physics

xXfJflH*TjJ8~i:1it,

845

847

*JE!'¥:tlt, 1955 fFm 11 ~m 4 ,jt)jm 299-316 In. A discussion on two-range nuclear force, Acta Physica Sinica 11 (1955) 299 (in Chinese). 2.

3.

4.

5.

6.

7.

8.

il'T:J:h~ ~ -1*jJ1ft~~,

865

~t~:A'¥'¥:t~ § ~f4~AA, 1955 fFm 1 Mm 53-66 In. Three-body potentials in meson fields, Acta Scicentiarurn Natura/urn Universitis Pekinesis 1 (1955) 53 (in Chinese). E8 Vlix.9J EEt. ~~J 1JflHll* 1~ x.9J EEt. ~:J:h 1Jf¥~ll ~ 1J¥t;:, ~t*:k'¥'¥* §~H'¥Jt&, 1956 fFm 2 Mm 211-218 In. Derivation of Maxwell equations from the microscopic Lorentz equations, Acta Scicentiarurn Natura/urn Universitis Pekinesis 2 (1956) 211 (in Chinese). ~T*DJPJ:T~JPJ::!i&, *JI=Jj['¥tlt, 1959 fFm 15 ~m 5 ,jt)jm 269-276 In. Mass difference between neutron and proton, Acta Physica Sinica 15 (1959) 269 (in Chinese). ~~ffl*~~t9]i 1'FJ:IHJIHtEqlil'TfrJffi: Tt~.L~1*gjt, (~ B. lbIf~1I:1JT~), 4PJE! '¥: tit , 1959~ 15~m7WHfB77 -388In. The universal Fermi theory of weak interaction and the capture of muon in atomic nuclei, (with B. Maevskii), Acta Physica Sinica 15 (1959) 377 (in Chinese). 3 11 il'Tfr He Jffi: T~.L~l$gjt, (~ **~, ~)*~), W£!~tlt, 1960 fFm 16 ~m 2 WHj~ 61-69 In. Muon capture in He 3 nucleus, (with Chu Chia-chen and Peng Hong-an), Acta Physica Sinica 16 (1960) 61 (in Chinese).

*

~TIl1¥gjt~lf?X:;z)JjHJF:I:~l31JilJN~t1%, (~ ~~'T), W£!~tlt, 1960 fFm 16 ~m 2 Mm 70-75 In. On the sign of the effective pseudoscalar term in 11 capture reactions, (with Huang Nien-ning), Acta Physica Sinica 16 (1960) 70 (in Chinese). 11 il'T~~Jffi:Tt*~r:Utst~&fP~:;Z&, (~ ~j[;:L1'q, W;E!~tlt, 1960 ~m 16 ~m 2 ,jt)j~ 76-80 In. The effect of recoil on the scattering of Il-mesons by light nuclei, (with Dai Yuan-ben), Acta Physica Sinica 16 (1960) 76 (in Chinese).

879

887

895

907

916

922

xxiv

7t frT~Tii~tiF~ 7t frTEt<J~1[t~~~, (~ ~7c*), ~W,¥;J=It, 1960 if 16 ~~ 5 :JtJ1~ 252-262 1ft. Dispersion relations for pion production in pion-nucleon collisions, (with Dai Yuan-ben), Acta Physica Sinica 16 (1960) 252 (in Chinese). 10. {ft~ 7t frT1tY:~tttflt~1HjtE p ~;t!;*Et<J-1'-~~1J¥tEt<JJti5(,

9.

(~ 1PJf'FR*), ~W,¥;J=It, 1961

938

17 ~tg 3 :JtJ1tg 133-134 1ft. A suggestion of experiment to detect the p-resonance in 7t-7t scattering, (with Ho Tso-Hsiu), Acta Physica Sinica 17 (1961) 133 (in Chinese). 11. 7t+p- A + n +K &L\IZ~*-JliI1XJ~j(L\IZ~®(Et<Jli}f~, (~ yj:*iJj(, r'2b*%), 940 ~W,¥;J=It, 1963 iFm 19 ~tg 10 WU!~ 649-672 JJr. Investigation of the correlation between resonance effect and near threshold effect in the 7t+p~A+7t+ K reaction, (with Su Zhao-bin and Gao Chong-shou), Acta Physica Sinica 19 (1963) 649 (in Chinese). 12. ~;J;JIlmM*Q,*;J;JIlmMEt<Jt§li 1'FmZ.rar1~Jxtf$tt~? (~ r\§J*%), 964 r\§J~Mmw~t~~w, 1980 iF~ 4 ~~ 4 M~ 536-538 1ft. Is there any symmetry between flavour and colour gauge interactions? (with Gao Chong-shou), Phys. Energ. Forties Phys. Nuc!. 4 (1980) 536 (in Chinese). 13. SU(3)9=' Et<J~~ It~-*~lli!, (~ r\§J*%), 967 r\§J~Mmw~t~~w, 1980 iFm 4 ~m 5 Mtg 609-622 1ft. Unified electro-weak model in SU(3), (with Gao Chong-shou), Phys. Energ. Forties Phys. Nucl. 4 (1980) 609 (in Chinese). 14. ~~q£f5{7t:m:T1tEt<J-fl19:%A, (~ J3'l'J~t~, ~jC[!]WJ, t±*~), 981 9='OO;r4'¥11*::k,¥,¥;J=It, 1980 iFtg 10 ~tg 2 Mtg 26-35 1ft. A general form of path integral quantization, (with Yin Hong-jun, Ruan Tu-nan and Du Dong-sheng), J. Univ. Sci. Tech. China 10 (1980) 26 (in Chinese). 15. 3tf" SU(3)XU(1)$£T;rQ~R:~~ 1t~-*jUt~Et<J-®i1i~, (~ ~*%), 991 r\§J~~~:EJIl.~~~W, 1981 iF~ 5 ~~ 1 :JtJ1~ 39-46 1ft. Some discussions on the SU(3)xU(l) electro-weak unified model ofleptons and quarks, (with Gao Chong-shou), Phys. En erg. Forties Phys. Nucl. 5 (1981) 39 (in Chinese). 16. Wess-Zumino-Witten 1f~j(1'Fm:m:Et<J;J;JIlm/G~tt, 999 (~ $~¥J.~, ~-ar, *1]'*), r\§J~~~W~~~W, 1984 iFtg 8 ~tg 2 M~ 252-255 1ft. On the gauge invariance ofWess-Zumino-Witten effective action, (with Guo Han-ying, Wu Ke and Song Xing-chang), Phys. Energ. Forties Phys. Nuc!. 8 (1984) 252 (in Chinese).

*-T

iFm

927

xxv 17.

1~xtf$'&'ffi;fI]1f~j(1iLf\;:I:, (l::j !J~¥J(~, ~PJ, J'i'ij~MmlIl::j~4mlJ. 1984:tFm 8 ~~ 4 :ltJ1~ 508-512 ~.

xtf$JX'ffi,

*11'*),

Symmetrical anomaly, unsymmetrical anomaly and effective Lagrangian, (with Guo Han-ying, Wu Ke and Song Xing-chang), Phys. Energ. Forties Phys. Nuc!. 8 (1984) 508 (in Chinese). 18. {± ~ 1~ ~1t at 2 a"rf=r ~j( 1-'F ffl ;fl] ~tiI.& 'ffi , (l::j .$~¥J(~, */J\~, ~PJ, *11'*), J'i'ijIlMmlIl::j~4mlI, 1985 :tFm 9 ~m 2 M~ 252-256 ~. Effective action and chiral anomalies in any even dimensional space, (with Guo Han-ying, Li Xiao-yuan, Wu Ke and Song Xing-chang), Phys. Energ. Forties Phys. Nuc!. 9 (1985) 252 (in Chinese). 19. tl'-iJ!Hl"J Chern-Simons lJ!;f1]'f::1fJi¥J@ffl, (l::j ~-ffi~, i~f~~), J'i'ij~~4mL!l::jt~4mlI, 1986 :tFm 10 ~m 2 177-185 ]}:t. The general Chern-Simons cochain and their application, (with Wu Vue-liang and Xie Yan-bo), Phys. Energ. Forties Phys. Nucl. 10 (1986) 178 (in Chinese). 20. -t~:8'JTi¥JLJll~":-TbZ.;1t@ffl, (l::j ~-ffi~, i~f~~), i\N:i~~4m1Il::j~4m1I, 1986:tFm 10 ~~ 3 M~ 283-287 ~. A new co-boundary operator and its applications, (with Wu Vue-liang and Xie Yan-bo), Phys. Energ. Forties Phys. Nuc!. 10 (1986) 283 (in Chinese).

Mm

Part IV

~jHt-~l1E.!:§i;~Hf~*~l1E

1004

1009

1014

1023

1029

Statistical physics and condensed matter physics

21. l~fijffJlX*~JE'ffi~i¥J Goldstone t~, (l::j ~*(JJ<), 4m1I'¢1~, 1980 :tF~ 29 ~m 5 M~ 618-634~. On the Goldstone mode in the stationary state of a non-equilibrium dissipative system, (with Su Zhao-bing), Acta Physica Sinica 29 (1980) 618 (in Chinese). 22. =~ffl~·M~**fEJ2J&i¥J5e~~*, (l::j -T~, ~;fat*), 4m1I'¢1~, 1980 :tF~ 29 ~m 7 M~ 878-888 ~. Transformation properties of three sets of closed time path Green's functions, (with Yu Lu and Hao Bai-lin), Acta Physica Sinica 29 (1980) 878 (in Chinese). 23. 1~fij~ittmi1::l::jII{ijJP9JjJ~ (I) ) )(~)tZ.. JJ1J*¥, (l::j ~jf{JJ<, ~M;f*, -T~), 4m1I~1~, 1980 :tF~ 29 ~m 8 :ltJ1~ 961-968 ~. Nonequilibrium statistical field theory and critical dynamics (I) generalized Langevin equation, (with Su Zhao-bin, Hao Bai-lin and Yu Lu), Acta Physica Sinica 29 (1980) 961 (in Chinese).

1031

1048

1059

xxvi

24. ~F-'jZ1.tjtJEit:J:~i'2"lJll{£WZ9JjJ* (II) ifLlxJ~i'2"*:ii, (lJ #1318*,*, T~), 1ml-!*ilt, 1980 ~m 29 ~m 8 969-977 !n. Nonequilibrium statistical field theory and critical dynamics (II) Lagrangian field theory formulation, (with Hao Bai-lin and Yu Lu), Acta Physica Sinica 29 (1980) 969 (in Chinese). 25. atrS]&.mx;H$*O~F-'jZjj#Eit!E'ffi~ (I), (lJ m:*il]<), 1mlJ.**, 1981 ~m 30 ~m 2 164-171 !n. Time reversal symmetry and non-equilibrium statistical stationary states (I), (with Su Zhao-bin), Acta Physica Sinica 30 (1981) 164 (in Chinese). 26. atra]&.mxtt$*o~F-'jZ1.tJ#EitJE'ffi~(II), (lJ m:*i7]<), 1mlJ.~ilt, 1981 ~m 30 ~m 3 401-409 !n. Time reversal symmetry and non-equilibrium statistical stationary states (II), (with Su Zhao-bin), Acta Physica Sinica 30 (1981) 401 (in Chinese). 27. !*-Tt~m*$1**#E-'jZ~:J:~:ili1l;J.~ffEl, (lJ m:*il]<, T~), 1ml!*ilt, 1984 ~m 33 ~m 7 999-1007 !n. The generalized mean field expansions for many Fermion systems, (with Su Zhao-bing and Yu Lu), Acta Physica Sinica 33 (1984) 999 (in Chinese). 28. ff~~-#cit~if*PlJ~f~il1itJUll, (Ej m:ffil]<, T~), 1mlJ.*ilt, 1984 ~m 33 ~m 6 M 805-813 In. On a set of coupled equations for the order parameters-statistical Green's functions, (with Su Zhao-bin and Yu Lu), Acta Physica Sinica 33 (1984) 805 (in Chinese).

:!tAm

Mm

:!t)jm

Mm

1067

1076

1084

1093

1102

Part I Field theory, particle physics and nuclear physics

This page intentionally left blank

3 LETTERS TO THE EDITOR

815

CONCERNING A SYMMETRY PROPERTY OF THE NEW GELL-MANN THEORY CHOU GUAN-CHAO Joint Institute for Nuclear Research Submitted to JETP editor July 6. 1957 J. Exptl. Theoret. Phys. (U .B.S.R. ) 33. 1058-1059 (October. 1957) RECENTLY Gell-Mannt has proposed a new theory of the interaction between elementary particles. In this theory all baryons have spin tit. and the same mechanical mass and parity. They form a supermultiplet which is split up only when moderately-strong interactions occur by way of K mesons. The Hamiltonian of the interaction With 11' mesons is written in the folloWing way: H"

= ig {(Pr,p -

+ (f+T,E+ -

V2 (PT.n,,+ + nT.p"-) + (2'1.:=:0 - 2-T.:=:-) r." + V2 (EoT.:=:-,,+ +.;. T.:=:o,.-) V2 (f+T.Yo,,+ + YOT,E+,,-)+ (ZOT,lO - f-T,E-) ,,0 + V2 (ZOT,E-,,+ + f-T,lO,,-)}.

nT.n) r." +

YOT,YO) r." +

(1)

(see Ref. 1 for the notation). In our note we shall show that in addition to the so-called global symmetry discussed by Gell-Mann. there is another symmetry property of his theory even in the absence of K coupling. In the Hamiltonian (1) we make the substitutions

where Ac denotes the charge-conjugation operator of the field of particle A. After this substitution the Hamiltonian takes the form H" = ig{(~T•.:E:!-a;T.:=:;)"O + V2(.E;T•.:E:!,,+ +~TA"-)+ (PeT,Pe -neT.ne)r.o + V2(ne T.Pe'lt+ + p;r.ne,,-)

+ (Z:T.~ -E;-T.E;)r:o + V2(E;-T.~,,+ + Z~T.E;1t-)+ (EtT.E:- - Y:T.Y~) """ + V2(Y~T.Et,,+ + 1:tT.Y~"-)}'

(3)

The follOWing relations exist between the field operators A and B:I

At! =

BeAto ifT~B = - BeT~Ato

AT~I.B

= - BeT~T.Ae. AT.I~B = B.T.T~Ato

AT.B = BeT.Ae.

(4)

With the help of (4) we can satisfy ourselves that the Hamiltonian (3) agrees with (1). In the same way it can be shown that the Hamiltonian of the interaction of baryons with photons (5)

is invariant under the substitution (2). with the addition of the substitution ~ -~. This symmetry property of the Gell-Mann theory provides us with selection rules for certain processes. Let us consider processes in which there are only 11' mesons and photons in the initial and final states. For every Feynman diagram there will be another diagram in which the lines of the inner baryons are replaced by the propagation lines of the baryons appearing in the right-hand sides of the substitution (2). For example. (i.e •• by the lines proton lines are replaced by the lines of the baryon described by the field operator of an anti S --particle). antiproton lines by the lines of a S - -particle. etc. If the number of outer 11'0 mesons is odd. then the matrix elements of these two graphs will cancel each other when they are added. Consequently in the new Gell-Mann theory the process of the decay of a 11'0 meson into two 1'-rays takes place only through a K interaction. If it is assumed that fJ interactions of baryons also have Gell-Mann's global symmetry. then we can obtain a few other selection rules. Consider. for example, a tensor fJ interaction:

So

H, = g~ {Pr~T.n + EOT~T.E-

+ f+T~T.Yo +

ZOT~T.E-}eT~T."

It is clear that this Hamiltonian changes sign under the substitution (2). Thus the matrix elements for the fJ-decay of a 11'+ meson. that is

through a virtual baryon-pair. cancel each other in pairs. These processes also are allowed only through a K interaction.

4 816

If only the interactions of

LETTERS TO THE EDITOR 11"

mesons with nucleons are taken into account, then the theoretical proba-

bilities of the processes ,,0 -+

2, (ReI. 3) and ..+ -> e+

+ +", 'I

(Ref. 4),

obtained from perturbation theory, are greater than those observed. The considerations above indicate that it is possible to reduce the discrepancy between theory and experiment by allowing also for the interactions of 11" mesons with all baryons.* Of course. without a study of the interaction with a K meson we are still unable to say with certainty that the new Gell-Mann theory gives better agreement with the experimental data for these processes. In conclusion I wish to thank Professor M. A. Markov, Professor Khu Nin, V.!. Ogievetskii, and M. I. Shirokov for their interest and for discussions of the work presented here. *This same result for the decay process sult was obtained earlier by Kinoshita.'

11"0 -

2-y was also obtained by Gell-Mann. t An analogous re-

1M. Gell-Mann, Phys. Rev. 106, 1296 (1957). 2R. H. Good, Jr., Rev. Mod. Phys. 27, 187 (1955). 3 T. Kinoshita, Phys. Rev. 94, 1384 (1954). 'So B. Treiman and H. W. Wyld, Jr., Phys. Rev. 101, 1552 (1956). Translated by W. M. Whitney 213

5 828

LETTERS TO THE EDITOR

SPINS AND PARITIES OF THE HYPERFRAGMENT Hi AND OF THE K MESON CHOU GUAN-CHAO and M. 1. SffiROKOV Joint Institute for Nuclear Research Submitted to JETP editor July 26. 1957 J. Exptl. Theoret. Phys. (U.S.S.R.) 33. 1072-1073 (October. 1957) IN this note we shall give a summary of the angular correlations in the cascades:

,,-+ He'~

H~

K- + He

+ K.lI~~ He'+". K~ ,,+ ,,;

t _

H~

+ n, H~ -

He·

(1 )

+ '/'C

(2 )

for several variants of spins in parities of H~ and K. We assume at first that the spin of the K meson is zero. If in cascade (1) the first reaction takes place near the threshold (kinetic energy of the '/1'- mesons in the laboratory is 620 to 640 Mev). then it is natural to assume that H' and K are formed predominantly in the s state if 'If (the product of the parities of '/1'-. He'. an~ K) is (- l)i (i is the spin of the hyperfragrnent), and in the p state if Tr = (- 1)& (in this case the production in the s state is forbidden). The distribution about the angle 'Y between the direction of the incident pions and the Spin variant. decay products (in the system where H~ is at Parity variant rest). given in the table. has been obtained under these assumptions. As can be seen. it is possible in principle to determine not only i, but also the prod"=(-1)' ucts of the parities of ~ and K (assuming that the product of the parities of the '/1'- and He' is - 1 ). "=-(-1)' Cascade (1) contains three reactions, and it therefore permits also the determination of the spin of the K meson by the Adair method I (true, if k F O. the product of the parities of H~ and K is no longer determined by the angular correlations). For this purpose one first selects such cases of the reaction Tr - + He' + K. in which the hyperfragrnent and the K meson make (in the center of mass system) small angles af'" 0 with the direction of the incident pions (aligned with the z axis). If the H~ decays associated with the K - Tr + Tr decays are further selected such that the pions make small angles a :S ~a with the sarne z axis, then the distributions about 'Y. which are chosen on the basis of this selection and which determine the spin i. will be the same as on the threshold in the variant Tr = (- 1 )i (see table). If one chooses instead the decays K - 'If + 'If, associated with the decays along the z axis, it is possible to determine k. The correlation relative to the angle between the z axis and the direction of the decay products of K (in the system where the K meson is at rest) is given as a function of k in the sarne first line of the table. 1 The value of the permissible intervals of small angles af and a (~af and ~a,) diminishes with increasing l~ax -the maximum important orbital momentum of the products of the reaction K- + He' ~ + K-Illld spin k respectively. If l~ax = I, we get ~"f Rj 20'. Cascade (2) was first proposed by Dalitz,' and Gall-Mann' gives a set of correlations for this cascade. If k = 0, these correlations have the same form as in the table, but relative to another angle 8, see Ref. 4. We emphasize here only that these formulas must be compared with the experimental distributions, obtained for the reactions K- + He' - ~ + '/1'0 without preliminary formation of a mesonic atom. If we verify somehow that the reaction took place "in flight,- one can expect that at K-meson energies up to

Hi.

Hi

H1

Hi

6 829 LETTERS TO THE EDITOR 4 20 Mev (but above 0.01 Mev) the He nucleus captures the K meson in the s state [or in the p state, if 1T = -( -l)i 1 by some specific (non-electromagnetic) short-lived forces. At greater K-meson energies the correlation formulas are the same, but are valid only for reaction products making small angles with the direction of the incident K meson. Cascade (2) does not permit determinat ion of the spin of the K meson, and the effect of k oJ 0 on the discussed correlations relative to Ii can be investigated only qualitatively. If k oJ 0, the correlation corresponding to a given i becomes less anisotropic, smoothes out, and thereby yields too low values of the hyperfragment spins. I express my gratitude to Professor M. A. Markov for discussions.

lR. K. Adair, Phys. Rev. 100,1540 (1955). I. Shirokov, J. Exptl. Theoret. Phys. (U.S.S.R.) 32,1022 (1957), Soviet Phys. JETP 5, 835 (1957). 3 R. Dalitz, Proceedings of the Rochester Conference, 1956. 4M. Gell-Mann, Phys. Rev. 106,1298 (1957). 2 M.

Translated by J. G. Adashko 223

7 Nuc/eay Physics 6 (1958) 10---19;

©

Noytll-Holiand Publishing Co., Amstudam

SPINS AND PARITIES OF THE JiB"' HYPERFRAGMENT AND K-MESON CHOU KUANG-CHAO and M. 1. SHIROKOV Joint Institute 01 Nucleaf' Research, Laboratory 01 Theoretical Physics, Dubna, U.S.S.R. Received 24 July 195i Abstract: If K-mesons. hyperfragments or hyperons are created in the n-+He' reaction near the threshold or in the K-+He' reaction at low K--particle energies (but without mesic atom formation). the spins of these unstable particles (and in some cases their parities) can be determined from the correlations between the directions of the momenta of the particles involved in these reactions and the directions of emission of decay products of the unstable particles. Qualitative changes in the correlation formulae for the cascade K-+He' -+ AH'+nD. AH' -+ He'+n- are indicated which arise if one rejects the assumption that the K-meson spin is zero. The helium experiments together with the previously suggested hydrogen experiments (e.g. n-+p -+2'+K or K-+p -+2'+n) are the only ones which are of practical use for the determination of new particle spins from angular correlations.

1. Introduction The method employed in the present paper for the determination of the spins of unstable particles consists in the following. The cascade a+b ~ c+d, c ~ e+f is studied. The products of the first cascade reaction (particles c and d) are in general polarized in some definite way. The angular distribution of the decay products of a particle depends on the magnitude of its spin and on the nature of its polarization, that is, on the wave function of its spin state. Under suitable conditions this function may be known and then for each spin value of particle c a particular distribution will exist. If the experiment is carried out under prescribed experimental conditions the particle spin can be established by comparing these theoretical distributions with the experimental one. By correlation we understand the angular distribution of the decay products of c (and/or d) which depends on its spin state; the latter in turn may depend on the angle of emission of the created particle as a parameter. If only particle c is considered and the beam a and target bare unpolarized (or possess zero spins) the correlation parameters and arguments in the general case are three vectors: the direction of the incident beam a, direction of emission of the reaction product c, and the direction of emission of its decay products. In view of the in variance of all processes under threedimensional rotations the correlation will in fact depend only on the 10

8 SPINS AND PARITIES

11

relative position of these three vectors which can be specified by three angles (see below). Dalitz 1) earlier mentioned that measurement of the correlation for K-+He' -+ AH'+no, AH' -+ He'+n- should yield information not only on the spin of the hyperfragment but also on the parities of the new particles. The hyperfragment spin may give infonnation on the AO spin and on the nature of the interaction between AO and nucleon. TheO correlations for spins o and 1 of the AH' are presented in ref. 2) for various parities on the assumption that the K-meson spin is zero.

2. Correlation for the 7t+He' Cascade 2.1. GENERAL METHOD

General formulae for the angular distribution of the decay products of an unstable particle and for the spin state of particle c and/or d produced in a reaction of the a+b -+ c+d type have been obtained, for example, in ref. 3). We shall now obtain the correlation for the simple case of the cascade n-+He' -+ AH'+K, AH' -+ He'+n- and thus illustrate the general arguments presented above (and also the method used in ref. 3)). The wave function "P' for the products of the reaction n-+He' -+ AH'+K can be written in the form

pt = S(T, -T)po.

(1)

Of course this does not make "P' a better known function; so far only the dependence of "P' on the dynamics of the problem (i.e., on the S-matrix of the problem) has been fonnally singled out and in addition it has been emphasized that "P' depends on "Po (initial condition at time -T). Since the total momentum P of an isolated system is conserved, we get that p, = 0 if Po .:. . . O. The state of the system of particles AH', K or n-, He" can therefore be described in their relative momentum P and spin projection m representation (in the c.m.s.). Rewriting (I) in Dirac's notation ') we obtain the following expression for the probability amplitude that after the reaction AH' and K will possess a momentum pr and spin projections m i and m k (Pr; m i , mklf)

(Pr; m i , mkISlp)(plpo) = (PI; m i , mkISlpo)·

=

(2)

The symbol (plpo) signifies that prior to the reaction n- and He' had a definite momentum Po. Integration over P is implied. Note that we have employed the law of conservation of total momentum. We shall now make use of the law of conservation of total angular momentum (a.m.) J. Rewriting (1) for the case when the initial state "Po is characterized by definite values of the total a.m. quantum number 10 and

9 12

CHOU KUANG-CHAO AND M. I. SFlIROKOV

its projection Mo. we see that in order for "Pt to describe a state with the same definite I and M the S-matrix must be diagonal in I and M: (... j' M'ISI··· IM)=( .. . /51 1 . . . ) ' {)]']. ()M'M' (3) (It can be proved that elements that are diagonal with respect to M do not depend on M.) We now express the elements (Pt; mi. mklSlpo) in (2) in terms of the elements of (3):

(-&rCPtpr; mi. mtlSlpo) = C(Po) . L (t?rCPrl 1' ,a')(ikm,mkliks' m') . (s' I'm' ,a'is' l' j' M') (s' l' j' M' PtiSIlMpo) (JMIt?oCPo).

(4)

The unitary transformation function for transition from the representation in variables Pr. m •• mt to the representation involving I and M is obtained as the product of familiar transformation functions: 1) the wave function of a state with definite orbital a.m. l' and its projection p' in the representation in spherical angles t?r. cpt of the vector pr , that is. the spherical function Y /I' (t?r. cpr) === (t?tcpr/1' ,a'); 2) the wave function of a state with a definite " spin 5' and its projection m' in the representation in variables m. total and m k• i.e., the so-called Clebsch-Gordan coefficient (ikmj mkliks' m')5 ); and 3) the coefficient (5' l' m' ,a'is' l' j'M'). C (Po) is a normalization factor which is of no importance for the following exposition; i and k are the spins of particles AH', and K. ~ denotes the sum over 1', ,a', s', m', I', M', I and M. Employing (3), orienting the z-axis parallel to Po and taking into account that in this case Y'M(O, ({Jo) = [21+1/h]l. dM,o, we obtain the following final expression for the wave function of the reaction products:

(Pt; m" mtlf) = C'(Po)'

L

(t?r({Jrll',a')(ikmjtnkliks'm')

I' p' ,'m'

(5)

. (5' l' tn' ,a'is' l' to) (5' 1'15'11) (21+ 1)1. The symbol J has been replaced by I which is the orbital angular momentum of particles ;n- and He' (and is equal to the total a.m. before the reaction). The law of conservation of total parity (a consequence of the invariance of interactions under space reflection) is expressed by the equality of the total parities before and after the reaction. Thus I and I' should be such that II" ·IIHe (-I)' =IIH • IlK (-1)/ or II(-I)I+I' = +1 where II is the product of intrinsic parities of all four particles participating in the reaction. In a similar manner the angular distribution of the decay products of a polarized particle of spin i at rest can be obtained if the particle decays into two zero spin particles

10 13

SPINS AND PARITIES

(6) m

'Pm is the spm wave function of the particle. 2.2. K-MESON SPIN ZERO AND LOW .7l-MESON ENERGY

Let us temporarily assume that the K-meson spin is zero; then s' = i. Suppose the reaction n-+He4 --+ AH4+K is being studied near the threshold, that is, at n-meson kinetic energies of 620-640 MeV in the laboratory system. (n-mesons of such energies can be obtained with the Birmingham proton synchrotron.) We assume that the forces between AH4 and K are of the short range type (that is, these particles are created in a certain finite volume and do not interact after escaping from this volume). Since we know nothing more about these forces it will be most natural to assume that slow (in c.m.s.) AH4 and K-particles are created predominantly in the s-state (I' = O)t. Disregarding all matrix elements (il'IS!II) with I' = I, 2 ... compared with (iOIS!II) we find that t can assume only a single value I = i. However, this value can be forbidden by the law of conservation of parity if ll(_I)i = -1. In the case when II = (_I)i, (5) takes the form

(milf)

= bm"o' C"· (iOISili)(2i+l)l.

(7)

Inserting (7) m (6) and denoting {} by I' we get ~i(Y) oc:

(8)

[1\ (cos y)J2

where F;(x) is an unnormalized Legendre polynomial (see ref. 6)) and I' is the angle between the direction of emission of the hyperfragment decay products (in its c.m.s.) and the chosen z-axis (that is, the direction of the incident n-mesons). The correlations for spin values i = 1 and 2 are presented in table I. TABLE

~

i

Parity

II = (-I)' II=-(-I)'

=

I

3 cos' y

!

I

(1- cos 2 y)

i

1(I ¥

=

2

6cos'y+9cos'Y) (cos' y- cos' y)

If II = -- (-- 1) i the value I' = 0 will be forbidden, because in this case l = i and ll· (_I)i = -1. Assuming that AH4 and K are chiefly formed in the p-state (l can again then assume only one value l = i) we obtain t Of course it is always possible to imagine an interaction [or which this will not be true (even if I' = 0 is not forbidden by the parity conservation law; see below). However the correlations derived below are valid for arbitrary short range forces although the angular (tt r) range in which the correlations are valid may be smaller (see section 3).

11 14

KU~NG-CH~O

CHOU

~NP

M. J. SHIROKOV

(PI; milf) ex:: Yl.-m/~I' !PI)(li-m i m t iliiO).

Besides y, the correlation depends on two other angles Integrating over them we can obtain the correlation

(9) (~I

and !Pl-!p).

i

Ii(Y) ex::

I

[Yim(y, 0) (li-mmlliiO)J2

m=l

(see table I).

3. Correlation for Restricted Directions of Emission 3.1. K-MESON SPIN ZERO AND HIGHER n-MESON ENERGIES

Adair 7) has indicated another approach to the problem of determining the spin from correlations. Mathematically this method is based on the fonnula Y',I'(O, !p) = [2l+1/4.nJ1. ~I'. 0 (the quantal expression of the perpendicularity of the angular momentum to the corresponding momentum.) We shall again assume that the K-meson spin is zero but shall not impose the restriction that the n-meson energy is close to the threshold value. We shall consider the decays of only those hyperfragments which are emitted at angles ~I ~ 0 or ~r ~ n, i.e., along the axis zl/Po- From (5) it immediately follows that (0, !pr; milf) ex:: 15 m ,. 0, i.e., the spin wave function of these hyperfragments is the same as (7)t. Thus the correlation fonnulae are the same as those near the threshold for the parity case II = (_I)i. What is the angular range of ~r in which these formulae are valid? From the deduction it follows that in this interval we should be able to neglect spherical functions Y"'I"(~r,!p1) with f.l' =1= 0 compared with Y ". 0 (~r, !pc). A calculation shows that for I' = lone tenth compared to unity is neglected in the ~r interval (00 - 200 ). This interval should decrease with increasing l'. A consequence of this is that one is obliged to assume that the forces are of short range and that the threshold should not be considerably exceeded in the experiments. For each given energy the maximum essential value of I' should be found by assuming some range of forces (usually ~ 10-13 cm) and the permissible angular range should be estimated. At the threshold and for II = (- 1) i formula (7) is valid over the whole sphere of directions of emission of the hyperfragment (for all angles ~r). t This result can easily be explained in a descriptive manner. The projection of the total a.lll. on the axis z Ilpo is zero before the reaction as the ;rc- and He' spins are zero and the projection of the orbital a.m. on the momentum itself is zero. Thus after the reaction the sum of the projection of the AH' spin and the orbital a.m. l' should vanish. For hyperfragments ejected along the z-axis J.l' is again the projection of the a.m. on the momentum and must therefore vanish. Therefore for such hyperfragments m, can assume only one value: ml = O. It can be seen that it is sufficient to use only conservation of the projection of the total a.m. providing that m, = 0 and emission in the direction of the z-axis is not forbidden by the laws of conservation of the total a.m. and parity. (This prohibition just occurs near the threshold for JI= _(_I)'; see below).

12 SPINS AND PARITIES

15

As the n-meson energy is increased, the region of applicability contracts towards the poles. When II = - (-1) i the angular distribution as well as the spin wave-function of hyperfragments emitted along the z-axis vanish at the threshold (see (9); (Pr; Olf) = 0, as (liOOlliiO) = 0). This means that in this case and near the threshold no region exists in which (7) is valid. As the energy increases, hyperfragments begin to appear also at the poles and correlation (8) is valid for them. 3.2. K-MESON SPIN NOT ZERO

The assumption that the K-meson spin is zero may be abandoned if one applies Adair's method (although, admittedly, the statistics are impaired in this case). If k 0 the most general correlation will be a function of five angles (a function of the relative position of four vectors) :F (1ft; f), cp; f}K' CPK) where f}K' CPK are the angles of emission of the K --+ n+n decay products in the K-meson rest system; these angles, as well as f} and cP, are measured in a coordinate system whose axis is zllpo t. Let us construct the correlation :F(O; f}, cP; 0, CPK)' i.e., construct the angular distribution for decay products from the following ensemble of hyperfragments: first of all hyperfragments (and K-mesons) those emitted along the z-axis are chosen; after this from these events we choose those in which the decay n-mesons from K --+ n+n are emitted parallel to the z-axis. Formula (6) for the K --+ n+n decay shows that only the components (0, cpr, pr; m i , Olf) of the ¥'ave-function (Or, cpr, pr; m i , mAolf) are required for this correlation. From (5) we then deduce that (0, cpr, pr; m i , Olf) occ5 m ,. 0' that is, the correlation :F(O; 0, cP; 0, CPK) is identical to (8). The correlation :F (0, 0, cP, f}K' CPK) has also the same form and permits one to determine the K-meson spin. It should be noted that in the cascade n+p --+ E+K, E --+ N +n, K --+ n+n this correlation can yield a spin value k or k-l and hence the measured value of the K-meson spin may turn out to be smaller than the true value 7). In conclusion it should be stressed that the parity can be determined experimentally only in the vicinity of the threshold and only if the Kmeson spin is zero. If the n--meson and helium parities are known it should be possible to determine the product of the K-meson and hyperfragment parities. It may be mentioned that the law of conservation of strangeness (not mentioning the laws of conservation of electric charge and number of baryons) does not permit one to determine separately the parity of each new particle.

"*

t The so-called polarization correlation o[ AH' and K is used. In this connection it should be mentioned that formula (1.5) in ref. '), p. 1033. and the corresponding paragraph are wrong.

13 IG

CHOU KUANG-CHAO AND

M. 1. SHIROKOV

4. Correlation for the K-+He' Cascade The K-+He 4 -+ AH4+n O reaction is of the same type as that discussed above (if k = 0 it is symbolically written as 0+0 -r i+O) but it is accompanied by release of energy. The most frequent events are those involving electromagnetic capture of K- mesons on one of the helium atom orbits (formation of a mesic atom) and subsequent (non-electromagnetic) interaction with the helium nucleus. However, this subsequent capture may proceed from the s-orbit (1=0) as well as from the p-orbit (1= 1); according to theoretical estimatesB) the probabilities for each of these events are approximately the same. The presence of I = 1 in the initial state is explained by the fact that a system (mesic atom) with a.m. equal to 0, I, 2 etc., is initially formed by the long range electromagnetic forces. The probability for mesic atom formation rapidly decreases with increase of K-meson energy 9). At an energy ~ 0.1 MeV the cross section for mesic atom formation is approximately a million times smaller than the nuclear geometric cross section, i.e. ~ 10-' mb. Moreover the mesic atom is predominantly produced in the s-state when the K-meson energy exceeds the ionization energy of the mesic atom (~ 0.01 MeV). Thus if it is ascertained in some way that the reaction takes place "in flight" t, that is, without mesic atom formation (or mesic atom formation in the s-state) the effect of long range electromagnetic forces will be negligible. Employing the generally accepted concept of a finite K-meson-nucleon interaction range it seems most natural to consider the element (s'l'IsoIO) in (5) to be larger than all other elements (1 =1= 0) providing that the energy of the incident K-meson does not exceed 20 MeV (the interaction range in this case should be assumed to be of the order of the helium nucleus, i.e., 1.6X 10-13 cm). For II = (_I)i and k = 0 we then obtain from (5) for the hyperfragment wave function (milf) oc Yi, mJ&r, !prj and for the angular distribution of the hyperfragment decay products !F(DI, !PI; D, !p) oc I.!Y im ({}, !P)Yim({}l, !prl1 2 oc [Pi (cos 0)]2 (10) m

where 0 is the angle between the directions {h, !PI and D, !po The correlation is the same as that described by (8) but refers to a different angle. Suppose now that comparison of experiment with theory indicates that the parity is II = _(_I)i. The statistics can then be expanded by considering events involving mesic atom formation, since capture from the s-orbit is forbidden. The same correlation formulae as those in the second line in table 1 are obtained if capture from the d-orbit is neglected. (It should t For example the reaction K-+He' occurs before the K-meson track becomes perceptibly curved or the reaction products in the laboratory system are observed to emerge in directions which do not differ by exactly IRO".

14 SPINS AND PARITIES

17

be noted that if the hyperfragment and K-meson spins are zero the reactions .7c+Het -+ AH'+K and K-+He' -)- AH'+:rcO will in general be forbidden for II = _(-1)0 = -1.) At low K -meson energies (1 = 0) and II = (-1) 1 Adair's procedure for the reaction under consideration is identical with that presented above (indeed for 1= 0 the z-axis can be chosen parallel to Pr and from (4) and (3) it then follows that for the projection on this axis (milf) ex: 15m ;o and (10) is immediately obtained); for II = -(-1)' it does not have any region of applicability. At higher energies Adair's procedure should be slightly modified: the z-axis is chosen parallel to pr (instead of z//Po) and the region of applicabilityof (10) is then restricted by the largest significant value of 1 which is the orbital angular momentum before the reaction: those events should be chosen in which the angle between the incident K-mesons and the axis zllPr is small in the c.m.s., or, what is the same, in which the hyperfragments are emitted at small angles with respect to Po. rt should be noted that the statistics obtained for the x-+He t -+ AH'+Ko and K-+He' -+ AH'+:rcO reactions can be plotted on a single graph (if k = 0). The possibility k i= 0 can be accounted for only qUalitatively for the cascade K-+He' -+ AH'+:rcO, AH' -+ He'+x- (one additional assumption being made.) If k i= 0 one obtains I' = li-kl, li-kl+l, ... i+k (see eq. (5).) Since the energy release in the reaction K-+He' -+ AH'+:rcO is not very large (R:I 150 Me V) we shall assume that of all these values of l' only the lowest value occurs, that is I' = li-kl (or l' = li-kl+l in the other parity

2

2

~~0.-::-8--:06-:---0.-4-Q=-2--'0 COS 8 '<---o.-e--0.-"'e....:::...-O-.4-0-.2-=OCOS 8

Fig. 1. Correlation in K-+He' ..... AH'+:71, AH' ..... He'+:71- cascade for various cases (i, k) of the spin values of the hyperfragment and K-meson (the product of their parities being - I.)

15 18

CHOU KUANG-CHAO ANI> II. I. SHIROKOV

case.) Clearly, the larger /i-k/ is, the better founded this assumption is. The calculations are now more complicated (the K-meson beam is assumed to be unpolarized.) A summary of the angular (0) distribution for various spin values of the hyperfragment and K-meson is presented in fig. 1. (The product of their parities is -1, which corresponds to II = + 1. It should be noted that in table I the method of classification with respect to parity is. different; thus the cases i = 1, n = -1 and i = 2, II = +1 are united in the same row.) Qualitatively the effect of k =1= 0 can be expressed as follows: /i-kl or li-kl+ 1 can be determined from the correlation. Thus k =1= 0 smoothes out the correlation (10) for hyperfragments (and hence lowers the measured spin value). The same conclusion can be drawn for the cascade K-+p -+ 1:'+K, 1:' -?- N +n. Thus even if the 0 correlation is set up for Alvarez events 10} "in flight", isotropy, strictly speaking, would merely indicate that the hyperon and K-meson spin values are approximately equal. 5. Other Reactions Besides the reactions discussed above, two other reactions of the 0+0 -?- i+O type exist: the interaction of :n; and K-mesons with helium accompanied by formation of a AHe' hyperfragment. However at present no two-particle AHe' decays are known. Other modes of interaction of nand K-mesons with helium are possible, such as K-+He' -+ AH3+n, AH3 -* He3 +n-, and K-+He' -* .E"(AO)+H3 or K-+He' -+ 1:'-+He3 , followed by 1:' -+ N+n. The correlation formulae for such reactions (of the 0+0 -+ i+t, i -+ !+O type) turn out to be the same as those for the K-+p -+ E-+n+ reaction 11) so that, for example, cases of the K-+He' -+ 1:'-+He3 reaction can be easily plotted on the (j correlation graph of the K-+p -?- E-+n+ reaction. Besides the reactions (of the 0+0 -+ i+O, i -+ 0+0 and 0+0 -+ i+!, i -+ !+O types) analyzed above and those of the O+! -+ i+O type t previously suggested 7.8,11), no other reactions exist which are suitable for the determination of the spins of new unstable particles produced by bombarding hydrogen, deuterium, H3, He3 or He' targets with n mesons, K-mesons or nucleons. For other types of reactions (e.g. K-+D -+ L'+p) the correlation fonnulae inevitably contain unknown parameters. In some cases certain assumptions regarding these parameters can be introduced which are similar to that made in section 4. However, in this case the t Examples of such reactions are n'F +p -+ Y +K where Y is a I

K-+p->-~+K~.:7r+Hel-+AHI+KO; and

or AO particle;

n+ nucleus (spin I) -+ hyperfragment + K. hyperfragment -1+0.) Reactions involving energy release: K-+p-+Y+n. K-+H"-+AH'+n-, p+He'_..tH"+KO (p is an antiproton.) in general reactions of the type

16 SPINS AND PARITIES

III

correlation fonnulae cannot be expected to yield more than qualitative results and the respective reactions cannot be a reliable means of detennination of the spins of new particles, although they may yield a non-trivial (> t) lower limit for the spin. The authors express their appreciation to Prof. M. A. Markov for interest in this work and discussions. Note added in t"~ ~oot: After the pr~nt paper was sent to print we learned that an identical result regarding the determination of the K-meson spin by Adair's method has been obtained by J. J. Sakurai (Phys. Rev. 107 (1967) 1119). From a point of view different from ours, Dr. Sakurai also emphasized the importance of the n+He' ->- AH'+K reaction.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

K. Dalitz, Proceedings of the Rochester Conference (1956) M. Gell-Mann, Phys. Rev. 106 (1957) 1298 M. I. Shirokov, JETP 3l (1957) 1022 P. A. M. Dirac, The Principles of Quantum Mechanics A. Simon, Numerical Table of the Clebsch-Gordan Coefficients (ORNL-1718) E. Jahnke and F. Emde, Tables of Functions R. K. Adair, Phys. Rev. 100 (1956) 1540 R. Gatto, Nuovo Cimento 3 (1966) 1142 M. I. Podgoretsky, Ups. Fiz. Nauk 51 (1953) 253 L. W. Alvarez et al., Nuovo Cimento 5 (1967) 1026 S. B. Treiman, Phys. Rev. 101 (1966) 1216

17 Nuclear Physics 6 (1958) 669-671;

©

North-Holland Publishing Co., Amsterdam

ON NON-UNIQUENESS OF NUCLEON-NUCLEON SCA TTERING PHASE SHIFTS L. G. ZASTAVENRO, R. M. RYNDIN and CHOU RUANG-CHAO Joint Institute at Nucleal' Research, Laboratory at Theoretical Physics, Dubna, USSR Received 23 December 1957

The cross section for scattering of mesons by nucleons does not change upon application of the phase shift substitution proposed by Minami 1). The two phase shift sets which are obtained from each other by means of this substitution can be differentiated either by carrying out polarization experiments 2. 3) or by studying the energy dependence of the cross section at small energies. Below we derive similar transformations for nucleonnucleon scattering. Elastic nucleon-nucleon scattering can be completely described by the matrix M(k, to; u 1 , u 2 ) which defines the amplitude "Pr of the scattered wave in terms of the initial spin state "PI: (1)

Here u 1 and u 2 are Pauli matrices of the two nucleons and to and k denote unit vectors directed along the direction of motion of the incident and scattered nucleons. In order to derive the transformations under consideration we note that the scattering cross section for unpolarized nucleons 0'0 = t tr MM+ is invariant with respect to replacement of M (k. to; u 1 , u 2 ) by any of the following matrices: Ml =U1 ,kMo1 ,ko,

M2=02'kM02'ko. M3 = 1 ' koz . kMo l 'ko0 2'

°

(2)

ko·

Expanding M in terms of the spherical functions Y{;'(k) which describe states with definite values of total angular momentum i. its projection m, orbital angular momentum l and spin s, we get (3)

The values of sand l are detem1ined by the summation rule for angular momenta as follows: 669

18 670

L. G. ZASTAVENKO

et al.

for s = 0 (singlet), 1=

i,

for s = 1 (triplet), l = j,

i±l.

For a given value of j the R~; j'." on account of the reversibility of motion, form a symmetrical four-row matrix Ri which satisfies the condition (4)

this condition follows from the unitarity of the S-matrix. Consider, for example, the first transformation (2). Since the operator (/1 • k commutes with the total angular momentum operator we have (5)

where £1 1 )1 is a unitary matrix. Thus (6)

where (7) RU)I therefore satisfies condition (4) and possesses the same synunetry properties as RI. The elements of RIl)l can thus be considered as elements of a new scattering matrix which yields the same cross section as M. The same also applies to the matrices RIZU = £1 2 )1 RI VJ)/+ and RI3H = £I'll Ri £13)/+ corresponding to the second and third transformations in (2). The matrices LUll have the form

Ill,au =

$

0

0

0

0

V2i+ 1 ±Vi+1 ±V1 0

0 1

0

0

0

0

-I

L(3)1

=

=Fvi+ 1 =fV1

V1

-vi+ 1

0

0

0

0

-VJ Vi+l 0

0

0

0

1

---

2i+l 2Vi(1+1) 2i+I

(8)

2Vi(i+ 1) 2i+ 1 1

2i+I

The first column (row) refers to a singlet and the remaining ones to a triplet with the follOwing order of l : j, i+1, i-I. The upper sign refers to index 1, the lower to index 2. Since, in contrast to (/1 • k (/z . k, the operators (/1. 1I • k do not commute

19 ON NON-UNIQUENESS OF PHASE SHIFTS

with the operator of the square of total spin,

t

671

(a 1 +a 2 )2, the matrices

Ml and M 2 lead to singlet-triplet transitions. The first two transformations

therefore cannot be valid for collisions between identical nucleons for which singlet-triplet transitions are forbidden by the Pauli principle. This is also true for n-p scattering, providing that the hypothesis of isobaric invariance is correct. The operators at . q are operators of rotation of the ith nucleon spin through an angle :rr; about the direction q. This permits one to determine the transformation properties of various spin characteristics on replacement of Ri by R(3);. A result of this substitution, for example, is a change in the sign of the polarization Po arising in collisions involving unpolarized nucleons. In conclusion, it should be noted that the quantities measured in a complete set of experiments, sufficient for the construction of the nucleonnucleon scattering matrix 4), do not change if along with replacement of Ri by R(3); the signs of all phase shifts are reversed (complex conjugation of Ri). The non-uniqueness of the phase shifts which arises in this case can be removed by investigating the energy dependence of the cross section at low energies or by replacing one of the experiments of the complete set (e.g. Knn) by a triple scattering experiment in mutually perpendicular planes.

References 1) 2) 3) 4)

S. S. R. L.

Minami, Progr. Thear. Phys. 11 (1954) 213 Hayakawa, M. Kawaguchi and S. Minami, Progr. Theor. Phys. 12 (1954) 355 Ryndin and Ya. Smorodinsky, Doklady AN SSSR 103 (1955) 69 Puzikov, R. Ryndin and Ya. Smorodinsky, Nuclear Physics 3 (1957) 436

20 363

LETTERS TO THE EDITOR

PHASE INDETERMINACIES IN NUCLEONNUCLEON SCATTERING L. G. ZASTAVENKO, R. M. RYNDIN, and CHOU GUAN-CHAO

(4)

which arises from the unitartty of the S-matrix. Consider, for example, the first of the transformations (2). Since the operator (aik) commutes with the total angular momentum operator,

Joint Institute for Nuclear Research

("lit) yl:'
Submitted to JETP editor November 26, 1957

Thus

J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 526-527 (February, 1958)

THE

cross section for scattering of mesons by nucleons remains invariant under the phase substitution Indicated by Minami. 1 The two sets of phase shifts, obtained from one another through this substitution, can only be distinguished either by means of polarization experiments,:t-I or by analyzing the euergy dependence of the cross section at low energies. We obtain below a similar transformation for the case of nucleon-nucleon scattering. The elastic scattering of nucleons against n1:lcleons is completely described by the scattering matrix ~ (It, lEo; crl' G1), which determines the amplitude Xf of the scattered wave in terms of the initial spin state Xi: X, = M (k. k.; ~" ~.) XI.

Ml (k. ko;

~" ~.) = ~

I. m

We expand now the matrix M in terms of the spherical function y:(It) describing a state of given total angular momentum j, its z-component m, orbital angular momentum ! and spin s. Then M (k. ko; "1>

~.) = ~ ;; I. m

~ yl:' (k) y{::,+ (ko) RI.; I'r' I,.;

I'. r

(3)

The values of s and f are determined from the rule for adding angular momenta, and are as follows: for s = 0 (singlet) != J; for s = 1 (triplet) f =j, J:I: 1. For a given value of J, the quan-

tities R~,f'S' form a symmetric (reversibUity of the motion) four-rowed matrix R! satisfying the condition

~

Yi:' (k) Yl~t (ko) R!!~ ~""

(6)

I ... I' •• '

(7)

The matrix L(I)j is a unitary, antisymmetric, Hermitian matrix. Therefore R(t)j satisfies E~. (4) and has the same symmetry properties as R. Therefore the elements of R(t)j may be considered as the elements of a new scattering matrix which leads to the same cross section as M. All this al?plies as well to the matrices R(2)j = L(2)j x RjL(Z)J+ and R(I)j =L(I)jRJL(I)j+, corresponding to the second and third transformation of (2). The matrix L(i)j has the form

=V2/+1 (

o

0

0

0

±Vi+1 -YT

=FYi::f- 1 =FYI ) Vi -VJ+T

±VT VT+T

a.

(2)

i-

where

(1)

Here at and a2 are the Pauli matrices'for the two nucleons, and lEo and It denote unit vectors along the incident and scattered nucleon directions. In order to obtain the transformation of interest, note that the scattering cross section for unpolarized nucleons = Y.Sp MM+, is invariant to an exchange of M (It, lEo; O:t, crz) with one of the following matrices Ml = (~lk) M (~lko). M. = (~.k) M (~.ko). M. = (~lk) (~.k) M (~~ko) (~.ko)·

(5)

-Io 0 0 1

L (S)} _ -

(

0

0 0

o

0

1

-21+1

-

2l'7V"+1i

2j+l

0 0

0 0

• (8)

0) 0

_ 2 Jf7V+T)

2j+l

,

_1_

2/+t

The first columns (row) correspond to singlet states, the remaining to triplet states in the following ordering of f: J, j + 1, j - 1. The upper sign corresponds to superscript 1, and the lower sign to superscript 2. Inasmuch as the operators (crl,tIt) in contradistinction to (crtlt)(atlt) do not commute with the square of the total spin operator Y. (crt + cra)I, the matrices Mt and Mz lead to Singlet-triplet transitions. Therefore the first two transformations cannot take place in the case of identical nucleon collisions where singlet-triplet transitions are forbidden by the Pauli principle. This also is true for n-p scattering if isotopiC invariance holds. The operators (crill) represent operators which

21 364

LETTERS TO THE EDITOR

rotate the spin of the i-th nucleon by an angle 1r about the direction q. This ailows to determine the transformation properties of various spin characteristics when Rj is replaced by R (3}j For example. this exchange leads to a change in the sign of the polarization P which takes place in the collision of unpolarized nucleons. We finally remark that changing the sign of all the phase shifts (taking the complex conjugate of Rj) leaves the cross section unchanged. and changes the Sign of Po. Thus a simultaneous application of this transformation with the transformation Rj into R(3)j leaves unchanged the cross section as well as the polarization. Therefore the two sets of elements of R obtained from one another by means of the indicated transformation. cannot be distinguished through the Simplest polarization experiments (double scattering). IS. Minami. Progr. Theor. Phys. 11. 213 (1954). Kawaguchi. and Minami. Progr. Theor. Phys. 12. 355 (1954). 3 R. Ryndin and la. SmorodinskU. Dokl. Akad. Nauk SSSR 103. 69 (1955). 2 Hayakwa.

Translated by M. A. Melkanoff 101

22 710

LETTERS TO THE EDITOR (1)

AHe i

with the assumption that the spins of and K are 1 and 0, respectively. This process is completely described by a spinspace matrix M (n, n') which gives the amplitude of the diverging wave.' The most general form of the matrix M (n, 11') is M (n, n') = a + be [D x D'l,

(2)

when the product I of the parities of all four particles is equal to + I, and M (n, n')

=

aan

+ ben',

(3)

when I = - 1. Here n and n' are unit vectors parallel to the momenta of the incident and emerging particles, respectively; a and b are certain functions of the energy and of the angle between n and n'. The density matrix Pi of the initial state has the form Pl=A(1

+cP),

(4)

where P is the polarization vector of the incident particles, If the reaction takes place at threshold, or if we select only the AHe i particles emitted forward (i.e., n Hn'), then we can neglect the second term in Eq. (2). At the threshold, Eq. (3) takes the form M = aan, and for nun' we have M = (a + b ) an. The polarization vector P' of the AHe i particle in the final state is calculated by the formula I P' = Sp (Mr'IM+c) / Sp (M~IM+).

(5)

Substituting Eqs. (2), (3), and (4) into (5), we get

ON THE DETERMINATION OF THE RELATIVE PARITIES OF ELEMENTARY PARTICLES CHOU HUAN-CHAO (CHZHOU GUAN-CHZHAO) Joint Institute of Nuclear studies Submitted to JETP editor January 16, 1958 J.Exptl. Theoret.Phys. (U.S.S.R.) 34, 1027-1028 (April, 1958) SINCE parity is not conserved in the weak interactions, it is of great interest to determine the parities of elementary particles by means of the strong interactions. We consider below several reactions which can be used for the determination of the relative parities of strange particles. We have in mind the following type of reaction:

P'=P,

when I

=

+1.

(6)

P' = (2(1>n) n - P),

when I

= -I.

(7)

If the parity is not conserved in the decay of the AHe i , then from the angular asymmetry of the decay one can measure the direction of polarization of the AHe i and distinguish between the possibilities (6) and (7). We emphasize that the Incident beam must be polarized, and in such a way that the polarization vector is neither parallel nor perpendicular to the direction n. The other reactions of this general type are as follows: );I + He' ~ .\He'+ ,,± at threshold, or when the .,He' emerges forward; (8)

E± p (n)

+ He' -> A +

He' + ,," at threshold;

+ He' --+ A + He' + Ii- + (/(") at threshold.

(8') (1')

In the last two reactions He" can be replaced by

any other nucleus with spin 0 (for example, C 12 ).

23 LETTERS TO THE EDITOR We note that, in the reactions (1) and (1'), the polarization vectors of the AHe i and A depend only on the vectors P and n and the relative parities of K and A. Therefore, a case of the reaction (I') can he simply added in with the cases of reaction (1). The writer expresses his gratitude to M. 1. Shirokov and L. G. Zastavenko for valuabl~ advice and a discussion of the results. I L. Wolfenstein and J. Ashkin, Phys. Rev. 85, 947 (1952); R. Oehme, Phys. Rev. 98, 147 (1955).

Translated by W. H. Furry 213

711

24 SOVIET PHYSICS JETP

VOLUME 34 (7), NUMBER 5

NOVEMBER, 1958

THE RELATIVISTIC THEORY OF REACTIONS INVOLVING POLARIZED PARTICLES CHOU KUANG-CHAO and M. 1. SHIROKOV Joint Institute for Nuclear Research Submitted to JETP editor December 6, 1957 ,J. E:xptl. Theoret. Phys. (U.S.S.R.) 34, 1230-1239 (May, 1958)

It is shown that, in the rest system, the relativistic formulas for the angular distribution and the polarization vectors and tensors for a reaction of the type a + b - c + d are essentially thc samc as the nonrolativistic formulas, if thc spin of a particle is defined as its internal angular momentum around its center of mass. The square of this internal angular momentum is Lorentz invariant. The spins of the particles are arbitrary, and their rest masses are nonvanishing. The main difference from the nonrelativistic case is that the description of the spin state is not the same in different Lorentz reference systems. Therefore for cascades of reactions (for example, for experimcnts on double scattering) corrections must be applied to the nonrelativistic formal theory. The relativistic changes in the angular correlations are indicated [or successive reactions of the type 7r + P - Y + K, Y - N + 7r.

INTRODUCTION FOR reactions of the type a " b - c "d, formal theories are known which express the angular distribution and the state of polarization of the products of a reaction in terms of the states o[ polarization of the incident beam and the target and unknown parameters which are the elements of.the S matrix for the process a " b - c "d. The simplest example is the well-known formula for the [unction f ( 0) which appears in the expression

'f (r)

'C."

c'"

+ f (Ii) Col" Ir

theory of the scattering of spiniess particles is essentially relativistic. In order to obtain the angular distribution in any Lorentz system, one has only to transform (J (e) = 1f (e) 12 from the centerof-mass system into the desired system by using known formulas. What is therefore required for the relativistic generalization is the definition of the relativistic spin operator. The spiro operator which we introduce satisfies all the requirements which can be demanded in terms of the concept of the spin as the intrinsic angular momentum of a particle.* To obtain the relativistic formulas we use the method of Coester and Jauch in a form presented in a paper by one of us. 3 We emphasize that in this method one needs only the ability to describe the state of a free particle possessing spin; we do not need relativistic equations (for free particles) like the Dirac equation (which plays an essential part in Stapp's relativistic theory' of the scattering of particles with spin It2).

[or the wave function of a stationary scattering process of particles of spin zero. In this case the unknown parameters are called the scattcring phase shifts. These theories are based on the use of the conservation laws (in particular the law of the conservation of the total angular momentum). Coester and Jauch 1 were the first to obtain formulas for the ,mgular distribution and polarization in the case of particles a, b, c, d of arbitrary spins; their 1. THE CONSERVED PHYSICAL QUANTITIES IN starting point was the explicit formulation of the A RELATIVISTIC THEORY conservation laws in terms of the diagonal propThe conservation laws are simply an expression erty of the S matrix with respect to the conserved of the fact that the physical processes in an isolated quantities. These same formulas have been obsystem must not depend on the means used to detained by Simon and Welton, but by a different method (cf., e.g., Ref. 2). *Iu. M. Shirokov has informed us that he has employed this These formulas are nonrelativistie, but only besame description of the spin state (which was obtained by him cause the spin state of the particles is described in from the theory of the irreducible representations of the inhoma. the Pauli approximation (so that it has the same geneous Lorentz group) to the formulation of a similar relativisappearance in all Lorentz reference systems). The tic theory of polarization and correlation effects. 851

25 852

CHOU KUANG-CHAO and M. 1. SHIROKOV

scribe it, in particular on the choice of the reference system. Here it is of course assumed that space-time is homogeneous and isotropic (we can suppose, however, that this assumption is contained in the concept of an isolated system). In quantum mechanics this fact is expressed by the requirement that the S matrix of a physical process must commute with ten operators, the infinitesimal displacements of the origin of the space and time coordinates providing operators P Ii' and infinitesimal space-time rotations giving operators MtLv • The fact that an operator commutes with the S matrix means that the S matrix is diagonal with respect to the eigenvalues of this operator, * and consequently that the corresponding physical quantity is conserved, i.e., remains unchanged in all internal processes. Four conservation laws have the clear physical meaning of the conservation of the total momentum and energy P tL {px , P y , P z , iP o }. Of the six other operators MtLV ' three operators Mk( (k = 1, 2, 3) do not have an immediate physical meaning' and instead of M tLV we shall introduce six other operatbrs, which have the physical meaning of the coordinates of the center of mass of the physical system and its total angular momentum around its center of mass. The properties of the center of mass follow directly from its conceptual meaning: the motion of the system as a whole can be characterized primarily (in the very first approximation) as the motion of a material point with mass equal to the rest mass (or energy) of the system and with momentum equal to the total momentum P of the system. The center of mass of an isolated system must therefore move uniformly in a straight line. Furthermore, in quantum mechanics we must require that the center of mass R actually be the coordinate operator of a certain particle, i.e., in particular, that the well known commutation relations hold between Rx , Ry , R z , P x , P y , P z . We can obtain such an operator R in the following way. It is well known that the following are the commutation relations which must be satisfied by the operators Pi and MtLv (cf. Ref. 6 and Sec. 3 of Ref. 5): [Pi, Pi] = 0; [Pi, E] = 0; [M i , Pd = i.iihP.;

(1.1)

*Let us write out AS - SA = 0 as a matrix product. In doing so, we choose a representation in which the operator A is diagonal (Le., we label matrix elements with its eigenvalues). Then

A"S,,- SimAml = (Ai i.e.. Sil must be equal to zero if i

*

A,) Si/

l.

=

0,

(1.2)

IN i , PJl

[M i , EJ = 0; [M i , M i ] = i.iikM.; tOijE; [Ni , E] = iP . ; [M . , Nil = isiikll'k; (1.3)

=

(1.4)

Notations: [A, B 1 = AB - BA; i, j, k take the values 1, 2, 3; and {M 1 , M 2 , M 3 } = {M 23 , M 31 , M 12 }; ilv'j = M i4 ;

Eijk is a tensor antisymmetric in all its indices, with E123 = 1. It is understood that we confine ourselves to those state vectors I/!o which describe states with a definite rest mass m o, i.e., for which PtLPtLl/!o = - m3l/!o, or (Po - E) I/!o = 0 (the time displacement operator Po is equivalent to the factor E). We note that since [Ni, El = iPi the average value of Ni is a linear function of the time. Therefore Ni is "conserved" in the sense that internal processes have no effect on this time dependence. We introduce three new operators R x , R y , Rz , for which

(from which if follows that [Ri, E 1 = iPi IE). We represent M in the form Mk = ~.'ijRiPj

+ 10,

i.f

and find from Eq. (1.1) that [Ji, Pj 1 = o. If we require that Rx , Ry • R z be the components of a spatial vector (as must indeed be the case), i.e., that [Mj, Rj J = iEijkRk, then [Ji, Rj J = 0, and J x , J y , Jz also form a three-dimensional vector. It then follows from Eq. (1.2) that [Ji, Jj 1 = i€ijkJk· In like fashion, representing N in the form

1/, (RiE + ER .. ) + Ki == R;£ - iPi /2E + Ki, we find that [Ki, Pj 1 = O. Therefore also [Ki, E 1 Ni =

0, and the average value of Ki is constant in time. We now pose the problem of expressing M and N in terms of the operators R and J which we have introduced. For M this has already been done. It remains only to express the spatial polar vector K in terms of P and J. It can be shown that if Rx, By. R z are the first three components of any four-vector [i.e., for example. [N, R 1 ,: if i;o' j). then K cannot be expressed in terms of J and P only so as to satisfy all the commutation relations for N. This means that if K is constructed from P and J alone, then [R, P lk and Jk are not the spatial components of a fourdimensional tensor of the second rank. =

°

26 THEORY OF REACTIONS INVOLVING POLARIZED PARTICLES The simplest K (namely one linear in J x , J y , Jz) satisfying Eqs. (1.3) and (1.4) has the form

= O.

K = [PJI(E ±mr'

(cf. Refs.. 7 and 6). * Since the problem has a solution, it follows that: (1) R is • conserved" (in the sarne sence as N), since it can be expressed in terms of the conserved operators M",v (see Appendix). R can be called the center-of-mass operator. It is the same as definition (e) of the center-of-mass in the papers of Pryce. 8 (2) J is also conserved and, what is particularly important for our purpose, J2 = J~ + J~ + J~ is Lorentz invariant, since [N, JZ) = O. We emphasize that this is true for arbitrary K = K(P, J).

2. USE OF THE CONSERVATION LAWS. RELATIVISTIC DEFINITION OF THE SPIN OF A PARTICLE The four conservation laws for the total momentum and energy and the three conservation laws for the center of mass can be expressed very simply. The argument is usually carried through in the Lorentz system of reference Ks in which the ( conserved) total momentum is zero (the socalled center-of-mass system). The origin of the coordinate system can be taken at the center of mass (more precisely, at the pOint given by the average value of the operator Rt for the particles a and b (or c and d). Then J is the total angular momentum. Since the commutation relations between J x , J y , Jz are the same as for the total angular momentum or for the Pauli spin matrices, the eigenvalues of .1 2 and jz are respectively equal to hZJ(J+ 1) and M=J, J-l, ... , -J. The conservation law for J is expressed by the fact that the S matrix is diagonal with respect to the eigenvalues of .1 2 and jz: ( ... J'M'IS/ .. . JM) = ( ... ISiMI ... )ill'Jil'WM

853

and also the fact that ( ... I SJM I ... ) is independent of the value of M, which follows from [Jx • S)

(2)

*We have not been able to show that no other K's exist. Beginning with different considerations, L. G. Zastavenko has evidently proved the uniqueness of K. We are grateful to him for a discussion of this question. tThe conservation law for R means something more than the conservation of the average value. The requirement [R, S] - 0 means that if the system is in a state with a definite value of R (we note that in the interaction representation the wave function for the external behavior of the system does not change with the time), internal processes in the system will not take it out of this state. This property does not get used explicitly. but an operator R of this kind is required for the definition of the conserved angular momentum J of the system (and of the spin of a particle. see below).

To find, for example, the angular distribution of particles c and d, we need to know the explicit expression of the elements of the S matrix in the representation of the particle momenta. In order to express these elements in terms of the elements (2), we must first of all enumerate the remaining variables of a complete set (denoted in Eq. (2) by dots), which commute with each other and with JZ and Jz. The initial and final states of the process a + b - c + d are states of systems consisting of two free nOninteracting particles. From the meaning of the S matrix, its elements are the transition amplitudes between such states. Therefore to label the elements of the S matrix we must take as our variables a complete set of quantum mechanical quantities describing the free particles a and b or c and d. The total angular momentum J (in the system Ks) is represented in the form J = 11 + h. where 1 is the total angular momentum of a Single particle in Ks. The procedure stated in Sec. 1 for obtaining the conserved angular momentum relative to the center of mass can be applied to a system of arbitrary physical nature (for example. to a system of fields). One needs only to know a concrete representation of the operators P and M. Therefore it is natural to apply this procedure to an "elementary" particle, whose physical nature is in general unknown (by the definition of "elementary"). Besides the coordinates r of the center of mass and the momentum p we then get just one conserved external characteristic of the particle. its angular momentum 8 relative to its center of mass r. In defining the spin s of a particle, we are only fixing precisely the concept of the spin as the intrinsic angular momentum of the particle. Accordingly, J = [ r x pI + I, and in the system Ks, in which PI = -P2 = p, we get J

= [r, x ptl + 5, + [r 2 xp.1 + 52

~[(r, -") x

pI

(3)

+ 8, + '" "I + 8, + "'.

In the system Ks we can now proceed in complete formal analogy with the nonrelativistic treatment to introduce the total spin operator 8 = 81 + 82 (the square of which. however, is not a Lorentzinvariant quantity). The eigenfunctions of the square of this quantity, S2, and its component Sz can be expressed in terms of the products I/>ilml x l/>i zm 2 of the eigenfunctions of the squares and components of the operators 81 and 82 (the eigenvalues of si are denoted by h2id i l + Since

1».

27 854

CHOU KUANG-CHAO and M. 1. SHIROKOV

the commutation relations for s, st> S2 have the usual form, [sx, Sy 1 = is z and so on, the coefficients in these expressions will be the well known Clebsch-Gordan coefficients (ili2mlm21 i li 2sm), which are also the transfurmation functions for the transformation from the representation in the variables it> i 2 , mt> m2 into the iI' i 2, 5, m -representation (and inversely). A similar meaning attaches to the coefficient (lsjJ.m IlsJM ). We can now take as the variables denoted by dots in Eq. (2) 5, l, and the absolute values of the momenta in Ks (or the total energy of the system, which in Ks is equal to an invariant, the rest mass of the system). 3. FORMULAS FOR THE CROSS-SECTION AND THE POLARIZATION VECTOR AND TENSORS. RELATIVISTIC ROTATION OF THE SPIN We can now express the elements of the S matrix in the representation of the momenta of the particles and their spin components in terms of the elements (Pc, s', l', J', M'lsl'Pa, s, l, J, M). The transformation function from the representation in the variables p, s, l, J, M into the representation of the momenta and the spin components is the product of three transformation functions. We can write out the transformation as follows (cf. Ref. 9):

X (i,.i dl1l,l1ld

I i,ids'm') (/'s'!,-'I/l' I

/'s'JN/)

(s'/'I

(4) ",J,np"}1 sl)

We have used Eq. (2) and the law of the conservation of the total energy. Pc and Pa are the momenta of particles c and d and a and b respectively in K; J c ,
ll,,; +-~+Vl'~ + x; = V,T---;;! + V,;; + x~. Summation over the labels l', jJ.', s', m', J, M, t, jJ., s, m is understood. (lhl' \I!,-I' .. ) = :bl,

V"t --'--;- Y,"

lems of normalization and of deducing the crosssection in the system Ks are solved in just the same way as in the nonrelativistic case (cf. Ref. 3). One can also introduce in just the same way the statistical polarization tensors instead of the density matrices. All the formulas will have the same form as those of the nonrelativistic case. 3 The difference lies in the fact that the quantities rna, mb, m c , md (or T, iJ), and also the total spin and other variables are relative to the system Ks. The same spin state has a different form in a different Lorentz system K (for example, in the laboratory system). In order to learn how a spin state specified in Ks is described in K, we must find the transformation function from the representation in the eigenvalues of S2 and Sz in the system Ks to the representation in the eigenvalues of §'l and sz, which are the square and component of the same operator, but in the system K. In view of the fact that s2 is Lorentz invariant, the spin operator B is a vector rotated as compared with s. Consequently, the transformation function is the same as is obtained in the solution of the problem of describing a given spin state in a rotated system of spatial axes:

The matrices P;"m (cos e) are defined in Ref. 10 [Eq. (22) on page 77; we note that the matrix P~m written out explicitly on page 78 docs not agrce with Eq. (22) and is incorrect J. If the rotation is interpreted as a turning of a vector in a stationary coordinate system, then it consists of (a) a rotation of the vector around the z ~L'ds by the angle
pl. '?) (" II',,),

where YtjJ. (J,
«I'" .

n, - 'r). +- '( + Yr~ + o()

0,
slnU = (C+-;;),T-+y;)(1

n')

(6)

'(e, ,

where

w' is the energy of a reaction product in the laboratory system, and J and

28 THEORY OF REACTIONS INVOLVING POLARIZED PARTICLES angles of its momentum p in Ks , measured in an axis system with the z axis parallel to fJ (the .~ and y axes are chosen arbitrarily). In Stapp's4 Eq. (48) for sin g there is a mistake (or a missprint): it does not have the factors YYf3 ly(a)y(b) in his notation \. If one repeats Stapp's calculations (in accordance with his arguments), the result is just the present Eq. (6). The rotation by the angle g must be applied counterclockwise around the vector fJ [p], fJ being the velocity of the laboratory system relative to the center-ofmass system of the reaction (see Appendix). This relativistic effect of rotation of the spin state of course does not show up at all in the trans[ormation of the angular distribution into the laboratory system (since the angular distribution is a polarization tensor of rank zero). One needs only to carry out the ordinary (kinematic) relativistic transformation of the angles from Ks to the laboratory system. The nonrelativistic theory of the angular distribution in reactions with unpolarized beams and targets remains valid also in the rela-. tivistic domain (except for changes or increased precision in the meanings of the quantities involved in the formulas). As for the polarization vector and the polarization tensors, they are not directly measured in the experiments. In order to measure the polarization vector for the product c of the reaction a + b c + d, we must scatter c from a target e and measure the asymmetry in the angular distribution of the scattered particles c. Then we obtain information about the polarization vector pI in the center-of-mass system Kg of the reaction c + e - c + e. The desired polarization vector is obtained from pI by a rotation. The angle of this rotation is found from Eq. (6). In fact, in the successive Lorentz transformations from Ks to the laboratory system (by means of the known velocity /3) and then to the system K' (velocity /3') a rotation occurs only in the first transformation, since the momentum pIc of particle c in the laboratory system is parallel to fJ', so that sin!1 z ~ sin (tfp'c) = O. This question is analyzed in more detail in Ref. 4, and the treatment carried through there is valid for arbitrary spin. Quite generally, the relativistic rotation of the spin is seen to be of importance only for the treatment of cascades of reactions. In the following section we shall deal with the problem of the relativistic changes of the angular correlations in successions of reactions of type a + b - c + d, c e + f. In conclusion we point out that in the change from Ks to the system Ko in which a particie is

855

at rest the description of the spin state does not undergo any changes, since here fJ II P and !1 = O. Therefore we can regard the quantities ma, mb, and so on as describing the spin states of the particles in their rest systems Ko. This interpretation is preferable to the preceding one: the spin states of particles are described by quantities whose definition does not depend on the system Ks , i.e., on what target the particle is interacting with, on what its energy is, or on the energy balance of a particular reaction.*

4. THE RELATIVISTIC ANGULAR CORRELATIONS IN CASCADES OF THE TYPE a + b - c + d, c - e + f We shall consider first the cascade "11"- + p Y + K, Y - N +"11", which is well known in the literature. If the first reaction occurs near threshold, the correlation in the angle y between the directions of the incident "11"- mesons and the decay nucleons provides a way of determining the spin j of the hyperon Y. This correlation can be found if one substitutes into the expression 2/-1

q

Q-O.2...

,--q

F(&,
for the angular distribution of the decay products of the hyperon in its rest system Ky the concrete expression for the statistical polarization tensors p (q, v) of the hyperon (cf. Refs. 12 and 13). In the center-of-mass system Ks of the reaction "11"- + p - Y + K we have near threshold (the z axis is directed along the "11"- meson beam) p, (q, v) - Q (j, q) 3,.0'

(8)

The nonrelativistic correlation in the angle y is *In connection with this interpretation, however, the following misunderstanding can arise. Since there is only one system K. in which a particle is at rest, in any reaction the m values mean always the same thing: the spin components in the rest systems. Consequently it might seem that nO transformations of the spin state are actually necessary. The point is that if we are liven the velocity .. " of a system K. relative to K. and the velocity ~, of K, relative to K" then the velocity "" (which is of course a function of v" and "") turns out not to if [ .... v"l"O (cf. Ref. 11, Sec. 22). The be parallel to transformation from K. to K, must have the form of a Lorentz transformation with a spatial rotation [see Eq. (58) of Ref. 111. H the particle was at rest in K., then in K, it has the velocity v.., and by using this velocity we can go over to a system K, in which the particle is again at rest. Calculations show that the product of the transformations from K. to K, and then on to K, has the form of a pure spatial rotation: s(,) = D-'_(.» if DY•• = ...; .... (the space axes of the Lorentz systems K... K,. K., K, are of course assumed parallel).

''is

29 856

CHOU KUANG-CHAO and M. 1. SHIROKOV

obtained by simply substituting Eq. (8) into Eq. (7): 2J-l

Fnr(&.
(9)

q-o 2/~1

~ ~

Q'U,q)Pq(cosy).

are large enough (> 0.1 Mev) so that mesic atoms are not formed (for more details see Ref. 9). In the center-of-mass system of the reaction K- + P - Y + 7r, with the z axis parallel to the direction of [nk x ny], where nk is the direction of the incident beam of K particles,

q~O

In actual fact one must substitute into Eq. (7) not the Ps (q, II), but the statistical tensors of the hy-

peron referred to the system Ky:

Ps (q, ,) ~ Q (j, q) 0'.0'

Relative to this same set of axes, but in the rest system Ky of the hyperon p (q. ,) = ~ D~" (0, Q, 0) r. (q, ,')

p (q, v) = ~ D:'", (
"

" =

(10)

V 4"/(2q + fj V;, (n, '!'.) Q (j. q).

Here
2/-1

Fr(&''?)~}; (2q+ l)-lQ'(j.q)

L;

Yq,(lt.'f')Y;,(Q,,?,)

'--q

q-o 2/-1

(1/4,,)

=

L;

Q'(j. q) Pq(COSYr),

(11)

q-O

where 'Yr is now the angle between the direction of emergence of the decay products and the direction {Il ( J c ),

,

0

%

=

V 4,,/(2q + I) Y;., (0, 0) Q U, q).

(12)

The difference between the relativistic correlation Fr< J,
Since the most general case of the cascade a + b - c + d, c - e + f, in which all the spins are arbitrary and the correlation depends on unknown parameters, is of no practical interest, we Simply note without proof that the nonrelativistic form of the angular correlation can be retained. To do this one finds for each case of the cascade, using the measured angles of emission of the particle c, a particular system of coordinates belonging to this case. The angles of emission of the products from the decay of c are calculated relative to this system. The distribution in these recalculated angles has the old, nonrelativistic, form (but one has, of course, changed the rule for constructing the angular correlation from the experimental data). APPENDIX l. Let us express s in terms of Mltll and Pw Let K be the rest mass of the particle and w = (p2 + K2 )l/l, and M = [r
N = rUl - ip/2Ul

+ s,

+ [p x sl/(w + "l.

(A.!)

30 THEORY OF REACTIONS INVOLVING POLARIZED PARTICLES The four-dimensional vector

r. =

rotation of the vector around the y axis in the counterclockwise direction through the angle fl must have the form

(lj2i) ".v.,M.vp,

(EJ.lV(JA is the completely antisymmetric tensor of the fourth rank with E12a4 = 1) then has the form:

r =

SX

+ (ps) p/(
x),

r, =

857

i (sp).

(A.2)

Noting that (rp) = w (sp), we find from Eq. (A.2): s=rjy.-(fp)pjx
(A.3)

All these operator equations are to be understood as being in the momentum representation. From the second of the relations (A.1) we now get r
(AA)

2. The vector S in the new system K, which moves relative to Ks with the velocity {J (in units of the speed of light) can now be found in the follOwing way. Substituting into the right and left members of the equations [cf. Ref. 11, Sec. 18, Eq. (25)J

(A.5)

r

the expressions (A.2) for r and in terms of s, Pf.L and S, Pf.L' respectively, and replacing the Pf.L by their expressions in terms of PJ.l (which have the same form (A.5», we get the expressions for S in terms of s. First of all we establish the fact that B is a linear combination of the vectors s, {J, and p. This means that the vector is obtained from s by a rotation around an axis perpendicular to {J and p. There remains only the determination of the magnitude and Sign of the angle of rotation around this axis. For this purpose we choose a convenient set of three space axes (it is obvious that the angle of rotation cannot depend on the choice of the axes): z I {J, y I [{JxpJ. A

s

Sx

= cos Qs x + sin Qs z

Sz

= - sin Qs x + cos ns z ·

(A.6)

Representing the expression for B in terms of s (in the chosen set of axes ) in the form (A.6), and finding the coefficient of Sz in the expression for Sx (as that having the simplest form), after cumbersome calculations we get the formula (6) of Sec. 3 for sin fl. IF. Coester and J. M. Jauch, Helv. Phys. Acta 26, 3 (1953). 2 A. Simon, Prob. Sovr. Fiz. 1955, No.6, p. 21 (Russian translation); Phys. Rev. 92, 1050 (1953). 3 M. I. Shirokov, J. Exptl. Theoret. Phys. (U .::Uj.R.) 32, 1022 (1957), Soviet Phys. JETP 5, 835 (1958). 4 H . P. Stapp, Phys. Rev. 103, 425 (1956). 5 C. M~ller, Dansk. Mat-Fys. Medd. 23, No.1 (1945). S L. L. Foldy, Phys. Rev. 102, 568 (1956). 11u . M. Shirokov, Dokl. Akad. Nauk SSSR 94, 857 (1954). 8 M. H. L. Pryce, Proc. Roy. Soc. AlSO, 166 (1935); A195, 62 (1948). 9 Chou Kuang-Chao and M. I. Shirokov, Nuclear Phys. 6, 10 (1958). lOr. M. Gel'fand and Z. la. Shapiro, Usp. Mat. Nauk 7, 3 (1952). 11 C. M~ller, The Theory of Relativity. Oxford 1952. 12M. 1. Shirokov, J. Exptl. Theoret. Phys. (U.S.S.R.) 31, 734 (1956), Soviet Phys. JETP 4, 620 (1957). 13R. K. Adair, Phys. Rev. 100, 1540 (1956). Translated by W. H. Furry 247

31 543

LETTERS TO THE EDITOR

the production plane of particle a. The second of these systems has Zc parallel to nco and Yc in the direction of the cross product na x nco Here ni is the unit vector along the direction of motion of particle i. and q is the rank of the statistical tensors. The spin indices Tare defined in terms of these particular coordinate systems. With this choice of coordinate systems. the Euler angles for the rotation gcg;t! are {- JT. ec • JT - CPc}. where ec and cP c are the spherical angles of the unit vector nc in the za. Ya. xa coordinate system (see Shirokov 2) • Let the state obtained from the initial one by space reflection be characterized by the statistical tensor Under the reflection the za and Zc axes. chosen along the momenta of particles a and c. change direction. while the Ya and Yc axes remain invariant. The spherical angles ocI and cP cI of the reflected - nc vector in the reflected {za. Ya. xa} I coordinate system are

pi.

(2)

SELECTlUN RULES IN REACTlUNS INVOLVING POLARIZED PARTICLES

PI

CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor April 23. 1958 J. Exptl. Theoret. Phys. (U .S.S.R.) 35. 783-785 (September. 1958) SIMON and Welton! and Shirokov 2 have obtained the selection rules for a reaction of the type a + b - c + d in the form of relations between the polarization vectors and tensors. They assume that the initial state is not polarized. The present communication gives a derivation of the selection rules for any arbitrarily-polarized initial state. We shall use Shirokov's notation2 and assume that all the particles have nonvanishing rest mass. Consider the statistical tensors of the final state in the a + b - c + d reaction.

p'(qe,

Te.

qd' Td ; geg;;l),

The spin operators remain invariant under reflection. If e c and CPc are replaced by ecI and CPcI in Eq. (1). we obtain the statistical tensors from p'; the spin indices T of the new PI tensors must be quantized with respect to the old nonreflected Zc. Yc. Xc system. Since the reflected {zc. y c. xc} I coordinate system differs from the initial one only by rotation through an angle JT about the Yc axis. the transformation properties of the statistical tensors 2 lead to the equations p;(qe, Te, qd' Td ; {Je, -CPe) = D~\,(O, n, O)D~d<, (0, n, 0)

( 1)

which depend essentially on the parameters of the rotation gcg;;:! which carries the za. Ya. xa coordinate system into the zc. y c. Xc coordinate system. The first of these systems is associated with the initial state. the za axis being parallel to na. and the Ya axis being perpendicular to

L I

'Cc'C

,CC

d d

d

Here the spin indices T are quantized with respect to their own proper coordinate systems. Since D~T'( O. JT. 0) ~ (- 1 )q+T 6 T,-T' (see Shirokov 2 and Gel'fand and Shapir0 3 ). Eq. (3) leads to

p; (qe,

Te,

qd' Td ; {Je, -CPc)

= (-I)q<+«+qd+
-Te ,

qd' -Td ; {Jc, CPc) .

(4)

The law of parity conservation may be stated in the following way: if the initial statistical tensors P are replaced by the reflected tensors PI. the tensors P" of the products of the reaction are the tensors PI which are obtained from p' by Eq. (4). In other words. if the statistical tensors P' are written p' ~ F ( P ). then (5)

32 LETTERS TO THE EDITOR

544

Equations (4) and (5) together give the most general selection rules in the form of a relation between the statistical tensors. Let us now consider some simple examples. If the initial state of the a + b - c + d reaction is unpolarized, then P = PI' Then according to (4) and (5) pi = PI, or p' (qe'

'c.

qd' 'J; &e)

'c.

= ( - I)Qe+Te+Qd+T"p' (qe' -

qd' - ~d; &e).

(6)

In our case the p' tensors do not depend on
&,. -

'P,)

= (_ l)Q,+T,+Qg+T gp ' (q,._ 'I' qg. _ '.;

For the special case in which becomes

q{ =

&,. 'P,).

(7)

qg = 0, Eq. (7) (8)

Since
Translated by E. J. Saletan 154

33 SOVIET PHYSICS JETP

JUNE, 1959

VOLUME 35 (8), NUMBER 6

THE SHAPIRO INTEGRAL TRANSFORMATION CHOU KUANG-CHAO and L. G. ZASTAVENKO Joint Institute for Nuclear Research Submitted to JETP editor May 5, 1958 J. Exptl. Theoret. Phys. (U.S.S.R.) 35,1417-1425 (December, 1958) We give an integral transformation which is related to the decomposition, into irreducible representations of the proper Lorentz group, of the representation according to which the wave function of a particle with mass M and spin s transforms.

k' /k' by

THE problem mentioned in the abstract was stated and solved in 1955 by 1. S. Shapiro. I Unfortunately, he used a wave-function transformation law which is incorrect for s > 0 (the transformation law for the spin under a pure Lorentz transformation2 ), and thus the equations he obtained can be used only for spin zero. In this paper we solve Shapiro's problem for arbitrary spin using a wave-function transformation law previously obtained;2 the principles by which the integral transfs>rmation is obtained are the same as those used by Shapiro. In the first two sections we state some known facts which we shall need later. 2,3 In the third and fourth sections we derive the integral transformation.

n' = So;

I = rK, (1.3) where Kno = Ilo in the sense of (1.2), and l' is a pure rotation such that rllo = tno [again in the sense of (1.2)J. Here the choice of no is arbitrary; (l.3) obviously determines l' up to an arbitrary rotation about the Ilo direction. We shall choose no as the unit vector along the third axis; we define r by the condition r = R3 (
1. IRREDUCIBLE REPRESENTATIONS OF THE PROPER LORENTZ GROUP Let (1.0) be the 4-momentum of a particle with mass 0, and let k • . c (UO kk).

K

(1.0')

Obviously k = R (n) ko, where R (n) is the rotation* which carries the third axis into the direction given by n = k/k; if e and 'I' are the spherical angle coordinates of n, we may write R(n) =R,(l'

+ 1t/2) Rd6),

(1.2)

then n' = Sin and n" = S2n' implies that nN = S2Sln. Arbitrary transformation t of the Lorentz group can be written in the form

=

(kllo k'2). k.,

(l.4)

We now define the transformation K (S, n) by applying (1.3) to the matrix t = S-IR (n) (the definition of R (n) is given in (1.1»:

S-' R (n)

(1.1)

=

R (S-'n) K (S, n)

(1.5)

[the vector S-In is defined in (1.2)J. Applying both sides of (1.5) to no, we see that in the present case I' = R (S-In ). Further, making use of the uniqueness of the factorization given by· (l.3), we find that K{S, n) has the property that

where R3 ( Q) is a counterclockwise rotation about the 3-axis through an angle Q. Let k' = Sk, where S is a transformation of the Lorentz group. We shall denote the relation between n = k/k and n' = *10 what follows we shall always denote by R a pure rotation (or the matrix in the appropriate function space corresponding to such a rotation); L will denote a pure Lorentz

K (S" 0) K (S" S.'n) = K (S,S .. n),

which we shall need later.

transformationa

990

(1.6)

34 THE SHAPIRO INTEGRAL TRANSFORMA TION

"'mp(5, n)

Consider the association

f (n) _s_ f' (n) =

0;

(5, n) f (5-'n).

(1. 7)

Let us find the properties that o! (S, n) must satisfy in order that this association form a representation of the Lorentz group. We have

f' (n) ~ f" (n)

=

0;

(5" n) f' (5;-'n)

(5" n) '" (5, S;-'n) [(5-'5;-'n). It is thus necessary that =

0;

'" (5" n) '" (5, 5;-'n) = '" (5,5, n).

(1.8)

It is seen from this that we may choose as (S, n) any matrix element [K (S, n )122 to any power [see (1.4) to (1.6)]. Thus o!

0;

=

(/" [,)==~f;(n)f2(n)dQ(n),

Now (1.4) implies that if k' ~ S-lk, then I k2212 ~ Denoting the association k - n ~ kli k I by

arg k'2 =

[k

(1.9a)

'1,9 (5, n).

'Let us study the angle ",(S, n) in more detail. (a) S = R. In this case the definition of ",(s, n) in (1. 9a) is equivalent to

R-I R (n) = R (R-' n) R, (? (R, n)).

(1.9b)

(b) S = Lp [the definition of Lp is given in (2.1)]. Let the rotation R(L, n) be defined (clearly not uniquely) by (1. 9c)

L-In = R (L, n) n.

Then

=

R (L, n) R-I (L, n) L

J' dQ(n)f,(5 . -, n)f,(5-, n).

Indeed, we note that

(S, n) becomes

L -IR (n)

•• \ [ k (n) (/" [,) = k(S-ln)

(1.9)

we may write I k (S, n)221 2 ~ k (n)/k (S-ln ). Further, let us write*

o!

which remains invariant under (1.7), i.e., which is equal to

J

Ikl =k(n),

(1.10)

Equations (1.7) and (1.10) define the complete set of representations of the proper Lorentz group with p taking on all complex values, while m takes on only integral and half-integral values. 3 It can be shown that all these representations are nondecomposable and that all except those for whic ip is an integer are irreducible; they are all inequiva'l.ent, except for pairs one of which has param eters p and m, and the other of which has pararr. eters - p and - m. It is easily shown that when p is real, (1. 7) gives a unitary representation of the Lorentz group in the sense that for such representations one can define a scalar product

Ik"laexp{iargk".b}.

I kill k'i.

so that

= [k(n)/k(5-'n)]'-ip/'e'm.(s.n).

991

(n)l' dQ (n)

= [k

(5-'n»), dQ (5-'n).

Then writing S-ln ~ m, we obtain (f;fz) = (f 1f 2 ). These unitary representations form the so-called first class of unitary representations of the Lorent group. The Lorentz group also has another set of unitary representations (the second class), but for our purposes only the first will be necessary. 2. THE TRANSFORMATION OF THE SPIN UNDER PROPER LORENTZ TRANSFORMATIONS

Let Lp (or Lp) be a pure Lorentz transformation such that

-'f? (n)

(2.1)

= R (L, n) R (n) {R-I (n) R-I (L, n) L -IR (n)}. The transformation in curly brackets does not alter the direction of the 3-axis. If we take advantage of its arbitrariness and choose the rotation R(L p ' D) to be about the axis parallel to P x n, the k" matrix element of the transformation in the brackets will be real. (To see this we note that the expression in curly brackets is equal to R- 1 (Lpo, Do) L~'.. Here Po is obtained from p by the same rotation as leads to 0" the unit vector in the direction of the 3-axis from n. The proof is completed by a simple calculation involving the matrices of the 2 x 2 representation.) The angle q>(L, D) therefore satisfies the relation

R (Lp' n) R (n) = R (R (L p' n) n} R, (
where p = (p, E ), and Po ~ (0, M). Let S be any Lorentz transformation. We shall write the transformation SLp as a product of a pure Lorentz transformation and a rotation, i.e., SLp = LR. It is easily seen that here L ~ Lsp. One can thus consider the formula 5Lp = LspR (5, p)

(2.2)

to be a definition of the rotation R (S, p). It is easily shown that R (5,S,. (5,5,)-' p) = R (5" 5;:'p) R (5" (5,S,)-'p).

(1.9d)

(2.3)

From this one easily finds that the association The angle involved in the rotation R(L p' 0) is given in the Appendix.

'YsM

(p,) ~ 'Y~M (po) = R (S, 5-'p) ...'YSM (S-'p, 0') (2.4) S

35 992

CHOU KUANG-CHAO and L. G. ZASTAVENKO

(where RS is the matrix whi~h represents the rotation R (S, S-Ip ) in the (2s + 1) -dimensional irreducible representation of the three-dimensional rotation group') gives a representation of the Lorentz group. This is the representation according to which the wave ·function wsM (p, a) of a particle with mass M and spin s transforms. 3. INTEGRAL TRANSFORMATION

and we obtain E - p.n)-l-iP /2

Y pmn (p, a) = ( ~ ,

(L

)

p'

nY pm

L -1 p n

(0, a),

(3.4)

!':IQting that in (1.9c) R ( L p ' n) n = Lpln, we can rewrite (3.4), using (3.3), in the form E _P"")-t-iPI2

Let us now break up the representation sM defined by (2.4) int~ irreducible representations pm [see (1. 7) and (1.10»). For the representation sM we can define the invariant scalar product

.

e-,m.

Yamn(pa) = ( ~

R!.·(L., n)R"m(n).

(3.5)

Let us now prove that the Y kernel as defined by (3.4) satisfies (3.2). We do this by inserting (3.4) into (3.2). The left side then becomes R (S, S-'p) •• ,R"m (Ls-\., S-'D)exp {- imtp (L s -'" S-ID)}

~ ~ 'I" (p, 0) 'Y (p.,)d'p / E p ,

x {k (S-'n) / k (Ls-\., S-'D)}l+i'/l,

so that in decomposing sM we,can obtain only unitary representations.* Thus the desired decomposition leads to the integral transformation 'l'p.=

±rdP~dQ(n)Yamn(p,a)famn,

(3.1)

m--sO

tamp

=

~

r Ed'p Yamn . (p, a) 'Yp ••

£.J ~ 17--9

(3.la)

P

The kernels Ypmn(P, a) and Y' must satisfy the following condition. If wpa = fpmn according to (3.1) and (3.la), and if Wpa.§.¥pa according to (2.4) and fpmn.§. !pmn according to (1. 7), then wpa=fpmn according to (3.1) and (3.la), for all S. We may write this as the following condition on Ypmn(p, a): R' (S, S-l p)•• ,YpmS-'n (S-'p, a') =

{k(n)/k(S-ln)J-l-ia/2e'm~

(3.2)

(S.nIYpmn(P, a).

The inverse kernel Ypmn (p, a) satisfies a similar condition if we write

and if (3.2) is satisfied. Let us first set p = 0 and S = R in (3.2). Then RS (S, S-Ip ) = RS (R) is simply the matrix corresponding to the rotation R. In this case (3.2) gives

R' (R)"'YamR-'n (0,

a') = e im • (R. nlYamn (0, a).

Yamn (0, a) = R' (D).m.

= Lp

in (3.2). Then

*In fact, only representations of the first class. 1

exp {imtp (S, n)} [k (D) / k (5- 1 n)]-1-<0/2 [k (D) / k (L;ID)j'+iO/1

x exp { - imtp (L., D)} R (L;'D).m According to (2.2), SLS-Ip = L R(S, S-Ip ). Therefore k (L;,.,.s-'n) = k [R-' (S, S-'p) L;'DI = k (L;'D),

since a rotation does not change the magnitude k. Thus the factors k on both sides cancel. Further, R (S, g-'p).. ,R(Ls-"pS-'D).'", = R (S, S-'p) ..R [R-' (S, S-'p) L;'D]"m

=

R (L;'D)...

X exp (-im'l'[R-'(S, S-'p), R-'(S, S-l p)L;'oll.

Thus the rotations on the right and left are also equal; we are left with the angles cpo We use (2.2) to write the last angle obtained in the form 'l'[R-' (S, S-'p), Ls:"~'D].

To this we must add cp (LS-IP, S-In ); according to (1.8) and (2.2), their sum is tp [Ls-'pW' (S, S-'p), SID] = tp [S-'Lp, S-'D].

Taking the angle

4. CALCULATION OF Cmp

Comparing this with (1.9b) and recalling that R3 ( cp) au' = e -iacp oua', we find that

Further, let us set S

while the right side becomes

We have thus arrived at the mutually reciprocal integral equations (3.3)

'1'.... (p, a) = ~) dpdQ (D)l(Ep ---pon) I M]-H'/I

.

X R.." (L •• 0) R"m (0) t ... (0);

(4.1)

36 993

THE SHAPIRO INTEGRAL TRANSFORMATION

these functions, * this equation is easily transformed to the form x

R•• ,(Lp,

n)

R"m (n) 'Y'M (po)

1

(4.2) (l

! 2;:fo (p -

[see Eqs. (3.1) to (3.5)]. To find the still undetermined factor C mp , let us rewrite (4.1) and (4,2) in the form

e"" \,

o(p-p')il(n- n')om",' =

it' p

em, ~

?') =

o

x.h u~"a [0: (p,

[(Ep-p·n)/ Mj-Hi'/'

n)

dl" [(Ep- pt) / Mj- H i'!2

-1-1

(4.4)

+ OJ u~a (oc (p, 1"»

I)

i/,'na (8),

or (p -

(21rf2i)

R•• ·(L p ,

1

~

dl

[(Ep _ p/") 1Mj-I-iP'!'

x

x [(Ep - p·n') I Mj-I-i,'!2 x

P~~ ~

R"m (n) R•• • (L p ,

n') R"m,(n'),

p') :::

, f

£; ~ F a;p's () {J f maps () P •

Gu,p ) p'dp ~

(4.3)

"

a

where We now multiply both sides of (4.3) by Ram'(n'), sum over m', integrate over dQ (n'), and choose n = no along the 3-axis. We thus obtain

°(p-p ')

C mp (d'p Ji:; [(E p-p.n u) / Mj-I+i,/,

I) ma --

I

Fop" (p) = ~ [(Ep - pr) / Mj-I-iP'/2U~a (oc (p, I"» dt";

-,

I

i maO' (p)

~ [(Ep - pi) / Mj-I+i,/,U;,. [oc (pi)

=

In this expression we write R ( L p , n') = R (p/p) x R(Lpo,n")R-1(p/p). Here Po is directed along the 3-axis, and n" is obtained from n' by the same transformation which carries p into Po. Since p'n' =po·n" and dQ(n') =dQ(n"). we have

We now calculate F aps (p). We do this by using the power series

where x = 1 + cos

2q (q -

' - I j ipl!!

-mp ,\

x [(Ep X

R. m (L p ,

R,p.

and

+ o)! (s -

o)! (q

q)!

The calculation gives

pu'n") / Mj-I-ip '!2

:10) Ro. (p I p)

0',

u~. = _----'-(----'-)'_-q-'(~s..:..+-'q"-)! _ _

, (p - p' ) Om" . = c (' d:'p i:; [(E r -p·no)1 Mj

o

+ 8j U~a (0) dl.

-I

x [(Ep - p.n') / Mj-I-i'·/'R.". (I,P' no) R o" (Lp, n') dQ (n').

(p / p) R,r. (L.,n")

tin (n").

F

()

a~s ,p

= ~

p

J ~ q 2q+1 1.:::.J a~a.

q cos (Ap / 2)

q1.

+

(p / 2) sin !Ap /2)

+ (p / 2)2.

q

(4.6)

We shall first perform the integration over the azimuth angles, writing dQ (p)

-c

sin 0 dO d'~, dll (n") = sin 0"
According to (1.1), R(p/p) = R3 (
I"» R,'(,!,"

R(Lp,n") = R,(,," +"'/2) Rdo:(p,

+ "'/2).

Here cosh A = Ep /M, and Q aps (p) can be represented in the form A (p, p) cos (Ap/2) + B (p, p) sin (Ap/2), where A and B are functions bounded for all p > 0 and approaching zero as p - 00 as l/p uniformly in p. We now evaluate fmaps (p). To do this we make use Df the series

Here t = p. n/p = cos Ii and t" = p. n" /p = cos Ii", while the angle a(p, t) is defined [see Eqs. (A.2) to (AA)]. Since R3(
u'mo

(0) = (I

+ x) 1 (m -I- a) /21

(1 _ x) 1 (m

- a)/21

x = cosO.

(2~)'" (p 00

X

I

p')

oma

=

e

mp

Using (A.5) and (A,6), we find that for 1m ± 01.1 .. 0

I

~ P~dP ~ dl ) dl"[(Ep_pl)/Mj-HiP!' o

p -1

*

-1

x a .•

x U~a (oc (p, 1")) U;a (8) u~. (0) oam; here the u~n (x) are functions defined by Gel'fand and Shapiro. 4 By making use of the properties of

X (I

[(s

1I~"n (6)

= (_I),-m i n- m [2' (s -

m)!]-l

+ n)! (s -m)! /(s- n)! (s + m)!I'/' (I _

+ x)-(m-l-n)/2 (d / dx),-n (I _

x),-m (I

+ x),+m,

U~n (6) = u~m (6) = u~n_m (6);

~ U~iCl (Od u!n (62 ) =

u:nn(6 + 6 1

x){m-n 1l2

2 ).

x = cos 6;

37 994

CHOU KUANG-CHAO and L, G. ZASTAVENKO

Ifmap. (p) I < canst / pl+a, where a is the smaller of 1m the sum in (4.4) in the form

±

(4.7)

ex I. We rewrite

I.)

+ 0] a:"a (0) u!. (IX')

= ~u:'. [ot (pi)

+ 0] a~,a (~) U;nm (IX')

] u:'a[IX (p,

a.,-m

Because umm = u-m-m, we see that ex = -m does not enter into the last sum. Using the summation theorem for the umm functions, we can write (4.4) in the form ~

/) (r - r') = (Z,,)2C".,

+ ]

t

~ P~dP F mp's (p) F m _ o

p

f ma.. (p) [F ap', (p) -

os

(p)

F .. ,., {P)]}.

(4.8)

1Q.1+!11I1

It follows from (4.6) and (4.7) that only the product of the principal parts of Fmps and F m - ps contributes to the 0 -function. All the other parts of the integral in (4.8) can only give a finite contribution (which must cancel in the sum). Thus

a(p_p') _

co

(21r)'C

\'

J o

mp

X

~ q'tos().p'/2)

.t.J

(q')'

q'

Since

+~ =

(f/.. ~ q cos (Ap/2) q

APPENDIX A Let so that

+ (p'/2) sin ()p'/2) zq+q'+1 + (p'/2J'

L;ln = R(L p , n)n and
q

q'

cos ()..p/2) cos ()..p' /2) dA

~ sin IX= ~ a

(f/..

=

a(p -

p')

Ep

1 + cos IX =

~ cos ()..p/2) sin (Ap'/2) (f/.. ~ 0,

+M

[1 + __

M_ _ ] Ep-(p-n)'

(E.-p..,+M)' (E M)(E p,oj' p

+

I - cos IX = [qq'

(A.3)

I

It follows from this that (see Appendix B)

_

p'+(2m)'

mp-~.

(Ep

+

M)(Ep

(A.4)

pon)'

It is now easily shown that

+ (p/2)'j ~marm2q+q'

~\q. + (pl~)'j \(q')'+ (pI2)'j'

qq'

C

(A.2)

p

p'-(p.n)'

o

we arrive at ""

,

This leads to

co

16

I

which is implied by (1.2) and (2.1), we easily arrive at

sin ()..p/2) sin ()..p'/2)

-

+ M) -p][Ep- (pl1W

Lpin = [nM +p(p'n)/(Ep

a.ma'm·

o

emp

(A,I)

which determine R (Lp, n), and from the formula

and

_1_ _

R (Lp, n),

From the conditions

o

+r

a be the vector of the rotation

R (Lp, n) n =cos
+ (p 12) sin ().p/2)

q' + (p 12)'

~

by (lol), and R ( L p ' n) is defined by (lo9c) and by specifying the direction of its axis perpendicular to both p and n. By R!b we denote the matrix elements of the (2s + I)-dimensional irreducible representation of the three-dimensional rotation group which correspond to the rotation R( Equation (4.1) gives the expansion of the wave function "'sM (p, a), transforming like a wave function of a particle with spin s and mass M according to the s, M representation [see (2.4)] in terms of the functions fpm (n), transforming according to the irreducible representations of the proper Lorentz group. Equation (4.2) gives these irreducible components of "'sM (p, a) in terms of the function itself. The authors express their gratitude to Professor M. A. Markov for his interest in the work.

+ cos (IX + 0) =

l-cos(1X

(4.9)

+ 0) =

where t = cos

(J

(£p

+ 1)/(£ -

pi),

(A.5)

+ p)(I-I)/(£ -pi),

(A.6)

(Ep-p) (I

= p.n/p.

APPENDIX B 5. CONCLUSION Thus our final result is that if a relation of the form (4.1) exists, the inverse relation is given by (4.2) with C mp given by (4.9). In these equations Ep = (M 2 + p2)1/2, the rotation R(n) is defined

We shall here prove (4.9). We have " q q+q' LiO a¢2 !q'

qq'

qq' + (p/2)' _ + (pI2)'lIq" + (p/2)'j -

1

q q2 tf1

'Q 2Q'tf1'

2 ~ q' + (p/2)' ~ q+q' . q

q

Comparing the equation Aq == '" I-J 2q' a q' / (q + q') q'

38 THE SHAPIRO INTEGRAL TRANSFORMATION with u~m =

L

a q ' Jj:q', we see that

q' 2

Aq =

I \ d s zq J XUmmXq-l o

When q > m, integration by parts shows simply that Aq = o. When q = m, we easily obtain

Am

= (_)s-m (s-m)I(2m-I)1 {s + m)!

2ma:m2 m

1 I. S. Shapiro, Dissertation, Moscow State University (1955); Dokl. Akad. Nauk SSSR 106, 647 (1956), Soviet Phys. "Doklady· 1, 91 (1956). 2 M .!. Shirokov and Chou Kuang-Chao, J. Exptl. Theoret. Phys. (U .S.S.R.) 34, 1230 (1958), Soviet Phys. JETP 7, 851 (1958); Iu. M. Shirokov, J. Exptl. Theoret. Phys. (U .S.S.R.) 34, 717 (1958), Soviet Phys. JETP 7,493 (1958); E. P. Wigner, Revs. Modern Phys. 29, 255 (1957). 3 M. A. Naimark, Usp. Mat. Nauk 9, 19 (1954). 41. M. Gel'fand and Z. Ia. Shapiro, Usp. Mat. Nauk 7, 3 (1952).

Thus

~= m 2 +\p/2)2.

q,q'

995

Translated by E. J. Saletan 302

39 {

S.B

N uclea,. Physics 9 (1958/59) 652-654;

© North-Holland Publishing Co .• A mste,.dam

A NOTE ON THE I>ECA Y OF THE X-HYPERON AND ITS ANTIPARTICLE CHOU KUANG-CHAO Joint blstilute of Nuclear Research, Dubna, USSR

Received 17 July 1958

Abstract: This note examines the information which may be derived from mea.surements of the branching ratios and angular correlation of E- and ~-decays.

1. Branching Ratio Recently Okubo in a very interesting paper 1) proposed an experiment on the measurement of the branching ratio in the decay of the };+ hyperon and its antiparticle ~+. A difference between their branching ratios will indicate the violation of invariance both under charge conjugation and under time reversal. To obtain his results Okubo utilized the lowest order perturbation theory for the weak interaction. In view of the great importance of this experiment as a test of the validity of invariance under time reversal we shall show in ~his note that Okubo's results are quite general. They can be derived from the Liiders-Pauli theorem together with the requirement of unitarity of the S-matrix. The probability of the decay process is characterized by the following elements of the R-matrix (i,

i,l. mlRli, m)

where i denotes the total isobaric spin of the nucleon-meson state; i the total angular momentum of the system which is equal to the spin of the I-particle; m the eigenvalue of the operator i.; 1 the orbital angular momentum of the nucleon-meson state; and R is connected with the scattering matrix S by the relation S = l-iR. Okubo's derivation rests on the relation (7) in his paper 1) which can be expressed in the form (i, i.l,

mlRli, m>c = Io(-l)l exp

(2i<5 t • I .,)
1, mlRli, m>'"

(1) where the subscript c on the left-hand side denotes the matrix elements for the antiparticle process; 10 the product of the inner parities of the I, N particles and the 3l-meson; "t,I" the phase shift of the 3l-N scattering in the 652

40 A NOTE ON THE DECAY OF THE X-HYPERON AND ITS ANTIPARTICLE

fi53

corresponding state. The relation (I) is proved in reP) by considering only the lowest order perturbation in the weak interaction. Applying the leT-theorem to the transition matrix elements we obtain according to reP) i.l. mlRll. m)c = -Io(-I)I(i. i.l. -mIR+li. -m)*

= -

I.,(-l)~(i.

i.l. m\R+li. m)*

(2)

where R+ denotes the hennitian conjugate opl!rator of R. The unitarity condition for the S-matrix leads to the relation -i(R+-R) = R+ R.

Multiplying both sides of (3) on the left by the state (i, right by Ii, m) we get -i(i, i, I, mIR+-Rli, m)

=

(3)

i, I, ml

and on the

[exp (-2ic5 i • J,,)-IJ(i, j, l, mlRlj, m).

(4)

In obtaining (4) we note that the only possible intennediate state which gives a contribution to the matrix element (i, i, I, mlR+ Rlf, m) i"; the nucleon-meson state if the electromagnetic and other weak interaction effects are neglected. This neglect does not correspond to the lowest order perturbation theory, since we are now dealing with the R matrix, whose elements are connected directly with the observed probability of the corresponding process. A simple calculation shows that the contribution of the neglected intennediate states (e.g. nudeon +meson +photon) is at least of the order ~ = 1/137 times smaller than the main contribution. Similarly for the antiparticle process we have -i(i, j, I, mIR+-Rli, m>c = [exp (-2ibi • 1.,)-IJ(i, j, I. mlRli. m>c.

(5)

The phase shifts for the n-N scattering are taken equal to that of n-N scattering. Applying (2) to the left-hand side of (4) and (5) we get (i, 1·, l, mIR+-Rii, m)c

= Io(-l)'
(6)

and the relation (I) follows directly from (4), (5) and (6).

2. Angular Correlation Measurement of the angular correlation in the decay of the L'-particle and its antiparticle will also yield valuable information on the validity of in variance under charge conjugation and under time reversal. The simplest case will be the decay of 1:- and its antiparticle. The asymmetry parameter IX defined in ref. 3 ) has the form

=

Re A* BI(IAI2+IBI2}

A = B =

<'J, :1, o. mlRli, m)

~

with

(7)

41 654

CHOU KUANG-CHAO

where, for simplicity, the spin of the I-particle is assumed to be t. The asymmetry parameter for the antiparticle is denoted by OCc' From (1) and (7) we get cos(ds-dp-Js+J p ) (8) oc cos(ds-dp+Js-J p ) where we have denoted the phase shifts of the s-state and the p-state by 6. and 6p respectively; J s ::md J p are defined by the relation A =

IAI exp [i(ds+J s)],

B =

IBI exp [i(dp+J p )].

The :'t-N phase shifts are given by ref. 4 ): ~B-dp =

_17°

(at E

R::I

120 MeV).

If the degree of polarization of };- and its antiparticle can be determined independently and the deviation from I, T and C invariance is quite large, the deviation of cx.c/cx. from unity may be detected by experiment. A deviation of occ/oc from unity will indicate violation of both T and C invariance.

Formula (8) retains its form when the spin of the 1: particle is t. In this case as' op' J. and J p should be replaced by the corresponding quantities . th e sa t t es 1,. =2", 3 . 1y. III 1. = ~2' l = 1 and 1,' = ~. 2' 1 = 1. 2' l = 2• respective A similar but more complicated formula holds for the deca) of E+ and its antiparticle. Finally, we note that the lack of asymmetry in the decay of the 1:-particle 5) may be due to the accidental equality os-op+Ja-J p ~ ±90°. In this case we may expect quite large asymmetry in the decay of its anti· particle.

References 1) S. Okubo, Phys. Rev. 109 (1958) 984 2) S. Watanabe, Revs. Mod. Phys. 27 (1955) 40 3) T. D. Lee, J. Steinberger, G. Feinberg, P. K. I(abir and C. N. Yang, Phys. Rev. 106 (1957)

1367 4) H. A. Bethe and F. de Hoffman, Mesons and Fields, Vol. II (1955) 5} International conference on mesons and recently disc(.Jvered pal ticIes, Padova-Venice (1957)

42 1106

LETTERS TO THE EDITOR The latest experimental data show that the probability of the {J decay or (I'-, v) decay of hyperons is apparently considerably smaller than that given by the universal (V -A) theory. 7 But even if these data are confirmed and we are forced to abandon the universal interaction including the hyperons, nevertheless the universal interaction between (p, n), (e, v), and (I'-, v) will remain an extremely probable hypothesis. Since it already seems that the {3 decay and the decay of the I'- meson can be explained within the framework of the (V -A) theory, it is of particular interest to examine the process of I'- capture and test the idea of the universal (V -A) interaction in this case. In the present note we present expressions for the probability of the capture, the angular distribution, and the polarization of the emerging neutrons in the case of capture of polarizec! 1'-- mesons by protons, based on the assumption of the universal ( V-A) coupling with conserved vector current as proposed by Gell-Mann and Feynman. The capture probability and the angular distribution are (in units Jj = c = 1): w = (2"r'

("a~r'2-1G'1

1 = 1 +31-'

1,,=

[I

+ ~(I + A') +

+ "p (nos)J p'dQ., ~1-'(21- + ~1-'/2),

I--A'+~(I +A')-~1-'(2)'+~1-'/2).

The total probability for capture is

THE UNIVERSAL FERMI INTERACTION AND THE CAPTURE OF MUONS IN HYDROGEN CHOU KUANG-CHAO and V. MAEVSKII

,-1 = 2-1G2,,-2a;;:3p'l. The nolarization by ~~~ formula I [I

<

of the neutrons is given

+ "P(nos)] = [a + b (n oS)J n + cs, + ~A + ~I-'(I- + ~1-'/4)J,

Joint Institute for Nuclear Research

a = - 2 [},(I, + I)

Submitted to JETP editor September 13, 1958

b =-~P[A(I-+ 1)-1-'(1 +A+~1-'/2)J,

J. Exptl. Theoret. Phys. (U.S.S.R.) 35,1581-1582 (December, 1958) THE theory of the universal Fermi interaction with (V -A) coupling proposed by Marshak and Sudarshan l and by Feynman and Gell-Mann2 is supported by all existing experimental evidence on {3 decay. To explain the equality of the Fermi {3 -decay constant and the constant for the I'- decay, Feynman and Gell-Mann have put forward the hypothesis of a conserved vector current in the weak interactions. This hypothesiS leads to the appearance of an anomalous magnetic-moment effect in the {3 decay ("{3 magnetism"). Calculations of the effect have been' made by Gell-Mann and others 3- 5 and have apparently already received experimental confirmation in {3 decay. 6

c

=

2P (I- - I) [A +

~

(A + 1-') /2J.

Here we have used the notations: pn is the momentum of the neutron, and Ps is the polarization of the I'- meson (n and s are unit vectors); A = - CA/CV, {3 = p/M; G "= 2 1/ 2 Cv is the universal coupling constant; M is the nucleon mass; I'- = I'-p - I'-n is the difference of the magnetic moments of the proton and neutron; al'- = (ml'-e 2 ) -I is the radius of the mesonic K orbit in hydrogen. Terms of the order (v / c)2 have been neglected in the calculation (v is the speed of the neutron). In addition it has been assumed that in the pseudovector coupling the higher moments do not give an additional renormalization. A numerical computation (taking G = (1.01 ± 0.01) x 10- 5 M- 2, I'- = 4.7, 11.=1.24, {3 = 0.1) shows that the effect of the anomalous magnetic moment increases the total

43 LETTERS TO THE EDITOR probability of capture by "" 17 percent, and the angular correlation coefficient by a factor of about 2.4; the polarization of the neutrons is changed only slightly as compared with the results of the ordinary theory. At present calculations are being carried out on the capture of mesons by nuclei accOl"ding to the Feynman - Gell-Mann theory. It can be expected that in this case also the correction introduced by the anomalous magnetic moment wiU be large. 1 E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109, 1860 (1958). 2R. P. Feynman and M. Gell-Mann, Phys. Rev.

109, 193 (1958). 3 M. Gell-Mann, Test of the Nature of the Vector Interaction in {3 Decay (preprint). 'J. Bernstein and R. Lewis, The Vector Interaction in {3 Decay (preprint). 5 Weinberg, Marshak, Okubo, Sudarshan, and Teutsch, Phys. Rev. Letters 1, 25 (1958). 8 Boehm, Soergel, and Stech, Phys. Rev. Letters

1, 77 (1958). 7 F. Eisler, R. Plano, et aI., Leptonic Decay Modes of the Hyperons (preprint).

Translated by W. H. Furry 329

1107

44 SOVIET PHYSICS JETP

VOLUME 36 (9), NUMBER 1

JULY, 1959

CHARGE SYMMETRY PROPERTIES AND REPRESENTATIONS OF THE EXTENDED LORENTZ GROUP IN THE THEORY OF ELEMENTARY PARTICLES V. I. OGIEVETSKII and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor July 15, 1958 J. Exptl. Theoret. Phys. (U.S.S.R.) 36, 264-270 (January, 1959) The extended Lorentz group, including the complete Lorentz group and charge conjugation, is considered. It is shown that the use of irreducible proje,ctive representations of this extended group requires the existence of charge multiplets. Charge symmetry and associated production of strange particles follow from the invariance under reflections and charge conjugation and from the conservation laws for the electric and baryonic charges. The PauliGUrsey transformation holds for free nucleons. The extension of the condition of invariance under this transformation to the case of interactions leads to isobario invariance for strong interactions of all particles. 1. INTRODUCTION

I

T is known that the strongly interacting particles form charge multiplets (p, n, 11'+, 11'0, 11'-, K+, KO, etc.). Particles belonging to the same multiplet have almost identical masses and identical spins, but differ in their electric charges. In agreement with experiment one adopts the hypothesis of charge symmetry and the stronger hypothesis of charge independence. In the conventional theory this is expressed by the invariance under rotations in a certain formal isobaric space. The particles of a given multiplet are considered as states of the same particle with different projections of the isobaric spin. For example, the proton and the neutron form the nucleon. The description of the nucleon makes use of a reducible eight-component representation of the full Lorentz group. We have a similar situation (reducibility of the representation of the full Lorentz group) for the other strongly interacting particles. Now the question arises: if it is required that the elementary particles be described only by irreducible representations, would it then be possible to extend the Lorentz group and to find irreducible representations of this extended group which automatically lead to the existence of charge multiplets and charge symmetries? The solution of this question is the subject of the present paper. We extend the Lorentz group in the following fashion. The wave functions of quantum theory are complex functions. The operation of charge conjugation C, which takes a particle into its antiparticle,

is always represented by the product of a linear operator (matrix) and the antilinear operator of complex conjugation: C:

C.rV,

(1)

where Co is determined such that 1/10 transforms aocording to the same irreducible representation of the proper Lorentz group as 1/1. Besides the proper Lorentz group L, the spatial reflections I, and the time reversal T, we also inolude the charge conjugation C in the extended group. Together with the conventional irreduoible representations of the extended group, we also consider its irreducible projective representations. * The importance of using the projective representations of the full Lorentz group was pointed out by Gel'fand and Tsetlinl in connection with the theory of parity doublets of Lee and Yang. The possibility of using projective representations is connected with the indeterminacy of the phase factor of the quantum theoretical wave function. Subsequent to Gel 'fand and Tsetlln, the projective representations of the full Lorentz group were discussed .We are given a projective representation of the group G, if an operator R(g) Is given for each element g of the group G such that the operator R(g,lo) - a(g" Io)R(g,)R(Io) corresponds to the product of group elements g,lo. If a(g,. 10). 1. the projective representation reduces to the conventional one. In general. a (g". 110) can also be equal to - 1. Anticommuting operators of the projective representation may thus correspond to commuting elementa of the group. In particular. the conventional spinor representation is a projective representation (the operations I - Y. and T - Y.Y. antlcommute ••whereu the spatial and time reflections commute).

179

45 180

CHARGE SYMMETRY PROPERTIES McLennan. 2

by Taylor and Taylor notes the connection between these representations and the isobaric invariance. As only the full Lorentz group is conSidered, protons and neutrons, and n± and ?To mesons, differ with respect to their spatial parities. Salam and Pais, at the Seventh Rochester Conference, also discussed the need for a new definition of the operations of space-time reflections from which the charge symmetries would follow. In this paper we do not consider all irreducible projective representations of the extended group. We restrict ourselves to those necessary for the description of the strongly interacting particles. We shall show that multiplets, charge symmetries, and associated production of strange particles are immediate consequences of the standard conservation laws for the number of baryons and electric charge, and of the invariance with respect to the full Lorentz group and charge conjugation, if the nucleons, :=: particles, and K mesons are described by the new projective representations of the extended group, while the remaining particles are described in the usual fashion. In our theory the Pauli-Glirsey transformation holds for free nucleons. This transformation is connected with the isobaric invariance in a natural way. If the requirement of invariance under this transformation is extended to the interaction Lagrangian for the nucleons, the isobaric invariance for strong interactions follows for all particles. In this theory the Lagrangian for the interaction with electromagnetic fields can be easily written down with the help of the charge operator. It appears that the electromagnetic interactions are invariant only under Wigner time reversal, but not under Schwinger time reversal. The case of weak interactions, which do not conserve spatial parity, is more complicated and will not be discussed in the present paper. To be definite, we shall assume that the relative parities of all baryons are identical and that the reflection of the conventional spinors is performed with the help of the operator y,. All bosons are considered as pseudoscalars. We start with the discussion of the nucleons.

f=,,~,

a) I: b) T: c) C:

f

It is easily shown that the requirement I'

= T' = C· = I,

(2)

leads, for the case of four-component spinors, to the following expressions for the operators I, T, and C:

iI41'~'

(3)

where T is the Schwinger time reversal for spinors,3 and the matrices Yi are expressed in the Pauli representation. The following commutation relations hold between the operators I, T, and G: a) IT=-TI,

b) IC=-CI,

c) TC=-CT.-

~)

We retain relations (2), but we require that, in contrast to the conventional theory, the sign in relation (4a) is changed for nucleons, i.e., we demand that a) IT = TI,

b) IC = -C/,

c) TC = -CT.

(4')

The commutation relations (2) and (4') can only be satisfied by 8 x 8 matrices: a) b) c)

~'= t, X I'~' y' = 1 x I'~' ~,= it, X I'~',

/:

T: C:

(5)

where T are the Pauli matriCES. These operators, together with the operators of the proper Lorentz group (in which YIJ should everywhere be replaced by 1 x YIJ)' form the irreducible projective repre:sentation of the extended Lorentz group. The spinors 1/J = (

~~)

have eight components.

In this representation the Lagrangian for the free field is uniquely determined:* L =

f(l

x I.a;+

it,

x l,mH,

(6)

where 1/J = 1/J*T (1 x y,). The field equations have the form (7)

The Lagrangian (6) as well ~s Eq. (7) are invariant with respect to the two one-parameter groups of transformations ~' =

exp (i).) y,

f =

exp [itl X I"]~'

(8) (9)

and with respect to the three-p&rameter group

4' = 2. THE FREE NUCLEON FIELD

=

~,= iI2~',

a~

+ /;t, X I'~"

(10)

where 1a 12 + 1b 12 = 1. The transformations (9) and (10) are the analogs of the Pauli transformations' for the neutrino. They are different only in that Ys is replaced by Tj X Ys and T3 x Ys, respectively. *According to Schwinger L ~ L T, where the superscript T signifies transposition of the operatOls of the Hilbert space. 3

46 CHARGE SYMMETRY PROPERTIES

We introduce the new four-component spinors o/P=(~1+'rso/,)/V2,

o/n, =

o/p,=

(- 'rS0/1 -I- <j>2) / Vi,

(11)

('rSo/Cl- of'2) 11/2,

they satisfy the ordinary Dirac equation 'r"a"<j> = -m<j>,

(12)

Under the transformation (9) we obtain <j>~

=

exp (i).) <j>p,

<j>~ = exp (i).) of., (13)

o/~, = exp ( - ii,) ofpc,

only consider interaction Lagrangians without derivatives ):

(o/cl+'rSo/'2)/V;i;,

Yn =

Lo = igOt:3 X 'rsho

Transformation (9) may thus be viewed as a gauge transformation connected with the conservation law for the number of baryons. iJ!p, I/!n, ¢pc, and iJ!nc refer to the proton, neutron, antiproton, and antineutron fields respectively. The following transformation should be related to the conservation law for the electric charge:

=

igo (P'rsP - n'rsn) 1'0 = igoiiN~3'r'
(19) the meson field
L~

<j>;" = exp (- i).) <j>",.

181

=

g~>~~2 X l>f'f'~ = ig; (P'roP

+ /i'ron) 'P~'

(20)

'f"

=

T: C:

;0'

='f'> = 't'

;Oc

-;0,

(14)

we similarly obtain the unique expression Indeed, under this transformation: F: <j>~ = exp (ii) h,

L = ig (~<j>,'P' - ~,<j>1') = 2ig (p'r,n'P' -+- "'r,P'f')'

Y~" . exp ( - ii.) <j>'>I' (15)

The three-parameter trnnsformation (10) is isomorphic to a rotation in the isobaric space. Indeed, if we form the conventional eight-component nucleon field iJ!N

=

(~~),

we have, under the transformation

(21)

We may thus assume that

(10), (16)

where A is a vector with real components AI' AZ' A3; T are the usual 2 x 2 Pauli matrices, and a = cos

i).1

sin

i';(i'l

If we now require that not only the free nucleon Lagrangian, but also the interaction Lagrangian be invariant under transformations of the three-paramo eter group, then g = go/12 = grr/12 , and

(17)

An analogous isomorphism arising from a purely formal doubling of the number of components was pointed out by GUrsey. 5

(23)

Under the transformation (10) the meson fields transform according to (~'''i'

..

e,'pli("A))(~,,,)expl-j(~'A)j,

(24)

3. INTERACTION' OF NUCLEONS WITH ORDINARY where the masses of the rr+, 71"-, and rro mesons BOSONS must be identical. We first consider the interaction of nucleons with We have thus arrived at the conventional isoa neutral pseudoscalar field
c:

(18) 'toe = Yo'

The requirement of invariance with respect to I, T, C, transformation (9), and E lead to a unique interaction Lagrangian (in the following we

4. FREE K MESONS

The conventional representation of the extended Lorentz group for bosons is exhausted by the 7r mesons. We shall describe the K mesons by the projective representation in which

47 182

V. I. OBIEVETSKII and CHOU KUANG-CHAO I' = I,

C' = I,

T' = - I , (25)

IT=TI,

IC=CI,

TC=CT.

The simplest irreducible representation consistent with these commutation relations is twodimensional,

(:~).

'P =

The operators I, T,

and C have the form

I:

'fl' = -,~,

T:

'jl' =

C:

1'c= cp'.

i'll'.

(26)

We identify the ~ meson with 'Pt, the K meson with 'P2, the K- meson with 'Pt, and the KO meson with 'Pt. The conservation law for the electric charge is then connected with the transformation =

exp

L = ig[1-(1 x"-~Jxl)(1 xl +"xl)1'xYo - ;j;,(1 x,. +"1 X I) (I x 1-"3 X I) 9' xYul + Herm. conj. (33)

O

E: 'P'

equations (31) and (32), since the transformation T for the nucleons anticommutes with the transformationS related to the conservation of the baryonic charge. If we choose (31) for T, then the only form of the Lagrangian invariant under the transformations of the extended Lorentz group and the transformations (9) and E is

[+ (1 -I- "3) ),] 1"

(27)

Indeed, with this transformation,

In going from I/J and 'P to the operators of the nucleon and K meson fields, we obtain the usual form for the Lagrangian for the interaction of nucleons with Ao particles:

L=

igA

(p,.AoK+

+ !1"AoKO) + Herm.

conj.

(34)

The transformation law (32) for T leads to a Lagrangian that differs from expression (33) only by a plus sign between the two terms of expression (33). It corresponds to the I: particle:

°

KO' = KO.

1(0' = 1(0.

(28)

The conservation law for the hyper charge corresponds to the transformation 'P' = exp (i T3A) 'P. so that

K+'

=

K-'

= exp(-i}.)K-,

exp (i}.) K+ ,

KO' = exp (n.) KO. 1(0' = exp(-i}.) Ko.

(29)

5. INTERACTION OF K MESONS WITH NUCLEONS. A AND 1:: PARTICLES We now turn to the investigation of the interaction of K mesons with nucleons. Since both the K mesons and the nucleons transform according to the projective representation of the extended Lorentz group, and since the baryonic charge is conserved, an additional baryon must necessarily participate in the interaction. This requirement inevitably leads to the law of associated production of strange particles. We first consider the case of a neutral baryon. We already said earlier that the relative spatial parities of all baryons are, for definiteness, assumed to be identical: I:

(30)

For the nucleon transformation T there are two possibilities for the unknown baryon: y~ = -

",.y

oc,

(31)

T: (32)

We have to introduce the antibaryon Yoc into

(35)

We thus arrive at the conclusion that the transformation laws for Ao and I:o corresponding to the transformation T for the nucleons, differ by their signs. We now consider the interaction of nucleons with charged baryons. We can find a Lagrangian which is invariant under charge conjugation, space inversion, and time reversal, and which is consistent with the conservation laws for the electric and baryonic charges, only if we require that, under T, T:

~+ -+ -"~i. E;-. E- --> - "~i. Ed'

(36)

This implies that the masses of the I:+ and I:particles are equal. The Lagrangian has the form L = - ig [;;;(1 x I - ~I x,.) (I x 1- '3 X I) 1'* xE+ +

.fc(1 x I +

~ J X ,,) (I x I +"3 X I) l' x E-l + Herm. conj. (37)

In conventional notation this can be written in the form L

= ig~± (p,.~+Ko

-I-

nT, ~-K+) + Herm.

conj.

(38)

The charge symmetry is obvious. If we now require that the interaction Lagrangian also be invariant under the Pauli-Giirsey type transformation (10), we obtain (39)

The masses of I: +. I: -, and I: 0 must be equal.

48 V. I. OGIEVETSKII and CHOU KUANG-CHAO We arrive at the usual isobaric-invariant Lagrangian L = ig,,[(p'I'.toK' - n'l',toKO)

+ V-"2 P'I'o t+ KO + V2n-r.t-WI + Herm. conj. 6.

:=:

(40)

PARTICLES

Still another possibility remains within the framework of these representations. In the projective representation (5), we can change the sign between the terms 1 x 1 and TI x 'Y5 in the transformation (14) connected with the conservation law for the electric charge: E:

.p'= expH-(1

x l-~l x T6P]
(41)

At the same time we retain the transformation (9) connected with the conservation law for the baryonic charge. In all formulas we then simply have to replace p by ::::0, n by Z-, ~ by KG, and KO by K-. We again have charge symmetry. If we require invariance under the Pauli transformation even in the case of interaction, we obtain the usual isobaric-invariant Lagrangians.

the new irreducible projective representation of the extended Lorentz group. We showed that the existence of charge multiplets, ch'U'ge symmetry, and associated production of strange particles are consequences of the standard conservation laws. The isobaric invariance follows from the invariance under the Pauli-Giirsey type transformation for free nucleons. This transformation is applicable, since the number of components of wave functions transforming according to projective representations necessarily.had to be increased. We did not discuss the weak interactions from this point of view. This task is much more difficult and less unambiguous due to the violation of the parity conservation laws. The authors express their sincere gratitude to Prof. Gel'fand for. valuable comments. 11. M. Gel'fand and M. L. Tsetlin, J. Exptl. Theoret. Phys. (U.S.S.R.) 31, 1107 (1956), Soviet Phys. JETP 4, 947 (1957). 2J. C. Taylor, Nnc!. Phys. 3, 606 (1957). J. A. McLennan, Jr., Phys. Rev. 109, 986 (1958). 3J. Schwinger, Phys. Rev. 91, 713 (1953). Paull, Nuovo oimento 6, 204 (1957). 5 F. Giirsey, Nuovo cimento 7. 411 (1958).

'w.

7. CONCLUSION We carried out the program which we set ourselves in the beginning of this paper. We introduced

183

Translated by R. Lipperheide 37

49 SOVIET PHYSICS JETP

VOLUME 36 (9), NUMBER 3

SEPTEMBER, 1959

REACTIONS INVOLVING POLARIZED PARTICLES OF ZERO REST MASS CHOU KUANG-CHAO Joint Institute of Nuclear Studies Submitted to JETP editor May 31, 1958 J. Exptl. Theoret. Phys. (U.S.S.R.) 36, 909-918 (March, 1959) In this paper the group-theoretical point of view is used for the description of the spin states of particles of zero rest mass. Complete sets of operators and of their eigenfunctions for a system of two particles are found in the representation of the momenta and spins. The statistical tensors for the particles produced in a reaction of the type a + b - c + d or a - c + d are obtained in the case in which one of these particles has no rest mass. The most general selection rules are derived for the reaction a + b - c + d in the form of relations between the statistical tensors, under the condition that the space and time parities are conserved. The wave functions are calculated for a system of two identical particles of zero rest mass.

space reflections. Thus if a y-particle has a definite parity, then for each momentum there IN reference 1 a study was made of the relativistic exist two spin states with eigenvalues ± i of the operator i; If, on the other hand, it does not have theory of reactions involving polarized particles a definite parity, then it can have only one spin with non-zero rest masses. This theory is not apstate (of longitudinal polarization). plicable to reactions involving photons and neutriIn the present paper we use the operator i; for nos. In what follows we shall call a particle (a the description of the spin states of 'Y -particles. photon, neutrino, or graviton) with zero rest mass As can be seen from Eq. (1), the operator t is a 'Y -particle. The purpose of the present paper is closely connected with the direction of the momento study the theory of reactions involving 'Y -partum. Therefore we shall work directly in the moticles. The general theory of reactions involving photons mentum representation. We emphasize that a spin has already been extensively developed in papers of vector does not exist for a 'Y -particle, and the Simon2 and of Morita and others;3 these authors de- total angular momentum j cannot be represented in the form of the sum of an orbital angular moscribe the spin states of photons by a vector potenmentum and a spin. Because of this the complete tial, using the Lorentz supplementary condition to sets of operators for the description of the states exclude the longitudinal and scalar components. of a system containing 'Y -particles, given in Sec. Such a procedure is complicated, and it is difficult 2 of the present paper, differ from the sets generto extend it to cases of higher spins. ally used for a system without 'Y -particles. In The group-theoretical description of the spin Sec. 2 we shall find the complete sets of operators states of 'Y -particles has been given by Wigner and of their eigenfunctions for systems of two parand by Yu. M. Shirokov. 4 Starting from the theory ticles. To obtain the formulas for the statistical of the irreducible representations of the inhomogetensors (s -tensors for short) of reaction prodneous Lorentz group, they showed that the spin ucts, and to obtain the selection rules under the states of free 'Y -particles are completely detercondition that space and time parities are conmined by the operator served, we use a method developed by M. 1. Shiro(1) kov. 5,S where j is the total angular momentum of the 'YLandau and Yang have established the existence particle; n is the unit vector parallel to the moof specific selection rules for the decay of a parmentum of the 'Y -particle. i; is a Lorentz-inticle into two photons. 7 Generalized selection variant operator with integral and half-integral rules of this type have been obtained by Shapiro. 8 values of 1. Since the total angular momentum j Because of the peculiarities of a system of two is an axial vector and n is a polar vector the identical 'Y -particles it is of interest to calculate operator t transforms as a pseudo scala; under the explicit form of the wave functions of this sys-

1. INTRODUCTION

642

50 REACTIONS INVOLVING POLARIZED PARTICLES OF ZERO REST MASS

643

tem for a definite ·total angular momentum, angular momentum component, and parity; these functions can be used to obtain angular distributions and polarizations for the decay of particles into pairs of identical 'Y -particles. It is shown that certain selection rules remain valid when parity is not conserved in a decay.

zero,10 so that the corresponding transformation function approaches the form (3). Let us consider the state of a system of two free particles 1 and 2. The rest mass of the system of two free particles is in general not zero. * As is shown in reference 1, for such a system of particles there exists a reference system in which the total momentum is zero and the total angular 2. COMPLETE SETS OF OPERATORS AND OF momentum equals the angular momentum relative THEm EIGENFUNCTIONS to the center of mass of the system of particles. In what follows all results are given in the centerWe denote state vectors of 'Y -particles characof-mass system. They can be expressed in another terized by the momentum p and the spin operator coordinate system, if need qe, by means of the f by symbols I p, /J>, where /J = ± i is the eigen- transformation function (3). value of the operator f. Since f is invariant Let us denote the total angular momentum of under the proper Lorentz group, I/o remains unthe system by J, and the total angular momentum changed by a Lorentz transformation. Therefore of particle 0/ by JO/. We suppose at first that the state vectors in different coordinate systems differ rest mass of particle 2 is not zero. In this case only by a phase factor h can be represented as a sum of an orbital anguL lar momentum la and a spin S2. 1 We introduce Ip, ",)-Ip, ",)' = "I.. {L, p)\L-lp. "'). (2) a new vector J in the following way: where L is a Lorentz transformation; "'/J ( L, p) (4) J = j, + J, = j + 5., j = j, + I,. satisfies the group relation The spin operator of particle 2 commutes with the "I.. (Llo p) "I" (L" L;lp) = "I.. (L,L .. pl. operator J, by definition. Since J and S2 satisfy The explicit form of the factor "'/J depends on the the usual commutation relations for angular mochoice of the coordinate system in which the total menta, J also satisfies these relations. angular momentum is quantized. If we choose the In the center-of-mass system a complete set of axis of quantization of the total angular momentum operators for the two-particle system consists of in such a way that the z axis is parallel to p and the following operators: the x axis to [vxpl, where v is the velocity of J', J., ji, Elo p. (5) the new coordinate system relative to the old for the Lorentz transformation L, then by means of If particles 1 and 2 have definite parities and parity the formalism developed in reference 9 it can easis conserved in the reaction under conSideration, then it is better to replace the operator f 1 in the ily be shown that in this case "'1/0 = 1. * From Eq. (2) we easily find the transformation set (5) by the space-inversion operator I. function between the state vectors of a 'Y -particle To express observable quantities in terms of in two different coordinate systems 1 and 2: diagonal elements of the R matrix (R = S - I, where S is the scattering matrix), we must find (Plo ....1 P.,I'.) the eigenfunctions of the complete set of operators (5) in the representation of the momenta and spins • =VPl/P,'6' (P'I-~ ail 0..,,,,, i = 1,2,3, (3) of the free particles. We write these eigenfunc1=1 tions in the center-of-mass system in the form where aij is the matrix of the Lorentz transfor(6)
p,,)

*1£ the coordinate system is chosen as in reference 9, then '11'(L, p)-exp{i,,
(1.91) of reference 9.

M' =

(V P~ + M: + V P~ + M: ). -

(P.

+ PI)'.

sod is zero only when the rest masses lit. sod M" of the two particles are zero sod their momenta are parallel (P. II Pa).

51 644

CHOU KUANG-CHAO

The functions (6) are determined by the following conditions. 1. If we choose the z axis in the direction n, then
i,

I'-~, p'>,=Co

,o",+""MOpp" IlIiJ.l

(7)

where k is the unit vector in the direction of the Z axis, and C is a constant which depends on J, j, and 1l0!. The relation (7) follows from the equation Jz = ~I + ~2' if the z axis is parallel to n. 2. Under a three-dimensional rotation the functions (6) transform in the following way (like spherical functions): (n, =

L
1'-1. {'2, plJ, M. i. 1'-;. p')

1'-1, 1'2. plJ, M'.

i.

1'-;, p') D;"'M (g), (8)

M'

if the transformation of the unit vector n under the

rotation is written n' = gn (cf. Note 4 in reference 5). In what follows we shall use the Euler angles {
R. 5. The time-reversal operator T is represented by the product of a unitary operator K and the operation of complex conjugation. The phase factor of the function (6) is determined' by the condition 11 K(n, 1'-" 1~2' pIJ. M. i. 1'-;, p')" = 71T

<-

n, I~" I~"

= (_l)1+M (n.

pi J, M, i, 1'-;, p')" pi 1. -M. i, 1'-;. p').

i~', I',.

(9)

Taking n' = k in Eq. (8) and substituting Eq. (7) in the right member of Eq. (8), we get (n, 1'-,.

I',. pi J,

M, j, I';' p') =

Co

,opp' D~,+",.M (gn). (10)

',l,lll

we find the phase factor C from Eq. (9). We thus get (11)

Under the reflection I the function (10) with the constant C from Eq. (11) transforms in the following way: I (n, 1'-1' 1'-,. ~=

I>,[ <- n.

= I~"[ (- I)-H, (n.

I""

j, 1'-;. p')

pi J.

M. j. 1'-;. p')

1'2. P 11, M.

i. -11;.

p'),

(12)

where Io is the product of the intrinsic parities of the two particles. We assume that I2 = 1 in Eq. (12). This condition determines the factor 1)1 apart from its sign: 1)1 = ± (_1) S 2+ i , or I in. Ill. I',. I'll. JH. j. I';' 1") ±(-I)i-i,I;,(n.IJ 1 • 11,.1'11. M.

=

i. -11;. P'~,.

(13)

where i l = 1III I. The eigenfunctions with definite parity I' have the form (n, {1" c

2-'/'(1

1""

1'~1. M.

+ f'I)(n,I"\.

i. /',

p')

:',,1'1.1. M. i. it, p'>,

(14)

where I' is an eigenvalue of the operator I. We introduce new variables t: I'

.=

± (-

1)/-1" r •

Here t = 0 0 r t = 1. In reactions with photons one value corresponds to electric radiation, and the other to magnetic radiation (cf. reference 3). The lack of uniqueness in the sign affects only the value of t, but cannot affect the final observable results for the density matrix. Therefore in what follows we can consider the expression (13) with the plus sign only. Substituting Eqs. (10), (11), and (13) in Eq. (14), we get the eigenfunctions of the complete set of operators J2, J z ' j2, I, and p in the representation of the momenta and spins: in.

where gn is the rotation of the coordinate system zyx into the system zlYlxl (zi axis parallel to n). If we choose the YI axis in the direction of [k x n 1, the Euler angles for the rotation gn are {-tr, J., tr - 1l2' p> so that 1)T = (- 1 )1l1+1l2, and using the relation

pi 1. M,

-1'1' -1'-2.

1"" 1",.

C"' Vi? (2jf 1)'/' PVV----;;~

I'll. M. j. 1'.1''>

.J", ""

J Clll15zl-tl DIJ.,+J.lI,M

C({n)

0

pp.:J

(16)

I t

where

If the rest masses of both particles are zero, we cannot introduce the operator j. In this case we use the following complete set of operators:

(17) The eigenfunctions of the operators (17) are found in a similar way, and have the form

52 REACTIONS INVOLVING POLARIZED PARTICLES OF ZERO REST MASS (n, 1'" 1'., pi J, =

M, I'~, I'~, p')

Y"R (2J + 1)'1' J pYV ~ D•• +."

2d

M

(g.)

, o.,••' o.,.,oop"

(18)

The complete sets (5) and (18) can also be used for a system of two particles witb nonvanishing rest masses. which is usually described by tbe set J Z, J z , (1-, S1. p, where 1 is tbe orbital angular momentum of tbe relative motion, and S = 11 + 12 is the sum of tbe spins of tbe two particles. Between tbe eigenfunctions of tbese sets of operators tbere exist unitary transformations (transformation functions)

645

J2, J z • (1-, S2. p. where 1 is tbe orbital angular momentum and S is tbe total spin of tbe two particles. We get the following results: 1. y + b - c + d. p' (qcXcl!dXd; nc) = [N,/(41t)') [(2id- I) (2id

(J,/;S~,

X }; Y;c XcOdXd

Jq'X'.

x <S;I;",'I RJ'I i.t.",.)'

+ I) 2-' J,I;S;)

Y"yx,. ..

(2i.

+ w'l'l'

<S;I;", I RJ'I j,/,,,,,)

x. (J,j,t,.

JqX. J,M,)

x D~'x (gcg:;') p (qyXyq.X •• ny),

(21)

where

(J. M, j, p.. pIJ'. M'.I. S. p') opp' (21 + 1)'/. (2S + 1)'/,cf~s. W (lSji" Ji,)

= oJJ'OMM'

and (J. M, 1'" 1'•• pIJ'. M', I. S, p') = OJJ,OMM'Op.' {(2/+ 1)/(2J

+ 1)}·I'C7.~:t::cft~:,~".. (22)

where W (abed. ef) is the Racah coefficient.

3. GENERAL FORMULAS FOR THE ANGULAR DISTRIBUTIONS AND THE POLARIZATION VECTORS AND TENSORS FOR THE REACTIONS a + b - 0 + d AND a - 0 + d

Y Oy'

x

Let us first consider the density matrix of tbe y -particles, <~y I p I ~y>. Since ~y takes only tbe two values ± i, we cannot construct tbe stensors from <~y I p I ~y> in the ordinary way. Fano 12 has shown tbat for photons it is convenient to use tbe stokes parmeters. This idea is easily generalized and is adopted in the present work. The Stokes parameters are related to tbe density matrix of tbe y -particles in tbe following way:

where

p

(qy, Xy) are the Stokes parameters,

(20), do not transform like a vector under rotations. The physical interpretation of the stokes parameters given in Fano's paper for photons is also correct for any y -particles. We shall not repeat it. In this paper tbe Stokes parameters will for convenience also be called s -tensors. The calculations of the s -tensors of the reaction-product particles is made by tbe metbod of M. I. Shirokov.s We use the notations introduced in reference 5. If the masses uf particles 1 and 2 in the initial and final states of tbe reaction are not zero. we use the complete set of operators

L (2J, + I) (2J, + I) (2j, + I) (2i. + I) (2J + I) (2q + 1))'/'

(J,i,t,. JqX. J.i,t.) =

F II (J't J't 1 _ I) -- (_I)i.+i,+h+I'C h2Iy . 1 It 2 ,!, /,i'fi1iy ,

II.

~y = iyO"y, and ~y = iya'y. We emphasize that p (1. Xy) defined by Eq.

yO. ' .

= (-

I)'·-Iy ' ' ' ' N,

= (21th)' R' (V'p~P~l-' (P.;

Pd)'.

The sum is taken over q'. X', J'. li, Si. J z• jz, t 2• J. q. X. qy. Xy. qb. Xb· 2. a + b - y + d.

lZ. 82. h. tlo

=

[N,j(41t}'] (2 (2id

x

+ I) (2i. -i- If' (2i. + If']'"

LY;yXyqdXd

(Jli;t;. Jq'/, J,i~t,)

x X Yq.X."b X• (J,/,S" Jqx. J,[,S.)

x D~'x (gyg;;') p (qa/A.Z.; n.).

3. Y + b -

y' + d.

(25)

53 646

CHOU KUANG-CHAO

p' (qy'Xy', qdXd; X

ny')

=

+1)(2ib + It']'!'

[Nd(4rr)'] [(2i d

~ Y;y'Xy,qdXd (J,j't~, Jq'X', J,i~I~)

X

X Y QyXyqbxb (J,i,t" JqX, J,i,t,) X

D~,x (gy'g:;') p (qyXyqbXb; n y ),

then we get in the final state the s -tensors

p' (qyxYqdXd, ny) = (N,/4,,) [2 (2id

X ~ Y;yXyqdXd (iai;I;, M'l,

p; = ( - l)q,+xc+qd+x d p' (qc - )C,qd - Xd; &, - 'P)'

+ I) (2ia + It'],I,

iai;t;)

(i;I;", I R'a Irt., )

D~~Xa (gy) p (qa, Xa),

(27)

where N2 = (21rti)3 R ( Vp2)-1 (Eyl Ea). The sum is taken over ii, tl' j2, t 2, qa' Xa ' q/, X'. In particular for qy = qb = qc = qd = 0 the formula (21) gives the angular distribution of the products of the reaction y + b - c + d for unpolarized incident beam and unpolarized target. In this case our formula agrees with the result obtained by Morita et aI., if we eliminate certain phase factors by means of a redefinition of the elements of the R matrix; Our results confirm the conclusion that Simon's formulas are erroneous. 4. SELECTIOl'j' RULES if parity is conserved selection rules exist in the form of relations between the s -tensors. These selection rules have been obtained in references 2, 5, 6, 13 for the case in which all the particles have nonvanishing rest masses. The results are easily extended to reactions involving y -particles. As an example we shall consider a reaction of the type y + b - c + d. The conservation of spatial parity gives the follOwing equation: (28)

From the properties of the D -function and Eqs, (22) - (24) it follows that:

Yq,X,qdXd (J,I;S;, Jq'z', J,/;S,)

=

(_I)[p[,+J+Q,+'d

XYq"_X,,qd,_xd(J,I;S;, Jq'-z',J,I;S;), YqyX,qbXb

PI = (_1)qy+2iyXy+qb+Xb P (qy, -Xy, qb, -Xb; ny),

(26)

4. a - y + d.

X

(30), and (31) in Eq. (21), we get the most general selection rules holding when parity is conserved; these rules are expressed as follows: if we replace the s -tensors p (qyXy%Xb, n y) of the initial state by

(J,i,I"

(30)

JqX, J,i,I,) = (_ I)J+Qy+qb

/ 1[, It, YqrX.yqb-Xb

(J,i,l"

Jq -/., J,i,I,),

(31)

Equation (31) is obtained by the use of the equations (-1) yq = (-1)Xy for Xy = ± 1 and (-l)h = Itl1t2 (-1 )qy for Xy = O. Substituting Eqs. (29),

The results obtained here for the reaction involving y -particles and the writer's results 13 for the reaction without y -particles are consistent with each other. We emphasize that the spin states of the y -particles are described not by s -tensors, but by Stokes parameters. We shall pot consider here the special cases dealt with in reference 13, for which the results have just the same form. The conservation of time parity gives a relation between the s -tensors of the direct (y + b c + d) and inverse (c + d - y + b) reactions. Such relations have been obtalned by M. 1. Shirokov 6 for the reaction without y -particles. His formulas require only a slight change for the reactions involving y -particles. In what follows we suppose that spatial parity is also conserved. Repeating the calculations of reference 6 step by step and using the same notations, we get the relation for the reaction with y -particles that corresponds to Eq. (8) of reference 6: (qy, XY' qb, "1.6 I W, (9" if, - 9,) Iq" Xc. qd, Xd)

xl Wo (-" + :p.,

if, "-1',) I qy, - XY' qb , - X6),

(32)

where the Euler angles {-'If + . The angles J and
'1',

=

'1','

54 REACTIONS INVOLVING POLARIZED PARTICLES OF ZERO REST MASS F = [(21.

+ I) (21d + 1)]'1. [2 (21. + 1)]-'1.

(33)

and (-1 )
+ I) 00(&) =

p~ (21.

+ I) (21d + I) 0 1(&)'

I I J, M,I";, 1";' p) =

(34)

o~(&, '1'1) -

° (&, 'P.) = + p:",p. (-I)"y] = 0

-

F'p~p-;' [(-I) "yp;P_. + P;Po

-

F'p~p-;' (p' ('1'" &, - 'P.) p), (35)

where P_I> Po, and PI are the Stokes parameters characterizing the spin state of the incident yparticles in the direct reactions; P'-I, Po, and PI are the Stokes parameters for the spin state of the y -particles produced in the inverse reaction with uripolarized initial state. 5, SYSTEM WITH TWO IDENTICAL y-PARTICLES Let us construct symmetric or antisymmetric wave functions with definite parity, total angular momentum, z -component of angular momentum, and energy, for a system of two identical y -particles. The state vector 1J, M, iiI, Ii~, p> transforms under reflection in the following way: I I J, M'I"~' 1";. p)'= 1;"1; IJ, M, -I"~. -1";, p).

where I6 is the product of the intrinsic parities of the two identical particles, which is always equal to unity; 1)1 is a phase factor. Since the state vector 1J, M, iiI, Ii~, p> transforms under rotations according to an irreducible representation of the rotation group with the weight J, a rotation through the angle

I~ ~.I J, M.1";, 1";;

p)

D~;+.;.;+.;

(0, rr, 0)

"lIJ. Z

aD (

The difference J,

647

of the reaction. After reflection the z and x axes change their directions, but the y axis remains unchanged. The "reflectedn coordinate system differs from the original system only by a rotation through the angle 71' around the y axis. Therefore from the relation D~'Ii( 0, 71', 0) =' (-l)J-lioli',_/J we get

,

= 1.(-1)

1-;-;' M \ • 'IJ, .-I".,-I".,p.

(36)

Let us now consider the operator S that replaces the first particle by the second and vice versa. In the chosen coordinate system the operator 8 can be expressed as the product of the operator I, which changes the direction of the relative momentum, and the operator that interchanges the spin indices iii and J.12. Thus Sp, M, I";'~' p') =(-1)

J-IL·-IJ.· 1

.,

•

'IJ,M,-I",,-l"l'P).

(37)

From Eqs. (36) and (37) we get the symmetric or antisymmetric wave vectors with definite parity I', total angular momentum, z -component of angular momentum, and energy

+ 1'I +- S'S + I' S'IS] I J, M.I";, 1";' p) 1i1.;+.;I .• ' IJ, M, 1", 1', p) = A [I

(38)

where J.1 = 2i or 0 is the eigenvalue of the operator 1 ~ I + ~ zl, A is a normalization constant, and 8' = + 1 for particles obeying Bose statistics and S' = - 1 for Fermi statistics. Multiplying Eq. (38) from the left by liz, pi and using Eqs. (19), (36), and (37), we get the wave functions for this system in the momentum-spins representation. l. 1i=2i, lil=liz=±i. (n,l"l,I"" p I J, M, 21. 1', p')

+ S'I') (2'-;h VR / p VV) (2J4~ T' lipp' [D;i.M (gn) 0.,+.,.21 + /' (- l)lWD~2i.M (gn) 0.,+.,,_21]·.(39)

=(1/2 V2") (I x

2. Ii = 0, Ii! = - 1i2 = ± 1.

(n, 1".. 1"2' pi J, M, 0, 1', p)

=(1/2 V2) (I x

+ S' ( - 1/)(2"h VR / p VV)[(2J+ 1)/4,,],"1ipp' DtM (gn) (0",,0..-1 + /' (_1)1 0.,_,0.,1), (40)

As can be seen from Eqs. (39) and (40), the wave functions are nonvanishing only when 8'1' = 1 for 1i=2i, and 8'(-1)J=1 for 1i=0. These are just the specific selection rules obtained by Landau

55 648

CHOU KUANG-C HAO

and by Yang for photons and by Shapiro in the general case. We note that when parity is not conserved the selection rule S' ( - 1 )J = 1 for 1.1 = 0 remains valid, but the factor 0l.lli01.l2-i + I/( -l)J x 01.l1-i01.l2i, which characterizes the polarization of the system, is changed. Therefore in the decay of a particle with spin J < 2i into two identical 'Yparticles a measurement of the correlation of the polarizations of the 'Y -particles not only will give the parity of this particle when parity is conserved (cf. Yang, reference 7), but also will give information about the nonconservation of parity in the decay. The wave functions (39) and (40) can be written together in a single formula: (n,l-'l,1-'2' p I J, M,I-', f', p') = _1__ F. 2d ~ 2V2 pVv 2J +

1)'/' DjA.l+IJ. J I, •. M(gn) a 011',+1',1.",

X (~

(41)

where ai = 1.11> I' = (-1 )J-2i+t,

= (1 + S'I') for I-' = 2i, = (1 + S' (- 1)1) for I-' =

F. F.

O.

By means of Eq. (41) one can easily calculate the angular distributions and the polarizations for the decay of a particle into two identical 'Y -particles. We shall not do this here. In conclusion I express my gratitude to Professor M. A. Markov, M. 1. Shirokov, and L. G. Zastavenko for interest shown in this work and a discussion of the results. 1 Chou Kuang-Chao and M. 1. Shirokov, J. Exptl. Theoret. Phys. (U.S.S.R.) 34, 1230 (1958), Soviet Phys. JETP 7, 851 (1958).

2A. Simon, Phys. nev. 92, 1050 (1953). Sugie, and Yoshida, Prog. Theor. Phys. 12, 713 (1954). 4 E . P. Wigner, Ann. of Math. 40,1490 (1939). Yu. M. Shirokov, J. Exptl. Theoret. Phys. (U.S.S.R.) 33, 1208 (1957), Soviet Phys. JETP 6, 929 (1958). sM.!. Shirokov, J. Exptl. Theoret. Phys. (U.S.S.R.) 32, 1022 (1957), Soviet Phys. JETP 5, 835 (1957). 6 M. 1. Shirokov, J. Exptl. Theoret. Phys. (U.S.S.R.) 33, 975 (1957), Soviet Phys. JETP 6, 748 (1957). 7 L. D. Landau, Dokl. Akad. Nauk SSSR 60, 207 (1948). C. N. Yang, Phys. Rev. 77,242 (1950). 81. S. Shapiro, J. Exptl. Theoret. Phys. (U.S.S.R.) 27, 393 (1954). 9 Chou Kuang-Chao and L. G. Zastavenko, J. Exptl. Theoret. Phys. (U.S.S.R.) 35, 1417 (1958), Soviet Phys. JETP 8, 990 (1959). 10 E. P. Wigner, Revs, Modern Phys. 29, 255 (1957). 11 L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 25, 729 (1953). R. Ruby, Proc. Phys. Soc. A67, 1103 (1954). 12 U. Fano, Phys. Rev. 93, 121 (1954). 13 Chou Kuang-Chao, J. Exptl. Theoret. Phys. (U.S.S.R.) 35, 783 (1958), Soviet Phys. JETP 8, 543 (1959). 3 Morita,

Translated by W. H. Furry 163

56 663

LETTERS TO THE EDITOR

SOME SYMMETRY PROPERTIES IN PROCESSES OF ANTlHYPERON PRODUCTION WITH ANNIHILATION OF ANTINUCLEONS

where p±.o are the momenta of ,r±,o mesons. From invariance with respect to C it follows that

t. (p" =

CHOU KUANG-CHAO

Submitted to JETP editor November 20. 1958 J. Exptl. Theoret. Phys. (U.S.S.R.) 36. 938-939 (March. 1959)

13 1

LET us consider the reaction (1)

We denote the amplitude for it by f (Pi. Pf. ITp. ITa). where Pi and Pf are the relative momenta in the initial and final states. and ITp and ITa are the Paull matrices of the particles and antiparticles. From invariance with respect to charge conjugation it follows that f(p,. PI' ap' aa)= f(-p" -PI' aa' a p)'

(2)

If the initial state is unpolarized. then it is not hard to prove by Eq. (2) that the polarization vectors of the hyperon (PI:) and of the antihyperon (P~) in the final states are given by

where A is a function of the scalar (Pi' Pi) . Measurement of the angular asymmetries in the decay of the I: - and ~ - produced in the reaction (1) gives the ratio

pyJp" PI' pt. PiI'~) = py.(-P" -PI' Pi, Pt. ~).

where aI: and a2 are the antisymmetry coefficients of the decays. As has been shown in reference 1. measurement of the ratio aI: / a2' is of great significance for testing the conservation laws associated with time reversal T and charge conjugation C; this ratio differs from unity only if T and C are not conserved in the decay. Let us go on to the consideration of the two cases

p+ p (ii + n)- YI + Y, +m,,+ + n"- + I-rr:"

(5)

+ Y, + n,,+ + m"- + 1,,0.

(6)

The amplitudes for the reactions (5) and (6) are expressed in the form at = I. 2, ... m. ~=

PI"pt, P;-, ~, ap ' aa)'

I, ... n.

T = I .... I,

(9)

n+ pCp + n) ..... YI + Y. +m,,+ +n,,- + 1,,0 _

Y~

+ Y~ + m,,+ + n,,- + l-rr:",

(10)

where Yi and y~ are obtained from Y1 and Y2 by means of the operator G. which is the product of the charge-conjugation operator and a rotation through the angle 71' around the x axis of the isobaric space. 2 For example

E- = m;+, EO = Gr.D, E+ = Gr.-.

and so on. We denote the respective amplitudes of the reactions (10) by (4)

PEatE/P'llat'll = at E /at1'.'

t, (p"

(p" PI' Pt. Pi" p~) = 13,(- P" - PI' Pi, Pt. p~),

Analogous relations exist also for reactions in which K mesons and nucleons are produced. There' are also a number of selection rules for reactions of the types

(3)

~. apt aa)'

(8)

P y, (p" PI' pt, Pi" p~) =Py,(-P" -PI' Pi' pt· p~),

p + P->E-r- E-.

f, (p,. PI' pt, Pi.

fl(-p" -PI' P;, pt, P~. aa' ap)'

Using the relation (8), we get not only equality of the total cross-sections and the angular distributions of these two processes (IT! and IT2). but also equality of the polarization vectors of the hyperons and antihyperons in the final state (Py and Py ). When the initial state is unpolarized we have

Joint Institute for Nuclear Studies

..... VI

PI' pt. P;-. P~. apt aa)

(7)

'I

(p,. Pf' Pt. PiI'~' apt aa)and ,,(p,. PI' pt, Pi'~' apt aa)'

From invariance with respect to G it follows that fl (p" PI' Pt. Pi. ~. apt aD)

= 'fJ'.(-P,. -PI' Pd. Pil. P~. aa' ap)'

(11)

where 1J = ± 1 is a phase factor. It is easy to show from Eq. (11) that for an unpolarized initial state a1 (p,· PI' Pt. PiI'~) = a.(-p,. -PI' pt. Pil. P~), P y, (Pi' PI' Pd. Pi'~) = PY~ (-P,. -PI' pt, Pi· p~), pY.(pi , PI' Pt. Pi'.~) = Py; (-P,. -P" Pd' Pil. p~). (12)

I Chou Kuang-Chao. Nuclear Phys. (in press). 2T. D. Lee and C. N. Yang. Nuovo cimento 3. 749 (1956). Translated by W. H. Furry 176

57 SOVIET PHYSICS JETP

JANUARY, 1960

VOLUME 37 (10), NUMBER 1

ON THE PROBLEM OF INVESTIGATING THE INTERACTION BETWEEN

'If

MESONS AND

HYPERONS L. 1. LAPIDUS and CHOU KUANG-CHAO Joint Institute of Nuclear Studies Submitted to JETP editor February 27, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 37, 283-288 (1959) It is shown that use of the unitary property of the S matrix makes it possible to obtain some information about the scattering of 'If mesons by A and E hyperons from an analysis of the data on the interaction of K mesons with nucleons. The possibility of studying the 'If-A and 'If-E interactions by examining peripheral collisions of hyperons with nucleons is discussed. THE study of the interactions of 'If mesons with hyperons is of special interest in connection with the determination of the symmetry properties of the interactions of 'If mesons with various baryons. 1. Let us consider the reactions R:+N~K.+N,

K+N~r.(A)

(la) (lb)

+",

E(A)+"~r.(A)+,,

(1 c)

=

(a~

ar.K

a: aI:

E

),

T'

=

( \

ak ak

akI: aKA) ab

aEA,

aAJ(

aAl:

aA,.

(3)

where 0 (0') is the unit vector parallel to the momentum of the particles in the initial (final) state. in the center-of-mass system; Aa and Ba are two complex functions of the energy and of 0·0'. The reaction ampUtude aafl has the form a.~ = A.~ +iB.~(a[nxn'J),

in a range of K -meson energies in which one can neglect channels in which two pions are produced. Since the elements of the S matrix for the reactions (1) are connected with each other by the condition of unitarity, the question arises as to what information about the scattering amplitudes E ( A ) + 1f - E ( A) + 1f can be obtained by studying the cross sections and polarizations in processes (la) and (lb). The first part of the present paper contains an attempt to answer this question. In what follows we assume that the spin of the K meson is zero and that the hyperon spin is !. We further assume that the interactions are invariant under space inversion, time reversal, lind rotations in isotopic space. The reactions (1) are described by elements of the T matrix (iT = S -1) diagonal in the isotopic -spin quantum number, To

aa can be represented in the form

(4)

when the product of the intrinsic parities of all four particles in the initial (final) states is II = + 1, and the form (5) a.p = A. p (an) + B. p (an'), when II = -1. Here Aafl and Ba/3 are two complex functions of the energy and of 0·0'. Let us turn to the analysis of the conditions for determining the T matrix from the experimental data. It can be seen from Eqs. (2), (3), and (4) that the number of real scalar functions involved in the matrixes TO and Tl is 13 x 4 = 52. The invariance of the .interaction under time reversal means that the S matrix is symmetric, and this reduces the number of functions determining the T matrix from 52 to 36. It can be shown further that when the conditions for the S matrix to be unitary are taken into account, the number of independent real functions is decreased by a factor of two and becomes 18. The same result is obtained if we use the general formulas obtained in reference 1. Let us now consider what information can be obtained by studying only processes (la) and (lb). The number of real functions characteriZing these processes is 5 x 4 = 20. They satisfy four relations of unitarity. Therefore only 16 of them are independent.

(2)

whure ak(ak) is the amplitude for scattering R ~ N - R + N in the state with the indicated Villus of the isotopiC spin, 0 (1)~ ab (ab) is lh. amplitude for the reaction K + N - E + 1f in lh. atate with the isotopic spin 0 (1), and so on. Tho spin structure of the scattering amplitude 199

58 L. 1. LAPIDUS and CHOU KUANG-CHAO

200 Reaction

(a) K-

+ p~K--I-p

K--I-p-Ko+n K~ -I- p-K~-I-p K~ + p~K~+p K- -I-p-.A+"o (f) K- +p-.:<-+"+ (g) K-+p-.:
(b) (c) (d) (e)

isotopiC spins 0 and 1, we need to determine in addition two more real functions of the energy and n • n', and two phase factors. For each state with total angular momentum j and orbital angular momentum I = j ±!, the TO matrix can be written in the form

Amplitude

';' (ak + a'k) '/,(ak-a~) 'I,(ak - a~) 'I, (ak -I- ;Z~) aKA

P~ exp (2io;() - 1 i ( _° -,0 'PKl: exp (,oK~)

-

- (a'k~/V6+ab/2)

a~~/Y6 - (a'k,,/V6--ab/ 2 )

The table shows 8 reactions of types (la) and (lb) and their amplitudes. The symbol K~ (~) denotes the long-lived (short-lived) KO meson; ak is the scattering amplitude of the ~ mesons, which is determined in the analysis of the scattering of K+ mesons by nucleons. In what follows we assume that the amplitude ak is already known. In reality the reactions (c) and (d) in the table are the same process. By studying the time dependences of the scattering cross section and of the polarization after scattering (Le., the dependences on the distance to the target), one can determine the amplitudes of reactions (c) and (d) separately. By measuring the differential cross sections and the polarization of the nucleons in reactions (a) - (d) as shown in the table, we can completely fix the scattering amplitudes ak and ak. The experimental data on the cross sections and polarizations of the hyperons in reactions (e) - (h), together with the four relations of unitarity, enable us to determine the reaction amplitudes aia;, ab, and aKA' apart from a common phase factor. Since the expressions for the cross sections and polarizations, and also the unitarity relations for reactions (la) and (lb) are invariant under the replacement

Po: exp (",oE) -

(7)

where the Pa are certain positive functions of the energy, and the 6 a are the phases of the corresponding processes. From the conditions for unitarity of the S matrix it follows that 0"Xl:. =0°",8° K E'

fJo1:. =poK={I-·(PK",.)'l';'. ...

(8)

pk,

The quantities .ok, oia; can be determined apart from a common phase factor by studying processes (la) and (lb). The quantities p~ and o~ are then determined to the same accuracy from the relations (8). Thus for 1T-~ scattering the difference of the phases in the various states with zero isotopic spin is completely determined by the study of the reactions with K particles.* For the states with isotopic spin (1) we have instead of the matrix (7)

_ i

PKe"'K -1

pI(Eei&K~

PKr:./ 5K r:.

p.t.e21&~ - 1

(

i5KA

(9)

i5.EA

PEA e

PI(Ae

Here and in what follows, we shall write instead of ph(6h) simply Pa(6 a )· From the unitarity conditions we get (10)

ph+p~+ph= I, COS

(20"

+ 20 K - 20K~) (PKPK")' + (PKO:P,,)' -

(PKAPO:A)'

(11)

2Po:PK (PK")' COS

a~~ -» ei5~ (E)akI:'

ip~" exp (ioh) ) ° 0 -,0 l'

(6)

(OEA

+ 20 K -

OKlo. -

(PKAPE!.)'

-+-

0KE)

(PKPKE)' - (PKE PEl'

(12)

2PKAPEAPKPKE

we cannot determine two phase factors e i6 0 and e i6 1, which are functions of the energy alone. This last follows from the relations that the amplitudes (4) and (5) satisfy by virtue of the unitary property of the S matrix. Since the number of independent real functions involved in TO and TI is 18, and 16 of them are determined apart from two phase factors through the study of processes (la) and (lb), for the complete reconstruction of the scattering amplitudes of pions by A and ~ hyperons in the states with

COS

(20A ,j- 20 K (PKPKAJ'

20 KA)

-+-

(PKAPAI'- (PK"PE")'

~P"PK (PKA)'

(13)

*It may turn out that in carrying out an unambiguous analysis it will he helpful to take into account Coulomb effects and the ene rgy dependence of the S matrix at low energies. We note that the Minami ambiguity exists for the reactions in question. Some possibilities for detennining the parity of the K meson relative to the hyperons through the analysis of the reactions (1) have recently been discussed by Amati and Vitale.'

59 INT"ERACTION BETWEEN

7r

It is easy to convince oneself that even when PK, Pia:, PKA, o~, 0la:, and 0KA are known, the unitarity relations (10) - (13) are insufficient for the reconstruction of the matrix TI. For this we need to know one more parameter in each state (for example, P1:). We note that the relations (10) - (13) lead to some interesting inequalities. Noting that Pet > 0 and I cos () I < I, we get from Eqs. (10) and (11) O
(14)

I (PI(PI(E>' + (PI(EP,), - PkA (1- ph - p~) 1< 2p"P K (PI(")"

(15)

Let us introduce the new notations ph + ph = a,

(PKPJ(")' - ph (I - ph) = c.

PKPk" = b,

Then Eq. (15) can be rewritten in the form

Iapi: + c I < 2bp".

(15')

From Eq. (15') together with Eq. (14) we get max{o;

+

~-+ Vb'-ac}

< P" < min {VI-Ph;

a

+Vb'- ac}.

(16)

The inequality (15') holds only for b 2 - ac 2: O. Consequently, the observable quantities PK, PKA, and Pia: must satisfy this inequality. Similarly we have I }< P max { 0; b, -01 - -al b'1 - alcl VEh

Q2

where a, =0 Pk"

+ ph =

a,

+ p -> r. + p + 0

+ E± --+ ~o + E±

IT+

+ p --+;t' +;to + p

bl~PX"PI(PI('"

b, =oPx (PI(")"

(Pxpx,,)' - ph (I - rid, c,=o(p"P KA ) ' - ph(l-ph)· Cl =0

2. Recently Chew 3l1d Low, and Okun' and Pomeranchuk,3 have suggested that peripheral collisions be studied as a method for determ:.ning interactions between unstable particles. We shall assume that this method can be used for the determination of the scattering amplitude for 1: ( A) + IT - 1: (A ) + 7r through studies of the processes l; + N - ~ (A) + N + IT and A + N - ~ + N + IT. The key point of

(20)

and E-+p--+r.o+P+IT-, and A+p--+r.o+n+IT+, and ~-+p--+IT-+ITo+p etc.

b) Near the pole the amplitudes for the reactions E± (A)

b,

(18)

al=op~,,+ph=a,

ITO

under rotations in the isotopic space. Similarly it can be shown that the amplitudes for the following pairs of processes are equal:

IT+

(17)

+ ~ Vbi-a,c,} ,

corresponds to the pole term, whose residue is proportional to the amplitude for IT - A (~) scattering. Assuming that in the physical region near the pole the reaction 1: + N - ~ (A) + N + IT is determined by the process (19), one can extrapolate its amplitude into the nonphysical region and separate out the residue of the pole term. To estimate the effect of other terms in the physical region near the pole, we shall formulate certain rules that must be fulfilled if the contribution of the pole term in actually predominant in this region. a) In the region near the pole the amplitude of the reaction l; + + p - ~ + + p + ITo is equal to that of the reaction 1: - + p - ~ - + p + ITo. This rule follows from the invariance of the virtual process

A + p --+ r.' + n + ITO

+ ~Vb:-alcl}'

201

the method is that the amplitude for the reaction 1: + N -1: (A) + N + 7r, regarded as a function of (PN-PN)2, where PN(PN) is the four-"Vector momentum of the nucleon in the initial (final) state, has a pole in the nonphysical region, (PN -PN)2 = 11 2 (Il is the mass of the IT meson). It is shown that the virtual process

L+

<min{VI-pl('O; ~. /J.t. ... Xh' 01

max{O; a2 ~-.!..Vb2-a,c,}

MESONS AND HYPERONS

+ p --+ E± (A) + p +

nO

and };± (i\.) + p --+ r;± (A) + p + nO are equal to each other. This rule follows from the invariance of the virtual process (20) under charge conjugation. In our case it is not of any great practical importance, but it can be of interest in other cases. For example, it can be shown that the amplitudes for the reactions K± + N K± + N + ITo are equal. This equality is useful for the determination of the interaction of K mesons with IT mesons, and has been noted by Okun' and Pomeranchuk. c) If the 'nucleons are unpolarized in the initial state, then they remain unpolarized in the final state also. Let us consider a reaction of the type (21)

in the region near the pole (p~ - PA )2

=

1l

2

•

We

60 202

L. I. LAPIDUS and CHOU KUANG-CHAO

assume that the dominant process in this region is E+ -;.

A

+ ~+,

rr+

+ p -;. IT+ + p,

(22)

the amplitude for which is proportional to U(PA)ru (PEl tp A

. PrJ}.

p.'l ar;p,

(23)

where r = 1 if the re1ative parity II of the }; and A particles is -1, and r = Y5 if II = + 1; a71p is the amplitude for the scattering 7r+ + P 7r+ + p. The amplitude for the process (22) does not contain any dependence on the spin operators of the hyperons for II = - 1 (more exactly, it contains a term proportional to rTy, but with a small coefficient); on the other hand, if II = + 1, the amplitude is proportional to rTy' k, where k is the unit vector parallel to the difference

PEI(E"

+ MI:)-PA/(E A + M A).

If in the initial state the ~ + is polarized (polarization vector P), then it can be shown from Eq. (23) that in the final state the polarization vector pI of the A particle is given by

P' P'

=

=

P for

n=

2 (Pk) k - P for

-

evaluate the effect of the non -pole terms, but also to get information about the relative parity of the A and ~ hyperons. In a number of cases the study of the polarization of the products from peripheral collisions can be a source of information about the parities of unstable particles.* The writers express their deep gratitude to Professor M. A. Markov for helpful discussions. t

Bilen'kil, Lapidus, Puzikov, and Ryndin,

J. Exptl. Theoret. Phys. (U.S.S.R.) 35, 959 (1958), Soviet Phys. JETP 8, 669 (1959): Nucl. Phys. 7, 646 (1958). 2 D. Amati and B. Vitale, Nuovo cimento 9, 895 (1958). 3 G.

F. Chew, Proc. CERN Annual Conference,

1958, Geneva, page 97: preprint, 1958. L. B. Okun' and I. Ya. Pomeranchuk, J. Exptl. Theoret. Phys. (U .S.S.R.) 36, 300 (1959), Soviet Phys. JETP 9, 207 (1959). G. F. Chew and F. E. Low, preprint, 1958. (J. G. Taylor, Nucl. Phys. 9, 357 (1959); preprint, 1959.

I,

n = + 1.

(24)

Thus if in the region where the pole term predominates one could measure the polarization of the A particles produced in the reaction (21) with polarized ~, then it would be possible not only to

Translated by W. H. Furry 41 *The possibility of determining the parities of particles by the study of peripheral collisions without considering the polarization has been discussed recently by Taylor.'

61 616

LETTERS TO THE EDITOR

ELECTROMAGNETIC MASS OF THE K MESON CHOU KUANG-CHAO and V. I. OGIEVETSKII Joint Institute for Nuclear Research Submitted to JETP editor May 28, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 37, 866-867 (September, 1959) IN the recent experiments by Rosenfeld et al. l and Crawford et al. 2 it was established that the mass of the neutral K meson exceeds that of the charged K+ meson by ~ 4.8 Mev. On the face of it the sign of this mass difference appears to contradict the concept that the K+ and KO mesons are spinless particles belonging to the same charge doublet. Indeed, if the KO meson has no electromagnetic interactions and the mass difference is of electromagnetic origin then the electromagnetic self-mass of the charged K meson should make it heavier than the neutral one (see, e.g., reference 3). On this basis the above-mentioned authors are inclined to interpret their results as an argument in favor of the Pais hypothesis, 4 according to which the K+ and KO meson do not form a charge doublet and may have different intrinsic parities.

62 617

LETTERS TO THE EDITOR It is shown below that there is not as yet sufficient basis for this conclusion since the mass difference can be explained, within the framework of the Gell-Mann-Nishijima multiplet scheme, by the electromagnetic interactions of the KO meson. Indeed, as noted in an interesting paper by G. Feinberg. 5 a spinless neutral particle which is different from its antiparticle. e.g .• KO• can interact with the electromagnetic field. This interaction results from a virtual dissociation of the K O meson into strongly interacting particles. for example a nucleon 'lnd an antihyperon. As a consequence the KO meson will have an electromagnetic structure. In the general case the gauge-invariant electromagnetic interaction Lagrangian can be written as L = -

i.(x) A.(x)

(1)

where jJ.! (x) is the operator for the total current of all interacting particles. In the f3 -formalism of Duffin and Kemmer the matrix element of the current taken between single K -meson states will have the form* (p'

I i. (x) I P>K c

q

=

v

+ ~f'K (q')) V (p).

if' (2rrr 3e-' qx \P')~. [FIK (q')

-

p'- P.

ii (p') =

u+ (p')

(2[;; -

I).

(2)

where p' and p are the K -meson four-momenta in the fill.al and initial states, v(p') and v(p) are the corresponding wave functions in the f3 -formalism. and F (q2) is the form factor satisfying

+ F2K (q'j,

F K + (0)

=

= FIK (q') - F2K (q'j,

FK , (0)

= 0,

FI{+ (q') = FIK (q') F K' (q')

I, (3)

since the charge of the particle is eF(O). Due to interaction (1) and by taking into account (2) we find for the self-mass of the K mesont .1.m =

v( p ) ' - - -~- ", \ d'q j,.

(21t)t

vu

tv

J

or __ -"",

".,

\"

IFK(q')I'( ("P--q)'

-"(C")'III~dq~-I(p

I

___ 4)' [-",-4 1 ,

F K+(q') = 16m'j(q' ;- 4m')", F K' (q') = -

4Aq'm'/(q2

(6)

then from (5) and (6) we obtain for the mass difference inK' -

inK'

= (m 'Hrr')e'('/3A' - I) ('1:.'-'- I).

= (m/2rr) a

(7)

Comparing with the experimental value of 4.8 Mev we deduce that A "" 2. We note that it will be difficult to observe experimentally other effects due to the interaction under consideration.t Consequently it is not necessary to give up the idea that K+ and KO form a charge doublet in order to explain the observed 1 • 2 mass difference. Both the sign and the magnitude of the difference mKO - mK' could be a consequence of electromagnetic interactions. *As remarked by Feinberg in the case of the,,o meson, which is a truly neutral particle, such a matrix element would vanish as a consequence of invariance under charge conjugation. tExpression (5) for the self-mass may also be derived from the usual theory in which the K mesons are described by second order wave equations and the electromagnetic interaction is introduced in a gauge-invariant manner by the substitution: ~ ieF(-O')A".(x). IThe absence of bremsstrahlung and the difficulties involved in separating the electromagnetic and nuclear scattering for KO were discussed by Feinberg. 5 The most characteristic experiment would involve observation of fast 8 electrons from KO mesons. However the KO -e scattering cross section is very small at low energies. Consequently the effect will be vanishingly small since even a 1-Bev K meson in the laboratory system will have an energy of the order of only a few Mev in the KO-e center-of-mass system.

a/ax". a/ax".-

1 Rosenfeld, Solmitz, and Tripp, Phys. Rev. Lett. 2, 110 (1959). 2Crawford. Cresti, Good. Stevenson, and Ticho. Phys. Rev. Lett. 2, 112 (1959). 3 S. Gasiorowicz and A. Petermann. Phys, Rev. Lett. 1, 457 (1958). 4 A. Pais, Phys. Rev. 112, 624 (1958). 5G. Feinberg, Phys. Rev. 109, 1381 (1958).

(5)

The FK(q2) as a function of q2 can be determined only from an as yet nonexistent exact theory or a full analysis of future experiments, For our purposes it is sufficient to take. for example.

+ 4m')'

Translated by A. M. Bincer 166

63 SOVIET PHYSICS JETP

JUNE, 1960

VOLUME 37 (10), NUMBER 6

DISPERSION RELATIONS FOR THE SCATTERING OF 'Y QUANTA BY NUCLEONS L. I. LAPIDUS and CHOU KUANG-CHAO Joint Institute of Nuclear Research Submitted to JETP editor July I, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 37, 1714-1721 (December, 1959) Dispersion relations for the scattering of 'Y quanta by nucleons with one subtraction are considered. For forward scattering six relations have been obtained which do not contain unknown constants or infrared divergencies.

1. DisperSion relations for the scattering of

y

,.({t, ...

,""'s' of the two invariants Mv Q'

M'v'- Q;(Q'

+ M') [(e'P') (eN)

x [(e'P') (eN)+(e'N) (eP')l i,I~. + ik.,({,l.

(e'~) (n')

Q' (e' P') «P')

M"" - Q' (M'

(1)

+

Q') =

(e' [kxk'])(e[kxk'l)

+ Q')

-[(e'P') (eN)

=t= «'N) «P')]

P' = P-(PK)KIK', K = -i-(k +k'),

p=o, (2)

The scattering amplitude can be written in the form

(.'k) (ep)

=

--.-ri'i'9 '

=t= (e'p)

(ek')

(6)

,in'O

P'=-(PK)K/K'=-MvK/Q'.

I" I = ko =

Mv I y Q2 + M' ,

2Q' = k~ (1 - cos e),

N/l V can be expressed in terms of invariant func-

tions,

k' _ Q' o

(4)

00'

where TIer are the four basis vectors of (2). Gauge invariance requires that e'k' = ek = 0, k~N/lv = 0, and N/lvkv = O. As a consequence, N/l V consists of a sum of eight invariant functions,

(7)

The following formulas are easily seen to be correct:

(3)

NIL" = ~ "1l~ CcJcJ'"1J:",

(e'p) (ep)

=

sin' 9

where K = I K I k; (J is the angle between k and k'; k and k' are unit vectors along K and K': P = k x k'. We prove (6) in the Breit system, where

Following Prange,S we choose the following four orthogonal vectors as baSis vectors: (k- k'), N ~ = iE~,).o P~K,Qo.

(e'k) (ek')

I x 11"'1 sin' 6 = Sfiii9

M'v' - Q' (M'

M'v'-Q'(Q'+M')

P = +(p+p'),

f

(5)

The normalization factors Q2/! M 2 v2 _ Q2 (Q2 + M2) I have been introduced for convenience. It can be shown that the following relations hold in an arbitrary system (thus, in particular, in the Breit system and in the center of mass system):

We introduce the notation

Q=

and

,

+ ikolf.J + Q'[M'v'-~' (Q'+M')] (e'N) (eN) [.,({. + lkolf.l

e~N~.,e, = M'v'- Q'(Q' + M') (e'P')(eP')[.,({,

(e'N) (eN)

k+p =k' +p'.

= - PK

Q2:

quanta by nucleons have been considered by a number of authors. I - 6 The presence of infrared divergencies, however, limits the applicability,of these relations for the analysis of the experimental data. In the present paper we derive dispersion relations in a form which is convenient for practical application. In a forthcoming paper 7 we shall use these relations in the discussion of the scattering of y quanta near the threshold for meson production. Before discussing the dispersion relations for the scattering of y quanta by nucleons, we consider in somewhat more detail the kinematics of the process. Let k and k' be the momentum four-vectors of the incident and the scattered photon, and p and p', those of the incident and scattered nucleons. These quantities are related by the conservation law

= M'v' - Q' (Q' +M') = Q' + M'

12 (1 + cos e).

(8)

(9) (10)

Multiplying (9) and (10), we obtain k~

T

.•

SIO

e=

Q'[M'v'-Q'(Q'+M'lI Q'+M'

With the help of (8) we write (11) in the form 1213

(11)

64 1214

L. I. LAPIDUS and CHOU KUANG-CHAO k~sin'9

4Q' [M'v'- Q' (Q'

=

+ ,11')11 M'v'.

(12)

From (5) and (7) we obtain, finally, O='---='~;-'·"'.-;-i M'v'

,

Q (Q. +M

J

M~'.!. (e'x) (ex') = Q

4

It is easily seen that for () - 0 we obtain (E = M. ko = v)

+

R, + R.I._ o = [oK, - oK, - v (oK. - oK,)1. R.I,=<> = v'oK ,12M, R,I.~. = - v'oN ,12M, R. + R.I.~o = {v'(oN, - oK,) + MvoK.I /4M.

(0' .. ) (0.. ') = (o'k) ~ek') •

k~sin' 9

Sin

9(13)

Using the formula N

= -

YQ'+ M' }

(14)

[kx'k'I.

(20)

l

the remaining equations of (6) can be proved in an o.
RIc (ee') + ROc (ses~) + iRac (a [e'x en + iR" (a [s> sel) + iR .. [( ak,) (s~e) - (ak~) (see')!

=

t

iR .. (ak~) (sA - (ake) (e's,)I,

(15)

where R I• ~. and R5 describe the electric, and ~, R,. and Rs describe the magnetic transitions; e and e' are the polarization vectors before and after the colliSion; s = k x e. s' = k' x e', where k and k' are unit vectors in the direction of the momentum of the 'Y quantum before and after the scattering; the label "c" refers to the center-of-mass system. The expression for oK containing the terms not higher than those linear in the energy of the 'Y quanta l2 ,13 can be written in the form oK = - e'M-I (ee') + ieM-' (21'- - e 12M) Ve (a [e'x

en

+ 21'-"'e (a {sc"s~IH ieM-'I'-'e \(ek',)(as;)- (e'k,)(as,)]. (16) With. the help of the relation (as') (ek') - (as) (e'k) 2(a(e'x~l)

= -

+ (ak') (es') -

(17)

(ak) (e's),

we can bring (16) into the form (15). Here R~

= -e"l M, R~ = 0, R~ = - 2(e/ 2M)'v" R: = -21'-"''' R: =

o.

R~ = (e/ M) I~v,.

(18)

We write the scattering amplitude in the Breit system in the form (15), and obtain from a comparison of (15) and (5) the relations R,sin"O = E(oK,cos9 -\- oK,) I M -ko(oK,COSO R,sin"O = - E (oK, cos 0

+ oK,),

+ .... ,) I M + ko (oK, + oK, cos 9),

R.=kgoK ,/2M, R.= -kgoK ,/2M.

(I +

COSO)oK._~~

(2M

I

QI

< Qmax =

+ mw) (6M' + 9Mm. 4- 4m!) 4M (M + m.)' m!;::::: 3m~.

where m7r is the mass of the 7r meson. Regarding the forward scattering amplitude. one usually restricts oneself to two dispersion relations for the functions RI + R2 and ~ + R, + 2R5 + 2Rs. It is seen from formulas (20) that for () = 0 we actually have four dispersion relations for RI + ~. ~, R" and Rs + Rs separately. 2. The retarded causal amplitude for the scattering of photons can be written in the form a (P') N~~I u(P)= -2~'i(pop~/ M')'I. x ~d'ze-i"
rj~ (T) ,[j~- i)] I P).

il (p') N~~u u (p) =

x ~d'ze-'"

(2,,·) i (Pop~ / M'l'/.

-

x oK._~ (l-cosO)oK •.

(22)

D~, ~" (N~':

+ N~~V) ! 2,

A~,

(N~'; - N~·) / 2i.

=

(23)

Taking the complex conjugate on both sides of (21) and recalling that j", is a Hermitian operator. we get ~N~~"t (p' k' pk) ~

= N~~t (p - k' p' - k).

(24)

Changing the order of j", and j v in the commutator in (21) and replacing the integration variable z by - z, we obtain N~,~ (p'k'pk) ,= N~~u (p' - kp - k').

(25)

Substituting (5) in (24) and (25). we have

.,{f;., (v, Q') =

"""1.3

(-v, Q'), """;..('1, Q')

=

,If •.• ( -

v, Q'),

,If

= --"""2.,

,= 0#. =

(-v, Q'),

O.

(26)

With the help of (26) the dispersion relations can be easily written in the for~

(l-cosO)oK,.

k: k =-2M(oK.+oK.cos6)+2~(1

j,(-T)]lp).

We define the quantities DJ,Lvand A,.Lv by

?

. '0 R .sm

(21)

Analogously. we can write for the advanced causal amplitude

o#~ .• (v, Q')

k~ . '0 =2M(oK,COSO+oK,) RIsm

-2~

It can be shown -' from the general principles of quantum field theory that the """i. as functions of II. are analytic for () = 0 (Q2 = 0) and for

D 1.3.5,6, (v Q') = ..:. p 'it

+ cos 9)

~. v' A

(v' Q')

1. 3. 6. • ' \1'2_,,1

d V,'

o

(19)

D2.4 (v,

'1

,,' Q'

Q')=z.v,P\~f' 1t J ,,'2 _'J'J. £". co

o

(27)

65 DISPERSION RELATIONS FOR THE SCATTERING OF 'Y QUANTA Let us consider the disperSion relations for Q2 when the Breit system cOincides with the laboratory system and II becomes the energy of the 'Y quantum in the laboratory system (1.s.). We obtain two dispersion relations by simply setting Q2 = 0 in (27). Four more relations follow if we first differentiate with respect to Q2 and then set Q2 =O. Since the dispersion relations for oK2., in the e 2 apprOximation contain infrared divergencies of the type 1/11 - 1/110. we find that for Q2 = 0 the only dispersion relations of practical use are those for the combinations

= O.

R. + R.=L..

R.+ R,,,,,,L ••

'llI 3• and 'llI2 the amplitudes for the magnetic dipole and quadrupole transitions. We must further introduce the amplitudes C' (!Illa• lEi). C' ('2. !Ill3). C'(16'3. !Ill2)' and C'(!Ill2.1E3) correspondingtothe transitions from the states !Illi into IEk. Invariance under time reversal requires that C' (!Ill,. lE2 ) = C' (IE,. !Ill.). C' (IE•• !Ill,) = C' (!Ill•• ',). (31) Finally. using the technique of projection operators. we find for states with j :s %

+ 21E. + 216. cos 6 - !Ill,. Roc = rol, + 2ro1. + 2ro1. cos 6-IE•• IE, -IE, + 2IE, cos 6 + f !Ill, + Vtl c' (',. 'llI,). RIC

R.. = -1£, which do not contain terms which become infinite for 11-0. For Q2 = 0 we have L = L(II). and it can be shown by comparing (20) and (26) that L ..... (-v)=-L •. , .• (v).

(28)

We can therefore write down the following dispersion relations for these quantities

Re L. (v) _ Re L. (0) = ~ pC'" !m( ~l (V'!) dv'.

l "

n

•

Roc =

rl

2...

tt+ 2~= -e'/ M. If. = O.

1m L ••••• (v') (y" _ y')

v"

dv'.

We shall derive two more disperSion relations in the following section. From (18) we find e'1 vo"

ReL.(O)=-e'/M. vReL.(O) = 'Mf£v= xr:r'M).."· e, f£= 2M /0.,

+ R, (v) \ =

-

..

y'

'"\'. (v') dv'

M + 2,,' P l

y"-v' ;

(29')

v,

it then coincides with the relation obtained by GellMann. Goldberger. and Thirring. I 3. In the center-of-mass system the quantities R tc ••.•• Rsc can be expressed in terms of the scattering amplitudes in states with definite angular momentum and parity. Let us denote the amplitudes for the electric dipole transitions with total angular momentum %and % by lEI and 1E3• respectively; let 1E2 be the amplitude for the electric quadrupole transition with total angular momentum and likewise 'llIt.

%.

roI~ = - 4f£'Yc / 3.

!D/~ = 2f£'vc/ 3. !Ill~ = O.

Co (IE•• !Ill.) = - ef£vc/ M.

(33)

R. = k~oK2/2M and R. = k: oK ,/2M with respect to Q2 and then setting Q2 = O. The factor IG in R3 and R, goes to "z for Q2 - 0 and thus compensates for the possible infrared divergency in oK2 and oK, • Let us now consider the quantity k~

R,= 2M

(30)

The first of the relations (29) is brought into the usual form by using the optical theorem: Re [R1 (v)

1E';-1E~= -[2(e/2M)2_eJ.L/MJ.c.

sion relations by differentiating

. ie'vov vRe L.(O)= -2 (')~ 2M V= -2'MMv.'

n L' (0) 2.. 1.,. VoMYo"v v"e. = - 2J.L--=-2'M)..

C (rol •• IE,). C (IE•• !Ill.). (32)

In addition to (29) we can derive two more disper-

(29)

~

V6 C' (roI,I£,) os -IE, VB C' (1E•• roI,) "'" -!D/. -

co (IE•• !Ill.) = o.

o

.

-!D/, -

USing the relations (16) and (18). we can separate out the energy dependence of these quantities for "C - 0 in the form

v v -v

Re L •. •.• (v) - v Re L •. •.• (0) = -;- P

= IE,

Rae = R.. = roI. - roI. + 2!1ll, cos 6 + fIE, + Vii C' (rol•• ',).

and for the quantities

Ld-v)=L.(v).

1215

oK, =

Mv' oK. (v.Q') 2 YQ'+ M"

If M2

(II. Q2) is an analytic function of II for Q2 < Qfuax. then Rs and 8Rs/aQ2 will also be analytic functions of II. Since the contribution of the one-nucleon state to R3 has the form

Dw~ / (v~ - v,).

where D is some constant and lib at once clear that

= Q2/M.

it is

i.e .• the contribution of the one-nucleon state to the dispersion relation for 8R3/aQ2 for Q2 = 0 reduces to zero. If we restrict ourselves to the states included in formulas (32). we have (34)

66 1216

L. I. LAPIDUS and CHOU KUANG-CHAO

For these same states ~z (v)/v 2 will be an analytic function of v whose crossing symmetry agrees with the symmetry of ~. Then the dispersion re-· lations for ~2(V)/Vz can be written in the form "

(v)

Re ---;r

2v

= -;;

rJ

1m

~. (v') dv'

,,'2 (v'S _ "I) =

o

r n~

2v

1m

~. (v') dv'

.,,'2 (v'l

1t

r

1m l. (v') dv'

~ '1//1(,,'1_,,2) .

(35)

dQ (q.) [T:;_n+ (q+, k', e', a) Ty_r.+ (q+, k, e, a)]

Combining (33) with the dispersion relation for separately the dispersion relation (35). Analogously, by considering the derivative of R" we can show that \1llz also satisfies the dispersion relation (35). For the states included in formulas (32) it was possible, in the final result, to obtain six dispersion relations for the eight quantities characterizing the scattering of y quanta. We did not succeed in removing the difficulties connected with the presence of infrared divergencies in the other relations. The repeated differentiation with respect to Q2 gives rise to the appearance of unknown constants, which have been calculated by some authors with the help of perturbation theory and which can be determined experimentally if sufficient experimental data are available, in analogy to the procedure used in the scattering of mesons and nucleons. In the present paper we shall not employ this method. To carry out calculations with the help of the dispersion relations for the scattering of y quanta by nucleons, we require rather detailed data on the amplitude for photoproduction. From the available data we may conclude that 1m \1ll2 = O. If we further use \1llg = 0, we find O.

- M,{2 ([kxe] q)

-1- E, {(ak)

Rs+Rs. we see that 15 2 (v) and C(l8 Z,\1ll3) obey

=

(k', e', k, e, a)]

k, e, a)],(38)

T y_ n (q, k, e, a) = iE, (ae) - M, WIoxe] q) - i (a[[kxe] q])}

".

\1ll.

=;; ~

_01(

where

or, finally,

2

i[o/t'+(-k', -e', -k, -e, -a)

+ -i;;~dQ(qo)[T:;_",(qo, k', e', a) Ty_",(q"

\12)

'.

Re ~ (v) = 2v'

Neglecting the terms which are quadratic in the electromagnetic interaction, we obtain the relation

(36)

With the assumptions

+ ita [[kx e] q]) + (ae) (kq)}

(eq)

(39)

is the amplitude for the photoproduction of pions on a proton. For the lowest states EI corresponds, as is known, to a transition from a state with angular momentum J = % and negative parity accompanied by the creation of a meson in the sl/2 state; MI corresponds to a transition from J = % with creation of a meson in the PI/2 state; M3 and and E z correspond to a transition from J = + with creation of a meson in the P3/2 state. It follows from (38) that above the threshold

%

{I E,I' + +1 E,I'cosO} =

1m R,c = v,

1m R" = v,{ I M,I' + 21 M,

1'-1 E,I'/6)

which do not contradict the available experimental data on photoproduction, the number of dispersion relations will agree with the number of functions introduced. To calculate the imaginary parts of the amplitude we make use of the unitarity of the S matrix. For y quantum energies below the threshold of 11" meson production the imaginary parts of the quanHties R I , ... , Rs are small. Above the threshold the imaginary parts of R I , ... , Rs are determined by the unitarity requirement on the S matrix.

=

vcA"

1m R" = 1m RIC,

1m R" = vc{ I MII'-IM3I' + i E,l'/ 12 -~- (E;M3

-1- E,M;) / 2) ~ v,A" ImR,,=-v,{[E,I'/6

+ (E;M, ,;- E,M;) / 2) =

1m Roc c= O.

v,A"

(40)

It is easily seen with the help of (40) that the total

interaction cross section for the y quanta is '/"0(4;:/-,,) 1m [R,,(OO)+R 2C (OO)] -=4rr [ I E,l' -I- I MIl' -I-

21 M,l' -I E,I'. 6],

(41)

this agrees, as it should, with the total cross section for the photoproduction of 11" mesons. I ' Using (40) we obtain '" () - .t.. ~\00 ~ I £+,\I"I' _+ '\12I £0 I' ' Re (!)2 v -- 6 1t j 'OJ' I

C (\1ll., 18,) = C (~" \1ll.) = CO (iS 3 , \1ll.) = - £'In 1M,

'1,A"

,

"

~

Re [18, (v) --t 218. (v)] = 00

v2 \

(0 0

+ 2.. ' J d·lv"

+ 0+) v'

0

Re 18,('1) - 2Re 1\1ll,,(v)-\1ll,I.

-

Re (iS l (v) -18, (v)]

= _ [2 (-"-)' _ 2M

ReC(\1ll~,

r

~] ~ j ~ I £i I' + 1£11' M (L v + 'It

18,)=.ReC (~, \1ll,)= •

•

V'

'01'2

'01 2

'

'.

~f ~ (Re£;M,)++(Re £;M,). 1t

.} Yo

v*

v'!!:

,,2

(42)

•

67 DISPERSION RELATIONS FOR THE SCATTERING OF 'Y QUANTA The contribution from multiple 1f meson production and from the prodUction of other particles is neglected. If it is shown by further analysis that 1m ro/z .. 0, it is not difficult to take this into account.

I Gell-Mann, Goldberger, and Thirring, Phys. Rev. 95, 1612 (1954). zN. N. Bogolyubov and D. V. Shirkov, Ookl. Akad. Nauk SSSR 113, 529 (1957). 3 A. A. Logunov and A. R. Frenklin, Nucl. Phys. 7, 573 (1958). 'A. A. Logunovand P. ~. Isaev, Nuovo cimento 10, 917 (1958). 5T. Akiba and J. Sato, Progr. Theoret. Phys. 19, 93 (1958). sR. H. Capps, Phys. Rev. 106, 1031 (1957) and 108, 1032 (1957).

1217

1 L. I. Lapidus and Chou Kuang-Chao, JETP, in press. sR. E. Prange, Phys. Rev. 110, 240 (1958). 9V. I. Ritus, JETP 33,1264 (1957), Soviet Phys. JETP 6, 972 (1957). 10 L. I. Lapidus, JETP 34, 922 (1958), Soviet Phys. JETP 7,638 (1958). 11 M. Kawaguchi and S. Minami, Progr. Theoret. Phys. 12, 789 (1954). A. A. Logunov and A. N. Tavkbelidze, JETP 32, 1393 (1957), Soviet Phys. JETP 5, 1134 (1957). 12 F. E. Low, Phys. Rev. 96, 1428 (1954). 13 M. Gell-Mann and M. L. Goldberger, Phys. Rev. 96, 1433 (1954). U M. Gell-Mann and K. M. Watson, Ann. Rev. Nucl. Science 4, 219 (1954).

Translated by R. Lipperheide 334

68 8.B

Nuclear Physics 10 (1959) 235--243;©Norlil-Holland Publishing Co., Amsterdam

CHARGE SYMMETRY PROPERTIES AND REPRESENTATIONS OF THE EXTENDED LORENTZ GROUP IN THE THEORY OF ELEMENTARY PARTICLES V. I. OGIEVETSKY and CHOU KUANG-CHAO joint Institute 0/ Nuclear Research, Dubna USSR Received 17 July 1958

Abstract: The extended Lorentz group, which includes the complete Lorentz group and the charge conjugation operation, is considered. It is shown that use of irreducible projective representations of tne extended group requires the existence of charge multiplets. Charge symmetry and pair production of strange particles follow from invariance under reflections and charge conjugation and from the laws of conservation of electric and baryon charges. The Pauli-Giirsey transformation is valid for free nucleons. The requirement of in variance under this transformation in the case of interaction also leads to isobaric invariance for all particles in strong interactions.

1. Introduction As is well known, strongly interacting particles can be combined into charge multiplets (p, n; n+, n-, nO; K+, KG etc.). Particles belonging to a given multiplet have almost equal masses and the same spin, but possess different electric charges. The hypothesis of charge symmetry and the more stringent hypothesis of charge independence are then postulated in accord with experimental data. In the usual theory an expression of this fact is invariance under rotations in a certain fonnal isobaric space. Particles of a given multiplet are treated as states of a particle of a given isobaric spin, the isobaric spin projections of these states being different. The proton and neutron, for example, comprise the nucleon. The nucleon can be described by the reducible 8-component representation of the complete Lorentz group. A similar situation (reducibility of the complete Lorentz grcup representation) also holds for other strongly interacting particles. The following question now arises. If one requires that elementary particles be describable only by irreducible representations, will it then be possible to extend the Lorentz group in such a way, and find such irreducible representations of this extended group, as to lead automatically to the existence of charge multiplets and yield the charge symmetry properties? This problem is examined in the present paper. The Lorentz group will be extended in the following way. 235

69 236

V. I. OGIRVRTSKY AND CHOU KUANG-CHAO

In quantum theory the wave functions are complex. The charge conjugation operation C which changes a particle into its antiparticle can always be represented as the product of a linear operator (matrix) and complex conjugation antilinear operator: (1)

where Co is defined in such a way that "Pc is transformed according to the same irreducible representation of the proper Lorentz group as 'P. Besides the proper Lorentz group L, space reflections I, and time reversal T we include in the extended group the charge conjugation operation C. Furthermore, along with the usual irreducible representations of the extended group we shall also consider its projective irreducible representations t. The importance of utilization of projective representations of the complete Lorentz group has been pointed out by Gelfand and Tsetlin 1) in connection with Lee and Yang's theory of parity doublets. The possibility of application of projective representations is connected with the uncertainty of the phase factor in the quantal wave function. After Gelfand and Tsetlin, projective representations of the complete Lorentz group were also considered by Taylor and McLennan I). A relation between these representations and isobaric invariance is indicated in Taylor's paper I). Only the complete Lorentz group was considered, and hence protons and neutrons and n± and nO-mesons differ only with respect to space parity. The idea that new definitions of space-time reflections are required which would lead to charge symmetries was also expressed by Salam and Pais at the 7th Rochester Conference. In the present paper we shall not attempt to study all the irreducible projective representations of the extended group and confine our attention to those which are required for description of strongly interacting particles. It will be shown that if nucleons, E-particles and K-mesons are described by the unusual, projective representations of the extended group, and all other particles are described in the usual manner, then multiplicity, charge symmetry and pair production of strange particles follow from the standard laws of conservation of baryons, electric charge and invariance under the complete Lorentz group and charge conjugation. In the present theory the Pauli-Giirsey transformation is assumed to t If to each element g of group G there corresponds an operator R(g) such that the product of the gropp elements is associated with the operator

R(g,g.) = cx(g,g.)R(g,)R(g.) it will be said that a projective representation of group G is given. If cx(g,. g.) ;!! 1. the projective representation is of the usual type. In the projective representation anticommuting operators of the representation can be associated with the commuting group elements. In particular. the customary spinor representation is projective (operations y, and Y4Y. anticommute. whereal the space and time reflections commute).

70 CHARGE SYMMETRY PROPERTIES AND REPRESENTATIONS

237

hold for free nucleons and is related in a natural manner to isobaric invariance. The requirement that the interaction Lagrangian be also invariant under this transformation for nucleons leads to isobaric invariance in strong interactions between all types of particles. It is not difficult to write down in this theory the interaction Lagrangian for electromagnetic fields with the aid of the charge operator. It is found that Schwinger time reversal is not valid for electromagnetic interactions and only Wigner time reversal holds. Weak interactions in which also space parity is not conserved are more complicated and will not be considered in this article. For the sake of concreteness we shall assume that the relative space parity of all baryons is the same and that reflection of the usual spinors is performed with the aid of operator Y~' All bosons are assumed to be pseudoscalars. We start our discussion with nucleons.

2. Free Nucleon Field If one demands for 4-component spinors that

12

=

T2

= C2 =

1,

(2)

it can readily be shown that the operators I, T and C can be expressed as follows: (3a) I: 'P' = Y~'P, (3b) T: 'P' = iY4.Y6'P, (3c) C: 'Pc = iY2'P*, where T is the Schwinger spinor time reversal operation 3) and matrices y are expressed in the Pauli representation. The following commutation relations hold for I, T and C:

IT = -TI, IC = - CI, TC = - CT.

(4a) (4b) (4c)

In contrast to the usual theory, in retaining relation (2) we require that relation (4a) change sign for nucleons, i.e., that the following condition be satisfied: IT = TI, (a), IC = -CI, (b), TC = -CT, (c). (4') Only 8x8 matrices satisfy commutation rules (2) and (4'): I: T: C: where 1" is a Pauli matrix.

'P' =

1"3Xy~'P,

'P'=IXy~'P,

'Pc = iT3X Y2'P*,

(5)

71 238

V. I. OGIEVETSKY Ar;D CHOU KUANG-CHAO

These operators together with the operators of the proper Lorentz group, in which Yp should everywhere be replaced by I X Yp' fonn a projective irreducible representation of the extended Lorentz group, the spinors tp being 8-component. In this representation the free field tp has a uniquely defined Lagrangian t L

=

°

'ii(l X Yp l'+iT2 X Y5m)tp,

(6)

where 'ii = tp*T 1 X y" and the field equations have the fonn (7)

Lagrangian (6) as well as equation (7) are invariant under two singleparameter transfonnation groups tp' = exp (iA)tp,

(8) (9)

tp' = exp (iTI X Y5A)tp

and the three-parameter group (10)

tp' = atp+lrrs X Y5tpC'

where lal + Ibl = 1. Transfonnations (9) and (10) are similar to the Pauli transfonnations for the neutrino ') and differ from them only in that Y6 is replaced by 1"1 X Y6 and TsXY6' respectively. If we introduce the new 4-component spinors 2

2

1

1

tpp =

y2

(tpl+Yr;¥'II)'

1pPc=

1

tpnc

= y2

y2

(tpCl+Y/i¥'ca),

(11 )

1

(-Yr; '1'1 +tpll),

tpn=

y2

(Yr.¥'Cl-tpCIL

they will be found to satisfy the familiar Dirac equation YI'0I'¥' = -mtp

(12)

and under transfonnation (9) ¥,'p =

exp (iA)tpp,

(13)

Thus (9) can be regarded as a gauge transfonnation related to the law of conservation of baryons, and ¥'p, tpn, ¥'Pc and ¥'nc refer to the proton, neutron, antiproton and antineutron fields respectively. The transformation E: '1" = exp r!i(lXl+1"l XY/i)A]¥' should be related to the law of conservation of electric charge.

(14)

t According to Schwinger, under time reversal one has L ..... LT, where the transposition sign refers to Hilbert space operators 0).

72 CHARGE SYMMETRY PROPERTIES AND REPRESENTATIONS

239

Indeed, under transformation E E:

1p'p = exp (iA)1pp, 1p n = 1pn,

1p/Pe

= exp (-iA)1pPe'

I

1p' ne =

1pnc·

(15)

The three-parameter transformation (10) is isomorphic with rotation in isobaric space. Indeed, if we form the usually employed 8-component nucleon field 1pN = (~:) we obtain under transformation (10) / 1p N

= exp [i(T . A)J1pN

(16)

where A is a vector possessing the real components AI' A2 , As and the T are the usual two-dimensional Pauli matrices and

iA3 ( 17) IAI Giirsey 6) has pointed out that an analogous isomorphism appears when the number of components is formally doubled. a = cos IAI+ -sin IAI,

3. Interaction Between Nucleons and Ordinary Bosons Consider first the interaction between nucleons and a neutral pseudoscalar field CPo with positive time parity

T: cp'o = CPo' C: CPoc = CPo. (18) The condition of invariance under I, T and C and transformations (9) and E uniquely yields the interaction Lagrangian (in the following only the interaction Lagrangian without derivatives will be considered) I:

cp'o = -CPo,

L

igoVTr3XY6WO = igO(V;PY61pp-V;nY61pn)CPo (19) = igO(V;NYr. T 31pNCPO), i.e., the meson-proton and meson-neutron coupling constants have different signs and hence CPo can be assigned to a neutral Jto-meson. If the neutral meson fP' 0 were a space pseudoscalar but would possess a negative time parity, the interaction Lagrangian L

=

=

g'01pT2 X I1pcp'o

(20)

which can be ascribed to the questionable po-meson, would follow in a unique manner. For the interaction Lagrangian between nucleons and a charged pseudoscalar boson field cP T:

and we again uniquely obtain

cP' = cP;

C:

fPc = cP*

73 240

v.

1. OGIEVETSKY AND CHOU KUANG-CHAO

L

=

ig(ip'PclfJ*-1[Jcf/XP)

(21 )

= 2ig(1[JpYs'PnlfJ*+1[JnYs'PpCP)·

Thus it may be considered that cp(cp*) refers to :rc(n+) mesons. Charge symmetry (i.e., the possibility of simultaneously making the substitutions 'Pp:o:± 'Pn, n+:o:± n-, n°:o:± _nO) of interactions (19) and (21) is evident. The general Lagrangian for interaction with n-mesons has the form L

= igo(1[JPYr,'Pp-1[JnY5'Pn) + 2ig(1[JpY/i'Po n ++1[JnY/i'Pp n -).

(22)

If we now require that not only the free nucleon Lagrangian but also the nucleon interaction Lagrangian be invariant under transformations of the three-parameter group we find

and

(23)

L" = ig,,'ijiNYr,(-r' n)'PN· Under transformation (10) the meson fields transform as follows: (-r. n)' = exp [i(-r . ..t)](-r . n) exp [-i(-r . ..t)].

(24)

The n+, n- and nO-meson masses should be the same in this case. We have thus arrived at the usual isobarically invariant theory of interaction between n-mesons and nucleons.

4. Free K-MesoDs For bosons the usual representation of the extended Lorentz group encompasses only n-mesons. We shall describe K-mesons by the projective representation in which

12 = 1, IT = +TI.

T2 = -

I,

IC = +CI,

C2 = I,

TC

=

+CT.

(25)

The simplest irreducible representation in which these commutation rules are valid is a two-dimensional one: cp = (:!) and the operators I, T and C have the form C:

lfJc = lfJ*·

(26)

Let us identify the K+-meson with CPI' the KO-meson with lfJa, K--meson with CPI*' and KO-meson with CP2*' We can then relate the law of conservation of electric charge to the transformation (27)

74 CHARGE SYMMETRY PROPERTIES AND REPRESENTATIONS

241

Indeed, under this transformation K+' KO'

= =

exp (iA)K+, KO,

The transformation q/

=

K-'

=

exp(-iA)K-,

(28)

exp(iTaA)!p for which

K+' = exp(iA)K+, K-l = exp(-iA)K-,

KO' = exp(iA)KO, KO' = exp(-iA)KO,

-

-

(29)

corresponds to the law of conservation of hypercharge.

5. K-Meson-Nucleon Interaction. A and I Particles We now pass to a study of the interaction between K-mesons and nucleons. Since K-mesons as well as nucleons transform according to the projective representation of the extended Lorentz group and the law of conservation of baryon charge is valid, one other baryon describable by the usual representation should necessarily be involved in the interaction. This condition inevitably leads to pair production of strange particles. We shall first consider the case of a neutral baryon. As mentioned above, for the sake of definiteness the relative space parity of all baryons is assumed to be the same: (30) Under the T transformation for nucleons two possibilities exist for the investigated baryon: (31 ) T' y'o = -Y4Y6 Y oe , / . Y O= YcYr;Y oe . (32) The antibaryon Y Oe should be introduced in equations (31) and (32), as transformation T for nucleons anticommutes with the transformation of conservation of baryon charge (9). If the law (31) is chosen for T, it will be found that the only type of Lagrangian consistent with invariance under the extended Lorentz group is

L = ig[1ji(IXYr;-T1 xl)(IXl+T3 Xl)!pxYo

(33)

Changing from tp and !p to the nucleon field and K-meson field operators we obtain the familiar nucleon-Ao particle interaction Lagrangian L

=

igA(PYr;Ao K++iiYr;AoKO)

+ Herm. conj.

(34)

The law (32) for T yields a Lagrangian which differs from expression (33) only in sign between the two terms in expression (33) and corresponds to the Eo-particle: (35)

75 v.

242

1. OGIRVRTSKY AND CHOU KUANG-CHAO

We thus arrive at the conclusion that under transformation T for nucleons the transformations for Ao and Lo differ only by sign. Considering now the interaction between nucleons and charged baryons we find it possible to set up a Lagrangian which is invariant under charge conjugation, space and time reflections and the laws of conservation of electric and baryon charges only if l:+ --+ -Y4YSl:C-' (36) T: l:---+ -Y4y 5l:C+' which implies equality of the l:+ and l:- particle masses. The Lagrangian has the form L = -ig[1ji(lxl--rtXYr;)(IXl--raXl)tp*Xl:+ (37) +1jid1x 1+-rt X Yr;)(1X I+Tax l)tpxl:-]+Herm. conj. In the usual notation it can be represented as follows: L

=

igx±(PY6l:+KO+nY6l:-K+) + Herm. conj.

(38)

Charge symmetry is evident. If we furthermore require that a transformation of the Pauli-Giirsey type

(10) also leave the interaction Lagrangian invariant, we get (39)

where the l:+, l:- and l:o masses should be equal and the usual isobaric ally invariant Lagrangian L = ig.ECpy6l:oK+-nYr;l:oKo +V2PYs l:+KO+V2 nYr;l:-K+)+Herm. conj.

( 40)

obtains. 6. .s'-partic1es

The ideas presented above admit of one other possibility. Thus, retaining the same transformation (9) for the law of conservation of baryons one can, in the projective representation (5), change, in transformation (14) which is related to the law of conservation of electric charge, the sign between I Xl and Tt XY5: (41)

In this case one should simply replace in all formulas p by EO, n by E-, K+ by K-o, KO by K-. We again obtain charge symmetry properties and the requirement of invariance under Pauli-Giirsey transformations in interaction again leads to the usual isobarically invariant Lagrangians.

76 CHARGE SYMMETRY PROPERTIES AND REPRESENTATIONS

243

7. Conclusion The plan set out at the beginning of this paper has been carried out. By introducing a new irreducible projective representation of the extended Lorentz group for nucleons and proceeding from this new representation we have been able to demonstrate that the charge symmetry properties, pair production of strong particles and multiplicity follow from the standard conservation laws whereas isobaric invariance follows from a transformation of the Pauli-Giirsey type for free nucleons. This type of invariance can be incorporated in the theory as the number of components of the wave functions which transform in accord with the irreducible representations is necessarily larger. Weak interactions have not been studied from this viewpoint in the present paper. This is a more difficult problem and less unambiguous owing to violation of space parity. The authors are sincerely thankful to Prof. I. M. Gelfand for valuable discussions.

References 1) I. M. Gelfand and M. L. Tsetlin, JETP 31 (1956) 1107 2) I. C. Taylor, Nuclear Physics 3 (1957) 606; J. A. McLennan Jr., Phys. Rev. 109 (1958) 986 3) ]. Schwinger, Phys. Rev. 91 (1953) 713 4) W. Pauli, Nuovo Cimento 6 (1957) 204 5) F. Giirsey, Nuovo Cimento 7 (1958) 411

77 SOVIET PHYSICS JETP

VOLUME 11, NUMBER 1

JULY, 1960

INTEGRAL TRANSFORMATIONS OF THE I. S. SHAPIRO TYPE FOR PARTICLES OF ZERO MASS L. G. ZASTAVENKO and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor May 28, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 134-139 (January, 1960) An expansion in terms of the irreducible representations of the proper Lorentz group is given for the representation which speCifies the transformation of the wave function of a particle of zero mass and of arbitrary spin.

1rHE correspondence

Ypmn(P,:;);-;-::: CmpYpmn(P, 0).

'1" (p, a)~'. 'F' (p, J) ~ exp (io'fl (S, kilT (S-'p,J),

(1)

where 8 is a transformation belonging to the Lorentz group, p transforms like the momentum vector of a particle of mass 0, k = pip, (J is an integer or a half.:.integer, and

Both here and later a bar above a letter denotes taking the complex conjugate. We note that the function Y ,mn (P, 0) = (1 /

2~)

omao (1 -

(nk» p-Hi",

(5)

satisfies expression (4). Thus, from (2) - (5) we obtain 'I:" (p • m)

.=

~ dp ~ dU (n) 0 (n - k) p-' H,I' f ,mn,

iPl 0.. ( n - k) p -1-/1'/2 'Y (p, m). f ,mn = C'J\ d:1p

(6) (7)

Here we have taken into account the fact that oll-(nk») = 21To(n-k). On substitutJng (7) into (6) we find that C p = 1/411' while the integral over p in (6) should be taken from - "" to + "". Thus, the formulas 'F (p, m) =

'r

dp

~ dU (n) a(n -

k) p-'+i,/'J f,mn,

(8)

-00

1. INTEGRAL TRANSFORMATIONS FOR PAR-

f ~mn

TICLES OF MASS 0

f ""n

--

~ dp'~ dU (n) Y "nn (p,

=

1\/i3p'

~

HPi

0) fpmn,

I'

Y,mn (p, J) l (p, 0),

(2) (3)

where fpmn transforms according to the irreducible representation (p, m) of the proper Lorentz group, and we obtain the following conditions for determining the functions Y and Y': V,"'5-'n (5 'p, 0)

= exp lim;:; (5, n)

- i~:r (S, k)} I/«n),

K (5-' n)I-H"'Y,,,,n (p, ;;),

k) p-H,,, Iy (p m)

(9)

I

give a solution of the proposed problem; it turns out to be simpler than in the case of non-zero rest mass.*

In a manner similar to the way this was done in I we obtain the following system of mutually inverse integral transformations 'l' (p, 0)

I 'J' = _ .. I-P 0 (n 4n .ll p I

(4)

where K(n) is defined by formula (1.9), I. An analogous condition for YfJrnll (p, fT) is satisfied if we take

2. COMPARISON OF THE RESULTS OBTAINED HERE WITH THOSE OF I We compare the results obtained here with formulas (4.1), I and (4.2), I for M = 0 (M is the particle mass). A direct transition to the limit M - 0 in the formulas indicated above is impossible. Instead of this we shall carry out the follow ing formal manipulation of those formulas: we *In the case m = 0 the same representation (p, 0) is in fact contained twice in the result obtained, since the representations (P,O) and (0, p) are equivalent. The transition between these two representations is given by formulas (14) and (15) with m = O.

97

78 INTEGRAL TRANSFORMATIONS OF THE 1. S. SHAPIRO TYPE

98

carry out the transition to the limit M - 0 in the factor R (Lp , n), we introduce a new variable of integration p/M (retaining for it the old notation p) and we. replace the factor h + p2 - P • n by p - P • n. If we now introduce the components of the wavefunction having a definite component of the spin along the direction of the momentum we shall obtain o

'1' (p, Ill)

~

=

df'

The function U is discussed in Appendix A and satisfies the following unitarity condition:*

~ Upm (I,

n) U,,,, (I, k) dD. (I) = 0 (n - k).

m T -Po

-m,n

_';"_JJ dP'P p-HP2 U,m (n,

=

l I

tUt

k) 'E' (p, III).

'"

~ lin (n)

(17)

On substituting (14) and (15) into (8) and (9) we obtain

'1' (p, m) = ~ dp ~ dD. (n) Upm (n, k) p-I "c" '1'_ p,

-m,n.

(18)

(19)

-00

(10) -;

;.'!.~

(:Lrn\2 \~ (h)3 Jlpl

=

'-,~-lIm

We substitute into (18) and (19) the following formula derived in Appendix A:

U.", (n, k)

x [p _ pnj-I-i", e-""O(P.

n)

(_),+mi"j>' (p, 11l).

(11)

if (p, Ill)

2ll;", (k) 'I"

=

(p, 0),

(12)

A. m [I - (nk»)-I-i,;1 Qm (n, k),

(20)

where

A,,,, =

Here

=

2

Hi

r (m + 1 -+

.'.

Qm(n,k)=m ~

I

~

ip /2) / 4-::f(m-ip / '2),

(21)

2[+ 1 - I I 1(l+I)D.,_m(n)D. m (k).

(22)

1:>lml a. __ [

where ~, (p, u) is the wavefunction utilized in I; the notation D~(k) is also explained there. In the derivation of formulas (10) and (11) the following equation was used:

rf)' (k)-' D' (Lpn)D' (n)j",,, where () (p, n) is defined in Appendix B. We shall compare the expressions (8) and (9), obtained above with (10) and (11).* From the function fpmn which transforms according to the (p, m) representation we go over to the function 'P -p-mn, which transforms according to the ( - p, - m) representation. We make use of the fact that the representations (p, m) and (- p, - m) are equivalent. Therefore the function 'P-p,-m,n may be obtained from the function fpmn by the following unitary transformation:

)

~ U pm (n,

Upm (n,

k)fp"'k 1m (k),

k) 9_ p, -m,

ndD. (n),

4n j! piP

-I-ip,~

A pm

x[I-(nk)l-l-iP~Q",(n, k) 'Y(p, m),

""

~ dp ~ dD. (n) p-HiP!'

(23)

A.m

x[l-(nk)l-l "'Q",(n, k)9_ p ,_m,

(24)

Since I 1pm:,,I'

I ((J' -+(4r:)'01 ,

4m')

and Qm (n, k) = rim. (n. k) ( - l)+m

(25)

(cf. Appendi~ C), (23) and (24) differ from (10) and (11) only in that the integral over p in (23) and (24) is taken between the limits from - 00 to 00. In particular, for m = 0 each irreducible representation (p, 0) occurs twice in the expansion under consideration in contrast to the case M .. O.

(14)

APPENDIX A (15)

DEFINITION OF THE FUNCTION Upm(n, k)

where U,m(n, k) =

X

_ ..!.. \" d'p

n -

(13)

M-·O

f,mk =

Cfl_ p , -m,

'1' (p, m) =

___ .>o""_o(_I),+,,,/mo(p.o),

?_,. _"'. n =

We then obtain

r r

(I (I

~

+1 ~~ I

21

+ 1 + ip! 2) - I + 1 _ ip 12) D"

I

-m

(n) D,,,, (k).

(16)

·It is proved later that (10) and (11) are not equivalent to (8) and (9) (and are therefore incorrect). We emphasize that this by nO means indicates that the results of the present paper contradict those of I; it merely means that the formal manipulation which leads to (10) and (11) is not justified.

It may be easily shown that from the fundamental relations (14), (15) and the transformation law for fpmk and 'P-p,-m,n the following functional equation for Upm(n, k) may be obtained: ·In particular, for m = 0 we obtain the foUowing simple integral representation for the Il-function:

[ef. also formulas (A.4) and (A.6»).

79 99

INTEGRAL TRANSFORMATIONS OF THE I. S. SHAPIRO TYPE Upm

(5- 1

1

n, 5- k)

XI [I (l + 1)-1'(/' + I)-ip] + (_)I-/,+,X/,[I' (I' + I)

= Upm (o,k)[K(o)K(k)/K (S-' O)K(S-lk)]-I-ip.'\l X

exp{im['l'(S,nl+'I'(S,k)J)

- I (I + I)-ip] (21 + 1)/(2/' + I) = O.

(A.1)

(the notations K(D)/K(S-I D ) and cp (S, D) are defined in I). Since the functions D~m (k) for 1 = 1m I, 1m 1+ I, . .. and for fixed m form a complete ~ystem, U may be represented in the following form (A.2) On taking in formula (A.!) for S the pure rotation S = R we obtain 9n taking into account formula (l.9b), I,

Further, we take in (A.!) for S the infinitesimal pure Lorentz transformation L:

From this it follows that

+ I) r (l + I + ip /2) / r (I + I -

X I = C (21

ip /2).

On utilizing the unitarity condition (17) already mentioned in the main text we obtain. finally. formula (16). In order to obtain formula (20) we note that the function Qpm (0, k)

== [1- (nk)]HiPI·Upm.(o,

k)

(A.4)

satisfies the same functional equation (A.!) which is satisfied also by Upm(D. k). only we must set in it 1 +ip/2 = O. From this it follows that (I

«m

(

n, k) -- A pm

V

21

1 (I

"-'

+ 1)1 +

I -Do-m ( ) D' (k) n om •

1~lml

In order to find Ap. we set k = -D in (A.4). Cln = (n+ep)!!l + (oep)J.

Since Vi (n) = R.('!'+ ,,/2) R,(O),

It can then be easily seeD that

DI (-n)

K (Clo) / K (D) = 1 + (cpO). Since according to (l.9d). I

then

/m.IL. k)D,;m (L -Ik) = ~ D~y (R (L, k» D~m (k)

we obtain from (A.l) and (A.2)

}J X ;15~_m (n) D!m (k)

=

[K (n) K (k) / K (L -I n) K (L -I k)]'~ ip/'

x}J XIl5~y(R (L, oJ) 15~_m (n) D~~ (R (L, kJ) D~m (k). (A.3) The parameter of the rotation R occurring in the above is defined by formula (A.4). I: DI (R(L, k»

=

O-iIH.)."" l-i(H~),

~

= R.('!' -I 3rr/2)R I (rr-&),

=

[kx:;;].

[DI (n)-'D I (-o)J-m.m = [RI(-&)R.(!t)R,(rr-6)]~m.m im imn = [R, (- rr) R, (")]~m. on = e- " (- 1)'1 (_ l(t-m e-

Since ~

_~_~ (_ 1)I+m

1;:01",11(1+ 1)

=

~ !'.,l.lml

(_1_ (_I)I+m+~(_I)I+m) = l+1

(_1)2on~

1

~ (- 1)1+"1 (21 -i- I) r (l + I + ip I 2) /

r (I + I - i:-

We then obtain from formula (A.3)

}JX;15~._m (n)D!m(k)

= [1- (l

x}J XI {[I - i (HI [n

x

x

+ ip/ 2) ep (n + k)J

'I'll] D' (o)} •. _m

([I-i(HI[k x'I'J)JD I (k)) •.

m.

Here we must express the cyclic components of the vectors D and k in terms of the generalized spherical harmonics OhO(D). Oho(k). and we must then eliminate products of the D -functions in accordance with the following rule Db. (n) D;d (n)

= (I lac I LM)

By equating to zero the coefficient of cp we obtain (in the intermediate steps of the calculation we make use of Racah's rule for combining three Clebsch-Gordan coefficients into one):

m

and. moreover.

= ( - I)''''

r (m -i-

I -i- if' I ~) /

r (m -

i:. /2),

the proof of formula (20) is complete. APPENDIX B PROOF OF FORMULA (13) According to formula (l.9b). I. DS (Lpo) V S (D) = DS (L;' D) R, ('!' (L p, oJ).

Further. from formula (A.2) of Appendix A in it may be easily seen that

L;IO-_>-k. \/---'0

Finally. we have,

I

2)

80 100

L. G. ZASTAVENKO and CHOU KUANG-CHAO

[R (kWl R (- k) =

Rl (_rJ) R,,(-;;- 0.2) R, (;;- 0 -,. 0/2) Rl (;:-~)

=

R,(- 0) PI (0).

Qm (S-l n , S-lk) -, (;." (- k, k)

= (-

1)5+",

Thus from (A.5) and (B.1) we obtain formula (25).

s [D (k)-1 D-'(- k)l,,,,, = [R" (-;:) Rl(r-)l~n

= 0",_", (_1)'+"'. 1 Chou

Therefore, 0(p, n) = lil11?(L p , n).

(B.I)

,\1 ....... 0

Kuang-Chao and L. G. Zastavenko, JETP

35, 1417 (1958), Soviet Phys. JETP 8, 990 (1959). 2 Chou Kuang-Chao, JETP 36, 909 (1959), Soviet Phys. JETP 9, 642 (1960).

APPENDIX C PROOF OF FORMULA (25) In formula (A.1) we set 1 + ip/2 = 0, S = Lp, with

Translated by G. Volkoff 21

81 SOVIET PHYSICS JETP

JULY, 1960

VOLUME 11, NUMBER 1

SCATTERING OF GAMMA-RAY QUANTA BY NUCLEONS NEAR THE THRESHOLD FOR MESON PRODUCTION L. I. LAPIDUS and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor July 9, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 201-211 (January, 1960) The elastic scattering of y -ray quanta near the threshold for single meson production is treated by means of dispersion relations. It is shown that when one takes into account meson production in the s state there are appreciable departures from monotonic variation with energy of the scattering amplitudes, cross sections, and other observable quantities near the threshold of the reaction. On definite assumptions about the analysis of photoproduction in the range of y -ray energies up to 220 Mev, calculations are made of the scattering amplitude and the differential and total cross sections for elastic scattering of polarized and unpolarized y -rays by protons, and also of the polarization of the recoil protons above the photoproduction threshold.

1. The study of the scattering of

y -ray quanta by nucleons is especially interesting near the threshold for single meson production. The region near the photoproduction threshold is of interest not only for comparisons with the predictions of dispersion relations, but also in particular in connection with the studies of departures from monotonic variation with the energy of the cross-sections (and polarizations) near the threshold of the reaction. 1 From this latter point of view the scattering of y -ray quanta by nucleons and nuclei near the threshold for meson production is of especial interest as an example of a process going with a comparatively small cross section and being strongly perturbed above threshold by the process of intense meson production. Thus marked effects can be expected in the region near the threshold. It is clear that a suffiCiently accurate experimental study of the anomalies near the threshold can be useful in understanding the process of pion production near threshold. As a more detailed examination shows, the polarization effects are especially sensitive to the parameters characterizing the photoproduction of pions. Our main purpose here is a detailed examination of the effect of meson production on the cross section, the polarization of the recoil nucleons, and the polarization of the y rays near the photoproduction threshold. Phenomenological analysis and dispersion relations are used to obtain formulas useful for the analysis of experimental data. The results of the numerical calculations, which are based on definite

147

assumptions about the analysis of the photoproduction, must be regarded as preliminary. In making the numerical estimates we have completely neglected fine -structure effects associated with the mass difference of the mesons (and of the nucleons ). There are many well Imown papers in which the scattering of y -ray quanta by nucleons has been treated by various methods (see literature references in our previous paper 2 ). In the present paper we have tried to manage with a minimum number of assumptions, without resorting to approximate methods, whose use is hard to justify. We consider not only the scattering cross sections for unpolarized y rays, but also the polarization effects in the scattering. In this connection we have also considered the polarization of the y rays. 2. Let us represent the transition matrix in the form

Let us choose two coordinate systems x, y, z and x', y', z' in which the z and z' axes are parallel to the initial and final momenta of the photon, and the y and y' axes are in the same direction. In these coordinates the functions for the spin eigenstates of the photon with the eigenvalues Sz = ± 1 have the following form: ~,

=-

(h -

ij)/V2.

~-l = (h

+ ij)/V2,

(1) = - (h' - ij)'Vi, ~:"', = (h' + ij)/V2, where h, J, and k are unit basis vectors directed along these coordinate axes. In the general

~;

82 148

L. I. LAPIDUS and CHOU KU ANG -C HAO

case the polarization state of the photon will be a linear combination, i.e.,

2A

(2)

where c1 and c-1 are the respective probabilities (sic) of the photon states with Sz = + 1 and Sz = -1-

Using the spin eigenstates as the basis of the representation, we can write the transition matrix in the form (3)

Let us further introduce the density matrix of the photon in the form (4)

The density matrix Pf of the final state is connected with the density matrix Pin of the initial state by the relation (5)

Although in Eqs. (3) and (4) the transition matrix and the density matrix are written as three-rowed matrices, they have only four independent nonzero elements. Consequently we can represent them by means of two-rowed matrices and use the well known apparatus of the Pauli matrices. 3 (6)

P=

(elel ele:"!) = -}(I + "yP). C_1C~

c_1c_ 1

+

2A = (~';N~,) \CIN~_,) = SpoK, 2Bz = (~;'N~,) - (C,N~-,)=Sp(O~oIf), (~;'N~_,)

+ (C,N~,) =

+ R,) sin O("n),

2iB y = [R a - R.-(I-cos O) (R,- R.)] " (k- k'), 2B. = (R, - R.) (I - cos 0) + i (R, + R,) sin 0 ("n).

(9)

where n sin IJ = k x k', cos IJ = k· k' . It is easy to calculate the density matrix of the final state: P, = (A + "yB) (I + ",P) (A+ + "yB+) = {- {AA+ + BB+ + (AB+ + BA+) P - ; ([BB+] Pi) + "y {AB+ + BA+ + ; [BB+]+ (AA+ - BB+) P +[B (PBT) + (BP) B+I + iA [pB+]-; [PBI A+}. (10)

+ +

By means of the expression (10) one can calculate all observable quantities. For the interaction of unpolarized 'Y rays and nucleons the differential cross section will have the form d~/da,==/o(~) = +Sp(AA+

+ BB+).

(11)

where the spur is taken over the nucleon variables. Substituting Eq. (9) in Eq. (11), we get 4/0(0) = IR, + R,I' (I + cos' a) + IR,-R.I'(I-cosO)' + I R. + R, 1'(3-cos'a + 2cos a) + IR. - R.I' (3- cos·a - 2 cos 9) + 21 R, + R.I'(I + cosO)· + 21 R, -R.I'(I- cos 9)· +4Re (R.+R.)" (R.+R.) (l +cos 6)' -4Re (R.-R.), (R,-R.) (I-cos 9)'. (12) The expression for the polarization of the nucleon after the interaction of an initially unpolarized photon and nucleon can be represented in the form* 2/. (9) <,,>, = sin 9 n 1m [{R.+ R.)(R, + R,}, (l +cos 9)

{R1R; - R;R.+ R,R; -R;R, +[R,R;-R;R.+ R,R;-R;R.] cosO}.

=2; [kx k')

(13)

The well known fact that the cross section 10 ( 11) does not change when one replaces electric transitions by magnetic appears in the fact that Eq. (12) is invariant under the simultaneous interchanges: (14) It can be seen from Eq. (13) that the expression

Sp (o~.,K),

2iB y = (~L'N~_,) - (~:,N~,) = i Sp (o~.i{)

(R, + R,)(I + cos 0) - i (R.

-(R,-R,) (R,-R.nl-cos~)J (7)

where P x , P y • and P z are the Stokes parameters. Nonvanishing P x and P y correspond to linear polarization of the photons along the x and y axes, while P z " 0 corresponds to circular polarization of the photon. From Eq. (6) it is not hard to get

2B. =

=

2B. = (Ra + R.)" (k + k') -I. (I + cosO) (R. + R.)" (k + k'),

(8)

where the spurs (traces) are taken over the photon variables. The quantities A and Bi can be connected with the quantities R 1, ••• , R s, which were introduced in reference 2 (hereafter referred to as I) and which determine the matrix oIf [cf. Eq. (I, 15»):

for the polarization of the recoil nucleon also remains unchanged by this transformation. 3. Let us now establish the relations between the Stokes parameters and the statistical tensor moments. As is well known, the statistical tensor moments are defined by the relations *Eqs. (19). (23), and (24) in reference 4 contain errors.

83 149

SCATTERING OF GAMMA-RAY QUANTA BY NUCLEONS

Too = 1/ Va, T" = -}

T,D = S,/y2", T,o =

IS; - S!

V+ (-i-S~

+ i (SxSy + SySx)),

T,_, = +[S;- S; -

I (SxSy

(15)

Sp T1MT;'M' ,- "JJ'''MM'' (16) By means of these tensor moments the density matrix can be represented in the form

-+ PloT 10 ~. p,,,T,o + p"T" -+- p,_,T,_ " (17) = 21/2 P20 = 3- 1/ 2. The parameters PJM

PI = PooT"o

I/ere Poo are connected with the Stokes parameters:

P,_. =

y'FP"

Px

';-

IP g • (18)

In virtue of time-reversal invariance,5,4 the expression for the cross section 1(0,
where 2/"(6)

(T,,>, = sin'O(l R,I' -\- I R, 1'_1 R,I' -I Ral'), (20)

i is the initial polarization of the y-ray heam. We note that the expression (20) changes ~ign under the transformation (14). 4. For practical calculations we use the results o[ reference 2. The deviations from monotonic variation of the cross-section for scattering of y rays in the immediate neighborhood of the mesonproduction threshold are due to the production of mesons in the s state. According to the available experimental data, the cross section for production of rr+ mesons in the s state is much larger than the cross section for production of rro mesons in this state. Difficult experiments had to he done even to establish the fact that ,,0 mesons Indeed are produced in the s state. The energy v of a photon in the laboratory syskm (l.s.) and the energy Vc in the center-oflIlass system (c.m.s.) are connected by the well known relation

',,='1/Vlf-LV/M.

qd'l,

=

('I . i m~ / 2/V1)

I V I -+ 2'1 I /III

1M; [' I- I M~ 1'= 61 M" I' 1-

fI!; =

+ vu-m; 12,11)! (I ~~ lIle (I -+ lIl e /2M)

(v - '1 0 ) (v '/ 0

+I M~~

2oMi~ I' "" 61 M,,:'

-

We have calculated the dispersion integrals by using simple expressions to interpolate the energy dependence of 1 El 12 and 1 M3312 and then integrating directly. Setting Vo = 150 Mev, and hereafter measuring energies in terms of vo, in the range 1 '" v '" VI = 2.20 we approximate the energy dependence of 1 El 12 by the following expression:

IE,I'z IE;[' = AY;'A

=

1/'1,

(3.3. 10- 10 cm/sr 'I.), '10 = 0.54 e'jM.

(22)

It is just the contribution E~ in the dispersion integrals that leads to the nonmonotonic behavior in the energy dependence of the real parts of the amplitudes. As can be seen from (I, 42), the contribution of I EI 12 is characterized by two integrals

~v3f~~ 1t

.)

\1'('01'2_\1 2 )

.

(23)

I

Substitution of Eq. (22) in the expression (23) gives

we find without difficulty that the expression for square of the momentum of the meson produced = "l~ -

"~)'I'! v.

I

the

'I;

re' (v' -

Because there is a mass difference between neutron and proton and between ,,+ and ,,0 mesons the effects near threshold in the scattering of y rays by nucleons have a "fine structure. n To make reliable numerical calculations one would need a much more detailed analysis of the data on photoproduction than we now have available. Wishing to get an idea of the scale of size of the effects near threshold, we shall confine ourselves mainly to a consideration of rr+ -meson production. The quantities E 1 • E 33 , and M33 are taken from the analysis of Watson and others. 6 The production of ,,0 mesons is taken into account only in the resonance state (through M33 ). The connection of the photoproduction amplitudes with the rr-N scattering phase shifts is well known (cf., e.g., references 6,7).* In the range of energies where we can neglect the difference between the Vo (,,0) and Vo (rr+ ) thresholds. when we sum the contributions from rr+ and ,,0 mesons the terms containing the rr-N scattering phase shifts cancel each other. For example

UHing the expression for the total energy of the meson in the c.m.s. '0.

(21)

This then gives

+- SyS.)].

They are normalized so that

PlO =

q;~ ('1'-"i)!(1 -i- 2'1/ M).

I),

-

'- 2'1/ M);

"an with some accuracy (better than 7.5 percent) hI' replaced by

*In the more general form of the problem one requires the

parametrization of a three-rowed S matrix, which describes bot the photo-production and scattering of

11

mesons and also the

scattering of y rays by nucleons. For the scattering of y rays effects of deviations from isotopic invariance can give additions to the scattering phase shifts (and to the mixing coefficients) that are by no means small.

84 150

L. I. LAPIDUS and CHOU KUANG-CHAO

~? !.E,I',dv' = ~Ax 1tjv~-v

,

VI

2 a\

-; v,J

,

2

,,=

,,>1

(24)

I E,I v' (v"

( v--;; v I

I

v: - I ' ". (v'-I)'I. v (v; -1)'1. --) ---In

'I.

d'_ v')

2

2"

..,1

A' X

( v' _ 1 ) '/,

-'--

vi

_

+ v, (v' -1)'/.,

,,(,,2 _1)1/1 _ \/1 (,,2 - 1)111 •

,

(' I-v' ) '/.

---

From Eqs. (24), (25), (22), (I, 42), and (I, 32) it can be seen that at the meson-production threshold the derivatives of the quantities RI and R3 go to infinity (approaching threshold from the side " > 1), and the derivatives of the real parts of these quantities also go to infinity (on the side ,,< 1), whereas on the other side of threshold the derivatives are finite. This result is very general. Thus the dispersion relations turn out to contain specific effects near the reaction threshold like those discussed and analyzed without use of the dispersion relations by Wigner, Baz', Okun', Breit, Capps, Newton, and others.* The use of dispersion relations makes it possible to examine in more detail the effect on the elastic scattering (or on the reaction) of the inelastic processes that occur in a certain energy range. Moreover, the interesting effects that occur in the immediate neighborhood of the reaction threshold ("local effects· which could be discussed when one does not use the method of analytic continuation given by the dispersion relations) are only a part of the total effect of the inelastic processes on the energy dependences of the quantities that characterize the elastic scattering. From the example of the scattering of y rays by protons we can see how the presence of the inelastic process of meson photoproduction in the energy range ,,> 1 affects the characteristics of the elastic scattering, including also effects for ,,< 1 (deviation from the Powell formula, or from Eq. (1.16) for y < 1). The deviation from monotonic variation in Eqs. (24) and (25) is characterized by a sharp drop from the value of the function at 1 in the region ,,< 1 (with an infinite derivative at 1) and a slow drop in the region " > 1 (with a finite derivative at 1). 5. In the range of energies 330 - 500 Mev (2.2 < " < 3.34) the quantity I EI12 is represented in the form

,,=

(v;-I)'I·_(v2 _1)'I••

1t

and

.

, (.... -1)'/. I(V~-Ij'/'+(V'-I)'I" tan-l(v2-t)/r _ _ _ _ ln

v>1

_ __

tan- l

{v'

(v' _ 1)

(25)

v<1

--'--

"~(1_v2)'·

'1'1.

IE,I'= 1.27(I-O.175v)'e'jM.

(26)

The contribution from this energy range to the values of the real parts of the amplitudes is small, if for the scattering of the y rays we consider the energy near and below the threshold. The analYSis of the photoproduction made previously, and particularly the results of Akiba and Sato, indicate that

I Mal' = 61 Mas I' "'" I E.I' "'" Re (E;M a).

(27)

For our estimates we adopt Eq. (27). The polarization of the recoil nucleons is especially sensitive to this assumption. In the energy range 1 < ,,< 2 the quantity I M3312 can be approximated by the expression

IM.. I' =

B ov(v 2 -

1)'/"

Bo = O.OOge' j M.

(28)

Consequently,

I Mal'= 61 M" I' = Bv (v' -

1),/.,

B = O.054e' j M.

and the contribution of this expression, which describes the production of mesons in the p state, to the dispersion relations is given by the integrals

, I

~~~ ,;,;~I>iv' = I

+ 2~

v'

": v' [}(v: - I )'/'+ (v: _ I )'/, (v' -

(1 - v')'h

and

'I

- , (v'--I)/'ln

tan-'

l

(V;-I('+(V'-I)'I" (v:-I)"'-(v'-I)'I, (v; - 1) i (I - v')

I)]

. •

v<1

(29)

,,=

*The writers plan to tum to the application of dispersion relations to this .problem in another paper.

(30)

which have the characteristic feature that the second derivative with respect to the energy goes to infinity (again on the side II < 1).

85 SCA TT ERING OF GAMMA -RA Y QUANTA BY NUC LEONS In the energy range 2 < v < 3.34 61 M •• I' = 2.17 (I - O.244v)'e' / M.

(31)

The contributions of the expressions (28) and (31) are given by integrals of the forms

+ ~v' + p" dv'

J (v) = 2v' ('" 1

j

1t

"

= ~ In i( ::" -

"

J, (v) =

\I.!)

v' (\1'2

vyH'+Y" ("'.±.:'.)a-B'+Y"(~),a} , \ v, + v , v,

1\ v, - v)

2v' (' 1t

J

;Jv'-c1:-Jv~

d::"( " __:

v '1_

'Iv

'"

~ ~

(32)

,.,

(V2~-V VI";

v)

+ [1'1 In (,\Iiv;=--V;l} - v~

processes on Re (R 3 ) is very strong, although the contribution of Re (R 3 ) to the observable quantities is small, so that the experimental study of the energy dependence of Re (R 3 ) is a difficult problem. The energy dependence of Re (R() and Re (Rs) is given with great accuracy by the general relation (1,18). The departure from zero of Re (R 2 ) and Re (Rs) is entirely due to inelastic processes, but the production of mesons in the s state does not contribute to these quantities. The differential scattering cross section (in the c.m.s.) (12) can be written in the form 1.,(0, ")

- ---\21"('1,-,,) f «I'I- '(V-) In \ I I - II 'lit j v

-~

A,,('I)

1-

/1,('1) cos 0

+- 11, (v) cos' 8 + A • ('I) cos' 0, (33)

6. The energy dependences of the real parts of the amplitudes R I , ... ,Rs (in the I.s.), calculated by means of dispersion relations, are shown in Fig. 1, a, b. The half-widths of Re (Rd and Re (R 2 )

151

(35)

The results of calculations for the scattering angles 90° and 0° are shown in Figs. 2 and 3. We at once note the marked difference between the energy dependences of the cross sections at (J = 0 and at 90°. J

RoN

o

FIG. 2. Energy dependences of the differential cross section 10 (90") (curve 1), the total scattering cross section (curve 2), and the differential cross sec· tian with the dispersion part not taken into account (curve 3) (the values of the functions are expressed in terms of (e'/M)' as a unit). Experimental data from reference 9.

OJ

I

i,K

b

Zr R'KS

FIG. 1. Energy dependences of the real parts of the amplitudes R, and R, (a) and R" R.. R, , and R, (b) (the values of the functions are expressed i~ terms of el/M as a unit).

are voi10 and voi20, respectively, and are mainly due to the square of the ratio of the real part to the coefficient A in Eq. (22): (34) In a general analysis of the nonmonotonic behavior near the threshold A. I. Baz' has given for the width of the peak restrictions of the form 1'0 (1 - v 2 )1/2 «1 (where ro is the interaction radius). The more detailed treatment of the present paper has automatically given the more accurate criterion (34). The effect of the inelastic

FIG. 3. Energy dependence of the differential cross section 10 (0"): curve 1 is for the cross section in the I.s.; curve 2, for the c.m.s. (values of the functions in units (e'/M)'. 0

05

The function 10 (0°, v) has been calculated earlier by Cini and StroffolinL 8 We have improved the accuracy in the region near the threshold. Outside this region there is good agreement between the two calculations. Our results relating to 10 (90°, v) in the energy region near 200 Mev also agree with other published calculations. 9 A new

86 152

L. 1. LAPIDUS and CHOU KUANG-Cf/AO

contribution is the careful treatment of the region near threshold, in which there are effects not discussed previously. Figure 2 shows the energy dependence of the total cross section for elastic scattering, and also shows for comparison the energy dependence of the cross section calculated from Eqs. (16) and (18). The effects near threshold are practically imperceptible, but the difference between the two curves shows the general effect of inelastic processes on the elastic-scattering cross section. The local effects are much more prominent if we calculate the difference

ing by polarized protons does not differ from 10 (II). Above the threshold for production of 7r mesons there is a nonvanishing polarization of the recoil nucleons. The values of the imaginary parts of the amplitudes above threshold are shown in Fig. 6, The results of calculations on the dependence of the polarization at II = 90° (angle in c.m.s. ) on the photon energy (in the 1. s .) are shown in Fig. 7. It can be seen that over a rather wide range of energies, 180 - 220 Mev, the polarization reaches 20 to 25 percent.

aD [

ImR

os/4rr -/0 (90°, v)

0.6

or the dependence of A2 on the energy II (Figs. 4 and 5). To get experimental data on Aa one

FIG. 6. Energy dependence of the imagin- 04 ary parts of the amplitudes (in units e' /M).

ImA't

1.5

FIG. 4. Energy dependence of 2[0./417-1,(90")] (in units (e' /M),).

fmA'S

FIG. 7. Energy dependence of the polarization of the recoil protons at Ii = 90°.

04r--6p)gO' o.Z U '--_ _ _ _ _ _ _ _ _--;.~-:

I

FIG. 5. Energy dependences: 1 - of the photon polarization 2
e

(e'/M)').

villo

/5 vl'IJ

The values of the polarization are rather sensitive to the assumptions made in the analysis of the photoproduction data, and in particular to the assumption (27). Consequently, the experimental study of the polarization of the recoil nucleons could give valuable information about the photoproduction of mesons. In the expression (20), as compared with Io (II). there is a decided decrease of the contribution of I R412, and I R312 occurs with the negative sign, so that the dips near the threshold are particularly marked in the energy dependence of < T 22 ( 90 0 ) (Fig. 5). 7. A detailed examination of the scattering of y rays by nucleons in the region near the mesonproduction threshold, made by the use of dispersion relations, has made it possible to see what effect the production of mesons in the s state has on the anomalies near the threshold. The scattering of y rays by nucleons and by nuclei is an example of the sort of process in which the energy dependence of the amplitudes is especially strongly affected by inelastic processes and the effects extend over a wide range of energies. In y-N scattering the local effects on a number of observable

>

needs only to study the cross sections Io (II, II) at II = 45', 90', and 135 with sufficient accuracy to find the energy dependence of the difference 0

/.(45°) +- lu(135°) -/0 (90°). It is interesting to note the energy dependence of the polarization of the recoil nucleon. Below the meson-production threshold the imaginary parts of the quantities R t , .•• , Rs vanish in the e 2 approximation, the right member of Eq. (13) is zero, and there is no polarization of the recoil nucleon. Below threshold, in virtue of invariance under time reversal, the cross section for scatter-

87 SCATTERING OF GAMMA-RAY QUANTA BY NUCLEONS quantities are quite appreciable, but rather severe requirements are imposed on the procedures for experimental studies, expecially as regards resolution in energy, since the widths of the dips in question are of the order of 5 to 10 Mev. The treatment given in the present paper shows that the effects near threshold are sometimes masked by the strong energy dependence of the scattering amplitudes. Therefore it seems that the most favorable conditions for the experimental study of such effects should be found at small energies, and also for the interaction of particles with small spins. In the case of y-N scattering, besides the contribution of the "peak" amplitudes R t and R 3, there are large effects from other amplitudes, particularly from R •. The effects of these" smearingout" factors may be smaller in the scattering of y rays by helium nuclei (cr other spinless nuclei), since in this case the transition matrix will have the form AI =

R; (ee') -+ R; (ss').

A treatment of the scattering of y -ray quanta by deuterons near the threshold for the photodisintegration of the deuteron, where local effects will evidently be large, will be presented in another paper. From the point of view of the general effect of some processes on others it is interesting to analyze the photodisintegration of the deuteron in the energy range near and below the threshold for meson production. Noting the results of the calculations on the y-N scattering, we can evidently suppose that the well known" resonance" energy dependence of the cross section for the photodisintegration of the deuteron is due to meson-production processes above threshold and can be treated by a method using dispersion relations. It is commonly assumed that at quite high y -ray energies the y-N scattering cross sections will be almost entirely due to inelastic processes, Le.,

15

to the imaginary parts of the amplitudes. In this connection it may be very interesting to study y-N scattering, and especially the polarization of the recoil nucleons, near the thresholds of reactions of the production of new particles, such a, "(

:~

N

->

Y -'-- K,

and a number of other processes. In this case the difficulties associated with the size of the cross section and the low energy of the recoil nucleon may very probably be smaller. The writers are deeply grateful to B. Pontecorvo and Ya. Smorodinskil for helpful discussion 1 E. Wigner, Phys. Rev. 73, 1002 (1948). A.1. Baz', JETP 33,923 (1957), Soviet Phys. JETP 6, 709 (1958). G. Breit, Phys. Rev. 107, 1612 (1957) A. 1. Baz' and L. B. Okun', JETP 35, 757 (1958), Soviet Phys. JETP 8, 526' (1958). R. K. Adair, Phys. Rev. 111, 632 (1958). 2 L. 1. Lapidus and Chou Kuang-Chao, JETP 37, 1714 (1959), Soviet Phys. JETP 10, 1213 (1960 3H. A. Tolhoek, Revs. Modern Phys. 28, 277 (1956). • L. 1. Lapidus, JETP 34, 922 (1958), Soviet Phys. JETP 7, 6S8 (1958). 5 L. Wolfenstein and J. Ashkin, Phys. Rev. 85, 947 (1952). L. Wolfenstein, Ann. Rev. Nuclear Sci. 6, 43 (1956).

6Watson, Keck, Tollestrup. and Walker. Phys. Rev. 101, 1159 (1956). 7 E. Fermi. Supp!. Nuovo cimento 2, 17 (1955). (Russian Trans!.. IlL. 1956). 8 M. Cini and A. Stroffolini, Nuclear Phys. 5, 684 (1958). 9

T. Akiba and J. Sato, Progr. Theoret. Phys.

19. 93 (1958). G. Chew, Proc. Annual Internat. Conf. on High Energy Physics at CERN. 1958, p.93.

Translated by W. H. Furry 33

88 477

LETTERS TO THE EDITOR

ON THE PRODUCTION OF AN ELECTRONPOSITRON PAIR BY A NEUTRINO IN THE FIELD OF A NUCLEUS A. M. BADALYAN and CHOU KUANG-CHAO

could be expected that the cross section for this process would be smaller than that for scattering, since it contains the factor (Ze 2 )2, and the phase volume gives an additional numerical factor (211')-2. On the other hand, the phase volume is proportional to since there are three particles in the final state. This process is described by two second-order diagrams. The calculation of the contributions of the two diagrams to the cross section leads to extremely cumbersome formulas. We shall, however, get the right order of magnitude for the total cross section if we confine ourselves to the contribution of one diagram. The differential cross section for the process then has the form

w1,

Submitted to JETP editor November 26, 1959 J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 664-665 (February, 1960) PRESENT experimental possibilities have allowed a rather close approach to a measurement of the cross section for scattering of a neutrino by an electron. t This process is a very important one for testing the theory of the universal weak interaction. In the laboratory system, in which the electron is at rest, and for incident neutrino energy Wt »m, the cross section for scattering of a neutrino by an electron is (1)

i.e., a linear function of Wt. There is another process, II + Z - II + Z + e+ + e-, for which the laboratory system coincides with the center-of-mass system. On one hand, it

d~ =

16g' (le')'

2

w}w:!e-;-t_

dp_dp_Jk, (k,k.) q
x [ 2a+"--(p~P_)T2fp.

~<:'~m'-Up_)l

m'

t'

J (2)

where

t = k, -

k2 - P-.

q = k, - k2 - P. - p_ .

Here kt, k2' p+, and p_ are four-vectors that refer respectively to the neutrino in its initial and

89 478

LE TT ERS TO THE EDITOR

final states and to the positron and electron; WI' €+, and €_ are the corresponding energies. For high energies of all the particles involved in the process the differential cross section dO'2 has a sharp maximum near the direction of the momentum of the incident neutrino. All of the emerging particles are concentrated in a narrow cone around this direction, with angular aperture .J '" m/wl. This follows from the fact that the denominator of the expression (2) contains the factor w2'

[w,w,(1 - cos i),,) - w,o+ (I -

+ w,s. (1- v+ cos ~,+)

v. cos i),.)1

(.Jik is the angle between the momenta of the i-th and k-th particles, and v. is the velocity of the positron), together with the fact that the effective recoil momentum of the nucleus is q ~ m. The reduction of the "effective" solid angle sharply lowers the degree of the energy dependence of the total cross section. Apart from terms of second order in m/WI the total cross section is (3)

where a, fJ ~ 1; 1 < {3 < 2. Comparison of Eqs. (1) and (3) shows that for Z/137 '" Yz the cross section 0'2 becomes comparable with 0'1 only for incident neutrino energy WI "" 10 Mev. It is only at energies higher than this that the process of production of an electron-positron pair may become observable. The writers express their gratitude to Ya. A. SmorodinskH for his interest in this work and for a discussion of the results. IC. L. Cowan, Jr. and F. Reines, Phys. Rev. 107, 528 (1957). Translated by W. H. Furry 135

90 730

LETT ERS TO THE EDITOR Let us denote pion-K-meson scattering amplitudes by f (11' + K - 11' + K). Then from the above symmetry properties we obtain the following selection rules: 1) The following scattering amplitudes are eqQal to each other:

f (,,± + ~

1(± ---> ,,= + 1(±) =

f (,,± +1(0.-",±

= f(". +

f (,,0 +

1(± --->,,0 + 1(±)

+ 1(0) = f(1t± + I(O-'>,,± +_1(0)

1(0_+lt0 + J(D)

=

{("O +j(o ....... ,,0 + 1(0). (2)

2) The cbarge-exchange amplitudes vanish:

f(,,++ 1(- ....... ,,0+1(0) = f(1t-+ 1(+--->,,0+ 1(0) =

f (1t+ + 1(0 _itO

+ 1(+) = f (It- + 1(0 ---> ,,0 +

1(-)

= o.

3) The K + K - n1l' annihilation process proceeds only through the isoscalar state. To obtain experimental verification of these selection rules, one can study the angular distribution of the products in the reaction K + N - K + N + 11', for which the one-meson term in the cross section is proportional to

A"(AI+I'-")-llf(:+ 1(--->: + 1()I",.

POSSIBLE SYMMETRY PROPERTIES FOR THE 71-K SYSTEM

(3)

where A' is the square of the nucleon momentum transfer. Expression (3) has a maximum for At =".' in the physical region. Z A measurement of CHOU KUANG CHAO the form of this maximum would provide information on the amplitudes f ("Jr + K -11' + K). Joint Institute for Nuclear Research According to the theory of Okun' and PomeranSubmitted to JETP editor January 27, 1960 chuk3 and Chew and Mandelstam' the scattering phase shifts in high angular momentum states are J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 1015-1016 determined by diagrams with the smallest number (March, 1960) of exchanged 11' mesons. If the K+ and KO have the same parity then the K + N - K + N scatterTHE Hamiltonian describing the 71'-K system has ing phase shifts in high angular momentum states the form are determined by diagrams with two mesons ex(1) changed. Consequently a phase shift analysis of the process K + N - K + N would give certain where H1I' is the pion Hamiltonian including the 11'71' information about the amplitudes f (11' + K - 11' + K). Interaction, HK is the K -meson Hamiltonian, and A violation of these selection rules would imply g is the coupling constant of the 71'7I'KK int-eraction.! that the Hamiltonian contains te~ms with derivaIt is assumed in (1) that the 71'-meson and K-meson tives of the form interactions with baryons can be neglectl'
91 LETTERS TO THE EDITOR The author is grateful to Prof. M. A. Markov and V. I. OgievetskiI for their interest in this work and valuable discussions. IS. Barshay, Phys. Rev. 109, 2160; 110,743 (1958). 2 C. Goebel, Phys. Rev. Lett. 1, 337 (1958).

S L. B. Okun' and I. Ya. Pomeranchuk, JETP 38, 300 (1959), Soviet Phys. JETP 9, 207 (1959). 'G. F. Chew, Report at the 1959 Kiev Conference (in press).

Translated by A. M. Bincer 205

92 966

LETTERS TO THE EDITOR

ON THE DECAY OF ~ HYPERONS CHOU KUANG-CHAO Joint Institute of Nuclear Studies Submitted to JETP editor January 27, 1960 J. Exptl. Theoret. Phys. (U.S.S.R.) 38, 1342-1343 (April, 1960)

THE

experimental data on the probabilities and asymmetry coefficients of the decays of ~ hyperons by various channels evidently satisfy the rule I ~I I =~. If the I ~I I = ~ rule receives final experimental confirmation, it will be necessary to renounce the theory of the universal weak interaction between charged currents. 3 At present it is desirable to have more data to test this rule. Let us denote the amplitudes for the processes ~+_P+7r°, ~+-n+7r+, and ~-=n+7r- byA+, Ao, and A_, respectively, where A = a + ib (uk); k is the unit vector in the direction of motion of the nucleon. The absence of asymmetry in the decays ~± - n + ~ means that for these processes Re(ab *) =

o.

(1)

There are three ways to satisfy the condition (1): 1) a = 0, 2) b = 0, 3) the phases of a and b differ by 90'. Since the interaction of pion and nucleon in the final state is small, the third possibility violates the conservation of time parity.

93 LETTERS TO THE EDITE>R Many authors 2,3 have shown that the rule I AI I =! holds only for ao = b_ = 0 or a_ = b o = O. To choose from among the three cases the one that exists in nature, one could use measurements of the polarization of the nucleons from the decay of polarized 1:% particles produced in reactions 1r± + P -1:± + K+. - Denoting the polarization vectors of nucleons and 1: hyperons by P and P1:' we get
. ,. 1a I' .,. I b I'

Ik P I x

~ .

(2)

Inparticular P=2(P1:k)k-P1: for a=O; P1: = P for b = 0; and for the third case P has a component along the direction of k x P1:. It is obvious that a measurement of the diraction of the polarization vector of the nucleons will not only give information to test the rule I AI I = !, but can also help to choose one solution from the two that are possible (ao = b_ = 0 or a_ = b o = 0 ) if this rule holds.

967

If there is no transverse polarization of the neutrons from 1:- decay, this means that the initial 1: - particle is unpolarized. In this case the absence of asymmetry in the 1:- decay does not lead to Eq. (1). The quantity Re (ab*) can be determined from a measurement of the longitudinal polarization of the neutrons.

~~Sudarshan and R. E. Marshak, Phys. Rev. 109, 1860 (1958). R. P. Feynman and M. GellMann, Phys. Rev. 109, 193 (1958). 2 F. S. Crawford, Jr. et aI., Phys. Rev. 108, 1102 (1957). F. Eisler et al., Phys. Rev. 108,1353 (1957). 3R. E. Sawyer, Phys. Rev. 112, 2135 (1958). G. Takeda and M. Kato, Progr. Theoret. Phys. 21, 441 (1959). B. d'Espagnat and J. Prentki, Phys. Rev. 114, 1366 (1959). B. T. Feld, Preprint. R. H. Dalitz, Revs. Modern Phys. 31, 823 (1959). M. GellMann, Revs. Modern Phys. 31, 834 (1959). Translated by W. H. Furry 256

94 VOLUME

SOVIET PHYSICS JETP

12, NUMBER

JANUARY,

1

1901

DISPERSION RELA TIONS AND A.NALYSIS OF THE ENERGY DEPENDENCE OF CROSS SECTIONS NEAR THRESHOLDS OF NEW REACTIONS L. 1. LAPIDUS and CHOU KUANG-ClIAO Joint Institute for Nuclear Research Submitted to JETP editor February 5, 1960 J. I<:xptl Theoret. Phys. (U.S.S.R.) 39,112-119 (July. 1%0) The application of dispersion relations to the analysis of the eneqw dependence of scattering (and reaction) amplitudes near thresholds of nl'W reaetinns is discussed. General expressiems are obtained which characterize the nonmonotonie behavior of forward-scattering amplitudes as [unctions of the energy. The eneq~y dependence of one of the ampliludcs for elastic scattering of y-ray quanta by deuterons is examined near the threshold for photodiSintegration of the deuteron.

1

An examination of the scattering of y-ray quanta by nucleons near the threshold for pion production! has shown that the dispersion relations automatically lead to the appearanee of discontinuities of the derivative of the real part of the amplitude, if one takes account of the energy dependence of tne reaction cross seelion near threshold.* Within the framework of the dispersion relations the problem of the appearance of discontinuities of the deri vati \'(! of the forward scattering amplitude involves the analys is of integrals of the form

incident particle in the laboratory system (l.s.j, I'

J

k

w:::L

'''I) (w +w,-o) 1(""'-1"),

-\1') ( .\ 1

(Ii)

the very threshold, which in some eases lead to sharp "peaks," "dips," and "steps," and also the general influence of inelastic processes "ccurring in sorne cner!!."y ran!4c ()n the proc..;esses at a given eneq,y. We recall that the unitarity relations for the S matrix make it possible to lake in-

(Z)

where qc and kc are the momenta before and after the collision (in the c.m.s.) and B is a constant. It is not hard to see that (,,'

~ \1

The applicalil)n Ilf dispel'sinll l'e1atil)l)s makes

involving particles with masses J.L (incident), M (target), m andlJJl is given by the expression

(1/,,'1:,)'

I 1(1' II

it possible to examine both "lucal erreets" near

wu '

+d

I"

III'

where the usual notation is used, and the total cross section (J (w) includes both the elastic scattering cross section as (w) and the inelastie interaction cross section a c (w). The behavior of (Jc (w) near the tlHeshold of the binary reaction " -\- b -. c

(\1

III)'

is the threshold enerl-(Y of the l'eaelion (4), and

'l~P\!!-.u:~~J_ tlT-":il

I(IJJI

(·1)

to account the influence

where w ~ (k 2 + J.L 2) ! '2 is the total energy of the *The nonmonotonic behavior of the cross section near the threshold has been treated phenomenologically in a diploma research by G. Ustinova, and also by Capps and Holladay.2

t)fl

:t

g-iven process of other

processes oecurrin!!; at the same ener;\y. 2. The study of y-N scatterinl-( has shown that there are "local effects" in only two of the six scalar amplitudes l1l'l'(\t,(\ to describe liw transition matrix in this case. The presence of oliwr strongly energy-dependent amplitUdes hinders the analysis. For a detailed analysis of the inelastic processes it is necessary to examine the dispersion relations for nonvanishinl-( momentum transfers Q2, and possibly also double disperSion relations. In the present paper we confine ourselves to the examination of dispersion relations in the tolet! energy for Q2 ~ () for a scala l' function A ( w ), which is the trace of the scalle ['in!!; matrix, Sp .11 (,",,1/'

Ii),

(7)

and whose imaginary part is related to the total cross section. The contribution of inelastic processes to 0 ~ Re A (w) is charactcrized by the two 82

95 83

ENERGY DEPENDENCE OF CROSS SECTIONS integrals (8)

(9)

where
in these integrals the range of integration is from the threshold Wt to wI' the limit of the region of the s state in
It follows from (14) and (14') that the first derivative has a discontinuity at the point woo The resulting energy dependence has the characteristic feature that the derivative is infinite on the side Wo < Wt and has a finite value when we appj:oach the point Wo = Wt from the region Wo < Wt· It might seem that the quantity k~II ( - wo) obtained as the result of substituting (3) and (4) in (9) would also have a discontinuity of the derivati ve at Wo = Wt - <5. On changing the sign of Wo in (14), however, we easily verify that this is not so. The values of the deri vati ve of k~II ( - Wo ) calculated with approach to Wo = Wt from the two sides are identical. Thus although in a relativistic treatment cross-symmetrical inelastic processes indeed contribute to the real part of the scattering amplitUde, they do not lead to nonmonotonic energy dependence of the amplitude. The application of dispersion relations enables us to get detailed information about the magnitude and half-width of the anomaly near the threshold. The half-width € of the drop in the region Wo < Wt can be estimated roughly in the following way. Near Wo = Wt (wo < Wt) the argument of the arc tangent is large; using this fact, we find 1jl (w o) =

Defining the half-width (12)

it is not hard to show that k;fl(wo)=-{q,(wo)-}(I

D (w. -

V=a €

ll.

(15)

by the condition

e)=

i- D (WI)

(16)

and using Eq. (15). we then get

+"),,11")4(1-')

D(lUl)-(BI4ll)V(WI-W o)(O>o+lUl-/) .,,+D(lUl),

--HI-woil-')~(-i"»),

(13)

(17)

from which we have

where

e"" i1(wl - /)/2(' [4llD (WI) IBP.

(18)

In the limiting case in which the contribution to D (Wt) not associated with the expressions (10) and (11) can be neglected, -tan-I

(2f'+0j("'1+1k)-2a(--I') 2¥-a(-Ik)R

o (WI)

J .

V-

a (w o) [~

J (WI) =

+ tan- 1(2",o- 0) ("'1- "'0) +~.~" (wo) I,

2

0'0

<

BJ (0)1)14ll 2 ,

(19)

and from Eq. (13) we have the result that

For the function ¢ (w) at the point "-\) we get the expressions 'f(wo) =

.~

(13')

2 y=-a("'o) R

0)1;

'

(14)

+1(1

f· w , /I-')1jJ (I-') HI--U>I/I-') ~(- Ii)].

(20)

3. Let us conSider the photoproduction of neutral pions "(+p ..,p+"o

(21)

near the threshold of the reaction "( +p

~n

+"+.

(22)

In this energy range it suffices to consider the

The expression for k~II ( - wo) is obtained from (13) and (14) by the replacement Wo - - woo

electric-dipole transition. We denote by EO and E+ the transition elements for neutral and

96 L. I. LAPIDUS and CHOU KUANG-CHAO

84

charged mesons. respectively. From the condition of unitarity and the experimental fact that Re EO :::: O. we get the result (23)

where 0'3 and O't are the phase shifts for rr-N scattering. Substituting the experimental data for 0'3. at, and E+, we get 1m EO = V 2/3 (a, -all q~'q~'V qJv·3,3· 10where

15

cm,

(24)

(II is the energy of the photon, and as, at are the

scattering lengths). The anomalies at the threshold are determined by an integral of the form

P~'

dv

(v'-v;tl'!'(V2-v~tl'/'/v'/'(v-vo),

(25)

which undoubtedly gives "peak" singularities. In analogy with this, in the general case we can consider the cross section of the reaction (26)

a+b~c+d

near the threshold of the reaction (27) If the threshold of reaction (27) is far from that of

reaction (26). we can always find an energy range where 1m M (ab

~cd)

= M (ab

~ef) M+(ef

-·cd)

+ ... =

Aq

+ ... ;

(28) here A is a weakly varying function of the energy and q is the relative momentum of the system ef. The other terms in the sum (28) are also slowly varying functions of the energy if there are no thresholds of other reactions in the neighborhood. In this case the dispersion integral has the usual form, and we can determine the magnitude and half-width of the "peak" or "dip" in the same way as for the case of scattering. In the case of such a process as the photoproduction of pions 1m M (ab ~cd) = Aqq;{'

+....

the derivative from the side Wo < Wt is infinite. That it is precisely the first derivative that is infinite is due to the form of the relations (3) and (4). The behavior of the cross section for the reaction (2), when its products are in states with nonvanishing angular momentum 1, is given by the expression

a;n =

B(I)

{(ill - w,) (0)

+ lO"

_ 0) / (w 2 _fl2) \1+'/..

Substitution of (3) in (8) makes the lth derivative infinite. It is perhaps interesting to note that, unlike the nonrelativistic treatment. this use of the dispersion relations has not required the assumption that the partial amplitudes are analytic. It has turned out that it is enough to use only the analytic character of the scattering amplitude with respect to the total energy. with a bounded value of the momentum transfer, Q2 < Qkax. As Baz' has pointed out, the unitarity of the S matrix has the consequence that as the number of channels increases the effect in each channel decreases. An analysis of y-N scattering near the threshold for photoproduction of pions, for which there are "peak" effects in only two out of six scalar functions. has shown that there is also a smearing of the effect with increase of the spin of the particles. An important feature of the theory of dispersion relations is the discussion of the convergence of the dispersion integrals at high energies or, what is the same thing, of the number of subtractions. The main calculations in the present paper are made for dispersion relations with one subtraction. In the case of dispersion relations without subtraction one must make the replacement (31) With suffiCiently high experimental accuracy the difference between Eq. (8) and Eq. (31) can give information about the number of subtractions. Let us note briefly what sort of singularities can appear near the threshold of the reaction a+b~c-ld+f·

(29)

since the threshold of the reaction ab - ef is close to that of the reaction ab - cd. In these cases the dispersion integrals are rather complicated and we have not been able to carry out the integration. 4. It thus follows from the conditions of causality and unitarity. together with Eqs. (3) and (4). that the first derivative of the real part of the scattering amplitude has a discontinuity. and that

(30)

(32)

By substituting in Eq. (8) the cross section of the reaction (32) in the form a, = B'k;;-lp~max= r(w -

WI)'

(reaction products in the s state). we get

f} WI

od",

"'_
rf'dw(
=}

"'_
{I

= r T{(w l -wO)2

WI

-(WI-wo)'1 +2(wo-wl) (WI-OlI)

+( rn o-wl)' ln l(wl-wo)/(wl-wo) I),

(33)

97 85

ENERGY DEPENDENCE OF CROSS SECTIONS which leads to a logarithmic infinity in the second derivative of D(w) with respect to w. For a reaction with four particles in a final s state the quantity ( Wo - Wt)2 In 1Wt - Wo 1 is replaced by (wo - Wt)5 In 1Wo - Wt 1 . Similar behavior of the real part of the scattering amplitude appears at the thresholds of all reactions. An example of the application of dispersion relations that is well known in the literature is the analysis of the coherent scattering of photons in the Coulomb field of a nucleus 3 (cf. also reference ,I). It is not hard to convince oneself that near W = 2m (y == w/2m = 1) - the threshold for prouuction of an electron-positron pair - the real part of the scattering amplitude has an energy dependence of the type xk In x (x = y - 1). To see this we have only to examine the expression for the real part of the amplitude:

~

/) (w) =

(::r {~n

+ 2~n

[(109

-(67-

;.)F.(T)l-\~-+},

-

e;r'ke•.~ +:-

(34)

o,W)

~

H

where y

C, (1)

=

Ae'el-iBS[e'el

Re ~ arc :in x cosh-I ( ; ) dx, C, (I) "~ 1,621176;

CI(Se)(Se')+-(Se')(Se)] (36)

IA +i-(C+O)I'+,i-IC+DI' i B I' +

+ID -

kal =

C I',

(37)

and we have

o

D,(I)

+-J

D [IS [kxe\)(S !kxe' 1)1- (S [k~'J)(S (kxe1) 1.

The cross section for scattering of unpolarized y rays by unpolarized deuterons then takes the form

(2C, (1) - 0,(1)1

+ ~nEdT)

~.)(I

the threshold of the reaction y + e - 2e + e+ has the characteristic dependence x2 In x. 5. The elastic scattering of y rays by deuterons near the threshold for photodisintegration of the deuteron is an example of a process for which use of dispersion relations is necessary for the analysis of the anomaly near the threshold. The nonmonotonic behavior near the threshold in this case comes from the magnetic-dipole disintegration. The electric-dipole diSintegration leads to appreciable changes in the energy dependence of the amplitude for elastic y-d scattering in a certain relatively wide range of energies. The amplitude for forward elastic y-d scattering can be represented in the form

= 4ft 1m (A +- } C

+ 1- D).

By means of the dispersion relation for the quantity L = A + %C + %D,

1,83193;

ReL("')=_~-I-2W'pf imL(w') Md ' ~ J W' (00'2 _ (1)2)

d' (.tl

I

(38)

Wd

and K(y) and E(y) are the complete elliptic integrals of the first and second kinds. As is well known, for 1_y2« 1[A =In(4{1_y2)-1/2]. K (y) = A f

+ i-(A -

I) (I - ,')

:! (A - ¥.-)( I -

,,(')3

+ k(A -

+)(1 -,')'

+ ' .. , (35)

which indeed shows that the dependence is of the (orm In x. It is not hard to check that the scatloring of light by light near the threshold of the 1'l)llction y + y - e+ + e- is a process, well known in quantum electrodynamics, for which the amplitude is characterized by a "local" anomaly (cr. Figs. 2-4 in the paper by Karplus and Neu5 1ll1ln )*. The amplitude for the Compton effect near

0

'Regarding effects of Ihe Coulomb interaction see papers hv Ilaz" and by Fonda and Newton.'

where wd is the threshold for photodisintegration of the deuteron, let us examine the effect of inelastic processes on the energy dependence of the real part of the amplitude L. In the calculation of the dispersion integral it is convenient to use the theoretical expressions for the cross sections for photodisintegration of the deuteron (cf., e.g., reference 8). Let us begin with the examination of the "local effects." The expression for the cross section for magnetic-dipole diSintegration is (m) _

ac

-

(2.)' (f'p _ I'·)' (r -1)'/' (1 + rB'fT'j)' rlr 1+0'/1011

~ 3 ~c Me

2n

(39)

where y = w/I E I, W being the energy of the photon; 1EI = 2.22 Mev and E' - 70 kev are the binding energies of the np system in the 3S1 and ISO states; and the rest of the notation is as usual. Because of the factor [y - 1 + E' / IE 1]-1 the expression (39) does not admit of the simple analytic continuation

98 86

L.1. LAPIDUS and CHOU KUANG-CHAO x=

Y 1- 1 ..... i Ix I.

t.p (io) = 2Mc' Zp (ro) / e'

since in this case 'Yuc (m) goes to infinity below the threshold at 1K 12 = €' /1 d. Substitution of Eq. (39) in the dispersion integral i

L

0

-

'0 p \) •

Z ( )_ £

00

2n'

d

,a,(m) ( , ) 0

e'

2

•

="3 2Mc' Mc' (!'-p - !,-.)' (I

x{YI-T09(1_

~-To

io

(40)

el-

I:

__

+VE/IEj)'

X{Vf+&/2&-2/& +2 YE7TE!/3 & +

YI ElIE' 12},

(41) The dependence of the quantity .a. (Yo) = L ZL (Yo )/(e 2/2Mc 2) on the 'Y-ray energy is shown in the diagram (curve 1)

For extreme values of 'Yo we get from Eq. (40)

E

(./<')2 +v ~

2 t.dio) ="3MC' (f'p - f'n)2 1

3 E' .. / jtjl+2jtj ..-( E')} To• x {S+2jtj.-V To~

t.dio)=-f~c.(f'p-fLn)·(l+V'I:'I)·'

io>!'

I; (42) (43)

At the photodisintegration threshold with 1 - 6 = 1/30. Zdl) = 0.24e'/2Mc·.

On the side of energies smaller than the threshold energy the half-width of the peak is somewhat smaller than t'. i.e., about 50 or 60 kev. The contribution of the cross section for dipole absorption o(d)=41t~~ (T-l)'l. C Me' £ l'

at D =Re L is of the form

~~

The quantity .a.p = .a.p ('Yo) is shown in the diagram by curve 2. In the limiting cases io~ I; io>l.

(46) (47)

t.p (1)= 0.156.

where 6 = 1 - f' /1 €I ,0 (x) = 1 for x'" 1. and o(x) = 0 for x < 1; for 'Yo = 6 2

2]-'/.).

At the threshold for photodisintegration

-_

+ YEll E I)'

)+ Y1+To_3._ 2T~Y27T
ZL(&) ="3 2Mc'MC' (!,-p-I'-n)2(1

+ (I .f 'To),l. -

t.p(rp)=--i-,

for Yo '" 6 gives ZL (io)

o

t.p(io) =~i~.

12 _ 12

1

= 2 {T-' [(I - io),/.9 (I - io)

(44)

The total effect of the dipole and magneticdipole diSintegrations on tI:u> real part of the amplitude L is shown in the diagram by curve 3. Right at the threshold the effect of photodisintegration leads to a change of the amplitude by about 40 percent. The contribution of photodisintegration of the deuteron to the polarizabilit-j of the deuteron can be seen from Eqs. (42) and (46). Since the value of the cross section for photodisintegration of the deuteron at high energies is larger than the sum of the expressions (39) and (44). the estimates obtained here can be regarded as lower limits on the quantities, although the contribution of high energies is small. The treatment carried through hel'':: for one amplitude of the y-d scattering can serve as an indication that inclusion of the effects of inelastic processes, and primarily those of the photodisintegration of the deuteron. in th.e analysis of elastic 'Y-d scattering can be important over a wide range of energies. Similar effects must naturally occur also in the scattering of y rays by heavier nuclei. A study of the elastic scattering of 'Y rays by nuclei shows' that for quite a number of elements the cross section for nuclear scattering of 'Y rays near the threshold of the reaction ('Y. n) is characterized by a peak of considerable height with an energy width of about ± 2 Mev. which is evidently- due to nonmonotonic effects near the threshold. Further improvement of the accuracy of the experimental data on elastic scattering of y rays and on the energy dependence of the cross sections of (y. n) reactions near threshold is necessary for a more reliable analysis of this effect. I L. 1. Lapidus and Chou Kuang-Chao, JETP 37, 1714 (1959), SOViet Phys. JETP 10, i213 (1960); JETP 38, 201 (1960), Soviet Phys. JETP 11, 147 (1960). 2 R. H. Capps and W. G. Holladay, Phys. Rev. 99, 931 (1955), AppendiX B.

99 ENERGY DEPENDENCE OF CROSS SECTIONS 3 F . Rohrhch:J.l1d R. L. Gluckstern, Phys. Rev. 86,1(1952). 4 A. 1. Akhiezer and V. B. Berestetskil, KBaHToBaH qAeKTpOAHHaMHKa (Quantum Electrodynamics), Fizmatgiz, 1959, Sec. 55. 5 R. Karplus and M. Neuman, Phys. Rev. 83, 776 (1951). 6 A. 1. Baz', J ETP 36, 1762 (1959), Soviet Phys. JETP 9, 1256 (1959). 7 L. Fonda and R. G. Newton, Ann. Phys. 7, 133 (1959).

87

8A. 1. Akhiezer and 1. Ya. Pomeranchuk, lleKoTopble BonpocbI TeopHH HAP a (Some Problems of Nuclear Theory) , Gostekhizdat 1958, Secs. 11 and 12. 9 E. G. Fuller and E. Hayward, Phys. Rev. 101, 692 (1956).

Translated by W. H. Furry 23

100 VOLUME 12, NUMBER 2

SOVIET PHYSICS JETP

FEBRUARY, l!lill

INELASTIC FINAL-STATE INTERACTIONS AND NEAR-THRESHOLD SINGULARITIES L. 1. LAPIDUS and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor, February 23, 1960 J. Exptl. Theoret. Phys. (U .S.S.R.) 39, 364-372 (August, 1960) It is. shown that. non-monotonic energy variations can occur in the energy spectrum of particle a from a reaction of the type A + B - a + C + D in the neighborhood of the threshold for the reaction C + D - E + F. As an example, we analyze the spectrum of K mesons from the reaction N + N - A+ N + K in the region of the energy of the 11. - N pair close to tile thresilold for the process A + N - I: + N. For the process p + p - 11. + N + K we find the cncrgy spectrum of the K mesons when the incident nucleons are unpolarized, and the polarization of the baryons when the incident nucleons are polarized. We discuss the non-monotonic energy variations in the spectra of particles for some other reactions. In the Appendix we analyze the production of Y -K pairs in np collisions and discuss the case of a scalar K particle.

1. INTRODUCTION

I

T is known that in processes of production of particles an interaction between two of the particles formed affects the energy spectrum and angular distribution of the third particle. In certain cases, the effect of final-state interaction can be separated from the primary mechanism for production of the particles. This occurs when the effective radius for the primary interaction is much less than the radius of interaction of a pair of particles in the final state. In addition, if the interaction of the pair of particles with other emerging particles is weak, the interaction of the two particles in the final state can be characterized by a two-particle scattering length. The theory of final-state interaction was applied by Migdal, I Brueckner and Watson,2 and Paruntseva 3 to meson production in NN collisions. Recently Henlei and Feldman and Matthews 5 applied it to the analysis of the reaction N

+N

->

Y

+N

~I

K.

(1)

They showed that the energy spectrum of the K mesons is strongly distorted by the effect of the YN interaction. Karplus and Rodberg6 generalized the theory of final-state interaction to the case whe re the strong interaction in the final state can lead to an inelastic process. In the present paper we shall show that in the neighborhood of the threshold for production of the I: hyperon certain anomalies occur in the energy spectrum of the K particles formed together with

the 11. hyperons. They are a new example of near-threshold anomalies which have been extensively studied in recent years. T In addition to the cross section for the new inelastic process, the shape and appearance of near-threshold anomalies depend on the spin and parity of the particles. The study of these anomalies with sufficient accuracy can enable us to determine properties of the produced particles. On the assumption that the final state of reaction (1) is described by singlet and triplet s waves of the Y-N system, we analyze in the second section of the present paper the kinematiCS of the reaction and obtain expressions for the energy spectrum of the K mesons and the polarization of the 11. particles and nucleons when the incident beam of nucleons is polarized. In the third section, starting from the unitarity of the S matrix and the analyticity of the reaction amplitude, we give a general formulation of the theory of inelastic final-state interactions. In Sec. 4 we conSider local near-threshold anomalies in the energy spectrum of K mesons in the reaction N + N - 11. + N + K in the neighborhood of the threshold for formation of the I: hyperon. In conclusion, we mention some other similar processes and discuss the possible generalization of the method developed here to these processes. 2. KINEMATICS. PHENOMENOLOGICAL ANALYSIS. We introduce Jacobi coordinates in the final state of the three-particle system: 258

101 259

INE LAST IC FIN AL-ST ATE INTERACT IONS R=

MNrN+Myry+Mxrx MN +My+MK

do _ (2n)' E dfl"dflqdT 2 (E'-4M;v)'/,

(2)

where MN, My and MK are respectively the masses of the nucleon, hyperon and K meson; rN' ry and r K are their coordinl'.tes. The momenta conjugate to R, p, and r will be PR, Py and q respectively. The total energy E in the new variables is equal to (c.m.s.) E = p}/2my

+ q'/2p. + MK + My -

p. =

MN ;

MK (M N + My) MK+MN+My·

X

(3)

my

pydDyq'dqdDq ,

(5)

where dfly and dfl are the solid angles for the momenta Py and qq respectively. To be specific, we conSider the reaction P+p-->!\.+p+W

(6)

(7)

(8)

If the K meson is a pseudoscalar particle, the

spin structure of the T matrix has the form (ApKITlpp)=A.da,+a,. k)

+ B,\ {(a + CA {(al l -

a,. k)

+ i ([a,a,J k)}

a,. k) - i ([a 1a,J k»).

(10)

where T = q2/2J.1 is the kinetic energy of the K meson with respect to the center of mass of the A-N system. If the protons in the initial state are polarized (with polarization vector p). the polarization vector of the A particle in the final state, P A'. will be

- i CAI'J (kP) k + [i AA - C,\!'

-1 A.\ + C"I'J P.

(11)

The expression for the polarization of the nucleon in the final state differs from (11) by the sign in front of C A-

Let us look at the unitarity condition (ApKIT-r+lpp)

The admissible energy for the final state of reaction (6) in the c.m.s. does not exceed 80 Mev, so that we may assume that the particles which are formed are in an s state. Let us represent the S matrix element in the form (t\pK' I 5 I pp) = - 2nio (E, - E,) (/l.pK' IT i pp).

A\ - C.\I' + 21 C.\I'].

3. ELASTIC FINAL-STATE INTERACTION

below the threshold of the reaction p+p->EO+p+W.

[I AA + CAl' +!

P" [1 AA+ CAI'+ I AA-CAI' + 2ICAI'J = 2 [i AA+ CAi' (4)

The phase volume of the final state is expressed in terms of Py and q as follows: dJ =

x (2mA[L)'/' [T (T max - T)J'/'

= Zrri

L; (.\pK ITin) (n I r+ I pp) 0 (Ei

- En).

(12)

where 1n> is a possible intermediate state lymg on the same energy surface as the initial state. Let us assume that in the region of energy considered the imaginary part of the T matrix is related mainly to strong interaction in the A- p system. Then we may neglect on the right side of (12) all intermediate states except for ApK states, and approximately replace <ApKI T I A'p'K' > by <ApITIA'p'> . This means that we are neglecting the interaction between the K meson and the A-p pair. In the low energy region the matrix element <Api T I A'p'> is equal to

(9)

where a is the spin matrix, k is a unit vector along the direction of the incident proton: AN B/I.' and CA are scalar functions of the total energy E and the relative momentum P A of the A- N pair. Since there are two identical particles in the initial state, the elements of the T matrix must be anti symmetrized with respect to the two initial protons. It can be shown that this results in BA = O. The expression for the cross section for reaction (6) with unpolarized particles has the form

= (4n'p.\mAf 1

[+(3 + ala,) "3 + +(1- al<:l')"I]'

(13)

where (14)

and 01 and 03 are the scattering phases in the Singlet and triplet states respectively. Using all these assumptions and taking account of invariance under time reversal, we find from (12)

102 260

L. I. LAPIDUS and CHOU KUANG-CHAO

Illi ii"

ReA,

Re,,, 1-

1m -- .-,.,_.

Re'l:t

!m=<:!

LJ l'\t' :

1.\ -~-::: t an 0:1 ' I ',e j-1.\.,

.1.\=(1

i tan""lRe.i.\"",(1 ·-iu"I'.\)Re.h,

CA = (I

i tan

lid ReC.\

~(I

.:- iu,p.\)

l~eC.\.

(15)

From (15) we see that for {j - 0, i.e., in the absence of final-state interaction, the quantities A A and C A are real functions. In the energy region we are considering, the matrix elements of the reaction matrix are functions of two quantities: E - the total energy, and w- the total energy of the A- p system. If all the singularities of the amplitude a,e associated with physical processes, then A A and C A as analytic functions of wand E are representable in the form (p"ar'

,'''''I ,ill ,; (",) f (,.,) F.\

(l').

who neglected the dependence of the reaction matrix on spin. From (17) and (18) we see that the investigation of the energy spectrum of K mesons and, in particular, of the polarization of A particles and nucleons is very important for the determination of the Ap-scattering lengths.

4. INELASTIC INTERACTION. NEAR-THRESHOLD SINGULARITIES. As the energy is increased, the I: channel is opened, and we may expect a change in the spectrum of K mesons and other quantities for the A Kp channel. In this case, in the unitarity condition (8), we must consider as a possible intermediate state the state I I: NK>. We shall restrict ourselves to interaction in s-states. As in the preceding section we assume that*

(16)

(ANKITI"f.N'K') "" (ANITII:N')(KIK'), where f ( w) is an entire function which. for small and use the fact that values of the energy, can be replaced by a constant. Thus we finally approximate A A and C A by (ANITlI:N) = [4nZp~2p~2m~Zm~Zrl expressions x [~(3+ 0"1 O"Z){33 + ~ (1 - 0"1 O"z){3ll ' (19) i AI\ = - p la e 03 sin 03 'A~, CI\ = _I_e io , sinol·C;Z, (16') A 3 PAol where the indices A and I: denote quantities in the where a3 and at are the triplet and Singlet corresponding channels, while Ap-scattering lengths in the s state, while AOA pl;=[2ml;(E'-T)]1/2, E'=E-Ml;+MI\' (20) and C~ can be regarded approximately as real functions of the total energy E alone. ConseAssuming that there are no bound states of the quently, taking account of the unitarity of the S p-I: system, we represent the energy dependence matrix and the analyticity of the reaction ampliof [33 and [3t in the low-energy region in the form tude leads directly to the main result of the theory (21) of final-state interaction (cf., for example, the paper of Gribov 8 ). if the internal parities of I: and A are the same. By using (16) the expressions for the reaction The influence of the I: channel shows itself for cross section and the polarization of the A parthe A channel not only as an additional term in the ticles can be represented as unitarity condition (8), but also as an additional term in the matrix element of the Ap scattering !f!f = (2n)4 Z(EL;MM'/' (4n)2(2ml\J!)3/2[T(Tmax - T)j1/2 matrix proportional to PI::

x

[2 (p"a

sin'\ 3)

IAo lz +4 1\

2 Sin \

(Ph a,)

IC<W] 1\

(22) (17)

'

where (23)

and crr!A is the total cross section for the reacJ tion I: + N - A + N in the state with angular momentum j. Using (19)-(23), we find from (8) _ 2AOd A

J\

sin 0, Sin? cos(01-03)

p

(18)

PA,0 30 1

p~ = 2m A (T max - T). If we change the sign in

front of C~ on the right of equation (18), we obtain the expression for the polarization of the recoil nucleons. Expressions (17) and (10) can be considered as a generalization of the results of Henley,

lIIlC,,=(lmC,,)p~ _,,+C~PE.

Im.4,\ ,= (1m AA)p~=o -;- A~PE.

(24)

"The inclusion of terms of the type , which are small for this reaction, but are necessary in other cases, complicates the expressions but

does not change the fundamental result.

103 INELASTIC FINAL-STATE INTERACTIONS where (0'" rr/2)

l, = A~ (pt.';4Tt) O;·A (PE = 0)

tan2

0a

+ A~ bal COS

2

C~ = C~ (pt.j4Tt) o~·" (P E = U) tan2 01 +C~ bl l cos 2

0,. 01'

(25)

The relation (24) is valid when the kinetic energy T of the K meson is less than E'. For T > E' the production of a real l; particle becomes impossible, and we must replace Pl; by ikl;' where kl; =,j 2ml;(T - E'), T > E', .so that the term which depends linearly on kl; appears in the real part of the reaction amplitude. The presence of terms proportional to Pl; (T < E') and kl;( T > E') causes the derivative with respect to the energy to become infinite both in the energy spectrum of the K mesons and in the energy dependence of the polarization of 11. particles (and nucleons). The order of magnitude of these anomalies is given by (24) and ,(25), and their shape depends on the relative sign of A~, A~, b 3,t and O. All four cases of anomalies which have been discussed in the literature for binary reactions can also occur in this present case. All of the expressions in Secs. 2, 3, and 4 were gi ven for the production of particles in pp collisions. It is not difficult to generalize them to the case of np collisions. This is done in the Appendix. We also discuss there the case of a scalar K particle. We note that, in the general case also, the quantities which replace AA and CA have terms which are directly related to the final-state interaction, as well as terms which are not caused by it. We emphasize that the expressions obtained in the present section refer to interaction in an s state of the final system. The relatively large mass difference of the 11. and l; hyperons makes it difficult to apply the theory of inelastic interaction to the analysis of reaction (1), but this does not change the basic assertion that there is a non-monotonic behavior in the spectrum and the causes for its occurrence. It was shown earlier 9 that the direct analytic continuation Pl; - ikl; can not be carried out when there is a resonance in the neighborhood of the threshold. In this case, it is necessary to make use of dispersion relations. Since the analytic behavior of the reaction amplitude as a function of w is not known, we have not carried out such an analysis. However, even if such a resonance occurs, we may expect non-monotonic variation with energy for a relative energy of the

261

11.- N pair equal to the threshold for the new channel. If l; and 11. have opposite parities, the first term of the expansion in (22) starts with P~ and only the second derivative with respect to the energy becomes infinite. Consequently, the study of threshold anomalies in the energy spectrum of K mesons with sufficiently high accuracy may prove important for determining the relative parity of the l; and 11. particles. 5. DISCUSSION

Thus, endothermic inelastic interactions of the type C + D - E + F in the final state of the reaction A + B - a + C + D can give rise to nonmonotonic variations with energy in the spectrum of the particles a, whose form can be determined from the condition of analyticity and unitarity of the S matrix. To investigate these singularities experimentally requires, of course, good accuracy and high energy resolution, but as a result of discovering them and studying them one can obtain information concerning the interaction of unstable particles, their spins and parities. Earlier we have treated the production of hyperons and K mesons in NN collisions. We mention various other processes in which similar anomalies can occur whose study may give information concerning the interaction of unstable particles. In the spectrum of mesons from the reaction

,,+

(26)

in the neighborhood of the threshold for (27) there will occur an anomaly whose magnitude and character will be related to ~p scattering at low energies via the reaction amplitude (27). In the spectrum of protons from the process for production of 7l" mesons by K mesons (28)

an anomaly may occur for an energy corresponding to the threshold for the reaction (29)

if there exist forces leading to such a reaction. If one attempts to construct a Lagrangian for the 1f K interaction and does not consider interactions containing derivatives, the expression obtained

104 262

L. 1. LAPIDUS and CHOU KUANG-CHAO Lint

= g('P'.. ''P~) ('P~'
there will also be energy non-monotonicities.*

is invariant with respect to rotation of the isotopic spin of each of the particles, and all processes for K Tr scattering with charge exchange are forbidden. Under more general assumptions one does not obtain a forbiddenness for reaction (20), so that the observation of a non-monotonic variation with energy in the proton spectrum from reaction (28) would be of interest from the point of view of the study of the symmetry of the Tr K interaction. Among reactions in which two Tr mesons participate, it is interesting to note that in the distribution of nucleons from the reactions (30)

there may occur similar anomalies for a relative energy of the Tr° mesons exceeding 9 Mev, where the charge exchange reaction (31) becomes possible. Including threshold phenomena in the reaction (31) can have significant effects in the theory of Tr Tr interaction at low energies. The existence of a threshold in reaction (31) can lead to a non-monotonicity in the spectrum of charged Tr mesons from T' decay.

Analogously to the reactions (30), in the spectrum of nucleons from the reactions

APPENDIX A. PRODUCTION OF A PSEUDOSCALAR K MESON IN np COLLISION In np collisions there are two possibilities for production of A particles (A.1)

We denote the reaction amplitudes in the singlet and triplet isotopic spin states by To and T j respectively. The reaction (A.l) is then described by the amplitude II:! (To + T I ), and the reaction (A.2) by the amplitude 1/2 (TI - To). The spin dependence of the isotopic triplet amplitude is given by (9) with B A ~ 0, while ,.\NI\!T"iN.V,·

(33)

there may occur energy anomalies associated with KK interaction. Moreover, in the final state of reaction (33), there is no Coulomb interaction which might mask the non-monotonicity (cf. the paper of Newton and Fonda lO ). In the spectrum of protons from the reaction I'

+ I' -, p + p + ~o

(34)

-t- p ---+- n

and in the spectrum of action

Tr+

(AA)

where B~ is a function of the total energy E. The expressions for the cross sections for production and polarization of the A particlcs have the following forms: do = (2lt)'c

IE".~'M'U"~' (4r.)'(~m':L)"'IT(T,,,ax 1l,1' : 1.,1"

C, F1i,1'

TJ 1"

dT

21(', f' 8,

'J, (A.5)

PAIIA,+CA±RAI"'I/!' ·C" H3", ;·2IC,,+13,I'1 =21IA,,!·C,:l-B.,I'

'C,F13,I'I(kP)k

";'-IIAA-C"TB.,I'- lA, :C"±B,,I'IP.

(A.6)

The plus sign in front of B~ holds for reaction (A.l), and the minus sign for (A .2). From (A .J) and (A.6) it is easy to obtain the "intensity rules": do (np

->

.'l.pK") = do (tll' -. AnK'),

forPllk.

(A. 7)

(A.8)

These relations are obtained on the assumption

mesons from the re-

p~d

iklcr l "cr,ll. (A.3)

-~- 1't'+

p+p-n+p+~

near the threshold for n +-

cr"k)

P.dnp->.\pK")-~Pdnp--+,\nK")

near the threshold for ITO

Hd(cr ,

Under the assumptions made earlier we can take account of final-state interaction by setting

:< II A,\ + c"":

in the neighborhood of the threshold for the reaction

(A.2)

tl '.1\".

tl+I'->.\

-:-1t

0

~~

*The scattering lengths for low energies of the 1T o.p system differ from those obtained on the assumption of isotopic invariance because of the presence of non~monotonicities which violate isotopic invariance and are related to the reaction An estimate using dispersion relations gives a correction - 5%.

105 INELASTIC FINAL-STATE INTERACTIONS that one need only consider the s wave in the final state. They can be used for an experimental check of this assumption. B. PRODUCTION OF A SCALAR K MESON IN NN COLLISIONS In this case I.\NK

AA.

'.\N K I Tol N N)

A.\~' .4~\ (p., a,r' e'" sin 0"

=

B, = HO\ (p, a 3

B" (<1,<1,); (B.1)

t' e'"

sin

°

3,

(B.2)

If we introduce .

do (NN~l\N.K)·8 (£2 - ;'M~) ", -=-:-=--:::---;-;----:-----"---;, dT IT (T max - T)J'/' (2n)' E (I.n)' (2m" 1')'/, •

the cross section and polarization of the A particles in all three reactions are given by

f (pp

1 A. B. Migdal. JETP 28, 10 (1955). Soviet Phys. JETP 1, 7 (1955). 2 K . M. Watson and K. Brueckner, Phys. Rev. 83. 1 (1951). K. M. Watson, Phys. Rev. 88, 1163 (1952). 3 R. P. Paruntseva, JETP 22, 123 (1952). (E. M. Henley. Phys. Rev. 106, 1083 (1957). 5 G. Feldman and P. T. Matthews, Phys. Rev. 109, 546 (1958).

I T,I NY; =

t (N N - •. \ N K)

263

• ApK') ,-

/I~

I' (p,\ a,r' sin' 0,.

PA(pp-+.'\pK) = P;

t (np ~ .\NK) = f (pp -+ ·\pK+)· r 31 H. I' (PA a,t' sin' 03.

8 R.

Karplus and L. S. Rodberg, Phys. Rev. 115,

1058 (1959).

7E. P. Wigner. Phys. Rev. 73, 1002 (1948). A. I. Baz', JETP 33. 923 (1957), Soviet Phys. JETP 6, 709 (1958). G. Breit, Phys. Rev. 107, 1612 (1957). R. G. Newton, Ann. Phys. 4, 29 (1958). R. K. Adair, Phys. Rev. 111, 632 (1958). L. Fonda, Nuovo cimento 13, 956 (1959). 8 V. N. Gribov, JETP 33, 1431 (1957), Soviet Phys. JETP 6, 1102 (1958); JETP 34, 749 (1958), Soviet Phys. JETP 7, 514 (1958); Nuol. Phys. 5, 653 (1958). 9 L. I. Lapidus and Chou Huang-chao, JETP 39, 112 (1960). Soviet Phys. JETP. 12, ((1961). 10 L. Fonda and R. G. Newton, Ann. Phys. 7, 133

(1959).

(B.3)

P, (np- •.\NK)~, (I .1\1' -IB,I')( I AAI' ;.31 HAI't'P. (B.4)

Translated by M. Hamermesh 74

106 SOVIET PHYSICS JETP

MARCH, 1961

VOL UME 12, NUMBER 3

ON THE PSEUDOVECTOR CURRENT AND LEPTON DECAYS OF BARYONS AND MESONS CHOU KIJANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor April 7, 1960 J. Exptl. Theoret. Phys. (U.S.S.R.) 39, 703-712,(September, 1960) By the use of the analytic properties of a certain matrix element it is shown that the result of Goldberger and Treiman regarding the decay 1f - /.I + " is valid for wider classes of strong interactions than those found by Feynman, Gell-Mann, and Levy, and in particular for the ordinary pseudoscalar theory with pseudoscalar coupling. A formula is obtained which can be used for an experimental test of the assumptions that are made. Lepton decays of hyperons and K mesons are also discussed.

a.p. (x)

1. INTRODUCTION

AT the present time the theory of the universal V - A interaction given by Feynman and Gell-Mann and by Sudarshan and Marshak is in good agreement with all the experimental data on f3 decay and the decay of the /.I meson. 1 The experimental ratio of the probabilities for the two types of 1f -meson decay, R (1f - e + ,,)/R (1f -- /.I + v), also agrees with the theoretical prediction. It may therefore be supposed that the universal V - A theory is also valid for processes of capture of /.I mesons in nuclei. One of the most important problems is the calculation of the probability of the decay 1f - - /.I + " according to the universal V - A theory. This problem has been studied in detail in a paper by Goldberger and Treiman (G. T.)2 by means of the technique of dispersion theory. Despite the fact thatG. T. made many crude approximations, the numerical result of their work agrees almost exactly with the experimental result. Quite recently Feynman, Gell-Mann, and Levy (F.G.L.) have reexamined this problem in a very interesting paper. 3 They have shown that the G.T. result can be obtained rigorously in certain models. Namely, let us write the Hamiltonian for f3 decay and /.I -meson capture in the form H = (goIY2}(P.

+ V.) L. + Herm.

adj.

(1)

where (2)

PO! and VO! are the pseudovector and vector currents for the weak interactions. F.G.L. succeeded in finding three models of the strong interactions in which the following equation holds:

= ian (x) /

Vz.

where a is a constant parameter and 1f (x) is the pion-field operator. By using the equation (3), F.G.L. obtained the G.T. result in a simple and elegant way. F.G.L. stated that their results would be extended later to any theory of the strong interactions. In their new theory, it is said in reference 3, there appears a form factor

is an analytic function of the variable s = - (Pn _ pp)2. 2. If the matrix element of the commutator for equal times is equal to zero, then we can write a disperSion relation Without subtraction. 3. The contribution of the nearest singularities predominates in the dispersion relation. From our point of view the form factor

492

107 493

PSEUDOVECTOR CURRENT AND LEPTON DECA YS gA{3. and f separately. a test of this relation between the constants gives a sensitive ~riterion for the correctness of the assumptions made about the universality of the pseudovector coupling in the weak interaction and the analyticity of a certain matrix element. In Sec. 4 the lepton decays of hyperons and K mesons are treated in an analogous way. From the data on the lifetime of K mesons the result is obtained that the pseudovector coupling constant gA Y for the {3 decay of hyperons is smaller than the coupling constant gA for the {3 decay of neutrons. 9

T(s) = - Y2GFm'j(-s +m') + T(s),

where G is the renormalized constant of the strong interactions of pions with nucleons, and T' (s) is a function analytic in the region 151

< 9m'.

'1'(5) = I :=

0 (x).

Applying this equation to the decay get (0 I 0 (0) In) = -

(4)

JI. +

1T -

II.

we

qa (0 I p. (0) In),

(5)

where qa is the four-momentum of the pion. The matrix element < 0 I P a (0 ) 11T > can be expressed in the form

(OIP.(O)ln)

= -q.F(m')j~

(6)

where m is the mass of the pion and F (m 2 ) is a constant parameter. which is determined by the lifetime of the pion. Substituting Eq. (6) in Eq. (5). we get (0 I 0 (0) In) = - m 2 F jY2qo.

(7)

Let us now turn to the consideration of (3 decay and II capture. In the general case the matrix element is of the form (nIP.(OJlp)=un{gAr.r,+if(pp-p fI )r}u, 6 p

(8)

II;

where gA and f are invariant functions of

s = -(pp - Pn)2. Applying the relation (4) to fJ decay and II capture, we get (n I 0 (0) Ip) = -

(pp -

Pn). (n I p. (0) I p).

(9)

+ fs] unr,up.

(10)

The central problem is to find the connection between the matrix elements < n I 0 ( 0) I p> and <0 I 0 (0) 11T>. This can be done if we use the analyticity properties of the matrix element (n

10 (0) I p)

= iunr,upT (5),

(14)

+ ex. (5- m 2) j m'.

(15)

Comparing (11) with (10). we get 2Mg A

+ fs = - Y2GF 'I' (s)m' j(-s+ m').

(16)

A very important fact is that the relation (16) holds for all s. Setting s = 0, we get (17)

This is the fundamental result of F.G. L .• which was first obtained in the paper of Goldberger and Treiman. 2 For II capture s.

=-

Mm;j(M

+ m.) = -0.9m;.

From (15). (16). and (17) we get* 2MgA.+f.s.=m'2MgA~j(-s.+m2).

(18)

Equation (18) is an exact relation between gAo f for II capture and gA for fJ decay, which can be tested experimentally. It must be pointed out that the derivation of (18) has been carried out in the most general way. for an arbitrary value of a. As will be shown in Sec. 3. this holds for almost any theory in which the matrix element is analytic. Substituting the experimental values of G. gA/3 and F in (17). we get '1'(0)

=

0.8.

(19)

From this and Eq. (15) we find

Substituting Eq. (8) in Eq. (9). we have (n I 0 (0) I P) = i [2Mg A

+ m'),

where

Let us write WaP. (x)

(13)

The derivation of (11) - (14) is given later, in Sec. 3. It is also shown there that T' (s) is in fact a slowly varying function for small s. In the region I s I < m 2 the function T' (s) is approximated with good accuracy by a constant. Let us rewrite (12) in the form T (5) = - Y2GF 'I' (s)m'j (- 5

2. THE RESULT OF GOLDBERGER AND TREIMAN

(12)

(11)

ex. = 0.2.

(20)

We emphasize that the G.T. result is valid only for those theories in which the condition a« 1 holds. This question is discussed in the following section. *This relation is contained implicitly in a formula of Gold· berger and Treiman.'

108 CHOU KUANG-CHAO

494

3. THE ANALYTICITY OF THE MATRIX ELEMENTS Let us now turn to the calculation of the matrix element . Using the standard method, 5 we write (n

10(0) I P)

{un ~ d'ze- iPn ' (0 I T ('1 (z) 0 (0)) I p)

= -

-u:~d'ze-iP"~(zo)(OI[1j>n(z), O(O)1lp),

(21)

where 17 (z) = is+6S/61Jin (z) is the current operator for the neutron field. Hereafter the equaltime commutator will be omitted; it would give an additive constant in the final expression and would not affect the analytiC structure of the matrix element, for example, the locations of the poles and their reSidues, the branch points, and so on. We note that T (l1(Z)O(O» = 6(-z) [0(0), '1 (z)}

+ '1 (z) 0(0).

where (J (z) = 1 for Zo > 0 and (J (z) = 0 for Zo < o. The second term makes no contribution to the matrix element. Thus we bave (n

I 0 (0) I p)

= -

iU n ~ d'ze- I - n '

a(- z) (0110 (0). '1 (z)] I p).

(22) In the coordinate system Pp" 0 It II easy to show by the method of Bogolyubov' that the function T (s) in Eq. (11) bas a pole at 8 • m 2 and a cut that begins at the point s • 9m 2• At other points it is analytiC, If the {ollowlng Inequality holds:

IImP•• 1> 11m V p~.- Mi,

(P n . ' " M- s/2M).

(23)

Unfortunately, the inequality (23) la satisfied only in the case of Imaginary nuoleon masa. We assume in what follows tbat the analyticity of tbe matrix element in the variable 8 doe8 not cbange on analytic continuation with respeot to the maS8 variable. The residue at the pole 8 • m 2 18 ea81ly caloulated and is equal to 21/20 0 I 0 (0) Ill' > (2qo) 1/2. Thus we have

<

T (5) =

}iTO. (010 (0)/ n) -s+m

+ T' (s) =

-

V ~q-:

V~OF"'~ t- T' (s). +mt

-oS

(24)

The corresponding Feynman diagram for the term with the pole is shown In the drawing. In Eq. (24), T' (s) 18 an analytic function with a branch point at 8 '" 9m2 • The spectral resolution of the function T' (s) If of the form 00

T'(s)=a.+ ~ \) ~ds' 11: 5'(S'-S) • • m·

(25)

where p ( II) is the spectral function. For small s we can expand T' (s) in a power series in s, which has the radius of convergence 9m 2 • Setting T' (s) =

~ansn,

(26)

one can easily show that for large n lim' a.+1s rt--+oo

I

an.

/<4. 9m

(27)

Therefore the series (26) converges rapidly in the region I s 1< m2. 1f the spectral function does not change sign, the inequality (27) holds also for small n. In this case, for arbitrary n> 1 we have

We note that for f3 decay and II capture the distance between the points s = 0 and sll = - 0.9m~ is much smaller than the radiUS of convergence 9m 2. Therefore with good accuracy we can replace T' (s) by a single constant both for (3 decay and for II capture (the error is of the order of 0.9m~/9m2 '" Y20)' Thus we get the final result given by the formulas (14) and (15). Let us now go on to the consideration of the quantity ct. If the matrix element of the equaltime commutator Is not zero, then in general one must use a dispersion relation with a subtraction. In tills case the quantity ct is proportional to the subtraction constant ao, which can be very large. If, on the other hand, the matrix element of the commutator is zero, than there is a disperSion relation without a Bubtraction. Then it is reasonable to suppose that the contribution of the nearest singularity predominates and the quantity ct is small (ct'" 0.2« 1). Thus it is reasonable to assume that l{J (s) is slowly varying for any theory in which there is a dispersion relation without subtraction. Let us consider the ordinary pseudoscalar theory, with the Lagrangian

109 495

PSEUDOVECTOR CURRENT AND LEPTON DECAYS L=

-N(ii+ Mo - iGo(~")T.)

fJ decays.T Many authors have expressed the

)( N -m~'" /2-(iJ.,,)"/2-'J...,,'.

(2S)

By means of the gauge transformation

"-.,, + v (4M. + 2M)/3G. we get by the standard method, explained In reference 3,

+ 2M) / 30., + i} (Mo - M) N~T.N (4M. + 2M) / 3G..

p. = N-.r.T. N - ia." (4M.

o (x) =

(29)

ia.p. = 2G.N N"

+ (m~" + 4')..0,,2,,)

(30)

We shall show In this case that the equal-time commutator for the operator 0 makes no contribution to the matrix element < n 1 0 (0) 1 p >. Let us examine the matrix element of the commutator 1 = (012G.N,,+ i +(M o - M)~T.NIN).

(N I P. (0) I Y> = UN {gAyT.T.

From symmetry properties we have 1,= iA-.r.UN.

i3Au N = -2i(01'1(0)IN),

where

+ (M -

+ i~y [(P N -

py)

x T. -T. (.oN - py)] T. + ify (p y - PN).T.) U y, (31)

Multiplying Eq. (31) on the left by the matrix TY$, we get

'1 (0) = iG. (~,,) T.N

opInion that the universality of the weak Interactions evidently does not extend to strange-particle decays. Nevertheless, It is reasonable to assume the existence of a limited universality [a lepton current in the form (2)8]. In what follows we assume that the K meson is pseudo scalar and the V and A Interactions exist for the lepton decays of strange particles. In this case the Hamiltonian for the weak decays of strange particles is of the form (1). FollOwing the example given in Sec. 2 for the pseudoscalar theory with pseudoscalar couplIng, we can construct the pseudovector current In such a form that a dispersion relation without subtraction holds for the matrix element ' Generally speakIng, the matrix element for hyperon decay consists of three terms:

M.) N

is the current of the nuC'l.eon field. It Is known that the matrix element < 0 111 (0) 1N> Is equal to zero, and therefore 1=0. Thus we have shown that In the ordinary pseudoscalar theory there exists the pseudovector current (29), which satisfies all the necessary requirements. l! the pseudovector current is of the ordinary form

then the matrix element of the commutator is not zero, and In general there is no dispersion relation without subtraction. Even In this case there is hope that the G.T. result is valid. This question will be discussed In the Appendix.

4. LEPTON DECAYS OF HYPERONS AND K MESONS The experimental limit for the probabilities of lepton decays of A and E hyperons is an order of magnitude smaller than the theoretical value calculated on the hypothesis that the effective coupling constants In hyperon decays are equal to those In

(32)

from which we have (N I 0(0) I Y) = i (N

= i

[(MN

liV.IY>

+ My) gAY + fy5] u-NT.Uy,

(33)

where s = - (py - PN)2. Repeating one after another the arguments presented In Secs. 2 and 3, we easily get the following equation: [(MN

+ My) gAY + fy5] (34)

where GKY is the renormalized coupling constant for the KYN Interaction, and FK is a constant parameter associated with the decay of K mesons. We have further <0IP.(0)IK>=-q.F K /V2q •.

(35)

We can determine FK from data on the lifetime for the decay K-" + II. In Eq.'(34) Ty(s) is a function that Is analytic in the region

I s I < (mK + 2m)'.

(36)

Let us denote by TN the kInetic e!lergy of the nucleon recoil in the rest system of the hyperon. Expressing s in terms of TN, we get (37) In the present case the values of s that correspond to fJ and " decays are very close together, as compared with the distance between the s given by Eq. (37) and s = (mK + 2m)2, Therefore with good accuracy we can replace Ty (s) by a constant ay.

110 496

CHOU KUANG-CHAO

Thus we have [(MN

Substituting (46) in (45), we get

+ My)gAY + tysl =

-

GKyFKmk / (-s

+ mk) + ay.

(38) The relation (38) can be used to test the universality of the pseudovector current in lepton decays of strange particles. Applying the dispersion theory of Goldberger and Treiman, we find for the function fy: fy = - GKYFK j(- S

+ mk) + T~ (s),

(39)

where Ty (s) is a function analytic in the region (36), which with good accuracy can be replaced by a constant aY. Substituting Eq. (39) in Eq. (38), we get (MN+My)gAy=-OKyFK+ay-Say.

(40)

The relation (40) is a generalization of the formula of Goldberger and Treiman for the decay of straqge particles. The experimental data on the llfetimes of K and IT mesons show that Fie« F~. Therefore It can be seen from a comparison of Eqs. (40) and (16) that to accuracy ay- sa'y g~y
(41)

even for the case in which GKY and G are of the same order of magnitude. This fact was first pointed out by Sakita. 9 We emphasize that our method can also be easily extended to the case of a scalar K meson and to other types of weak Interactions (for example, S + P). In the case in which the relative parity of K and YN is positive, we have to deal with the divergence of a vector current

=uN{RVyr. t -r.(,oN-.oy)I-1

idl·(fl y

«() I Y> Is

Il;y[(;'"

of the

Comparing (47) and (16), one sees that to accuracy ay-say

(~)'=(~ gA~ MN-My

FKGKY)' =5C(G Ky )'

oft

f1f,G1f,

'

where C is of the order of unity. Therefore in the case of the scalar K meson the small probability of lepton decay of hyperons could be explained only by having the coupling constant GKY for the KYN interaction be smaller than the pion-nucleon constant G lT • We note that A and ~ can have different relative parities. Let us consider this case. For simplicity we call the K particle a scalar, if the relative parity of K and A N is positive, and a pseudoscalar if it is negative. In the case of the pseudoscalar K meson, Eq. (40) holds for the decay of A particles, and Eq. (47) holds for the decay of ~ particles, if we write in the form (43). In the case of the scalar K meson, conversely, Eq. (47) holds for the decay of A particles and Eq. (40) for ~ particles, if we write in the form (32). We note that the relations (38) and (45) can be used for the determination of the renormalized coupling constants GKY, if precise experiments are made on the decays of strange particles. The writer expresses his hearty gratitude to Professor M. A. Markov, Ya. A. Smorodinskir, and Chu Hung- Yliang, and also to Ho Tso-Hsiu and V. I Ogievetskil for their interest in this work and a discussion of the results.

1',.).I"y'

.lI y ) I!". I '[,'"'1 ""lIy'

APPENDIX In the usual theory the pseudovector current has the form

,;,:) ••

(43)

From this we have

= il(M N

(47)

(42)

iaaVa = () (x).

The matrix element
(MN-My)gl"y=--GKyFK+ay-a~s.

(44)

p. = N'r.r,N.

for which the divergence has been calculated in reference 3 and is given by iiaPa = 2M oN'r,N - 2100N Nte.

It is easy to repeat the remaining arguments, and

the final formula will be of the form

(A.l)

(A.2)

In order to calculate the matrix element

(MN-My)gVy+dys-~ -()~yFKIII~/( -.' IlIIk) t-ay.

(45) Applying the dispersion theory for the function dy, we find (46)

<0 I ilaPa I IT >, iiP •• = -

we write Eq. (A.2) in the form

.~M •. +.2M·I(·

I

G;;- I

- 2iO,JVNte,

I

Go mo - m, ) te -

, 4"o1t'te] (Ao3)

where J is the meson-field current. Using the

111 497

PSEUDOVECTOR CURRENT AND LEPTON DECAYS fact that the matrix element < 0 1 j ( 0 ) 1 7r > is equal to zero, we get

In this case the divergence of the pseudovector current is

(Ola.p.(O)ln) = i2MoG;-J~m'VZ3/V2qo .

- 8Mul.oG;;-1 /0 In'n In) - 2iG o (0 IN Nn In),

ia.P. (A.4)

where Z3 Is the renormalization constant for the pion wave function. We now go on to the consideration of . We rewrite Eq. (A.4) In the form . 4Mo + 2M •

0

a•• P =-I~J+

(x),

(A.5)

where the operator 0 is that of Eq. (30). In the present case we can write a dispersion relation without subtraction for the matrix element , with the term with the pole defined by Eq. (12). Thus we have i (ni a.p. (0) I p) = +(4M o

+ 2M) GG u-

J

d (s) F (s)Vl,iun,ou p

(A.6)

+
where d ( s) and F (s) are the respective form factors for the IT -meson propagation function and the vertex part. If the first term in Eq. (A.6) Is small in comparison with the term with the pole, then the G. T. result would hold also for the usual theory. We assume that the first term in Eq. (A.5) predominates. Comparing Eqs. (A.5) and (A.6), we get 2M.&nt

F':::::-G'" o m

VZ•.

(A.7)

By means of Eq. (A.7) the first term in Eq. (A.6) can be expressed in the form 2M

+M

i -3~ G

Fm'

MI'

-

unTou p d (s) F (s).

(A.8)

Since in perturbation theory the quantity 6m 2 diverges quadratically, it is very probable that 2M.+ M m' ..... 1 3M,

MI.-....·

(A.9)

In this case the first term in Eq. (A.6) is actually small in comparison with the term that has the pole. Therefore it seems to us that the G.T. result is also valid for the usual theory. It is interesting to note one more example, in which the pseudovector current has the form

iP. (x) =

a." (x).

(A.10)

= me" -

iGoN~,oN - 41.0n'"

(A. 11)

From this we get (0 I 0 (0) In) = i (0 I a.p. (0) In) = m'

VT./ V2qo, (A.12) V Z;u.Tou

i (n Ia.p.1 p) = (n I 0 (0) I p) - i V2G d (s) F (s)

The term with the pole is of the form ViG VZ,m' /(-s

+ m').

p•

(A.13) (A.H)

Comparing the expression (A.14) with the second term in Eq. (A.13), we verify that they are of the same order and cancel each other. From this example it can be seen that for those theories in which a disperSion relation without subtraction does not hold for the matrix element the G.T. result is in general not a good approximation. IR. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958). E. C. G. Sudarshan and R. E. Marshak, Phys. Rev. 109, 1860 (1958). 2M. L. Goldberger and S. B. Treiman, Phys. Rev. 110, 1178 (1958). 3 Feynman, GeU-Mann, and Levy, The Axial Vector Current in f3 Decay (preprint, 1960). 4 L. Wolfenstein, Nuovo cimento 8, 882 (1958). M. L. Goldberger and S. B. Treiman, Phys. Rev. 111, 354 (1958). 5N. N. Bogolyubov and D. V. Shirkov, BseAeHKe s TeopHIO KsaHTosaBBblX nOJleli (Introduction to the Theory of Quantized Fields), GITTL, 1957. [lnterscience, 1959) Lehmann, Symanzik, and Zimmerman, Nuovo cimento 1, 205 (1955). 6 Bogolyubov, Medvedev, and Polivanov, BonpocbI TeopKH AKcnepcHoHRblX cooTHomeHKlI (Problems of the Theory of Dispersion Relations) Fizmatgiz, 1958. TF. S. Crawford, Jr., et aI., Phys. Rev. Letters 1,377 (1958). P. Nordin et aI., Phys. Rev. Letters 1, 380 (1958). 8 Chou Kuang-Chao and V. Maevskil, JETP 35, 1581 (1958), Soviet Phys. JETP 8, 1106 (1959). R. H: Dalitz, Revs. Modern Phys. 31, 823 (1959). 9B. Sakita, Phys. Rev. 114, 1650 (1959).

It is easy to show directly from Eq. (A.10) that

the matrix element for f3 decay is equal to zero.

= m'" (x)

- ] (x) = O(x)-j(x).

Translated by W. H. Furry 133

112 SOVIET PHYSICS JETP

VOLUME 12, NUMBER 4

APRIL, 1961

ELASTIC SCATTERING OF GAMMA QUANTA BY NUCLEI L. 1. LAPIDUS and CHOU KUANG-CHAO J oint Institute of Nuclear Research Submitted to JETP editor May 12, 1960 J. Exptl. Theoret. Phys. (U.S.S.R.) 39,1056-1058 (October, 1960) The energy dependence of the cross section for elastic scattering of y quanta near the photonuclear threshold is investigated with the help of the dispersion relation for forward scattering. The first peak in the scattering cross section is attributed to dispersion effects. Experiments required for a more detailed analysis are discussed.

1.

A recent investigation of the scattering of y quanta by deuterons below the threshold for pion production with the help of dispersion relations! shows that photonuclear processes have a very strong effect on the elastic scattering of y quanta in a wide region of energies (up to - 100 Mev). In the present paper we discuss the scattering of low-energy y quanta by nuclei. Fuller and Hayward,2 as well as Penfold and Garwin,3 have already used dispersion relations for the analysis of the elastic scattering of y quanta by nuclei. However, they considered only the scattering above the threshold for the (yN) reaction. Taking the formation of particles in the S state into account leads to the known non-monotonic behavior near the threshold for the (yn) reaction. The disperSion relations not only allow us to discuss the Wigner-Baz' effect, but also to take account of the general effect of inelastic processes on the energy dependence of the elastic scattering amplitude in a wide region of energies. It appears to us that, within the framework of the rlispersion relations, the first peak in the scattering cross section of y quanta 2 is related to the near-threshold effects in a natural way. We restrict the discussion to forward scattering. In the dipole approximation, which, apparently, does not contradict the experimental data on the absorption of y quanta by nuclei up to - 30 Mev,3 our results are also valid for other scattering :lngles and for the total elastic scattering cross section <Js ( J) ). 2. The scattering amplitude for y quanta has the form I -~

R, (e'e) -I Ro ([kxejx[k'xe'j) ·1 (e'e) [R, - R2 (ke') (k'e)

for a spinless nucleus and

+- (kk') R,J (1)

T

=

R,(e'e), l?,([kxe][k'xe'1)

i/(,(a[e,e\)

11<.(a[[kxe'\x[kxe[])-tll?.[(ak) (e[k'xe'l) (ak') ([kxeJe')I

IR" [(ak') ([k'xe'] e). - (ak) (e' [kxellJ

(2)

for the scattering from a nucleus with spin ~. The corresponding expressions for nuclei with larger spins are more complicated. In the dipole approximation only R! is different from zero. DispersiCl.\l. relations can be written down for all scalar functions Ri. The imaginary parts of the amplitudes are related to the cross sections for absorption of y quanta in different states by the unitarity condition. The scattering amplitude for y quanta at low energies is known, so that the real and imaginary parts of all Ri can be calculated from the data on the absorption of y quanta in different states with the help of the dispersion relations and the unitarity condition. However, up to the present time no detailed analysis of the absorption of y quanta by nuclei has been carried out. Therefore we restrict ourselves to the dispersion relation for R! + R 2 , which has the usual form Re (R, (v, 0 = 0°)

+ R2 (v, 8 = OO)J (3)

In the literature accessible to us (see the new review article of Wilkinson 4 ) we did not find detailed information on the energy dependence of the cross section for absorption of y quanta by nuclei. Only in the case of He~ (and the deuteron) is the photodisintegration cross section known in a wide region of energies. 5 Experimental data on <Jt (J) for aluminum up to - 30 Mev were obtained recently by Mihailovic et al. 6 These authors found evidence of a fine structure in the dependence of

735

113 736

L. I. LAPIDUS and CHOU KUANG-CHAO

the cross section in the region of the giant resonance. Data for the region near threshold are not available. By smoothing out these experimental data, we obtained with the help of relation (3) the energy dependence of the scattering amplitude shown schematically in Fig. 1. This behavior of the scattering amplitude is apparently characteristic for a whole group of nuclei with large ITt ( v). The photodisintegration of the deuteron leads to a decrease in the cross section for yd scattering near the threshold, as compared to the Thomson cross section. However, for other nuclei with a large absorption cross section the picture is different. Since the dispersive part of the amplitude below and slightly above the threshold is positive, the total amplitude vanishes at some energy below Vt and then becomes positive (in the case of the deuteron! the amplitude does not change its sign). Above the threshold the real part of the amplitude decreases rapidly, goes through zero, and becomes negative, increasing in absolute value with increasing energy. Above the threshold, the amplitude has, of course, both a real and an imaginary part. All this leads to an energy dependence of the scattering cross section which is shown in Fig. 2.

ULc "t

FIG. 1

"

FIG. 2

In its general behavior, it agrees with the experimental data. 2 For the aluminum nucleus the scattering cross section reaches the value ITs (v) '" 2 X 10-il8 cm 2 in the region of the first maximum, which is close to the experimental value. It is clear that, if the dispersion effects are not so large as to change the sign of the scattering amplitude in the threshold region, the first maximum will not appear. The ratios between the cross sections at the maxima for different nuclei are conne$d with the relative role of the absorption of y quanta in the region near threshold and in the region of the giant resonance. The fact that not only the region near threshold, but also the giant resonance, gives a contribution to the first maximum leads to an appreciable widening of this peak. The half-width for aluminum exceeds 2 Mev.

In the evaluation of the data on the absorption of y quanta by aluminum we did not consider the effects connected with the difference in the thresholds for the (yp) and (yn) reactions, and we also neglected the resolving power of the apparatus. It ,seems to us that a more detailed analysis would lead to little change in the basiC result. 3. The interpretation of the energy dependence of the cross section for scattering of y quanta by nuclei proposed earlier7 is thus apparently confirmed. Owing to the relatively poor accuracy of the experimental data on the absorption cross sections and to the absence of data on the absorption cross sections for a whole series of nuclei, it is not possible now to conduct a reliable analysis for all nuclei whose scattering properties are known. The fruitfulness of using the dispersion relations in the analysis of the scattering of y quanta by nuclei makes it feasible to conduct a whole series of investigations with y quanta. First of all, it appears necessary to obtain information on the energy dependence of the absorption cross sections of y quanta in the region near threshold as well as in the region of high energies. The consideration of the cross sections for photoproduction of pions in the calculation of ITt ( v) may become necessary in the energy region of ~ 120 to 150 Mev and above. Knowing the absorption cross sections in a wide region of energies, one can obtain information on the polarizability of nuclei. For Al the polarizability d

~ "" [ dv' Re (R 1

+ R,)

J.~. =

nc

f at (v) uV

2,,- P } ~ "

turns out to equal ~ 2 x 10-,39 cm 3 (the error may be as large as 50%). For Het the value is a = (0.70 ± 0.05) x 10-10 cm 3. A more detailed phenomenological analysis of the absorption of y quanta in a wide region of energies becomes inevitable if one wants to obtain information on the spin dependence of the disper.sive parts of the scattering amplitudes for the 'Y quanta. On the other hand, the dispersion relations and the unitarity of the S matrix can be used to obtain information on the absorption cross section from the experimental data on the scattering of y quanta by nuclei. For this purpose the inverse dispersion relations may prove to be convenient. The authors are grateful to Ya. A. Smorodinskil for useful comments. ! L. I. Lapidus and Chou Kuang-Chao, JETP (in press).

114 ELASTIC SCATTERING OF GAMMA QUANTA BY NUCLEI 2 E. G. Fuller and E. Hayward, Phys. Rev. 101, 692 (1956). 3 A. S. Penfold and E. L. Garwin, Phys. Rev. 116, 120 (1959). 4 D. H. Wilkinson, Ann. Rev. Nucl. Sci. 9, 1 (1959). 5 A. N. Gorbunov, Tr. FIAN (Proceedings of the Physics Institute, Academy of Sciences) 13, 145 (1960).

737

6 Mihailovi6, Prege, Kernel, and Kregar, Phys. Rev. 114, 1621 (1959). 1 L. I. Lapidus and Chou Kuang-Chao, JETP 39, 112 (1960), Soviet Phys. JETP 12, 82 (1961).

Translated by R. Lipperheide 196

115 SOVIET PHYSICS JETP

MAY, 1961

VOLUME 12, NUMBER 5

THE ELASTIC SCATTERING OF y RA YS BY DEUTERONS BELOW THE PION-PRODUCTION THRESHOLD L. I. LAPIDUS and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor May 12, 1960 J. Exptl. Theoret. Phys. (U.S.S.R.) 39, 1286-1295 (November, 1960) Dispersion relations and the conditions for unitarity of the S matrix are used for the analysis of the elastic scattering of y rays by deuterons below the threshold for pion production. The low-energy limit is examined for the scattering of y rays by nuclei of arbitrary spin. The energy dependence of elastic yd scattering is deduced on the basis of the experimental data on the photodisintegration of the deuteron. The result differs decidely from that of the impulse approximation over a wide range of energies. It is found that it is not important to include the influence of photoproduction of pions from deuterons in the range of energy considered.

1. INTRODUCTION

THE scattering of y rays by deuterons is an example of a process whose amplitude is decidedly affected by inelastic processes, such as the photodisintegration of the deuteron and the photoproduction of mesons. Inclusion of the influence of pion photoproduction, which is important at y-ray energies near and above the photoproduction threshold, requires a rather detailed analysis of the processes r+d-.NNlI,

Yi-d-+d+no

and is not dealt with here. The influence of the photodislntegration of the deuteron on the elastic yd scattering near the threshold for photodlslntegratlon, and the departures from monotonic variation with the energy, which lead to a sharp decrease of the cross section, have been considered previously. 1 The purpose of the present paper is to make an analysis of yd scattering on the basis of dispersion relations over a wider energy range, in which meson production still does not have much effect. The experimental data 2 on the scattering of y rays by deuterons in the energy range 50 -100 Mev do not fit into the framework of the impulse approximation, S and this forces us to carry through an analysis that does not involve this approximation. On the other hand, the contribution to the scattering amplitude from meson-production processes falls off rapidly below the threshold for photoproduction of mesons.

We shall confine ourselves to the forward scattering. In the calculation of the dispersion integrals we take into account the cross sections for the electric-dipole and magnetic-
2. THE PHENOMENOLOGICAL ANALYSIS As is well known, the formulas for the electric and magnetic multipole waves Y~~ (k) of a photon are (A = 0,1)

yj~

=

~ ci~-e llXI m-e (k) ~e,

Yi:.! = - i [kxyj~l.

(la) (lb)

where k is the unit vector along the momentum of the photon in the center-of-mass system, YZ m (k) are normalized spherical functions, and ~+ ~o

=

= k,

-(I

+ ii)/Y2,

~_ = (I -

ij)!V2

- the eigenfunctions of the photon spin - satisfy the transversality condition 898

(2)

116 899

THE ELASTIC SCATTERING OF Y RAYS BY DEUTERONS kY\~(k) = O.

ao(OO)=IA +HC+D)I'+ !sIC

If we write

+ D I' + i-I B I' + i-I D '1- =

'1+=-:Zm·

1 (,r-

, 2

f) i

0,

(3)

for the spin functions of the deuteron, then for the yd system we can construct eigenfunctions of the total angular momentum J2, the component J z , and the parity from Eqs. (1) and (3): (4)

In the center-of-mass system (c.m.s.) all quantities in the final state are denoted by symbols with primes; for example, k' denotes the direction of the photon momentum in the final state. By means of Eq. (4) we can write the scattering matrix T in the form (e'Te) =

~ Y}~·1(k') Y}~1 (k)a;,\:,

.,.

(5)

C I',

(11)

and we have

+ ~D) =

4nlm (A +~C

(11')

qat,

where O't is the total interaction cross section, including both elastic and inelastic interactions;

q = I qlk. Let us turn to the phase-shift analysis. We include the amplitudes for electric-dipole and magnetic-dipole transitions. The magnetic-dipole transition is characterized by the matrix

FJ = -

~

y}01M (k) y}OlM (k)

3 "dM

4n .LJ

IM-r lr

C IM-f CIM-I'

CIM

1M-r'lr'

°r

,0

10 IM-r'T), T]r'~M-r':,M-r'!

101M-r

(12) where we have used the fact that for forward scattering

!MU'

where e and e' are the respective polarization vectors of the photon in the initial and final states. Parity conservation requires that

a;,\: =

0 for (- 1)/+'

+ (- J)l'+".

(~+° n+)

(6)

Time-reversal invariance leads to the symmetry condition U' A'I. ajU'= ajrJ.

From Eqs. (12) we easily obtain

(7)

The usual arguments show that for forward scattering the spin dependence of the matrix T is of the form

+ (~_° FC) =

o

(~+ F~+) -

(CF~+) =

8~

-

Here S is the operator of the spin vector of the deuteron; its components Si satisfy the DuffinKemmer commutation relations:

j

(~~ T~±) = A =F B (Sk) + ~ (D + C) [2 - (Sk)'J.

PI -} (Sx

I

+ is.)'.

(13)

'10 - )/2

_1/'1.

~/s

.'

" _1/3

(14)

In obtaining the relations (13) we have used the relations

'1+'1: =

+

(S;

+ S,),

'1_'1~ =

'1+'1~ = T", =

+

T,,_,=

+

[5; -

+

'1o'1~ = I -

S~

+ i (SxSy + SyS,)].

(S; - S,).

[S~ - 5; -

i (S,Sv

+ S.S,)].

5;,

(15)

The case of the electric-dipole transition is obtained from the relations (13) by replacing t+ by i[t+ xkJ = - t+ and t- by i[t- xkJ = t-. Comparing Eqs. (13) and (14) with Eq. (10). we get

i,. h, (etl + 2P/) (at) + aj').

2A =

(~~ T~+) = ~ (D - C) {(Si)' - (Sj)' =F i [(Si) (Sj) + (Sj) (Si)]). (10)

By means of Eq. (10) and the method developed previously' we can construct the density matrix of the final state and calculate all observable quantities. The unpolarized forward scattering cross section is given by

'S y). '

(Sx -

0 0

=

u/=

'1_'1: =

Using the Stokes parameters to describe the polarization of the photon, as was done in our paper on yN scattering,' we get without difficulty from Eqs. (2) and (8):

8~

PIT", = -

~/= Ii =

(8)

3 ~I 21 an

For i = 0, 1, 2 the quantities ai' f3i' Yi are

(e'Te) = A (e'e) + iB (S [e'~eJ)+ ~ C [(Se) (Se') + (Se') (Se)J+ ~ D [(S[k~eJ)(S [k'xe'P+(S (k'xe']XS [kxe])].

3 (~-n-) = 8" liS"

3 snPIT,.-, = -

(~+° F~_)= -

+ PIS; J ,

3 [etl 8;t

°

2B = - ~ "r/' (a(m) 8n.LJ /

+

a(I). I

j

C=

-YhOa(<) 8n t-'1 J '

D= -

P 8li Z; J la, 3

(m)

'

(16)

I

The condition for unitarity of the S matrix reduces to the relation

117 900

L. 1. LAPIDUS and CHOU KUANG-CHAO

211i [T'(- k', - k, -e', -e, - S)yd_yd -

T

(k', k, e', e, S)yd_yd J= q ~ dO n+p T~d~pTyd-'np,

=

(17) where q is the relative momentum of the I'd system. Let us represent T 'Yd-np in the form T,d_oJ'

= 2}Y;'·M(n)YI)~.\(k)dit·I'

(18)

where j is the total angular momentum, l' is the orbital angular momentum in the final state, and s is the total spin of the np system. Parity conservation requires that

(- 1("'H =

(--I)".

The quantities djih are connected with the partial cross sections for photodisintegration: 24ltoj}!) = (2j -I- I) I dih

12. (19)

The total cross section for photodisintegration is given by 24ltO',d_,p

= 2}(2j+lHidihI2+ldihl'j. jl's

(20)

Substitution of Eqs. (18) and (5) in Eq. (17) gives 4lt

1m a~~·. (q)

= q

2} (d;~"lrdi~:I"

(21)

I"'

and USing Eqs. (16) and (14) we get the results

1m

[A

J

(30 0 +.~ 0,) ql4lt,

1mB =

H-oo + fo, - f 0,) ql411, ( - 30~') + foi') - &-0~I»qI4lt,

ImC =

ImD=(-30~m)-!oo(;'»q/411,

e 1 fl)

;2:1'; ~ d'ze-iqZ;p', fl' I H (- zo) [e'j (z/2),

ej(-zj2)llp, fl).

For the case of forward scattering the deuteron momentum p can be set equal to zero. Considering the relations complex conjugate to Eqs. (23) and (24), we get ': fl' i e'N,,'(adu) (q) e I fl

>. =

Interchanging the order of (a' j( z/2 » and (aj( - z/2}) in Eqs. (23) and (24) and changing the sign of the variable z, we arrive at the relation
Let us represent Nret(adv) in the form (8). The conditions (25) and (26) reduce to the following symmetry properties of the scalar functions A, B, C, D: A,,'(adu) (v)" = A,,'(adV)( _ v), e"(adv)(v)'

=

C,,'(adv) (-v),

=

A'" (- v),

B",(adv)(v)' = _ B",(adu)( -v), D,,'(adu) (v)' = [f,,(adv)( _ v);

. A adu (v)

Badu (v) = -

ca
DadV(v)

Re Lt. •.• (vo) - Re L ..... (O) =

2vg p

3

(22)

where CTj denotes the partial cross section for photodisintegratlon in the state j, including the factor (2j + 1).

3. CROSSING SYMMETRY AND THE DISPERSION RELATIONS The retarded amplitude for I'd forward scattering can be written in the form (fl'ie'N"'elfl)

= - 2lt2j ~ d'zc-,qz

z/2)11 p, fl),

where 14 and 14' are the spin indices of the deuteron. For the advanced amplitude we have the analogous expression

rJ

(27)

B'" (- v),

= D'"

(- v).

dvlm L1.2 .• (v) V(VI_V~)

"d

It

Re L, (vo) - Vo Re L~ (0) = 2vo P 11

ej (-

(24)

(28) Denoting hereafter the quantities A (II), C (II), and 0(11) by L 1 (1I), Lz(II), and Ls(II), respectively, and B (II) by L. (II), we write the dispersion relations for the scalar functions in the form

+ -i- (C-i- D) 1 = l~lt 2} (ttl + -i- ~/)(lma\m)

1m A =

adv

S dv 1m L. (v)

co

"d

,

v'(vl-v:>

' (29) (30)

where lid is the threshold for photodisintegration of the deuteron, approximately equal to the binding energy of the deuteron. In order for it to be possible to use the relations (29) and (30) for an actual analysis, it is necessary to know L 1• 2,3 ( 0) and L, (0), i. e., to calculate the I'd scattering amplitude in the energy region close to zero. The result of the calculations carried out in the following section is that ReL,(O) =

-e' 1M ,

Re L~ (0) = (11.0 -

ReL... (0) = 0,

e 1M d )'.

where 140 is the magnetic moment and M
(31)

118 THE ELASTIC SCATTERING OF 'Y

+ b(~ + p~) + d (p,p,) + if (S [p."p,ll + h [(Sp,) (SPl) + (Sp,) (Sr.) + g [(Sp,) (Sp,) + (Sp,) (Sp,)L

4. THE LOW-ENERGY LIMIT FOR 'yd SCATTERING

Thirring, Low, Gell-Mann, and Goldberger S have shown that the limiting values at Vo = 0 of the scattering amplitude and of its derivative with respect to the photon frequency are determined by the statistical properties of the system, for systems with spin '12' Following a method developed by Low, we shall show that analogous results are also valid for systems with arbitrary spin. The S matrix for the scattering of photons from the state (q, e) into the state (q', e') is given by the expression

qij.=

S' = - e;qijej (4qoq~)-'I.,

(32)

~ P [j, (x), ii (y)] eiqx-iq'x dxdy.

(33)

Using the technique of Low, it is not hard to get the results*

(39)

where e is the total charge, j.Ls = j.Lo is the total magnetic moment, and the quantities a, b, c, d, f, h, g are invariant constants. Under Lorentz transformations il behaves like a component of a four-vector. Being an irreducible representation of the inhomogeneous Lorentz group, the wave function I p, j.L> transforms in the following way:

I p, I-').!:,. j p, f')' =

+ B (S [q'xq])+ D [(Sq') (Sq) + (Sq) (Sq')] =

qoq~C,

(35)

C = (2,,), 61') (p'

+ q' _

p _ q) ~ [

I

+

i, 1- q' )-q'J ~,J 0) E (q') - E (0) qo

(q - q'J

+

g l')= (2,,1' ~(')( 'I

I

+

'+ ' __

P

q

P

(q- q'J

+

(36)

Eq p,o = M

[E

)~[(q-q'liiJq)
J.

+ c (S (S, p,

+ p,) + i~Sx[p, + p,) + (S, p, + p,) S},

(37)

q [(pq)

""Ii' Eq

p

(

q

E ) Eq 1- M +·M Ep ] ,

(pq)] . + E;

(41)

The second system moves with the velocity - q/Eq relative to the first. For the Lorentz transformation from the first system to the second we have to accuracy vo/c2 (42)

R (L, p) = I + i (S [pxq])/2M'.

We also have to accuracy vic (p + qlijq)' = e (p+2q)/2M + if' [Sxp1 + c (S (5, P + 2q) + (5, p =(p I j

I 0) +

+ 2q) S}

(q / M) (p I j

10)

= ep/2M + il-' [Sxp]+c [S (Sp)+(Sp) S]

+ aq/M

(43)

and to accuracy vo/c 2 (p

+ q I io I q )' = a + b (p' + 2pq + 2q') + d(pq + q') + if (S [pxq])+ h (S, p + q) (S, p + q) + (Sq)(Sq)} + g «S, p + q) (Sq) + (Sq) (S, p + q)} = [I -

i (S [p"-qj)12M'] EqM-l [(p I i. j 0

+ (qM-', (p Ii 10»]

>

= a - ia (S[pxq])/2M' + aq'/2M'

+ bp' + h (Sp)(Sp) + e (pq) /2M'

+ if' (S [pxqj) /M.

From Eq. (43) it follows that

p,] (38)

*ThIs Is the most general expression for an arbitrary S, if we are Dot concerned with terms with energy dependence higher

than linear,

PlO=Eq=Vq'+M',

Y"

The summation in Eqs. (36) and (37) is taken over the spins of the particles involved in the ;reaction. Let us consider the case in which the states I q > and so on are eigenstates of a system with spin S. For the calculation of Eqs. (36) and (37) we need the expression for the current matrix < P21 j I PI> in the low-energy region to accuracy vic, and for < q' I jo I q> to accuracy v 2/c 2• It turns out that these matrix elements can be determined with the required accuracy on the basis of general principles. Since j and io are Hermitian operators and the interaction is invariant under three-dimensional rotations and time reversal, the most general form of the matrix element of the current is, in the approximation in question, (p,1 j I p,) = (e/2M) (p,

p,=q,

p, = p +

'

(q-q'Ji/J-q')<-q'~iil 0) E (q') - E (0) qo

and in the other

E (q) - E (0) --q,

1

(40)

p,=p,

p, =0,

J0)

io J q) (q 1 i,

Rpp·(L, p) I L-'p, 1-">,

where Rj.LIl' (L, p) is the rotation of the spin in the Lorentz transformation, which has been treated by a number of authors, 6, 7 Let us consider two coordinate systems. In one

(34) A (q'q)

901

RAYS BY DEUTERONS

c= 0,

and from Eq. (44) that d + 2b = e/2M',

f = f'/M - e/2M',

g= h=

o.

(44)

119 L. I. LAPIDUS and CHOU KUANG-CHAO

902

Finally we have the covariant expressions (P21 j I PI)

=

e (P,

(p, j io I PI) = e

+ p,) /2M --:- iflSx [P2 -

+ i (fl/M - e/2M') (S(P. + 2b (P, -P.)',

--:- e (P,P,) /2M2

5(0"1/("'/ 100

pd,

...

.__-------

a?S

050

Pd)

0;:5

(45)

and, as must be so, the first of these is the same as the matrix element of the current of a nonrelativistic particle interacting with a magnetic field (cf., e. g., the book of Landau and Lifshitz 8 ). It turns out that the term contained in the expression (45) makes no contribution to the final result. By means of Eq. (45) one easily gets

I In

1,5

711

l,S

JfJ

FIG. 2

the imaginary part of the quantity L! + (%) ( L2 + La)· fu the energy range considered, 1'0:>' 100 Mev, the dominant contribution is that of photodisintegration with I' :>, 75 Mev. For the other amplitudes a more detailed analysis of the photodisintegration 2 S = i(2n)W<) (p' +q' - p-q) (4qoq;)-'/' (e ( e'e) iM is required. If we assume that the contribution -2ie (S [e'~]) M(fl- e/2M) - (iIL2fqo) (S [[e"'llxle'xq'JI) from photodisintegration with "0 80 Mev is also deciSive for the other amplitudes and use the anal- (iefl/ Mqo) [(eq') (S (qxe'lj- (e'q) (S [qxe])]) (46) ysis of de Swart and Marshak,!O it is possible to or for the matrix T estimate the dispersion parts of all the scalar amplitudes. But for yd scattering the spin-de- T = e2 (e'e) / M - 2i (e/M) v (fl-e/2M) (S (e'xe] pendent amplitudes playa much smaller part as - i (fl2/vXS[!e..qlx(e'xq'lJ)- (iefl/Mv) [(eq') (S [q'xe']) compared with the case of yN scattering. Figure - (e'q)(S [qxe]}j. (47) 2 shows the energy dependence of the yd forward scattering. With increase of the y-ray energy For forward scattering, in particular, we have the yd scattering cross section at first shows a - T = e' (e'e)/M -iv (S (e'xeJ)[!lo/S - efM]', (48) marked decrease as compared with the Thomson from which the result (31) indeed follows for S = 1. limit, and then (I' ~ 4 Mev) rapidly rises, and in the range 20 < I' < 80 Mev reaches values larger In the energy range below the threshold for than (e7'Mctc 2 )2 bY'a factor four. photoproduction of mesons from deuterons the Inclusion of the magnetic-dipole absorption, terms that depend on the spin make an insignifiespecially near the threshold, leads to an additional cant contribution to the cross section, since the sharp "dip" of the cross section,! with a total mass of the nucleus is doubled in comparison with half-width - 200 - 300 kev. The width of the total that of a nucleon, and the magnetic moment is decrease of the cross section is considerably much smaller. larger. The large influence of inelastic processes that 5. RESULTS OF THE ANALYSIS. DISCUSSION involve the deuteron as a whole, in addition to the Experimental data on the photodisintegration of processes involving the individual nucleons of the 9 the deuteron are available right up to - 500 Mev. deuteron, makes it impossible to apply the impulse The results of the calculation of Re [ Ll + (%) ( L2 approximation to elastic yd scattering over a + L3 »), for which it is sufficient to know the total wide range of energies. * The presence of the incross sections, are shown in Fig. I, where the elastic process of photodisintegration of the deuvalues of the real part of the amplitude are repreteron has an especially strong effect on the polarsented as fractions of e 2/Mp c 2• The photon energy izability of the deuteron. If, as A. M. Baldin has is measured as a multiple of the threshold for shown, the polarizability of nucleons is entirely photodisintegration of the deuteron: I'o/I'd = Yo· due to the process of meson prodUction, on the The diagram also shows the energy dependence of other hand the main contribution to the polarizability of the deuteron, and of nuclei in general, comes from photonuclear processes at much smaller energies.

:s

FIG. 1

*In a preprint received very recently, Schult and Capps have made a new examination of the corrections to the impulse approximation for yd scattering and have come to a similar conclusion.

120 THE ELASTIC SCATTERING OF I' It follows from Eq. (29) that the polarizability of the deuteron is given by

ad

==d dv

[Re(L 1 +2/3L2 +2/3L)] 3 v=Q

= in! fie pJ",(V)dV v2' Vd

(49)

An analogous formula is also valid for other nuclei. Dipole absorption plays the fundamental role in the total interaction cross section (Tt ( v). When this is taken into account Eq. (49) goes over into the well known formula of Migdal!! (cf. also reference 12). Substitution into Eq. (49) of the expressions (·n) and (46) of reference 1 gives for the sum of the electric and magnetic po!arizabilities of the deuteron

ae + am

=

ad

=

h

tz

("f)2 {if + (! + Vr;;)2~e E pc

p.

x (!lp -!In)2}

= 0.64

x 10- 39 em 3 ,

(50)

which agrecs with the rcsult of Levinger and Rustgi (cf. refcrence 12). The presence of sizable contributions from photodisintegratioll in the I'd elastic scattering amplitude and in the polarizability of thc deuteron prevents our obtaining reliable conclusions about the polarizability of neutrons from the experimental data on the scattering of low -energy I' rays by deutcrons. To get information on the magnetic polarizability of the deuteron one must evidently have a much more detailed analysis of the photodisintegration of the deuteron and of processes of photoproduction of mesons from deuterons. * Strictly speaking, thc treatment carried out in the prcsent paper is valid only for forward scatteri ng. In the dipolc approximation, however, the main results remain valid for other scattering angles also. But we have not ·made a direct comparison with the experimental data, since in the experimcnts 2 inelastic scattering of I' rays by deuterons, ~(

_ r

(I --)- Ii

_I_

I) :

~(,

was observed along with the elastic scattering. Recently A. M. Baldin (private communication) has examined the corrections to the impulse approximation in the inelastic scattering of y rays and has arrived at the conclusion that for this process also there are appreciable corrections *All conclusions concerning the magnetic polarizability of the proton are very sensitive to the assumptions that have to

be made in the analysis of the photoproduction of pions. When one uses the analysis of Watson it follows from the results

4

thflt the magnetic polarizability of the proton is small. (In the case of the analysis of Watson it goes to zero.) This conclusion evidently is not in contradiction with the experimental data,U

RAYS BY DEUTERONS

903

associated with the photodisintegration. Thus we can evidently conclude that the results of an analysis that takes into account the photodiSintegration of the deuteron (and the production of mesons), and the experimental data on the scattering of I' rays by deuterons in the energy range ~ 50 - 100 Mev are in agreement with each other. For a more reliable comparison of calculated results with experiment one first needs an analysis of the inelastic processes over a wider range of energies. The writers arc grateful to A. M. Baldin, V. 1. Gol'danskii, and Ya. A. Smorodinskil for numerous discussions. ! L. I. Lapidus and Chou Kuang-Chao, JETP 39, l12 (1960), Soviet Phys. JETP 12, 82 (1961). 2 Hyman. Ely, Frisch, and Wahlig, Phys. Rev. Letters 3, 93 (19;'9). 3 R. II. Capps, Phys. Itev. 106, 10:31 (1957); 108, 1U:l2 (1957). • L. I. Lapidus and Chou Kuang-Chao, JETP 37, 1714 (1959), Soviet Phys. JETP 10, 1213 (1960); JETP 38, 201 (1960), Soviet Phys. JETP 1l, 147 (1960). 5 F. E. Low, Phys. Hev. 96, 1428 (19:34). M. Gell-Mann and M. L. Goldberger, Phys. Rev. 96, 1453 (1954). 6 Chou Kuang-Chao and M. I. Shirokov, JETP 34, 1230 (19:38), Soviet Phys. JETP 7, 851 (1958). 7 L. Zastavenko and Chou Kuang-Chao, JETP 35, 1417 (19;)8), Soviet Phys. JETP 8,990 (1959). 8 L. Landau and E. Lifshitz, Quantum Mechanics, Pergamon, 1958. 9 Barnes, Carver, Stafford, and Wilkinson, Phys. Hev. 86, :3:39 (1952). J. Halpern and E. V. Weinstock, Phys. Hev. 91, 934 (1953). Lew Allen, Jr., Phys. Hev. 98, 705 (1955). J. C. Keck and A. V. Toliestrup, Phys. Hev. 101, 36U (1956). Whalin, Schriever, and Hanson, Phys. Rev. 101, 377 (1956). D. R. Dixon and K. C. Bandtel, Phys. Rev. 104, 1730 (1956). Alcksandrov, Delone, Slovokhotov, Sokol, and Shtarkov, JET P 33, 614 (1957), Soviet Phys. JETP 6,472 (1958). Tatro, Palfrey, Whaley, and Haxby, Phys. Rev. 1l2, 932 (1958). !O J. J. de Swart, Physica 25, 233 (19:)9). J. J. de Swart and R. E. Marshak, Physica 25, 1001 (1959). 11 A. B. Migdal, JETP 15, 81 (1948). 12 J. S. Levinger, Phys. Rev. 107, 554 (1957). 13 G. Bernardini, Report at the International Conference on the Physics of High-Energy Particles, Kiev, 1959.

Translated by W. H. Furry 240

121 LETTERS TO THE EDITOR

1032

ON THE PION-PION RESONANCE IN THE P STATE HO TSO-HSIU and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor September 30, 1960 J. Exptl. Theoret. Phys. (U.S.S.R.) 39, 1485-1486 (November, 1960) QUITE recently there has been great interest in the question of the existence of a p resonance (isobar) in pion-pion scattering.! From a study of the structure of nucleons by the method of dispersion relations, Frazer and Fulco have concluded that an isobar with mass 435 Mev and halfwidth 10 Mev must exist in the p state of pionpion systems. 2 Similar results on the presence of a p resonance in pion-pion scattering have also been obtained by other authors. 3 On the other hand, the integral equations for pion-pion scattering have been derived more accurately by means of the ordinary theory of one-dimensional dispersion relations.' Preliminary results on the solution of these equations, obtalned by means of an electronic computing machine, show that the amplitude of the p wave is very small. It is as yet unknown whether there is another solution with a large p amplitude. Therefore a direct experimental test of the presence of a p isobar in the pionpion system is of great importance. For this purpose Chew and others 5 have suggested the reactions

studies of which could help to provide information about the interaction of pions in the p state. These processes are interesting because the theoretical interpretation of the results is simple and clear. But because of the lack of high-energy clashing electron and positron beams, it is difficult to conduct such experiments at present. In the present note we suggest the study of the following processes: It± + Hec _ He' + It± + It°, (2a) It±+d-d+lt±+lt°,

(2b)

p+p-d+lt++lt°.

(2c)

For all of these processes the initial isotopic spin is I = 1. Consequently, the pair of pions in the final state has the isotopic spin I = 1 and is in a state with odd orbital angular momentum. In the

low energy region these pions are mainly in the p state. Let us assume that there is an isobar with mass 435 Mev and half-width 10 Mev in the p state. Then in the reactions (2) the pairs of pions come from the decay of isobars that have been produced together with nuclei He 4 or d. Because of this it is to be expected that there will be a sharp maximum in the spectrum of the He4 (or d). Let us consider, for example, the reaction (2a). Suppose the energy of the incident pion beam is 700 Mev in the laboratory system. (l.s.). Then in the center-of-mass system (c.m.s.) one should observe a maximum in the spectrum of the He 4 at energy 11 Mev and with half-width 2 Mev. In the case of the reaction (2c) with incident beam energy 1.4 Bev in the 1.s. the deuteron spectrum in the c.m.s. must have a maximum at energy 36 Mev and with half-width 3 Mev. If the p isobar does not exist, then the shape of the spectrum of the He' (d) varies smoothly and is determined mainly by the statistical phasevolume factor. Therefore measurements of the spectra of the nuclei in the reactions (2) will give information about the existence of a p resonance in the pion-pion system. We note that the process d + d - He' + 11'0 + 11'+ + 11'- is also useful for studying the isobaric structure of pion-pion systems in the iso-scalar state. IS. D. Drell, Proceedings of Annual International Conference on High Energy Physics, CERN, 1958. 2W. R. Frazer and J. R. Fulco, Phys. Rev. Letters 2, 365 (1959). 3 F. Cerulus, Nuovo cimento 14, 827 (1959). (Hsien Ting-Ch'ang, Ho Tso-Hsiu, and W. Zoellner, Preprint D-547, Joint Institute of Nuclear Studies. 5 G. F. Chew, preprint, 1960; N. Cabibbo and R. Gatto, Phys. Rev. Letters 4,313 (1960); L. M. Brown and F. Calogero, Phys. Rev. Letters 4, 315 (1960).

Translated by W. H. Furry 272

122 SOVIET PHYSICS JETP

VOLUME 14, NUMBER 1

JANUARY, 1962

ON THE ROLE OF THE SINGLE-MESON POLE DIAGRAM IN SCATTERING OF GAMMA QUANTA BY PROTONS L. 1. LAPIDUS and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor February 27, 1961 J. Exptl. Theoret. Phys. (U.S.S.R.) 41, 294-302 (July, 1961) It is shown that when the sign of the yN-scattering pole diagram connected with rro-meson decay is correctly chosen, the contribution of the pole to the cross section for the scattering of y quanta by protons decreases considerably. In order to obtain information on the lifetime of the rro meson, the precision of the experiments must be appreciably improved.

1. INTRODUCTION

AI

few years ago Low[IJ called attention to the presence of a pole diagram connected with the decay of the neutral pion, in the amplitude of elastic scattering of y quanta by protons. An account of this diagram, from the point of view of the double dispersion relations for yN scattering, is equivalent to an examination of the nearest singularity in Q2. Several interesting considerations in connection with the double dispersion relations for yN scattering are contained in the paper by N. F. Nelipa and L. V. Fil'kov (preprint).* Zhizhin[2] considered a contribution of this amplitude in different states. Recently, Hyman et al [3] and in greater detail Jacob and Mathews[4] noted that the addition of the one-meson pole amplitude greatly improves the agreement between the theoretical and experimental results in the y-quantum region from 100 to 250 Mev. This problem is considered in detail in a recently published paper by Bernadrini, Yamagata, et al. [5] It is known that an analysis based on dispersion relations [6, 7] leads to scattering cross section values greater than the experimental values in this energy region. In the present paper we wish to call attention to the sign of the pole amplitude, which is very important, since the interference terms play the principal role. From the results of Goldberger and Treiman[8] for the decay of the neutral pion, and from the dispersion relations for forward scattering, which we used previously, [7] it follows that the (relative) sign of the pole diagram differs from that used by Jacob and Mathews. Thus, the addition

of the pole diagram does not improve the agreement oetween the theoretical and experimental results, and the discrepancy calls for a different explanation.

2. SCATTERING AMPLITUDE We denote by p and p' the nucleon momentum vectors in the initial and final states, respectively. and by q and q' the same quantities for the y quanta. Since they satisfy the conservation law (1)

q +p = q' +p',

it is convenient to introduce the following four orthogonal vectors: K =Hq

-c- q'),

Q = "(q' - q).=+ (p -

P' = P - K (PK)IK',

pO).

N~ = ie."pP:K.Qp,

(2)

where P = (p + p' )/2. From these four vectors we can construct two independent scalars: Q',

(3)

Mv = -(PK).

The lengths of the vectors introduced in (2) are connected with Q2 and Mv by the relations P"

K' = - Q', P' = - Q' - M', P' - (PK)'IK' = Q-' IM'v' - Q' (Q' M")l, N' = - pO'K' Q' = Q' IM'v" - Q' (Q' M") 1. (4)

=

+

+

The S-matrix element for yN scattering can be represented in the form (p'q'ISlpq) = (p'q'lpq)

+ 2~

1\(4)

(p'

+ q' -

p -

q) (

';1\ 'I. '

(5)

PoPoqgqo

where N =

U(p') e~N ~"e"u (p)

= 2lt'i

*The authors are grateful to Nelipa and Fil'kov, and also to Dr. Yamagata (see below), for acquainting them with their results prior to publication.

x 210

(P:'~

f ~ d4Ze--iIKZ
(e·j(_+))lp).

o

(6)

123 211

ON THE ROLE OF THE SINGLE-MESON POLE DIAGRAM In the center-of-mass system (c.m.s.) the differ-

ential cross section is given by the relation

t!= do =

I.!!!. N I"

"" L.J

IV

.pin.

w (7)

'

where W2 = - (P + K)2 is the square of the total energy in the c.m.s. The scattering amplitude N can be written as a sum of six invariant functions (K = "YJ.lKJ.I): 'N p..ve." --

e"",

-

+

[T l+'K'T 1+ ' 2

(e'P') (eP')

p'"

i

(e'P') (eN)-(e'N){eP') (P"N'),/,

+

(e' P'){eN) (e'N) (eP') (P"N'),/,

(e'N)(eN)

NZ

.T.

[T.

The probability of decay of the neutral pion is ~ (2n)" (qq'

=

"(4) (q _ _ q _ q') ri'q ri'q' = _1(2,,), ..

~ I(es') +(e's)] " , F ",

X

+ iK'T,l

K.T

(8)

y".

In some cases it is also convenient to represent

Summing over e and e' and integrating over the angles we obtain in the pion rest system

(lk') (se')J

+ iRe [(lk') (s'e) -

(15)

't' = 64nfm~IFI·.

(16)

(ok) (se'»),

(9)*

where s = k x e, s' = k' x e'; e, k and e', k' are the polarization of photon-momentum unit vectors before and after scattering, respectively.

T.

is

Using the dispersion technique, Goldberger and Treiman have shown that gel

F (0) = -

R,(ee') +R,(s's) +iR. (0 [e'e) + iR, (0 [s's))

+iR. [(ok) (s'e) -

I',

w = (m!/64n) ' F

the amplitude as an operator in spin space in terms of six non-covariant functions ~:

~ e~N ",e,=

(14)

,,'

The pion lifetime

Y

,S, q.) ,"IVT

4n'm. (I +flop) 1

p = [2flo~- (flo! -

10 + pi, + (g'/4n)I,'

(17)

+ flop),

flo~)J/(1

(18)

where J.Ip and J.In are the anomalous magnetic moments of the proton and neutron, while 10 and II are positive integrals. It follows from (17) that F (0) g

< 0,

(19)

This sign is of importance for what is to follow. 3. MATRIX ELEMENT OF NEUTRAL-PION DECAY 4. SINGLE-MESON DIAGRAM FOR THE SCATThe S matrix for the decay'of the neutral pion ha's the form

TERING OF GAMMA QUANTA BY PROTONS

/ (2n)'6 14)(q. (z,.) , r 2",.

The S-matrix element of the pole diagram is

x (q'ql/

-q-q')

(0) 10),

where q and q' are the photon momenta; qrr is the 4-momentum of the pion; J (x) is the current of the pion field: / (x) = i

6'i6~x)

S· =

ig.~ (x)

Y'"(31\> (x)

(11)

Ilpq) = ig

X 6 1') (p'

+ q' -

x (2n)4

1 (p' -

pi'

(;n),ii(p,)

p 2

+ m.

Y.

u (P)

q) (20)

(q"/. (0) 1 q),

It can be shown that
q)"]. (21)

(2,,)' (4q.q'o) I,

[rp(x) is the meson-field operator, >p (xl is the nucleon-field operator, and go is the non-renormalized constant of the pion-nucleon interaction J • The Heisenberg equation for the meson field can be written in the form

(- O·

+ m;) '" (x)

= / (x),

Since the matrix element (q'IJrr(O)lq) is taken for the pole at (q'_q)2=_m~, Eq, (21) contains exactly the value of F encountered in the rr o decay. Substituting (21) in (20) and going to the c.m,s" we obtain

(12)

(p'q' 'S _ I

and in the notation of Goldberger and Treiman(8] ..K == (?n)' = -

JI4i7q'
0)

i£""Ae~ e, q,q~F [(q

-

+ q')'l,

lee1=exe'.

q)

igF, (lit)

~q' " (4q,qoPOP.)

(2n)46 1'\p' I,

+ q'

«

2~ [i 'k) (es') - (Jk') (e's»

(13)

where F (q2) is the form factor. The expression for the decay S matrix contains F ( - m~). ·(ee')~e.;,'

p -

II pq> =

- i «(ok') (es') - (ok) (e's»)],

(22)

Comparing (22) with (9) we obtain for the contribution of the pole diagram

124 212

L. I. LAPIDUS and CHOU KUANG-CHAO R,c>

=

+ m~/2q2) is replaced by

R,. ,,0 R,p = RIP = O.

Rsp = - R6P

= ,gF q' 8:tW (p __ p'r -+-

_.

fir;

I

from this we conclude that the contribution made to the amplitude by the pole diagram due to the exchange and decay of the pseudo-scalar neutral meson reduces to the combination

It is important to note that by virtue of (19)

Roe - R6P <0.

(25)

if it is assumed that F (0) and F (- m;) do not differ greatly. In the cxpression for the cross section [formula (16) in [5) J the pole term enters in the combination +IRo-R,I'(I--cosl)' -

Re (R, -R.)* (Rs - R,) (1 - cos 0)'.

(26)

The contribution of one pole diagram has the form

It (0) =

=

+IRo -

R.I' (1 - cos 0)3

" (q)'g'(I)' m;t"t

-W

/'it

tfln

(I-cosO)'

(1

+ m~ I ~qt-cos O):.l

y;-' - ~ In I (Yo

(23)

(27)

which agrees with the result of Jacob and Mathews. We can expect the cross section of scattering by 90° to be reduced by addition of the pole term only when the second term in (26) is negative. Since R, is large and negative, owing to the large anomalous magnetic moment of the proton, Re (R 3 - R,) is a positive quantity in the region of energy under consideration. Thus, the second term in (26) is positive if R 5p - R 6p < O. Consequently, assuming the analysis of Goldberger and Treiman to be correct, the pole diagram does not decrease the theoretical value of the cross section, but increases it. If we use the results of our own analysis,[7J we find that Re (R 5 - R 6) is determined not only by the limit theorem, but also by the amplitudes of photoproduction of E2 and M 3• Since in this case the "isotropic" part of the contribution of the pole amplitude is automatically taken into account, it is necessary to add to the previously-obtained amplitude not all of expression (24), but only the contribution of (24) to the higher states, i.e., the difference

As a result of this procedure, which is necessary in order not to violate the unitarity of the S matrix (when e = 90°), the quantity yil l (where Yo = 1

+ I) I (y" -

I) :.

which leads to replacement of % by - 0.14 when q2 = m; (Yo = %). Thus, the contribution of the amplitude is decreased by a factor of 5, and the sign of the contribution changes. By virtue of this, a much higher accuracy is necessary before the connection between the amplitude of the neutral-pion decay and the amplitude of the scattering of y quanta by protons can manifest itself. It was recently shown that the lifetime of the neutral pion is (2.0 ± 0.4) x 10- 16 sec, [9) which also decreases the contribution of the pole diagram. The indeterminacies in the analysis of the photoproduction cannot influence the conclusion regarding the sign of the interference term in (24), since this sign is determined by the well known theorem for low energies. The scattering amplitude at low frequencies, first obtained by Low(IO) and Gell-Mann and Goldberger, [II) is reviewed in the appendix, where it is obtained as the contribution of the single-nucleon terms (see (6). We note, in particular, that TO - ~ (1 - A) s - NI --t

'-I',

(Y

--

(28)

iI'---,'

Let us give another, less rigorous but more illustrative proof of the correctness of the determination of the sign of the pole diagram. * The matrix element (q'l J,,( 0)/ q) can be represented in the form

c_~ iE,,,",<,e,, '10 q~ \-~\i,F 1('1 -

/q': J" (0) I 'I)

,(n (29)

so that
x iii

I!

"'I; ~ i (2n)-' gt/') (p'

+ 'I'

-

P - q)

(p') Yo" (p)

2 X I'Q _,Q_ '_Fr-"'"'_ 1

""P"

(30)

(,-N) - (,"N) ('P') (I'" Iv')'i,

hence T

5P

Q' - gF- - --

-:rt

~Q2

+ rll;

•

(31)

We now introduce the function

f (v, Q')

=

To (v. Q')/Q'.

(32)

If we regard f ( ", Q2) as an analytic function of Q2 at fixed ", we obtain from Cauchy's theorem and from (31) *An analogous approach was used earlier ll to obtain the Goldberger-Treiman relations.

125 ON THE ROLE OF THE SINGLE-MESON POLE DIAGRAM

- gF f (V, Q.) "

I 4Q'+

m~

(33)

+JQ,

where JQ is the dispersion interval, the lower limit of which is 4m~. In the region Q2 « 4m~, the integral in (33) is small and we can approximate f (v, Q2) by the expression

213

where the difference in the common phase factor is taken into account, and from which it is clear that the sign used in [4] for the pole term differs from that proved in the present paper.

APPENDIX f(v, Q,):::::;E _ _ I_. n 4Q2+ m!

(34)

On the other hand, f (v, Q2) is also an analytic function of v for fixed Q2. By Cauchy's theorem with account of (28) we have

SINGLE-NUCLEON TERMS IN THE DISPERSION RELATIONS

Recognizing that

T(e,.j(~))(e.j(-+))

+ (e'

(35)

where J v is a second dispersion integral. In the region 2v ~ m 1P the pole term will predominate and . n1. e' (\ + ~) t f(v, 'l):::::;--M-Q'/M'_V"

a (Zo) [e.j'(+),

=

e. j (-+)]

(A.l)

j(+))(e.j(--i)) ,

we determine the retarded and advanced amplitudes: N'·· = ± 2n'i (Pop;;M')'/'

x ~ d'ze±i(K<)

(36)

j( +), e· j( -

-i;]

Ip) . (A.2)

The vertex part of the current has the form

M 2v 2 '"

It is obvious that (34) does not hold near Q2, and (36) does not take place when 4Q2 = - m~. It is (p'lej(0)ip)=(2~)3 u(P')[e+ i4~(e(p'-p) still possible, however, that expressions (34) and - (p' -p)e)]u(p) = (2i)~ u (P')[(l + ).)e+~ (eP)] u(P), (36) are valid simultaneously near certain values of v and Q2. Equating these expressions for 2v (A.3) where € is the charge of the nucleon. = m 1T and Q '" 0, we obtain The pole term has in the region of positive freF = - 4ne' (I + },,)/gM, (37) quencies the form which is very close to the formula of Goldberger A' = - G:~L (k - p+ p.,) (I" .e .i (O)! 1'., , .L;6"1 and Treiman, obtained by an entirely different ., method. • <1'" I e'j (0) I p) = ~ d'p" 6(1} (k - p +-",) Actually, from (17) we obtain for (g2/41T2) I,

f

»1

which coincides with (37), apart for a numerical factor. The literature reports two different choices of the common phase for the yN scattering amplitude, one with a Thomson limit +e 2/M, the other 2 with - e /M. The error in the published papers lies in the fact that the choice of the common factor in the one-meson amplitude does not correspond to the choice of the sign of the remaining amplitude. A direct comparison of the amplitude used by Jacob and Mathews[4J with (9) shows that the functions fi introduced in [4J are related with Ri by the equations

t,

=

- f, = R, + R2 cos 0, f. R3 + R. cos 0 + (R. + R.) (1 + cos 0)

=

R.,

=

R..

,. = R.+R.,

X[(l

f. =R.,

+;..)~. '~e'pJu(p).

(A.4)

U sing the relations

I

u (P -

K)

u(P -

(21'''0)-1 d'p"

=

(A.5)

K)

d'p"O (p",,) b (p~ + M,),

(A.6)

we obtain

+ M') u(p') [ (I + A) e+~. (ep') J x[- i (p - k)+M1l(1 + A) e" +~ (e'p) j u (P).

A' =

~

b (I';

(A.7)

We can express AD in terms of the fundamental invariants: AO _ (e'P') (eP') A O + (e'N)(,N) A' -

P'/.

1

, (e'P') (eN) -

N"!.

(e'N) (eP') AO

(p"N'),/,

T

"-

+ (e'P') (eN) + (e'N) (eP')

3

A~.

(A.8)

(p"N'),/,

Comparing (A.7) and (A.B), we obtain

A1 P"

- (R. - R.) (I - cos 6).

f.

+A)e+~ep'lu(P-K)u(P-KI

xu(P')[(1

F = _ 4lt e' (I +~) I. + ph g h

= ;;.. 6

(P~ + M') u(1") [(1

x [- i(P-

+ A) P' + i

kJ + MI [(I +A)?' + i

(P'

p')]

(P'p)]u(p). (A.9)

126 214

L. 1. LAPIDUS and CHOU KUANG-CHAO

It is easy to verify that

Ii (p') Ii>' (- i =

A~

Q) = (P'P) = P".

(P'p') = (P', P -

(A.I0)

=

-

o

f'(~~A) 6(v-~)

e' (I

+),) •

~u

A. =

(v- Q 1M) u(p) YsK.u (p).

+2P" M

KP"}

Ii (P')i>' 1-i(P-KH Ml u(p) = U (p) - i (P2 + (PK.)) + i (,(Pi<:)

=

Ii (p')

K) + M)

II

(p).(A.ll)

1M l,iM -

+ (~'ilKP) J11 (p),

P'I u

(PK~)

(ilW -

[M

o

K) -

K) + MIl/

(;~l K) (A.12)

(p)

i (P'

+ (PK.»

( " (PKI' ')] u(p), + i,K.P+ i0PK.

u(p') [ - i (P -

(A. 13)

= ii (p')

(p)

W<' + 2,WI u (p).

(A.14) Using (A.I0) - (A.14) and noting that at the pole we have

KJ'

(P -

+ K.' -

= P'

= 21\' - 2 (P K) -

2(PK.) M'

= - M'

or that (A.15)

K.' = (PKJ,

we obtain

A1 =

+- ,'<'I (21(' -

2 (PK.)) U (p) (2M

+ if<:) u (p) (A.16)

(Q2)_ , =~MI\\v -M u(p')(2M+iKJ u(P). e2

Analogously,

A~

8~1 <'I-(v - ~;) Ii (p')

= -

iK u (p) (I

+ A)',

(A.17)

--'~' "(1+A),(. Q'\-( 'l[P"N'( 'M'-: K'l]U(P), U '-M,iu P M -I (A.18)

e'l1

+- A)

(

Q') -

I P'"

=~I\'''-~l lI(p')I'M(K -

']

iM)N u(p).

(A.19) From (A.18) and (A.19) we find A:(p"N')'I. = A·(P"N')" ,

"(~~ ),) =

A.21

Finally from (A.16). (A.17), and (A.21) we obtain

+2i(PK~\

e2

Q2

T, = "nM Q'/M'-v"

(i>-

()

f<:) +M) P'] u (p)

(P -

u(p') 1- iJ(P"

Ii (p') 1(- i

iu (P') y, u (p),

, - , '

<'1(" -~)(- iP") u(p') N U (P),

"Il-±,) HM

<'I ( v

•

To = 0,

•

0"

0

. , ' (I -'- A)'

T. = (I

+ A)

4nM Q'

T5 = MT,= ~Q'/M'-'"

"

"'/M'- "" I ) \4n = 137'

( "

(

A.22

1 F. E. Low, Proc. 1958 Ann. Intern. ConI. on High Energy Physics at CERN, p. 98. 2 E. D. Zhizhin, JETP 37, 994 (1959), Soviet Phys. JETP 10, 707 (1960). 3 Hyman, Ely, Frish, and Wahlig, Phys. Rev. Lett. 3, 93 (1959). 4 M. Jacob and J. Mathews, Phys. Rev. 117, 854 (1960). 5 Bernadrini, Hanson, Odian, Yamagata, Auerbach, and Filosofo, Nuovo cimento 18, 1203 (1960). sT. Akiba and J. Sato, Progr. Theor. Phys. 19, 93 (1958). 7 L. r. Lapidus and Chou Kuang-chao, JETP 37, 1714 (1959) and 38,201 (1960); Sovie't Phys. JETP 10, 1213 (1960) and 11, 147 (1960). B M. L. Goldberger and S. B. Treiman, Nuovo cimento 9, 451 (1958). 9 Glasser, Seeman, and Stiller, Bull. Am. Phys. Soc. 6, 1 (1961). 10 F. E. Low, Phys. Rev. 96, 1428 (1954). 11 M. Gell-Mann and M. L. Goldberger, Phys. Rev. 96, 1433 (1954). 12 Chou Kuang-chao, JETP 39, 703 (1960), Soviet Phys, JETP 12, 492 (1961). Bernstein, Fubini, Gell-Mann, and Thirring, Nuovo cimento 17, 757 (1960).

(A.20)

u(p') NKu (p) = (P"N,)'I. iu (p') y,u (p). iu (p') Nu (p) = I('u (p') y,K.u (p). if we now take into account the fact that (P'z NZ ) l/Z = p'2Q2 by virtue of (4), we obtain from (A.20)

V

which coincides with the previously-obtained results and has the correct signs. In all the calculations of the single-nucleon terms it is assumed that parity is conserved in the electromagnetic interactions. The results obtained remain valid also in the presence of CP invariance.

-g'\p', -( ')NK.' () MJ M u p u p .

It can be shown that

e'

T, = 4nM ,,'/M'-v' '

Translated by J. G. Adashko 58

)

127 SOVIET PHYSICS JETP

VOLUME 14, NUMBER 2

FEBRUARY, 1962

LOW-ENERGY LIMIT OF THE )'N-SCATTERING AMPLITUDE AND CROSSING SYMMETRY L. 1. LAPIDUS and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor February 27, 1961 J. Exptl. Theoret. Phys. (U.S.S.R.) 41, 491-494 (August, 1961) The low energy limit for the yN scattering amplitude is derived with the aid of single-nucleon invariant amplitudes. Subsequent terms in II for Q2 = 0 and the expression for the limiting value of the first derivative in Q2 as Q2 - 0 can be obtained by taking into account the conditions of crossing symmetry.

1.

Low, Gell-Mann, and Goldberger showed [I] that the condition of relativistic and gauge invariance makes it possible to express the limiting value of the amplitudes for the scattering of low energy y quanta on spin- I/2 particles and the limiting value of the derivative of the amplitude with respect to the frequency as 11- 0 in terms of the charge and magnetic moment of the particle. This result was later generalized[2J to the case of elastic scattering of y quanta by particles with other spins and also to the case of bremsstrahlung. [3J The res ult for elastic scattering also holds when only CP invariance is assumed. Consideration of the singlenucleon terms in the dispersion relations for yN scattering [4-8J also leads to the limit theorem. (A similar result holds for bremsstrahlung. C7J ) In the present note, we derive the limit theorem for yN scattering on the basis of the single-nucleon terms. The requirement of crossing symmetry for the invariant functions Ti(v, Q2) (= 1, . . . , 6) makes it pOSSible to obtain additional terms for the limiting values of the functions Ri(v, 0), which characterize the yN scattering matrix in the center-of-mass system, and also the limiting values of the derivatives of the amplitudes with respect to Q2 as v - O. (For the definition of the quantities Ti and Ri see, e.g.,[6J.) 2. The invariant functions Ti(V, Q2) are related to the scalar functions Ri (v, Q2) (i = 1. ... , 6) in the following way:

+

lW

I

16W'Q' M)(W'

T, +T. =(:,~~,),rli ~:(W~'M),J(R3--R.) 16W'Q' (W t- M)' (W' __ W)' Mv

+ Q'

HW'Q' (W' __ ,11')'

• _

-w-,- 1 , -

TO __

T, -

T.

~ (W'- M')'

+ iW I.W + M)' r1·-

(W

4Q'W' ] (W' _ M')'

Q'

+ M')

(R.

]

(Ra

__

Rs)

_ , __

,

(R.

R."

Tg

=

0,

MTo _ c' (I t- A) _ _Q_'__

I) -

6 -

(2)

--M--Q'/M2- v;&'

where we have used the system of units in which 1\ = c = 1 and the magnetic moment is J1 = e (1 + A )/2M.

For Q2

=

0, it follows from (1) that

=

2WZ Mv'

(Ra

'JW 2

,~W

(R,

T~

(T. +T,). = ~v (Ra -

III' hf

(R. - R,)'

where W is the total c,m,s, energy and II and Q2 are two invariants characterizing the kinematics of the process; W2 _ M2 = 2Mv + 2Q2. The pole terms for Ti(v, Q2) have the form [6J

(T. - T.).

RMW'

(RI-R2),

M')'

+ R.)

(T.). = -

+ R2),

(Ts). 352

=

~w

+ R.). + (w + M)' (Rl + R,)., R.)., (T,

+ T.).

=

')W 2

MVi (Ra -

W'

Mv (Ra - R.).,

w Mv 12 (R, + R.) + Rs + R.1.

R.).,

128 LOW-ENERGY LIMIT OF THE Differentiating the relations in (1) with respect to Q2, we obtain, in the limit Q2 = 0, '2W' = Mv (Rs

(T. - Ts). -

+

+ R.).' -4 WW + M (R. + R.).'

+

2 (2W' M'W M') M'v' (W M)

+

W' + W (W 4+ M) [ M'v' -

'4W (W

+ M)' (Rl

4 [W' M'J (R, + M'v' (W+M}.-M--V (Tl

(W

+4 M)' [W' M'v' +

2 (2W'

+ (T,

+R.).

R.).

(R. - R.).

-(W+M)'~

, = Mv W' [2W' M'v' (R. - R.). - (R, -

+ W~ (R. ,

W [ (T.). = Mv -

+ (2R. -

(2R,

(R, +R.).

=

± e'

(I +).) v/2W.

(7)

=

-2~' [I +(1 +A)'[v +a..v'

+ ...

(8)

and take into account the fact that W = (M + 2Mv)l/'l '" M(l + viM) for smaIl v, then from the condition that there is no linear dependence on v we obtain the relation

+ Ra).

R.r

ct3M -

+ 2R. + R, + R.) ] .

ctl =

follows from (1) that (R I + R 2)o and (R3 ± R,)o/v are finite when v - O. Since the functions (T 2 'f T,)o have a singularity of the form

a,M

=

(Rl +R2).

± (I + A)'[

•

(9)

(e'/M)

If + (I + ).}'].

(10)

From (8)-(10) we have

it then follows from (1) that 2M'

+ A)']

It follows from the requirement of crossing symmetry that the quantity v(T 2 - T 4) should be an even function of v. The absence of a linear dependence of the terms on v leads to the relation

l-~'f~(l + A)'H· (R.± R.). = _ ~ [I

(e'IM) [~+ (I

(4)

It is seen from (2) that the terms TI - T3 and T2 + T, do not contain poles for Q2 = O. It then

V

+ A) 12M'.

2

+ 2R. + R. +

M+v W'v

,

R.)o

R.).]. 2W' M'v' (R.

= - e' (I

const as

-

' , + T.).' 2= W Mv' (Rs - R.). + 4(R,-R.). [~-M-~] M'v' (W + M)' v

(T,).

M(T.)~.

=

-

and (2Rs + 2Ra + R3 + ~)o const as v - O. We see that formulas (5)-(7) obtained from consideration of the pole terms (2) contain the results of the limit theorem for Q2 = O. 3. It is of interest to note that with the aid of the conditions of crossing symmetry one can obtain additional information on the low energy limit. It follows from crossing symmetry that, for example, the quantity TI - T3 should be an even function of v. If in the first relation of (3) we make the substitution e' +alv + .... (Rl + R2). = -;\1

+ M) M'v' (Rl- R.) •.

4W'

(6)

-e'IM.

const and v(R I ± R 2)6

(R. ± Ra).

4W'

(W

=

we conclude that

,

+

-

(T.)~.

R.).

WhI' M'} M'v' (W + M) (Rs -

-

+ R,).

+ M +2W] (Rl + R.) ••

1 W

' + T,).' 2= W Mv (R. -

and that v(T 2 - T 4)6 does not contain a pole, it follows that

and (R3 ± R4)6 v- O. Since

'

353

T.)~. = -2e'IMv'.

(R.±R.).

+ R.). M] W + M (Rl + R.) •.

'2W'

±

(TI

(R,

(T. - T.). = Mv' (R, + R.). +

-

'Y N -SCATTERING AMPLITUDE

(5)

as v - O. which is in accordance with the limit theorem. Since T5 and Te do not contain poles when Q2 = 0 .. the quantity (Rs + Ra)/v should remain constant as v - O. Similarly. from the condition that (T I ± T310 contains a pOle of the second order

(R3

= -

~ (I-~)

+0 (v').

+ R.). = -2~,(I-~)v -2~,(1

+ A}'(I-~)V + 0 (v'}.

(11)

The functions TI + T 3, T 5, (T 2 + T 4 ). and Te should be ev-en functions of v. Similar considerations lead to e' (I - (I (R. - R.). = -"'M' ~

2V) v + 0 + A)'[ ( I - M

(v,).

(12)

129 354

L. I. LAPIDUS and CHOU KUANG-CHAO _

and

(R, - R.). - -

(2 (R, + R.) + R. + R.I. e' '2'1 -_ -2M'" \

(R, +R.).

(13)

+

3 -

2 (I

The function (T I

-

+ I.)'] v' + 0

(v')

(13')

T 3 )o is an even function of

v. Inserting in (4) (R.

+ R.)~

=

a~

+- ....

we obtain from the condition that there is no term proportional to 1/ v 2Ma;, -

2a',

c.

(e'/A-1) II -- 2 (I

fA)'].

A similar condition for the even function v(T 2 - T 4 )o leads to 2MC(~

Then

0';

~

-

c_

(e'IM) ]3

-

,

(R. + R.).,

+ 2 (I + I.)'].

2e 2/M2, and therefore

(R. + R.). =

f,2

=

+ A)

~v

e' [ + 4',\l''' -

--

1

2 M,,, +0 (I).

" ]3 + 2 (I + A)'] -i'M'

+- 0

(v).

(14)

The condition that the poles of the first order in the even functions (T! + T 3 )o and v (T 2 + T4 ) vanish leads to

=

2~'

]-

,

T

8(

I

+"')

(2R. + 2R. + R. + R.). = - i~ (2/.' - 21. - I), (16) It should be kept in mind that the expression for the limiting energy is valid for amplitudes in the center-of-mass system. The result obtained can be useful for analysis of the scattering of 'Y quanta by nucleons with the aid of the dispersion relation technique.

I F. Low, Phys. Rev. 96, 1428 (1954); M. GellMann and M. L. Goldberger, Phys. Rev. 96, 1433 (1954). 2 L. I. Lapidus and Chou Kuang-Chao, JETP 39. 1286 (1960). Soviet Phys. JETP 12. 898 (1961). 3 F. E. Low, Phys. Rev. 110. 974 (1958). 4 N. N. Bogolyubov and D. V. Shirkov. Doklady Akad Nauk SSSR 113. 529 (1957). 5 T. Akiba and I. Sato. Progr. Theoret. Phys. (Kyoto) 19. 93 (1958). a L. I. Lapidus and Chou Kuang-Chao. JETP 41. 294 (1961). Soviet Phys. JETP 14. 210 (1962). 7 S. M. Bllen'kii and R. M. Ryndin. JETP 40. 819 (1961). Soviet Phys'. 13. 575 (1961).

(R.-R,). =-~(I-~)+O(v')' (R. - R.)~

2

+ (I + I.)'J v' + 0 (v').

V) v + 0('v). -M

"(I A) =~v

+ 4M' ~ [I.' -

e' (I

3 + 2 (I + A)'J + 0 (v).

Similar conditions for the functions (T 51O and (TaM require that

(15) Translated by E. Marquit 90

130 SOVIET PHYSICS JETP

ON THE

K+

N-

VOLUME 14, NUMBER 4

APRIL, 1962

A(L) + 'Y PROCESS

L. 1. LAPIDUS and CHOU KUANG-CHAO

Joint Institute for Nuclear Research Submitted to JETP editor May 18, 1961 J. ExptI. Theoret. Phys. (U.S.S.R.) 41, 1310-1314 (October, 1961) Some information on A (1:) + 71" - A (1:) + y processes can be obtained by investigating the A(1:) + y reaction. A detailed phenomenological analysis of these processes in the s state is performed. The Kroll-Ruderman theorem for photoproduction of pions on hyperons near threshold is considered.

K + N-

1.

One of the most important problems in elemen- where p is the density matrix of the phase volume tary-particle physics is the study of interactions for the intermediate states with fixed total energy. between unstable particles, where for lack of an For two-particle (binary) reactions with a defiunstable-particle target it becomes necessary to nite angular momentum, the matrix p is diagonal. use indirect methods for this purpose. In the relativistic normalization of the wave funcWe have shown earlier [IJ that by using the unitions, the diagonal elements of the p matrix are tarity condition for the S matrix we can establish (3) certain relations between the matrix elements for where k is the relative momentum of the particle the processes K + N - K + N, K + N - A(1:) + 71" and A (~) + 71" :-- A (~) + 7l" for states with arbitrary in the c.m.s., Mn is the mass of the baryons in the values of the angular momentum. It is therefore nec- intermediate states, and E is the total energy of essaryto obtain ctlrtain information on the processes the system: A(l:) + 71" - A(l:) + 71" by analyzing the cross secE = (k' + M~)'/' + W + m2)'/'. (4) tions and polarizations of the baryons in elastic IT we introduce the notation scattering and in reactions involving K mesons (5) and nucleons. Similar conclusions were reached later by other authors. [2,3J In [2J and [3J there then Eq. (2) can be rewritten as is a detailed analysis of elastic scattering and inK' .~ T' - iK'T'· T' - iT' K'. (6) teraction of K mesons with nucleons in the s state. Existing experimental data allow us to esFrom (6) we obtain tablish the phase difference of the s waves in 7I"l: T' ~ (1 - iK')-' 1(' • K' (1 - iK'rl. (7) scattering with isospin I = 1 and I = O. In order to obtain certain information on the The cross section of reaction (1) expressed in electromagnetic and strong interactions of hyperterms of the T' matrix, in a state with definite ons, we consider in the present article the procangular momentum J and with definite parity, is esses (8) a (i ~ j) 4ttki2 (J + 1/2) I <j I T' I i) 12. (1)

K + N ->.\(1:) + r.

0

The S-matrix unitarity conditions cause the matrices for the processes 1T + A(l:) - A(l:) + y to be related with the matrix elements of processes (1). 2. For simplicity we consider the reactions (1) in the s state only. We use the K-matrix method developed in [3J. For our problem it is convenient to use a symmetrical and Hermitian K matrix, expressed in terms of a T matrix with the aid of the relation K

=

T-ittKpT

=

T-ittTpK,

Let us consider the submatrices of the introduced K and T matrices, which we denote by a .•~ (RN IKeRN), ~ ~+

= =

r = £= £+ = '1 = '1+ = 1,; =

(2)

932

Txx .~ (RN ITIKN), T·J(Y =. TJ(y'=
(9)

131 ON THE K + N We denote the submatrices of the K' and T' matrices by the corresponding primed letters. We neglect the matrix t. which is at least one order of magnitude smaller than the other matrices. If we introduce

A(E) + y

a

(11)

=

From (5). (7). (10). and (11) we readily find that T~K = (I -

iX')-' X',

T~y = (I -

iX)-'

= (I -

T'n

= (1 -

W(I

T~I( = (I -

=

C

(1 -

(16)

1 - inpK (aO -;- ibO).

From (14)-(17) we readily find that T~TK = "P~APk' [£~K + i1lhn'/'Pt'(b')'I'eiJ.E] 1'..', T~TK = "P~i:pk' [;h -i- i1]'h,,'I'p~' (b')'I'e",- ] 1',.1

s' + i (I -

iX)-'~'

iX')"

+ i1]'T (I

(I -

s' + (I -

iZ')" 1]',

iy')-' ~'T (I -

-

in') -, ~' ( 1- iZ')"

X' = a'

-+- W (1

Z' = y'

+ i~'T (1

+ 1]'T (I -

iX')-',

iZ')",

_ iy')" ~'T, -

in')"

W.

d:.

r&.

-

i"pI:Y) = a

+ ib,

We note that TAyK, TEyK, and TiK have almost the same energy dependence in the low-energy region. where (assuming the relative parity of the hyperons to be positive) the energy dependence of PE, PA, PyA, and PyE can be neglected. Let us proceed to examine the channels with isospin I = 1. In this case y and fJ are matrices, y.= (

(13)

3, In our discussion it is sufficient to take into account the electromagnetic interaction in firstorder perturbation theory. considering separately the contributions from the iso-scalar and isovector parts of the electromagnetic interaction. We start from the iso-scalar current. In this case the total isospin is I = 0 for the A + y system and I = 1 for the E + Y system, We denote by ~~. 1)~. and TJt the matrix el~ments with iso-scalar current for the processes K + N - A(E) + y and A(E) + 71' - A(E) + y. respectively. In the case of the iso-vector current. the total isospin is I = 1 for the A + y system and I = 0 or 1 for the E + 'Y system, The corresponding matrix elements will be denoted by ~~. ~±. ~~1, 1)~, and ~1, Let us consider the channels with isospin 1= O. In this case the submatrices a. fJ. and yare simply numbers, Expressions (13) are then reduced to

+ i"~'pI:/(l

(18)

iy'r'll',

(12)

X = n

"'~',

O )

Formulas (16) for the processes K + N - K + N and K + N -E + 71' were obtained by many authors. [3: Let us write

iX')-',

where

where

+ ib

in')-'

T;.y = i (1- iZ;)-' ~'T (I - in')" T~I(

npl( (aO

T~K = n'/'?,?: (bO)'I,/',;t,;',

iZ')" Z', T~y

T: y = is'T (I -

=

iX')-' X'

(I -- iZ')",

iy')" ~'T (I -

iX')-'

(15)

iy')-'

-

iZ')" ~'T (I -

~ (\ -

> O.

Substituting (14) in (12) we get

1',0 =

. (Koo~ 00) .

K

T;I( = (I -

+

where

then we can write

P'

+

:t 2ph21,

n - :t2~'rp1/[ 1 :t 2ph 2]

=

b = ,,~2pI:![ 1

(10)

= (I - in')-'

933

PROCESS

rA'\

lEA)

r,\~

hI:

~

,

=

(19)

(~AK' ~"K)'

It is easy to verify that in this case X is simply a complex number

X

=

a'

-+-

(20)

ib"

where a

,

= n -

R 'I

nppy' 1

+1r"

_, '/.p,T I

b'

PY p ,

I'

= ,,~py' 1

I

+ 'r"

1/ T

Py'~

(21)

From (12). (13), and (19)-'(21) it follows that IKK

="P K (a'

+ ib') t.~', T~K = n':'p~' (b~K)'I'e'>'AKt.~', ,,'Vi< (bh)'I'e"I:Ki\~', (22)

T~I( =

where n'V/li/:'l(e"AK,= (A I (I n'I'p'flitK"'r:.K == (E I (1 1'" = 1 -

inpK (a '

iy')_'~'T IK), iy,)-,~'TI K),

+ ib'),

(23)

and the quantities bAK and bEK are related with b by the equation bAK'+ bEK = b. If we represent the matrices ~ and 1) in the form

(14)

£ = (~AI(, £r:.I() ,

1]

=

'lAA ( 'lAE

'1~A),

'1';1;

(24)

132 934

L. 1. LAPIDUS and CHOU KUANG-CHAO

then the matrix elements TYAK and Tyl:K become

T~h.K = "p~}.p'kt..~' [SAK + i'lAh. ,,'V~b~'Ke"AK (25)

(26) To simplify matters we introduce new symbols

+ i'l~'J:,,"'p~'(b'('ei1::J, + iT)~En','Jp~ (bO)'/:en.!:), =tl,'~p~'~ [~~K + iT]~A:tl::p~ (b~J()'/~/AhK

a~, = rr.'/'p~~ [~~K cx~ = :tl/lp~'E [~~}( :J.~ =

photoproduction of mesons and hyperons. In the present paper we confine ourselves to a generalization of the Kroll-Ruderman theorem for photoproduction of pions near threshold. [,] Let us assume that the A and l: hyperons have a positive relative parity and that the K meson is pseudoscalar. If the created particles have low energies account of the electric dipole radiation is suffiCient. The generalized Kroll-Ruderman theorem states that, accurate to m7l' /M "" 15%, the matrix for the electric dipole transition is determined completely by the pion-hyperon coupling constant. Let us write the Hamiltonian of the pion-hyperon interaction in the form

-T i'l~~ ,,';'P~' (bh )'/'e"'EK J.

xb =

:tl/~p~~ (E~l(

+ ifJ~h. :t1."p~1 (b~J()"I/),/\.I\

J{ =

+ i'li,,!t'."p~· (b~K)'/·eo"KJ. a i = !t'/'P~i:!~iK + i'l'~f>.n"·p~(b~K)'VA/I.K

'l~,,-

(27)

with which the cross sections of the processes (1) can be written in the following form:

1(-

+p

-

AO

+. I

1<.0 + n _ AO +. I K- + p --> 2:0 + • \ 1<.0 + n --> 2:0 + y J K- + n --> 2:- + Y)

K,0 + p --> 2:+ + Y

Cross section: 2.-rm K / EKk

z:ttnx

Exk

C1~

I'

~±"K; .

1- ~~

Ex k I Z1ttnx

a~

I

V3 +:.l:

~o

-

I'

<'>11 '

Ia\: ±a<'>1ill"2/'.

Thus. the experimental investigation of the processes K + N - A(l:) + Y in Kp and Kd collisions can yield certain information on the matrix elements OIA and Oil:' Naturally, this information is not sufficient to reconstitute the matrix elements and TJ, which describe the photoproduction of mesons on hyperons. Nonetheless they may prove useful for a study of the interaction between hyperons and mesons or photons. 4. A powerful method for the analysis of strong interactions is the method of dispersion relations (d.r.), the use of which yields in many cases interesting results in the low-energy region. It can be assumed that the d.r. method is applicable to the

s

+ ig"" ([4" .,<J>,,] <J>,) + Herm.

conj. (28)

Following Low's method [5] we can obtain

+ i'li'f.""·p~' (b~X)'/'e'AJ:KJ,

Process:

ig~A 4" ',<J>,-\. <J>"

m,1 M,

'liA = 'l~"l=

11i" = a"'f",,!1

11\:,\-

m"1 M.

11h -m.1 M,

V2' a"'t",\ II + 0 (me 1M)]. +O(m,/M)J. 11h""m e IM.

Here m7l' is the pion mass, M is the hyperon mass, 01 = (2/471' = %31' and f2 = g2/871'M. lL.1. Lapidus and Chou Kuang-chao, JETP 37, 283 (1959), Soviet Phys. JETP 10, 199 (1960). 2 Jackson, Ravenhall, and Wyld, Nuovo cimento 9, 834 (1958). R. H. DaIitz and S. F. Tuan" Ann. of Physics 8, 100 (1959). M. Ross and G. Snow, Phys. Rev. 115, 1773 (1959). 3R. H. DaIitz and S. F. Tuan, Ann. of Physics 10, 307 (1960). P. T. Mathews and A. Salam, Nuovo cimento 13, 382 (1959). J. D. Jackson and H. Wyld, Nuovo cimento 13, 84 (Hi59). 'K. M. Kroll and M. A. Ruderman, Phys. Rev. 93, 233 (1954). 5 F. E. Low, Phys. Rev. 97, 1392 (1955).

Translated by J. G. Adashko 224

133 SOVIET PHYSICS JETP

VOLUME 14. NUMBER 5

MA Y. 1962

SCATTERING OF PHOTONS BY NUCLEONS L. I. LAPIDUS and CHOU KUANG-CHAO Joint Institute for Nuclear Research Submitted to JETP editor May 18. 1961 J. Exptl. Theoret. Phys. (U.S.S.R.) 41. 1546-1555 (November. 1961) An analysis of elastic scattering of photons with energies up to 300 Mev by protons is carried out by making use of the dispersion relations method. Six dispersion relations are utilized to estimate the real parts of the amplitudes at Q2 '" O. Photoproduction of pions is taken into account in a larger energy region than was done previously. Five subtraction constants are determined from the long wavelength limit and expressed in terms of the nucleon charge and magnetic moment. Differential cross sections and polarizations of the recoil nucleons are estimated. Photon-nucleon scattering at high energies is discussed.

1. Followin~

the work of Gell- Mann. Goldberger. and Thirring'-I] dispersion relations for photonnucleon scattering. whose validity in the e 2-approximation has been rigorously proved by Logunov.[U have been applied to the analysis of experimental data by a number of workers.[3-7] Cini and Stroffolini[3] were the first to calculate forward scattering cross sections for photons with energies up to 210 Mev. Certain qualitative peculiarities in the energy dependence of the forward scattering cross section were indicated earlier inCtJ • and also in[S] . Capps[4] has considered yN scattering through an arbitrary angle by taking into account a minimal number of states. In so doing he made use of some unpublished results of Gell-Mann and J. Mathews. Akiba and Sato[6J considered scattering through nonzero angles. In· order to evaluate the subtraction constants in some of the dispersion relations they made use of perturbation theory. The authors have previously[6] considered in detail dispersion relations for all six invariant functions. that characterize the yN-scattering amplitude. and have carried out a dispersion analysis in the energy region up to 200 Mev in an approximation in which certain recoil effects were ignored. It was shown that if photo production of pions in S states is taken into account significant modifications are introduced in the near threshold region. These changes are such as to improve the aggreement between the dispersion analysis and experiment. Near threshold the energy dependence of the amplitudes and cross sections becomes nonmonotonic. Aside from certain differences. connected with what assumptions were made regarding the num-

ber of subtraction constants in the dispersion relations and the maximum angular momentum of the states taken into account. all the published papers turned out to have in common the inability to obtain good agreement with experimental data in the energy region near 160-200 Mev. In a number of papers[9.7.l0] an attempt was made to eliminate this discrepancy by taking into account the contribution from Low's diagram.[ll] However a direct measurement of the lifetime of the '/1"0 meson[lU together with an analysis of the question of tbe sign of the pole amplitude[lS] have led to the conclusion. that the inclusion of Low's amplitude cannot substantially affect the results of the analysis. In connection with these discrepancies between the analysis and the existing experimental data we carry out In this work an analysis of yN scattering based on dispersion relation. In which we take into account in addition to photoproduction of pions In S states the contribution from the high energy regions In a more careful manner; we also analyze the question of the number of subtractions In the dispersion relations and. taking nucleon recoil fully into account. estimate the previously introduced quantities RI (II) at Q2 = O.

2. The connection between the Invariant functions Ti(lI. Q2) and the amplitudes Ri(lI. Q2} in the barycentric system is given by Eq. (1) otl14] (in the followlni 14] will be referred to as A). For the definitions of Ti (II. Q2) and Ri (II. Q2) see[13] (in the folloWingC13] will be referred to as B). The notation in the present paper is the same as the notation in A and B. By Ri without additional marks we will understand here the amplitudes in the barycentric frame.

1102

134 SCATTERING OF PHOTONS BY NUCLEONS Since according to the optical theorem 1m (~,

+ ~.) = 'II,a,

= wI-M' ~

4"

(1)

then one obtains from the dispersion relations for Fl' ... , F.

4"

2w

[j,_ (v ) _ D

1..

(w is the total energy in the barycentric frame) it follows that under the assumption at (w) - const as w _

we have asymptotically as w ~,+~,

-w',

(2)

T.- T.

-w,

II,

T. _w.

(3)

Vw.- (['/M

= -i- [T, -

F. (v.) = '110 (T,

= Wo (~, + ~.)/M, + 2R. + 2R.l/M,

T. - v. (T. - T.)1 [~. +~.

+ Ta)/2M

= (w./M)' (~. -

~.),

F. ('110) = t(T,- T.) = w:(~. +~.) /M'II.-2wo(~, +~,)/(M

+ WD)'

(4)

It is clear from the discussion above that the dis-

persion relations for the functions Fl' ..•• F. should contain one subtraction. All quantities on the right side of Eq. (4) are in the barycentric frame. If one takes into account that (for Q2 = 0) the amplitudes in the laboratory system are connected to the corresponding quantities in the barycentric system by (~,

+ ~.)' =WD (~, + R.)/M,

(~. -

~.)~ = (w.,lM)' (~. -

[~. +~.

+ 2R. + 2R.I· =wD[~.

= -

(5)

+~.

+

2R. + 2R.l/M,

'

Au (v) dv

v"(v'-v~ ,

(6)

Re (~. -

D~"'=

Re

~.)'~

[~.

+ R. + 2R. +

u-....= Re F. (v);

D~ (0)= -

2 [,,' - (e/2M)"l,

2R.r~

2J.'.,

D. (0) = - (e'/2M) A (2

+

A),

%-.

1m

R, = v,{IE,I' + 21E.I' + tlEII' cos 9 - tIM,I'}, R. = 'II,{IEII' +~IE,I' cos 9

- IE.I' + ~IM.I' +R.e (E;M.)} , 1m R.

= -

vert IE,I' + Re (E;M.)} ,

(7)

which represent the ~nerallzation of the corresponding equalities inCs]. The expreSSions for 1m R z differ from those for 1m Rt by the exchange Ei;:: Mi. Analogously. the expression for 1m ~ may be obtained from that for 1m and for 1m Rs from 1m Rs. In Eq. (7) we mean by the modulus of the· amplitude on the right side the sum of the contributions from photoproduction of 11'+ and w' mesons. We note that if the mass differences. between mesons and between nucleons are ignored then a cancellation in the interference terms. for example in 1ElfO 1Z + I E lr + IZ. occurs as a consequence of isotopiC symmetry in the pion photoproduction process. At that

as.

A, ('II) = vat/4#. = '\I {IE,I' + IM,I'

+ 2I E.I' + 2IM.I' + t IM.I' + -i IE, I'}, A. <'-) =

~.),

V

and where Ai (110) stands for the imaginary part of the corresponding amplitude; 11. = e (1 + A )/2M stands for the magnetic moment and I'a for the anomalous magnetic moment of the nucleon. If the elements of the amplitude for the photoproduction of pions in states with J :s 3/Z in the barycentric frame are denoted by E l • E z• Es (electric transitions into %+ and %- respectively). MI. M2• Ms (magnetic transitions into l/Z+. %- and %+ respectively). then the unitarity relations lead to the equalities 1m

as,

F, (v.)

D: = Re (~1 + ~I)'~

D~ (0)

A, .• (v)

'/

D, (0) = - e'/M,

becomes. as is well known. the photon energy in the laboratory frame IIIab (denoted In; the following by II). s. As can be seen from A, Eq. (i). for forward scattering the functions T, and T2 + T. reduce to R. so that at Q2 = 0 the dispersion relations for T5 and T z + T. are equivalent. Let us consider the functions

F, (v.) = v.T. = w.

2V: r =" l

.

D~"'=

Consequently, under the assumptions here made, the dispersion relations for T I, Ts and T, should contain one subtraction. whereas the dispersion relations for the quantities T 2• T. and T. may be written with no subtractions. In order to estimate the amplitudes RI + Rz, R,. ~. and Rs + Rs it is sufficient to write dispersion relations for T 1. Ts and T, at QZ'= O. At QZ = 0 the invariant '11=

do""

we

T,

.

vi-vi

:II

where

00

+ T. _ w', T, + T. _ const,

rl ~

2V:

(0) = 1..

•.• ('II.) - vDD,.• (0)

Assuming further that as w - 00 all Ri get from A, Eq. (1) that as w - 00 T,-T.

0

'/

00

-'II.

-wi

1103

v{IE,I' + IMII'+ t IM,I' +tIE.I'

+

-I M.+ tEl I' -I E. + M,I'}.

135 L. I. LAPIDUS and CHOU KUANG-CHAO

1104 A. (v)

= -v (wIM){IE,I'-1

MIl'

previously one studies properties of the functions· F, ("n)

+IM.-fE'I'-IE. - +M,I'}, A, (v) =

+ (w -

w{IE,I'+ IMII'+fIM,I'+~IE'I'

4. As was shown by Goldberger,CO the sum rule that follows from the nonsubtracted dispersion relations: .

r

el/

(v)dv>O

(9)

"/

is in contradiction with the long wavelength limit Rl

+ R,- -e'jM < O.

+ T,),.

(12)

F, (vo) = T~,

(8)

-IE.--i-M,I'-IM.-f E.I'}.

+ R,)-> + 2~'

(11)

(T, - T,Y.

F.(vo) = (T,

M) (7//41"1

Re(R,

=

(10)

Consequently, nonsubtracted dispersion relations for the amplitude Rj + R2 violate the requirements of relativistic and gauge invariance on which the long wavelength limit is based. Let us remark that possible sum rules involving the square of the magnetic moment are not in direct contradiction with the long wavelength limit when nonsubtracted dispersion relations are assumed for F 2 (v). As can be seen from Eqs. (6) and (8), of particular importance here is the contribution of the resonant state, proportional to I M312. The result is unchanged if one takes into account the (numerically important) contribution from photoproduction in S states, which decreases the effective contribution of I Ms12. The sum rule for the square of the magnetic moment is very sensitive to the ratio of the photoproduction amplitudes E z and Mil.' For certain ratios (for example for E2 = M,[5J) one can arrive at a contradiction. At the present time. however. the analysis of photoproduction is not sufficiently precise to permit the assertion that the experimental data are in contradiction with the sum rule. An increase in the accuracy of the photoproduction analysis. aimed at obtaining information about the amplitudes E 2• Mz and Es. would be most welcome. The fact that unsubtracted dispersion relations give rise to definite sum rules may be of particular interest in certain processes. Thus. in the case of 11"11" scattering analogous considerations (applied to dispersion relations at Q2 = 0) lead to the conclusion that the S-state scattering lengths ao and a2 are positive at low energies. The same holds for 1I"K and KK scattering. 5. If in addition to the functions introduced

(13)

one concludes that F 5,6 (v) are odd functions of v and contain no poles, whereas F7 (v) is an even function of v with a second order pole. As v - 00 F5.6.'~V-'/.,

so that the dispersion relations for these functions need no subtractions. These dispersion relations may turn out to be useful since when photoproduction in states with J ::s % is taken into account the angular dependence of the amplitudes Ri (v. QZ ) in the barycentric frame takes the form (cf.W )

+ 21£, cos 9 +fm. -+- c (1S,m.). R, = mi - m, + 2m, cos 9 + +;e, -:- C(m.IS,), R, = i€, -IS.

R. = - 1£, -

R.

C (m.If.),

=~ -

m. - C (If.m,),

(14)

and is characterized by eight functions of energy ll",a, ml,l,a, C (1S.m.) C (m.IB.). which can be ex~ pressed in terms of Ri (v. 0) and Ri (v, 0). It follows from Eq. (14) that if we restrict ourselves to contributions from states with J::S %

R', = R:,

= 21S,

(il cos a/ilQ')Q'~<>= -

41S~/M'v~,

so that (R.

+ R,Y

= (R,

+ R,),

=

IR. +

R,

+ 2 (R. -+-

R.)I'.(15)

In the long wavelength limit[t4J

(R.

+ R.Y

= -

2e'/M'v

+0

(I),

+ R,)' = - e' 13 + 2 (I + A)'1/2M' + 0 (v), (R. + R, + m. + ml)' (R.

= - e' (2A' - 2A -

1)/2M'

+0

(v).

(16)

The fact that Eq. (15) is in contradiction with the long wavelength limit (16) means that the restriction to states with J::S % is not a good approximation even in the low energy region. The crossing symmetry conditions introduce kinematic corrections of the order of viM. which corresponds to inclusion of states with higher values of J. The carrying out of the analysis with this high a precision requires the introduction of additional functions of energy and disCUSSion of a larger number *The prime denotes differentiation with respect to Q' and subsequent passage to Q' ~ O.

136 SCATTERING OF PHOTONS BY NUCLEONS of dispersion relations. Introduction of the Low diagram does not resolve the indicated contradiction. All estimates of the amplitudes given here were obtained with the neglect of Rj (", 0). 6. The results of the calculations of the amplitudes Ri ("0) at Q2 = 0 are shown in the figures. The energy of the photons "0 is given in units of the threshold energy "t = 150 Mev, and the values of the amplitudes in units of eo/Mc 2. For the calculation of the forward differential scattering cross section ~ (0') ~

i R. -r- R, \' + i R, -j R, + 2R. + 2R. \'

the amplitudes Rj + R2 and R3 + R( + 2R5 + 2~ are sufficient. To estimate D j ("0) use was made of the data on the total cross section for the interaction of photons with protons, including the second maximum and the cross section for pion pair production. The dependence of Aj ("0) is shown in Fig. 1. Previously we have neglected contributions from the energy region above 500 Mev. The result of estimating the amplitude Rj + R2 is shown in Fig. 2. The main difference between this and previous results appeared in the region 1 < "0 < 2, where as a consequence of a cancellation between the long wavelength limit and dispersion terms the value of Dj ("0) is significantly decreased. Let us note that this is precisely the energy region that is sensitive to a change in Aj (vo)' The second maximum in Aj ("0) corresponds to the second maximum in photoproduction.

1105

For estimating real parts of the amplitudes, other than Rj + R2, which require much more detailed experimental data on photoproduction, we limit ourselves to the energy region up to 300 Mev. For the amplitude R j + R2 it turns out to be possible t6 go much further, although with increaSing energy the indeterminacy in the contribution from photoproduction of pairs (and larger numbers) of pions becomes appreciable. In a number of papers[j5,16] the yp scattering at 300-800 Mev has been looked upon as a diffraction process with Re Ri « 1m Ri' The experimental study of yp scattering in the region of the second resonance is of interest as a sensitive method of investigation of the maximum itself. If, ignoring all Re R i , we restrict ourselves to the imaginary parts of the amplitudes alone and consider only the contribution proportional to I E31 2, then we find immediately from Eq. (7) that

R, ' - R.

=

R.

=

R.

=

0,

R, ,lmR, =-:2lmR,=2",!E 31 ', whereas the differential cross section[6] is equal to 6

(q)

= -:- R; (7 + 3 cos'S) =

+R; (7 +

3 cos'S),

(17)

in agreement with the results of Minami. [t6] The same result for the form of the angular distribution remains valid if in Eq. (7) only M3 (Rj - R 2, R3 - R() is different from zero. If simultaneously E3 and M3 (with Re Ri = 0) are different from zero then we have u (0)

'0'

-;,

(R:

+ R;) (7 + 3 cos' 0) + 10 R,R. cos O.

(18)

However, as our estimates indicate, the quantities Re (Rj + R2 ) are large in the region of the second resonance and cannot be ignored. From this point of view the second resonance differs drastically from the resonance, in whose energy region

%, %

I

;

J

5

"

b

7

Re (R. -:- R,) --:-~ 1m (R 1

The results of the calculations for R3 ± ~, R3 + R( + 2R5 + 2~ and ~ + ~ are shown in Figs. 2-4. In the evaluation of dispersion inte-

FIG. 1

5Jl

.J{/

'er,'''')

"l"IR,."'.2. . ,.\ . \ -2

e'lHe'

./

~

"/H,'

-J

FIG. 2

+ R,).

FIG. 3

137 1106

L. I. LAPIDUS and CHOU KUANG-CHAO dispersion relations (6) are not sufficient. Let us consider the function F(v) =w-'
+ T.)' -

viTo

+ T.)']. (19)

As can be seen from B, Eq. (4), we have

0.5

(20)

FIG. 4

grals 1E I 12, 1M312 and 1E312 were assumed to be different from zero, and the enerEY dependence of 1EI12 and 1M312 was taken from[6], whereas 1E312 was assumed to be different from zero in the energyregion 3.1 < 1'0 < 5.8. Let us remark that even in the absence of an imaginary part for Its + Its the real part of this quantity differs from its long wavelength limit, since the dispersion relations are satisfied by the invariant functions Ti(V, Q2). The values of a ( 0·) are shown in Fig. 5, where we give for comparison the results of Cini and Stroffolini[S] for aC-S ( 0·) in the barycentric frame. A significant difference can be seen in the near-threshold region. 7. For an estimate of RI - R2 and R6 - Its the

A study of the dispersion relation for F (v) makes it possible to estimate RI - R2 if Rs - ~ is known. In the energy region under consideration the coefficient of (Rs -. Re) in Eq. (20) is of the order of viM, however since the value of Rs - R. is large (in comparison with RI - R 2 ), the second term in Eq. (20) cannot be ignored. The function '1'(1') introduced in Eq. (19) is an analytic function of v with a cut along Vt < v < 00, satisfying the crossing symmetry condition: (21)

Thus, for I' «I't the function cp(v) is a real function and (22)

M,~(iMv~ij.(l -~)+

bv'

+ . ..

(23)

We see that the linear term in F (I') is' fully determined by the first term in Eq. (22). It therefore follows from Eq. (20) that for small v R, - R. = -

(e"IM) (1 - 3v1M)

+0

(v»

and the linear term in Rt - R z and in F(JI) are fully determined by the requirement of crossing symmetry, as is discussed in detail in B. The function F(v) introduced in Eq. (19) is an analytic function of v with cuts along I't < I' < 00 and _00 < v < -Vt and a (kinematic) pole at w'

=

M'

+ 2Mv =

O.

The requirements of crossing symmetry lead to the relation M'+ 2Mv • F (-v)

=

M'- 2MvF (v),

and for small v F (v)

~

-

(e"/M) (I - 2v/M)

+0

(v».

Applying the Cauchy formula to F ( vo), for p - .. , along the contour shown in Fig. 6 and writing a dispersion relation with a subtraction we obtain

F(v)=_~(1_2V')+ v~.\ F(v)dv =_::'(I_~) o M M 2", jV'(v-v,J M M v~ 7 M' + 2Mv 1 ]dv + "P )1m F [1 v-v, + Mi"=2Mvv+ v. :yo '/

FIG. 5

138 seA T T E R I N G 0 F PH 0 TON S BY N U C LEO N S 2v'

Re1jl(vo) -1jl(0) =

-,l-p

1107

';0 Im1j>(v)dv

~ v(v'-V~)

,

(27)

'/

where, according to B, Eq. (2), 1jl (0) = - e' (2

1m1\'

and e' (

v~

K(vo) = "

2v ) M'

+ K(vo) +

4v: Re F (M/2) M(v.+M/2) ; (24)

(25)

'/

Since 0,

=

Re F (M/2) cannot be determined from Eq. (24), and this quantity enters as a free parameter, which must be determined starting from the experimental data. Under the restriction to photoproduction in the states with J:S % only we get 1m F (v)

=

-IM,j')

y {~(lEII'-1 MIl')

+ 2(IE.I'-1 Mal') (I + } w-: M)

+ i-~(IE.I·-IM'I')( 1 + i- w-: M)}.

The results of estimating Re (R5

-

Be) at

(29) ~

= 0 for Re F ( M/2) = 0 are shown in Fig. 4. Es-

7J 1m F (v) [t M' + 2Mv t ]dv v-v. + M'- 2Mvv+ v. VI' K (M/2)

(wIM)'{w[-i"(IEol'

(28)

+ Re (E;M. - M;Es)] + My (M + ~tl [3 (lE.j" -IM,jO) + +(IE,I' -IMol") + Re(E;M.-M;Ea))}.

FIG. 6

Re F (vo) = -M 1-

(v) =

+ A.)/2M.

(25')

In Fig. 7 are shown the results of estimating Re (Rt - ~) with the help of Eq. (24) when the contribution proportional to Re F ( M/2) is ignored.

timates of the quantities Rs ± Be and Rs - Re, which playa dominant role in the differential cross section for Vo ;::. I, do not differ appreciably from those obtained previously.[8] The results here obtained are of interest from the point of view of the study of the energy dependence of amplitudes near the threshold of a new reaction. [8] In that case all estimates can be carried out to the end. Let us call attention to the dependence of the amplitude Re (Rt + R2)' whose value continues to fall off also above threshold. This result indicates that a sharp energy dependence of the imaginary .parts of the amplitudes above threshold may also for other processes lead to a displacement of the near-threshold minimum (or maximum) of the cross section relative to the reaction threshold. In Figs. 5 and 8-11 are shown the results of the calculations, with the help of Ri (v, 0), of angular distributions

1=0

for the angles 8 = 90, 135, 139 and 180·, and also of the total elastic scattering cross section a,/41f.

FIG. 7

For an estimate of Rs - Re at Q2 = 0, as can be seen from B, Eq. (4), it is sufficient to consider the function 1jl (vo) =

x

-i- v~ [T~ + ~ (T I + T.)l' = (~)"

{~(R. - R.) + w.~ M [RI

-

R. - (R, - R.)l} , (26)

for which the dispersion relation has the form

=

Bo + B./2

and of the polarization of recoil nucleons for 8 = 90·. The experimental data are summarized in[to] and[U]. The coefficient

B. ('\10)=2 [ I R. + R.IO_j R. - R.I"] is near to zero in the entire energy region Vo ~ 2. The experimental data, apparently, indicate that the quantity Re (Rs - Be) is positive. We were not able to achieve this by introducing Re F ( M/2) .. O. The requirement that Re (Be -He) be positive leads to large (negative) values for Re F ( M/2), which at the same time Significantly

139 L. I. LAPIDUS and CHOU KUANG-CHAO

1108

the dispersional analysis and experimental data is obtained. In the region 1 < 110 < 1.3. which is particularly sensitive to dispersion effects. it is apparently necessary to take into account contributions from higher states. for which it is necessary to have information on pion photoproduction in a larger energy region. 1 Gell-Mann. Goldberger. and Thirring. Phys. Rev. 95. 1612 (1954). M. L. Goldberger. Phys. Rev.

99. 979 (1955). FIG. 8. Energy dependence of the coefficients in the angular distribution. The experimental points are from ['.".17]'

~ (el/Mcz;Z

6

]I( eZ

a

I

I

(1958).

«R. H. Capps. Phys. Rev. 106. 1031 (1957); 108.

fMc')'

1032 (1957).

b

;f

H

I

·f Jf

J

2 I

'~!4-dJ o

---"L

o.s

1.0

I.S

\

1,J) .~

t.P 0

as

I

(5

1.0

FIG. 9. Energy dependence of the scattering cross section: a - for (J ~ 135°. b - for (J - 139°. The experiments! data are from[·, lO l l1J.

us 180

r ....

LL:s~:::::::=-.:2.J'E!!.0o

•,

0.1 o.s o.J

0

2A. A. Logunov. Dissertation. Joint Inst. for Nucl. Research (1959). 3 M. Cini and R. Stroffolini. Nucl. Phys. 6. 684

FIG. 10. Differential cross sections at different photon energies (indicated on the curves).

·o.J-0.5 -1J.7 -/ r.asH

FIG. 11. Polarization of recoil protons.

increases the contribution of I RI - R212 to the cross section and does not lead to an improvement in the agreement with the experimental data. It is necessary to remark that outside the region 1 < "0 < 1.3 a satisfactory agreement between

5 T. Akiba and I. Sato. Progr. Theor. Phys. 19. 93 (1958). 6 L. 1. Lapidus and Chou Kuang-chao. JETP 37. 1714 (1959) and 38. 201 (1960). Soviet Phys. JETP 10. 1213 (1960) and 11. 147 (1960). 7 M. Jacob and J. Mathews. Phys. Rev. 117. 854 (1960). 8

M. Gell-Mann and M. L. Goldberger. Proc.

1954 Glasgow Conf. on Nucl. and Meson Physics. Pergamon Press. London-N. Y. (1954). 9 Hyman. Ely. Frisch. and Wahlig. Phys. Rev. Lett. 3. 93 (1959). 10 Bernardini. Hanson. Odian. Yamagata. Auerbach. and Filosofo. Nuovo cimento 18. 1203 (1960). II F. E. Low. Proc. 1958 Ann. Intern. Conf. on High Energy Physics at CERN. p. 98. 12Glasser. Seeman. and Stiller. Bull. Amer. Phys. Soc. 6. 1 (1961). 13 L. I. Lapidus and Chou Kuang-chao. JETP 41. 294 (1961). Soviet Phys. JETP 14. 210 (1962). U L. I. Lapidus and Chou Kuang-chao. JETP 41. 491 (1961). Soviet Phys. JETP 14. 352 (1962) 15 Y. Yamaguchi. Progr. Theor. Phys. 12. 111 (1954). S. Minami and Y. Yamaguchi. Progr. Theor. Phys. 17. 651 (1957) . 16 S. Minami. Photon-Proton Collision at 250800 Mev (preprint). 17 Govorkov. Gol'danskii. Karpukhin. Kutsenko. Pavlovskaya. DAN SSSR 111. 988 (1956). Soviet Phys. "Doklady" 1. 735 (1957). Gol'danskll. Karpukhin. Kutsenko. and Pavlovskaya. JETP 38. 1695 (1960). Soviet Phys. JETP 11. 1223 (1960). V. V. Pavlovskaya. Dissertation. Phys. Inst. Acad. Sci. (1961).

Translated by A. M. Bincer 261

140

451

PI-IYSICS

A Suggested Experiment to Determine

th.

Spin of

Y:

and the Parities

of

,1 - I, ,1 - Yf and I - Y:

rn this nOte we propose an experiment In observe the final state ,1 - " resonance and lhe cusp arising from the near tbreshold effect of the l:-production, and from this to determine the spin of yt' (]).~) and the relative parities of .1 - J:, .1 - y~ and !: - y~ (P(,l - S). PCl and Pl.); - Yn). The centre oi mass energy of the initial " - P system ~hould be around 1900 MeV (this correspond. to 1305 MeV of the incident pion ill the laboratory system). Choose the evellts in which the energy of the kaon lies below 90 MeV and observe the correlation between resonance and ~ne cusp. This energy region will iust cover the y~ resonance due to Y ~ and the cusp due to

yn

141

452 I-production. Furthermore, since the energy of the knon is reiatively 10\\' and much below the mass of K* (MK .. -BBOMcV,QK*-250MeV), only the I-wave of kaon need. to be considered. Obviously, the cusp will appear in the J 1/2 state with Pd. = 0 or Pd. = t according as peA - X) is even or odd. Since tbe p()sition of the 1:: -". threshold lies below the A _ " resonance only by 55 MeV, and the background of the rC."nOlle" i. rather small (around 1 : 2 -+ 3 to the slope of this resonance). Thc cusp can only be observed when it appears in the same panilll wave state as tbe reso.nallce, and til is will lead w a useful information about the A - Z relative parity and the spin of Yi. Moreover, the relative angular momentum Pr~ of the A - n system can be determined from -' the angular disttlDucion and polarization measurements. From the above observations the spin of yt and the relative parities can bl uetermineu. We list below the diffcrent. con. elusion corrcspClnding to all possible experimental situations.

=

(1) Ii the cusp it observed togcther wid, the rc.~s()nallc('\ then we have Jr~ = ] :'2, P(I.· - y~) and if lhe P\"~ c,;n further he determined (sec (4) below), then peA - I)=

= -,

(-/}.~. (2") If only the resonance i. observed, and if the Jl't, P\'~ can further be determined (sec (3) and (4) hclow), then

i.

when

Jl'''' =

1/2, we shlllJ have P(I.-

Yf)=+. P(A':X)=(-/y~+I: ii. when J1' ,. = 3/2, we may have P(.1-

= -. since the Prt = 2 ~t:ltC can be excluded in this energy.

Yt)

(3) rized:

i.

When the initial if

nucleon is unpola-

r/u is isotropic and the hydIJ".,dE k

peron is unpolarized, then we have ii. if

1/2;

--~~I!.--. . . .- - a cos! 8 +bcosO + c and dIJ,h,dL1:

da

P II

Jrt =

_ sin 20 a" .. X a; ,

d!J"."dE k

=

In.b

then

we

y. nrl

e

Jy't 3/2, where cos = Di • D,h.. a;, and Q d.. being respectively tbe incident direction of ".-meson, the direction of the A-:r relative motion, and the solid angle of the .1-11' system. have

DA.",

(4) When the initial nucleon is polarized and its polarization is orthogonal to Di, then

in the case of Jr~ = 1/2: , do i. when P II -.-.- - , ' - = P • have Pl't

"

dQd",dL:.k

= II; h

II. W en

P

A

d() d!J",.dEk -

COlIst..

pO'

-~aA,,\nA'"

we

P

j

we have Py~ = I, where P

and P" ate the polariz:ttions of the initial nudeon :llld the final hyperon respectively. It will be noted that (1) and (3) ate two indepenuent observations for J},~; (2) and (4) arc ohservations On rei. ative parities. Two of the authors (Su Zhao-bin and Gao Chong-shou) arc indebted to Dr. Chou Kuangchao lor his kindly guidance, an'; .lls() to Prof. Hu Ninr; for his interest and support. Su Zhao-bin ("~) Gao Chong.-sh()11 (;flj~r;(n Chou KUan!Hhao UIiI:l'dD Peking Unit'f1TSUY

Jan.

7, 1963

142

SCIENTIA

Vol. XXlI No. 1

SINICA

.T::mnnr.'· 1979

THE PURE GAUGE FIELDS ON A COSET SPACE CHOU KUANG-CHAO

(nil7tE)

Tu TUNG-SHENG

(Institute of Theoretical Physics, Academia Sinica)

(t1:*~)

(In.sti~ltte

of High Energy Physics. ..:l.cademia Sinica)

AND YEAN TU-NAN

(~mm)

(University of Science and Technology of Chi·na) Received August 18, 1978.

ABSTRACT

The concept of the pure gauge fields on a coset space is introduced. By using gaug" fields on subgroup H, pure gauge fields on coset space GIH and the induced representation, a local gauge invariant Lagrangian theory on group G is constructed. The application of this theory to SUo X SUo gauge theory, the a model and the non·trivial t.opological property of the pure gauge field are discussed.

I.

INTRODUCTION

Since the emerging of the non-Abelian gauge field theory unifying weak anti electromagnetic interactionUl , the properties of the non-Abelian gauge field have been extensively investigated and the important progress has been made[·I. ~o\.s is well known, even with the symmetry of the vacuum spontaneously broken and the Goldstone bosons absorbed through the Higgs mechanism, the non-Abelian gauge field theory can still be renormalized. Though in the unitary gauge, only physical particles appear, the theory is not obviously renormalizable. While in the Landau gauge the theory is manifestly renormalizable, it still contains fictitious particles, which seem to destroy unitarity. The theory is in fact both unitary and renormalizable as can be shown simply in the R, gauge. The worst divergent diagrams are cancelled with each other. The undesirable effects of the fictitious particles are also cancelled. JUany low energy hadronic experiments showed that hadrons have not only SU3 (or BU.) symmetry but also SUs X SUo (or SU3 X SU.) chiral symmetry[31. The pion is a pseudo-Goldstone boson resulting from the spontaneous breaking of the chiral symme~ry. It was proved in [4] that when the chiral group is broken in Goldstone mode, the effective Lagrangian involving the pion field still possesses chiral symmetry in the approximation of the lowest order in breaking parameter and pion momentum. In this case, the pion field offers a non-linear realization[51 of the chiral group. If the pion field is taken as an elementary field offering a non-linear realization of the chiral group, the resultant Lagrangian has complicated non-linear terms. This Lagrangian gives some results which in the lowest order perturbation theory agree with experiments. But the theory is not renormalizable[OI. The linear a model can be constructed,

143 38

SCIENTIA SINICA.

Vol. XXII

using linear representation of the chiral group. This· model is renormalizable, but the pion field does not apparently have the characteristic of a non-linear representation of the chiral group. In this model, it is difficult to get results in the lowest order perturbation expansion in agreement with experiments. If a gauge degree of freedom like that of the non-Abelian gauge field theory is introduced to make the theory gauge invariant, the theory might be manifestly renorrnalizable in one gauge (which is to be called the renormalizable gauge), while in another gauge (which is to be called the physical gauge) good physical results might be easily extracted by tree approximation.

For this purpose we introduce the concept of the pure gauge scalar fields on the coset space. In the case the global topological properties are trivial, the pure gauge scalar fields are the manifestation of the gauge degrees of freedom and can be eliminated by choosing a suitable gauge. By using the pure gauge scalar fields the renormalizable gauge can be connected with the physical gauge. If the subgroup is U(l), the gauge field on which is the electromagnetic field, and if monopole exists, the monopole and its electromagnetic field can be described in terms of the pure gauge scalar fields. In this way we can avoid introducing singular strings in the expression describing the vector potential of the E. M. fields" of the monopole. Further the equation of motion of the monopole and the electromagnetic fields might be derived from the Lagrangian.

The plan of this paper is as follows: In section II we review briefly the induced representation of group G on its subgroup H, introduce the concept of the pure gauge field on the coset space, and discuss its transformation properties. In section III we discuss a physical system which is local gauge invariant with respect to the subgroup H. Using the pure gauge fields on the coset space we construct a local gauge invariant Lagrangian with respect to the whole group G. In section IV, for the sake of illustration, we construct an SUI X SUI gauge invariant theory, which coincides with linear a model in the renormalizable gauge and with the non-linear chiral model in the physical gauge except a few additional terms which account for the renormalizability. In section V, we discuss how to describe monopoles by means of pure gauge fields on the coset space. In section VI, we give the coset element parametrization with respect to the subgroup U... X Uft-.. of the group Uft. The corresponding expressions for the pure gauge fields on the coset space are also given.

II.

THE INDUCED REPRESENT.l.TION ON THE SUBGROUP A...'i[D THE

PURE

GAUGE FIELDS ON THE COSET SPACE

Consider a transformation group G underlying a physical system. Let H denote one of the subgroups of G, and g and h designate any element in G and H respectively, namely,

G={···g···},

H={···h···}.

tP denotes any representative element of the coset space with respect to subgroup H, namely,

144 PURE GAUGE FIELDS ON COSET SPACE

No .. l

39

GIH = {.. ·cP···}. According to the theory of Lie group, for any element g E G, there is the unique left coset decomposition,

g

=

(2.1)

(For right coset decomposition the discussion is similar.) Eq. (2.1) represents a de· compositiun of the bundle manifold G(GIH, II) into its base manifold GIH and fibre II. By choosing suitable local coordinate system in these manifolds we get the corresponding parametrization of (2.1). Similarly, for a specified t/Jo E GIH, we have (2.2) where t/J(g, CPo) and h(g, c/Jo) belong to the coset space and the subgroup respectively. We can take h(g, c/Jo) as the image of g on subgroup H through c/1o, that is, (~.3)

According to (2.2), there is

g' g<po = cP(g' g, cPo)h(g'g, (Po).

(2.4)

Multiplying (2.2) by g' from the left-hand side, we have

g' g<po = g' cP(g,cPu)h(g,
= cP(o', (p(g, cPo) )h(g',
(~.5)

Comparing (2.4) with (2.5), we have h (g' g,
= h(g',

(2.6) (~.i)

We have therefore the following mappings: g acts on

cPo to give the image g -+ h(g, cPo),

(2.8)

g' acts on g successively to give the image g' -+ h(g', cP(g, cPo)),

(2.8),

g' g acts on cPo to give the image g' g -+ h(g'g, cPo).

(2.8)"

Eq. (2.6) gives the multiplicative cumbination law of the images of G on H. Eqs (2.1), (2.2), (2.6), and (2.8) are the rules concerning the mapping of group G on its subgroup H through t/Jo E GIH. It is defined as the induced representation[1l. Assume that A~i U = 1, 2 ... nH, n" is the number of generators of subgroup H) are gauge fields on subgroup H and are transformed accurding to the induced representation of G on its subgroup H. Define (2.9) where A, are generators of subgroup Hand K is the coupling constant. Introduce the pure gauge field cpo which is any specified element in GIH. Let us further define

145 40

Vol. XXII

SCIENTIA SINICA

(2.10)

and discuss the transformation property of

B..

Making the local gauge transformation

g of the group G as usual, we have

BI' -

B~ = y(o"

+

(:.Ul)

8,,)y-l.

Owing to (2.10) and (2;2),

B,.. = g 01' A,

{

+

+

A

A:)cp'-I,

(:.D2)

where (:U3) ~Iaking

local gauge transformation g' successively, from (2.5), (2.1:3), we get B~ = g'(0l' =

+

B:)g'-l = g'c/,'(o"

cJ>(g',
+

l~)(g'
+ h(g',
. h-I(g, o»}
.4.1')

(~.1-l:)

If the product gf g is taken as one ~auge transformation, we have

Thus, making two gauge transformations g and gf successively is equivalent to making one gauge transformation gfg according to (2.6) and (2.7). In both cases we obtain the same potential B/'. In the sense of induced representation, the gauge transformation is closed. That is the basis of constructing a gauge invariant Lagrangian in terms of pure gauge fields on thE' coset space.

III.

COXSTRUCTION OF Loc.'\L GAUGE IXYARIANT LAGRANGIAN ON THE GROUP

G IN

TER~IS OF PURE G.WGE FIELDS ON COSET SPACE

Consider a Lagrangian which is global" gauge invariant on the group G and local gauge invariant on the subgroup H. (3.1) where

are field quantities providing a basis for a representation of the group G, and (3.2) (3.3) are the covariant operator and the field strength of the gauge fields .J.~ on subgroup H

146 No.1

41

PURE GAUGE FIELDS ON COSET SPACE

respectively. According to the global gauge invariant property of the Lagrangian, for any element g of G, there is

Notice that, if we substitute g by g(x), considering the absence of derivative of g(x) in (3.4), the identity will still hold. Put ¢o(x) as an element of the coset space G/H. Owing to (3.4), there is

Introducing the new field quantities,

= cPo(x)(p(x), Six) = c/Jo(x)(B,. + .1,.)c/JOl(X),

tP(x)

(:3.6)

we can easily verify

(3.7) b~

By using (3.6) and (3.7), Eq. (3.5) can

rE(tP(x), D~B'tP(.x), F~~l)

=

rewritten as

£.t"(rp(x), D~"rp(;L-). F~~».

(3.S)

Now let us prow that the Lagrangian

(:3.0)

is a local gauge invariant on the group G. In fact, by making local gauge transformation g(x) E G.

CP(x) - tP'(x) ]J,.(x) -

=

g(x)tP(.x),

fJ:Cr;) =

g(x)(B,.

+ S,.)g-'(;c),

(:3.10)

'One can easily verify that D~B'tP'(X)

= g(x )D<,!'tP(x),

F~~)' = V(x )F<,!.)g(.x).

(3.11)

Thus, we have

rE( tP'(x), D;B,tP'(X), F:<:')

= f,&'(g(x )tP(x), V(x )D<,!'tP(x), g(x )F<'!"g-'(X»,

(3.1~)

whf're there is no derivative 'Of g(x). According to (3.4), (3.S) and (3.12), we have f,&'( tP'(x), D<,!)'tP'(x), F<,!.)')

=

q( tP(x), Dt,!'tP(x), F~~».

(3.13)

'fhat is to say, the Lagrangian (3.9) is a local gauge invariant under the group G. Frem (3.5)-(3.8) it is seen that the Lagrangian (3.9) and (3.1) differ with only a gauge transformation. By making the local gauge transformation g(:1:) = ipo-' (:1:) ,

147 42

SCIENTIA SINICA.

Vol. XXII

the ~ in (3.8) wilI become ~ in (3.1). It is equivalent to taking the gauge condition as cpo (x) = 1 in the beginning. 'Assume that the numper of generators of the group G (H) is lIG (IlH), then we have nG-nH parameters describing the coset space G/H. Through the element cDo(X) of the coset space we have introduced nG-nll field quantities which provide a basis for a non-linear realization of the group G, and which are scalars under Lorentz transformation. (We assume that G is an internal symmetry group). In the usual gauge theory, IlG-nll .further v~ctor fields on the ·coset space are introduced for the gauge invariance on the group a, but in the present case, only na-nll scalar fields are to be introduced. It will be proved below that these scalar fields represent gauge degrees of freedom. They are not independent field quantities and can thus be eliminated through gauge transformation. For illustration let us evaluate the conjugate momentums cf 4>(x) Cho'ose a coordinate system in which c/Jo(x)a~¢o-l(x) can be written as

and B~(x).

"G-nH

cP08p.cPo' = ~ lj(C'i~x))8p.C'i(x),

(3.14)

j=1

where C'j(x) (j = 1, 2·· '1lc - IlH) are new' pure gauge fields introduced on the coset space. From F(,!/ = cPoF~~lcPOl it is easily seen that F~!l does not im'olve the derivatives of (x) is (:3.16) Thus we have (3.17) Eq. (3.17) shows that the new Lagrangian in (3.9) gives n,,-nH constraint conditions which are generated by gauge invariance of the Lagrangian (3.9) on the coset space G/H. These constraints make aj(x) not independent. Owing to the constraints or gauge inyariance it is necessary to introduce gauge conditions in the course of the quantization of the theory described by the Lagrangian (3.9). Eq. (3.17) makes necessary the introduction of nc-nll gauge conditions, while for 'Vector fields A.. on subgroup H, ilK gauge conditions must be illtroducf'd. Thus th('l'c are IlG gauge conditions altogether. If we choose IlG-n" gauge conditions as

(3.18) the Lagrangian (3.9) is restored to (3.1).

148 No.1

PURE GAUGE FIELDS ON COSET SPACE

43

However, both (3.9) and (3.1) describe the same physical system. It is justified to ask, since the Lagrangian of (3.9) is equivalent to that of (3.1), why do We introduce the concept of the pure gauge fields on the coset space? The answer is as follows: In studying the theory with the chiral symmetry of SUo X SU. or SUo X SF., we know that when elementary particles are basis of the linear representation of the chiral group, it is easy to construct a renormalizable Lagrangian. But the vacuum is not symmetrical under the chiral transformation. "When the symmetry of the vacuum is spontaneously broken, Jr, K mesons emerge as Goldstone bosons which offer the nonlinear realizations of the chiral group. It is difficult to prove the renormalizability of the theory with non-linear Lagrangian. The simplest Lagrangian of the non-linear realization of the a model has been proved to be non-renormalizable. If we start from linear representation of the group G, we may write down the renormalizable Lagrangian (for instance in the form (3.1)). Assume that the vaccum has only the symmetry of the subgroup H and that the spontaneous breaking happens on the coset space G/H, then there will be 71c-nH Goldstone scalar fields in the system[S). These scalar fields are collective e:!:citations of the system and should be described by the ccset space element rPo(x). In this case the pure gauge fields on the coset space aj(x) represent the Goldstone fields. They form a manifold on which the non-linear realization of the group G operates. They have a clear physical meaning. We can choosl' in (3.9) a gauge which keeps a;(x) and eliminates the surplus degrees of freedom. So a Lagrangian constructed of aj(x) and other field quantities is obtained. This Lagrangian differs from the apparently renormalizable Lagrangian only by a gauge transformation. Thus the theory described by the new La..,"Tangian built in terms of a;(x) might also be renormalizable. We call the gauge a;(x) = 0 the renormalizable gauge, and the gauge, in ,vhich a;(x) are kept as basic field quantities with the surplus degrees of freedom eliminated, the physical gauge. Notice that a;(x) have a real physical meaning only when the yacuum is broken spontaneously on coset space G/H. Only in such case the term "physical gauge" is valid. In the ne:!:t section we shall discuss the Lagrangian in both kinds of gauges for the case of SUo X SUo chiral group. IV.

SUo

X

SUo

PURE GAUGE FIELDS ON THE COSET SPACE ..\SD THE

a

MODEL

As an e:!:ample we discuss the connection between the SF. on the coset space and the a model.

X

SU. pure gauge fields

According to the general theory given in Section III, SUo Lagrangian can be written as

X

SUo gauge invariant

where

149 SCIENTIA SINICA

f~.

=

ol'B~ -

o.B~

+

Vol. XXII

(4.2)

lC/ijl,BiB:

is the gauge field strength LIn the group G;

HI' = ,PoCOI'

+

~4.o").pOl, 'Pu E H/ H.

(4.a)

BI'L and Bl'll i1.(·e related to BI" 1 + °rs BI' = - -2- B,.... .. , A

A

+

1 - rs - -'2- BI'R' A

(4.4)

It is easy to prove that the Lagrangian in (4.1) is gauge invariant under the following gauge transformation:

c/J - c/J'

=

gc/J

=

ei3"e i ;'c/J,

(JJ - l/J' = e-iaei&(JJe-i;'e-i:, B - B = e,ar, elb(o A

A,

..

'0'

1'''

I'

+

1 + r 5 B + ___ 1 - r5 B B )e-,b e-" ', = ___ I' ' J , . L 'J ,.1/, ii BI'L)e-ibe- , A.'.

A,

A,

BpL - B~c. = eii eib(81' + B/'R - B~R = e-i3 ei;'(81' + BI'R)e-i;'e i3,

a=

a(x) • -r/2,

b=

b(x) • -r/2, -r are Pauli matrices,

If we take the subgroup H { ... (] ... }, where

(4.5)

= 1, the identity element, the coset spal!e G/H == G = (4.6)

There are then:

11'=0, HI'

=

B I'L A

=

ei~·'eiii8I'e-iae-iji." ei~eiii8 e-iae-i~

I'

,

BI'R = e-iPeia8I'e-iiieiP.

(4.7)

we get

Starting from (4.1) and making the gauge transformation

(4.8)

where we have used (4.7). After gauge transformation,

-

1-'

(1- T (JJ'+(JJ') - f/r (8 + if )c/J' - G;jj' ( 1 + r, (JJ' + 1 \~

p

r

~

iJ.

\

2

2

1'5

(JJ'+) c/J', . (4.9)

- Gifj' (" 1 + rs (JJ' \ 2

+

1 - rs (JJ'+) c/J' 2 '

(4.10)

150 PURE GAUGE FIELDS ON COSET SPACE

No.1

45

where (4.8) has been used. The Lagrangian (4.10) is just that of the linear a model. Thus SUo X SUo gauge invariant Lagrangian in the renormalizable gauge becomes the usual linear a model. Starting from (4.9) and making the gauge transformation (4.11)

we have then "p' -+-"p"

= y"(x)"p',

rp' -+- rp"

=

e- i8 rp' e- i8

B'~". -+ B" ==

A, -+- BA" B,.L ,.

=

BA ",.

=

A

B ,.R -+,

= ,0 =

rp"+,

ei3r,a e-idf's "" eiOa ".e -ie ,

(·U2)

;8 e-i8a".e,

• [a".,o - ,o(eifta".e-iB - e- itl a".e i6 )]} - Veo Z)

_

J,"r".(a"

-

,7." 'r.". (a ".

'I-'

+

+

eiiir,a".e-iOr,)"p" - GJ,""p",o

iOr a -iOr''I-' ).1." - G'7." I" e'".e 'I-' r.;J ,0.

(4.13)

(4.1-:1:)

We call the gauge transformation (-±.12) the physical gauge because in that gauge the () is chosen in such a way that 4>" is diagonal. Here we assume, like the usual a model, that

rp' = a'

+

iT: • ,,:',

(4.15)

where a' is a scalar field, n' is a posudoscalar field and both are real; p is a real scalar field .. Lagrangian (4.14) can be obtained from another point of view. Applying the canonical transformation "p'

= e- i8r,,,p",

rp'

=

,oe ZiO

(4.16)

to (4.10), we can get the same Lagrangian (4.14). Thus the Lagrangian (4.14) can be obtained from (4.1) through gauge transformation in one way and from (4.10) through canonical transformation in the other. There is

rp ,

=

3

pe 2.",

=

pe'".

8

=,0 (

Assume that V(p2) has a minimum at p = po.

cos

6 SIn . e) . e + '~T:• • e

( 4.17)

151 Vol. XXII

SCIE)l'TIA SINICA

Put P

=

Po

+

(4.]8)

p',

e.

p0e-sm e =:tr,

(4.]9)

then there is sinl e = :trl/ pL cos2e = 1 - :trl/pL

(4.20)

:tr1/ p~

( 4.21)

pC-"; 1 -

€P' =

+ ·iT • :tr/po)·

Comparing both sides of (4.21), we have / :tr ' =-:tr. P u'=p ( 1_:tr2/p~yl,

(4.22)

Po

Substituting (4:.17)-(4.22) into (4.B), we obtain

with fC'p'

= - ~

fC' = :r

££'x

=

-

(8.u ,O')l_1'((PO

1.. (8 2

:trY _ p.

~r/'(1"p,8p,

+

1.. (:tr 2

+

( 4.24)

P'Y),

. 8,,:trY.

p~-nl'

Jl)c//', J[

=

(4.26)

Gpo,

(4.2i)

:;t'"x = - ;j'J"/·p,B/po· :tr)c//',

,i ( ( :trl\,!l\ T· (:tr X a,,:tr) , - 1- 1 - - / 1 2 p~1 / :trl .

(4.28)

(4.29)

Expanding (4.25) in terms of :trl / p~, we have 1 1 1 , :trl ... :;t',,=--(a :tr)2_-(:tr·a :tr)l_-(:tr·a :tr)-_

.

"p. ~

.)-Po.2

p.

-J 2

-Po

,"

.2

Po

( 4.30) The Lagrangian (4.30) is exactly the same as the Lagrangian of the non-linear a model. But in (4.23) there is the additional term q p' .. which also gives a contribution to the Ir-1r scattering. Besides, Lagrangian (4.23) is gauge invariant and can be changed into a renormalizable one through a gauge transformation. Thus the theory described by the Lagrangian (4.23) might be renormalizable. Now, we shall derive the Goldberger-Treiman relation in the lowest order approximation from our model.

152 47

PURE GAUGE FIELDS ON COSET SPACE

No.1

According to (4.10), the axial vector current is (-!.:~ I )

Substituting (4.16)-(4.22) into (4.31) we hUYe

(4.32) ~

a - '!.-" c/J r ~ Ts -or" 2 c/J + Po ~ 11: + ....

(4.33)

The f3 decay of nucleons should be determined by (;.d~5' (4.33), the corresponding coupling constant of :r decay is

f"

=

.As can be seen from

PuG",.

(-1:.34)

Substituting JIN = poG into the above equation we haye (4.35) As will be shown ill the following, G is just the ."(-X-X coupling constant, so that (4.35) is just the Goo T relation. In fact, in the lowest order approximation there is; ( 4.36)

Noticing that al'[CJjI1·p.TS'Z'c/J"]

.11:-

al'[CJjlrl'rs'Z'c/J" • :r]

• al'11:CJj"Tp.T5T: • al'11:c/J"

= -

=

-

CJjlrl'l"s'Z'c/J"

(4.37)

2p OGCJj"r5T: • 11:c/J",

we get (4.38) So, G is just the 1'C-N-N coupling constant. V.

THE PURE GAUGE FIELDS ON COSET SPACE AXD :\Io~oPOLE

In this section we shall discuss the example with G = S,[-", H gauge potential as the electromagnetic potential. Define 1

.al'

=

•

16

..4.I'IA0,

= U(l)

and U(l)

(5.1)

where 10 is the generator of the subgroup U(l). Introduce

(5.2)

153 Vol. XXII

SCIEXTIA SINICA

48

where SU(N) "'E - -. U(I)

'I'

When the monopole exists, A. has a Dirac string. It is convenient to use

B/L and

l/J as the dynamical nriables. Define 1

A

,.

gjJ/L = -.-. T,(B/LI(x»,

(5.3)

·!e.\

where (5.4)

For simplicity, we choose

(:5.6)

then we haye

I~ = ala

+

b, a.

=

2- X

V X-I'

b

=

l.

(5.G)

For the other choice of lo, we can deal with it in the same way. Multiply both sides of (5.2) by I (x) from the right·hand side and take trace, and we obtain (5.7)

'We may introduep differential forms: 0·8)

According to (5.7), w

=

Q -

_.1_ T,[(d4>-I)4> I aL ·!eN

(5.9)

FA = clw = 8.A/Ldx. 1\ dX/L (5.10) It can be proyed b,r using (5.6) that

(5.11) Substituting (5.11) into (5.10) we haye

154 PURE GAUGE FIELDS ON COSET SPACE

No.1

49

(5.12) Thus we obtain

-- 8 P.~II vi:!

-

"',;;7>' 1 -r -:--,..

OJ,lp7/"p,

T r {(8 .,,
lB.

In the above equations, cp = cfJ(x) can be regarded as a mapping of a spaee point x to a point in the coset space SU(X)/U(l). Take a sphere S' with infinite radius and consider the mapping

,/>(:0) :

'Ir( Y) Sl -+ -'-' - '-

(:j.H)

reI) .

If cp(x) is equh'alent to non·identity element of the homotopy group rr.(SU(lV)/ U(I», there will be monopole in the system. The contribution of the monopole to the

field strength is described by the second term on the right.halld side in (5.12) and (5.13). Integrating F .. over the space surfacL'. we cbtain the magnitude of the magnetic charge, namely,

f

FA

=

(5.15)

4:n:[j.

,\Ye illustrate the details for the special case of G

= SU('2) .

•-\.ssume that the monopole is at the origin of the coordinate system.

c/>(x) =

6 i9 ."./l,

n

=

sin
Choose

(5.16)

cos
where (8, cp) are the polar and azimuthal angles of the space point r respectively. i, j are the unit vectors along the positive direction of x and y axes respectively. Then we have ~

T

r=-. r

(5.17)

It is easy to prove that

Tr(dI(x) AdI(x)I(x)) = 4-i sin 8d8d
(5.18)

155 50

SCIENTIA SINICA

Vol. XXII

Then there is

" /\ "" f FA = - - g; TT(clI(x) f FA

.)~ dI(x)I(x)) = =-. e

-·i

4eN

Combining the above with

(5.19)

4/10, we obtain the quantization condition,

=

eg = -1.

(5.20)

2

If the monopole is at the point yet), we should choose iP(x) in some other way.

In summary, in the presence of the monopole, we can divide ..:!~ into two parts: ~ p' the electromagnetic field potential not originating from magnetic charge, and iP(x), the pure gauge field on the coset space describing the contributicn of the magnetic charge. y (t) here can be taken as a dynamical variable. 'C'sing (5.13) and the Lagrangian,

rv -_

;c.

-"41 F.4 FA pov

_

POll -

-

"41 (Ps

'(,'_T,lll: )(pB1"1' I ' ('r

I"JI -;-

/£:·

oJ

•

where

we can derive the equation of motion of~ p(.l') anu y(t) and thereby the interaction of the magnetic charge with the electromagnetic field. Howeyer the self-energy of the monopole is diYergent. Thus the monopole is point-lil;:e in this theory.

VI.

THE DECOl'IPOSITION OF THE GROUP

F.

,AXD THE CORRESPOXDIXG PURE

GAUGE FIELDS ON THE COSET SPACE

= {... g . .. },

We represent the group U. g = ei9 ,

e+ = e, e =

with

(0:p+a,l' fJ) D:t+ = 0:. o:t = 1

I.

IJl,

(6.1)

where a. are square submatrices, a.nd (3 is a rectangular sub-matrix. In that cnse we can choose i(G

h= e

0

I

0)

G, =

(e 0) . E H, iG

,

o

e'G:

(6.2)

where cosine and sine functions are defined as the Taylor expansions. Thus, we have cosv'fJr

=

(

o

(6.3)

156 So. 1

PURE GATGE FIELDS ON COSET SPACE

51

Define

= i tan v""iJi1 P = .ip tan vi+P.

'P

Vpp+

'f

T

=

Vp+p

·8+ tanV pr = - I. --:::::::::=-'tanVP+p p+

-I

'vpr

vrp·

(6.4)

Substituting' (6.4) illt" (6.a) WI' obtain

Thus, the left coset decompositicn of the group U. is

(6.6)

Put g =

(~ ~),

(6.7)

then there is :

(6.8)

Comparing the two matrices in (6.8), we have

rp~ cp' =

(Acp

+ B)

Ccp

~D'

,p+~cp+'.=-(C-Dcp+)

1

+.

A -Bcp

(6.9)

157 52

SCIENTIA SINICA

Vol. XXII

Thus the action of G on the coset space is equivalent to fractional linear transformation on the field cpo The gauge illYariant Lagrangian is still of the form (3.1). It is easy to show that

:A,. =

¢(a,.

+ AI')¢-l =

(b: ~~+

:

VI

+

)

rp+rp

+ rprp+ a,.Vl + rprp+ + A~l) + rpA;:)rp+, -al'cp-A~)rp + rpA;:)) a,.rp+ - cp+.-t~I) + .4~)rp+, rp+al'rp - V 1 + cp+rp a,. VI + rp+(P + 1;) + rp+ A~)rp

• (rpa,.rp+-'V1

VI

: ),

+

(6.10)

rp+rp

where (G.ll)

The existence of expressions

1 and 1 VI + rprp+ VI + rp+rp plicated non-linear interaction terms into the theory.

in (6.10) will bring com-

REFERENCES

[1] [ 2] [ 3] [ 4] [ 5] [6] [ 7] [ 8]

Weinberg, S.: Phys. Rev. Letters, V19 (196;), 1264; Salam, A.: Ele11lentary Particle Tlleary (cd. N. S'I'"nrtholm.) (1968). For instance, Abara, E. S. & Lee, B. W.: Gauge Theories, PlIys. Reports, V9c (1973). Baton & Laurens.: N'UOZ. Phys., B3 (1967), 349; Malamud, E. & Schein, P.: Proc. of Argo'IIIM Ocmference (1969), 108; Deinet, W. et a1.: Pllys. Letters, 30B (1969), 359. Lee, B. W.: Ohiral Dynwmic8, Gordon and Brench. Weinberg, S.: Phys. Rev. Letters, V18 (1967), 188. Taylor, K.: Phys. Rev., D3 (1971), 1846. Coleman, S., Wess, J. & Zumino, B.: Phys. Ret'., 177 (1969), 223£1. Goldstone, J.: )t1tOVO Oimento, 19 (1901), 15.

158 SCIENTIA

Vol. XXII No. 3

SINICA

March 1979

SOLITON-SOLITON SCATTERING AND THE SEMI-CLASSICAL APPROXIMATION FOR THE PROBLEM OF SCATTERING IN THREE-DIMENSIONAL SPACE CHOU KUAN-CHAO

(Ji!ilJ'tB),

AND DAr YUAN-BEN

(~j[;*)

(Institute of Theoretical Physics, Academia Sinica) Received March 20, 1978.

ABSTRACT

The soliton-soliton scattering amplitude in three-dimensional space is obtained in the semi·classical approximation.

The results obtained are also applicable to the problem of the

scattering of ordinary particles und generalize the previous results in this respect.

1.

INTRODUCTION

The quantum theory of the soliton has attracted much attention in particle physics in recent years. However, so far few researches have been carried 'Out for the problem of soliton-soliton scattering. For one-dimensional scattering, applying the W. K. B. approximation Jackiw and Woo[lJ obtained a relation between the phase-shift o and the time-delay .dol in the classical solution. In the W. K. B. approximation one has

2o(E)

=

lim %1-= %;--~

fxXI

J

dx' (V2rn(E - U(x') - V2mE),

I

where E and m are the energy and the mass of the particle respectively, and U is the potential acting on it. It then follows that,

(1)

where v(x', E) is the velocity of the particle with energy E at the point x', v(E) is the velocity of the free particle with the same energy. So far as we know, no concrete result for the three-dimensional scattering problem has been published in the literature. The meaning of the soliton-soliton scattering solution of the three-dimensional classical field equation in quantum theory remains to be clarified. The present article is devoted mainly to the problem of the soliton-soliton scattering in three-dimensional space. In Sec. 2 the needed formulas of the canonical quantization for the problem of the scattering of solitons are described and the general form

159 282

SCIENTIA SINICA

Vol. XXII

of the scattering amplitude with the neglect of ·the effect of the exchange of mesons is given. In Sec. 3 the three-dimensional soliton-soliton scattering amplitude is obtain. ed in the semi-classical approximation. There are two relations between the phase of the scattering amplitude and the dynamical quantities of the corresponding classical orbit, which are generalizations of (1) in the case of th~ee-dimensional space. Results obtained there elucidate the meaning of the classical scattering solution in quantum theory. The formulas obtained in this section are also applicable to the scattering of ordinary particles. Since the assumption of central symmetry of force is not needed in the derivation, the main result is more general than that in previous semi-classical researches. In Appendix B it is shown that for the scattering in one dimension a generalized form of the formula (1) can be derived without appealing to the W.K.B. approximation.

II. THE

AMPLITUDE OF THE SOLITON-SOLITON SCATTERING

Suppose that there is an N component sl'lllar field rp (rp., i = 1, 2,. .. N) and the classical equation of motion of which,

Dcp -

V'(rp)

=

0,

has solutions rpCI(X -Z(t), z(t» corresponding to the soliton-soliton scattering, in which X, - X" where X, and x. are the coordinates of the centers of the two solitons. Suppose that

Z = Ya (Xl + X.), z =

as t

-'>' -

00,

and that fPc/(x - Z(t), z(t»

as t

-4

+ 00,

-'>'

cpc/(x - X';(t)

+

fPc/(x - .K;(t»

(2)

in which fPcl(X - x(t)) is a single-soliton solution and

+ Uj t, X;(O) + u; t.

X,(t) = X,(O)

X;(t) =

(i = 1,2)

(3)

'" According to the scheme of canonical quantization E.] one introduces operators Z(t) and Z(t) corresponding to respectively with z(t) and Z(t) with th and 1£0 considered as parameters. Quantized field operators cp(x, t) can be expanded as cp(x, t)

= fPc/(x - tet), Jet)) + ~ §.(t) c/J,,(x - t(t), £(t),

(4)

" where §,,(t) terms describe the production and the annihilation of the "mesons". In this article we shall neglect these terms corresponding to the excitation of the meson freedom. Substituting cp(x, t) = rpc/(x - t(t), J(t» into the expressions

160 No. II

SOLITON·SOLITON SCATTERING

H =

28B

Jd'x [1-2 (Ocpy + 1- ~ (ocp)l + V(cp)] at 2; ox;

of the Lagrangian and the Hiuniltonian of the field , one finds that " H

where

P;

1

A

A

2

(5)

2

and Pi are canonical momenta conjugate to Zi and Zi respectively,

v= Jd x[V(cpc/(x 3

+ ~ (o;cpc/(X -

i(t), z(t)))

i(t), z(t)))Z],

;

M;; = Jd x [a~;

CPcl x - 2(t), z(t))]

[a~j

m;j = •f d~x [aaZ;

CPclx - 2(t), z(t))]

[-.L CPc/(X OZ;

3

1I1;;(Z) ~constant,

as

1

= - P;Mi/(z) P; + - P;'Yni/(z)P; + U(z),

JI;j(z)

CPc/(X - 2(t), z(t))

~constant,

J,

2(t), z(t))] ,

(6)

Fo

(7)

U(z)

-+

Izl- 00.

Formally, the formula (5) is not relativistic. Nevertheless, with V, M;; and m;; depending un the parameters UI and Uo, it can be shown that, after taking matrix element the relativistic formulas of the momentum and the energy can be satisfied with appropriately chosen UI and Uo, similar to that discussed in [2]. Now we turn to consider the quantum-mechanical scattering process of two solitons. Let PI, p. and P.', P.' be the momenta of the two solitons in the initial state and the final state respectively. Let IZ, z, t). denote the eigenstate of the operators 2(t) and z(t) in the Heisenberg representation and IZ, z, 0) == IZ, z). Apart from a possible phase-factor irreleyant to the final results, Lfi.t

IZ, z, t) = on

IZ, z).

Therefore, the propagator is

(Z ' , Z "IZ , t , z, t) -- (Z' ,z'1 e-~B("-')

IZ, Z ) •

(8)

Let /PI, P. t) denote the eigenstate of the operators PI (t), Pl(t) in the representation correspunding to the eigem-alue PI and Pl and IPI , P l , 0)

Heisenberg

== IP" P l ) . (9)

.1"

where ellis a phase-factor. In the problem of the scattering process, t - ± oc. It then follows from (7) that d .... ·i ." .... -p·(t) = - [H p.] - 0 dt fi' I

so

IP p" I,

t) can differ from

I

IP P" t + .M) l,

i ,

Iz(t) 1- 00

as

= 1,2.

at most by a phase-factor and can,

161 Vol.

SCmNTIA SINICA

284

therefore, be chosen to be independent of t when t ..... ± that one should take .f.~(p .• ,)

6~

•

00.

xxn

It can be seen from (9)

-{E(P.)I

=

6

B

(10)

•

It follows from (8)-(10) that the scattering matrix element can be written as

exp.i (P Z - P'

o

0

0

Ii

o

Z'

+p

0

Z - p'

Z') exp.i. (E't' tz

0

EO

(Z', z'lexp.i [bet - t')] IZ, z) d3Z' d3Z d 3z' d3z,

(11)

tz

where P, p and P', p' are the total momentum and the relative momentum of the initial state and the final state respectivel.", E and E' are the total energy of the initial state and the final state. Let

K(Z', Z'j Z, Zj t' - t)

':[P,ll] =

== (Z', z'lexp.i Ii

(12)

0

:. -4-"'§' [f(Z', z' j Z, Zj t' 't

[bet - t')] IZ, z).

az;

= (Z',z'lexp

!

t)

= (Z', z'l P;expi.. [H(t Ii

[H(t-t')]P;IZ,z)=-

-

n] IZ, =)

~ a~/i:(Z"Z'jZ,=;t'-t)o

+ Z.

Making

Z)] d P"'

(13)

The above equality shows that K(Z', Z' j Z, Z j t' - t) is independent of Z' the Fourier transformation

K(Z', z' j Z, Zj t' - t)

=

_1_

rK(P",' " z' zo t' -

(2nli)3 J

t) exp r.iP"

0

Ii

(Z' -

3

and substituting it into (11), 'One finds that lim (P;, P;, t'IP" Pzt>

= lJ3(P _ p,)_I_ rr exp.i (p z 0

(2nh)3JJ

o

Ii

p'

0

z'

+ E't' -

Et)

(14)

[{(Pj z', Zj t' - t) d3z d3z'o

It can be shown from (26) that K(Z', Z' j Z, Zj t' - t) satisfies the Schrodinger equation

lii~K(Z' z'o Z z· t' - t) at' ""

= [-

-2.,

liZ Mil 2 az;

~ - ~~miil-2, + U(Z')]K(Z', z' j Z, Zj t' aZI 2 az; aZI

t),

(15)

162 No. S

285

SOLITON·SOLITON SCATTERING

and the initial condition K(Z', Z': Z, z: 0)

=

(16)

lJ3(Z - Z')lJ3(Z - z').

Substituting (13) into (15) and (16) one finds that K(P: Z', Z; .t' - t) satisfies the equation of motion -

.!.~K(P· z' , z· t' i at' "

t)

= Hff(P z' z· t' e , z') K(P·7"

t) ,

(17)

3:. -E, mijl(z') .2, + U(P, z'),

HOff(P, z') = -

2 az;

U(P, z') = U(z')

az;

+ 1. P;Mijl(Z')Pj ,

(18)

2

and the initial condition Ii.(P; z', z: 0) = lJ3(Z - z').

(19)

It follows from (Ii) and (19) that

=
K(P: z', Z: t' - t)

III.

Ii

(20)

SEMI-CLASSICAL ApPROXDI:ATION

Now we proceed to solve the equation (17) to the first order term in Ii with the semi-classical approximation. Let us try the solution of the following form: K(P: z', z; t' - t) = At(z', z, t' - t)exp

[! Sc/(z', Z: t' - t)),

(21)

where t does not depend on 1L and S.. is thE' action of the classical orbit which is expressed by

Loff

=P

•

.

Z -

Hoff

="21 Z;mij(z) Zj -

-

U(P, z).

Comparing the coefficients of AD one obtains the equation

1..

7: 1(z')

2 m'l

as
+ U(P

,z

I

')

+

aSci = O. at'

(22)

Since this is a classical Hamilton-Jacobi equation, Equation (17) is satisfied to order AD. Comparing the coefficient of Ii. one obtains the equation -

at asci at at' = mijl(z') az; az;

The above equation can be written as

1 [

+ 2"

azscI mi?(z') az;azj

+

ami/ asci J az; az; t·

(23)

163 286

SCIENTIA SINICA

Vol. XXII

a (v;fz) _- 0, at' + az:

.2..f...

where v is the classical velocity of the particle. Written in this form the equation is. the same as that in the corresponding semi-classical formula for the scattering of ordinary particles. For the problem of ordinary particles, lll;j is a constant diagonal matrix. In this. case, it is already known that Eq. (23) has the solution t31 :

f =

IlaZS
(24)

j

In Appendix A it is verified that (24) remains to be the solution of (23) in the general case. The constant A in the formula (21) can be determined by means of the condition (19). When

E=

t' - t is small, Sci ~ ~(z' - z), mij(z' - z)j. From (19) one obtains. 26

Therefore,

Taking account of the fact that there llllly be more than one classical orbits connecting the points z and z', one obtains finally

K(P; z', Z; t' - t) =

1.

3/

(2nlif,) z

~ II azsc/, lit exp [~Sd(Z"

az;az

Ii

j

Z; t' -

t)].

(:!5)

Substituting (25) into (14) one finds that

. PZ, t >= H.r:!-.. "-+.,

- Et

+ E't')

I az;az; lit aZS
(_.i) f

exp

(P

3

(2n1iY/2 8

-

i... S,/(z', z; Ii

P') JJ ff expi -j; (p' t' - t)d 3zd3z'.

Z -

"

P •

Z

(26)

We now apply the saddle point approximation to carry out the integration with respect to z and z'. Integrating first with respect to z, one obtains a factor,

. -aZS
Itt

II az;az;·

i -, exp -':';c/(z,p; t , - t), Ii

(27)

wpere (28)

(29)

164 No. S

281

SOLITON-SOLITON SCATTERING

Since the relations (28) and (29) constitute a Legendre transformation, one has

88c/(z', p; f' - t) -_ 8p/c

D( z,p,.~'~) - •.

(30)

Zk

On integrating with respect to z', account must be taken of the condition of conseryation of energy of the classical solution. It follows from (28) and (29) that

(31) Recalling that m;; --+ constant, fj --+ oonstant, as t_-oo and t'--++oo, one obtains from the Hamilton-Jacobi equation and (29), (31) that (32) Choosing the third coordinate axis of z' to be parallel to the direction of p', one may determine the saddle point of the integral with respect to z,' and z,' by the following conditions:

t) =

8S
88,I(Z', p; f' - t) = 0

(33)

8~'

which fixes the direction of Pel' to be parallel to the third axis. From (32)

88 d (z', p; t' - t) _ ~ , -po

(34)

uZ3

'raking deriYath-es of the above equality one obtains that

8 Z8c/(z', p; t' - t)

8z; 8z;

=0

i

,

=

1,2,3.

So the saddle point method is needed only for the integration with respect to z,' and zo'. In the neighbourhcod of the saddle point 8 .. (z', p; t' - t) is linear in za'. Therefore, after carrying out the integration with respect to za', one obtains, for the 8 matrix element, a factor 2;rlid(p - p') which guarantees the conseryation of energy. So one obtains that

Jexp ! [8
.I

=

- t) - p' z;]

(211:ni) (2nn) 8(p - p')

"818
1 8 8
. exp

8z i 8zj

!

t)I'I- f • II 8 8c/(z'; p; ~' 8Z.l.i 8z.l.j 2

t)

ILo . II 818c/t~';i~Z:' -

t)

of)

Itof

W

1

t)lr

f

]

0

z.

'0

'0

=, '''•

8
(35)

where

8
(36)

165 288

SCIENTIA SINICA

8Se/ (Z', p, t' - t)

8 zi'

Vol.

x:xn

, =Pi'

(37)

Substituting (27), (35) into (26), collecting all the factors and using the Legendre transformation relations (28), (2~), (36) and (37), one obtains z

P). =_i_. lJ3(P-P')8( _ ')""[118 Se/(Z',Z;t''0.(P'I, P'ISIP 2 I, Z." ')" P .p LJ 8 ~ 8 . _1m Z1 Z.

O llt .

where S
= Se/(%', Z; f -

t)

+

p . Z - p' . oz'

+

E(t' - t),

8Se/(z', Z; t' - t) _ ' -Pi' 8 ' Z;

8Se/CZ', Z; t' - t) _ - - Pi, 8z i 8Se/(%', %; t' - t) 8(t' - t)

= _

E.

·(39)

Since the above relations constitute a Legendre transformation, one has 8Sc/Cp', p, E)

8Pi'

=_

Z~ l'

8Se/(p', p, E) 8Pi

=

z. .,

8Sc/(p', p, E) 8E

=

t' - t.

(40)

In the formula (38) 1: denotes the summation of the contributions of all classical orbits with the initial momentum p and the final mementum p'. Now we calculate the product of the determinants in the formula (38), 2

Z

1== 118 8"(%',, Z; t' - t)II'118 8,/(%', %; t' 8z j ,8zi 8z i 8z j

= 8(pi, z~) • (_ 1) 8(zj, z~). 8(Zi' Zk)

8ePi, z/J

t)lr

8(Pil z~) 8(Pi, p~j, z;)

l

.11 8z8e/(~" p, ~' 8Z1.i 8Z1.i

0lr

l

8(pi, z~)

= - 8(Pi, p~;, z;)

= _ 8(p;, z;, z;) 8(PI, Pz, P3) . Let PI denote the three components of the vector p in the direction of the coordinate fiXE'S e/(i = 1, 2, 3) of z', then

BCPI, Pz, Pl) = 1 BCPI' Pz, P3) ,

166 No. S

289

SOLITON·SOLITON SCATTERING

:. 1=- 8(p;,

z;, z;) .

(.n)

8(PI, Pz,~) Let () denote the scattering angle and b denote the collision parameter as t' -+ can be seen from Fig. 1 that

e,

PI

b cos cP,

z;

P3 = P cos

z; =

= - p sin e cos CPp, = b sin cp,

00.

P2 = - psine sin CPp, p;

=

It

(,l:?)

(3)

p.

Classical orOi t

", '" Fig. 1

Therefore,

1=- 8Cp;, z;, z;) = _ 8(z;, z;, p;) BCcose, cp,. 'li) = _ 1 8(z;, z;) 8(PI, Pz, P3) B(cose, CPp, p) 8(PI, Pz, P3) p2 8Ccose, cp,)

(-:1:4)

Substituting the formula (44) into the formula (38), one obtains finally that

]t

. (P' P'ISIP .P). = __1_lJ3(p_ P')lJ(E _ E')l.. , , [ 8Cz;, z;) I, 2 1. Z In ?_'1
,.

(45) Formula (45) is the general formula in the semi-classical approximation. It is also applicable to the scattering of ordinary particles. If the classical orbits satisfy the condition cp,

=

cP,

~=O 8cp ,

(,lli)

(45) can be reduced to

. exp i.. S
When there is only one classical orbit having contribution, it follows from (47) that

167 Vol. XXII

SCIENTIA SINICA

290

=bl~l. aeose .

du

dQ

(48)

which is just the well-known formula of thE' scattering cross-section of classical particles in a centrally symmetric force field. We now turn to discuss the phase-factor in (45). In the ease in which rotational invariance holds, one can write S.,(p', p, E) = S,,(p', p, E, 1"), where 1" = (p' - p)' is the momentum transfer squared. Noting that p', p and E are considered as independent yariables in (40), one finds that _

aSe/(p', p, E) _ (Pi ~_ api p' ap'

~)

'7.'

=

z.

= aScI(P', p, E) = (Pi ~ -2 (p'. - p.)~) B (p' P E 7:')

.-i

api

I

t' _ t

=

+ - (Pi C)

'_

pap

Pi) a7:'

, I

a7:'

,

Se/(p, p, E, 7:'),

cI

"

,

,

ascl(p', p. E) aE .

(49)

In the center of mass system, z'

=

'l ' ( .6. . + '--p t' - ~) . E :2.

%.1

+

z = z., ..

"!.p E

(t + !l
(50)

"'here Lt, is the time-dela~-, llamel~-, the difference of the time elapsed for the classical particle trayelling along the orbit and the time for the free particle with the same energy llloying along the asymptotics of the orbit. Taking into c·onsideration the relation (p' p) 'p = p'. (p ± p'), it can be found from (49) and (50) that

=

aSri ') - -..,, - ( Pi, - Pi ) (as api api

= -2(p' (p;

+

Pi)

p) •

(aS~1 + api

=

2p'· (p

E

+

= -1 (p'

~

%A -

aSel) api

rI ) as- p ) . p ,(asci, - - , -t- ap ap

p

=

(p' -

p) . p'

p' (p

+

p

p') (_ aSri aE

+

(~~l

p') (aSel ap

-

t::.el),

+

aSel) ap'

+ -:tT_ aSci f

or:

t::. ). cI

From these two fllrmulas it follows that aSel a7:'

= _

aS el aE

+ ~ (asci + 2p

(p' - p) . 27:' ap'

(51)

%A

(52)

aBel) = t::.-1. ap .

Writing S.,(p', P, E, 1") as SoI(E, 1") by means of the relations p

= p' = )

~z -

M

Z

168 No.3

SOLITON·SOLITON SCATTERING

291

in the center of mass system, (52) is reduced to (53) which is the generalization of formula (1) in the case of three dimensions. IV.

DISCUSSION

For the scattering of ordinary particles, the formula (47) was obtained previously in the case of centrally symmetric foree field[41. It was obtained in [4] after a series of approximations with the W. K. B. approximation to the partial wave amplitude as the starting point. Since the conservation of angular momentum is used in the derivation, it is not applicable to the general case of non-centrally-symmetric force field. It should be noticed that in the derivation of the formula (45) from (11) in the present article the assumption of centrally symmetric force field is not needed. In so doing, Formula (45) is more general than the pre,·iotls results, and the approximations used in the deriyation are simpler than those used in [4]. The results obtained in this article is applicable only to the case in which there is at. least one classical orbit with the initial momentum p and the final momentum p'. If 110 such classical orbit exists, one must use other methods, for example, the method of complex path. The factor 8(z;,z;)/8(costl, C{Jp) in Formula (45) has, in general, singular lines or singular surfaces. If the classical orbit passes these singular points, there will be an additional phase-factor exp i(mr/2) in the scattering amplitude as has been discussed in previous works. In the present article the effect of the exchange of "mesons" on the scattering ·of solitons is neglected. The exchange of "mesons" also causes the fluctuation of the solitons around the classical orbit and can in general contribute to the scattering amplitude in the order of approximation taken. This problem needs further investigation. In the adiabatic approximation its effect can be represented by an additional term of the order A in the potential U. The soliton in the three-dimensional space usually has an internal quantum number, for example, the charge. Introducing the collective coordinate of the U(I) group and the corresponding canonical momentum, it is not difficult to generalize the results obtained above to this case. For the soliton-soliton scattering without exchanging the charge the same formula (45) is obtained in this case.

Appendix A Taking derivatiyes of the Hamilton-Jacobi equation (22) one finds that

(A.I)

169 292

SCIENTIA SINICA

Vol. XXII

Let (A.2) then (A.1) can be written as

..,. sk/ == (a VI:

at

aS a) ;'k/ '"' =

+

d -I mij --, -, i j

-

aZ aZ

Tki S

(A.3)

i/'

From (24)

With (A.3) and (A.2) the above equation can be reduced to (AA)

This is just Eq. (23).

Appendix B Let there be classical soliton-soliton scattl'ring" sulutions q:'el(x - z(t), z(t)) one-dimensional space. In the system of Yilnishing total momentum, 'Pel ( x

_ Z(t)

, Z(t))

-+-

'Pel

(

x - X1(t) ) J 1 - ul

+

X;(t) ) , '\/ 1 - I£"

+

'Pel ( x - Z() t, z ( t )) -+- 'Pel ( x /-

'Pel

(

x/

'\/

CfJd

(

J.: -

/

'\/

TzCt)) '

as t

111

-+- -00,

1 - ul

X~(t) ) , 1 - I,l

as t-+

+00,

where

XiCO =

X iA ±

1£ (t + Ad), 2

x;Ct)

=

X iA

±

1£ ( t -

~d).

(B.1)

According to the scheme of canonical quantization of solitons as described in Sec. 2, after neglecting the terms corresponding to the excitation of the meson freedom, the field operator rjJ(x, t) can be expressed as rjJ(x, t)

=

'Pd(X -

2(t), z(t)).

(B.~)

On the analogy of Formula (5), in the case of one-dimensional space one finds (B.3)

where P and p are the canonical momenta corresponding to Z and z respectively. In the center of mass system, the term. containing P can be dropped. get) and {J(t) satisfy

170 No.

a

SOLITON-SOLITON SCATTERING

293

the quantum-mechanical Hamiltonian equations of motion, '" Z

When

1%1-00,

= I

H, Z],

'["

A

p = i[H, pl.

(BA)

m(z)-rn·o, U(z)-Fo,

(B.5) As is usually done in theories of scattering, one can introduce "in" field, "out" field and the eorresponding S operator with the relation

(B.6) £i"(t), ZOUI(t) and tlw eanonical momenta pin, pOu, conjugate to them satisfy equations

pOU' =i[H~U', pOU']

= o.

(Ri)

+

(B.S)

From th(·Ne· t'quatiol1s and (B. 5) we goet

iin(t)

= iin(o) + pin t,

zOut(t) = £OUI(O)

From tIll' conseryation of

pOU't. 1110

1110

ener~', L~+HA

ins _ HA in _ HAOut

00-0-0,

Corresponding to the classical solution (B. 1), we postulate that

(B.9) This equalit.y is equh-alent t.o neglecting the quantum-mechanical refleetive W
So ZOU'(t) can be 'Hitten as

ZO"'(t)

=

zin(t) _

pin A(pin). rna

(B.IO)

Comparing this formula with (B. S) it can be seen that Ll( p;n) is the time-delay operator. It follows from (B. 9) that S commutes with .p'n and, therefore, is a function of i/" only. Taking into account the unitarity of the S operator, S can be written as

S=e It follows from (B. 10) and (B.ll) that

Zi6(pin)

(B.ll)

171 SCIENTIA SINICA

294

Vol. XXII

(B.12) Sandwiching this equality between the eigellstates of the momentum, one obtains that d 2 -B(E) = ~(E). dE ~(pi")

(B.13)

is determined by the quantum-mechanical Hamiltonian equations of motions (B. 4). If the semi-classical approximation is used, .d(E) =::: .dol(E), where .dol(E), is determined by the classical Hamiltonian equat.ion~ of motion. Eq. (B. 13) then becomes identical to (1).

REFEREXCES

I: I] [2] [ 3] [4]

,Jaekiw, R. & Woo, G.: Phi/S. Bev., D12 (197;),1643. Christ, X. H. & Lee, T. D.: PlIys. Bev., D12 (1975), 1606. Gutzwiller, :M. C.: J. Matll. Phys., 8 (1967), 1979. Ford, K W. & Wheeler, J. A.: .:1,/1/.. PlINN., 7 (1!l59), 28;: Berr~', ::\1. V. & Mo;unt, K. E.: Rep. Prog. Ph !IS., 35 (1972).315; Knoll, .r. & Shaffl'I", R.: All". Phys .. 9i (1976), 307.

172

SCIENTIA SINICA

Vol. XXIII No. 1

January 1980

THE NON-TOPOLOGICAL SOLITON WITH A NON-ABELIAN INTERNAL SYMMETRY ZHOU GUANGZHAO

(CHou

ZHU

(CHU

ZHONGYUAN

DAI ¥UANBEN

(~j[;*)

KUAN-CHAO

NilJ't-ej),

CHUNG-YUAN

*:Sz),

AND

Wu

YONGSHI

(~i7kJl;j')

(Institute of Tlleoretical Physics. ~emia Sinica) Received September 9. 1978.

ABSTRACT

The non-topological soliton with a non-Abelian internal symmetry in 3 + 1 dimensional spacetime :is examined. The ease of S1J(2) internal symmetry is diseussed in detail. Existenee and stability of the classical single-solitonl solution are investigated with the example of a concrete model. The quantization of the s:ing~liton solution is carried out with the method of colleetive coordinates. It :is pointed out that the quantized soliton may possess. :in addition to ordinary isotopie-spin quantum numbers oF and .h. a new quantum number ,,;. A new Lorentz covariant method for the qUllIltization of the moving soliton :is proposed.

r.

INTRODUCTION

There are soliton solutions in many nonlinear field theories. Physically, they represent extended objects. The possibility of using these soliton solutions to describe hadrons has been explored in many literatures[lJ. Early investigations were concentrated on 1 + 1 dimensional nonlinear scalar field theories. In several models of this type, static soliton solutions can be found -out in analytical form. In physical 3 + 1 dimensional space-time, according to Derrick's theorem, there is no static solution for a system of scalar fields. However, if the system of nonlinear scalar fields has an internal symmetry G, it is possible that soliton solutions of the form exp {ia;(t)TJ • cp.(x) exist, where T;'s are generators of the group G in the representation to which the scalar field cp belong and cJ,(t) are group parameters which are nevertheless timedependent. The case of an Abelian group G was investigated carefully by Friedberger, Lee and Sirlin12J. The case of a non-Abelian group G is more complicated. In this article we shall examine the soliton of the system of scalar fields with a non-Abelian internal symmetry, talcing the iso-spin SU(2) group as an example. In Sec. II, equations determining the classical soliton solution and the general form of these classical solutions are discussed. The cases of 1 = 1/2 and 1 = 1 representations of the group SU(2) are discussed in detail. In Sec. ill, a G = SU(2) model analogous to the U(l) model discussed in [2] is -considered as an example. Existence of classical single-soliton solutions is proved and stability of these solutions is discussed for this model. It is shown that only sol:utions in which cp;(z) has merely components corresponding to 13 -= ± 1_", are stable.

173 No.1

NONTOPOLOGICAL SOLITON WITH NONABELIAN INTERNAL SYM},IETRY

41

In Sec. IV, the collective coordinate method {taking group parameters il;(t) as collective coordinates) proposed by Christ and Lee[3] is used to quantize the classical single-soliton solution. Consistency of the perturbation theory starting from the semi-classical approximation is discussed. A new treatment of the problem of Lorentz covariance for the moving soliton is proposed. It is pointed out that in the semiclassical approximation the collective motion of the soliton in the iso-spin space in a theory of iso-spinor

GENERAL DISCUSSION OF CLASSICAL SOLITON SOLUTIONS

1.

General Formulatian

Let us consider a system of scalar field cp(x) which i~ a representation of the group G. Generally speaking, the representation may be reducible. Under the operation of the element g of the group G, cp(x) - cp'(x) !p+(x) - rp'+(x)

= U(g)cp(x) == gcp(z), = !p+(X)U-I(g) == CP+(X)g-l,

(1)

where U(g) is the unitary matrix representation of g. In the following, U(g) will be denoted simply by g. The LagI'angian of the system is (2)

where the nonlinear potential Y(ep, cp+) is invariant under the group G. We now may seek the classical soliton solution of the form cp(x)

= g(t)cp«x),

(3)

where get) == g(a;(t)) in which a;'s depend only on t, while !p«x) depends only on the Space-coordinates x. Substituting (3) into the Euler-Lagrange equation,

a,. a,. cp(x) == ayealp+' p, p +)

(4)

corresponding to the Lagrangian (2) and multiplying the both sides of the equation by g-I(t) , one obtains the equation,

l iQ(t) - (IJ(t)y]cp«x) = 'i1 2cp«x) - _~r~p<: p:), cp,

Where

!J(t)

(5)

== dQ(t) dt '

(6)

174 42

SCIENTI.!. SINIC.!.

Vol. XXIII

The requirement that Eq. (5) has solutions, leads to certain conditions satisfied 'by get). These conditions can also be derived from the condition that the solution (3) should satisfy the principle of a least action. Substituting (3) into (2) one finds (7)

where (8)

(9)

Noting that, from the inval'iance of the potential 17 under G, we have

aj(t)'s are considered as generalized coordinates whose corresponding canonical moment.'t are (10)

III order to see the physical meaning of P';), we may write

ay

- - = t'1 ),;g,

(11)

aa),;

where h is the generator corresponding to the group parameter dlfficult to prove that when rp(x) takes the form of (3),

P(c) = _ i 10

rd x

J

3

{rp+(X)I/c apex) -

at

a/.:.

aq;+(x) I/ccp(rr.)}.

at

It is then not

(12)

Therefore, P';) is nothing but the component of the conserved isotopic-spin corresponding to the generator 1/c. The Hamiltonian corresponding to (7) is (13) Eyidently, H(c) is just the energy E. when cp(x) takes the form of (3), and then (14)

Noting that r(IPe, tIle form,

Thi'l is the necessary conditioll for y(t)IPe(X) t.o be a solution of the equation of motion.

175 No.1

NONTOPOLOGICAL SOLITON WITH NONABELIAN INTERNAL SYMMETRY

43

2. The Case oj G = SU(2), I - 1/2 For definiteness, consider the case of G = SU(2) with

_ 1 I. - -u•. 2

Using the equality a-Ida

= i{[ eosel3 + sine(cos
+

+

(-sin
and the anticommutation relation for I

=

cos
sin
+

(11)

13d
1/2 representation, (1S)

we ean easilr find from (8) t.hat J[~~

M~l

=

i1l~J

=

= JIlocosa,

J.lI~l

= .1I lo ,

.M~J

=

JI~J

on)

= 0,

where

(20) Therefore the Lagrangian (7) can be written as

which is formally the same as the Lagrangian of a spherically symmetric top. generalized momenta (10) corresponding to rp(t), aCt) and
=

]llo(rjJ

+

cbcosa),

p~)

=

p~)

Mli),

= .i.lflo(cb + cpcosa),

The (22)

respectively. The Hamiltonian corresponding to (21) is

H") = _1_ [ped ?

M

:"1' 10

9

+ _1_ (ped + • 2a '" SIn

pee)' - 2P!<)pe
'"

oJ-

+

Jd ,xV(m 3

T<1

rp+). <

Recallin~ that the ordinary isotopic-spin components of the field rp(:z;) are

we can readily verify that

(23)

176 Vol. XXIII

SCIENTIA SI:-l"ICA

Introducing the moving frame in the L'!IO-Spin space in connection with Euler anglf's (rp, 0,
In the present case, Eq. (12) is reduced to

e+ rprb sin f) = 0, q; + J,

J,cosf) -

orb sine =

0,

+ q;cosf) -

Ocpsine =

o.

(~8)

Substituting (17) into (6) one finds the expression of Q(t) as 3

Q(t)

=

~ w;l;,

(29)

;=1

where WI

= - 8sinrp + cpsinecos
W2

= 8cosc/J + cpsinf)sin
W3

= rb + cp cosO~

(30)

The Euler equation, Q(t) = 0 of the spherically symmetric top can be easiiy derived from Eq. (28). With this result, Eq. (5) is reduced to the form

[Vl _ 8V(~:(p,) + Wl] rp,(x) = 8(rpc rp.) 4

0,

(31)

3

where wl = ~

w1 is constant.

;=1

If rp(x) is the direct-sum of I = 1/2 and 1=0 representations, similar results can be obtained for the 1=1/2 part of rp(x). In the next section we shall prove that Eq. (31) has soliton solutions in a concrete model.

3.

The Case of G = SU(2), 1=1

Let us consider the real field belonging to the I = 1 representation of G = SU(2) , in which the components of rp(x) in the rectangular basis in the iso-spin space are real. As an Ansatz, rp.(x) in (3) is r.equired to have the form (cf. the next paragraph), (32)

where 9'0 is a constant iso-vector.

Suitably choosing the basis in the iso-spin space (or

177 No. 1

NONTOPOLOGICAL SOLITON WITH NON ABELIAN INTER X AL

SY~DrETRY

4;;

making suitable SU(2) transformations to "0) rpo can be brought int.o the form,
=

rpXx)

=

0,

(!'l3)

f/x).

For the real field, L

=

Jd xQ = Jd x 3

1 =?

-

where (

Mj~)

From (33),

Olle

3

u(.).·

LUjkaja" -

[- ;

op.rp· o!.rp -

nip)]

r d'·x.'f"(CP•.) ,

.I

(:1·0

or;-I or; '). = -1 rI d 3xlp.(x) (0,,-1 00 + - __ Ip/X), _J_ --"-

2 •

,oa,

oale

Oak

O($j

.rinds that (:37)

Using (17) we can easily obtain that

MW=O

otherwise,

(:~i')

where Ci!J)

'l'herefore, (40)

(.n)

where (42)

Let 5.(a = 1, 2, 3) be ordinary components of the isotopic-spin of the tp fi!'ld, it can be easily proved that

It folIo,vs from (37) that P~)

== 5; =

0 in thi'! case.

In the present case Eq. (15) is reduced to

e- cjllsinecose = 0, E.. (cjl sin2e) = o. dt

(44)

178 46

Vol. XXIII

SCIENTIA SINICA

Starting from (44) one can pro'\lle that WI (t)

wlt)

= - W2(t)W3(t), = WI (t)wlt),

(45)

where Wi are defined by (31) and (32). On the other hand, (44) can be derived from (45) independent of the value of W3(t). The reason is that, because of (37), t/J(t) in (16) is not a real d:y-namica.l variable of the system, so W3(t) can take any value without changing rp(t) and B(t) in Eq. (44). Without any loss of generality we can take wlt) = O. Therefore (45) is reduced to Q(t) = 0 and Eq. (45) is reduced to (46)

where £02 is a constant. soliton solutions.

For the model considered in the next section this equation has

Xote that in the present case that is different from the I of the soliton in the iso-spin space is a plane rotator. 4.

= 1/2 case, the motion

Disc1lssians of the Case of High-dimensianal Representatians

With the case of high-dimensional representations, the general discussion is more complicated. However, the discussion can be simplified in cert.."l.in special cases. Recall that in the hvo cases discussed above we have .Q(t) = O. Generally speaking, we can only ascertain that (iQ(t) - (Q(t»)Z)rp«x) on the left-hand side of (5) is independent of t. If we require furtller that Ui(t) - (Q(t»)Z

= constant matrix,

it can be derived from the Hermitian property of Q(t) that Q(t) finds get) = g(O) exp (iQt),

(47) =

O. From (6) one (48)

where g(O) is independent of t. Therefore, Eq. (5) is reduced to (49) Note that the solution rp(x) = g(t)rp«x) is an invariant under the transformation get)

-+

g(t)gol,

(50)

where go is independent of t. But under (50) Q -+ g.,Qgol. Therefore we can transform Q to a diagonal matrix by go suitably chosen. Denoting the componentB of rp($) in an irreducible representation of BU(2) by rpi and diagonal elementB of Q by coi(i= 1,2, "', N), Eq. (49) takes the form (51)

179 No.1

NONTOPOLOGICAL SOLITON WITH NONABELIAN INTERNAL SY::\UlETRY

47

in which some cp~(x) may be zero. H there are M indices i(J.ll ~ N) for which cp~(x) ..,:.. 0, and c.o~ are not all equal, the system of Eq. (51) L'I difficult to be satisfied. Even if it can be satisfied, the energy of the corresponding soliton solution is expected to be not the lowest. In the case of equal £01'.'1, nonzero cp~(x)'s satisfy the same equation in (51). It is expected that for the lowest energy solution, nonzero cp~(x)'s are all proportional to the same function ,«x). In the model considered in the next section, the situation is indeed so. 'l'herefore, for the lowest-energy stable soliton solution, we shall confine ourselves to the case in which ,Ql c.o 21M' i. e. Ql is proportional to a unit matrix in the subspace in which cp!(x) -j- O. In this case, Eq. (51) is simplified to

=-

(5~)

For reasons stated above we shall consider only solutions of this equation of form cp~(x)

where cp~ is COllstant for i

= 1, :!,

=

f«x)r:p~,

... , JI and cp~

(53)

= 0 otherwis.·.

For 8U(2) group, in the representation in which 13 is diagonal (spherical basis), being di.1gonal, can only have the form ,Q = 001 3 • Because of the condition QZ c.o l lM' rp!(x) can onJy have two nonzero components corresponding to 13 = ± J 3i • In the following, we shall consider mainly classical soliton solutions of this form.

=-

,Q,

III.

PROPERTIES OF CL.-I.SSIC.\L SOLITON SOLt;TIO~S

Classical properties of non-topological solit~ns with all internal U(1) symmetry was discussed in detail in [1, 2] . In thL'! section we shall generalize them to the case of the iso-spin 8U(2) group. As an example, let us consider the following Lagrangian dt>llsity: (54)

1'2

= ~

gl(XZ - X:,y,

where cp(x) belongs to some representation of the 8U(2) group and X is a real scalar field. The equations of motion corresponding to (54) are

(56)

According to the discussions in the above section, we shall concern ourselves only with soliton solutions of the form, cp(x, t)

=

lex, t)

= l(x),

g(O)e+i01l"cpc(x),

(57)

180 48

Vol. XXIII

SCIENTIA. SINICA.

where 13 is diagonal. The isotopic-spin of tbis solution is

~

.. , 00(13; + 13/)e-i..crJl-r·j)'fd3XIp!(X)

J. =

X [a(0)-11.a(0) ]iiq>Kx),

where 13; is the eigenvalue of 13 corresponding to 1p~(X). Since 5., being conserved, must be independent of t, the above equation can be reduced to

j d3XIp:i(X)[g(OyI1.u(0)]iiq>~(X).

J. = ~ 2001 3; From the relation

rr(0)1.u(O)

= ~

.,

R..,]., ,

where R is a matrix of rotation determined by g(O), and the fact that only nonzero diagonal element'!! are those of 13 , the equation of J. can be rewritten as

J.

~

.

=

2oonR. 3

j d3XIp:j(X)Ip~(x).

Define the total isotopic-spin as

J

== .jJ.Ja = ~

200n

i

which i'!! evidently independent of yeO). (57) is

E

=

f

d3x {VIp:(x) • Vlp,(x)

+

!

f d 3xlp:i(X)cp!(x),

(58)

-

The energy of the field corresponding to

(VX)Z

+

VI

+

V2

+ oozlp:(x)1~1p,(x)},

(59)

which is also indt>pendent or !leO). Substitnting (57) into (55), equations satisfied by Ip, and X nre found to be

{ - V2 + { _ VzX

+

a( a~, ") Ip, q>,

ooz

n} Ip!(x) =

ay, + aY2} = o. ax ax

0, (60)

It can be seen from (58)-(60) that solutions of the form (57) with different !l(0),8 havc the same energy, isotopic-spin and equations of motion. Therefore, we can consider only the solution with g(O)~l as the representative of these solutions. Furthermore, the energy, isotopic-spin and equations of motion are all invariant under arbitrary rotntions ill the subspace spIlnned by cp~(x) and q>;i(X). Therefore, we need only to di'lCUS.'!! the case in which q>!(x) = 0 for 13; < O. It is conveni('ut to introduce dimensionles.'!! quantities by .

cp,(x)

=

g!' R(p),

X(x) =-

!:... A(p), g

181 !'ici.l

NONTOPOLOGICAL SOLITON WITH

NONABELIA~

INTER);"AL SY1fMETRY

49

'",u'

V=-.

= IX",

'In

p =

!t=gX.,

,U%,

(58-00) can now be written as $

E= £

gl

Jd

3

p

= ~~~ ~ (l3pB+I~B,

(61)

g •

1Y} + 1.. ",$,

{VpB+VpB + 1..(V pAY + I(z.A1B+B +..!.. (..P -

2

lr -

V~ +

8

:!.K1B+B

+ 1.. (.1 2 2

J) 1.:1 J

=

2

(62)

O.

.

(68)

It can be easily proved tlUlt (63) can be obt:lined from tIlt' yariatioll of respect to A and B at fixed $. 1.

(6~)

"'ith

"Free" Solutions-SolutiolM Independent of Spflce Coordinates

Putting A(p) = const:mt and B(p) = constant inside the cubic box of the VOIWlU~ V in Eq. (63) the condition for non-vanishing B' ir; fOlmd to be

It follows thnt other eomponents must be zero. has

Denoting this solution by B(i), one

and

Evidently, corresponding to each 131 , there is a solution with the total isotopic-spin

and the energy

In the limit Y

-+

co, 8* • B

-+

0, A. -+ 1 and ",13,

-

m. we haye (li4)

This formula tells ns that for fixed .~ "free" solutions corresponding 1:0 llifferi'llt 13,'8 differ in their energies. The larger the 131 the lower the energy. The solution with the least energy has only one component corresponding to 1 3max different from zero.

182

50

SI~ICA

SCIENTIA

Vol. XXIII

Are these solutions all classically stable? '1'o.answer this question, one must calculate the second variation of the energy. It is easy t{) prove that

Jd plJ!+HIJ! + ~~ [J d p(IJ!+IiB + B+nlJ!) 3

3

(lllE}, = ;

r.

(65)

where

_1- Vl + K2B+B + H=

(

2

3iP -1 4

p

'

(lili)

2KzAB, qf=

(8A) 8B '

(67)

Substituting in(65) the "free" solution of A and B and taking the limit y--OO, we find that only the diagonal terms of H do not vanish. The eigenvalues of H are continuous spectra starting from 1- (3A.z - 1) and (I[Z.d. z - vZn) respecth·ely. If for ~ome solu4

tion BwCl =F 13max) we choose ...LO 8 Bill { ..,=0 llA.

thpn we have (llZE), =

i= i =F

for for

[3m.. , [3

m,.,

= 0,

.!:.. \. cl3p(8B(;»(vl n gZ.

v Z1i max)8B(l)

<

O.

This means the "free" solution with l =F 13 ma. is classically unstable. For the "free" solution with l = [3 max, H has no negative eigenvalue. Therefore thi.~ solution is classically stable.

2.

Ezistence of Classical Soliton Sol-ut·ions

In this section we shall prove that stable SU(2) soliton solutions exist. purpose, take the following trial solution: p ~ p.R,

0, A

=

For this

1

1 _ e::tp [_ p -

p.R],

p

~ "R,

~,a

p ~ p.R, p~",R,

(68) This trial solution satisfies the equation of motion for p ~ ~ and has the correct behaviour for p »p.R. The isotopic-spin and the energy of the trial solution (68) are .., ==

~ b~,I~b(l),

183 No.1

NONTOPOLOGICAL SOLITON WITH -NONABELIAN I"'TER'''L ... L> .,.... S Y MMETRY

E

=

005

+ n:~,~Z {R3 + 6g

• (RZ

+

Rd

+ ~

11 • RZd 4

+

89 RIJ} 24

+

635 d l 288

+

51

_6_ p.2d

d2 )}

(69)

respectively. For solutions of the form (68), the larger IJ is, the smallet· c.) will be. lt can be seen from (69) thnt for fixed 5 the solution with 131 = II max ha"l the lowest energy. For thi<J branch of trial solutions, we have

It can be proved by the method used in [2] that at least for (R » cl- 0«(,-1)) and

00

sl1fficientl~'

small

1 (' 4n:u ')~

5

-3/l!

--~., l l3 max _0

I

'

the energy of the trial solutiQn is lower than that or thE' "free" solution. Thi'l llieans, when the isotopic-spin of the field becomes larger than cel·taiu critical value, stable soliton solutions must exist.

3.

Soliton Sollltions for co Value ","ear to mllll

In the U(l) case, the energy of the soliton solution approaches that of tht' "free" solution us CJJ -+ m. For the SU('.!.) case considered in this !!ection, there is a series of "free" solutions with co -+ m1I 31 • There is also a series of corresponding soliton solutions. If we solve the equation in the approximation CJJ -+ m/ III and substitute it inm the expressions of the isotopic-spin and the energy; we find 5 -+ 00 and

(70) as CJJ -,. rnII3/' where illz > O. It can be seen by comparing this formula with (64) that there is indeed a series of soliton solutions whose energy approaches to that of the corresponding "free" solution from above as

4.

CiJ

-+

StabiUty of Soliton

mlI31 and 5 -+

00.

Sol'ution.~

It can be easily shown that soliton solll.tions with 13/ =F 13max are classically unstable. In fact, if we choose

, {lJl • Bin, lJBII) = 0,

lJA in (65), 'we can find

(lJlE}~.B(I)'A. =

;

fori for

=

l3 max,

i =r= l3 mos,

= 0,

f

d'plJB+(- V 2 + IPA2 - vlIDlJB

= !:!:.. I" d3plJB+(vln - vlI~ max)lJB < O. gl

J

184 52

SCIENTIA

SINH~A

Vol. XXIII

This proves the above conclusion. As to the branch of the soliton solution

B(I=l

3max

)

it has b~en shown that (i) thpl'e

is at least one solution in thi'! branch whose ~nergy becomes lowel' than that of the lowest "free" solution when ~ becomes hlrg~r than c:'l'mill erit.ieal value and 00. becomes sufficiently small, (ii) energies of soliton solutions in this bmnch approach to that of fre~ solutions from aboy" as w -+ .17 / 13 ma £ and ...57 -+ 00. 'l'herefol'C', the E-...57 ~urve must have the form as shown in Fig. 1.

It is very similar to the E-l) curve of U(1)12J. For ~ < ~S7 the energy of the lower branch of the soliton solution in the; figure is the absolute ....~..------!':------f minimum. Therefore it is stable. For the segJ, Illent CS, the energy of the soliton is higher than Fig. 1. E-5 curve. that of the "free" solution. However, just like the U(l) case discussed in [21, it is the local minimum and is thus still cIassienlly stable. In fact, Theorems 1 and 2 in [~J can b,> directl~- brought hC'rc~ and only minor changes need to be made in the last part of Theorem :), which do not affect the final result'!. The details will not be described here. IV.

QU'\~TIZ"TION

1)1'

SINGLE-SOLITO:>i SOLUTIO:>iS

1. Canonical Quantization With Collectre Coordinates Let lIS assume that classical single-soliton solutions of the form (3) exist alld then examine their quantimtion. Generalizing the method of r~,:3], WI' may write (7]) where rp.(x)'s are orthogonal solutions of t.lw foll ow ill g' ,·igelleljllatioll.

corresponding to nonz~l'O eigenvalues (w. """ 0). lowing conditions,

They are required to sati'lfy the fol(j":!)

r

a cpcC:%) = ax

\ (txrp~(x) -

"

0,

(7:!')

in order that the zero-modes corresponding to If invariallce and translational inyuriance can be removed. 'J.(t) Cllll hI' llIade r~al by sui1:ably cJjol);;ing- cp.,(x). By using tIJt's~ formulas the singl~-!'101iton Lag"Tallgian is fonnd to be

(73)

185 No. I

NON TOPOLOGICAL SOLITON WITH NONABELIAN INTERNAL SYMMETRY

53

wllere

+ ~ q"rp,,),

• (rpc

(74)

(75) (76)

V(rp) = VrpT • Vrp

+ V(rp).

(77)

Then, the canonical momenta conjugate to ak(t) and q,,(t) are (78) (79)

Then, canonical quantizl1tion amounts to replacing generalized coordinates ak(t) (collective coordinates of the gl'oup parameters), Un(t) and corresponding conjugate momenta Pk(t) and "'n(t) by operators satisfying canonical commutation relations. According to the rule gh'en in [3], the quantum Hamiltonian corresponding to (73).is

(80) where

P"

=

k

)

H

to a

llOWN'

JIM! -

series of

JI k"')

.

ill nk' lJ ",,'

TC n

~.

]<}x:pand

_ (M kk'

(P. '

Perturbation Theory

q"

as follows:

H = Ho + HI + H2 + "', the lth order in qn. For definiteness,

where HI contains terllls of and consider the casP. of rp belonging to I that .11",.,

= II? representation.

= JI ss = .I.ll",,,, = iU r ,

lll"""

=

(81)

let us take G=SU(2) It is easy to prove

MrcosO,

(82)

wherf' (82')

Remembering that 51 Ion is linear in qn, we can take

Ho

and

H,

in (81) as, (83)

186 Vol. x.."TIII

SCIENTIA SIXICA

54

(84) where il-I ro is gh-en by (20) and

is the total isotopic-spin operator. Neglecting the quantum correction of absorption and radiation of mesons in the single soliton state us the lowest order of the semiclassical approximation, we can take only the fro term. It follows from (83) that

(86) Therefore, in the lowest order approximation , fro,'; 2, conserved quantities in the single-soliton state.

P",

and

P'"

are commutative

Let jEo; J, M', lrI) be the simultaneous eigenvector of the operators fro, 5 l , P,!, and P"" corresponding to eigenvalues Eo, J(J + l)hl , M'l;, and Jlh respectiyel~-. The wave function of this state denoted by (
J 2J + 1 Df".u(
(87)

satisfies the follo,ving eigen-equations

·[_1_ ~ (Sine~) + _1_ (L + L sine ae ae sinle a
2cos e ~)l arpac/J '

= - J(J + l)Dt,'.II(
- i :

= iJI'D!t'M(
-- .;; aac/J Dtl'.\I('p, 0, c/J)

=

JIDt,'M(
(88)

From the well-Imown result in the theory of angular momentum, the solution of these equations is

(89) 'rhus the wave function of the single-soliton state JEo; J, M', M) in the representation (
Eo

=

_1_ J(J 2M ro

+

l)1i z +

i d'xV(
J

(90)

We now proceed to prove that it is always possible to choose a classical single-soliton solution such that J Z, J 3 , J; and Ec of this solution are exactly equal to the corresponding eigenvalues of a given single-soliton state JEo;J, M', M). For this purpos~, note that the classical equations of motion (28) of get) are equivalent to

187 No.1

NON TOPOLOGICAL SOLITON WITH NONABELIAN INTERNAL SYMMETRY

55

82 + p2 + ,b2+ 2.jubcos8 -. (.02, cp + cbcos8 = (.0;, cb + pcos8 = (.03, where

2 (0 ,

(.03

and (.0; are constants.

It is easy to verify that (~)2)

It follows from (23) that the energ~' of the classical solution is Ee

= _1_ (.02 + 2M/o

Jd xV(rp,). 3

(93)

Therefore, if the classical solution is chosen to satisfy the relations Mio(.02 = }tf /0(.0;

111/0(.03

= =

5(5

+

1)1;,2,

.1.11'1;., ~111i,

(94)

its energy and isotopic-spin 52, 53 and 5; will equal corresponding' eigenvalues of the quantum single-soliton state in the lowest approximation. When (94) is satisfied, it ean be derived from (31) that

B~o

Brpd x)

lEo; 5, Ji', ill) = O.

(95)

It follows from this e4uatioll and (84) that

HdEo;

5, M', .Ill) = 0,

(!)6)

which effectively eliminates the meson qn tadpole lliagram in the single-soliton sector. This is just the condition for (''Ollsi'ltency of the perturbation theory discussed in [1,2]. For the real field rp(x) belonging to the 1= 1 representation the discussion can proceed parallel to that described above. In this case we have

(98) wherp JI[o i.:; givPIl" by (39), while the total isotopic-spin operator (99)

can be obtained from (85) by putting P.j. = O. Thus, if we put the corresponding quantum number iii and (.03 in the classical solution to zero, (85)-(96) can be taken over to thi.:; case. In particular, in the lowest (semi-classical) approximation the quantum single-soliton state can be characterized by the eigenvalues of Ho, 52 and P". and denoted as lEo; J, ..1['). The energy and wave function of this state are

188 56

ED =

L

2ilf 10

5(5

+

1)

+ \ d 3x'V(f<), •

Vol. XXIII

SCIEXTL\. S!XICA

(IOn)

Y."",(e, cp)

The latter is formally the samp. as the wave function of a plane rotator. 3.

Lm'entz Covariance

Having discussed the single-soliton state at rest, 've now proceed to examine the moving soliton state to illustrate the Lorentz covariance of the theory. A new method will be used to treat this problem. Let us take again the example of G = SU(2), 1=1/2. Let US assume that the soliton moves along the z-axis with velocity It. Denote the space-time coordinates in the rest system of the soliton and those in the laboratory system by (x', t') and (x, t) respectively. They are related by the Lorentz transformation:

,

y=y', x=x, t = r(t' + us'), z = r(z' + 'Itt), t' = r(t - us), s' = r(s - ut),

(101)

where r = (1 - 1£l)-t. Let us introduce new space-time coordinates deSCl'ibing the motion of the center of mass of the soliton. They are denoted by (X', T') and (X, T) in the rest system of the soliton and in the laboratory system respectively. In the rest system of the soliton, (102)

T' =t'

is the proper time of the soliton. transformed as

X=X', Z

Under the Lorentz transformation, (X', T') are

Y=Y',

= r(Z' + 'uT'),

Z' = r(Z - uT),

T = reT' T'

==

+

uZ'),

(103)

r(T - uZ).

Originally, T' and T are defined only on the world-line of the center of inass of the soliton. We have used (102) to extend the definition of T' to arbitrary space-time points not on the world-line of the center of mass of the soliton keeping T and T' related by the transformation (103). Thus T i'l also defined on arbitrary point'!. The relation between t and T so defined is

.

T = reT'

+ ·uZ') =

ret'

+ uZ') =

r2(t - 1tZ)

+

r1!Z'.

(104)

Therefore

aTI at. =r,2

aT a;= - r ZU,

Note that, from (104) and (101) we still have T of mass of the soliton.

aT = aT =0. ax ay

=t

(105)

on the world-line of the center

189 No.1

NONTOPOLOGICAL SOLITON WITH NONABELIAS I:-
57

In the rest system of the soliton the classical soliton solution is gCt')'Pe(X' - X). In the laboratory system we haw' (p(X, t) = g( ret - "IlZ))cp/,1' - X,

Y, r(z - lit) - Z').

,If -

(106)

Substituting l' for t and using (101)-(108) we finel that

= g(T/r)(PcC.I.: -

(p(X,1')

r,

X, y -

y-l(Z - Z(l')))

== VcpcC,,-I;X - XCT»,

(107)

where an irrelevant additional term - ItZ' has beE'll omitted in the argnment of g(T/r). Using (:1:, y, z. t) as coordinates, the action of the system can be written as A. = ) cPxdtfL' =

,,-2 J(PxclT St'(rp(x,

fdTL(,,(T), X(T»).

T) =

(lOS)

Using the relatioll!l

a=

8rp= y 8z

8rpcC 1'-1.. X

-

8z

1.~.

X(1'» , 2 . 8q ( -1 VC1'» . -Y--1'lla·-'-,p" ·X-.<'\.' 'aai

e

I

,

,

with . (lZ(T) Z=--. (IT .

dai clT'

.

a·=-

,

we can easily obtain . ,,[Ce). 1J -_1' ai'" ika" 2

+

'IiZ'I -l' il' 0

2

-

,11[0

-

21'

CJ)

l

-

-

1( 2

l'

+

1) ''1

l'

j.

Q.

(109)

In the derivation of (109), the fact ha.c; been used that Eq. (31) satisfied by rpe(x) is the extremum of the yariat.ion of the following functional,

It follows from the virial theorem that

(a = 1, 2, 3).

Mo in (109) is defined as

190 Vol.

SCIENTIA SINICA

58

xxm

Substituting "( = [1 - (Z(T»l]-t into (59), we obtain that (110)

It follows that the canonical momenta conjugate to Cli(T) and Z(T) are

(111)

(112)

where

,_ dCli(T ' ) dT"

ai-

From (110) the classical Hamiltonian is found to be

(113)

Denoting the Hamiltonian of the soliton at rest by H', ,ve have from (112) and (113)

p = =

J 'V 1

Z

H'

- Zl

(114) '

Therefore the quantum Hamiltonian (neglecting the meson terms) is

I(M~ + 1.. M10w Z + _1_J z)2 + P;,

fIo = "

2

(115)

2M 1o

where JZ is given by (85). Evidently, p., j2, Pcp and Po/> are commutative conserved operators. Therefore, the moving single-soliton state can be characterized by their eigenvalues and denoted by lEo, P=; 5, M', M). Evidently, the energy eigenvalue Eo and momentum. eigenvalue P 2 satisfy the relativistic relation, (116)

where E~ is the energy (90) of the rest ,.. soliton. ,. ,. Note that because of (111) the eigenvalues of the isotopic-spin operators 52, Pcp, Po/> in the rest system of the soliton are the same as those in the laboratory system. To take account of the meson terms, one needs only to replace IP« by

,,(-I

;z-X(T»

191 No.1

NON TOPOLOGICAL SOLITON WITH NONABELIAX IXTERNAL SYMMETRY

59

(111)-(113) remain valid up to terms that are linear in q., so also (114) holds true to this order. '1]1erefore, the tenlls linear in q. are eliminated from fl if they are eliminated from H'. Thus the perturbation theory is again applicable.

Apparently, this method to deal with the problem of Lorentz covariance of the quantum single-soliton state is general. It is not restricted to the I = 1/2 representation or the SU(2) group. V.

DISCUSSION

From the results obtained above it can be concluded that the method of collective coordinates and canonical quantization can be used to formulate a consistent quantum theory of the soliton in an SU(2) symmetric theory, at lcast, for the single-soliton sector. Because of the lloncomnmtative character of the group and the difficulty inherent in a relativistic forIllulation of the two-body problem, some problems remain to be solved ·to establi.'1h a relativistic quantum theory of t,vo solitons. Among the results obtained above an interesting point is that for soliton solutions of the I = 1/2 scalar field, apart from the total isotopic-spin $, the third component of tIle isotopic-spin $3, there is an additional quantum number $; so that the degeneracy of the energy level is (2$ + 1)'. This quantum number is related to the charge of the solitoll. III fact if the solution of (31) is assumed to be of the form f(x)

C),

the classical soliton solution can always be brought into the forIll, tp(x)

by a suitable SU(2) transformation. sion of the charge of field

f(X)) = get) ( 0 Substituting the above equation into the expres-

we get

Q =2P", = 2$;. After quantization, the charge of the soliton can take integral values 2$, 2$ - 2, ... - 2$ + 2, - 2$. So far there has been no experimental evidence for existence of multiplets of this type. Do they not exist in nature really? This is still a point worth noticing· when one looks at experimental results. In the center-of-mass system solitons with nonzero spin can be treated with the method similar to that used to treat the isotopic-spin in this article. However, the Wigner rotation of spin and other complications will appear when the Lorentz transformation is performed. Since the soliton is not a point, it may also have a third spin quantum number S; in addition to S,. Another interesting point is that in certain eases the soliton with finite mass may have a divergent moment of inertia. In

192

no

SCIENTIA SINICA

Vol. XXIII

these cases all spiu-statt·s are degenerrte. Therefore when the soliton interacts with other particles it Clln absorb arbitrary angular momentum without changing its state . •Just like that the momentum appears to be non-conserved in the case of an infinitely hl!u,'Y particle interacting with other particles, the angular momentum would appear to lit! non-conserved in experimental observations. Nevertheless, thi~ does Hot imply tllUt the isotropy of spuce is violated. REFERJ;NCES

[ 1] For:J. review of early works, see PlIys. RII'p., 23C (1976), No.3. [ 2] Friedberg, R., Lee, T. D. &:; SirIin, A.: Phya, Rev., D13 (1976), 2739. [ :I] Cbri~t, N. &:; Lee, T. D.: Phys. Rev., D12 (1975), 1606.

193 Vol. XXIII No. 4

SCIENTIA SINICA

April 1980

ON THE VACUUM OF THE PURE GAUGE FIELDS ON COSET ZHOU GUANGZHAO (CHOU KUANG-CHAO

mlJ'tB)

(I1I8titute of TlI80TllticaZ Physics, .J.cademia Sinica) Received Febru:lry' 8, 1979.

ABSTR.\cr

The topologic:tl properties of the v:umum states for the pure gauge fields on coset are studied. When the homotopy group 7r,(G/H) of the coset m:mifold is different from zero, a winding number operator e:m be constructed. It is possible to introduce n B.-'I":tcuum on coset. Tho constrnined conditions restrict the ,"nlue of B. to be zero in physie..'ll st:Itc~.

1.

INTRODUCTION

The study of topological properties of the vacuum in theories of non-Abelian gauge fields--has attracted a good deal of attention in recent years. For pure Yang-Mills fields in the temporal gauge Ao = 0, it has been sho1rn that there exists a topological non-trivial vacuum state A,.';" O. The Euclidean instantons are believed to be the realization of tunneling between different vacuum states. '1'he physical vacuum is the so-called 8-vacuum[t]. Although 8-vacuum is a gauge inyuriant coJlcept, its realization is quite different for different choices of the gauge conditions. In Coulomb gauge, for example, Gribov has proved that there are topological nontrivial vacuum states with half integer winding numberU ,3J. It is suggested in [8] that, in Coulomb gauge, the change of ,vinding number for the vacuum states should be described by a transition function on the intersection of two gauge patches that cover the wh01e compacted Euclidean space. In a previous paper (hereafter call 1)[41, the concept of pure gauge fields on the coset GIH of a compact Lie group G with respect to its subgroup H, is introduced. With the help of this concept, we can construct a local gauge invariant Lagrangian under the group G, which contains vector gauge fields only on the subgroup H. After the introduction of the pure gauge fields on the coset, a question naturally arises concerning the topological properties of their vacuum states. It is the aim of the present paper to study this problem. Our conclusion is: If the homotopy group 1t:3(GIH) ~ 0, there e:rist topological non-trivial vacuum states for the pure gauge fields on the coset. It is possible to introduce an analogous 8~-vacuum with a def'"mite value of 8. = O. The paper is organized. as follo'vs: In Sec. II, we study the gauge transformation for the pure gauge fields on the coset. An operator T(g) on the Hilbert space which induces the gauge transformation g, is constructed. The physical states of the system

194 432

SCIENTIA. SnllCA

Vol. XXIII

are subject to the constraint conditions derived from the gauge invariance of the theory. In Sec. ill, topological properties of the vacuum states are studied in the temporal gauge. The 8-vacuum is introduced and its value is determined for the physical states. Finally in Sec. IV we shall discuss the results obtained. II.

GAUGE TRANSFORMATION

Let G be a compact Lie group, H its subgroup, with elements denoted by g and An element g E G can always be decomposed into a product of two elements belonging to the left coset GIH and the subgroup H respectively: k respectively.

g = cPk,

cP E GIH,

kE H.

(2.1)

Let cPo(z) be a given function valued in the coset, then for an arbitrary element g E G we have (2.2)

where cP(g, cf>o) E GIH and h(g, cPo) E H are nonlinear realizations of the group G[7]. They satisfy the following relations, cP(g'g, cPo)

= cf>(g', cf>(g, cf>o)),

k(g'g, cf>o) = h(g', cP(g, cf>o))l!(g, cPo)

(2.3) (2.4)

for arbitrary g', g E G. It is more convenient to use local coordinates to parametrize the group manifolds and the coset space. For infinitesimal gauge transformation we write (2.5) where Ii, j = 1, ... , nG are generators of the group G. We assume that the first nH generators 11 belong to the subgroup H. ;J, j = 1, ... , nG; k;, j = 1, .•. , nH(k; = 0, j > ns) are infinitesimal parameters that characterize the group elements g and h respectively. For finite group elements the same letters ;/(ltt) shall be used to parametrize them. The local coordinates for the coset space will be denoted by cs;, j = nH + 1, .. " na, and the local coordinates for the functions cP(g, cPo) and h(g, cPo) by Ri(g, cPo) j = nH+l, •• " nG and k;(g, cPo), j = 1, .. " nH respectively. When g is the infinitesimal gauge transformation (2.5), we have

+ iIi;;, cPo) = au; + Ri,k(CSO) ;k, ki(1 + ii;g;, cPo) = h;,,.(cso) gk, Ri(l

(2.6)

s..

In (2.6) aoJ are the coordinates of the element cPo. In the following repeated group indices are summed from 1 to na, so we must stick to the convention that h; = 0 ¥ j > nH and RI = 0 ¥ j ~ nH'

to the first order in

As a nonlinear realization of the group G, the functions RI •• tmd kj,. satisfy the following reaJiza.tion of the commutation relations:

195 No.4

VACUUM OF PURE GAUGE FIELDS ON COSET

433

(2.7)

and (2.8)

where (2.9)

and

!i/.m

are the structure constants of the group G.

Consider now a system with fields CP(z) which form the basis of a linear representation of the group G. In. the absence of the gauge fields the La.,"Tangian has the form (2.10) which is assumed to be global invariant, and we must introduce gauge fields. In. I we have constructed a gauge field B,. on the group G which consists of a vector gauge field 1,. on the subgroup and a pure gauge field cPo(:I;) on the coset, B,.(x) = cPo(x)(a,.

+ 1,.(z»cPo1(z),

cPo(z) E G/H.

(2.11)

When 1,.(x) transform under g(x) E G in the following nonlinear Way, g(x): 1,.(x) -1:(x) = h(g, cPo)(a,.

it is easily verified that

13,.

+ 1,.(x»h- (g, cPo), 1

(2.12)

transform as a usual gauge fields under the group G, (2.13)

With the help of these gauge fields we can construct a local invariant La.,oorangian from (2.10), £t'(CP(x),

D~B)CP(X), F~~» =

£t'o(CP(x),

D~)CP(x»

-

1 4 a tr

{m.~)F(B)I'P},

(2.14)

where (2.15)

In. (2.14) a is the normalization constant determined by the condition,

tr {liid

=

(2.16)

aBjI..

In. the La.,oorangian (2.14) we can choose CP(x), l,.(x) and ao/(x), l = nH + 1, •• " nG, which parametrize the coset function cPo(x) as independent dynamical variables. The La.,"Tangian (2.14) is invariant under the infinitesimal gauge transformation y(n = 1

+ iliSi

with the corresponding changes of the fields,

BCP(z)

=

ilkCP(z)Sk,

BA~(z) = fiilA~(z)kt.k("o)gk - a,.(h,.k(ao)gk), 6lJo1(z) = iRJ.r.("o)sr.,

i

=

nH

+ 1, "', ?lG.

/-1," ',nR }

(2.17)

196 Vol XXIII

SCIENTIA SlNIeA

In (2.17), A.~(x) are the components of the gauge field l' .fJ.,.

=

.

I

AI"

A

(2.18)

-~A .. Il'

The operator which produces this gaug~ transformation can be written as (2.19)

T(g(x» = exp {iftPxI(g(x),x)}.

For infinitesimal gauge transformation g(x) in (2.5), the generator [(O(x), x) is equal to

[(1

+ ii/gi,

x)

=

iP~(x)I/ctP(x)g/c(x)

- P ,,1(X) f milA!(x)hm./c(cio)gle(x)

•

.

+ ,iPaol(X)Rl.k(cio)g/c(x) +

P ,,1(X )ap(hl.k(aO)gk(X»,

•

(2.20)

where P~(x), P a,leX) and P ,,1(X) are respectively the conjugate momentums of the quantized field., $(x), aOl(X) and 1;(x). The commutation relations are [tP(x), P~(Y)]%D=IID = i8 (x - y), [aOl(X), P aom (Y)]"D=II. = i8/ m 83(x - y), 3

)

[.J~(x), P"::(Y)]"D9IO = i8,.p8/ m 8 3(x - y). Actually these quantized fields are not independent. There are constrained conditions due to the gauge invariance of the La,,'"'l"angian. We shall regard these constraints as weak conditions which are satisfied only when a physical state is aeted upon. As we have mentioned in I that the derivative terms a,.ao/(x) appear only in aql os momentum Paol(x) must be related to P~(x) through the constraint conditions, D~D)~(X),

PaD/eX) = p~(x)ll(ao)cP(x),

l = llu

+ 1, ... , nG,

(2.22)

where I/(ao) is determined by the relation, cPo(x)al'cPo1(x)

= Il(aO)al'ao/(x).

(2.23)

From the group relation, (2.24) and (2.23), we get for arbitrary g E G, li(cP(g, cPo)aI'Ri(g, cPo)

=

cP(g, cPo)al'cP-1(g, cPo) = gcPo(h-1(g, cPo)al'h(g, cPo)cP01g-1 + gli(cPo)g-lal'aoi

+

ga,.u-l.

(2.25)

Equating terms proportional to al'§/c(x) on both sides of (2.25) we get li(ao)Ri.1«ao)

=

cPOllcP01hl.1«aO) -lie.

(2.26)

Using this relation we can rewrite the constraint equations (2.22) in the following form, (2.27)

197 :No.4

VACUUM OF PURE GAUGE FIELDS O:N COSET

435

When the generator (2.20) acts upon a physical state, it can be simplified with the help of the constraints (2.27). Thus we obtain 1(1

+ il"g" , x)lphys)

= [(h(l

+ il";,, , cPo),

x) Iphys) ,

(2.28)

where l(h(l

+ il;;i, cPo), x) = iP",l,
•

•

(~.29)

In (2.29) we have redefined the matter fields as
=

cPii1(x)cP(x),

P",(x)

=

P~(x)cPo(x).

(2.30)

It is clear from (2.29) that [(It, x) is a generator in the subgroup H ,vith
can integrate (2.28) to give l'(g(x))!phys) = T(hCu(x), cPo)) JPhys).

(~.31)

Eq. (2.31) just means that in the present theory a gauge transformation gCr.) on physical states is equivalent to a gauge transformation on the subgroup H with element 1L(g(x), cPo) which is a nonlinear realization of the group G.

m.

THE VACUUM STATES

After the introduction of the pure gauge fields on a coset, a natural question is how to define their vacuum states. .As the vacuum states are described by pure gauges in the usual non-Abelian gauge theories, we can transcribe most of the reasonings from the existing theories to the present case. For simplicity we shall work in the temporal gauge Bo = O. The winding number operator is defined as, (3.1) where a is determined by (2.16). transform'3 in the following way,

Under a gauge transformation g(x) the operator

=

g(x): N-N'

T(y)NT+(g)

=

N

+ lI(y),

lV

(3.2)

where lI(g)

=

_1_

12~a

+

rd xe;ik tr [ga;g-lgajg-lgakU3

J

_1_

.J. du;eil" tr [aiy-lgB,,].

4~a j

I]

(3.3)

In (3.3) the second term on the right-hand side is a surfaoe integral at the boundary of the· three-dimensional volume. lI(U) is the winding number induced by the gauge transformation g(x). If U(x) satisfies the boundary condition, (3.4)

198 436

SCIENT!.4- SINICA

Vol. XXIII

it may be regarded as a map of the compacted three-dimensional space 8 3 onto the group manifold G: (3.5) g(x): 8 3 -+G. In this case v(g) is the Kronecker index of the mapping (3.5) and takes the value of an integer. In the following we shall consider two examples that make our discussion clear. First, let us take the group G to be a compact simple Lie group and the subgroup H to be an Abelian group U(l). It is well known[S] that the homotopy group (3.6)

The vacuum states· corresponding to the vector gauge fields 1... on the subgroup are topologically trivial in the present case. However, we can still find a gauge transformation g(x) in the group G which has the winding number v(g) equal to one. Let In) be the eigenstate of the operator if with eigenvalue n, T+(g) In) will be an eigenstate of if with eigenvalue n + v(g). As T(g) is a gauge transformation which commutes with all gauge invariant observables, the physical states should be eigenstates of the unitary operator T(g). From states In) with definite winding number we can construct eigenstates, (3.7)

" of the operator T(g) with eigenvalue gau.,o-e fields on the coset.

oi8c oCg).

This is the Bc-Yacuum for the pure

In the present case not all values of Bc are compatible with the constrllint condition (2.30).

Since "JrJ(H) = 0, T(1~) cannot change the winding number n, normalize the states such that

therefore we can

T(h) Iphys) = Iphys).

(3.8)

From the constraint equations (2.31) and (3.8), we conclude that Bc = 0 on physical states. As a second example we take the group G to be the chiral 8U(3)L ® 8U(3)R and the subgroup H to be the parity-conserved SU(3). We shall express the elements g E G in the following form, g = cP(a) h(p),

(3.9)

where cP(a) h(P)

= exp {; =

exp

{~

7sliai (x)} ,

(3.10) liPi(X)}

are elements in the coset and the subgi-oup respectively. In (3.10) ).1, j = 1,

···,8

199 No.4

VACUUM OF PURE GAUGE FIEL»S ON COSET

437

are the usual Gell-Mann matrices of SU(3). We can also express these elements in terms of the right-handed and left-handed group elements, (3.11)

where gl.(g) gReg)

= exp = exp

{! iCl + rs)ijg {! rs)ij;j} . j} ,

(3.12)

i(l -

It is possible to define two winding number operators corresponding respectively to the group SU(3)L and SU(3)R'

(3.13)

where BpR and B"L are respectively the corresponding gauge fields on the right-hauded and left-handed subgroup of G. They are related to the gauge fields B" through the relation, (3.14)

Under a gauge transformation

a=

a: NL.R-N~.R

{h • gR it is easily verified that

= lVL.R + VL.R(a) =

NL•R + V(UL.R),

(3.15)

where V(aL.R) is given by the same formula (3.3). From (3.11) we easily obtain, vL(h)

= vR(h),

Vl.(c/»

= -VII.(c/».

(3.16)

Let InL. nR) be eigenstates of the operators NL and NR with eigenvalues nL and nR respectively.

~LlnL' nR) = nLlnL' nR), } NRlnL. nR) = nRlnL. nR)'

(3.17)

We can always find an element 11, E H such that vL(k) = vR(k) = 1 and an element c/>E G/H such that 'VL(cfJ) = -VR(c/» == 1. Then under the gauge transformations described by these 11, and C/>, we have

jr""(h)lnL, nR) = InL + 1, nR + 1), } jr""(cf»lnL' nR) = If'lL + 1, nR -1).

(3.18)

Let us now introduce two winding numbers corresponding to the gauge transformations h and c/> respectively: (3.19)

200 438

Vol. XXIII

SCIENT!.-\' SINICA.

In terms of these new winding numbers we can write the state InL,nR) as In, nc) and have

T+(h) In, nc) T+(e/»ln,ll e )

= =

In + 1, ?Ie), } In,ne + 1).

The eigenstates of the operators T(lL) and T(ef»

(3.20)

can be written as

le,ec) = ~ e.s:p{i(ne + 11eec)} In, l1e). This is the e-vacuum states for the present e.s:ample. Like the first e.s:ample, T(e/» is equivalent to T(Tt(e/>, e/>o)) when a physical state is acted upon. Since T(lt) cannot change the winding number nc, we conclude that Bc can be normalized to zero for physical states. In general cases, if H is a semi-simple Lie group, '%J(H) = 0, one can proye with the help of the e.s:act sequencers] -,.. '%)(H)

~ ,,"3(G) ~ -:r;(GjH) ~ -:riH) ~ "zCH)-1/

o that (3.21)

When '%;(GjH) -'-;- 0, the purc gauge fields 011 the coset will huye topolog·icul nontrivial yacnum. states. Ou account of the constraint conditions (2.30) the Bc-yacuum introduced on the coset has the yalue ee = O. Therefore it does not cause CF non-conservation. Before the conclusion of the present section we shall briefly discuss the problem of vacuum ttmneling". .As an illustration we choose G = SU(2) and H to be the identity element. In this simple case the coset is the ,vhole group. In the temporal gauge it is well known rl ] that the change of the winding number between the future and the past vacuum states is equal to the second Chern class or the Pontryagin number, q

=

v(t

= +00) -

v(t

= -00)

= __1_ rd\J; tr {jr(B)*jr(S)I'~} 32~a

J

,..

,

(3.22)

In our case the field

is a pure gauge field.

On those points where g(z) is regular we always have

P<.!!) = O.

(3.23)

The contribution to the inte.,aral (3.22) comes only from those points where g(z) is

sin.,oular. On these singular points b,.(z) may represent point instantons. One example

201 !\,O.

4

VACUU~I

OF PURE GAUGE FIELDS ON COSET

439

will be the Polyakov-'t Hooft instanton with vanishing size, which can bc expressed in Euclidean space as[6], o(x)

=

x.

+

ia . x -/ x 2

'I'his O(x) has point singularity at x = O. The field B.u with O(x) given by (3.::?4) will contribute unit Pontryagin number when substituted into the integral (:3.22). In the present theory one should allow the existence of such point instantons (i.e. gauge transformations with point singularities), which cause tunneling between vacuum states with different winding numbers.

IV.

DISCt:SSIONS

We have shown in previous sections that it is post;ible to define a ecvacuum for the pure gauge fields on a coset G/ H in the temporal gauge when the homotopy group 'Jtl(G/H) is different from zero. 011 the other hand, one can fix the yacuum uniquely by choosing, for instance, the gauge condition al = O. Is this fact contradictor)- with those analysis in the temporal gauge? Let us recall that similar situation happens in ordinary non-Abelian gauge field theories. In Coulomb gauge, when the vector poten tial AI' is subject to the boundary conditions lim loX IY2 -4; = 0, the classical l.zi-:a

vacuum is uniquely •.1; = 0[S1. .Jackiw, Muzinich and Rebbi hay€' analyzed in detail this question of how to describe yacuum in Coulomb gaugers]. They concluded that the Coulomb gauge description of a gauge field with non-yanishin~ Pontr.ntgill index cannot be single-valued. Sufficiently large potential would eyolvc di'3COntinuously in time a sudden transition betwcen gauge equi"mlellt transyerse configurations.

'Vc have mentioned in the end of the preyious section that even in temporal gauge lllust haye singUlarities corrcspolldin~ to point instantons when the Pontl'~'ugin index is non-yanishing. Transformation from temporal gauge to the gauge
EI'

If these singular configurations were not allowed in the theory, the prcsent formulation would be completely equivalent to the usual one physically. The YaCUUIll states with different winding numbers on the coset represent ground states of unconnected world in this case.

We have demonstrated in the present work and in I that point instantoll!'J and POint monopoles can be described by singularities of pure gauge fields on coset. The existence of such topological particles makes the present formulation different from the usual theory. It is not clear whether such theory of pure gauge fields with singularities is applicable to reality. However, it has some peculiarities. Besides the ec-vacuum discussed here, it may generate additional anomalies in the consel"Yation of certain currents. We shall discuss these problems elsewhere.

202 440

SCI ENTIA SINICA

Vol.

xxm

REFERENCES

Qillan, C. G., Dashen, R. F. & Gross, D. J., Phys. Lett., 638 (1976), 334; Jackiw, R. & Rebbi, C., Pllys. Rev. Lett., 37 (1976), 172; 't Hooft, G., Pllys. Rev. Lett., 37 (1976), B; Pilus. Ret'., D14 (1976), 3432. [ 2] Gribov, V. N., Nm:l. Pilus., 8139 (1978), 1. (3] Sciuto, S., Phys. Lett., 718 (1977), 129; Adcmollo, M., Xllpolit:mo, E. & Sciuto, S., lOl/CI. P1tys., 8134 (1978), 477. [4] Zhou Guangzhao, Dn Dongsheng & R1lIln Tunan, Scientia Sinica, 22 (1979), 37. [-5] Steenrod, N., Topology of Fibre BU'1ldles, Prin. Univ. Press, (1951); BoY-II, L. J., Carinena., J. F. & Mateos, J., Fort. d. Pllys., 26 (1978), 175. [6] B~lavin, A. A., PolykoY, A. M., Schwartz, A. S. & Tyupkin, Y. S., Phy.y. Lutt., 598 (1975), 85. 't Hooft G., PhY8. Ret·. Lett., 37 (1976), 8. [ 7] Coleman, S., Wess, J. & Zummo, B., Pilus. Rev .• 177 (1969). 2239; Callan. C. G. Jr., et al •• Phys. Ret' .• 177 (1969). 2247; Sala.ln. A. & Strnthdee, J .• Phl/s. Rev .• 184 (1969), 1750. [ 8 1 Jackiw. R., Muzinicb, r. & Rebbi. C., Pllys. Rev., D17 (1978). 1576; Kicole, D. A., Nuclear Phys., 8139 (1978). 151. [ 1]

203 Vol. XXIII So. ;3

SCIENTIA

SINICA

May 1980

A MODEL OF ELECTRO-WEAK INTERACTION IN SU(3) X U(l) GAUGE THEORY ZliOU GC.\NGZHAO (CHOU KU.\l'G-CHAO,

m1J'tfj)

(Institute of Theoretical Physics, ..:J.cadl'mia Sinica) AND GAO CHONGSHOU (f.Ij~~)

(Department of Physics, Beijing Unit·er.~ity) Rccei.ed _<\.ngust 20, 1979.

ABSTRACT

The hclicity mb:ed repl"esentation is used to eonstmct a model of electro· 'Wcak in tl'l":lction ill 817(3) X17(l) gauge theory. It reduces in the low energy region to the simple ..'i'U(::?) Xl7(l) model in the limit of sin'lIw = 1/4. When sin'lIw is slightly less than 1/4, there is a small correction in the neu tral current s~ctor 'Which c:m be tested.

1.

I:-.ITRODt:CTION

Recent neutrino-illlluceu ehL'.:tic and dl'ep inelastic nC'utral curl"C'nt data m'e in agreement with expectations ba£eu on the simple SU-(,:2) X C(1) g-:mg-e 1Il0del of Weinberg-Salam [l.ll. The Weinberg angle ew is shown to be slightly less than 30°. In a preyious note (hereafter call I) a model in SC(3) IUlo;1 been proposed for the electro-weak interactions of leptons. The essential point is to lll'l"an!:!,e the left-handed and right-handed leptons in a single triplet without the introduetion of superl-,'"l'OUp structure[J.11. sin 2e IV is shown to be 1/4: [JJ. In the present paper we offer another model in the same spirit which takes quark into account. The ga.uge group is chosen to be SU(3) X U(1). Althoulth thiq is not a simple group and a.nother coupling constant fJ' is introduced, one can show that sin 2 eIV ~ 1/4. In the limiting ca~e where sin 2 ew = 1/4, the charged and neutral currents are identical with that based on the simple 8[."(2) X ['(1) gauge model. For sin281V slightly less than 1/4 there is a small correction in the Jleutral current !':ector which can be tested in experiments.

As in I, a global F(1) symmetry is preseryed after spontaneous symmetry breaking. This symmetry provides a conserved quantum number called weak strangeness. There are nine gauge bosoI1S in the model, four of which are the usual W:!:, Za and the photon. The remaining five contain a neutral. Zl, a charged pair V:!: and a doubly charged pair U:!::!: bosons. Two triplets of Higgs are used to generate mas>es of the gauge bosons and the fermions. After spontaneous symmetry breaking foul' heavy Higgs with two neutral ones Xa, cpo and a charged pair X:I: remain. On account of the conservation of weak strangeness, V:!:, U:!::!: and X:!: mesons can be produced only in pairs and the lightest of these mesons might be a stable particle.

204 No.5

SU(3) xU(1) MODEL OF ELECTRO-WEAK I~TERACTION

567

The paper i~ organized as follows: In Sec. II tlie transformation properties of fields are studied. A representation of the SU(3) algebra, which mixes the rs and the internal group generators, i~ used for fermions. In Sec. m, the mas'! spe~tnlln of "Vector gauge bosons is obtained through Higgs mechanism and the conservation of weak strangeness i'! e"tablished. The eletro-weak interaction of the fermions i!'! clir!cussecl in Sec. IV. Deviation in neutral CUl'rents from the Weinberg-~alam model is obtained. Finally we "hull di~cuss the results obtained. II, TRANSFORMATION PROPERTIES

The generators Ii, i = 1,"',8 of the SU(3) group can be decomposed into two sets and There are many possibilities. One possible choice which will be used in the following- is It = 1, :3, 8, 4, 6 and a = ~, ;j, i. Other choices giyC similar results. Define

t

t.

f\..

l~r;} =

'"

eI.

"

I

.•

l~t. =

,...

(~.l)

II'.

where e commute!'! with 1; and satisfies the relation 13 1 = 1. One easily yernies that the Lie algebra for I~') i'! the same as that for t. Four possible choices for e : e = 1, -1, rs, -1'5 will be u!':ed below. The goeneratol' of the r(1) g-auge group will be u,'noted b~· j'"'.

+

Besides the con~;eryution of yarious fermion number!'! there is another global F(I) sYlllmetr~- whose generator will be denoted by .~. Thi'l global U(1) will combine with an Abelian subl,.'"'l'oup in SU(:3) X [\1) to giYe a new conseryation number Sw after spontaneous symllletry breaking. This new quantum number HII' is called weak strangeness in I. Each g"cllC'ratioll (11" C), (l'I" .!L) or (l'r ... ) of leptons forllls half of It triplet in SF(:l) with r = o. Quarks of each color and isotopic doublet form half of a r = 2/iJ triplet plus half of a r=~/a singlet in "C(:~). ')'ran~fornmti()n propertirs fo1' fermion!'! are

c/J --~ c/J' = rtS)(;i(Z) )eiYU(r)ei.~~c/J;

for triplet:

II.~U.'

for singlet:

=

(2.2)

eiYU(r)eiS'iu..

where (2.3)

= +, -, oj, -oj stand for e = +1, -1, +1'5, -1'5 rt>specth-ely. In Eq. (2.2) = 1, .. "'~, e(x) and 1/ are group parameter" for the local 8["(3) X F(I) and the global CO) respecth-el.v. The !7enerator .g will take yalues l.. rs _l.. 2 for lepton 0

Here e

gj(x), j

1 "'lt t rIp e , 6

6 • ' let. for quark trIplet and - -1 1's - -5 f 01' qual'ksmg 6 2 6

rs - -;3

There are nine gauge fields A.~(z), j Define

= 1, ... , S of

Sr(8) and BI,(r,) of UO).

(2.4)

205 SCIE~TIA

568

Vol. XXIII

SINICA

Then :A~' transforms under SU(3) in the following way, 1~"---- :A~"

= F<e'(§j(x))(al' +

A~·'(x))U<·)+(§i(X)).

Two triplets of Higgs fields q>1 with Y = 0 and q>2 with Y = transform a.';!

(2.5)

- 1 are used. They (2.6)

= - 1.- for

The generator S

3

q>1' S

=

5 for q>2 and S

3

=

0 for gauge yector bo:;uns.

The covariant derivatives for the fermions and the Higgs are easily constructed. are

The~r

for fermion triplet:

D,,'"

for fermion singlet:

a" + :A~) + ;

(

Du'lt ,

( a".

=

a/, +

D"q> -- (

for Higgs triplet:

III.

=

"(+' A,.

U'l-BI') cP;

~ ) + -2i( J, J:Bu .

(~.7)

U;

+ 2,i",,) g YBI' q>.

SYMMETRY BRE.\KING, CONSERVATION OF CHARGE .... ND WE.\K STR.\NGENESS

The vacuum expectation yalnes of the Higgs are taken to be

(3.1)

s.nllllletr~·

One local F(1) and one /!lobal F(1) are

remain unbroken.

Their generators (3.2)

for charge: .. 811'

for weak strangeness:

2

1\

,..

= ,.; 3 18 + }' +

A

(3.3)

S,

which are conseryed quantum numbers. After spontaneous symmetry breaking all physical particles are eigenstates of charge Q and ,veal;: strangeness § w • 1 . For triplet fermiolls we must replace' is by 2 1'5.1.s and get from (3.3): 1 ~

,:sw

1

= ,.;_ 3

.

1'5.1.8

+

-1 1'5 6

+

"1 Y = -,) 1'5

-

(

1

)

+ Y.

(3.4)

-1

We use the quantum number Sw to classify all three components in a triplet. This implies that the he1icity components with the same Sw be either (L, L, R) or CR,

206 Xo.5

SU(3) XU(l) MODEL OF ELECTRO·WEAK .INTERACTION

569

R, L). 'l'he now observed leptons and quarks are then chosen to be

(3.5)

The other half components are weak strange fermion.'!, which will get heuyy masses by a suitable choice of additional Higgs fields and an additional singlet lepton field. The problem of mass spectrum for fermions will be discussed elsewhere. The charges and the weak strangeness of the particles participating in lo\v-energy 'Weak interactions are given in Table 1. Table 1

The mass terms of the vector gauge bosons are easily derived as,

~ rll ulI (W+W-+Y+v--+! 2

+;

g2IulI2(v+r-

+ ..!. III vll l ( 4

+ r-++c--) ZO

1

ZOl)

.) 3

+ _._1_ Z'o)~,

(3.6)

Slll'P

where

v:l: = .)2"':>'_' _1_(.~ + ..1') , U:!::I:

= .)2 _1_ ( 4.' + iA7) . , Z'G = - sincp

(~

(.1 3

-

./3.4.')) +

coscpB,

(3.7)

and .

Slncp

= / 'V

g

gZ

+ g'2

'

coscp

=

g'

./ g2

+ g'2

.

(3.8)

From (3.6) and (3.7) we obtain: (i)

The photon field A = cos cp

(~

(A 3 -

./3.1

quired: (li)

The maJ!Ses of the charged vector bosons are

8

))

+

sin cpB is massless as re-

207 570

SCIENTL~

SINICA

Vol.

Xxnr

(a.!)) m~ =m~

Let us introduce

11

+

m~.

new parameter (:UO)

which will be useful in the following. The Z and Z' mesons are not eigenstates of the mass matrix. Let the tllle neu· tral vector bosons be Z, and Z2, which are related to Z and Z' b~' it rotation.

+

Z = cosaZ, Z'

sinaZ2'

(:Ul)

= - sinaZ, + eosaZ2'

Diagonizing the mass lllutrL"': we have

,

mz

, [1 + -'v 14 (1 -

=

nlz

=

, nlz, [tg'a

1

m~.

+

-1

4

L'

-1'3. t!? ' a )~] . SIll
,

(:3.1::?)

( tga.,-. --/;3 .- \)~1 , Sinep.

where

= -34

'1

lIlz'

.

'1

IIII'y

'1

cos' a.'

(:U4)

One interesting limiting' case is!.) sincp

« 1,

2 sin ep

«

1.

(8.15)

'U

In this limiting case, Eq. (8.14) becomes tga

=

1

-13

sinep

(1. + i..sin!q) + 0 (sin tp)). 8u '

(:3.11l)

4

Substituting Eq. (3.16) into Eqs. (3.12) and (3.13), we obtain,

(1 + -±.9v sin4cp + O(sin6 cp )), mi = m~ . ~ (1 + .! sin cp + O(sin4 cp)) »III~ for smull • 4 sin cp 3 m~ =

mi

1

2

2

1) We note tlmt tg a

= .}_ sin '1',

1

7R~1

= m;'/eos Bog

2

(:3.17)

sin cP , I:

and :mb, 1nt-, lIIb -+ oc in the Iimitillg

e:ISt'

v.... co

v 3 for arbitrnry value of sin rp. The neutral current IN,I' between feIlIIions can be mitten as IN,I' = J~ -

sin" 8."J ;... • Therefore. all observable results in this limiting ease are eDCtIy the some as those in the

Weinberg· Salam model.

208 BU(3)XU(1) MODEL OF ELECTRO-WEAK INTERACTION

No.5

571

(3.18)

Comparing the above with the mass formula in the Weinberg-Salam model, (a.lH)

we obtain to the order sin 2cp that

"e w = -cos 1 2 cp

sm-

4

___

1

(3.20)

"=::::-.

4

This is an encoura",oing result since recent experiments require that sin 2 ew .,;;;;; 1.., How4 ever, this result is model-dependent, one obtains different values of sin2 ew when different representations for fermions and Higgs are chosen as is shown elsewhere'S], After spontaneous symmetry breaking four heavy Higgs remain. to the triplets cI>1 and cI>z in the form,

, X-)

cPO) cI>1

= (

0,

They are rt'latE'll

cI>z

o,

=

(00 ' X ,

where cPo and XO are real fields and X- is complex. cI>z has no coupling with fermions Only cPo field in cI>1 has direct coupling with fermiolls and generates llla..c;ses of them after symmetry breaking, IV.

INTERACTIONS BETWEEN FERMIONS AND G,\UGE VECTOR BOSONS

The gauge interaction Lagrangian for the fermions can be written as

+ where

URY"

(a"

+ ;

a'iB,,) UR

- (a + ..4."(5»)". + cPIY"" ,. '1"1, cP, =

(·U)

' V'l.) ( 61. , etc, 6R .

As Ly"R = Ry"L=O, one can immediately see that A~5.6·7 do not couple directly to fermions. It is also easy to find that

209 572

where

SCIE~TIA.

SINICA.

Vol.

xxm

ii, i = 1,00', 8 are the usual Gell-Mann matrices for SV(3) and 1;

~ v's ('

1

J

(4.3)

i; is just the generator used by Neeman, Fairlie and others in their graded SV(211) gauge group formulation. It occurs here simply as a result of the difference in the action of r, on the left-handed and the right-handed fermions, i.e., rsL =L,

rsR = - R.

In terms of the physical gauge fields found in the previous section, it is easily verified that the interactions with W:I: and the photon have the usual form with e = -1 gcosrp :2 ' which indicates that sin1 8 w =

..!.. cosl rp 4

(4.4)

ill the charged current sector. This is in agree-

ment with that obtained from the mass of Z, meson in the limiting' case of small sin 2
~"Tangian

. - {z '''' [1 - ; - (2l.

llJ:/Jr'"

+

j

-

:2 V 3

ZI", [

1;-== (2i j 2V 3

-

with the neutral ....ector bosons has the form,

QA" + 1-) coso:

1(. sinrp Q"- - .1-

+ -:-

SIner

2

Q +:h sino: -

+-2 (siner Q.

YA) sino: ]

_.1_ f)coso:]} cPo

SIller

(Li)

In the limiting case of small sin
ft' off = 4

;F

V 2

[.l ~ a

1!"

.l~l + .±. .}N .},;;,J, 9t'

( ·!.6)

II'

\vhere (4.7) 0.8)

,nth sm- 8 II' = -1 cos-, rp 4 o

,

are two neutral currents coupled to the Z. and Zl mesons. In the case of sinz8 w=I/4, the first neutral current has exactly the same form as in the simple gauge theory, while the second one is a pure vector current for quarks which is difficult to observe owing to the interference with strong interaction. Recent experiments tell us that sin' 8 w is slightly less than 1/4. The present model provides a small correction (depending on the value v) in the neutral currents of fermionso This small effect could

210 No.5

SU(S) X U(l) ~roDEL OF ELECTRO-WEAK INTERACTIOX

be measured by more accurate experiments. V.

DISCUSSIONS

We have extended the previous SU(3) model for leptons to a SU(3) X [;(1) model for both quarks and leptons. Results like the mass spectrum and the Weinbergangle are model-dependent. However, there are some common features which make a whole class of such models different from the usual ones. Firstly, there are gauge bosons V and U which have no direct coupling with yet observed fermions. They only couple ordinary fermions with weak strange fermions which are assumed to be heavy in the present model. Thus in the low-energy region they have practically no influence on the electro-weak processes. These gauge bosons can be produced through virtual photons or Z bosons in electron positron colliding beams. The pair production cross section for these mesons in sufficiently high energy colliding beams will be of the same order as that for the W:!: bosons if the phase volume correction is taken into account. Secondly, it is possible to define a new quaJOltum number called weak strangeness in this class of models. 'rhe "V,U gauge bosons, some fermions and some Higgs scalars are weak strange particles. They are produced only in pairs and the lightest one will be stable if the conservation of weak strangeness is exact. The model formulated in the pre~ent paper could also be tested in accurate experiments on neutral current interactions. There are two problem'! that remain unsolved. The first one concerns the cancellatioll of trianbrular anomalies. It is not possible to cancel
[1] [ 2] [ 3] [4]

[5)

Weinberg, S., PlIys. ReL·. Lett., 19 (1967), 1264; Salmn, A., Proc. 8th Nobel S!!,mposil,m, Stockhollll, (1968); Glashow, S. L., Illiopoulos, J. & MlI.i:mi, L., Pllys. Rev., D2 (1970), 1:!85. Tittel, K., Baltay C. & Weinberg, S., 'fulks given in 19th international conference on high eneI"gY phyHics, Tokyo, (1978). Chou Kwo.ng-chao & Gao Chong-shou, Ke3:'U8 TOfIgbao, 2S (1980), 2l. Neeman, Y., Pl,ys. Lett., 81B (1979), 190; F:Lirlie, D. B., Phys. Lett., 82B (1979), 97; Squires, E. J., Phys. Lett., 82B (1979), 395; Taylor, J. G., Phys. Lett., 83B (1979),331; ibid., 84B (1979), 79; Dandi, P. H. & Jarvis, P. D., Phys. Lett., 848 (1979), 75. Lee, B. W. & Weinbet-g, S., Phys. Rev. Lett., 38 (1977), 1237; Lee, B. W. & Schrock, R. E .• Phys. Rev., DI7 (1978), 2410.

211 KEXUE

Vol. 25 No. 1-2

TON6BAO

.Tanuary 1980

ELECTRO-WEAK THEORY IN 5U(3) Zaau

GOANGZHAO (CHOO KWANGCHAO JI!iI~B)

(InatituM of TheortrtioaZ Physica, .J.oaaemia Sil7lica) AND GAO CHONGSHOO (~~~)

(Depart1llll'/l.t of Phy.ftca, Beij'1I.g U1I.it'Braity)

Received August 20, 1979.

Ne.'eman and Fairlieu •21 have recently attempted to embed the SU(2) X U(l) electro-weak groUp[31 inoo a supersymmet:dc SU(2Il) gauge theory. Fairlie and oth2Ts[J.Ol further extended the graded gauge fields over ,a space time manifold with more than 4 dimensions. Such an embedding would fix the Weinberg angle to be sin"9. = % and if it was performed in an extended space time or super space time manifold. the Higgs mass could thus be determined. The number of extra dimensions is equal to twice the number of lepton triplets. Otherwise, the lepton masses will be the same. The common feature of this kind of theory is such that it contains unphysical particles with unfamiliar statistical properties. In the present note, helicity mixed rt'presentation is u&ed to construct a model of unified eIectro-w4!ak theory for leptons in a gauge theory of simple group SU(3). E~b gen-eration of two component leptons forms half a triplet in SU(3).

sets

The generatOTS of the group SU(3) ii' i = 1, "·,8 can be decomposed into two i. and i,. where a and a are listed in Table 1. Tabl.l

Case I

n m IV V

VI

vn

a

IX

257 246 147 156 1238 3 678 34 58

13468 1 3578 2356'8 2 347 8 456 7 1245 1267

We now define new generators i~c) by

..

[AC.)

where

€

=- [I<..,

(1)

commutes with i~·) and satisfies the relation (2)

It can be easily verified that the Lie algebras for i!c) are the same as those for. FGur poarible choices for €:€ = +1, -1, + ')Is, - '')Is will be used in the following. They are denoted by +, -, 5, - 5 respectively.

t/.

212 22

KEXUE TONGBAO

Vol. 25

Besides the local SU(3) symmetry, a global U(l) symmetry with generator S is introduced in the present model. T.he lepton triplets are transformed in the fc,Uowing way

c/J - c/J' = U(5J(gi(:I:)) eiS'I c/J, iii -

iii' = iiie-iS'I U<-SJ+(g;(:I:)),

(3)

where U<')(Si(:I:)) = exp{iI~')S;(:I:)} are the transformation matrices in SU(3), ~(x) and 1/ are the group parameters for local SU(3) and global U(l) group respectively. The generator

S = 1.. y,

for lepton triplets. 6 Gauge invariant kinetic energy :001" the leptons has the form

iiiy"(o" + J~») c/J, where 1~')'

=

igA.~J~I) are the gauge potentials. 1~') -1l:)' =

UCI)(O"

(4)

They will be transformed as

+ 1l,'») U<·)t.

(5)

There are many possible choices of the Higgs fields. Here as an example we take it to be a 3 X 3 matrix

Their covariant derivatives are (7)

For concreten~ we shall use a and a li:sh:d in the fir:st case in Table 1. The first four cases give essentially the same results. In this case the Higgs field can be chosen as (8)

which is a six-dhnensional representation of the- gauge group. AllGther possible choice is a three-dimensional representation !P(3) = 4>"

J...

The invaTiant Lagrangian has the form

+ 1.. tr { (D ,,!P )t(D"!P)} + 1.. f iii!P(l + 2

+ 1.. f*i;j!P+(1 2

2-

y,)c/J - V(!P, !P+).

Ys)c/J (9)

If there are only SU(3) invariants of even power of

213 No. 1-2

KEXUE TONGBAO

Th'e vacuum expectation value of the Higgs scalar

cp(6)

is taken to be

(10)

=

where iii 1, ... ,8 are the Gell-lVIann matrices for SU (3). The electromagnetic gauge group UE.M(l) and a global UCl) with generator Sw remain unbroken. The conserved charge Q and the weak strangeness SJV COITE'Spo,nding to these two 1111br10ken U (1) symmetry are re,speetively, (11)

(12) The masses of the vector bosons are found to be

,4 2 3" mw,

rnz =

2

rny

=

2

'inw,

(13)

242

l1tu =

'mw,

where V± and U±± are four new vector b-rsons corresponding to

v± = .;\ U±± =

;2

(Ai ± iA5), (14)

(A 6 ±iA7).

The photon field A and the neutral vector boson Z· are related to the gauge fields

A 3 and A· in the following way: (15) The charge and the weak strangeness for gauge bosons are list'ed in Table 2. Table 2

Particles Q

S'"

w±

.d

ZO

V±

u±±

±1

0

0

±l

±2

0

0

0

"+1

"+1

We require that the weak strangeness for the yet to be observed leptons be ¥2. This ensures that the helieity of the lepton triplet be 'Of the form (L, L, R) after spontaneous symmetry breaking. There are S.. = -¥2 compronents of the form CR, R, L) in?p. It is possible to make their masses very heavy by a suitable choice of additional Higgs fields. We assign to the components for the first generation of leptens as

214 KEXUE TONGBAO

Vol. 25

(16)

The charge nperator for leptons now takes the form

(17) When a lepton triplet of the form (16) is acted UpOD, the charge operator becomes "

1,

Q= 2

-

"

0.3 - .j 3 18) =

0 (

-1 -1

)

'

(18)

where (19)

We note that i~ is just the generato r used by Ne 'eman, Fairlie and others in their graded gauge group formulatil()n. It occurs here simply as a result of the difference in the action of rs on left-handed and right-handed leptons, i.e.

rsL = L,

rsR

=

-R,

The weak and electromagnetic interactions for leptons are nnw easily obtained from the Lagraillgian. It:is easily verified that these interactions have exactly the conventional form in the Weinberg-Salam model with sin·8.. = lJa,. The weak strange vector bOBons dQ not interact with the present observed leptons directly. They only couple E", ~ leptons with the E.. -lh heavy leptons. Therefore, they have practically no influences on the low-energy weak interaction.

=

=

Five he.avy scalaTs remain in the p,resent choice of Higgs fields. They can have arbitrary masses when suitable self-interaction potential is chosen. One of the Higgs scalars is a real neutral field, with zero weak strangeness. Its vac.uum expectation value is v given in (10) which generates masses for both vector bosons and electrons. The other four Higgs are weak strange particles which do not cl()uple to leptoills directly. Two of these Higgs are neutral and the other two are doubly charged. Conservation of weak strangeness ensures the pair production of weak strange particles. The weak strange bosons I()f the lightest mass would be stable particles if this conservation law is exact. The V'" and U±= vector bosons can be produced through the mediation of other heavy vector bosons and photons. The production cross section for a V'" pair is the same as that for a W'" pair in an electron-positron annihilation reaction with sufficiently high energy. These characteristics may be of help in the experiments looking for these weak strange particles. When terms like detIP + detIP+ were allowed in the Higgs self-interaction potential, the conservatiOill of weak strangeness would be broken and all heavy bosons

215 No. 1-2

KEXUE TONGBAO

25

would be1!ome unstable. Nevertheless, eve!ll iill this case, the decay amplitude of the V-particle (assumed to be the lightest of the weak strange particles) is of the order of magnitude O(g3) in contrast with O(g) for the decay of W-particles.

If !p(3) is chosen t'O be the Higgs, a residual SU (2) symmetry remains unbroken and the double charged U-particIes remain massless together with the photon. A linear combination of !p(3) and !p(S) will not change the mass 'Of V-particle but left the U-particle mass in the range 0 ~ m. ~ 2m",. All 'Other c'Onclusi'Ons still hold in these cases. There is no difficulty in adding further generati'Ons 'Of leptons like (v., JL) and (VT, 1') in the present theory.

The extensi'On 'Of the present m'Odel to include quarks is a pr'Oblem yet to be solved. The difficulty lies in the different charge assignments f'Or leptons and quarks. There are tw'O possibilities: Either c'Omp'Osite model in terms of prequarks should be tried, or the gauge group SU(3) should be enlarged. Both of these approaches have their ,'Own problems. S'Ome of the preliminary results 'On this subject will be reported in 'Other papers. We would like t'O thank 'Our colleagues at the Institute 'Of The'Oretical Physics, and the Institute of High Energy Physics, Academia Sinica and Beijing University for their interest in this work. REFERENCES

[ 1] Ne'eman, Y., Phys. Lett., 81B(1979), 190. [2] Fairlie, D. B., ibid., 82B(1979), 97. [3] Weinberg, S., Phys. Bev. Lett., 19(1967), 1264; Salam, A., Proc. 8th. Nobel Symposiwm, Stockholm, 1968; GIashow, S. L., Illioponlos J. & Maiani. L., Phys. B.e~·" D2(1970), 1285. [4] Squires, E. J., Phys. Lett., 82B (1979),395; Jaylor, .J. G., ibid., 83B(1979), 331; 84B(1979), 79; Dondi, P. H. & Jarvis, P. D., ibid., 84B(1979), 75.

216

Vol.

~5

-"0.

KEXUE

8

TONGBAO

August 1980

DISORDER PARAMETER AND DUALITY ZHOU GUANGZHAO

(CHOU KUANG-CHAO

Jj!jJ:7tB)

(Institute of Theoretical Ph y,\'ics , Academia Sinica) AND XrAN

DrNGCHANG

(HSIEN TING-CH.\NG

i]H"~i§)

(Institute of High Energy Physics, Academia Sinica) Received August 17, 1979.

In studying the problem of quark confinement, t' Hooft introduced a loop dependent disorder parameter B(C'), which depends on loop C' and is related to the Wilson loop operator A (C) of the non-Abelian gauge field .{u by the algebraic equation[l] B(C')A(C)

A(C)B(C'):.

=

(1)

For the SU(N) group, z is the element of the center of the group Z(X). t' Hooft defined the disorder parameter B( C') by singular gauge transformations. The authors of Refs. [1.-3] luwe poilltt>d out the elctromagnetic duality property between E( C') and .1(C). By introducing dual potential and defining the Wilson loop operator i(C') as composed of dual potential, Halpern[2] proved that in the Abelian case, .I(C') is a realization of BeC') in the algebraic equation (1) and guessed that a corresponding i( C') could be introduced in the non-Abelian case, However, in all these works. no direct proof has been given for the explicit electromagnetic duality property between B( C') as defined by the singular gauge transformations, and A (C). The purpose of this paper is to discuss directly the (luality property bet"'een n( C') 'and A (C) from the definition of B( C'). In order to illustrate our method, let us discuss first the transformation, the gauge potential A" will bpcome A:

=

A,u

~\beliall

case.

By gauge

+ l.. a).,

(~)

e

whose value is path-dependent when 1 is singular. Now by defining the field strength corresponding to this singular gauge transformation as

F~t'"

=

l.. 2e

[a", au]}"

p,

lJ

=

O. 1, 2,:3-

and by integrating Eq. (3) over a surface L: bounded by a loop C, one obtains

!fa

,,1dy" =

ff P':udy" 1\ dyV.

(4)

I

Let us consider the case of two 3-dimensional loops C and C' at the same specific time, the coordinates of them are respectively

217 636

Vol.

KEXUE TONGBAO

C:

y=y(e),

C': z = z(e'),

Suppose Hi!

= ~

Ckj/F'j;(k, j, l

0~e~21<,

y(0)=y(21<);

0 ~ e' ~ 21<,

z(O) = z(21<).

~5

(5)

= 1,2,3) to be the magnetic field strength formed by

the ma,,"'Iletic force lines along the direction of C', then it follows from Eq. (4) ).(21<) - ).(0)

=

(6)

neiP,

where n is the number of times loop C encircling loop C', counterclockwise. For the Abelian case, the values of ). at the same point reached by different paths can differ by an integer factor of 21<, hence, ).(21<) - ),(0)

(7)

2n-mn,

=

From Eqs. (6) and (7) the quantization condition of the

where m is also an integer. ma,,"'Iletic flux

eiP

=

(8)

21<»1

is obtained. The above-mentioned field strength along C· may be expressed as

(0) And from Eq. (3) the following equation for)' can be derived:

!

8 j ).(x, C')

1

(10)

d3x'G(x - x')iYF'j;(x', C'),

=

wherei, j, l, assume the spatial indices 1,2,3, only; G(x) satisfies the following equation

In the gauge of Ao = 0, the disorder operator B( C') is defined as the following operator of singular gauge transformation B(C')

where

Fa;

=

exp { i 1FOj(x) 8 j

(! ).(x, C') )d3X},

(11)

represents the electric field strength in the Heisenberg representation and is

the canonically conjugate operator of the gauge potential Aj; 8 j

(! ).(x, C'))

is given

by (10). Carrying out the variation of Eq. (11) with respect to the coordinate z(e') of loop C', one obtains B-1( C')IJB( C')

=

i 1FOi8j

(! 1J)'(x, C') ) d x, 3

(12)

and from (9) and (10) it follows

8j

(

!

1J)'(x, C') )

=

1

d3x'G(x - x')8 1IJFj:(x', C'),

(13)

218 KEXUE TONGBAD

No.8

637

and (14) Substituting (14) into (13), and then the result into (12); by partial integration and using the Gauss's theorem akFok = 0, one finds

=

i Ad ( ) k I -I/1Fkl Z dz 1\ dz , 2

(15)

AI" where Ftl = 2 ckll'"FI'V is the conjugate field strength.

In the gauge of A. o = 0, the Wilson loop operator A (C) is A(C)

=

exp

fie f

(16)

Ai(y)dyi}.

C

Now from the variation of A. (C) with respect to the coordinate y( a) of loop C, it is easy to derive (17) and compariug (1;)

with (17), one finds the following electric and ma,,"'Iletic duality

properties:

e -1/1;

Fkl ~ Ftl;

yea)

~ z(8');

B(C') ~ A.(C).

(18)

It should be pointed out that this duality property holds only for a pure electromagnetic field system, since the sourceless Gauss's theorem has been used in the proof.

Now let us discuss the non-Abelian case and the gauge group is chosen to be SU(N). First of ail, we have to look for the singular gauge transformations which define the disorder operator. Many authors[~l have pointed out that in the non-Abelian case, corresponding to the gauge inyariant ma,,"'Iletic string, a unit vector lex) of Lie algebra should be introduced, which remains invariant under parallel displacement:

DI'I(x) = al'l(x)

+

e[A,,(x),

lCx)]

=

0,

tr

chx»

= 2.

(19)

And the field strength projection in the direction along lex)

Fl'v

=

.1 -

2

,," tr CFl'vI)

(20)

contains the gauge invariant magnetic string. The group parameter defined as

1.Cx, C') characterizing the singular gauge transformation is

lex, C') = 1.Cx, C')1. = l(x»).(x, C'),

(21)

219 638

KEXUE TONGBAO

Vol. 25

i.,

where a = 1, ... nG are the generators of the group G; lex, C') is a singular function defined by Eq. (10). In the Ao = 0 gauge, the disorder parameter is defined as the singular gauge transformation by the following expression: (22)

lex)

By using the invariance property of follows

under parallel displacement, from (22) it

(23) where FOi is the gauge invariant electric field strength defined by (20). It is easy to see that (23) and the corresponding operator in the Abelian gauge field theory possess the same form, hence, by variation with respect to the coordinate of loop C', one finds

B-I(C')oB(C')

=

;

tPFj,,(z)dz i 1\ dz",

(:2 -±)

where

(25) represents the dual strength. In the Ao

=

0 gauge, the Wilson loop operator is

.fl(C)

=

tr

(T exp{ie f .l;(y)dyi}),

(26)

c

where T represents the operator of ordering along loop C.

B(C')A(C)B-I(C')

=

It is easy to prover;]

_-1(C)z,

where

z

=

(27)

exp {ietPl(x)}

represents the elements in the center ZeN). By carrying out the variation of A(C) with respect to the coordinate of loop C, Nambu et al. !61 have obtained

(28) where (20) while

ci>(y, C)

=

A -I( C)1. tr {T (la(Y) exp

{ie f A/y)dyi})}.

(30)

c

The 1.(y) in Eq. (30) represents the 1. at point y of loop C. Generally speaking, ci>(y, C) is a functional of loop C. However, it has been proved[61 that ci>(y, C) is invariant under parrallel displacement along the direction of loop C, though it does not possess this property along other directions.

220 No.8

KEXUE TO::-i"GBAO

639

By comparing (24) with (28), it can be seen that there is some sort of duality property between B(C') and A.CC). But this duality property is not as perfect as that in the Abelian case because of the fact that lex) and t:/>(y, C) iLre not completely similar in character. This may not be as surprising as it appears since it is not always possible to find out an appropriate dual potential(7] in the non-Abelian gauge field theory. After the submission of this work for publication, we noticed that Mandelstam[B] had considered some problems similar to that discussed in the present paper. REFERENCES

[ 1] [ 2] [ 3] [4,]

[5] [ 6]

[ 7] [8]

't Rooft G., N1I.c1. Phys., B138 (1978), l. Halpern, M. B., Phys. Re1l., 019 (1979), 517. Yoneya, T., Nucl. Phys., Bl« (1978), 195; Englert, F. & Windey, P., Phys. Rep., 49 (1979), 173. 't Hooft G., Nucl. Phys., B79 (1974), 276; Polyakov, A. M., JETP Lett., 2(J (1974), 20; Ezawa, Z. F. & Tze, R. C., Nucl. Phys., BI00 (1975), 1; ~-±, ;§;!l!i*, ~f8+, «~llI!~f[l:o, 25 (1976), 514; Arafune, J., Frennd, P. G. O. & 000001, C. J., .1. Jlath. Phys .• 16 (1975), 433. Corrigan, E. & Olive, D., Nucl. P1IYs., B110 (1976), 237. Nambu, Y., Phys. Lett., 80B (1979), 372; Corrigan, E. & Ifussla.cher, B., Physl Lett., 81B (1979), 181; Gervais, J. L. & Neveu, A., Phys. Lett., 80B (1979), 255. Gn Chao·hao & C. N. Yang, Sci.. Sin., 18 (1975), 483; Brandt, R. A. & Neri, F., Nucl. Phys., BI4S (1978), 22l. 1Iandelstam, S., Phys. Rev., 019 (1979), ~391.

221 Reprinted with permission from AlP. Con! Proc. 72 (1980) 621. Copyright 1981 American Institute of Physics.

621 AXIAL UCI) ANOMALY AND CHIRAL SYMMETRY-BREAKING IN QCD K. C. Chou* Virginia Polytechnic Institute, Blacksburg, VA It is well known that the absence of the AIlJ anomaly is necesl sary for the corresponding gauge field theory to be renormalizable . This condition places severe restrictions on the choice of the possible gauge group,and the representation for fermions, as we have just heard from A. Zee.

I would like to report, on the

other hand, some consequences of the presence of ABJ anomaly in certain global current conservation equations.

This is a work done

in collaboration with L. N. Chang. One important question in QCD is the origin of chiral symmetry breaking.

This problem is related to the structure of the theory

for large distances,where perturbation theory cannot be used. It has long been conjectured that topologically non-trivial gauge field configurations play some significant role in explaining both confinement and chiral symmetry breaking

2

3 4 Recently, Coleman and Witten , and Veneziano , have analyzed the question within the context of liN

c

expansion.

In particular,

Coleman and Witten argue that the axial anomaly and confinement already imply chiral symmetry breaking.

Their argument makes

no essential use of the non-trivial topological configurations, but relies instead on the absence of analyticity structure in the axial vector vertex, brought about by the anomaly. reaches the same conclusion, using the liN

c

Veneziano

expansion and by

*On leave from Institute for Theoretical Physics, Peking.

222 622 consideration of the fluctuation of topological charge density in the pure Y-M field sector. In this talk I want to point out that their conclusion on the necessity of chiral symmetry breaking can be obtained without recourse to the liN

c

expansion, if proper attention is paid to

the quantum fluctuations in the topological charge density. We start by recalling that in the presence of non-trivial topological configurations, one could incorporate the 8-vacuum caused by the resultant phenomenon of tunneling by augmenting the conventional Lagrangian with an additional term

2

vex)

~ Fi *F illV D 3211"2 llV ' II

*F

2

1 ].J\i

d

II

(1)

gAll

FOP

£ ].J\iCp

In equation (1), 8 is the parameter characterizing the topological structure of the vacuum, while J (x) are external a sources coupled to various combinations of quark currents 0 (x). a Since we are interested in chiral symmetry breaking, our attention will be focused on the densities Ci(l±YS)q

==

O± and Cil(1± YS)q

==

O~.

In the following 8 will be considered as a function of x in some intermediate steps of calculation. The Lagrangian of (1) has an apparent D(N ) x D(N ) chiral f f symmetrY,with N flavours, if we set Ja(x) f

0.

However, due to

the ABJ anomaly in the axial current, 8 will change to 8 +/2N f ~S under the abelian chiral phase transformation q(x)

~ exp[iY5~5]q(x).

The best way to study the chiral symmetry structure of (1)

223 623 when J(x) #

° is

through the effective generating functional,

which we shall now define.

The generating functional, W, for

the connected Green's functions implied by (1) can be expressed as W[J,B]

tn

-i

=

Z = JDqDqDA

Z 4

II

exp{i!d x c£(x) + /l,i..(x)]

(2)

Here/l,~x) includes the gauge fixing and compensating terms

necessary to give meaning to the A integration. II

The classical

fields Vex) can now be defined by direct differentiation

v

oW

(x)

(3)

As a result of the axial anomaly, and the formal invariance properties of (1), the generating functional W has to satisfy a Ward identity of the form

a

II

oW oj +(x) ll-

= iJ

±

oW + N ~ oJt(x) - f oB(x)

(4)

-

This Ward identity can be satisfied by any functional with the following local invariance property

B(x) - ~ ~5(x)J

where

~5(x)

(5)

is an arbitrary function of x.

We define now the generating functional r by using a Legendre transformation on the sources of the scalar currents

o±

224 624

f[U±(x), JW±(x), O(x)] W[J,(x), J ,(x), O(x)] -

+ Jd

\J:'"

4

(J+(x)U+(x) + J (x) U_(x»)

x

(6)

Then the Ward identity for the axial UCI) symmetry implies that r is invariant under the local transformation

J + ~-

8(x)

J +(x) +3

7

w-

~N

w~Nf

8(x) - 12N

7

f

(7)

f,S(x)

~s(x)

Or in other words, r is a functional of the form +i

L

r[U±e

8

Nf

, J +(x) ~-

I

+ 3 (N

-

~

f

e)]

(8)

Note that the classical fields U±(x), which are the vacuum expectation values of the corresponding quark bilinear fields, can be determined through the relation

(9)

Any nontrivial solution to (9) when J

±

= 0,

J

~±

=0

and e(x)

= cQnstant,

would signal the existence of spontaneous chiral symmetry breaking. Since U

+

~

U

-

+if'f U

±

+ Ue-

1N"f

*

we can write them in the form

.'L LIT

(10)

where n' can be interpreted as the vacuum expectation value of the n'-meson field which corresponds to the axial U(l) pseudo Goldstone field in chiral dynamics.

225 625

fiNf

Notice that r is an even function of 8 - ---f- n' as a consequence 11

of (8), and the symmetry under space reflection. that the CP conserving solution of eq. (9) at J.

It is easily proved =

0, J+

-11

= 0,

e(x) = e is f

n'

e

11

12N

(11)

f

Thus all the physical quantity evaluated at this point will be

e

independent and CP conserving. Nevertheless, the nth order derivative of r with respect to

8 when n' is kept fixed is the Green's function of nv(x) 's where diagrams with one particle lines of nand U are removed.

::~ In"~

(_i)n fd4xl···d4xn<0[T(V(Xl)···V(Xn)) [O>I.P.r.

It has been shown by one of us the n' meson mass.

5

that

02~ I11 'u 08

(12)

is proportional to

This result is a generalization of the result

given first by Witten

6

in the leading liN

c

expansion approximation.

The point we wish to emphasize is that (12) can be nonvanishing only if U 1 0, so that if any of the moments defined in (12) were to be nonzero, chiral symmetry would be spontaneously broken.

This is the main conclusion of my talk.

Now the right

hand side of (12) represents a quantum correlation function

of

the topological charge density, and there is no general reason for (12) to vanish for n even.

The case for n odd can be

excluded, of course, in the chiral limit when CP is a good symmetry.

We therefore argue that QCD will. in general, induce

the spontaneous symmetry breaking of flavor

chiral symmetry.

226 626 The picture we are presenting may therefore be summarized as follows:

Owing to the presence of instantons, the QCD vacuum

acquires an additional parameter O.

In the absence of any

external spontaneous chiral symmetry breaking, like those induced by Higgs couplings, the chiral phases of the quarks will automatically refer themselves to 8. consequence of the axial anomaly.

This is the direct

However, large scale quantum

fluctuations of the topological charge density requires such phases to be defined globally, which can only occur if the chiral symmetry is spontaneously broken.

Hence quantum corrections to

topologically non-trivial gauge configurations induce spontaneous chiral symmetry breaking.

REFERENCES 1.

D. J. Gross and R. Jackiw, Phys. Rev. D6, (1972) 477.

C. P.

Korthals Altes and M. Perrottet, Phys. Lett. 39B, (1972) 546. 2.

G. 't Hooft, Phys. Rev. D14 (1976) 3432. Rev. Lett.

~

(1977) 121.

D. G. Caldi, Phys.

C. G. Callan, R. F. Dashen and

D. J. Gross, Phys. Rev. D17 (1978) 2717. 3.

S. Coleman and E. Witten, Phys. Rev. Lett. 45 (1980) 100.

4.

G. Veneziano, CERN TH-2872 (1980).

5.

K. C. Chou, ASITP-80-005 (1980).

6.

E. Witten, Nucl. Phys. B156 (1979) 269.

227 PHYSICAL REVIEW D

1 JUNE 1981

VOLUME 23, NUMI:IER 11

Possible SU(4), X SU(3)J X U(l) model Chong-Shou Gao StQnford Linear Accelerator Center, Stanford University, Stanford, Calljornia 94305 and Department of Physics. Beijing University. Beijing, People's Republic of China'

K uang-Chao Chou Institute a/Theoretical Physics, Academia Sinica, Beijing, People's Republic nlChina IReceived 14 October 1980) An anomaly-free model of strong and electroweak interactions involving leptons and quarks in the

SU(4), XSU(3)jxU(I) gauge theory is constructed. After spontaneolls symmetry breaking, it reduces to quantum chromodynamics for strong interactions and a broken SU(J) X U( 1) model for electroweak interactions. As a limiting case it gives the same results as those of the Weinberg-Salam model in the low-energy region. The Weinberg angle is bounded by sin'e" < 1/4 and becomes slightly less than 30' in the limiting case. Below the mass scale of SU(4), breaking there exists an inequality between the Weinberg angle and the strong coupling constant. which is consistent with experiments. A correction to the neutral current of the Weinbcrg·Salam model is suggested. A new conserved quantum number is introduced in this model and there exist several new fermions with masses lighter than 160 GeV. The Kobayashi-Maskawa expression of Cabibbo mixing for quarks may be obtained in the model, generalized to include several generations of fcrmions.

l INTRODUCTION

Recent neutrino-induced neutral-current experiments are in agreement with the expectations based on the Weinberg-Salam model. ' The Weinberg angle 8w is found to be sin28w = 0.230 ± 0.009, averaged over the various experiments. 2 Beyond the Weinberg-Salam model one may ask the following. (1) Is there any symmetry higher than SU(2)L x U(l) for the electroweak interaction? (2) Does sin28w being slightly less than t have special physical meaning? (3) How can the Weinberg-Salam model be unified with the strong interaction? If the answer to the first question is "no," the next problem to be solved is the grand unification of the Weinberg-Salam model with the strong interaction. In this way one may construct a model of grand unification, such as the SU(5) model suggested by Georgi and Glashow. 3 If one thinks that the answer to the first question is "yes," this leads to another question: What kind of higher symmetry might this be? There are several considerations which may become the motivations to choose the higher symmetry group: (i) left-right symmetry before spontaneous symmetry breaking, (ii) quark-lepton unification, and (iii) a sensible prediction for the empirical Weinberg angle. Among the many possibilities meeting these criteria is the SU(3)X U(i) group for the electroweak interactions. In a previous paper' a model with the SU(3)x U(i) gauge group was proposed. This model is left-right symmetric before spontaneous symmetry breaking and anomaly free. It gives the same

results as those of the Weinberg-Salam model in the low-energy region in a limiting case. The Weinberg angle is bounded by sin 2 8w "; t in this model and sin28w becomes slightly less than t in the limiting case. In this paper we discuss a way of unifying the strong and electroweak interactions by embedding this SU(3)x u(1) model into a larger one, SU(4)c xSU(3)x U(l), where the main results of Ref. 4, including the interesting property of sin 2 ew ,,; t, are preserved and several further consequences are obtained. Before discussing this model we will briefly analyze the construction of the SU(3)x U(l) group, which will be helpful in understanding the motivation of an extension to SU(4)XSU(3)XU(1). When one embeds the SU(2)LX U(l) model into an SU(3)x U(l) model, a naive requirement is that "L and e L correspond to the first two components of a left-handed triplet and e R correspond to the third component of the right-handed triplet of the SU(3) group. There are two pOssibilities. Case A. "L and e L belong to the representation ~L and e R belongs to the representation ~R of the SU(3) group. This case has been investigated by Lee, Weinberg, Shrock, Segre, Weyers, and many others" in detail. Case B. "L and e L belong to the representation l L while eR belongs to the representation ~j" the conjugate representation of~. This case is investigated in Ref. 4. These two cases lead to different consequences, as summarized in Table I. In the expressions for the charge operator, j 3 and j, are the third and the eighth generators of the SU(3) group, respect2690

© 19R1 TIle Americ;)n Physical Society

228 23

2691

POSSIBLE SU(4), X SU(3), X UtI) MODEL TABLE I. Comparison of the two possible schemes in the SU(3)x U(l) group for electroweak interactions. Case A

Case B

Charge operator

Q;i3 +(1/,r:nia+ f

Q;i3 -FJia+Y

Sin 201V

1 3 41+(3g'/g")

1 1 41+(g'/g")

Boundary

<.!4

<1-4

New conserved quantum number

No

Additional heavy particles

Yes

Yes

With exotic charges

No

Some of them

To be embedded into an SU(6) model

Easily

Cannot

ively, while Y is the generator of the U(1) group. Of course the Y assignment of the fermion multiplets are different in these two cases. In case B, there exists an additional conserved quantum number called the weak strangeness s." coming from an unbroken U(1) symmetry after spontaneous symmetry breaking. There are several heavy particles to be discovered in this case too. But most of them have nonvanishing values of S IV while the known particles in the Weinberg-Salam model have SIV; O. We may call these particles with Sw '" 0 the weak strange particles. Some of them have "exotic" values of charge, for example, Q= 2 for a heavy vector boson and Q =} for a heavy quark. The most interesting characteristic of the case B is that the upper bound on the Weinberg angle is close to the measured value. In addition, the existence of the conservation of weak strangeness gives many new physical. predictions for the highenergy electroweak interactions, and is therefore interesting in its own right. However, because the left-handed and right-handed fermions belong to different representations of the SU(3) group, this kind of model cannot be embedded into a simple SU(6) model of grand unification. So we have to study other ways of connecting this kind of model with the strong interaction. One attractive idea is that the color group may be an SU(4) group and the lepton number may be treated as the fourth color as suggested by Pati and Salam.6 Adopting this idea, we will extend this kind of SU(3)xU(1) model to an SU(4)CXSU(3)xU(1) model.

Weak strangeness

the corresponding gauge fields, and the coupling constants are denoted by

ii, ct,

j=1, ... ,15, g"forSU(4),

fo A~, i=1, •.. ,8, gforSU(3), F, B~, g'for U(1) ,

respectively. Besides the local symmetry there is another global U(l) symmetry whose generator will be denoted by S. This global U(l) will combine with an Abelian subgroup in SU(4)xSU(3) x U(l) to give a new conserved quantum number S w after spontaneous symmetry breaking. We will use four numbers in parentheses (m, n, F, S) to denote the representations for these four groups, respectively. For example, (~, :!.' -1, 1) means the representation ~ for the SU(4), the representation 3 for the SU(3), F= -1 for the local U(l), and 5 =1 for the global U(l) groups. For simplicity we discuss the model involving only one generation of fermions. It can easily be extended to the case involving several generations. The fermions form four left-handed multiplets and four right-handed multiplets: IjJL:

(~,

:!.'

1, 1), IjJR: (~, ~, 1, -1),

SL:

(~,

!,

1, -3), SR: (~,

1jJ"L'

(~, 3* , -1, -1),

S"L' (~,

!,

-1, 3),

Sf. R'

1jJ"R'

!,

1, 3),

('!.' :!.' -1, 1),

('!.' !, -1, - 3). (2.1)

II. FUNDAMENTAL STRUcrURE OF THE MODEL

The local gauge groups considered in this model are the SU(4) color group, the SU(3) flavor group, and the U(1) group. Their generators,

After spontaneous symmetry breaking, one SU(3) symmetry, one local U(1) symmetry, and one global U(1) symmetry remain unbroken. The unbroken SU(3) group is a subgroup on the first

229 2692

CHONG·SHOU GAO AND

three dimensions of the SU(4) group and becomes the color gauge group for quarks. The generators of the local U(1) and the global U(l) groups are the charge

Q=i, -{3 iB+ m'i2j;5 +~i

(2.2)

T ABLE II. Charge and weak-strangeness quantum number for fermions. Q

e

2' SW=ff IB-~F+tS, A

•

(2.3)

respectively. Since quarks are degenerate for three colors and the color index is unimportant in many discussions, we may omit it and express the fermion multiplets as

iJ!L

UL

gR

d'L

xR

dL

hR

U'L

wR

hL

dR

,

= ilL

NR

eL

ER

eR

EL

,

iJ!~ =

e'L ~'L

~L

-1

E

-1

u

~'

,5'

iJ!~

-1

N

2

"5" 1

d

-"3

g

3

2

-1

Iz

5

-1

"3

v'

0

e'

-1

u'R

we.

,

iJ!R =

Sw

~,S

and the weak strangeness •

23

KUANG·CIIAO CHOU

-2 -1

= ~R

~R ~'R

(2.4)

2

u'

"3

d'

1

-3

w

-"3

x

1

4

-"5"

Their transformation properties are where the symbols in the upper half of each column vector denote three colors of quarks while the symbols in the lower half of each column vector denote leptons. The quantum-number assignments of various states are listed in Table II. This model is anomaly free; the proof can be carried out in the same way as in Ref. 4. III. SPONTANEOUS SYMMETRY BREAKING

Six multiplets of the Higgs fields are introduced to realize the spontaneous symmetry breaking.

respectively. The self-interaction potential of the Higgs fields is chosen to be

F

V=

l: [- a, tr4> i 4>, + b, (trot> T4>, )2J + c(trot> ~ 4> D)(tr4> 1ot> A)

I=A

(3.2)

where a's, b's, c, d, e, andf> 0,

A' it leads to a stable minimum for V. The vacuum expectation values of the Higgs fields may be taken to be

4> B' ot> c' ot> D' ot> E' and ot> F' respectively. As shown in Appendix A,

230 23

2693

POSSIBLE SU(4)c X SU(3), X U(l) MODEL

<.')'""'~ ~; ;1°(:), <.")'" "[~ ~; ;1 <.,)."~ [~

;;: {

::}

<••)"".

0 (:)

m·(: :D'

(3.3)

<%),"~ ::: _}G). <.,),"",[:: +0 respectively, where all v's are positive and determined by the coefficients in V. Owing to the stability of the minimum, no pseudo-Goldstone particle appears after spontaneous symmetry breaking and all the remaining Higgs particles are massive. They may be rather heavy by a suitable choice of the coefficients in the self-interaction potential. One may easily verify that the color SU(3) symmetry, the electromagnetic U(I) symmetry, and the weak strange U(I) symmetry remain unbroken after spontaneous symmetry breaking. There are 24 gauge bosons in this model. After spontaneous symmetry breaking all gauge bosons other than eight gluons and the photon get masses. The mass terms of the vector bosons have the form

! g2(V /+vs 2+VC 2+VF 2) W+W- + !g2(V A2+ Vs 2+VC 2+ VD2)V+V- +! g2(4v s 2+Vn 2 +VF 2)U++ U--

(3.4) where 4

W'=k(A 1 'fiA 2 ), V'=;i(A ±iA

uH

5

),

=k(A 6 ±iA 7 ), C'12/3=Ji(C9'fiCI0) , (3.5)

C ,2/3 =_1_ (C"

2.f2

'f

iC 12 )

C .2/3 =_1_ (C 13 'f ic 14 ) '3 12 .

Three neutral gauge bosons will get masses from the last four terms in (3.4); we denote them by Z I' Z 2' and Z 3' respectively. Eight gluons and the photon remain massless. The electromagnetic field is A1 (.lA3 -(1+1/A 2 +11/l2)1/22 -

The masses of these particles are

13 A" 2

1 15 1 \ +~C +/"LBJ' (3.8)

where >..=.f6

mv2=!g2(VA2+VS2+VC 2+V D2) , mu 2= !g2(4vS2+VD2 +VF2),

(3.6)

respectively. From (3.6) we get the inequalities

L

g

,

/l=21L. g

(3.9)

The quantum numbers of these bosons are listed in Table III. The bosons indicated by an asterisk are new particles introduced in this model. The six C~2/2 are the so-called leptoquarks. They have fractional charges and couple quarks to leptons. The V and U bosons have nonvanishing weak strangeness. We will call this kind of particles

231 23

CHONG-SHOU GAO AND KUANG-CHAO CHOU

2694

TABLE II!. Charge and weak-strangeness quantum number for gauge bosons.

Gluons Q

0

SO'

,

,

e 213

e~213

'Y

, ,-

0

0

0

0

V·

W -1

Q

WS

WS

WS

V-

u"

Z3

o o

-1

(3.14)

.

The limiting condition (3.10) ensures that sin 28w is slightly less than ~ and is consistent with experiment. Both Z2 and Z3 are much heavier than Z" Their dominant components are Band C15. For two special cases the simple expressions simplify_ Case A. When }J.2VD' » ~(A2 + }J.2)VE" we get

-1

ws

the weak strange particles_ An interesting limiting case is g2«g,2, g1f2,

we obtain to the first order of approximation for the I/A2 and 1/1-12 expansion that

sin28W=O~[1-e2 +~)J

U

-2

-1

0

0

SO'

Z,

(3.13)

QCD W·

Z1

, ,-

to the Z boson in the Weinberg-Salam model. Comparing with the mass formula in the WeinbergSalam model

i.e., A" 1-1'» 1,

(3.15) (3.10)

vp2«vA2+vB2+vc 2 «vn2, VE2.

Z2

The physical meaning of the limiting case is that the coupling constant of the SU(3) group is much smaller than those of the SU(4) color group and the U(!) group; the mass scales of SU(4) and SU(3)t breaking are much higher than th~t for the secondary SU(2), breaking. In this limiting case the masses of the three massive neutral bosons are approximately

'"

15 C , Z3

(3.16)

B.

'"

Case B. When }J.2V D2« {(A2 + }J.2)VE 2, we get

2

A2112

-1

2

m z '" 2ti-2--2VD' 2 A +}J.

mZ32

",{g'(A2 +

2 112)v E

(3.17)

1

(

Z3 '" (A2 + }J.2)'J2

-}J.B + AC

15)

(3.18)

The masses of gauge bosons are shown in Fig. 1.

- [(1-1

,

VD

It manifests the existence of three mass scales, which are related to the breaking of SU(4)c' SU(3)" and SU(2)f' respectively.

,~ 2)2 + 4 vE

-A2}J.2V D2V

s' r/2}

The interaction Lagrangian among the fermions and the Higgs fields has the form (3.11)

Case

Case 8

A

Z3

m

Z3

A2+ 2 2 "''-g2 { }J.2V 2+~ 2 4 . D 4 vE ,,2+ 2 + [( 1l2VD2+~VE2

_A,21l2VD2VE2

Z

1

f2}

can be expressed as

z,,,,i(ff A3+AB) -

1 13

-

r

i

Z3

I

"

""

Z3 ---'/

( ~

•

e

/

/1

/

/

"" " "" " " /'

I

i

ISU(4)c

C

I breaking I

/

"

/'

""

Z, I

:SU(3)t

: breaking I

U

1 ) ~ B+"AC15

3 ) +2ff -1- - (1 - +1 A2}J.-2 (A _,{3 AS) .

V ZI

(3.12)

Its dominant component is just Z J3A 3 + A 8), so it may be treated as the particle corresponding

w

y,gluons

SU(2) breoklno

=0 { (

FIG. 1. Mass spectrum for the gauge bosons.

232 23

POSSIBLE SU(4), X SU(3), X U(l) MODEL

2695

(3.19) where
(3.20)

After spontaneous symmetry breaking the mass terms of the fermions are derived from (3.3), (3.19), and (3.20). They have the form

(3.21)

The terms involving v F lead to the mixing between 1', Sand ifJ', S'. For the masses of fermions the following remarks may be made. (1) In each generation of fermions, there are eight weak strange fermions and eight ordinary fermions. One may aSSume that the choice of coefficients ensures the weak strange fermions to be much heavier than the ordinary ones; thus in the low-energy region only the fermions with S.. = 0 can be observed. The eight ordinary fermions are doublet degenerate. In general, one generation of fermions in this model includes two generations of fermions in ordinary classification. The existence of the T lepton and b quark implies

that there are at least two generations existing in this model. This means that one may expect the existence of a fourth generation of fermions in ordinary classification. (2) We will discuss the generalized Cabibbo mixing of fermions. If we have n degenerate states d j , i = 1, ... , n, the mass term of these states can be expressed as (3.22) From mass matrix M = (m/j), one may construct two Hermitian matrices MM+ and M+M and diagonalize them by means of certain unitary transformations U and V,

233 2696

C H 0 N G· SilO U GAO

AND

K U A N G· C II A 0

23

CliO U

(3.23) This implies that the transformation dL

-

Utd L d R

-

+[(rv-W+(}+6)211/2,

V\t R

(3.24)

wili ctiagonalize the mass matrix At. Since the charged weak current is left-handed, only U relates itself to the generalized Cabibbo mixing. If one uses U- 1 / 3 and U 2 / 3 to denote the U matrices for - -} charged and { charged quarks, respectively, then the unitary matrix U=U_ 1 / 3 [f,/3 describes the Cabibbo mixing among quarks. There are 112 parameters appearing in the U matrix. 2n-l of them can be eliminated by the choice of relative phases and "(,, - 0/2 of them can be related to the rotation angles in an II-dimensional space. So, there are 11(11 - 3)/2 + 1 phases appearing in the general expression of mixing, which may lead to the CP violation in weak interaction for n ",. 3 and is just the description given by Kobayashi-Maskawa. 7 In this model 11 is an even number. For n = 2 if one denotes the mass matrix as

M=(: ~),

(3.25)

then the eigenvalues of the mass matrix are

and the mixing angles have the forms

2

tan

eR =

2(G'}+ (35) !3" _ Y + 52

a2 _

(3.27)

If one identifies the four ordinary quarks in this case as u, d, c, and s, the Cabbibo angle Be can

be expressed as (3.28) In this model, 5=0 holds and C/, i3, }'S can be obtained from (3.20. Since there exist at least two generations in this model, the mixing should be for the 11 = 4 case. (3) There exist right-handed neutrinos in this model. Since they do not couple with charged leptons via the IV boson, there is no contradiction with experiment. BeSides, there may be the Cabibbo-type mixing among massive neutrons, which will manifest itself in neutrino oscillations. This model predicts that the oscillation among different flavors of neutron os m;ty happen while the oscillation between 1) and j) is forbidden. This prediction is consistent with recent experiments. 8

IV. INTERACTIONS BETWEEN THE FERMIONS AND THE GAUGE 1l0S0NS

The gauge interaction Lagrangian for fermions can be written as

£ =1iL y~(a~ +igjiA~ +iK"I;C~ +ig' ~B~)
+SLy~(a~ +ig"I;C~ +ig'~B~)SL +SRy~(a~ +ig"l/c~ +ig'iB)SR + if~}~(a~ - ig it A~ + ig" Ii

+s~ y~(a~ +ig"J;C

C~ - ig'~ B~)
t - ig'i B~)S~ +Sk y~ (a ~ + ig"l;c ~ - ig' ~ B ~)sk·

(4.1)

We shall discuss the interactions between the Sw = 0 fermions and the gauge vector bosons first. For this purpose, the terms involving Sw* 0 fermions are neglected and the following substitutions are made in (4.1 ). I UL

d' L

dL

uL

,

,

dR

UR

,
,
, ,

llL

eL

eL

llL eR

~fl

Only the IV boson appears in the charged weak interaction; this is reasonable on account of the conservation of the weak strangeness. The charged weak interaction has the form

,
llk

s.-[] s;-[J s;-[::J

(4.2)

,..JL (-liLY ~~V+~eL +eLy ~lV-~llL +Z{2 (4.3) where the terms involving quarkS imply the sum over three color s.

234 23

POSSIBLE SU(4), X SU(3), X U(I) MODEL

The interactions involving C ~, j = 1, ... , 8 are just the color-SU(3) interactions of quantum chromodynamics (QCD). The interactions involving C ~, j = g, ... , 14 give the transition between quarks and leptons. Since the masses of Cj2/3 are very heavy, they can appear only at rather high energies. The neutral interactions consist of four terms involving a photon, Z" Z2' and Z3' respectively. The interaction involving the photon gives the electric charge to be

1

1

e

=" g

Comparing with another definition of the Weinberg angle in the Weinberg-Salam model

e =gsinOw

(4.5)

we obtain .

1

2

sm ew=

'4

1 1 1 + (1/,\,2l+1/f.l2 < '4

(4.6)

in contrast to (3.16). We can treat (3.15) and (4.5) as two definitions of the Weinberg angle. In the limiting case, when Z, is much lighter than Z" they lead to the same expression for sinew, as is evident from (3.16) and (4.6). From the expression (4.6) for ell" a lower bound on the strong coupling constant may be obtained: 2

as>

'3

However, a. is a running constant. It decreases as Q2 increases, so that inequality (4.7) should hold for any energy below the mass scale of the SU(4) breaking. Using the experimental value sin 20w = 0.230± 0.009, Cis>

a (1-4sin2ew)'

(4.7)

(4.8)

0.06,

consistent with the experimental estimation of Cis' We may use the inequality (4.8) to estimate the mass scale of SU(4) breaking. If we use the estimation of Cis'" 0.23 for Q- 30 GeV in the formula

(4.4)

(1 + (1/,\ 2) + 1//L 2)112 .

2697

as (Q

127T ) In(Q2/A2) , f

2) _

- (33 - 2N

(4.9)

the upper limit of this mass scale may be estimated as -10 5 GeV and -10 6 GeV for Nf = 5 and 6, respectively. Of course, the estimation value will increase as the number of flavors increases. Now we discuss the neutral currents in this model. The interaction involving Z, manifests itself very much like the usual neutral current. In the limiting case of (3.10), to lowest order it can be expressed as (4.10) where J,~ "'J~ - sin20wJ;m.

(4.11)

It is just the well-known formula in the WeinbergSalam model. To the next order of approximation J , becomes

(4.11')

Since (vA2+VB2+vc2)/vn2",mw2/mv2« 1 and ,\2» 1, the correction term in (4.11') is smaller in magnitude than the main terms. The effective Lagrangian involving Z2 and Z3 can be expressed as £,rr = 4

!If (r2J2pJ~ +r3J3pJ~)

(4.12)

with r3«r2« 1. When f.l2VD2»~(,\2+f.l2)V/, J 2p , r" J 3p , and r3 take the forms

J

JF

3~ = P'

r =f.l2 3

16

~-~ mZ3

2

-

16

mv

respectively, the contribution of Z 2 interaction to the neutrino-induced neutral-current experiment can be described effectively as a correction for the J ,p current J 'p - J,p -r 2J 2p =J ,p _r2(j)1/2J~15.

(4.14)

One may note that the correction term in (4.11') is much smaller than that from 22 current and only the Z 2 correction should be considered. Since the effective Z2 charges of quarkS and leptons are t and - ~, respectively, this correction can be observed in more accurate experiments. When Jl2 V D 2 « ~ (,\ 2 + f.l2)V e' we get

(4.13)

2'

Since r 3«r" the Z2 interaction is more important than the Z3 interaction. Since the effective Z, charge and 22 charge of the neutrino are ~ and - t

J 3p

_(~)'/'!:c

-

3

"5

f.l J p

-

.!.2,\,J!c J Fp' (4.15)

235 2698

23

CHONG-SHOU GAO AND KUANG CHAO CHOU

The correction term in (4.11') is much smaller than that from the Z2 current in this case_ Since the effective Z2 charge is of the form Q~ ~tF+(W/2I'15

(4.16)

2

which gives different values to two kinds of fermions, we may use them to distinguish these two kinds of fermions, I/! , Sand I/!', S'. The effective Z2 charges for different fermions are given in Table IV. Owing to the mixing between these two kinds of fermions, this effect might not be explicit. However, one may expect the existence of some difference between them. Two interesting remarks may be made. (1) In the tree approximation there is no restriction on the fermion masses. If one calculates the radiative corrections, the stability of the vacuum will give a bound to the fermion masses. 9 ,10 Using the formula given in Ref. 9 one finds that the masses of E, ~,h, andw are bounded by m< 2[% (1+

(4.17)

The masses of other weak strange fermions are bounded by 3(

1)]

1 + ( 123

1/.

mw

)'/4 21smilwl . (~)'/2 a (a )

27) 1/4 + ( 41 21 sinil wl ~

mw,

5'

1/2

(4.21) where s+ ~ ~ (sin2i1 L + sin2i1 R ),

s _ ~ hsin2i1L - sin2IJ R }.

c+~t(COs2i1L +cos2i1 R ),

c_~t(COs2i1L -cos2i1 R },

ilL and ilR are the left-handed and the right-handed mixing angles, respectively. For several special values of mixing angles, 1/ may vanish. But in general 1/ is of the order of unity. If one takes 1/ - 1 and identifies e 2 and e , as the muon and electron, respectively, the lower limit of my can be obtained as

(4.23)

(4.19)

mw'

lepton quark

3

lepton

-I

quark

1/ ~ (}s+ +c+s+ +c_sJ' + (s_ + ~ c_ s+ +} c+sJ',

from the experimental upper limit R < 3.6x 10- 9 • This estimation is consistent with the requirement of the limiting case discussed above,

TABLE IV. The effeclive Z2 charges for different kinds of fermions.

~'.

where 1/ is a function of the mixing angles. For the case B discussed in Sec. III, 1/ can be expressed as

(4.18)

These estimations depend on the value of as' If one takes Ols" 0.20 in (4.18) and (4.19), the upper limits become mN' m. < 210 GeV and m" m, < 440 GeV, respectively. However, (4.17) means that there exist at least four weak strange fermions in one generation lighter than 160 GeV. This prediction can be verified experimentally. (2) Though the quantum numbers F for I/! and S are different from those for I/!' and S I, the radiative decay between two degenerate states is al-

~.S

(4.20)

(4.22)

4CO~4i1JJ 1/. mw" 160 GeV.

mN,m,< [ '2 1+ 4cos4i1w

lowed in this model. For example, if e and e' mix with each other and form two leptons e , and e" me, < me" one may calculate the partial width of the decay mode e,- e , +y. In the tree approximation it is forbidden. But in the one-loop apprOXimation, it is allowed. Using the method given in Ref. 11 one may obtain the branching ratio to be

-,,

If one identifies e, as T instead of Il, the lower limit of my will be much lighter than that obtained

from Il. V_ ADDITIONAL CONSERVATION LAWS

In this model there exist several additional conservation laws. We discuss them separately. A. The conservation of quark number

Many models of grand unification predict that the proton decays. But in this model the proton may be stable. We will give a simple proof of this. One may introduce a new global U(l) symmetry in this model, whose generator is denoted by Ii, and the R aSSignments of the various multiplets being R~l

for I/!,S,I/!',S',

R ~ 0 for A~,B~,C~,
236 23

POSSIBLE SU(4), X SU(3){ X U(I) MODEL

One may easily verify that the invariance of this global symmetry holds before spontaneous symmetry breaking. After spontaneous symmetry breaking it is broken too. But it will combine with 1~5 and form a new global U(I) symmetry whose generator is the quark number

N =(~)1121~5+~R. N is conserved after

spontaneous symmetry breaking. Since the particles with nonvanishing N are, for N= 1, quarks, Gi /s , and several particles in
In this model there exist four weak strange vector bosons and eight weak strange fermions in each generation. They are listed below. Sw= -1

sw= 1

vector boson

V~,UH

V-,U--

lepton

N°,E+

C2 ,!;-1

quark

g2/3,h S/3

W

-4/3

,X

-1/3

All of them are massive and probably heavier than the known particles. The conservation of weak strangeness requires that (1) weak strange particles can be produced only in pairs; (2) they are weakly decaying and the decay chains end in a final state with the lightest weak strange particle; (3) the lightest weak strange particle is stable; and (4) the weak strange bosons do not couple directly with the ordinary fermion pair. Since the weak strange vector bosons are much heavier than the W boson while some of the weak strange fermions are lighter than 160 GeV, the lightest weak strange particle should be a fermion. It may be produced in high-energy e+ecollisions and manifests itself as either a stable lepton or a stable hadron. The weak strange fermions interact with each other via the right-handed current coupling with the W bosons. The transition between weak strange quark and lepton will be suppressed by the propagator of V or U. This means that, for example, if the lightest weak

2699

strange particle is a lepton, the width of the lightest weak strange quark ought to be smaller than that estimated in the ordinary way. It is interesting that the masses of all weak strange fermions with exotic charges (C-, h Si3 , and w- 4/3 ) are bounded by rn < 160 GeV, so that existence of these partic les can be observed in the e+e- experiment. However, owing to the weakstrangeness conservation, the existence of weak strange particles will not have Significant influence on weak-interaction processes below the threshold for production of the weak strange particles. It has to be noted that the existence of the global U(l) symmetry is not intrinsic, and is derived from the special structure of the Higgs potential. Otherwise, there might be several terms violating the weak-strangeness conservation, which will then become partial. VI. SUMMARY AND REMARKS

The main results of this model can be summarized as follows. (1) The electroweak interaction can be connected to the strong interaction via the SU(4)xSU(3)x U(I) model. In this model there exist the same number of left-handed and right-handed multiplets of the fermions as before spontaneous symmetry breaking. (2) This model is anomaly free. (3) As a limiting case, it gives the same results as those of the Weinberg-Salam model and is in agreement with experiment. The Weinberg angle ew is bounded by the relation sin2ew'" t and the neutral current for ordinary fermions reduces to that of the Weinberg-Salam model in the limiting case. (4) There are some deviations from the Weinberg-Salam model concerning the predictions in neutral currents. The additional terms in neutral currents are colorless and flavorless, may possibly be different between the two kinds of fermions, and can be verified by more accurate experiments. (5) A new conserved quantum number Sw, called the weak strangeness, is introduced in the present model. The model predicts many particles with nonvanishing weak strangeness. All of them are likely to be heavier than known fermions. They can be produced only in pairs and the lightest weak strange particle is stable. Some of the weak strange particles have unusual charges and can easily be identified experimentally. (6) There exists a relation between the Weinberg angle and the strong coupling constant. A more accurate value of the Weinberg angle can be used to get the bound of the strong coupling constant and to estimate the upper limit for the mass scale

237 2700

CHONG-SHOU GAO AND KUANG-CHAO CHOU

of symmetry breaking. (7) Quark number may be conserved and the proton may be stable in this model. (8) There are two types of fermions introduced in this model. They will mix with each other after spontaneous symmetry breaking, giving use to Cabibbo-type mixing of fermions. (9) The neutrinos may be massive and there may exist oscillations among different kinds of neutrinos. The oscillation between neutrino and antineutrino is excluded in this model. ACKNOWLEDGMENTS

23

We will use the indices i,i,k, ... for the SU(3) group and the indices IJ-,v,/.., •.• for the SU(4) group. In order to remove the degeneracy of spontaneous symmetry breaking the additional term VI of lhe Higgs potential must be introduced. We introduce a correlation term of cP A and D as c(trcP~ D)(trcP~cP A)'

(A5)

According to (A4) this becomes

CR2(cP A)R2(cP D)cos 2e = CV A2V D2 cos 2 e .

This work was supported in part by the U. S. Department of Energy under Contract No. DEAC03-76SF00515. We would like to thank Professor S. Drell, Dr. Y. S. Tsai, Dr. H. Quinn, Dr. W. B. Atwood, Dr. A. Ogawa, and Dr. C. L. Ong, for carefully reading the manuscript and for valuable discussions. We thank Dr. S. Wolfram for valuable discussion about fermion masses.

If c> 0 the minimum takes place at cos 2 e = O. This means that there are nonvanishing vacuum expectation values for cP at different components of cPA and 4> D' We may use a tranformation to ensure that the nonvanishing vacuum expectation values take place for the i =1 and i = 3 components of cPA and 4> D respectively. We make both v A and v D positive by suitable choice of the phases. We further introduce an additional term as

APPENDIX: THE FORM OF THE HIGGS POTENTIAL

d[ tp b, tp DltpB*1 i.1 tp AI E J>I + tp Ditp bltp B{r.} tp ~,E I>']

There are six Higgs multiplets introduced in this model. Their transformation properties are 4> A: (!.,~,o,-2), 4>B: (!.,~·,O,-2),

(A1)

tr1

This means that the minimum takes place at

We will discuss the form of the Higgs potential, which can realize the spontaneous symmetry breaking we need, and the stability of the breaking. The potential will have the standard potentials for every Higgs multiplet,

F

Vo=

It becomes a minimum as Re O. However, the se If -potential V 0 makes the restriction that

the components of cP B agree,

V=Vo+Vr,

where d> O. Since the minimum takes place at (tpDi)O=VDOi3 and (tpA;)o=VAOil' it becomes effectively

2 -2dvD v A Re'hI321'

D: (!.,~,-2,-2) ,

,

(A6)

(A2)

tpBlnl =v B > 0 which is just adopted in the model. Now we introduce the additional term involving cP A, cP c , and cP E as

e[tpJi"tpEvtp6"Vi tpAi+ tpE"tp;vtpC"Vitp~i]

(A7)

with the coefficient e> O. Using a transformation in the SU(4) group to ensure that (tp"")o=vEo",, with VE> 0 it becomes

2:l-a, tr,+ b,(tr;,l'], z=A

where a" b, > O. In Eq. (A2) the minimum of V 0 takes place at tr4>;
2ev E3v A RetpC441' The components of cP c can also be denoted by the index 01 = 1, ... ,15 instead of IJ- and v. We change the notation as tp"vi - tp I.)i, then we have tpc,,, - --I3/2tpCIlS)l and this additional term becomes

-13e 1i E'v A RetpCIlS)l' Using the same argument discussed above we obtain that the minimum takes place at

238 23

2701

POSSIBLE SU(4)c X SU(3)[ X U(I) MODEL

The additional term involving q, A, q, D' and q,F is chosen to be

-2fu AU DR e
(AS)

where f> O. Since the minimum takes place at (
and becomes a minimum at Re F 2= UF > O. In summary, if the self-interaction potential of the Higgs fields has the form

I B

V=

L: [-a, trq,i ':A

q"

+ b, (trq,iq,,)2 J+ c(trq,~q, D)(trq,~q, A)+ d( ril

+ e( AI + + A'

i.' (A9)

with a's, b's, c, d, e, andf> 0, then it leads to a minimum capable of generating the spontaneous symmetry breaking adopted in this model. Now we discuss the stability of the spontaneous symmetry breaking. There are eight-, six-, and four-dimensional degeneracies appearing in the choice of the breaking components for 4>E' 4>D' and 4> Ao respectively. These 7 + 5 + 3 =15 super-

fluous components can be removed by the choice of gauge, making 15 gauge bosons massive. So no pseudo-Goldstone particle will appear after the spontaneous symmetry breaking, and all remaining components of the Higgs fields will get mass. In other words, the spontaneous symmetry breaking is stable.

*Permanent address. IS. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A. Salam, in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No.8), edited by N. Svartholm (Almqvist and Wiksell, Stockholm, 1968); S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2,1285 (1970). 2j(. Winter, in Proceedings of the 1979 Tntemotional Symposium on Lepton and Photon Interactions at High Energies, Fe rmi/ab , edited by T. B. W. Kirk and H. D. 1. Abarbanel, (Fermilab, Batavia, IllinOiS, 1980), p. 258. 3H . Georgi and S. L. Glashow, Phys. Rev. Lett. 32,438 (1974). 4K . C. Chou and C. S. Gao, Report No. SLAC-PUB-2445, 1979 (unpublished); Sci. Sin. 23, 566 (1980). 5B. W. Lee and S. Weinberg, Phys. Rev. Lett.~, 1237

(1977); B. W. Lee and R. E. Shrock, Phys. Rev. D.12, 2410 (1978); M. Yoshimura, Prog. Theor. Phys. 57, 237 (1977); T. Moriya, ibid. 59,2028 (1978); H. Komatsu, ibid.~, 2013 (1978); K. Ishikawa et aI., ibid.~, 227 (1978); K. Inoue et al., ibid. 60, 627 (1978); G. Segre and J. Weyers, Phys. Lett. 65B, 243 (1976); K. Inoue et aI., Prog. Theor. Phys. 58, 1914 (1977). 6J . C. Pati and A. Salam, Phys. Rev. D10, 275 (1974). 'M. Kobayashi and K. Maskawa, Prog. Theor. Phys.~, 652 (1973). RF. Reines et aI., Phys. Rev. Lett. 45,1307 (1980). 's. Coleman and E. Weinberg, Phys-:-Rev. D 7., 1888 (1973) . !OH. D. Politzer and S. Wolfram, Phys. Lett. 82B, 242 (1979) . D. Bjorken, K. Lane, and S. Weinberg, Phys. Rev. D.!.§., 1474 (1977).

"J.

239 ~1l'ff ~ 2 .Jtrj

~

J981~GJj

•

* •••

~

M •

\' t I

~

.1

JOURNAL OF CHINA UNIVERSITY OF SCIENCE .... ND TECHNOLOGY

~ ~

1-:

\ \. 2 1 ~81

LillP.

On the Quantization and the Renormalization of the Pure-Gauge Fields on the Coset -Space Zhou Guang-zhao

Ruan Ttl-nan

(Chou K nang-chao) (['lStilllte

of

Theoretical

(i)epilJ'lmt'nt

PltY"ic,', Academia Sillica)

Chiua

of ModeI'll

University

of

Pllysic.,', Science

ilnd Tee/mology)

By using Faddeev-Pnpov trick, pure 13, R.

gauge

fields

on

the

S. transformation

space

coset

of

the

this

pa:h-intcgral is

theofY

Ylluntilation

The

realized,

of

invariance

is demonstrated,

from

the

under

which

the

\'i'ard-S Javnov identities are deduced. Since this theory is renormalizable

In

lhe !';allge cf>o=l, F.o{=O. it is also renormalizable in other gauges. This

15

veri fied by means 0 f the ga11iU' independence of the S-m;ltrix.

1. Introduction In fefereD"'>. [1 1 the

~nHcept

or

pmI'

g~llge

fi"lds

on

a

("oset

space

has

Leen proposed. First we consider a Lagrangian 1\'bich is f"auge inva:'ia!lt lln;lN the subgroup H of a g-rollp C,

(1.1) where cp(x) forms the basis of a linear representation

I':lllgf' fields on thf' subgroup H. L"t

i{

(l

of group

= 1,2 ... ·, n,,) 1)/,

G;

.1"

arp

th"

Ibe i!"f'lleratnrs of 1ili"

. 15 •

240

0.2) Then A~, D,:') and F ,;.~, introduced in Eg. (1.1) can be written as

AIL = -igIIA~, A

FILv Tt is possible to construct

' = olLAv -

IYi. = OIL + AIL' ,

(1.3)

"

ovAIL + [AIL,AvJ.

I.agrangiall ,,·bieh is g~lUgP- invariant on til" I"IOUp C

:l

with vpetor gallge field,; ollly on the subgrollp H.

.:£(B)

B 'B)) , = .:£((x) , DIL(x) , FILv(x

(x)

= l/Jo(x)

BIL(x) D~

= l/Jo(x)IYi. l/Jo-I (x),

P:v(x) 111 Ell· (1.5), ~o (x) IS a

= l/Jo(x)P:v(x)l/JoI (x). fUflct;oll

varned

0.5)

lion on the group C. Tllat i,;;,

y(B)

co;;"i

the

Coil

difference hetl"cell the Lagrangian Y"") ulld

with g(:,;)=~r/(.,,)

0·4)

is jllst

.y(B)

hecomes .<;1'

1..1)

is performed. In other words,

transformation to

::/(./)

field

\'ari-

abIes.

variahles

take

the

independent

field

tfitllsforma-

i6

is Hot

we

gauge

11

the

Ohviou.>Iy

Il'hen a gauge 5/'(8)

under tile gallge condition o/u(:.;)=1. So B" and

CjH.

equivalent

an independent to

be

$(:~)

and

A

AJ"). Tn order to restriet the gauge freedom definitp gauge condition. integral. then carry Ollt

"l/e the

first

in

choose

qU311ti',:ation

quantization;

the

in

gauge

an

one

0=1,

arbitrary

must

choose

writp

by

gauge

a

path

the

using

the

Faddeev-Popoy trl.ck. The effective action is invariant under B. R. S. transformation. and from it the Ward-Takahashi identities can be derived. If it

can

be

shown that the S-matrix is invariant under variation of the gauge c.ondition. then the renormalization in an arbitrary theory with Lagrangian

y,(.-I)

gauge

can

be

proved.

This

is

because

tbe

is renormalizable. ~o the illvariance of the S-matrix

implies that the theory with I.agrangian

5/(8)

is also renonnalizable.

2. Path-Integral Quantization in the Renormalizable Gauge ,.

If we

cllO05e th" g3u~e condition 'i"o(.'t')=l,lllcn cp(x) and A"(x) are ordinary

field variables. The Lagrangian • 16 •

y: (.'J clearly describt>5

HIl

ordin"ry

field

theory

241

Wilich can b~ quantiGcd by the path-integral lormalism llj and We cal! this gange the rcnormali7,ahle gauge.

gauge

IS

The generating

rcnormalizable..

functional

in this

IS

z=

J

[d

{i J 4X.!C'(A)},

(2.1)

d

where the gauge conditions arc' F.u(rp,A,.)=O, The rlllmbcr of gauge c:onditilJl1s tors of the subgroup H. i. ~.

(l·=1,2,···n/l).

(2.2)

ill E'l' (2.2) is equal to the Humber of genera-

!lil'

ror examplc.

F.fI=i3"A~,

U=1,2,"',IlIl)' is

the coulomb gauge. The Faddcev-Popov determinant 6.. satisfics

the

following

condition,

(2.3) with F,~ =F A (h
,,·here dp.(h)

1'\

the invariant measure

'.

+ A~)h.-l),

ull

(2.4)

the suhgroup H,

1.

e.

(2.5)

dr' (lJ) = df" (1,1,1) = d fl (It'Ii) .

Choosing a local coordinate system hi (l=1,2,''''''II) to parametri7.e the subgroup ('Iement lI,

(2.6) we bavc "

hdh- 1 = - il1dh",h 1m , h Im=h1m(h). Define the invariant lengLh

Oil

(2.7)

the subgroup H manifold as,

(2.8)

from which the invariant measure of the 5ubgroup H is ootained. (2.9) 1\'ith tbe help of the £'1' (2.5) we can verify that 6..,15 all 011

inyariant

'111antily

the :;ubgrollp H, 6.~=6.., and we have

LlA Let us choose

(l;,

= detM

(A)

,

A

M1m(x, y)

I5Fh (x)

= c5h~;(Y)

I

h=e'

(2.10 )

(i=tJl1+1,"',n(;), a.; lite local coordinate5 which parametrize

the coset element
(2.11) \I'bere

.-

ki

(i=nH+l,"',n(;) arc generators of the coset G/IL So

nds to (/;=0 (i=lIl-I+l,"',n,;) [rom which we

CQrt

dciinc

• 17 •

242 ,Ie 8(¢0-1)=

IT

;=~ll+l

O(Cl,.)-}

(2.12)

-.

V'g«(j.)

The invariant mea5urc d/J.(¢o) should satisfy the folloll-ing relatiolls

d I" ( cP 0) = d f'- ( ¢ (g , 1) 0)) ,

(2.13)

Jd P.(¢o)8 CQ'o-1)=1.

Inserting tlte left liand side of (2.13) illto I~(I' (2.1), the generatillg functional

becomes (2.14) where

dp

F. 4 =F.;(QJ,A,J, 6.-I=6.;(!1?,A,.). Lel

(rp 0)'

I [ d,u. (g) is all invariant

us

rOllsicier

flOW

how

to

choo;:c

measure of the group C, then for fixed g' Ill"

must have (2.15)

d/l(g)=dp(gg')=riIIC/g),

I'-utling g=cph,

we have 19=cp(t/,rp)h(g',CP)h. Furthermore.

\Ie

define riV(
by the relation d/J.(g)=dV(
where dll, (h) is the invariant measure of till'

g' and

l'

one

call

obtain

dl"

(2.16)

subgroup H.

(h(g', cp)h) =d/J. (h)

Therefore

and

from

hy

which

fixin~

it

is

deduced ri I) (
CCHllparing Ell (2.13) Iritll E'1'

= d 11. (

j" tlte invariant measme Oil the rosel spar'c,

(2.17) d/!(rp)=d,u.(g)ld!~(h)

conclude

that.

Because

of the

fact

thnt

eoset

i"

not closed under group 1l1111liplic3tion, it folloll's tbat

(2.18 )

f , «(I) = has

compoJlents

valued

011

f

i I

«(.i)

r + f , ) ((l) k i

I

tile algebra

of

iuvari;lnl

length

tbe

(2.19)

subgrollp

By

H.

means

of

thr

rej·atioll

C'lS. (2.7) and (2.18) the

UIl

tile

group

G

manifold

can

be

expressed in the form

(2.20) lI'ilcrc

• 18 •

243 (2.21) The invariant mca.ourc on group G is lilen gi\'cn by

dJ.,l,(g) = Jg(a) det(h1m)dh 1 ... dhnHdanH+l ... danG '

(2.22)

dJ.,l,(cJ» = Jg(a)da nH +1 ... danG ' 11'

h",rc

(2.23)

g(a) = det (gij glj

and

.Jf:

i, lite Lee-Yang factor.

3. The Path-Integral Quantization in Arbitrary Gauge ;\iow lel us tmnto

i1

galige di fferent from tile r'cnormalizable gauge

1'C = l,

f"j=O. Tile gauge conditions arc (3.1) lI,here n,; is

thr

llumber

fllnctiollal i)."C'i"o,tf>,A.),

of

genr:rators

of

liIr:

group

C;.

Thr

I,'adrleev-['npllv

',l'ilielt sali:;fi~~

(;).2)

F;=Fs(¢(g,rpo),g1),h(g,1)o)(.3.,-L~J,,)h-l(g,
hy a set olloral counlinales .:.-,' (rl=1,2,"',Il,;)

clemcnt g will ill

tlw

(3.3)

be repre~ente,d

form

(3.4) where X" (a= 1,2,"', n,,) are gt~ncral()rs of the group G. satisfyillg the COllllllutation rela! iuns (3.5) Introdllcing the invariallt length

Oil

the group manifold of G as

dS 2 =2tr(gdg- 1 )+(gdg- 1 ) =d~ad~becaecb'

(3.6)

we definc the illvarianl mcasure in thc followillg form

dJ.,l,(g) In terms

= d~ld~2"

.d~nG· det(eab )·

0f the local co()rdillalc5 <;., tbe

Faddccv-Pupnv

(3.7) determinant

lias

the

form (3.3)

, 19 •

244 Inserting the left baud side of Eq. (3.2) into Eq. (2.14), we have

J

z=

[d).1(C/Jo)d
{i Jd

4

J

d).1 (g) 8 (FI)tl. B .

X2(B)}

(3.9) Performing a transformation in the integralint: variables:

(!J-~t1J.

1>0--'¢(g, 4>0)'

;;~ -h(g, ¢o) (0,; + ..1

p

)h- 1 (g, c/>o),

(3.10)

wc obtain from Eq. (3.9)

,",: =

J •

Decomposin,!;

J

[d p. ( ¢ 0) d r!J d ,{ ,,] 0' (F B) c.. g cxp { j d 4... sri'

(B J }

rdf~(~-l)O(¢(g-l,¢o)_l)O(P~(K-I.+O)D._{.

dft(g-l)

inlo

(3.11)

d/I.(g-l)=dp.(¢([l, ¢D).)dp.(h(g-l,¢o»,

it

15

easily verified thaI

(3.12)

fdp. (g-l )8(4)(g-J., ¢o) _1)8(ph(r- " ""J).c..{=1.

which is sHh~tituted into E'l' (3.11) to obtain 7. = This

15

J[dfL (1) 0) dtJJdA,.] 8' (p }!)c.. B w {i Jd 4x Y' (RJ}.

(3.13)

e:

the generating fUIIctional

ill

an arbitrary gauge FA=O.

4. Properties of the Transformation Under the left multiplication of a group clement gEG,

the

element

0

Oil

the coset space transforms as

(4.1)

g<po =
.'1'= 1

by

US11lg

Eq~.

+ i X.~,"

(4.2)

(2.6) and (2.11) we get

h,(g,o/c)=hl"(a)q.,

hl.(('i.)=_§~'i~---±!lLI"_, a~"

(4.3)

_-0

The infinitesimal form of the group relations h (g' g,

cP 0) =h(g', 0/ (g,%)h(g, 1>0)'

(4.4)

a.nd

9 (g' g, 0/0) =.p(g', 0/(;:, 0/0»'

(4.5)

can be written respectively as

Jh/b(a) fimnhma( a)hab( a) + Rja (a) - : 1 - va)

• 20 •

-

Jh/a(a) Rjb (a) -:.-- = fabch/c( a), va j

(4.6)

245

and R ()aRib(CX) ()aRia(CX) _ ( ) ja CX a cx - Rjb CX a cx - fabeRic CX , j

(.t.7)

j

from which we can deduce the transformation of

the

field variables under

the

action of JieG,

ii1J (x)

= i.Y. ,p (x)~. (x) ,

(4.8)

oaj(x)=Ri.(x)';.(.\:) .

For

the

~gke

by Lee t >] and

uf Jet

concise 1>i

stand

presentation

let

115

take

([J~

or

Ili'

III

for

A/.,

the Ihis

notation 110tation

introduced Eq.

(4.8)

becomes (4.9)

D~(xi>xa)

= i(Xa) ap p(Xi)8(x; -xa),

(4.10)

Df(xi,xa) = R;a (x;) 8 (xi - xa). By using £15.

(4.6), (4.7)

(4.10),

and

the

following

relations

arc

easily

dedaced

(4.11)

where (4.12) The condition for the illvariance of the Lagrangian uuder the action of the group

G can bl>, rewritten a5

0:;[

-._-~-

0';-.

We now

introduce

and

thc

usc

I

=

.

aY lJ;---=O.

';=0

lhe anticol11lllulalive

techniquc

of

Lec

and

(4.13)

00/;

scaLH

fields

Zinn-Justin lS )

qa, to

17. (a=1,2, .. ·,nc)l3]

ohtain

thc

effective

action

(4.14) Finally the generating functional has the from

(4.15)

• 21 •

246 where .fa and 771t are tb~ Faddecv-POpO\T ghost fields.

The effcctive action. (4.14) can be 5holnl to be

iln-ari·ant under

the B. R.

S. transformalion l11 (4.lfi)

(..1.17)

wbcra

oj.

is
C-Ilumber.

Dy means of Eq.

(..I.U)

call

we

demon5lra Ie Iha t (4.18)

from which the invariance of the effcctiyc action LInder

B.

R.

S.

tran5forma-

lion follows wilhout difficulty. i. e.

8Seff [I/>,';,ryJ =0,

(4.19)

[dl/>d';dryJ = [dl/>'d';'dry'J.

(4.20)

Thus the ·generating functional Z

is

invariant

B.

llllder

I\.

S.

Iran;fnnllatillll,

namely [>;:;:

=

O.

(4.21)

5. Ward-Slavnov IdenHty Dc fille the generating functional as Z [n =

B)'

\lSlng

J[d.pd;dIIJ

ex))

{is, f j [.p,f, 11]

(5.1)

the il1\ct',ral form\lla for GraEsmann variablt:s

J

de ,C j = S i

\1"

+ i.l;
i •

(5.2)

+ iIi.p i} =0.

(5.3)

e have

J

[d¢d';dll]:; _,xr {is. II

liFon performing R.

r

R .. S. transformation, we gcl

:'r {i 8 .r; + i] i '" i

[dcpd;dlll (0' a +<1':=,,)

E([s. (5.3) and

[d¢d;d'il (i:;" J i Dr{>

i

+ 6';') cxr {is
R. R. S. - trall"rL-,rlll,·ll[·',>LI ~

lhe \Vard-S lavllo\' idcnli I y • 2.2 •

is,,, i'} =0.

(5.4)

= 0,

(5.5)

(5.4) lead to

r SllbslitutilL."".. .,

'-I iJ j

i "',}

(.f' t16) . t E-fl, C-~) . , (1'. 1-)· I llll) ;.J.;.J we

It

O ) a l'1}

247

(5.6) tfleories.

This was first derived hy S lavno;' and Taylor[3j for ordinary g3.uge

It

is also true for theories with pure-gauge fields on c(,sel. We de fine Z b. U]

==

if

[dcpd;dll]'; a 71 I, exp {is d

f[dcpd~dIIJ (H.-

=Mbd

1

[

,\,j .Ba -I

+ iJ 1> i}

I

i

[lp J CXP {is d

I

+ i.1 irp i}

1 i5 ]Z[1 iO] .j

(5.7)

or

,\.1/i [ i

;j ]

Z

b,

U]

= 04

,

Z

LJJ ,

(5.8)

then Ward-Slavnuv identity (5.6) call be rewritten in the form

~ p B.h~.J]Z Lrl-.riD~[

iL Jz,.. u]

(5.9)

=0.

6. The Gauge Independence of the S-Matrix In the rCIlurmalizable gauge the generating fUllctional is ljJ

Z.1 [K] =

J

[d
[j (F. 1)i'>.1

~Xl' {iS A + i K i 'P i},

(6.1)

For b"eneral

where rp i represents tile ordinary field variables 'P. (x) and AUx). gauge r.onditiOl1s, the generating funclio/lal is

(6.2) Substituting the func:.tiClnal (3.12) in to (6.2), we baye

Z 8 LT] =

f •

[dfI' (¢o)diJ)dA"J J(FIJ6 scxp {iSH

+ iJ iCP i}

fdP'(R'-1)6(rp(!I'-1,cpo)-l)l)(F~rg-"'I'(J)6A'

(6,3)

which upon using a transformiltion on the integration variables,

(6,4) become:;

Z E [.!l =

f

[d /1. ( 1> n ) 1i!P d A I' Fi (1> n - 1) ,l( F ../) 6.( e i

.< B

J11 ( g) d

0 ( F D i'> 5 0:;., r

ri J 1> n . i

(6.5)

If the gauge conditiun I<'i is ~ldv"d fur!? at

g=gOl

till'

itltegratioll

over g

23

in

248 Ell· (6.5) can be performed

ZB[Jl

=

=

and

we obtain

J[dJ.l( CPo)dCf>dA Jl1 8 (CPo - 1)8(FA)~A J[dcpdAJl18(FA)~A + iJiCPi(CP)}

exp{ iSB + iJiCPn

exp{iSA

= exp {iJiCPi [i:K] }ZA [K1IK=o' rl[

(6.6)

where cp.I (cp) = cP Igo I'1'0'" _ \. The formula (6.6) can be used to yield relationship between the Green's functions in the general gauge and that in the renormalizable gauge. On the mass shell the two generating functionals have no other differences except for changes in the renormalization constants of the external lines. So the S-matrix is the same when the gauge condition is changed. Thus we have demonstrated that the theory of pore-gauge field on coset space can be quantized and is renormalizable in arbitrary gauge provided that it is so in the gauge CPo = 1, FA = O.

References [1 J

ChOll

F\.lIilng-chao,

Til

Tung-sheng.

RUal1

Ttl-nan,

scielltia

Sinica,

XXH (1979), No.1, 37. [2 J [3] [4] [5

J

V. N. Popov and L. D, Faddecv. Phys. Letts, 25B (1967), 29. F. A. Bere3in, The method of second quantiza (Academic Press. :\c'" Yurk, 1966), 49. C. Becchi, A. Ruuct and R. Slora, Comm. Math. Phys., 42 (975), 127. R. \1/. Lee, :\lelhods in Field Theory (Session 28 of 1.1'.5 ·Houchos 1975), 79, and I he re f ercnccs quoted there.

*)(1'1j ffl F add ee v-P 0 pov t~ T55tnX: T RHjl: ~HJ) ~1~J!ire:#4(J9l1*t2ff(5} ii:-J-ft. ~1E SjJ 7 f:1I1 1~Ji';: B. R.

cp 0 =

s.

1, F .• = 0

J>f. ~K ~Jf HjJ .

~~"f~~(J~. F.l:!tll:~tt\Y W'ard-Slavnoytf~A. d!f-:fI~tEm1lt',*,H

"f ~ liT mJJ: It Jl~, rt; tE;fI~ fli1m m: TIlL ~ PT 1f! ~ it ag. j;3: nr dl

s ~€ ~41 fI~ mffi X

249 KEXUE

Vol. "27 No.2

TONGBAO

February 1982

THE U(l) ANOMALOUS WARD IDENTITIES AND CHIRAL DYNAMICS ZHOU GUANGZH.\O

(mIJ'tB)

(Institute of Theoretical Physics, Academia Sinica) Received August 20, 1980.

It is generally believed that the QCD Lagrangian has N1 conserved axial currents except for quark mass terms. In the world of zero bare quark masses for the first L flavours, the corresponding conservation laws are spontaneously broken and L' pseudoGoldstone bosons are thought to be generated. For three quark flavours the absence of the ninth light pseudoscalar in the real world is a well-known puzzle first pointed out by Glashow m and studied subsequently by Wienberg"\ Kogut and Susskind"] and many others'4- B]. In 1976 it was observed by G't Hooft'4] that instantons might help resolve the paradox through the anomaly in the axial U(l) channel. However, Crewther[5] has shown in a series of papers that instantons with integer Pontryagin number are impossible to satisfy the anomalous Ward identities in WKB approximation. Recently, a very interesting proposal based on the analysis of anomaly III the framework of the liNe approximation has been made by "\Vitten'6]. According to his opinion the U(l) problem can be solved at the next-to leading order of its liNe expansion, due to quantum fluctuations in the topological charge density. In this note we argue that the idea underlying Witten's proposal is a general one not necessarily connected with the liNe expansion. To P' order in the low energy limit both the mass and the wave function renormalization constant for the 1'(' meson are determined by the quantum fluctuations in the topological charge density. In thc world of zero bare quark masses, the Lagrangian has the form

st'

= - ..!.. F~.,Fal" + qil/Jq - 8(z)v(z) - J(z)O(z), 4

(1)

where

is the topological charge density. In Eq. (1) a set of hermitian composite fields O,(z) is introduced with J,(x) as their external sources. The Lagrangian (1) is invariant under U(L) X U(L) chiral group with L quark flavours, except for the source term J(x) O(x). We shall put 8(x) =8, J,(x)=O only at the final stage of the calculation_ The currents for the U(L) X U(L) symmetry are

250 148

KEXUE TONGBAO

.

JI" =

1 . 2 l'

.,.

i'J.yl"_,1,.q

_

Vol. 27

1·

(2)

J5'1 = qyI"Y5 -2l q 1 ,

where ii indicates the Gell-Mann matrices with

10

= .../ 2/ L.

We shall include the currents to the composite fields and write their external sources as Jl'i and JpSi respectively. Sometimes we use O(x) and J(x) to denote romposite fields other than the currents and their sources. The generating functional for the connected Green's functions is defined as

(3) where

OJ

indicates all the operators written in Heisenberg picture.

The vertex functional [[O(x), J~(x), JI'51(X)] is related to 1V [J(x), lui (x), JpS;( z)] by a IJegrendre transformation

[[O(z), Jl'i(Z), JpSi(Z)]

=

W[J(z), JI';(x), JpS;(z) 1

(4)

-1.f'(x)0;(:r;)d4x, where O,(x)

IS

determined by

O,(z) =~. lJJj(x)

(5)

In Eq. (4) the sources relating to the currents are not Legrendre-transformed. .Eq. (4) it is easily convinced that

Sf

8f

8W

F.rom

(6)

ojpSi (x)'

We assume that the composite fields O,(z) form a linear representation of the (!hiral group. The Ward identities for the generating functional have the forms 81'

81'

"11' "TV = iJ(x)Ti ~(y)' 8J I'lx) BJ x U

01~) = iJ(x)T5i

lJJ 1'5i X

SW

lJJ pSi

+ "'/2L(v(x)Oio'

(7)

In (7) , Ti and T5! are the representation matrices of the group generators. The axial U (1) anomaly term

V -2L (v( x»

.

can also be WrItten as

sW V -.2L -(-)-.

S8 z These Ward identities have been justified by Crewther and used in literature to study the U(l) problem.

It is more convenient to start from the Ward identities for the generating func"tional of the vertex functions. They have the form

251 KEXUE TONGBAO

No.2

149

(8)

(9) The Ward identities (8) and (9) are satisfied by any functionals of O(x), J~;(x), e(x) and are invariant under the following infinitesimal local gauge transformation

J "'; (x) and

O(x) - (1 + iakc)Ti + iaSi(x)TsJO(x), Jl'i(X) - Jl'i(X) - iJl'k(X)hiiCti(X) - Jp.six)dkiiaS/x) - BI'Cli(X),

(10)

Jpi(x) -J,,:;;(x) - iJplx)fkiia.(x) - Jl'k(x)dwas.(x) - BI'Clsi(X), e(x) - e(x) -J2LClso. Therefore the vertex functional formations.

IS

invariant under the above local gauge trans-

In the following we shall choose the composite field O(x) to be the '2Lz scalar and pseudoscalar biquark densities

q

Ai q and illY 5

2

1i q. 2

Tht'y l:an be representpd by

an L X L matrix

(11) and its hermitian conjugate, which transforms under [O(L)

X U(L)

into

(12) where Y L and VR are unitary matrices belongiug to the left-handed and right-handed U (L) groups respectively. The vacuum average yalue of fJ: will be denoted by U. For L = 3 one can construct only four SU(3)X S["(3) invariants from U and its hermitian conjugate U+. They are tr(UV+), tr(UV+VV+), det(VU+) and det V/det V+, the first three of which are automatically U(3) X U(3) invariants. Under the axial U(l), In ( (det U/det U+ 'j transform in the following way In (det U / det U+) - In (det U/ det [;-+) +i2 J 2LCtso and

e-

(13)

.2... In (det UI det U+) is an invariant. 2

Within the limit of long wavelength and low frequency, the vertex functional should be a function of these invariants only. Hence after putting the external sources· to zero and e(x) = e, we have

252 KEXUE TONGBAO

150

Vol. 21

r =F [tr(UU+),tr(UU+UU+),det(UU+),8 - ~ In(detUjdet U+)].

(14)

The symmetry U(L) X U(L) is broken spontaneously down to the parity conserving U (L) . It is argued in [9] that to the second order in energy momentum P of the pseudoscalar Goldstone bosons

uu+ bolds after a suitable normalization. masses we can always put

1-f2 2 •

=

(15)

In chiral symmetry limit of zero bare quark

(I6) "here

Jr.

is the 1( boson in this notation.

From Eq. (16) we see that 8 - i..ln(detUjdetU+) 2

=

8

+ ~~L t

(17)

1)'.

x

The gauge invariance conditions (10) and (13) then imply that the vertex functional for 7J' and e should be constructed from the inyariant 8(x)

+ ~2L

(18)

"I}'(x)

L

and the covariant derivative

(19) The effective Lagrangian L.u (x) is related to the vertex functional by

(20) and can be constructed from the invariance properties required by the Ward identities. it has the form To the order of

P"

rR e f _ f -1 ;;z, -

A.(8+~2L --"I}')D

1)

'Da' '1)

2

f"

+ 1f A. 2 " ~8

(8 + ~f""I},, 2L ')(a 8+ J 2L a ') D" ' f" ,,1] 1]

+ 1- f 2

- E

A. " 8

"

(8 + J f"2L 1]" ')(a 8+ Jf"2L a ,)2 ,,1]

(8 + J2L 1]'). f"

(21)

253 KEXUE TONGBAO

No.2

The wave function renormalization constant for the

z~, ;1nd the mass of the

7)'

= (A(e) meson is

+ .j2LA~eCe) + m',

151 1)'

meson is equal to

2LAe(e»-1

(22)

2L dIE Z ,

=

(23)

j!de2~·

?

From Eqs. (20) and (21) one obtains

r

J Be

(Br ) e-iP"rd"xle(x)=e x Be( 0

=

dIE _ PlnAeCe)

de 2

dZE

'rherefore, both dez

+

=

-i

r (T(iJ(x)iJ(o»)ee- iP .xd x 4

J

O(p4).

(24)

and Ae(8) are related to the quantum fluctuatuions of the topo-

.lugical charge density in the 8-vacuum. We also find from Eqs. (20) and (21) that

r

J BJ pSo =

r

t~

x Be( 0 )

e- iP-xd4x Ie(x)=e=-i

.l.... P'"f;A~eCe) 2

+

r (TcJ;o(x)iJ(o»)ee-iP-xd~x

J

(25)

O(P3),

-iP·xd4 I -·P'"f' x e(x)=e = t "

Blr

J BJpSo(x)B7!'(o) e =

iP'"fiA(e)

+

.jLj2A~8(e»

+

(26)

O(P3).

It is easily verified that in the liNe expansion

AeCe) ::::: O(ljNJ,

(27)

A~eCe) = O(ljND,

a.nd

f ~<e)

::::: f ~(1 + O(l/NJ).

Therefore to the leading order in liNe, Eq. (23) reduces to Witten's formula • = -2L (d- E)' = 1, m;, 2

z~,

f;

dez

(28) nO

Quarks, 0=0'

To the next order in liN. we may neglect A,. in Eq. (22) and find

(29) where A.(O) is related to the quantum fluctuations of the topological charge density b~r Eq. (24). It is interesting to note that in gener al Z'1 is not equal to f ,Jf ~'. A full analysis of the effective Lagrangian for low energy pseudoscalar mesons

254 152

KEXUE TONGBA.O

Vol. 27

with bare quark masses taken into account can be carried through without difficulty. Results on e periodicity and the like can be obtained in the general case in accord with those given by Di Vecchia, Veneziano and Crewther[··71. These questions will be discussed elsewhere. REFERENCES

( 1]

Glashow, S. L., in Hadro1l.ll and Their In.teractions, Academic Press Ine. New York, (1968). 83. l 2] Weinberg, S., PhY8. Rev., Dl1(1975), 3583. r 3] Kogut, J. & Susskind, It-. ibid., Dll (1975), 3594. [ 4] G't Hooft, ibid., 37 (1976), 8. [51 Crewther, R. J., Riv. Nuovo Cimento, 2(1979), 83; Phys. Lett., 70B(1977), 349; CERN, TH2791, (1979). [6] Witten, E., Nucl. Phy&., B156(1979). 269. r 7] Veneziano, G., ibid., B159(1979), 213; Di Vechia, P., Phys. Lett., 85B(1979), 357; Di Vecchia. P. & Veneziano, G., CERN TH-2814, (1980). [ 8] 1\ath, P. '" Arnowitt, R., CERN, TH·2818, (1980). [9] Lee, B. W., Chiral Dynamics, Gordon & Breach, New York, 1972.

255 Canlm. in Tlleor. Phys.

vol.

1, No. 1

(1982)

69-75

NEW NONLI NEAR a MODELS ON SYl"1[~ETR I C SPACES CHOU Kuang-chao ( J~!';; (Institute of Theoretical Physics, Academia Sinica) SONG Xing-chang ( 51: :H: ·(Institute of Theoretical Physics, Peking University) Received July 9, 1981. Abstract Two new nonlinear

a models,

defined on the symmetric coset

spaces GL(n,c)® GL.(n,c)/GL(n,c) and GL(n,c)/U(n) respectively, are

fo~lated

in this paper.

The latter may be useful in discus-

sing the four-dimensional Yang-Mills fields.

In previous papersl1,l], we have studied various classical nonli~ear a models taking values on symmetric coset spaces and have sho~vn generally how to use the duality symmetry to deduce the infinite sets of conserved currents, both local and nonlocal. In these papers, a unifying point of view is proposed to ~onnect varieties of integrable 0 models which have attracted our attention in recent years. The same point has also been noticed by Eichenherr and Forger(3] who have proved that, for compact global symmetry groups, the a model has the dual symmetry if and only if the field takes values in a symmetric space. The main feature of our formulation is to use the so-called "gauge-covariant current" ff,t. (3] as the central role to formulate the equation of motion, the duality symmetry and to deduce the conserved currents. The important advantage of this formulation is its unification and simplicity. In this note, we shall extend our study to other models, such as the GL(n,c) prinCipal model, and a new model Hn which takes values in the symmetric space G1(n,c) /U(n). For the purpose of application some of the notations used previously will be changed slightly. As mentioned previously(2,31, the U(n) principal field defined on the homogeneous space U(n)/{l} can be re-formulated as a o·model on symmetric space U(~ ®U(n)/U(n). Similarly we consider the model on symmetric space GL(n,c) ® GL (n,c)/GL(n,c). An element of the group G=GL(n,c)~GL(n,c) is expressed as (1) with both gL and rule+)is

gR

belonging to GL(n,c) independently.

The multiplication (2)

+)

++)

Xbe gR defined here is just the g; in the Ref. (2J. A factor -i appearing in Ref.(2] is omitted bere. J.l is suppressed.

Sometimes the Lorentz index

256 70.

CHOU Kuang-chao, SONG xing-chang

An element of the subgroup ~ E H = GL(n,c) is represented by . (3)

which is invariant with respect to the involution a (4)

Then G/H is a symmetric space with involution a, whose element is ~

= (

cp.

cpt-!).

(5)

Now any element of G can be decomposed into the product of a subgroup element and a left coset element g=
Following Ref. [1,2] for any

¢EGL(n,c) define++)

(7)

Here ~ and j{ take values in the Lie algebra of subgroup H and coset G/H respectively. i.e., (8)

and (8' )

Introducing the covariant derivative defined as

.s.p-~= a)'o~-~Jej<=(~"'-"'H,.... ClJ'-'I't-I_ H)1>+-'),

c8...... f-'

~~-l+JeJ'
(9)

and (10)

we have (11)

.md then

257 New nonlinear cr models on symmetric spaces

-tTrJC}LJC'''=

71.

+r,. (~;<- ~-I) JJ-""~

=-tt,. (I\... k-" t

k; I<~J' ) .

(12 )

with (12') Now the current can be written as+)

J = 2~J<. f-' =

2(;1<+-1 ,-cfl<+'I'+-')

=

(.h , :JR) ,

(13)

where

(14 ) So setting (15)

then

We can rewrite Eq. (13) as (16 ) It can then be easily shown that (17)

;lnd that ~3" -

a"3}L -

(3p. . 3~)

=2~ (J}.,...J{,,-J),,](... H-'= 0

(18 )

Here the covariant derivative of j( is defined as (19 )

The Lagrangian for this model is just Eq. (12);

£=-Trl(,.. R'"=tTr (J)...... ~-f )J)..... ~

=tTr(~Q-')d-""Q , which is invariant under the global group GL(n,c) GL(n,c). The field equation reads

e

+) In contrast with that in Ref.[2], a factor 2 is added here.

(20)

Gl(n,c) and the gauge group

258 72.

CHOU Kuang-chao, SONG xing-chang

(21 )

or equivalently,

JYX.,....=o.

(22)

This implies that

or (23)

This means tha.t 3J.l is conserved and J{J.l is covariantly conserved. I f we treat GL(n,c) ® GL(n,c)/GL(n,c) model as in Ref.[2l, we obtain the Lagrangian form L= "zTr1i;.1t =lttraJ.l q. aJ.lq-l, qE: GL(n,c) which can be considered as GL(n,c) principal model. But it can easily be seen that this L is not hermitian. If we add the hermitian conjugate L+ to L, we then obtain eq. (~2). So the method given here has taken into account the hermitian property by itself . For the case in which gL and gR are unitary, and so are h and cp , we obtain q+

=

(24)

q-:-l

and H+ = -H,

K+= -K.

(25)

Then the Lagrangian turns out to be (26)

and our model reduces to the U(n) principal chiral field as discussed before~21 Another important case we are interested in is a new model which takes values in the symmetric space GL(n,c)/U(n). As mentioned in Eq. (3), the element of the group G=GL(n,c) takes the form (27)

The elements of the subgroup 1J E: H=U(n) and the coset by

¢

t G/H can be represented

h E: U ("J

(28 )

and 1' respectively.

t,

cj>E~LC1t.c) •

(29)

The involution operator which makes H invariant is (30)

Now define

259 New nonlinear cr models on symmetric spaces

n=~-I

Cli!

=

73.

(<j>-Ita<j>t-I)

(31)

=Je+R=(H-t/< ,-HtK) for any

~ E

GL(n,c), and we have

H= Yz

(;.1 at 1"
K= ~ (cf'"'a.p-cp-+acp+-r). Introducing the covariant derivatives

(33)

and the hermitian matrices (34)

the Lagrangian for this model can be written as 'II -

,

~/

, (f\ ..:-1 ="2T,. a::1p.J: )(<0

1/P

G\..--TT~~.tt.

1/1.... ::. ) ':I:

I

=]'"T,.

(

d....... Q

-,

P

)J Q. (35 )

=-t-"k-,,"kP=ft~ (~r') d~l'

The rema~nLng procedure can proceed according to the standard strategy as before. All the statements and the formulae among 3,1£, and Q (i.e., Eqs. (16) through (18» remain to be true. But their component form has to be modified, since in present model, the second component of is <1>+ instead of ¢ and q is hermitian here. He list these formulae as follows:

J L=2tI< 1'-\: -li/+-cj>-at-I ) 1'-' =_(-'a+-tta+ t -') t =('I'io)-'

(16' a) (16'b)

a ('I'1't)= r'd'/ j

d .. t=2cf'(i'.. JS..,·t(k", k.}<-))-' ,

(17' a)

(17 'b) L

L

L

to.

dj'
= 2 q... k ... -lJ,,·J<.,...)4.,I=O

(18' a) J

and

a....... J..II. -a,Jj'
J

J ..R]

=2'f't,-, (D.,...I<,,-i'v l
Here

~he cov~riant

derivative of

~

is again defined as in Eq. (19)

(18 'b)

260 74.

CHOU Kuang-chao, SONG Xing-chang

and (19' ) The equation of motion, Eqs. (21), (22) and (23), also stand as before. Another equation which can be derived from the integrability of Q is

The left-hand side of this equation is nothing but the curvature (field stren~h) Fuv taking values in the subalgebra u(n). From Eqs. (19') and (22), it can be shown that for two-dimensional space-time, (37)

This means that Fuv satisfies the sourceless Yang-Mills equation. We can see from Eq. (32) that H+=-H, K+=K and then Eq. (16')· implies and L

R-I

~=U,..t

.

J~=JR

(3S)

Taking account of the curvatureless property (IS') we get

l

R

(R

11))-1

a"~=&(d,,J...... + J",J.I"-

~

=~(~I:)r'·

(39)

These equations can be combined to be written as

(40)

In this model, different from that in other models, the matrix q appearing in the Lagrangian is hermitian. rather than unitary. We may call it Hn model which has its great importance in connection with the four-dimentional Yang-Mills fields. [4] We would like to point out here that it is not necessary to write down our model in the present form. As is well-known. an arbitrary nonsingular complex matrix g can always be written as the product of the form ~ h. where ~ is hermitian and h is unitary. The decomposition is unique. Thus {h} forms the unitary subgroup U(n). and {~} forms the symmetric coset space GL(n,c)!U(n). Separating the antihermitian and hermitian parts from ~-1 a ~ and identifying them as q and K respectively. we obtain Eq. (32) once again. From them all other quantities. can be defined and relations among these quantities can be derived as well. But the formula tion described above has the advantage that pairs of quantities are grouped together and so the symmetry is more transparent. This property is very useful in practical applications. To sum up. we can see from Eqs.(20) and (35) and the earlier discussion[I.2] that the Lagrangian

261 New nonlinear

a models on symmetric spaces

75.

(41 )

in which q(x) belongs to some compact Lie group in matrix representation, describes a variety of nonlinear models.

Different models correspond to q(x) vary-

ing over various subsets of the group G, GL(n,c) or U(n). For the cases in which the matrix q(x) varies over the whole group G, we obtain the principal models. our Rn model.

When q(x) is restricted to be hermitian, it yields

A number of the so-called "second class" models[51, such as O(N)

models, CPn models, HPn models and Grassmann models, can be obtained when the matrices q(x) vary over some subset F of the group G in such a way that

(42) Here the star operation is an involution automorphism, qo is a given fixed element of G which satisfies qo q3 =const. and u is an arbitrary element of G. For all these models the Lagrangian can be brought into the form

(43 ) and the duality symmetry stands for two-dimensional space-time.

Then following

the standard procedure both local and nonlocal conservation laws can be obtained. [1,2]

References 1.

X.C. Chou and X.C. Song, ASITP-80-008, PUTP-SO-003.

2.

K.C. Chou and X.C. Song, ASITP-BO-010, PUTP-80-004.

3.

H.Eichenherr and M. Forger, Nuc1. Phys., 8155 (1979) 381.

4.

K.C. Chou and X.C. Song, ASITP-81-013, to be published.

5.

E. Brezin, S. Hikami and J. Zinn-Justin, Nuc1. Phys., 8165 (1980) 528.

262 Cgmmun. in Theor. Phys.

Vol. 1, No. 2

(1982)

185-203

THE Hn SIGMA-MODELS AND SELF-DUAL SU(n) YANG-MILLS FIELDS IN FOUR-DIMENSIONAL EUCLIDEAN SPACE CHOU Kuang-chao( JiJjt?J ) (Institute of Theoretical Physics, Academia Sinica) SONG Xing-chang( *1B: ) (Institute of Theoretical Physics, Peking University) Received July 5, 1981.

Abstract The Hn a-models. i.e., non-linear eJ-models taking values on the symmetric coset space SL(n,c)/SU(n) ,

both

in

two-

dimensional and four-dimensional Euclidean space, are formulated.

The relations with self-dual SU(n) Yang-Mills fields

are also discussed.

I. Introduction In recent years there has been considerable progress in the investigation of the two-dimensional non-linear sigma models, or chiral fields,[1-3] both on classical level and on quantum level. The main interest in these models comes from their remarkable similarity to four-dimensional non-Abelian gauge theories. Like the Yang-Mills theories, these Don-linear models are self-coupled systems which are renormalizable[41 and asymptotically free[S]. They also possess some kind of instantaneous solutions,[3,6] and exhibit dynamical mass generation[7]. All these properties are believed to be important characteristics of four-dimensional gauge theories. More recently, the parallelism between the gauge theories and the nonlinear a-models is much more strengthened by the discovery of the Backlund transformation[8,9] for the self-dual Yang-Mills fields, of a system of linear differential equations comprising the self-dual equations as their integrability condition[lO], and especially of the realization that the classical Yang-Mills equations can be formulated in the loop space as chiral equations[ll]. This parallelism could be put forward furthermore and may lead to new and usefu~com putation approaches to Yang-Mills theories by analogy with the a-models. One of the distinguishing features for the two-dimensional a-models is the existence of a new class of unconventional conservation laws, both local and nonlocal, and infinite in number. Generally the nonlocal conservation laws are constructed by using either the linear eigenvalue system[12], the zenrcurvature conditions for the conserved currents[13], or the dual symmetry[14]. Less is known about the local ones which can also be derived either from the parametric Backlund transformaUons[IS] or from the dual symmetry[16]. The physical mean-

263 186

in~

CHOU Kuang-chao,

of such conservation

la~s

SONG Xing-chang

bas not been completely clarified.

However it

has been shown that local as well as nonlocal charges can properly be defined as operators in the quantized a_models[17].

Acting upon the asymptotic states

these conserved charges give an infinite number of strong constraints on the S .

. .

.

motr1x and suppress the part1cle product10n 1n some models

[18].

.

Tt1S fact

leads to the factorization of the S matrix into two-body amplitudes[19] can be calculated exactly by N-1 expansion for O(N) a_models[20]. For high-dimensional gauge theories it has been conjectured that

which

analo~ous

nonlocal conservation laws will provide some relation between long- and

short-

distance behavior[21] and cause stro~g constraints on the structure of S matrix as well.

Pohlmeyer[8] and Prasad et

al.[22J have obtained such non local con-

servation laws for (anti) self-dual Yang-:dlls fields in four-dimensions "dual" symmetry and curvatureless condition[13] respectively.

from

Prasad et al.

l'ave also constructed a parametric Backlund trar:sformation for the self - dual Yang-Uills fields, but they could not deduce the local conservation la~s from these transformations as was done for the principal chiral fielcl[ 15 1. Meanwhile Pohlmeyer has indicated that, apart from infinite numbers of nonlocal continurty equation, there exists another set of continuity equations which are the analogs of the local ones in two dimensions. for the second set.

He has obtained the generating functional

But in contrast to the two-dimensional case, the

obtained involve non local expression of the field.

results

So, up to now, it still

leaves open the question whether there exist corresponding local currents

for

the self-dual Yang-Mills theories in four-dimensions or not. In this paper we will study the Hn a-models (which we have proposed sometime earlier[23]) further, and discuss the relation between these models and the four-dimensional self-dual Yang-Mills fields.

Our discussion shows that at

least for a restricted set of the solutions, the four-dimensional self-dual Yang-Mills fields do have an infinite number of local conservation laws. The plan of this paper is as follows.

In Sec. II, by a brief

revie~

of

the form of the four-dimensional Euclidean space, we fix our notations and conventions.~;Sec.

III is devoted to the ~ a-models, i.e., the models defined on

the symmetrical coset space SL(n,c)/SU(n).

For both two-dimensional case and

four-dimensional case, the equation of motion, the possible dual symmetry and the conservation laws deduced from this sy~metry are given.

We turn to discuss

the self-dual Yang-Mills fields in Sec. IV and show that it bears a close resemblance to the self-dual Hn a-models. is given in Sec. V.

A short discussion about the results

II. Self-dual conditions and SU(2)xSU(2) formulation In a four-dimensional Euclidean space, consider the transformation of coordinates from X)«.,Ll=1.2.3.4) to

,,{iJ=X, t .[2~=

iX z

~J=(!f.J."Jj.J)

defined by[24]

.J2j=x, -

iXz (2.1)

)(riX4

.[2"j:;;;x3 t;)(4

264 The Hn sigma-models & self-dual SU(n) Yang-Nills fields in four-dimensional EuCiLiean s!,ace

The invariant length of

187

x'" is

So the metric in the new variable has only the following nonvanishing components (2.2)

For any four-vector

A~,

the covariant components in the new coordinates are

-fiA y = ,fiA Y = A, - iA,

-v':?"A,Y=--12l =A, +iA"

...J2 A ;:: -fiA J = Aj +iA4 J

{2Aj =/2i=AriA+.

(2.3)

The same is true of the derivative vector

..[i d/=ff ad,! .fidJ=fi

= <1,-; J.

a~;::

.ffJy;:: ,.[2 Jay:;:: .1,

,

I2JJ =f2;j

J 3 tiJ+,

Any anti-symmetrical tensor

F... v

tid, ,

=;)3-;,14 .

can be decomposed into a self-

=-Fy,"

dual part and an anti-self-dual part

(2.1 )

where

F.... v

is the dual tensor of ~

~"v

1

f.. . . =T~"A.f FA!.

(£'234=1)

(:2.4' )

In the new coordinates

S)'J=

T YJ

O.

= FVJ

Syy=t( Fyy - FJJ),

Ty-y =t(Fly+ FJ}),

SyJ= FyJ

TYJ

SJ~

= FJy

=

0 ,

(2.5)

Ti'Y=O.

S,n =t(FJJ - Fyy).

TJ]

Slj = 0 ,

TJ~= FyJ

= -+ (Fyy

t FJJ ) ,

• [24,8]

So the self-dual condition

T",v= a (F.... oJ

Fn= FyJ = 0

Fyy

t

Fif =

0

= F""v)

irnplies that

(2.6a) (2.6b)

Now any four-vector A~ can be arranged into a matrix form by introducing (Y. Brihaye et al., Ref.[8]) I

A= .n~A""

(2.7a)

265 188

Kuang~chao,

CHOU

SONG Xing-chang

Then (2.7b)

where unit matrix.

are Pauli matrices and I is a 2 x 2

;

For instance, the coordinates x~ can be represented by the matrix

form

(J

~)=

I x = 5tr;..,X .... = y

-~

y

.

(2. 7a')

This notation exhibits clearly the lecal isomorphism between SO(4) and SU(2)xSU(2).

An SO(4) rotation

corresponds to the transformation (2.8)

y - y'= HYN, where M and N are two independent SU(2) matrices whose generators are (TJ

I

(-J

J

(2.8')

11(=z(Lj-ll(,,)

II(=2"(Iij+lK4);

are the respectively (i,j.k= 1.2.3 cycl.~ and 1,.. ... =-1v..... (jA,J}=1.2.3.4) generators of SO(4). In other words, Eqs.(2.8) imply that an SO(4) vector x~ transforms like a(i,i) representation under SU(2)XSU(2). From Eqs. (2.7), the scalar product of two vectors AI' and B", can be expressed as (2.9)

A=:JrvJAA,...

with

(A:=A+ for Ap real.

+ denotes hermite conjugation).

Sp denotes the matrix trace in ordinary space A

B

and

A

(SU(2 )®SU( 2) space).

Here

~.Ieanwhile

B transform under SU(2)xSU(2) ,as

AB respectively.

(2.10)

N' (A B) N •

This means that t'he traceless matrices

As - iSpAB

and AB - iSpAB

transform as (1,0) and(O,l) representations respectively. As is well-known, under SU(2)XSU(2), an anti-symmetric tensor F..... y forms as (1,0)9 (0,1) representation.

trans-

Here (1,0) and (0,1) are just the self-

dual and anti-self-dual parts respectively.

So if F,..~=

A....e. - A.e.... ,

we have

(2.11)

In general the field tensorccan always be written in matrix form as

F = S$ T

with Sand T defined by Eqs.(2.11) and transformed according to Eqs.(2.10).

266 11Je

lh sigma-models

&

self-dual SU(n) yang-Mills fields in four-dimensional Euclidean space

189

Therefore, the self-dual condition (2.6) T=O is a vector equation covariant under I (-) spin rotation. Now setting (2.12) we see from Eqs.(2.8) and (2.8') that tion in planes ( i , j ) and (k,4) while (i,j) and (k,4).

represents the synchro-rotathe contra-rotation in planes

For example, a rotation of angle

sented by the trans.f0rIllation with by one with

M=U k (8) N=Uk (8)

M=N=U2(6).

M=N

-1

=U 3 (8)

e

in (1,2) plane is repre-

and a rotation in (2,4) plane

Whereas the transformation (2.13)

implies that

(2.14)

III.

Hn Sigma-models In a preceding paper[221, we have discussed the Hn a-models, i.e., models

defined on the symmetric coset space SL(n,c)/SU(n), in which the induced MaurerCart an i-form is d 0.= 0.lJ dx lJ , and

.n,.. =

(.n;.

11;) (3.1)

where

~ ~

SL(n,c).

For real space, ElJ and KlJ are anti-hermitian and hermitian

respectively, so HlJ belongs to the subalgebra SU(n) and KlJ to its complement algebra on the coset SL(n,c)/SU(n).

Under the gauge transformation (3.2)

HlJ and KlJ transform as H",-

h- I

H}'

h + h-, o.,uit

(3.2a) (3.2b)

k},- 1,-' I<.,uh

This means HlJ is a composite gauge potential and KlJ an ordinary vector field which transforms covariantly. transform just as HlJ does.

We may notice that the combinations HlJ ± KlJ

The covariant derivatives for the fields

~

etc. are

defined as

~1'= ~"'-CPHj<

.

t..rf= ~f+ Hj
(3.3)

267 190

CHOU Kuang-chao, SONG Xing-chang

and then

K~

may be rewritten as

kr = cf'.6p 9'

= (.6p

= -(.1p

'1'-') cf'=_cj>'

1>-') cf .6p

(3.4)

'fH ,

whereas the covar:~nt derivative for Ku is (3.3 ' )

The hlaurer-Cartan integrability conditions (3.5a)

and (3.5b)

imply (3.6a)

t.,..

K,..

K" -
= o.

(3. 6b)

The currents defined as (~= q,q/) d

L

11)'-= (J)'-

R

. J)'- );

J;= 2,*,kp<j>-I=-j~~r'

,

(3.7a) (3.7b)

have the important property of being curvatureless d?AJ.L -J"Jp' -[ ~l

= ~

•

J"'J (3.Sa)

2
J: -J"J! (J; , J:J t

= 2tH(~/("-4J)Kfo)t'= 0 •

(3.Sb)

For two-dimensional models, the Lagrangian density can be written as etr is the trace sign for the matrices in the internal space)

_

Ii (

--;rr~d

-I

e 3.9)

f'Q

)<16.

Then the field equations read (3.10a)

or equivalently .P

A I<./"=

Q

(3.10b)

Introducing the light cone coordinates for the Minkowski case (~=O,l)

268 The Hn sigma-models

&

self-dual Su(n) Y3Ilg-Mills fields in four-dimensional Euclidean space

.[i~=x·+

191

,[27=X·-x '

X' ,

or the complex Coordinates for the Euclidean case (lJ = 1,2) 5F, =;X' - i

x!=JZy,

Eq.(3.10) becomes ~~k1+ 6'11<5=0.

(3 .10b' )

Since Eq.(3.6b) takes the form 6SI<~-A~k';=o

,

(3.6b' )

it follows from Eqs.(3.10b') and (3.6b') that 11~ K~

= A~ I(~ = 0 ,

(3.11)

and the dual symmetry

(3.12) holds from which the conservation laws, both local and nonlocal, can be found according to the standard strategy[ 25 1. The results obtained for the principal chiral fields can be applied to the present Hn models. (The only difference lies in the fact that KlJ is anti-hermitian in the former case but hermitian in the latter case.) The non local conserved currents were given by Ogielski et al. in Ref.[15]. BeSides, we obtained the explicit form[15,16] of the local ones which can be written as k;=D,I,2,'"

with

j"'= 0 ~

and

(3.13) where ,

J /(~)-zk~=

B.=(k

I

KS(k

J

k~fT.

and hi> 0 (i=l,2 ••• n) are eigenvalues of the positive definite hermitian matrix (K ~ K~)?t :: A • For four-dimensional Euclidean space (p=l,2,3,.4), Hp is still the SU(n) gauge field. By using Eq.(3.6b), the field equation (3.l0b) takes the form

269 192

CHOU Kuang-chao, SONG Xiang-chang

(3.14)

This equation together with Eqs.(3.6) describes the whole system of fourdimensional ~ models. Different from the two-dimensional case in which the gauge field ~ is sourceless[15], Eq.(3.6b) shows that H~ is not sourceless in four-dimensional case. Now we restrict ourselves to discussing the so-called self-dual En models, i.e., models in which the anti-self-dual part of the field strength vanishes. Then Eq. (3.6a) splits into the following form

Fr1=-{ I
(3.15)

a,

Combined with Eqs,(3.6b) and (3.14), Eqs.(3.15) describe a system of fourdimensional self-dual ~ models. From Eqs.(3.14), (3.15) and (3.7), we can show that

~:tJy+,,.rj= 2+(6}'kyt4~j(,+(Ky.I
(3.16a)

'yr;

(3.16b)

t

d.l:rj = 2*-' (.1}'/(y +.411<1-(K, • 1<,) - (K1 . klJ) '/' ~ ~ 0 •

This means that JL u and JR; (ii" y,z) are "quasi-conserved" and (3.8) imply that they are curvatureless. So according to the argument in Ref.[22], we can construct an infinite number of nonlocal continuity equations by the inductive procedure proposed by Brezin et al. [.13] Now we turn to the construction of local currents. Define a current

Ju = tl' ku B •

(ii=y.}l

(3.17)

which is a scalar in the internal space, and then we have d ... Jw. = i.,.. (d ... kiL) B t t ... k ... d .... S

=-:1.-,.. ([H ... • kit] B) t t-,.. kW.c11L13

=i." ku.

.d ... B .

So J u is "quasi-conserved" if and only if Il uB is trace orthogonal to Kii' This can be achieved by assuming that

I1I<;p bBKj"B te,

c.{ B. k::;} .

(3.18a)

AJB=-a.ky-bBKyBtC, (S. kl)- Cz{B. ky} •

(3.1Bb)

AyB =

CS. K,) t

The compatibility condition for B gives C,=

a

(3.19)

270 The Hn sigma-models & self-':'=l 5U(n) Yang-Mills fields in feur-dimensional azclidean space

193

but leaves a, band c;: :a::-bitrary. For simplicity we may choose c2 = 0, u=-a=1. Notice that Eqs.(3.6b}. (3.14) and (3.15) are invariant under the transformation

X,,-x ... x~

stands for each vector of

a~,

H~

and

K~.

(u= Y,~}

(3.20)

If we define the new current

corresponding to Eq.(3.17) and let B(y) satisfy the same set of equations as B does (with t~e replacement(3.20», Ju(Y) will also be conserved. Nowexpanding B(y) as the power series in y -

BO")=L: i=D

r -i

(3.21)

h.i'

substituting it· into Eq.(3.18a) and collecting all terms of the same order in we obtain the following set of equations

y

(3.22a) (~=o,

(3.22b)

/,2"')

from which an infinite number of local continuity equations can be derived •

. (~) J..

=J...

(b~ k.,J,

( j, =

0, f. 2 •... )

(3.23)

Eqs.(3.22b) can be rewitten as C3.2·2b') Eqs.~3.22a)

and Eqs.(3.22b') have already been solved in Ref.[16] under the assumption that the positive definite hermitian matrix K j Kj is invertible so a hermitian nonsingular matrix A can be defined as the square root of KzK z such that its eigenvalues Ai> O. The solution for be is (3.24a)

and in the diagonal representation of A the recursive formulas for bk (k=1,2 •••) are ( L. j

Then the first three local currents read

=

I. 2 •••

It ) •

(3.24b)

271 194

CHOU Kuang-chao, SONG xiang-chang

.IoJ

l'i

=.t...

ky '1"-' '(1)

Jj

=0 (3.25)

Now let us come back to the general case not satisfy the self-dual conditions. the current J u can also be defined as in of Eq.(3.l9) the compatibility condition

H

where the field strength Fll \) does The symmetry (3.20) still stands and (3.17). So do Eqs.(3.18). Instead for B now gives (3.26)

We may choose for simplicity

C,=o •

Q.=!.=C.=I.

and proceed with the same strategy as before.

Then instead of Eq.(3.22a) we

obtain

which gives (3.27) Now corresponding to k=O ,I c'ase of Eq. (3. 22b) we have 0= bokJb,t b, kfbo t{l., , k,} ,

4)'b,

= b.k,jb.tb.klb.tl..kjb. t {b•.

kJ} •

The first one is an identity while the second one gives (3.28) so the solution of b1 involves non local expression. And theq the continuation equations will be nonlocal for k :: 1. This result is similar to that given by pohlmeyer.[8] At the end of this section, we would like to show t~e relation between our Hn models and Pohlmeyer's model. As mentioned before, q= ~~+ is an nxn hermitian matrix with unit determinant. For n=2 we may "et

where

T are

Pauli matrices in internal space.

and unit determinant implies that

with

Hermitian means (qo'

q)

are real

272 The HI! sigma-models & self-dual SUln) Yang-gills fields in four-dimensional Euclidean space

195

The Lagrangian densidy (3.9) then becomes

This means that our H2 model turns out to be Pohlmeyer's SO(1,3)/SO(3) nonlinear a-models.[8J So Hn models can be considered as the generalization of H2 ~ S0(1,3)/SO(3) a-model.

IV.

Sel f-dual SU(n) Yang-rU lIs fields

Let us consider the SU(n) Yang-Mills theories in Euclic!ean four-dimensional real space which satisfy the self-dual conditions. The first two of the self-dual equations, Eqs.(2.6a), can immediately be integrated to give[24J A

~., '-1

Y=T 'fJ

(·,1.1)

where the reality of the gauge potential Au' i.e., (4.2)

has been taken into account, oj) = cjI(y,z,y,z) e: SL(n,c) transformation is the replacement

,/,-th. under which the gauge potential [Au,AvJ transform as

h=k A~

(y.J.Y.~) €

and cjIy=

dycjl,

etc. A gauge

(4.3)

SU("),

and gauge field strength

Fu v = auAv-

avA~

+

~- h- ' (~+ ~JJ,..

c,u.J1-A- ' Cu.IJA • It has been pointed Ollt that, by introducing a positive hermitian matrix[81

which has the very important property of being invariant under the gauge transformation (2.3), the only non-vanishing field strength can be expressed in the form

(4.4) and the third of the self-dual equations, Eq.(2.6b), now becomes

273 196

CHOU /{uaD.g-chao, SONG Xiang-chang

(4.5a) or (4.5b) or equivalently (4.5c) Since F"y, (F"y+ F ,,:z) 112 and Fiij are three independent components transforming like the I(-) spin vector T" (T_ 1 , To' T1 ), Eqs.(2.6j and then Eqs. (4.1) and (4.5) are invariant separately under the 5U(2)XU(1) subgroup of SO(4), i.e., under the transformation (2.8) with M

= U• •

N=J

(1.=/.2.3) •

and

(4.6b)

N= U,3

M= J ,

(4.6a)

Besides, as Eq.(4.5} is homogeneous, it is also invariant under the dilation of the coordinates (~ is a real parameter different from zero) (4.7)

Moreover, there exists a series of internal "Lorentz" transformations[8]

w= 'W ('y.}>

E SL (n. c )

(4.8)

under which the gauge potential ~, and therefore F~v' remain unchanged. Another transformation which leaves (4.1) and (4.5) invariant is the replacement[B] (4.9)

(T means transpose here.) For SU(2} case it can be considered as the special one of (4.8') by setting 'W" w+ .. T2, since 1:',1)..7:.

=

~T,

(,)L=o, 1.2.3)

T~=(I,T} and T~=(I,-T). The above description indicates an important fact that there exists a remarkable similarity between self-dual Yang-Mills theories and the Hn models discussed in the preceding section. The fields ~ are just those discussed in Un models and so is the gauge transformation (4.3). Only the definitions of the gauge potentials in these two theories are somewhat different. Instead of Eqs.(3.1} we have

where

A",==.n.:=H... t 1<.... from Eqs.(4.1).

(4.10)

Similar to Eqs.(3.3). the covariant derivatives should be

274 The Hn sigma-models

self-dual SU(n) Yang-talls fields in four-dimensional Euclidean space

&

197

defined as

(4.11)

whereas (4.12)

as can easily be seen from Eqs.(4.1) immediately. define

Corresponding to Eq.(3.4)

(-1.13)

=Il~- A;:c: =zka.. . and then Maurer-Cartan integrability conditions imply that

F.,;=])ycj -.D}C1 - (e1 _ CJJ= a, Fn =-])jCJ tlJjC,- [(y.

ell =

(4.14a) (4.14b)

0,

( ..... y=

(4.14c)

Y-5)

The currents take the form '1 .,.= +COLTd.-I=IJ"-.t

JL

=

(JJ..

J.,.' = 'P'e'~ .. 'P =

-I

,/,),r'=-.p.D..t-

1

8" 1~

J"

(4.15a)

,

+'

~ - .J. '-I C J."-r .. =tJ.,-1_ 3'" "

(4.15b)

Combining all these results, we get

Fn = cp-I"(~)'J;-CJJ:r;

- [.r~.

J; J)+

- tt Cilyrj-Cls J'). 1" (J; _ J; J) .-1 -

2 (Ay I
=

DyCrD}Cy-(Cy,CJJ -=0

Fyj =_

,-I

(4.16a)

y,Ji 1) t

(JyJi - diJ~ - (J

=-'f (~Jr;- iIJ r; t(J;' rpH+-

1

=-2 (Ay/<J-Aj /(y)

=-1)7 CltD] cy- r cy _ CJ-l = 0 • and

(4.16b)

275 198

CHOU .Kuang-chao, SONG Xiang-chang

F",,=oj>-I (dv 1~ ) '"

=.D... Cv

(4.16c)

Therefore, besides the f':)rm given in Eqs.(4.5) w.hich can be rewitten as (4.5a' ) or (4.5b' ) the third of the self-dual conditions can also be expressed as (4.17a) or (4.17b) or (4.17c) The last form Eq.(4.17c) can be considered as the shortened form of covariant Laplacian equation first proposed by Yang.[24] It is clear that in some twodimensional planes (e.g. 1,3 plane), Eq.(4.17a) reduces to Eq.(3.10b'). So self-dual gauge theories reduce to two-dimensional Hn models. Now we turn back to discuss the covariance property of the self-dual conditions under the spatial rotation 50(4). We have mentioned in the last section that T = [FzlJ'''~ (Fyy + Fzii )' Fiy] as a whole is a r(-J spin vector and the self-dual conditions T = 0 are covariant under SU(2)x5U(2) a 50(4). As indicated in Eqs.(4.6), out of the six independent ~otations of 50(4), four[~5U(2) x1J( 1)] do leave the equations "invariant". So it is sufficient to consider one· extra rotation, say+) CoS

M= I,

e

Si1t9 )

N=U.lz8J = (

C058

-$1\9

(4.18)

under which the coordinates transform as

y--/ =YC

0

59 t "ISinG

and the gauge potential A and F, should transform as A-A'=

i.e.

+)

T,

AN

,

J -~ '= ~coS' - Y$i... 8

(4.19)

the anti-self-dual part of the field strength

T-T

,

=

't

N TN

(4.20)

t

;J

SiDce Ul(29J .U3(~} U;t(29J U3 ( and U3 leaves the equation "invariant", the equation "'ill be covariant under the rotation Df U1 if it is covariant under. U2'

276 The Hn s.i.gma-models Ii self-dual SU(n} YBZlg-Mi.lls .f.i.elds .in four-d.i.mens:i.Dnal Erxlidean space

Ay- Ay' =

199

A, coss t Ai SillS, (4.20a)

and

F1 •1 ,= FlY FJ'Y't

cos's t Fj y .s;,,'8 t

-}( Fyy t

FiI'= (Fyy t Fil) (0528

FI' Y' = 1;c05'e

t FlY $in'lJ

T

FJj j 5;" UJ ,

(Fjy - FJy

-f (F,1

y -"

(4.20b)

)5;,,28 ,

FIt- ) Si" 28 •

In the rotated coordinates A' should behave just in the same way as A does in the original coordinates, so it can be expressed in the form

,.~ E SL

A.: = r'l~'

, ".

c-'

(.!.:::1)

Letting (4.22)

j= wf . we have

from which we can easily see that only if+) W

~f

Wy '

A

will transform as in Eq.

= - SllIeJ} L

•

W

-1

I

Wi

=

•

(4 .• 20a)

t

if and

(4.23)

S,ll 8 J"'j.

The converse statement is also true: The system (4.23) is compatible if and only if Eq.(4.5a') is satisfied. Therefore Eq.(4.23) is just the system of linearized equations for the self-dual gauge theories, which is equivalent to that given by Zakharov and Belavin,[lO] and to the "duality-like" symmetry equations given by pohlmeyer.[8] Just follow the procedure applied to the two-dimensional non-linear a models in a previous paper~14]and Eq.(4.23) can be integrated to give

where P denotes the anti-order operator along the path of integral. WoESL(n,c) is the function of y' and z' only and can be taken to unit rr.atrix by a propel' transformation of the type (4.8) in the new coordinate system. Now Eq.(4.23) can be rewritten as +)

Prom (4.23)

.I._I ( - 1 ) . . 1 .

T

.I.-I

L.I.

W Wy' T=-Sirt8T JjT

= Then

'-'3'1' = =

7-'3"

Cos, B

'f-'';'I + Sid 4>~

+s - Sine Cl

COSI Ay t Si"e AJ ;

= Co.s8 t-'t,-Sid 'r'fy;' Si.e Cy = c.oSSAS - Si,,8A y •

Sin.e c y .

277 200

CHOU Kuang-chao, SONG Xiang-chang

w'/= - ~ e ("'1 wt =

tJ'

T IN

('Ny t VI

Then the existence of W implies

J~

)

Jy )

Wyz - WZ!I = O. 1. e. , (4.25)

Expanding W in powers of para!lleter F; = tg e. an infinite number of non local continuity equations given by POhlmeyer[8] follows. Using an inductive procedure Ogielski et al.[22) have also constructed a set of nonlocal conservation laws. The following important fact can also be seen from Eq. (4.24). [nUke w- 1 Wu. which are local functionals of field q as in Eq.(4.23). w- 1 Wrr may involve some kind of compLicated integrals. This means that Cu and Crr will undergo nonlocal transformations under the rotation (4.18). So it is impossible to obtian the local conservation laws by applying to e u and Cli the "duality-like" symmetry linked with rotation (4.18). However. in spite of this nonlocal transformation property. the existence of W in Eq.(4.24) preserves the locality of the covariant derivatives of Cu and Cli' so that the Maurer-Cartan identities (4.14) and the self-dual condition (4.l7b) are still satisfied by themselves. This can be checked by direct calculation. From the above discussion we can see that great similarity exists between the four-dimensional Hn models and the classical Yang-Mills fields. Both theories deal with the same sort of field ~ or q which undergo the same kind of gauge transformations and are subject to the same set of constraint equations-Maurer-Cartan ider.tities, Eqs.(3.6) or (4.14). Their dynamical equations. the field equation (3.14) in the former and the self-dual equation (4.17) in the latter. are similar in form but different in content (since A~~H~). The maiD difference lies in the fact that ~ in the former is an exact scalar while in the latter ~ behaves as an ordinary scalar only for the transformations of the subgroup SG(2)xU(1). Rotations like the one considered in Eq.(4.18) induce local SL(n,c) transformations in internal space. Therefore the self-dual Sr(n) gauge fields contain more solutions than that of the corresponding Hn models. From the forlllal resemblance between Eqs.(3.14) and (4.17), we can see that the symmetry (3.20) holds and the scalar current j~ =

t,. Cwo 6

can also be defined in gauge theories as in Eq.(3.17). This current will be "quasi-conserved" (dujBU =0) if and only if DuB is trace orthogonal to Cu' Then similar to Eq.(3.18) we assume that DuB has the form

])y B =

a. C1 t b 8 CJ B + C, (6.

.])5 B=-Il.

e,l t c. {B, 'J}

ey-b B Cy sH,(S, Cjl-C.{s, Cy} •

278 The Hn sigma-models & self-dual SU(n} Yang-Mills fields in four-dimensional E.1Jclidean space

201

and the constraints imposed bi the compatibility condition are again

C, = o. Making the choice

a

= b = c2= 1,

expaning B(Y) in powers or Y

inserting it into Eq.(4.27a) and collecting the terms of the same order in Y, we obtain a set of equations to determine b k . Just like the situation in Eq. (3.28) the equations for b k (k ~1) yield non local solutions from which follows another set of nonlocal continuity equations. This result is similar to that given by Pohlmeyer[8]. Similar equations can 3.1so be derived from the parametric Backlund transformation.+) For those solutions of the self-dual SU(n) gauge theories which satisfy an additional condition

H

(and then FyE=O, by taking hermitian conjugate), we see that the compatibili"y condition gives C,

=

0

as in Eq.(3.19), but leaves a,b and c2 arbi'trary. Then following the sarne procedure as given in Eqs.(3.20) through (3.24), we obtain an infinite number of local conservation laws as in Eqs.(3.25) (with the replacement l\.u+D:z' 80: Ku+C u ' etc.) The self-dual Hn sigma-models discussed in the preceding section are the special cases for tbis kind of solutions.

V,

SUIrJr.a ry

In this paper the Hn a-models both in two-dimensional and four-~imensional Euclidean space are formulated, and the classical self-dual Yang-Mills fields are analysed in terms of the quantities defined in the ~ models. We have shown that a close resemblance exists between these two theories. In some twodimensional planes the self-dual gauge fields reduce to the En models in twodimensional space. And in whole four-dimensional Euclidean space, the selfdual gauge fields have a great similarity with the Hn models, partially in content and partially in form. The essential difference lies in the fact 'that the field ~ in the Hn models is an exact scalar under the spacial rotations oISO(~) while, in the self-dual gauge theories, • is a scalar only for the transforma+) From Eqs.(I4} of Ref.(9], the following Contizluity equation

D=aJ

tT

tt' J' t

i'-'lylt al t,. l 'Ii'}'

1"

r-' h)

..

can be derived, where q is a solution of the field equation and q' another solution lihich dependes 011 a par_ter Il i.JlIplicitly. Expanding q' in powers of y ('/'=£ ~(")yl"), yZ ~

-(l-;}/(l+))), tore obtain an infinite set of conservation laws. However the BT gives the solution of q(kJ (1c ~lJ being nonlocal functionals of q, ajJq, etc.

279 202

CHOU Kuang-chao, SONG Xiang-chang

tions of the subgroup SU(2)XU(1) aRd the rotations otaer than SU(2)xU(1) give rise to non-linear local SL(n,c) transformations on ~ in internal space. By using the known structures of two-dimensional Hn models as a guide, an infinite number of nonlocal conservation laws in matrix form has been obtained for both four-dimensional Hn models and the self-dual Yang-Mills fields. The "duality" symmetry often used in two-dimensional non-linear a-models gets a new feature of being coordinate dependent in four-dimensional self-dual gauge theories. Another set of symmetry can be defined in four-dimens~onal theories from which another set of infinite continuity equations in trace form,the analogy of the two-dimensional local conservation laws,can be cons~ructed. The meaning of this set of continuity equations has yet to be clarified. This set of equations turn out· to be local expression for a restricted set of solutions, e.g., the four-dimensional self-dual Hn mOdels. It might be worthwhile to study corresponding problems for the Yang-Mills theories in loop space form and to investigate the associated quantum theories.

References ~(1976)

1.

K. Pohlmeyer, Comm. Mach. phys.,

2.

M. Luscher and K. Pohlmeyer, Nucl. pnys.,

207. ~(1978)

V.E. Zaharov and A.V. Mikhailov, (Sov. Phys.) JcTP H. Eichennerr, Nucl. Phys., 8146(1978) 215; H. Eichenherr and M. Forger, Nucl. Phys.,

46.

~(1978)

~(1979)

~(1979)

1017;

544;

381;

V.L. Golo and A.M. Perelomov, Phys. Lett., B79(1978) 112; M. Dubois-Vio.1ette and Y. Georgelin, Phys. Lett., 3.

~(1979)

251.

A. d'Adda, M. Luscher and P. di Vecchia, HUcl. Phys., B146(1978) 63;

M. Luscher, Phys. Lett.,

~(1978)

465;

E. Witcen, Nucl. Phys., B149(1979) 285. 4.

A.A. Migdal, (Sov. Phys.) JETP £(1975) 743; E. Brezin, J. Zinn-Justin and J.C. Le Guillou, Phys. Rev., £!!(1976) 3615; 4976; S. Hikami, Prog. Theor. Phys.,

~(1979)

226;

A. Mckane and M. Stone, Oxford Univ. preprint (1979). 5.

A.M. Polyakov, Phys. Lett., B59(1975) 79;

6.

A.A. Belavin and A.M. polyakov, (Sov. Phys.) JETP Lett.,

7.

D. Gross and A. Neveu, Phys. Rev., DI0(1974) 3235;

E. Brezin, S. Hikami and J. Zinn-Justin, Nucl. Phys., B165(1980) 528. ~(1975)

245.

B. Brezin, and J. Zinn-Justin, Phys. Rev., Lett., 36(1976) 691; W. Bardeen, B. Lee and R. Shrock, Phys. Rev., £!!(1976) 985. 8.

E. Corrigan, D.B. Fairlie, R.G. Yates and P. Goddard, Comm. Math. Phys., 58(1978) 223; Y. Bribaye, D.B. Fairlie, J. Nuyts and R.G. Yates, Journ. Math. Phys., !!(1978) 2528; M.K. Prasad, A. Sinha and L.L. Chau wang, Phys. Rev. Lett., £(1979) 750;

9.

K. PohJueyer, Comm. Math. Phys., ZlJ1980} 317. For a review see L.L. Chau Wang, in Proc. of the 1980 Guangzhou Conf. on Theoretical Palticle Physics,

10.

~980,

pl082.

A.A. Belavin and V.E. zabkarov, Phys. Lett., !Z!(1978) 53.

280 The Hn sigma-TIIOdels

11.

&

self-dual SUrD) Yang-Hills fields in four-dimensional Euclidean sl=ace

A.M. Polyakov, Phys. Lett., BB2(1979} 249; S.Y. WU, Physica Energiae Fortis et Physica Nuclearis, I. Ya. Aref'eva Lett. Math. Phys.,

~(1979)

~(1979) 382;

241;

L.L. Chau Wang, talk at the xxth Int. Conf. on High Energy Physics, Wisconsin, July, 1980. 12.

M. Luscher and Pohlmeyer, Ref. (2); A.T. Ogielski, Phys. Rev.,

~(1980)

406.

13.

E.Brezin, C. Itzykson, J. Zinn-Justin and J.B. Zuber, Phys. Lett., B82(1979) 442:

14.

H. Eichenherr and M. Forger, Ref.(2};

15.

K. Pohlmeyer, Ref. (1):

H.J. de Vega, Phys. Lett., BB7(1979} 233. K.C. Chou and X.C. Song, ASITP preprint 80-008, to be published in Scientia Sinica. A.T. Ogielski, M.K. prasad, A. Sinha and L.L. Chau Wang, Phys. Lett.,

!!E..( 1980}

387:

K.C. Chou and X.C. Song, ASITP.preprint 81-009. 16.

H. Eichenherr, Phys. Lett., B90(1980} 121: K. Scheler, Phys. Lett., B93(1980} 331; F. Gursey and X.C. Tze, Ann. of Phys., !!!(1980} 29; K.C. Chou and X.C. Song, ASITP preprint 80-010. to be published in Scientia Sinica.

17.

M. Luscher, Nucl. Phys., B135(1978) 1.

18.

A.M. polyakov, Trieste preprint IC/77/122 (1977).

19.

D. Iagolnitzer, Phys. Rev.,

20.

A.B. Zamolodchikov and Al. B. Zamolodchikov, Hucl. Phys., B133(1978} 525.

~(1978)

1275.

21.

A.M. polyakov, Ref. (11).

22.

M.K. Prasad, A. Sinha and L.L. Chau Wang, Phys. Lett., B87(1979} 237.

23.

K.C. Chou and X.C. Song, ASITP preprint 81·-003, Commun. in Theor. Phys. 1 (1982) 69.

24.

C.N. Yang, Phys. Rev. Lett., 38(1977) 1377.

25.

K.C. Chou and X.C. Song, Ref. (14) and (16).

203

281 Commun. in Theor. Phys. (Beijing, China)

725-730

Vol. 1, No. 5 (1982)

ON THE DYNAMICAL PROBLEMS AND THE FERMION SPECTRA IN THE RISHON MODEL CHOU Kuang-chao ( JlIBOCD CERN

~d

Institute of Theoretical Physics, Academia Sinica Beijing, China

DAI Yuan-ben ( liUt:.fz ) Institute of Theoretical Physics, Academia Sinica

Received May 6, 1982

Abstract We propose to introduce a strong coupling

U(l.' y axial

gauge boson into the risbon model which may explain dynamical problems concerning preservation of chiral symmetries and existence of solutions to anomaly conditions.

Mechanisms through

which exotic particles gain masses are discussed.

It is found

that exotic particles such as color octet leptons and color sextet quarks may gain masses ranging from

10 1

to

10 3 •

Some

phenomenological aspects of these exotic particles are discussed.

Among various composite models proposed for leptons and quarks the rishon

model~] has several special interesting features including economy and fitting in well investigated gauge theories. Powever, there are embarrassing problems of dynamical nature for this model among which are the following, 1. Within leptons and quarks the color SU(3)c interaction is much weaker than the hyper-color SU(3)HC interaction. In the approximation Qc=O the theory is similar to QCD with 6 flavours. However, it is necessary to assume that some chiral symmetries are not broken by the SU(3)HC interaction in contrast to QCD wAere chiral symmetries are assumed broken spontaneousely. 2. There is no flavour number n independent solution to 't-Hooft anomaly conditions[2] for the Qc=O limit SU(3)HC gauge theory with SU(n)L x SU(n)R xU(l)v flavour sy~etry. On the other hand, it seems unlikely that the weak SU(3)c force can play an essential role in binding light fermions. 3. It is assumed that there are quarks in the configuration uL=(tLtL)VL but none in the configuration (tRtR)VL though space parts of these wave functions can have the same symmetry. Note also that the fundamental Lagrangian of the SU(3)HC x SU(3)c gauge theory has the symmetry t£o- tR with VL , VR fixed. 4. There are exotic particles (ttt) in 8 and 10 and (ttv) in 6* and 15 representations Of SU(3)c which are assumed to be heavy. How can it be achieved and how large are the masses of these particles? 5. In order to avoid contradiction with the observed long life time of the proton the scale A of the SU(3)HC interaction must be assumed to be

282 726

CHOU Kuang-chao and DAI Yuan-ben

larger than 10'GeV at least[3]. What can be the origin of the scale of the weak interaction GF1;o 300GeV i f A » G;l? In this note we shall suggest some ideas which may provide partial solutions to these problems. As a possible soluton of the problems 1 and 2 we propose to introduce an additional U(l)y gauge field interacting with the current

into the theory.

The local symmetry of the theory is assumed to be

We have used U(l)B_L to replace U(l)O in the original rishon model. The photon is assumed to emerge in the later stage when the B-L gauge boson is mixed with composite weak interaction gauge bosons in the low energy effective theory. Since contributions to U(l)y anomaly of t and V cancel each other ·the theory is renormalizable. The global symmetry of the theory is U(l)R x Zl:' where Zl~ is the unbroken discrete subgroup of U(1)x with

l)..= it~.r,s t

+ i. V~ l",s V.

We assume that the U(l)y gauge coupling constant gy is comparable to SU(3)nc coupling constant gHC. Since the. U(l)y interaction between the pair (tRt L } or (VRV L ) is repulsive, it may prevent the scalar composites condensating and thereby keep some of the chiral symmetries unbroken. In order to avoid contradicting the low energy phenomenology the U(l)y symmetry must be broken at a large scale A'. This can be realized by the condensate (VLVLVR )2 corresponding to a Major~na mass term of vR which is required also from phenomenology of the rishon model [4]. We assume that the scale A' lies within the range A> A' »G;I. To see that the U(l)y interaction prefers this condensation to (tt) or (VV) note thoat the average U(l)y interaction energy per rishon in a composite of n rishons is l.

I (2

:{!.

Z) ,

1y Tn" Y- ~ Yj

"

J

(1)

where y and Yi are Y charges of the composite and constituents respectively. This interaction energy is negative for the composites ,(VLVLVR )2 while positive for (tt) and (VV). The corresponding quantity for the SU(3)HC interaction is (2)

where C and C~ are the Casimir operators of SU(3)HC. From Eqs.(l), (2) and other considerations it seems plausible to assume that the condensate (VLVLVR ) 2 is fa~ored over other possible Y breaking condensates. This condensation also breaks U(l)B_L' U(l)R and Z12. Being vR Majorana mass term it transforms as (!, !*) under SU(3)CLXSU(3)CR. However, it is easy to see that effective interactions produced by this condensation in the rishon number conserving sector preserve Z12 and transform as representations of zero tri-

283 On

the Dynamical

P~oblems

and the Fermion Spectra in

th~

Rishon Model

727

ality with respect to both SU(3)~L and SU(3)2R such as <.!..!).(.§..~).(lO.lO*)+ (10*.10) •••• where SU(3)~ is the SU(3)c group restricted to V rishons. Therefore rishons are still kept massless at the scale ~'. Among the H.C. singlet composites of three rishons there are only some vvv states that gain Dirac masses from the sue 3 )CL x sue 3)CR breaking effects of this condensation (see below). Therefore. in the first stage of the discussion on the ~pectra of three rishon bound states in which U(l)B_L and SU(3)c interactions and the condensation at the scale A' are neglected the global hyperflavour symmetry of the theory can be taken as ;(3)

If we want to require the flavour number independence of the number of the multiplets of massless composites we are led to the anomaly conditions for the theory (4)

with fermions in the representations

tL = (11 VL = (!!

J

1.

J

1-. t) I

jt J

I )

L . 3'-3

This problem was first discussed by authors of[5]. But they did not give a convincing argument for using the symmetry (3). Possible representations for H.C. singlet composites consist of the following: ttt 8-L=l, ttv

Ii't =

( 0:0

'1,_ =

§

'1,.-=

0

)

.

). . o::J),

lJ+

=(0

• 1. ) ,

'I5t= ( 0", OJ),

'15_= ( 0*.

B).

':1 2 - = (a •

'16=

ell-{Uf'

'13 =

'17t=

(wo,{U . 1).

'17_=

ell-,{r .

8 ). ( ff1 , ).

D).

1).

Other composites correspond to representations ql which are parity conjugate to qi and complex co~jugate representations q! and ql* obtained from qi and qi by the replacement t ++ v (with opposite "Sign of B-L). Let us denote the indices corresponding to qi by 1 i • From left-right symmetry(accompanied by the change of sign of gy which is irrelevant for bound states) of the the... Here the SU(3} part is the unbroken subgroup of the sgsatry group of the Lagrangian.

SU(3}~ XSU(3}i XSU(3}r XSU(3}~

284 728

CHOU Kuang-ehao and DAI Yuan-ben

ory, we have ti=- 11.1.. The anomaly conditions for 3 SU(n)L and SU(n)L x SU(n)L "U(l)B_L triangle diagrams are readily satisfied with ti'" ti. Tile anomaly condition for SU(n)L)( SU(n)L x U(l)R triangle diagrams is reduced to i+ +(n

2 _

3) 13 T (11\ I)),

+Ft( 0 t 21 (nt3) i,t

-f-tn (11±3) (J~t t)s±) t

~>t(3nt IHoi

2)

in =

(5)

I .

In contrast to the situation in the original rishon model, r.here are a lot of n independent solutions for the anomaly conditions[5]. If we insist that the space part of the internal wave functions of light fermions should be symmetrical we should have

For the illustration of mechanisms through which the exotic particles may get masses let us COnsider th~ simple n independent solution

lu = 14= 3

(6)

This solution has the interesting property that 'Jnly composites with negative U(l)y interaction energy exist. Besides three generations of leptons and quarks with

e: =

(til tR ) tL

;{

=

( VII V~) VL ( 7)

the solution (6) contains particles (ttt), (vvv) in!, 10, 10* and (ttt), (vvv) in~, ~*, 15,1&*. These particles gain masses fro~QCD chiral symmetry breaking. From the investigations in strong coupling approximation of lattice gauge theories [6] and coherent state variation calculations[7] it seems that the chiral breaking condensation in QCD is a direct result of the strong interaction between the fermion pair which has nothing to do with the confinement [10]. If this is correct the scale of chiral symmetry breaking is not he but M determined by ( 8)

where C is the value of SU(3)c Casimir operator. From Eq. (f.), components of G 7 + in 15 and components of q~+ in 10 of SU(3)c representations reay gain masses of tbe order of 10GeV i f they are proportional to <~1j;>o '" M3. In this case the only possible way in which the rernainin~ exotic particles 6* of q 7- and 8 of q H can gain needed masses is the electroweak interaction. As an illustration of different mechanisms to make! and ~ massive let us relax the requirement of n independence and consider the following solution to tbe anomaly condition for n=3

285 On the Dynamical Problems and the Fermion Spectra in the Rishon Model

l,t=

0

12_= XI. = 3,

)21- = - 3

13 =

0

Rst =- t

Is_ = I , J7_ = 0

16= 0 ,

i 7+=-

2

J

729

J

( 9)

The interaction of two rishons throu~h the exchange of one color gluon contains the SU(3)CL x SU(3)CR breaking term

~"Ol~ ~ ~J\"~ '/'1(

(10)

which transforms as (!!.~) in SU(3)CL x SU(3)CR symmetry. This term causes mixinp; of components of different representations of SU(3)CL x SU(3)CR which belon~ to tbe same representation of SU(3)c and have the s~~e x char~er81. As a result of this mixing. among particles in tbe solution (9) tbree leftbanded!! in q2- (q~_) are mixed with three rigbt-handed !! in q:~.,. (qi+) and large masses of O(~c h) are gained. In addition. !! in q~+ and q~_ can be mixed by the cbiral breaking effec.t of tbe condensation (VL VL VR ) 2 mentioned previously and gain masses of O(h'). These massive particles form parity doublets. Therefore they give no contribution to mass corrections and magnetic moments of ordinary leptons. Similarly. the ~ and ~* components of q,% and q~% get masses of O(~ h). The ~* components in qs-· can gain masses from tbe color gluon correction as shown in the following diagram. From gauge invariance the gluon vertex in this diagram corresponds in tbe lowest dimension operator product to the effective Lagrangian (11)

wbere

F~v

is the color field and

f

is a constant.

Since the product of

left-handed qs_ and tbe complex conjugate of right-handed qs+ belonging to (~.~*) x Q. * ~*) contains (l.!!). f does lIot contain the suppressing factor due to SU(3)CL x SU(3)CR symmetry. Since qs+ and ·qs- have the same x charge. (11) is not suppressed by Z12 either. Therefore f is not much smaller tban 1. The self-energy integral formally div6rges quadratically. Tbe form-factor of composite particles provides a cut-off at the scale h. If the effective range of integration is II we sbould have (12)

On the other band, tbe mass insertion on the internal line of the diagram is expected to decrease as tbe momentum becomes larger than the scale M,S ' The estimation (12) can be invalidated if tbe mass insertion is switched off too fast so tbat the effective range of integration is nbt II by M1S ' Anyhow. it seems possible that exotic particles in ~ gain masses of tbe order of M1S ' Note tbat tbe above discussions on masses of exotic particles do not necessarily depend on tbe existence of the U( 1>1' yauge boson. It follows from these discussions that. in general. tbere are exot.ic particles not mucb heavier than 10 GeV.

286 730

CHOU Kuang-chao and DAI YUan-ben

Possible physical effects of colored leptons in low energy phenomenology was· discussed in [9). From SU(3) gauge invariance the effective interaction between colored leptons L and ordinary leptons t in the lowest dimension is of thE' form (13) where f' contains additional suppressing factors due to chiral invariances. Another low dimension effective interaction is of the form (14) In low energy proc.esses these couplings are very weak. It turns out that the existence of colored leptons with h > 10'GeV and M > 40GeV does not contradict known experimental facts. The colored leptons in ! and quarks in £ discussed above can be produced in e+-e- or p-p collisions. They decay into ordinary leptons and quarks by emitting a gluon with a narrow decay width of the order of (15)

For h > 10'GeV, r is extremely small. Before decaying to ordinary leptons and quarks they should form exotic mesons or baryons. Low lying exotic baryons should have long life-time also. Low lying exotic mesons decay mainly into gluons, just like J/W or T. The colored lepton can also combine with a gluon to form a color singlet. In our opinion, the most feasible test of the rishon model is to search for such exotic particles in experiments which can be carried out with accelerators of the next generation. The above discussions have not provided an explanation of the problem of the scale of the weak interaction. However, in view of the complicated spectra -l- ,,300GeV can emerge from of the theory, it seems possible that a scale of GF the theory.

References 1.

H. Harari, Phys. Lett. 86B (1979) 83. M.A. Shupe, Phys. Lett. H. Harari and N. Seiberg, Phys. Lett.

!!!

~

(1979) 87.

(1981) 269, the Risbon MOdel, WIS-81/38.

2.

G.'t-Hooft, Cargese Lectures (1979).

3.

H. Harari, R.N. MOhapatra and N. Seiberg. Ref. TH-3123-CERN.

4.

H. Harari and N. Seiberg, Phys. Lett. 100B (1981) 41.

5.

s. King, Phys. Lett.

6.

J-H. Blairon, R. Brout, F. Englert and J. Greensite, Nucl. Phys. B180 (1981) 439. J. Greensite and J. Primack, Nucl. Phys. B180 (1981) 170. H. Kluberg-5tern, A. MOrel, O. Napoly and,B. Peterson, Nucl. Phys. B190 (1981) 504. J.R. Finger and J.E. Mandula, Quark Pair Condensation and Chiral 5YI1lll/etry Breaking in QeD. S. Weinberg, Color and Electroweak Forces as a So'Jrce of Quark and Lepton Masses. Dept. of Phys., University of Texas preprint. 1981.

7. 8. 9. 10.

~

(1981) 201.

C.B. Chiu and Y.B. Dai, Phys. Lett.

l£!! (1982) 341.

J. KOgut, M. S~one. H.N. WUld. J. Shiqe1llitsu. S.H. Shenksr and D.K. Sinclair. The Scale

of Chiral 5Y1lIllletry Breaking in OUantum Chromodyna1llics, ILL- (TH) -82-5.

287 SC I ~NT IA SIN leA (Series A)

Vol. XXV No.3

Mareb. 1982

WILSON LOOP INTEGRAL AND STRING WAVE FUNCTIONAL ZHOU GUANGZH'O n

(CHOU K UANG-CiiAO

EEl.ll<.n) !
AND

LI

XIAOYUAN

( ) */J\~

(In.stil1/-te of Tllem-ctical Physic&, ..A.cadc·mia Sinica)

Received July 3. 1980.

ABSTRACT

In tllis paper we show that t.he Wilson loop integral cOI'responds to an external source term in the generating functional for Green's function witll II cli~ussion of its effects. This implies that. some approximations in the previous derivillion of the string. like equation may not be applIcable. TI1~ calculation of t.he field strength in the presence of this external sourcE' may be more useful 1lI the stu'}y of confinement.

Recently, many authors ll - 4] have discussed the possible connection behwl'n the Wilson loop integral r.] and the string wave function aIr.] , and attempted t.o show that under certain approximations the Wilson loop integral satisfies a string-like equation if the gauge fields meet certain constraints. :Meanwhile. some authors have argued that how to formulate pxactly a quantum field theory for sueh objects is far from clear['I, and there is still a great gap in the attempt to derive the string tlleory from a Yang-Mills theory. In this paper, we SllOW that the Wilson loop integral corresponds to an external source in the vacuum and its effect is disc ussed. We would like to point out that some approximations in tue previous derivations of the string-like equat.ion may not be applicable in the confining vacuum. And we also suggest that the calculation of the field strength in the presence of this external source may be 11SI'ful in discussillg the confinement prolllem. For simplicity, it is appropriate to consider t.he example of the Abelian gauge field at first.

In the usual Euclidean space path integral formalism the Wilson loop average ~4. [0]

can be defined as A[C] "'" N~[d.q>(z)]cJSi'(q'(·))J~%eI1Pc.4"(Z)J'!',

(1)

where £L' (qJ(z)) is t.he Lagrangian of the syste'JI1, cp(x) stands for all t.he fields inclucling tlle gauge field; A~(x) is the vector gauge potential, g is the coupling constant, X is a normalization factor and C denotes a Inrge close curve ,vhich is parameterized by

x"

=

:z;"(s),

So

~

S ,;;;;

s"

:<"u(so)

=

:z:F'('~1)'

(2)

By artificially introducing t.he 6 function A,,(x)

= Jd4yA,.cy)o(~)(y - xes»),

(:3)

288 No.8

WILSON LOOP INTEGRAL

&;

STRIXG WAVE FUNCTIONAL

265

the integral around the closed curve C, g tPc A'Cx) dx,.ld-s ds, can be converted into an integral over the whole space, then we have

(4) where j ,.(y,

C) == Pc d;; [J

il4l(y -

(5)

xCs»ds.

'I'hus the Wilson loop average can be \'I'1'itten as A[C] =

J[dlp]eJd411[5!':'I'!I/))+i,.(II.ClAI'(lIil .

It is nothing else than the ordinary generating functional for the gauge field Z[J~J evaluated at the specific. external source J,,(y) = JiY) = jp(y, C). Xamely,

A[C]

=

Z[J p ] I'p=i p '

(6)

It should be stressed that, in the usual generating functional approach, the external source J.(x) is only the auxiliary fields, because aU the Green functions are obtained from the functional derivatives of the generating functional by taking all J.(x) == o. But now, the external source j~(x) is properly introduced by the Wilson loop integral itself, so it may play a substantial role. Therefore the Wilson loop average can bl' considered not only as a functional of the curves C, but also as a functional of the extl'rnal so.uree J,(x). Now it is very easy to obtain the functional derivati.... es of A[C] with respect to the curve x"(s),

(7) wherl'

(8) is the vacuum expectation value of the vector gauge potential A~(x) in thl' presence of the external source .J~(y). And it. is not difficult to show that

8jP(y, C) = g {o:dx ~ oxp(s) ds /Jy. a

_

dx· ~} d,~ oyP

8(4l(y _

xes»~.

(9)

Inserting Eqs. (8) and (9) into Eq (7), we immediately find the first-or:der curve functional derivatives /JAr C)1 = yP,.lx(s), j(C»dx- .A[cL

lix" Cs

ds

(10)

where F,..C,,(s), j(C» """

~ Apex, j) - ..E.._ A.(x, j).

ax'

ax~

(11)

At first glance, Eq. (10) seems to be a well known result for a long time tTl • But this is not the case. In fact, the p •• (x, j(C) now is t,he vacuum expectation value

289 266

SCIEXTIA SIXICA

Vol. XXV

(Series A)

of the gauge field strength fo".v (x) in the presence of the specific external source jp(C) which is not vanishing pven for pure gauge field. And F".v (x., j(O)) obeys an equation with the external source

~ F"v(x, j(O» = - j"(x, C), ax"

(12)

as the average of the terms arising from gauge fixing eondition is vaDi.<;hing. In what follows, we will show that it is just this property that makes the result different from those of the previous deriYatioDs. Let us differentiate Eq. (10) with respect to Z

6 A[0] ,

8xis )8x"Cs)

_ - gl

again, we obtain

;r..(8')

(F",.(x ()s ,}.) d::c" F""('x (') d.r., s ,}') ds cU a

+

6

[J--,

6X/S)

[F,..(x(.~),

·]

A[C} '

dx· j)J -A.[OJ.

(13)

ds

and

+

r

8F

L

I

Bj"(y)_ d'y

" , J~ .j" ox,.(l) J BJU(y)

= - 8(S -

, s )j.(x(s»

I" " dx \ + [J \TJl /lOP"''' -;-) .' ds

(14)

lei

where ( )r; stands for the conneetcd parts of the Grt'en functions wit.h the external source i.(x). InsertiIlg Eq. (l4j into Eq. (13), we come to the final result

8'11[01

6xis')6x"'(s)

P/'''(x(l), .') dx" + (fo y,.. (x(s) .J.) dx· ds . J ds' - ga(1) - s')j.(x(s), C) liZ"} A.[C] , ds

dx· F.U" dx.~\

= {ll[J..,

I"

ds

d.s'lei

(15)

Eq. (15) is an exnct equation for t.he 'Vilsoll loop average. 'When s' -s, the secondorder functional derivatives of the A [C] involve singularities. In this case, it is appropriate to (lefille Z

8 A[C} Bxls)oJ'f(S)

=.!..Jr"h

BZA[e)

d.s'.

28 r-. 6xl'(S')8x"CS)

.

(16)

hrre e is an infinitesimal quantity. :-';ow let us give some comments. In order to derive the string-like equation, t.he anthors of R.efs. [1---'1] usually assullIe that .J.

(]) 'fhe right side of Bq. (15) is cOlllpletely indl'pendent. of any l'xlernal sour(·.l!S It implies that the gauge conne('tioD salisfies the full source· free field eqnations DI'F'''' = 0

or

dx.(sL D,.'}"" d.s

=

0,

(17)

lind mrans t.hat only the vacuum ('xpectation ,"!llue in the absence of the ext,>rnal

290 WILSON LOOP INTEGRAL & S'flUXG WAVE FTjNCTIONil.T"

No.3

:?67

sources are considered.

It is possible t.o use ratlwr speeial classical field cOIlfiguration to generate

(2)

automaticaIl~'

=

" " dx" d·r (Ji'I'VCX(S)F""'(X(S'). _. ~ ArC]' J-s tis

(18)

(fo ~ •.( xCs ))FI'a( xC s)) >f a~ ,

(HI)

(3) where

f

=

1/1(ffi!,"fol'v) is a COIlstiLnt or at

As a result of these approximations,

Ale J would ubc'y

2

a A[C]

=

o:r.,.(s)axl'(s)

slo\vly Y
1II0iSt it

11

string-like !'quatioll,

u~f (dX")l ACC]. \ ds

(:!O)

From tl1{' abon' dis('ussiolls. it is VE'ry clear that the assumption (1) may 110t bl' true. 'l'his is b{'eallse the Wilson loop iutt'gral itsi'lf illltomaticall;\' induees tile specific ext~rnal source j. (C) whith is not vanishing eY('n fo)1' full sourct'-fL'l'e gauge field. FurtheJ'ulCJre, it. for!'es us to consider the vacuum ('xpectiltion YHIut' of thf' fi(:'ltl strl:'1l~1 h in the PJ'I'Sen(~t' Ot this spl'cific external som'(·e.

Tn order to further illustrate the above point. We:' may see what would happen for t.he solution of the string-like !'quation in II ('onfining vacuum. Tn this ens!'

(21) is in lIC'cord with the 1Yilsnn ('onfinrnwnt ('ritt'I'j"n, HI1'.I!7' ill I~q. (21) i:; the least nrea enclos!'d b~- a dos!:'d ('urv!:' C and is parametrized by the eqnntioll :c." =

r,P(cT, -r); a is n constant a

=

!1.J7

By diffcl"l'lItiating Eq. (21) wit.h rl'sjJC'tt h' "ury!:' aA[C]

-_. = oX·"(.~)

and ;r~(s).

t > u. we obtaiu

- a .8Y _ - A(e]. 8xl'(.~)

(~2)

.

where

8.9'"' (/x" ---=n"SI-.' . ds

(23)

OXI'(8)

and

I

_ a(x!',x") fJCcT,.r) ..

IIp.-

thcboundor,pDint of the :Ut"3. i?

II

S'~(s)

I

a(xI' , x") a(cT.-r)·

(24)

.

C11(,l!iting l<Jq. (10) against 1<Jq. (22), we han' vP~.(.r(s),

.

in the· ('on fining

yaCUlIDI.

Liiseher hns shown tllat lO •

. dx'·

J) = tls

o."i"

-(l,-_.-

i5 J"( S )

=

dx"

--all,."
(~5)

291 268

SCIENTIA SINICA

Vol. XXV

(Series A)

= (dz. )2 (~)Z Bxl'(S) ,ds and

oZ9' -,---:-:ox,.(s) Bxl'(s)

=

. lim

.~O

(26)

29' \'+' oxis')oxl'(s) ds , = 0 . 8

•

(27)

r-.

From Eqs. (22), (26) and (27), we get the following equat.ion

olAe CJ oxis)ox"(S)

=

III

(llx")lA[ c].

(28)

\ ds

At first glance, Eq. (28) seems to be only a tOpy of Eq. (20) with a

=9

vi,

where

f stands for the vacuum expectation value of the field strength squared in the absence of external sources, and only depends on the fluctuation of the vacuum itself (as the vacuum expectation value of the field strength is zero in tlle absence of external sources). Also f may be relevant to instantOlls for the self·dual gauge fields. Actmdly this is not so. Eq. (25) bas already indicated that the field strength F}t" (x(s), j) should not vanish, otherwise there would be no solution in a confining vacuum. And this is possible only in the case where t]le external sources are not neglected. III other ,vords, in order to make the Wilson loop average A[O] satisfy Eq. (28), a is inevitably dependent on the field strength induced by the external source j,.(O) in the confining vacuum. And it is not right to consider only the fluctuation of the vacuum and to set P "" to zero. This indicates again that the assumption (1) may not be applicable. Let us noVl' cousider the case of no confinement. Wilson confineml'nt criterion implies that

For a large closed curve C, the

A[C] = e-6L ,

(29)

where L is the perimeter of the curve C, and (3 is a constant. (:!9) with respect to the curve z~(s) one has BA{C1

= _p ~

8X"(S)

8:&1'(5)

A[C] = _p

By differentiating Ell·

dXI'/d.s A[C]. [(dx"/dsYP'l

(30)

Therefore, without confinement we should have F ( () .) dx. !1 ,.. x s ,1 -;t;

{J =

-

dx~/d..~ [(dx"jdsY] 1/2

(:31)

Let us differentiate Eq. (27) again, we come to all equation satisfied by the Wilsoll loop average in the case of no confineml'nt

02.;1[OJ =p2[(dx")Z]11l.A[CJ. oX"(s)OX,.(s) ., as

(32)

It is very interesting to note that FJq. (:32) will be the same as Eq. (28) ill form if s is taken as the length of the closed curve C (i.e. (dx"/dsY ~ 1).

From the above, we conclude that the ealculation of the field strength F,.. (x(s), j) in the presence of the external source j (C) induced properly by the Wilson loop

292 No.3

W:J:LSON LOOP INTEGRAL & STRING WAVE FUNCTIONAL

integral itself may be useful in the discussion of the confinement problem. we have

-fJ

. da~

gF"ix(s), J)

d

269

In fact,

,without confinement

a,x,,!ds [(da v /ds)2]1/2

{J!7

=

s

\

with confinement {Jx" s a is reI event to the tension of the theory.

-a - ( )'

for a large closed curve C. The above discussions can be extended to the case of non-Abelian gauge fields. In that case let us introduce a set of auxiliary parallel field
dcP(s)

=

gA.(s)q,(s),

(33)

ds

d
[

(34)

.1(s) = A~(x(s» ax'" Ii.

(35)

ds

In Eq. (35) l;{-i == 1, ... 'FIe) are the generators of the group G. fields "p(s) is chosen to be 10) which satisfies the condition

cP(s) 10)

=

The vacuum for the

o.

(36)

Eq. (33) can be solved to give .t. ( S ) 11'"

=

- (') lV",.[s, s'J IP,. s,

-+( S ) IP"

=

-+( s') lV,.,,[s, ,s), IPy

(37)

where W"T[S,S']

=

{pexp { 9 tJ.(s")ds"}LT .

(38)

The Wilson loop can be identified as W[ CJ

=

(39)

WaAsl> so].

Consider now Green's function

(P(cPis)
=

(011'( 41,,(s)rJJt(s') /0)

=

e(s - s')W",[s, s'].

(40)

Hence the Wilson integral is p(Jual to Green's function (41)

It is easily verified that the equations of mot.ion (33) are derived from the Lagrangian

L~

=

c;p+ dIP - IP+ ..4.(8)11>(8), ds

293 :170

scn~STl

A SISICA

Vol. XXV

(Series.\)

with the parameter s acting as the proper time. Write it in a path integral form, Green's fUllction (41) becomE's W[C] = ) [dcp][dtJ)][dc]J+] exp {- ) se,lx)(rx -

t

Lq,(S)dS} (42)

. rJ>,,(s,)cp;;(so).

This form is analugnus to that given by GCf'\'ais and Nevcu,·I, 'I'hey take the

lY[C]

=

(43)

wherl~ Z[ it] is the generating functional for the giLuge fil'ld.A~ external sourl'e j~ (;(, C) induced by t.he Wilson intf'gral, J"'i( ,. .x, C) = g

~ (j~(4)( .1;

_

C)dx/'

.t.~

(ls

1< is, ( )

(U)

where In Eq. (44), the fields r.P(s) are written in the interaction pil?,ture and satisfy tho: fl'l~" field I'quation dIP(s)/ds = O.

The first variation of lV (CJ with respect to the curve C can be easily calculated and equals

_ (v~p.)} dx" ,

ds

=

9 (p(1h.CxCs»i;Cs)tJ)..(s,) &;;(So»\ dx" .. I (ls

C4U)

where F~. (x(~») is the non·Abelian gauge field strength. The nonlinear term jn F~. (x(s» ILrises because of the parallel nature ofthe fields 1..(s) = tjJ+(s)i,.tjJ(s) under displacement along the curve C. Eq. (46) muy also be expressed as o1\'[C] = r.]i' .7 ". OX/' ( S )

ex, C)lf{C] dx" d·s '

(4/ )

where

'rhe nonvanishillg of F' /'" (x, C) is intirnal.rly ronneeiet! 1.0 the prrsenep of thl.' external sources inducp.d by the Wilson integral just as ill the Abelian case. A second variation oflF(e] giye~ an ('quatioll similar to that of (28) and it j,.: also illegitimate to pya luate thr coefficient in front of W rCJ on the right side of that equation by nl'glecting' the exte-rnal sourel';; induced by Wilson intl'gml.

294 No.3

WILSON LOOP INTEGRAL &: STRING WAVE FUNCTIONAL

2'11

Aft(>r the completion of this work, we arc informed by Professor G. Parisi that Makeenlm has also stressed the importance of the external souree term induced by the Wilson integral in deriving the string-like equation from a Yang-Mills theoryUOI.

We would like to thank Prof. of Makel'nko.

G. Parisi for sending us a pl'eprint of the work

R·E}o'ERE~ eES

[1]

[ 2] [ 3]

l4]

e5 J [6] [7] [8] e 9] (101

Gervais,.1. L. & N~\"eu. A., Phys. Lett., 80B (t9?9) , 255; Nambu. Y., ibid., 80B(1979), 372: Gliozzi, F., Reggl), T. &; Yirasaro. M. A., ibid., 81B(1979), 178: Polyako,', A. M., ibid., 82B(1979), 247. Corrigan, E. &. Hasslaeher, B., ibid., 81B(1979), 18I. Durand, L. &; }trendal, E., ibid., 85B(1979), 211. Eguchi, T., ibid., 87B(1979), 91: Foerster, D., ibid., 87B (1971), 1l7; Weingarten, D., ibid., 8iB (979), OJ; Makeenko, Yu. M. & Migdal, A. A., ibid., 88B(19;9), 135. Wilson, K. G., Plips. ReL'., DlO(l974), 2445. Marshall, C. &; Ramoml, P., Nile!. Pllys., B85i1975), 375. MandelstllDl, S., .J'1In. PhY8., 19(1962), 1. Liiseher, M., PhYIl. Lett., 9OB(1980). 277. Gervais, .T. L. & ~e.cu, A., Nuel. PIIYS., BI63(1980), 189. Makeenko, Yu. M .. Institute of Theoretical an.l Exp~l"imental HTO-141,

Plly~ies

(Moscow),

Pl'eprillt.

295 Vol.

xxv

No.7

SCI E N T I A

SIN I C A (Series A)

July 1982

BACKLUND TRANSFORMATION, LOCAL AND NONLOCAL C6)NSERVA TION LAWS FOR NONLINEAR a-MODELS ON SYMMETRIC COSET SPACES ZHOU GUANGZHAO (CHOU KU .... NG-CHAO

Jilil:l'tB)

(I1I"titllte ot TT,eoreticlIl Physics, .:lcru/emitl Sillica, Brijillg) ... ND

SONG XINGCH.\NG

(* IT ~)

(Peki·ng r; nirersit!l)

Reecin'll .J:munry 9, 1981; revised .Jum' ":!.7, 1981.

~\BSTRACT Tl\""o-diml'll~ionnl nOlllim
It is "Well knowll that all two-dimellsional nonlillear u-models on symmetric coset spa(.'es have an infinite number of nonlocal conS€'rYation lawsl1-<1. In contrast. mudl less is known about the local ones. The existence of an infinite number of local COIlS€'rYation laws has been established for O(S) a-models byPohlmeyerl!'J, O(N) principal chiral fields by Cherednik[61, and CP n models by Eihenherr l71 and Scheler[SJ. Recently Ogielski, Prasnd. Sinha and Chau Wang have devt'lopt'd a parametric Blicklund transformation thpreby the local conservation laws can be obtained in prillciple f91 .

In the present paper the duality symmetry[7.8.10J is used to construct a generalized Backlund transformation for nonlinl'ar a-models defined on sYlllmetric COSl't spaces. Thl' transformation contains a continuous parameter 7. Conserved current dl'pt'liding on 7 is constructed which yields an infinite number of nonlocal conserved currents when expanded in powers of 7 around 7=1. A slight modification of the 7-dependent current expanded in powers of 7 around 7 = 0 or 7-1 =- 0 gives two S€'ts of infinite number of local conserved currents. The main adYantage of the preS€'nt formulation is its simplicity and generality. We shall give a brief account in the preS€'nt paper and details will be published elsewhere. Consider a compact Lie group G with subgroup H. Any element go E G can be decomposed into product of a left coset element cPo E G/H and an element of the

296 No.7

COXSERVATION LAWS FOR XONLIXEAH

f]·~roDELS

717

subgroup, ho E H. i.p..

(1) L'ndl'r thl' left :tetioll of

II

:,l'roup elelllent

{j.

th,' eost-'t element 'Po beeolllt-'s

{J.po = ,peg, 'Po)h(g, 'Po). } ,peg. 'Po) E (;;H. h(g. 'Po) E H.

(~)

Till' transformation 'i'1-
0(11).

The basie dynamical fiE:'lds of It nonlinE:'ar cr-model art' the loeal eoordinate functions niC.r), j = 1.~." '11(; - l1H of thE:' CORet manifold (;;H: ,/>(x) = ,/>(n:(.1:». Let Q"

=

-i.p-l(x)a"cP(.r.)

=

Hu

+ K".

(;3)

wherE:' H" and k" are valued in thl' Lip. alg-I'bra of the subgroup H aml its I'olllfjll'ment.. the ~ubSt,t of thl' Lie a\l,rl'bra eorrl'spondillll' to tht' cos!'t G/ lI. rt'specti\'ely. Wht-'Jl tht-' eosPT manifold (; . II is a symllll'tri,' spa,·p. H." and Ku ;;atisfy till' f()llowin~ integrability conditions.

aji,. auk.

a.H" + i[H".H ,.] = -i[k".i{.]. + i[HII.k.J = B"k" + i[H".i{"J. r

.\ La!!rau!!ian invi\riant. und\'r tl\l' :.dohal transformation form.

O[

(~)

till' )!rllup

(~

ha:> the

(;) )

l'llIll'r a 1')I'ai 11'i1l1sf"l'lIIatillll of till' !!I'OIlP (;. it is "asil~' \'t'rifi,'o! that t ransfllrlll as [,,11.)\\":;:

HII -- h( y. ,i,)( - ia"

k" -

+

Ii"

fill )/,,-I( u •.,,) - i( "',~-!(!rla"y )'i'/r.-').;,

h( y. 'i' )K,,/I-'( y . •/.) -

i( h.i,-'e a-10,,!! )",/I-I)!:.

and

K" (n)

(7 )

h: 1It'IlOtl' prCljl'dion ontll till·' Lil' alg-I'hra of th,> suh!!roup H anti its cCllllpit'mlm t l'orrl'sptlnding- to thl' l!Cl~wt (; .... 11 l'\'SllI,,·th,.'I,\'. For infinitesimal transformation

a=

1111\',>

WI'

1

+

(8)

ilia./.c).

(!J)

'I'll\' l!OnSerVI'u "lllIation

Cilll

alslI be \\"ritt('11 as

a"K,.(x) From Eqs. (:3) and (~)

OIU'

+ i[H",kuJ = o.

(10)

easily obtains

a"l. - a.l" III the ligh t COlle coorciinates"; 2 ;

=

2i[}",J.J = n. (t

+

x) and"; 2 7J

(11) =

(t - x) the conserYl"d

equation (10) and the integrability conditions (4) can be combinoo to give

297 718

Sl.'IEXTIA SIXI('.-\,

(Serie~

A)

a~K~ = i[K~.H~], ar,K~ = i[K~,H~], } a~fI~ - a,.fl~ + i[H~.H~] = -i[K~.K,;].

(12)

In terms of the curr~llt ;" we haYe

a~J,;+a~;;=o,

}

aJr, - a~;; - 2i[J~.J~] = 0. Eqs. (12) anti

(1:~)

will bt' uur starting equations.

It. can bt' easily verifieu that. Eq. (12) is inyariant. under tht:' .iual transformation,

K~

-+

K~(r) = rl{~. K~

fI~-+H~(r)=H;. where

-+

Klr) = r-IIl ... 1

fI,.-+H .. (r)=H~,

(14)

r

r is any parmueter uifferent from zeru.

TIll' dual SYlIlIIlI'try illlpiit's

illllllt'diat~ly

li/r)

=

that

fI/r)

+

(l.j)

Kir)

is a pure gaug't' ami llIay be set t'tIuul to - iy-I( cpo r )a,..y( .p, r) W}lt'!"· Y(.I,· r han lJt' decomposed into product, (lU)

Q~ = - i./,'-I( .p. r )8".:,'( ./ •. -r) = H~ + K~ = h( c/,. r)( -';8,. + fI / r ) )h- I ( '/', r) + h( ./,. r )K,.( r )h- I ( ,I)· r ). Eq. (17) tells transformation that HI' = H,.. fOl'e if "'i is a of Eqs. (12). A,

A

(

(17 )

us that H~ and K~ e
r»

,.,.

(

)

1\. conserved current depending on ",'( "" r) can thus be constructed and has t.he form,

J,,~~')

a"j"(,,,') Of course,

cp'K~.p'-'

=

=

J"(,,,') == ; /~, r)

=

y(.p, r)Kir)y-I«p, r). }

(18 )

O. satisfies the same set of Eq. (18) as

J"( "')

dues.

Rewriting Eq. (15) in the form,

-iy-I( ... , r)a"Y("', r) == -i

+ K/r) - it",

( l!l)

and integrating it, we get

(20) where

P is

an anti-ordering operator along the path and

J/.,) -= CPK,,(" )cp-l.

(21)

298 I'OXF;ERVATIOX L\ WS FOR XOXLTXEAR

(]·~IODELS

"119

:-;ubstituting Elj. (20) into Eq. (IS) we ()btain a eonSl>rveu (;urreut depenwng on u c:ontinuous parumetpT

7.

Elj. (22) expanded in powers of

7 arounu r = 1 yields an infinitE' number of I"':al consf'rvpd currpnts which eoincidl's with that obtained ill R ... fs. [l--!].

11011-

In onlE'T to get the local cons('rved (!Urrl'nts from Elj. (22) we shall study the strUl!tur'> uf the current conservation equation first. Let us 'define j~(c)

=

(~a)

= tr{K/x)B(x)},

tr{},.(x)l(x)}

wilt:'rt' ..leI:) is a function valued in the Lie al~ebra of G, while = uI:count !.If the equation of motiull (!l) Ill' (10). line cun t'asily obtain

B

,/>-I.1.

Takin~

.i~ will hi' 1CI}11""rwtl if au_I(./') is tril"'> IITrh"!!"llid Til Ju(.r) or equivalently a",fj(.J:)+ i[iI"'(..c). n(J)] is trace orthogonal til i'i,,(.r). Tht're are llIany pm;siblt' ,·h,)il·t>s lit .1( f) or D(..c) in which the conservatiun t!onwtion of j~ will b... satisfied. Fill' install
a",l}(.c)

=

iUJ(f).H,,(.r)] + (/~u..fl'·(J:) - b"p.~Ii(.J:)K'·(.e)B(.J:) + i!.'1 [Ii(.e). k,.(.r)] + l',t p..{ t:(.t), k,'C.J;)}.

(25)

Ii is "asily \,erifit'll ThaT for ill·hitrar.\- ClIlI:stallts II. b, 1.'1 auu !.',. tht' t:ousl'rvatioll ':lIl1llition- is satisfit'u. aml th,' iUT,>grauiliT.'- ,~t)JldiTiou for 1i(.J:) illlpost's a l!ullst.raint alHoUg thl's!' paramell'l's:

fib

( 1)

('I

=

1.

It

=

b

= (',

D.

=

+ c; +

C3

(:!(i)

1.

=

Then E,\. (:!.j) turns out to

btl

a/lex) = i[k Hd + iUi, K;1, } A

....

+

"

a.,B(J:) = i[J.J, II,,]

....

A

(27)

i[ n. Ii,).

This llleans

J(.c)

=

10 =

eOJlst,

anu then the fullowing normalization cunditiull mllst be satisfied:

trBZ(x) = trA2(.r) = CUllst.

(28)

Duality symmetry theJl implies that

j~(r)

= tr{Kir)B(7)}

(2~J )

is also a conserved current if B( 7) satisfies the following equations!

a~B(r)= i[B(r),H~]

+ i7[B(7),K~J,

(30a)

a/J(7) = i[B(r),H~]

+ i7-I[B(7),i~].

(30b)

299 720

SI~ICA (Scrie~

SCIENTIA

Now expanding B( y) ill powers of

A)

Vol. XXV

y-I

.

L: B. y-'.

B( y) =

(:31)

"=0

and substituting it into Eq. (aDa), one gets

[B.,Kd

-[B.-I.Hd -

=

ia}J.- I.

Evidently the first cot'fficient

Bo = satisfieos both Eq. (:32) with

II

(trKi{~)-tji:~

= 0 and the normalization condition

(2:;).

Explicit expressions for B.(n ~ 1) can be obtained from Eq. (:~::!) for C['l lIlodel or 0(3) chiral model. Fl)r thesp. models Eq. (2!l) f,riyt's un infinir,· num!J,.>r ilr local conserved currents When B( y) is expanded in powers of y-I as in E.!. (:31). aej.~

+

o.

a~j.~ =

j.~ = tr{KeB.L j.~

=

}

(:{..J: )

tr{K~B.-,},

with the first one equal to .

Jo~

;:.. )'" = (K'\. tr ~I1.~

.

Jo~

= I) .

Another set of conserved currents can be obtained if of y.

expanol~

OIl!'

T:r Y) irl

In general. Eq. (:32) does not lWI~I'ssaril:- ,\"ield lneal solmion for to the constraint equations,

8.,

Ilwiw.!

(:36) held for arbitrar.r

illtl~gpr III.

When the duality sYlllllletry with Y = -1 is applied to thi,; cast' Olll' will obtain t.he cllse ill which CI = -I,ll = b = C2 = O. (2)

CI =

C2

= 0,

·0

b = 1. Eq. (25) takes thl' form ...

a;B •

a~B

~

A

+

AI'" A

...

"

•

aK~

=

i[B, H;J

=

i[B, H~] - aK~

••

- a- BK.B, }

+

a-

I'"

BK~B.

(:17)

The llorlllalizat·ioll I;olldition (28) is compatibh' with Eq. (:37) if tht' addirional condition

(:38) is held simultalll-'ously.

For CPR models this conditiun coincides with Eq. (::!';).

The local cO!U;I'n"ed I;urreuts can be generated from Eq. (::!!I) with B( Y) satisfyimr the following equations:

a;B(y)

=

a~B( y) =

Expanding

+ i[B( 1'), H~] -

-i[B(y),Hd

Be y) in powers of

y-l

ayKe - a-lyB(y)KeB(y), ay-li(~

+

a-ly-1B( y )K~B( y ).

we get from Eq. (39a) that

(3!Ja) (a!Jb)

300 No. i

17·~rODELS

COXSERVATION LAWR FOR XOXLIXE.\R

7:!1

(40) (41}

It is t>vidl'nt that

Eo =

a is

it

solution of Eq. (4,0).

But this trivial solution iii Illlt,

uSl'ful sinct> it gives jol' zero identically.

n

For CPo models and their generalization, the C( 11 + m)!T( n) X III) mudds, or their orthogonal analogues, the Lie algebra valued function on coset spa\!t' ,an ~ tIt'composed in to two parts:

it

it =

Ki+) + k·-),

sueh that

j{.-> = (K'T»t. fCT>K'T)

=

!,' .j,:l)

k->K-·-j = O.

or

For The

r B'T>( r )lQ-lE(+>( r).

(-1:4)

with both l('+!k-) and j«-)K<+> bt>lon~inl! to thl' Lil' algebra the subgroup. tIlt'sl' IllIMl,,!:s. E'I. (:l~l) split into TWO bran.:hl~", a Pl)siTiw Oil€' and anegati\,t' ont'o positi n' bral11'h of E". (:l!la) rl'ud"

a/J'Tl( r) = i[EiTl( r). H<]

+

ar f'Q+! -

11-

l

From this ,'qllation and the normalization "ontiitioll lllt'llTiollt>d ahoy... the rl',;;uir;; of H,,£, [.'] ror (' J'. Illot!t'ls ean lx' rl'prlllitu'l'd,

Xo\\' t'xpantiill!! H( r) in powt'rs of r- I WI' gl't frlml £q. (4.j,) th;n

1/-1

. k.+)i'i'-.-'jJt+'· =

~

""--'

J

;'

"-1

(+.jJ i[B l.:),.

fl.,] -

a=k.:.~", '

(·W)

j=1

E", (-l;j) ,'all

whi,'h

~'ields

hI' SO\\'('11

to lEi \...

th,' first "onsl't'\'l,d t'nrrl'nt

jo~ = atr{ (j{~ T)it~-l)f }, jo~ 1!I at:t'ord with Rl'f. [!I], t:urrent

From Eq, (4ti) with

/I

= (),

= 1.

Wt'

,:UII

dl,dul'l' t.h .. S!'t'Olltl (.J.:J)

The t:ollstant (t is it St:aliug parameter which cau be put equal til 1. Highl'r order IOl!al currents call be obtained from Ell, (4:H) (or (-l:l) more gen€'rally) by sl)lving linl'ar matrix equatiolls. 'l'he solutions of theS!' equations for l!Ol1crptt> models will be di~cuSSl'd l'lsewhl'rt>, REFERID;CES

[ 1]

Golo, V. r. &; Pt.>rt'IOIllOV, A. M., Letts. in lCath. PIIYS., 2 (1978), 4i7; PolY-.llcov, A. ~r .• in "Collective Effeet.q in COlll1en~etl Media", Proc. of 14t:, Winter School of Theoretical Physi.cal i'n Karpucz, Wroc·

lat', 1978.

[2]

Zaharov, V. E. & )Iikhnilov, A. Y., (SOli, Phys.) JETP, 47 (1978), 1017.

301 722

SCIENTIA SINICA. (Series A)

&; Pohhnl~·,·r. K .• X/(cl. PhYR .• 8137 (HI'i8). 41i. Eicheuhc·rr. R. oS: Po/'g,'I'. :\1.. ibid .• 81SS (1979). ,:81; BI'l'zill. E .• Itz)·kson. C. Zinn·.Juslill •. r. & Zuht'r, J. B .• Phy~. LPN., 828 (1979), 44:?; Ogil'l~ki. A. T .• Phy.•. Be,. .• D21 (If1S0). 40(1. Pohlm('yt'T. K. COmllll/lI. J/ath. P1I!ls .• 46 (197(i). 207. L'hereuni.k, 1. Y.. 1·/'for..lft/th. Ph.'l.•.• 38 (1979). 1211, Eichenherr. R .• PIt!ls. Lftt .• 908 (1980). l:!l. Scheler, K .• fbi,l., 938 (1!lSO). 3~1. Ogi('lski. A. T .. Pl-asnd. ),1. K .. Sinhu. A. &: C1mu '''nllg, L. L .• ibid., 918 (980). 387. Flume. R. &: .\re~"t't, S., -ibid •• 8S8 (1979). 353. Coleman. S .• WCfS ••J. &: Zumino. B .• Phys. Rei' .• 177 (1969). :?:?39; Callun. C. G. .11'. ('1 al.. ibid., 177 (I!l69). :!:?4'i; 8.11 Sill , A. &: Slrathuee, J .• ibid., 184 (1969). 1750.

r. 3] LllsehN.)1.

l -1] [5] [6]

[ i] [8] [ 9] [10] [11]

v.ol. XXV

302 Vol. XXV Xo. 8

SCI EN TI A

SIN I C A (Series A)

Al1gu~t

1982

LOCAL CONSERVA TION LAWS FOR VARIOUS NONLINEAR a-MODELS ZHOU GL'ANGZIlAO (CHOU Kt:.\NG-CHAO

(II/.~titl/tp

"f 7'heoreliClzl Ph!lliics,

fa! it B)

':/'clIl!p,mia Sinka. B.,j>;"!I)

AND SONG XINGCHANG

(;f:f:j"'*)

(Peki'Jlfl Uni!"er.~ity)

Up("iWd .Jannar:· !I, 1.981; re~i"ed Jllnl' :!';, 19t11.

ABSTIUCT

On the b:L·i~ of the ronnllmtion giYen in a prceeding paper, we I1erive an iniiuiti'l"e >;eries or conEerwu iO":11 I'urrpllt~ e:\"'Pli,'itl~- for the two-dimensional elassh,'al a-nIl1,lel~ ,'II th.· ",",'l!pJr.,. Grn~Sl1lnlln manifold C;(m + /I ) / [ ' ( m)r&U(n) :n:r1 for prill.'ipal ~hiral fiel'l.

rOn

III a previous P<1p'>1'\lj, by l1leans of the dllu lit.\· ;;:\")nm"~ r.\-. \\",. "011;;1 l"1l1"l.'tl it BiiddtllHl trulI"formatioll dl'pPllding Oil a l:olltiuuOllS parumd"r r. illld ,!!:t\-P. It rlll"lllulalion til Li'''Ilu,!e both tIl,· local and nonloeai conservation laws fnr tWII-dilllensiollalllunlin.·ar C1-lJIodds on sYIllllletril: I'oset ~pacl>s. tiellL'rally th,' La!!r:l1l!!i.all ,'''r rhe nonlinear C1-IlIUt.l1'! has the form

,,, = J.. .' p-'.I'. ""-"\..' .)"

.:.L.

whieh is iuyariallt undl'l' the I!lubal trallsforlllation of a '~OIIl[lilCT l.i.· ~l'IJIIP (; wilil,· 11l\:al-iIlYilriant under tht:: gaul!l· trall:sfllrlllatioll of a closed :subgroup J1 of (;. lIer,· ~. > uenotes the U invariant illll\~r product. For tlw fidd !l(.c) taking values ill U (!l(x) is equiva.lent to cP(x) in [1] lip to a ~!iLul!e trnllsformatiou). we define !j ...

= -i!l-I(.c)a,.!I(.r) = iI" + ft..

(2)

wh~l"L· H" allli K" are valued ill the Lie algebra of t111~ subgruup II awl its orthogonal eompl~lIlellt l:orre:spouding to the coset (;jH respectively. Both flu ami K,.. art:: lrerillitian when y( x) is unitary. Define the covariant deriYiltives as

D"y(x)

a,..y(;;) - ·iy(x)H,..(x).

=

(D,,!l(x) = a"y+(x)

+

-iH,,(.c)y+(.c) for y(x) uuitnr.,-).

then

K" =

-'iU-1(x)D"U(.c)( =iD"y(x)y(.c»).

(4)

SO H,. is tht:: composite gauge field whl'reas X,. is a vector field whil:h transforms covariantly under the gauge transformation. The Lagrangian (1) tan be written a.~

~ = l:..
(5)

303 826

SCIE~TIA

SINTCA

(SP.rit'~

Yo!. XXY

A\

and the field equation can be cast into the form,

D,.D"'{} -

Du!JO-1n"O

=

(6)

fl,

or

WK,..... a4(,. + i[H",K .. l

{i,. giws

And the intl!l!'rability condition for

D,.K. - D.K,.

t

(i)

iOD"g),

(8)

0,

=

a,..H. - a.H,. + '"

(6')

o.

=

A

A

A

,i[H,..H.J

=

-i[K~.K.l, ""

A

J

The conserved current j,. takes the form

J,. =

O(;;)K,.. O-l(X)

=

--i(D,.!J)O-l(

=

and satisfif>S e1luations

l

a"J,.=(),

a..J,. -

0",,]. -

2i[.l,., .l.1

(!l)

0, I

=

In orul'r to get the local COllf-Oerved currpnts a scalar eurrr'Jlc

(10) is defined by introducing' the G valued function 1(;;), and B(.r;) = y-1A.(x)(j, j~ is consern'tl if a".4 is orthO'J'ollal to J,.(J;) or D,.iJ orthogonal tl) K,.c.';J. :;;I.lch iUlll!tion 1(z) or B(.x) exists ILnd two simplest casps were considerr'
In tilt> st>Cond casf',

D"B(r;) = <",.(aK: - a-1BK."B) ,

(1:3)

The solution B(x) may be restricted to the subset of the Lie algebra corrt'spondillg to the coset (fIH and the normalization condition (12) is compatible with (B) if additional condition (l-!) is

h(~ld

simultaneously,

It is evident that Eqs. (6) and (7) are invariant under the duality transforma-

. tion ( in light cone coordlllates ;

=

t+x -=-,

Ke - Ke( r)

=

rite, K~ - K.,( r)

-/2

iIe-iIe(r)=-iI~,

'Ij

=

t - X)' . --=-

-/2

=

r-lK~. }

(15)

iI~-iI.,,(r)=fI~,

Then duality transformation implies that

(16)

304 Xo.8

LOCAL COXSERVATIOX LAW~ FOR Y.\RIOrs XOXLIXL\H a·.,roDELS

·)~

8 -'

is also conserved if B(r) satisfies Eq. (11) or (18) "'ith ill' replacP.d' by k,,(r). 'I'hese equations C<1n br' solved by expanding B( r) in powers of r- 1 •

B(r)

=

I:'" B!,-i.

(17)

1"'0

For the second

CUSI~.

WI-'

obtain

(18a) /-1

Boil~BI

+ B/foBo = -aD;B1_ 1 -

L

Bjk;f~I-).

(ISb)

j=l

'rhen Ell· (16) giws an infinite set of local conserved currents.

it~

=

.il~

(K;.BI>.

=

(il~.BI-2>'

(10)

XOII" we discuss the solutions of Erj. (Jil) and deduce the conserved ,!llrrents (19) fur s€,yeral coner,·te molll·ls.

r

First l:ollsider tll'~ D"-muul'ls 1111 lit" l:ulIlpl,,:,: <';!'aSSllls values on thp ::;ub;:;t't of tIll' al!!ehrll corrp~p01Hlill!! rn th .. "IIS,'t u/If = ("(III + 11)"C(III)/2'["(n).

r"c

".. 1\."

..

= \. "holt

k.-: \ J' .'

(20)

wht>rl' j,·.,,(k;) is an II. X 111(111 X II) matris:. Thl'n k.;l{" tuk",; \"o1lnes llil rhe sllbalg-ebra of II = r( III )-~~){.'( it). (so is thl' (kJ\[JI.l by a pr"p,'r ,l"fillirion of ;:;qllarl' root) i. l;).

(21) with /itk; and ki',·tlll X 1/1. and n X n matrices resp,'Ctin~I.r. When·m"'-': II, aSSUlIlL' ktk; to he invertible (both !.-tk; and k)oJ lIlay he llssulllt,d til bl~ when 111 = n). and define a matrix,

\,;,~

may

ill\"~rtiblp.

which has the propertit>s,

S" (24) is a solution of Eq. (lSa) (with a=l) which also satisfies the normalization condition (12) and the additional condition (14). Since KiKi is a semi-positive definite hermitian matrix it can bi! brought into a diagonal form:

305 828

Vol. XXV

SCTEXTTA SIXICA (Series Ai

(25)

by an T.:(m)0C(II) transformatioll. anll i.• > U(a = 1, 2·· '111) as ktk; is ill\·,·rtihl,· by assumptioll. Some of the eigenvalues fta(a = 1,2· . ·n) must be zero. III this representation we have (sometimes WP. ignore the index ; ill unconrused cases). K.~K6c = .t;0"",

£.8

= i.;;III:.s'

(:!(j)

E"uEao = OJ,':

or in matrix forms kk+ = ,lt~'l f+

=

11- l k+.

=

I!

kA-

I

,

(2G' )

J

e+e """ [,

where I is an

In

X m

unit matrix, 11 = diag

0.1"

·i. m ) and

,It

= ding (UI"

',IL,),

:Now from Eq. CISb) /-1

BoK~B/

L

+ BJI:JJo= -aJ);B/_ 1 -

BJ\~B/-j

:t:_!:

=

(lEO

i=l

multipl~;ng

Ii; from righ t anti Ht respt'Cti vl"ly. "',. g'(·t (EK)(B/1) + (Bd{)(/Uo = (e/-/o· (Kl~)(KB/) + (KB!)(i(E o) = (i'aJ/_ I ) ; .t.(B/K).b + (B;ll)..i-., = (('/-I/l).b' i.• (IiB/L + (Il:iJ/).bi." = (lI:C/_ I)...

Therefore (KG/_I).;.

']') _ (I\.:J/~" -

.t.

+

•

(27 )

)'b

Substiiuting them into Eq. (ISb'), we obtain the solution, (R I ).,

=..;!- [CGI-I).8 I••

(('I-IE)..

_

(BI)ab = [( CI-I)ab - Eaa

1.

.t.

+

1b

+1b"),b E~8]' (EGI_I).b j ~_ 1.

( 283) (28b)

I'b

The local conserved currents can then be obtained from E4- (l!I) anll the first two are C29a)

jlYj"" O.

Solution (27) has another property, that is,

(:!!Jb)

306 ~).

8

LOCAL COXSERVATIOX LA\VS FOR Y.-\RIOT."S XOXLIXE.\P. a·~IOIH':Li;

=

L: f.- [D(I.'+")L.,

-

II

=

8:!!1

(:lO)

I).

_1";1

Be"Y)·

as r"lluired b.\· tho' llol'lllalizatioll condition (12) for

(l~/)

Taking account of th~ conditioTl (:30), Wp can deduce the second consPl··..ed current from (27) aftE'r smne calculation.

=

-

= -

L ~[(C~L- (('OE).i.(E('o)'.,]- L: (" +)".I".) /Co£).o(E('o) •• L ~ [(D~E)~. - (/)./~·E),;U·/).E)"Jl .rh

I' J

lib

I .. ~

,0

!'J

(:31 b)

n

Alt'·rnatiwl.\' th,' {"( /II + /1)/ 111)0 C( /1) :r-lIlolh'\ ~an also bt' o..lefillt'{i as a locaJl.\· 1"(111) invariant theory of an (m + '/1) X n matrix fipld z(x) = Zi"(X) (which is the first III columns of fipld V(x), 'i=l, 2..... m + II: 1. 2. "', m) with zz=l(z is

,,=

till' iWl'mitiall t·l)lljulrat,· of z). or till' cnrrt'spolldiulr projt'Ctor fi,·ld~~.I. (82)

1'h" La!!ranlrian (1)

~all

!>t. writtpll as

!L

=

-

1 i,K "" K"

2'"

=

1-

-

:!

=..l tr8 pa"l' = 2 'u

fr

D

VD'~!J

.11.

fr

DuoD";;. .

H"rp tht' covariant dprivative for .. fipld is defined as

D"z

=

a,,< -

-z:.• .I" ,

D"z

=

a"z + i.-l"z,

(34)

with tht' lJ(lIl) gaugt' field defined as

A"

= -.;za".,

For a matrix X transforming as D"X

zz, =

(A,,) •• == (II,,).;.

(:35)

the covariant derh'ative is defined as

al'x + i[A.".XJ.

The field equation is

[P, a,.8I'p]

== 0,

(34 )

307 830

Vo!. XXV

SCIEXTIA STXICA (Series A)

or

D"D"z

+

z{)"zD"z

=

O.

It can be shown after some algt·braic operatioll, that k:k~ = D;zD~z =

k:Dik~ Df,k~k;

Df,kii,Df,k~

=

(::l7a)

AZ.

= D~zDiD~z, = D~D~'ZD~z.

D~Df,zDiD~z

(:fib)

(:37 e) (37d)

- (Dii,zDf,zY,

and

D;AZ

=

Df,Df,z Diz

+

D;zD;D;z.

After a lengthy deduction, Eqs. (:31) can be rewritten as j~f,

= -t,A-3

[D~Diz

Dii,D;z - A'-

- L: i.;llb'(i.~ +

Df,D~zD;z/1-1D;zD;D;:

J

i.!,)-;(Df,D~zDf.z).JiDf,zD;D;z).;

"b

+

L

l;llb1(l.

+

lb)-l (D~A')b.(D;/l')b.J'

"b

For CP. model. A is a single compout"llt quantity rl"pt"llllill", I..nly I.·n g .Illll e:lll ht' taken equal to 1 by U~illg i! propt"r eOllforlllal transil'rnwtiuu:';. ~I) Uiil~ = U alit! Eqs. (:38) retiUet' tu (3!Ja) jz~ = A-l(Df,zD~z

+ D~zD;z).

(3!Jh)

Tl1t'n the l'('sult eoillcides with results given by Eichellherr:1J and by i-iehelt·r w . Xow we turn to discuss the CO'-1) principal chiral field[;J. G

_-\.11 d~IIlellt of grollp

= UL (.Y)0FlN) is expressed as a = (ar., YR),

yiYL = gtaR

=

1-

with the multiplication rule (-l.l )

Any element of A EH

(J

can be decomposed iuto a product of an dement

I .•

f

tht' subgroup,

= TJL+R(N), A

= (h, k)

with a left coset element 4' E G/ H,

Therefore g

= 4''' means (42)

308 LOCAL COXSERYATIOX L.-\WS FOR \'ARIOn; XOXLTXE.-\R (J·:.\lODELS

Xo.8

8m

Following Eq. (2), we define

·Zt" + ..x- ==

-ir/l-Iar/l = -!:(.p-Ia.p, .pO,p-I) = (H

+

K.

f{ -

Here

~ = (Il, Il).

..x- =

(Il, -K),

II = - ~ ( cjJ -lb.p + cpacp -I), l{ = - i.- Ccp -I Ocp - ,pa,p -I). ~

XI)W

K).

(43)

I

(48' )

~

the Cart an inller product of t\\'o f'1E:'mf'nts can be written as

=

T,gifh = t,giI.Y21..

+

t,giRY2R.

Then the Lagrangian of a principal model takes the form =:!I l' r.'Y(' ,,,'X~. = 21 T r f2J #1 r/lf'Z"r/l '

!;[

2 ,= = 1- fa .J,.za"·1.."u'l' 'I" ~'

l... t o~

4,'. "

=

tr rt," pl'

O~I'-I (1 •

== a"r/l - -ir/l,zt" = ca"c/> - i,pllu·oucp-I - i'p-IH.). EC"IP == o!.r/l+ + i,ZC r/l+ = (b •.p+ + iII ",p+ .o",p + iH.,p) . .i

!Z"r/l

(46)

and

lS

also

Ull inl 1'.\'.

(.~7 )

,p2

(1 =

Th,· "llllat iOIl of 1I1lItiuIl horn La!!l'all!!ian (-1::))

i~

(48) and the

illt'~rability

cOllllitioll implit's

C·HI) So the uuality SYllllw·try (15) stallus. Accurdillg to Ell. (8) the conserved eurrelll'

,~ l:UIl

bl' written as

,Y = cp,X'r/l-1 = C,p1lcP-I, _'p-l/{,p) = (.h,

(50)

,jF,),

anu

(51u)

~j R =

.i. c/> -I ( cP -lac/> - cpacp -I) cjJ = i.. cp -zacpz = ~ 11-1011. q q

-

~

-

By introducilll! (; valued fUllction ~r:/(.&) or t'y,uivulelltly 8I)(x) quantity J~

= T,J "od = T,X ,,81) = t,l1.,.(BL

will be cunserved if 5'lJ" fJI) is orthogonal to

{l/),.9iJ

-= ae ".X»

..x-".

(51b)

-

BR )

When we take

- a-Ie ".fJI) X»9IJ ,

=

cp-I,..d(.&)q;"

the

(52)

309 SCIENT!.<\' SINICA

(Scric~

Yol. XXV

A)

the solution may only take values on the subset of the Lie algebra (:orresponding' to the coset, i. e.

(53) with B satisfyin;! E1t. (l:l). Then Eq. (18) holds as hf'fore, alld au infinitivl' set of local conserved currents is a)!ain givpn b~' Eq. (1!1). In prt>st>llt ease we can make thl:! positive definit,· hermitian matrix T\~T\~ diagonal b.\- all r(X) transformatiou[61,

K,K, -

('i '; .. J. .

(5.)

.1.:\.'

aud assllme it to IJt. in\"l'rtibh~ so that .1.; > 0 (i = 1. :! ... - . .\"). solution of Eq. (lila) for Bo (tht> paraml'tl'r (/ is pnt into 1), Bo

(I{~K~)-!/W; =

=

Th"'l1

han' thl'

WI'

f{~(I{;K: )-U

(;j;j )

with tilt' prlJpt>rries,

Bo/{;

=

I(~BQ = (/{;K~ )1.'2.

m

=

(5H)

1,

and the rE'l:urrence forlUulas similar to Eqs. (27) and OS) are r.") . _ (C t ,IOi; (B~!l\. iJ ~,

.1.;

+

(KB)' _ (I(C 1.)/ ;} - .

I_j

I.;

+

(iij)

T ),;

.'

I.j

(.")8 )

The zeroth consern,d current follows from Eq.

0,"».

jo~ = trK~Bo = fr(K;K~)Ti',

(5!) )

jo~ = 0,

and thp first currl'ut can bf' dffiuced from Eq. (5;).

(60) .il~

= O.

These results coincide with those of Ref. [6]. model, \~P can also obtain an equation

Similar to Eq. (:3fJ ) for (; raS!;lllann

(Ii I)

as requirt.'
=

t,K~B~

(5;) with l

=- 1/2tr C t B o =

-1/2tr(D~Bt

+

B1K~Bt)Bo

=

(l8b') with l

2

=

2

310 ~o.

8.

~~~-

LOCAL ('OXSER\'ATIOX L.\WS FOR VARIOT:S XOXLIXEAR c,·,roDELS

= -1/2trDlBIBo) =

-1/2a~tlB1Bo)

+

1/2trBID~Bo -

- 1/2trBl(BoK~Bl

+

BIK~Bo)

=

-t.(BiIl~Bo

=

-1/2(i.;+.1.j)-l[(Co);lCo)ii+l!XCoBo)ilCoBo)i;] (5:3) and OJ) -1/2(.1. i + A.j)-!

+

1!2Blh'~B1Bo)

(1Sb') with l

= =

X [(D;Bo)u(D~Bo)i;

= -1!4(A.i jl~

i/2trBIIl;B1Bl

- 1!'2trBIK~B1B~

==

+

+

1!2(D~BoBo)iiCD;noBo)j,J

1

(61) and (56)

(18b') with l

=

1

A.i)-I(D;Bo);j(D~Bo)ii.

tr[(~Bo

= A.jl(T\<),lh';)j,.

(I):.!)

REFERE~CES

[1 j

Ch)11. K. C. &: Stmg, X. 1.' .• S,:i'!lIti,I Siniell (Seri~,~ A), 25 f[~lS:!), 711i; Iarthl'r rei~rellt~l!~ :JI~ li~tL',l tht:'rp,

~ ~;

Ei~hellh~rr, H. ,\; Forger, .\1., :Suel. Phy.•. , B155 (1D;9), ,:81-

I ['" I [.~ 1 ~ ,j!

Pohlme~'l!r, K., Commun. Jl
i :;

311 Volume 109B, number 6

PHYSICS LETfERS

I I March 1982

COMPOSITE GAUGE BOSONS IN A NONABELIAN THEORY -Cr Swce-Ping CHIA

I

and Charles B. CHIU

Center for Particle Theory, Department of Physics, University of Texas, Austin, TX, 78712, USA

and Kuang-Chao CHOU

2

Institute of Theoretical PhysicS, Academill Sinica, Beijing, People's Republic of China Received 22 December 1981

We contend that the recent Weinberg-Witten no-go theorem does not necessarily rule out the possibility of gauge bosons being composite objects. A dynamical model for composite SU(3) gauge bosons is presented.

Recently Weinberg and Witten [1] gave a surpris· ingly simple theorem. Among other things they show that, in a theory in which there is a conserved Lorentz covariant vector current, there cannot exist massless vector particles, whether elementary or composite. Their arguments are based on the Lorentz covariance of the current 12J. This no·go theorem has cast some doubts on the line of approach of many authors [3], wherein one considers the possibility of non-abelian gauge bosons as composite objects. We contend, however, that this theorem does not necessarily rule out the possibility that gauge bosons can be bound states of some elementary fermion fields. In fact, we find that if the massless bound state is neutral with respect to the charged Noether current defined in terms of the fermion fields, then the no-go theorem is bypassed, and this bound state can exist. In this paper, we illustrate this point with an explicit example. Consider a theory with four-fermion vector inter. action. At the fundamental level there is, for definite. ness say, a global SU(3)-color symmetry. The lagrangian dcnsity is given by .. This work is supported in part by the US Department of Encrgy. I On leave from Department of PhYsics, University of Malaya, Kuala Lumpur, Malaysia. 2 Present address: Theory Division, CERN, Geneva 23, Swit7.crland.

E= ~(i~ - m)..p - ~G(hl'~xa..p)(iii'Y1'4xa..p).

(1)

The corresponding SU(3) Noether current is i~

=-g~'Y1' ~Xa..p ,

(2)

where g = a..jG is a dimensionless coupling constant and "a" some energy scale. The quantity is a conserved four-vector current. For this theory, it would appear that, by applying the argument of ref. [1), it is not possible to obtain a massless octet-color·gluon as a bound state of the fermion-antifermion system. To arrive at this conclusion, one first observes that it is not possible to construct a nonvanishing covariant matrix element of the Noether current of eq. (2) be· tween two massless vector bound states. In the spirit of ref. [I], one would then assert that the massless vector bound state should not exist. However, we find that the following situation may also be admissible. As we shall explicitly show, the four·fermion theory of eq. (I) without taking any specific limit could be formally eq uivalent to a theory with a massive vector boson mediating the current of eq. (2). Upon taking the strong-coupling limit, the massive vector boson is converted into a massless vector boson and a massless scalar boson. In this limit, the massless scalar boson, which con· tributes to the breaking oflocal gauge invariance, de· couples completely from the rest of the system. In

i!

457

312 Volume 1098, number 6

turn the remaining system acquires a local gauge invariance and the massless vecior boson can be identified as the gauge boson. We will further show that the matrix clement of j~ of eq. (2) between two massless bound states is zero, showing that the hound states are neutral with respect to the current. This illustrates our contention that the neutrality of the gauge boson to the original current could be used as a dynamical mechanism to bypass the no-go theorem, and to allow the gauge boson to be a composite object. We now proceed to present the details of our arguments. We first write the generating functional for eq. (l) as: IV= JJ)[wla[~lD[AQJexp(ifd4X

X

[~(i~ -

(3)

m

I I March 1982

PHYSICS LETTERS

·-g~)w +~a2A~AQ" + ~7) + i'iWl).

where 7) and i'i are four-component Grassmann color sources. Here the auxiliary octet field A~ is introduced [4). And we have used the notation All =~A~}..a. Integrating over the fermion fields in eq. (3), one gets:

factor finite, where II is the dimension. For our dem· onstration below we could have also used the Pauli·Villars regularization procedure, keeping the mass of the 'ghost fermion tlnite. The remaining finite terms are collected in R(gA "v), Owing to the gauge invariant cutoff procedure adopted, the remainder is a function ofgA"•. Eqs. (4) and (5) differ from those of earlier work [4J ~ 1 in that, with our gauge invariant regularization prescription, the quadratic term of the vector field is absent in Eg of eq. (5). In earlier work, one assumes the cancellation of the quadratic terms, which could be motivated *1 by regarding A" to be a Nambu-Goldstone boson associated with the breakdown of Lorentz symmetry. Since this cancellation procedure involves a nongauge invariant cutoff prescription, we do not adhere to this point of view here. Without taking any specific limit, we assume that eqs. (4) and (5) together imply the presence of a massive bound state *2. More specifically we denote the renormalized quantities by:

(8) where

(9)

X lEg + i'i(i~ - m - g~ )-1 11 + ~a2 A!AQllj ),

(4)

where

(5) with

(6) and 10 = -

-~ Jd 4p (p2

4112

I

=2(4-n)+ ....

wp

_ nz2 + iEr l

(7)

In eq. (5), we have explicitly written down only the divergent term. In order to maintain manifest gauge in variance, we have used dimensional regularization with the.cutoff prescription of keeping the (4 .. n)

458

Here.2 3 takes into account renormalization with reo spect to the effects of fermions only. The theory now corresponds to an effective theory with a vector meson of rna ssM = aZY2. We shall now take the strong-coupling Iimit,g'" 00, and in turn Z3 -+ O. The theory now passes from the massive case to the massless case. At this point it is useful to perform some formal manipulation to dis· play explicitly what happens to the degrees of free· dom as the theory passes from the massive case to the massless case. Let us go back to eq. (3). We first introduce a set of new fields: B" = ~B~ AQ, Q> = ~Q>aAa. and and a new source 7)p. by means of the SU(3) local transformation U =exp( -igl/J/a). In particular,

A,.. = U\B" + /.Q(I/J)o"q,oj

11 12

ut,

oJ; '" Ut#pand 7) = UTiI' (10)

The original argument for the abelian case can be found for example in ref. (51. For the abolian case, sec ref. 161.

313 Volullle 10913, number 6

£-4 _~B~Jja/lv +R(gRB/lv) +iip(i~ -m -gRQ)-I Tlp

where (11)

Substituting Ihese into eg. (3), the lagrangian becomes

£= ~p(i~-

II March 1982

I'HYSICS LETTERS

//I .'

gQ) l/I p + ~pTlp + iipl/l p

+a 2tr[(B/l + LarV/l)(B/l + Laa/l rfl)] .

(\2)

The path integral measure is also changecYaccordingly:

CZ>= D[l/I ]D[iiI]D[A a]

=D[l/Ip]D[~p]D[Ba]D[¢a] 5W).

(13)

The 5(rpa) constraint in eq. (13) can be converted into a gauge condition on B-fields, for definiteness say the axial gauge condition: n/lB/l =O. where nil is a space· like four·vector. After some formal minipulation, the measure (13) can be written as

where ~(rpa) is a determinant involving only rp-fields. We record here that although the axial gauge is chosen explicitly. we could have also chosen the Lorentz gauge and obtained a gauge invariant measure which is also frame independent. Finally after integrating over l/I l' and ~ 1" the gen· erational function becomes

w~ JD/na]DW] ~W)5(IZ'Ba)exp(i

J

d4

x£l.

(J 5)

where

+ a2 tr(l.a L b) a/ll/la/lr:} .

It is to be noted that so long as Z 3 is nonzero, ¢ is not an independent field. It can always be transformed away through the redefinition of the field. On the other hand, in the limit Z 3 = 0, rp can no longer be transformed away. So among the three degrees of freedom for the massive vector boson. two of them are transferred to the massless vector boson and the remaining one to the scalar boson. Furthermore, the cou.pling term between jj~ and cff1 is proportional to zjl-. In the limit Z3 =0, cff1 completely decouples from B~. Also in this limit the a2 tr(La Lb)a/lcff1 a/lrpb term is the only term in eq. (18) which is not local gauge invariant. Owing to the rp-decoupling mechanism, the remaining effective theory has now a local gauge invariance. One can explicitly check that for the QeD theory with a local gauge symmetry, the generating [unc· tional after integrating over the fermion fields gives, in the strong-coupling limit. an expression which is identical to eq. (I 8) without the rp-term. Thus the four-fermion interaction theory defined by eq. (1) has at least in a formal sense generated a gauge boson. We now turn to the Noether current of eq. (2). From eq. (3),

x exp(i Jd 4x ~ A*Ab/l) X

+iip(i~ -m·gQrlT/p

+a2 tr[(B/l + Laa/l¢a)(B/l + Laa/lrpa)] . Using Z3 and gR defined earlier and jja we can rewrite the langrangian as

(16)

=lJaZ 31/2,

£ _ 1 -a -a/lv -. -I - -..l1/lV B +R(gRB/lvl+T/Jl(I~ -m -KRiI) TIp +a 2 tr[(ZY2 B/l +L aa/lrpaHzY2 B /l +Laa/lrpa)j. ( 17) Taking the limitK -400. which implies 7.3 40,

(18)

-~. exp(i (

aAo /l

.

d4x [\ji(i;l- m -g·f)j l/I +iil/l

+\1171)'

J

(19)

Integrating over'" and. iii, and taking the derivative with respect to A~, we obtain (~O)

We have used the Euler-Lagrange equation to arrive at eq. (20). The same result can also be obtained for· mally through integration by parts. In eq. (20) A~ is the field operator in the A /l sector with a fermion field having been averaged out. The relation of eq. (20) can be traced to stem from the original constraint associated with the auxiliary

459

314 Volume l09B, number 6

II March 1982

PHYSICS LETTERS

field AI" After our integrating over fermion fields and the identification of the massive bound state, this constraint is now promoted to a current·field identity. It can be shown thatj~ is still a conserved current, as it should be p . Eq. (20) also enables us to calculate the matrix element of j~ between two massive A-bound states. For instance, to lowest order in perturbation theory,

ever, the important point is that it is now decoupled from Rand the fermions, "'p' ~ On the other hand. in the effective theory of B and of the non-l/> sector, there is a new Nocther current:

"'p

,.al'

non

= __ g :,.

",1'1 Aa .,.

R 'l'p'

2

'l'p

+g fabc7Jb7JcI'V R

v

•

(25)

In general, the matrix elements of j~~~-

B-statcs do not vanish. But now this current is no (21) X l(P1

+P2)1'e!·€I-€rpI·E~-·e!I'P2·€d.

This expression displays its Lorentz covariance explicitly. Eq. (20) together with eq. (10) gives:

,.a I'

2Z I /2 (tr(Ut"Aamb)'ifb) = _la 2 3 I' (22)

Now we consider the limit Z3 =O. In this limit, the first tenn in eq. (22) vanishes t4 . So j~ becomes a function of pure r/>-fields. On the other hand, from eq. (l7) one sees that in this limit r/> decouples from B. Due to this r/>-decoupling mechanism, the second term of eq. (22) will not contribute to the matrix elements of j~ between two 'if-states. We have succeeded in demonstrating that in the strong-coupling limit: (23) for all band c. In other words, the massless bound state 'if is indeed neutral with respect to the original Noether curren t. It is interesting to note that in the limit Z 3 = 0, from eq. (22) (24) which is still a conserved four-vector current /7J and it is associated with the global SU(3) invariance. 1I0wB

This can be verified using fermion equations of motion together with a generalization of the current-field identity and the antisymmctric properties of AsZ 3 approaches ?ero, taking into account the rapid oscillation in the phase factor of U, wc find (tr(llhaU~h X B*> .... ~(11~>. So the first term on the right-hand side of eq. (22) is proportional to Z~12 .

F.

14

460

longer a Lorentz covariant four·vector. So the no-go theorem of ref. [II does not apply. In closing, let us reiterate the main points here. Our starting four·fermion theo!)' has only the global SU(3)-color symmetry. The corresponding Noether current is well defmed. It is a four-vector and is conserved. After taking into account the effect of fermion loop in renormalization, this theo!)' in its strong-coupling limit, formally contains a massless vector boson 'ifI' together with a massless scalar boson cp. In this limit the original Noether current can be identified with a pure r/> contribution as in eq.(24). But in the strong coupling limit, r/> decouples from the rest of the system_ Consequently RIA is neutral with respect to the original Noether currenL On the other hand, in the non-l/> sector, we have a local gauge invariance with the massless vector bound state acting as the gauge boson. A new Noether current can now be defined. We have thus demonstrated our contention that the vanishing of the matrix elements of the Noether current defined by the original fermion theory, taken between massless vector bound states, does not necessarily preclude the possibility of the existence of composite gauge bosons. We wish to thank Professor E.e.G. Sudarshan, Professor S_ Weinberg and Dr. Xerxes Tata for stimulating discussions and for their invaluable suggestions. Two of us (SPC and KCC) would like to thank Professor E.C.G. Sudarshan and colleagues in the Center for Particle Theory for the hospitality during their visit.

References [11 S. Weinberg and E. Witten, Phys. Lett. 96B (1980) 59. [21 Sec also: E.C.G. Sudarshan, Phys. Rev. 024 (1981) 1591.

315 Volume 109B, number 6

PHYSICS LElTERS

[3) See for examples: H. Terazawa, Y. Chikashige and K. Akama, Phys. Rev. DIS (1977) 480; T. Egllchi, Phys. Rev. 017 (1978) 611; F. Cooper, G.S. Guralnik and N.J. Snyderman, Phys. Rev. Lett. 40 (1978) 1620; C.C. Chiang,C.B. Chill, E.C.G. Sudarshan and X. Tata, Is the QeD gluon a composite object?, Phys. Rev. D, to be published; and references quoted therein. [41 Sec for examples: K. Kikkawa, Prog. Theor. Phys. 56 (1976) 947;

II March 1982

T. Eguchi, Phys. Rev. D14 (1976) 2755; H. Tcrazawa, Y. Chikashige and K. Akallla, Phys. Rcv. 015 (1977) 480. [51 J.D. Bjorkon, Ann. Phys. (NY) 24 (1963) 174; G.S. Guralnik, Phys. Rev. 136 (1964) B1404. [6] I. Bialnynicki-Birula, Phys. Rev. 130 (1963)465. [7] A.T. Ogielski, M.K. Prasad, A. Sinha and L.L. Chau Wang, Phys. Lett. 91 B (1980) 387.

316 Volume 11011. number 3,4

PHYSICS LETTERS

I April 1982

MASSLESS BARYONS AND ANOMALIES IN CHlRAL (QCDh D. AMAT!, Kuang-Chao CHOU I and S. YANKIELOWICZ CERN. Geneva, Switzerland Received 25 January 1982

The spectrum of chiral SU(N) gauge theories in two dimensions is shown to consist of free massless mesons and baryons that may be visualized as composites of free massless quarks and antiquarks. Baryons do not contribute to anomalies which are saturated by the mesons. The singular nature of the chirallimit is recognized.

A dynamical framework is needed in order to understand how a confining theory treats chirality and anomalies: whether by realizing chirality and saturating anomalies with massless bound fermions [I] (baryons) or by saturating them with Goldstone bosons if chiral symmetry is spontaneously broken. In a lIN approach the second scenario is realized [2), but this is no surprise because that approach is unwarranted for a dynamics with massless fermion bound states. Attention was then directed to two-dimensional models [SU(Nh] where the dynamics is accessible, hoping of course to be able to distinguish peculiarities of two dimensions (particularly pathological as far as symmetry breaking is concerned) from dynamical features expected to survive in four dimensions. 't Hooft did [3J the first analysis of SU(N}z with a liN expansion in a light-cone gauge. He identified a colour singlet meson bound state that in the chiral limit (m =0 where m is the quark mass) becomes massless and free. This massless meson saturates the U(l) anom· aly as expected in a spontaneously broken scenario. But as previously mentioned, the liN expansion is unsuit able for detecting massless fermions. For m =0 and for any N, the conservation of both vector and axial currents implies Oil' = 0 and thus the existence of a corresponding massless decoupled meson that saturates trivially the appropriate anomaly. In a recent paper [4J these states have been explicitly con1

On leave from Institute of Theoretical Physics. Academia Sinic., Beijing, J'eoplc's Republic' of China.

0031-9163/82/0000-0000/$02.75 © 1982 North-Holland

structed and it has been argued that no massless baryon states could exist because they would spoil the anomaly condition. On the other hand, working in a gauge first proposed by Baluni [5] the spectrum for finite m was identified [6] and shown to contain both bound mesons and baryons with masses going to zero for m -+- O. This is a common feature of two-dinlensional models [7J. It was also recognized [7] that these baryons would necessarily spoil the anomaly condition (as a function of N) in the chirallimit if baryon form factors would satisfy some smoothness conditions. We therefore find a few apparent contradictions. On the one hand the baryons present for arbitrarily small m could not disappear from the Hilbert space and therefore it should be possible to find them also in the strict m = 0 chiral theory. On the other hand, if they are there they should not contribute to anomalies and this implies a singular behaviour of form factors such as m -+- O. We shall indeed show that both statements are correct and that there is no contradiction left. The picture that arises in the chiral limit is very trivial (free massless mesons and baryons) and is totally due to the two dimensionality that allows free massless quarks to produce free massless bound states. These two·dimensiunal gauge theories reali;le chirality in such a trivial way that they shed no light on dynamical mechanisms for chirality realization or breaking in four dimensions. In what follows we shall discuss the single f1avour 309

317 Volume llOH, number 3,4

I April 1982

l'fn'sICS LETTERS

case, the generalization to arbitrary N f being straightforward. Let us first identify the massless baryon for the m =0 case following a method analogous to that used in ref. 14] to identify the boson. We shall therefore work in the.4.. =0 light-cone gauge in which the gauge field ri+ may be totally eliminated in terms of a Coulomb interaction and in which only the left-handed (actually + 'YS) Iji is dynamical a left mover) quark field Iji+ = in the chiral til =0 case under discussion. In this gauge, the role of the hamiltonian is played by the generator of displacements of x+ 1/..Ji(x O + xl) given by

hI

fined by bj(p_)IO) = Ji(p_)IO) = 0 with the vacuum energy normalized to zem. An eigenstate of P+ with eigenvalue P+ and fixed P will have mass

(6) The exact zero-mass boson state found in ref. [4] has the form

=

J

p+ =- ~ dx- dy-

with a normalization condition

dy+ ra(x)

X S(x+ - y+) Ix-

(7')

.- r I J+acv) ,

(1)

with

Similarly, the massless baryon state we are looking for will be given by

=

where la, a I, ... , N2 -- I are the SU(N) group generators. Energy momentum conservation ensures that P+ is independent of x+ and one may take x+ 0 in studying the bound-state spectrum. We expand Iji+(x) in terms of the creation and annihilation operators:

(8)

=

I Iji+(x) = ~lt4 ~

~

dP

J -v.:.~ [c-iPf b(P_) +eiPy J+(p_)](3),

0

11

P+=

Indeed, using the identities

(fa )ij (Ia)ml!

and write P+ in the form: ~

with the normalization constant

=ASi/)mn + BSinSjm

'

(10)

with

~

J dq_dp_ J V(k1:Jdk .. lb;(q.Jbj (q_ +k_}

(II)

NA +B=O,

()

and

+b;(q _)dj(-L -k_)+dj(q.)bj( .. q_ + k.) + dj(q

J

A - B

dt(q .. - k_>] (Ill)j/Ia)mn

X [b:(p _)bn(p_ .. k_.) + bin(P _) +dm(P.Jbn(-p_

d:< p _ + k_J

kJ+dm(p )d:(p_ +k.)] .

(4) The specific form of the Coulomb potential

(5) where P stands for the principal value, plays no role in the following calculations, It is easily shown that the physical vacuum is the usual perturbative state 10) de-

310

= -(N -

I t I C2(N) ,

(12)

where C2 (N) is the second Casimir invariant, one can easily prove that J\IB,p.)=O.

(13)

Therefore IB, p_) represents an exact massless baryon state. For quarks not in the fundamental representation of the SU(N) group, it is still possible to have massless bound baryon states provided the relations (I)) and (12) arc satisfied. A direct calculation for the matrix elements of the

318 Volume IIOB, number 3,4

single t U{l) curren t yields = -IN q_/i(L - P _) /i(q+) ,

(14)

and (B, P'JJ+(q)IB, p_>

=N(min(p_, p'-)/max(p_., p,-rl'/2

X (p_p'j/2 /i(q_ - p. + p'-) /i(q+).

(IS)

The spectrum as well as the matrix elements found above are not a surprise. In order to see why, let us first recall that the light cone gauge .4 _ =0 implies that the right-handed quark field 1/1 _ = ~(I - 15) 1/1 remains free. Moreover, in two dimensions a system of free massless quarks moving in the same direction has zero mass. Had we chosen the opposite light-cone gauge condition.4+ =0 the roles would be reversed, 1/1+ would be free and the gauge invariant states (7) and (8) would appear as massless mesons and baryons constituted of free quarks. Their singlet current matrix elements are those of a free theory and being gauge-invariant quantities should coincide with those calculated before in the ..L =0 gauge. And, indeed, eqs. (14) and (15) represent free theory matrix elements. Moreover, BB will also appear as a massless bound state satisfying
I April 1982

PHYSICS LEITERS

(16)

as it is also easy to verify in the.4 _ =0 gauge. Eq. (16) implies of course that baryons do not contribute to the anomaly. By further exploiting this separation of left and right worlds it is easy to see that the chiral theory consists wholly of free colourless mesons and baryons. This very simple result leads to a fact that at first sight is puzzling. Usually the matrix elements of (I 5) and (16) are considered to be the same, each obtained by a different analytic continuation. We will show that this concept of analytic continuation is untenable in this two-dimensional chiral theory and we will identify the origin of this apparent contradiction. As already stressed, the left and right worlds are completely separated in the chiral theory and the physical states are composed of purely right movers or purely left movers, massless quarks of antiquarks. Therefore all physical states have zero square momentum and there is no invariant momentum transfer other than zero entering the theory. Thus the very concept of analytic continu-

ation loses meaning. This is weIl seen by considering the theory for m *- 0 where analytic continuation of form factors has been established, and witnessing how the limit m -+ 0 is nonanalytic. Indeed, for mIg <0: I and working in the Baluni gauge [5] it has been established [6] that the spectrum is described by a sine-Gordon theory with a hamiltonian given, for the single flavour case we are discussing, by

H

= ~7T2 -

~(oxop)2 - a(l- cos«(3op)] ,

(17)

where a=4m 2N e ,

(3o=(47T/Nc)1/2.

(18)

The baryon is identified with the soliton state with mass Ms = 8..jQ/(3 = (16/"';;) mNe '

(19)

that vanishes for m -+ O. The U(1) current is given by

il' =-«(3/27T) €I'''o''op,

(20)

and its matrix element between soliton states dermes a U(l) form factor which has been exactly calculated [8]. It depends analytically on a parameter 8 related to the momentum transfer t by t = 4m 2 cosh 28/2. For t = 0 (m "" 0) the form factor is equal to N (i.e., the charge). For m -+ 0, t -+ 0 irrespectively of 8 which obviously implies a singular limit. We see therefore that for the chilal case the matrix elements (15) and (16) are not analytically related. The limit m =0 in the Baluni gauge is indeed subtle_ The hamiltonian (17) loses its non-linear term which generates the topological excitations. It acquires, however, a fermionic zero mode [9] whose contribution to the current matrix elements is proportional to /i(q). It ensures that the baryon charge isN [Le., eq. (IS)] that does not contribute to the anomaly [i.e., eq. (16)] which is a /i(q 2) effect. In everything said before we considered N > 1. The singular behaviour of the limit m -+ 0 is also manifested if we consider the continuation to N';;; I. In this region, corresponding to 47T .;;;~2.;;; 87T [cf. eq. (18)], the sin..Gordon theory of eq. (17) is equivalent to a repulsive massive Thirring model whose coupling g is related to (3 thro ugh [10]

I + 2g 17T == X = 87T/{32 - I . In this region (X < 1, g < 0) there are no bound states and the soliton of the sine-Gordon equation corresponds 311

319 Volume II Oll. number 3,4

PHYSICS LETTERS

to the Thirring fermion. This is reOected in the observation that the time-like form factors of the sine-Gordun theury in this region have a cut starting at 4m 2 • unlike the mesun pole which was present fur A> I which goes into the unphysical sheet [8]. Note that for N" I (i.e., A '" I) we obtain a free fermion theory. Thus for A 0;;;; I the intermediate states contributing to the current correlation function will be fermion-antifermion pairs. Therefore, in the limit In -+ 0 the anomaly is trivially saturated by the original quark. This apparent change of responsibility in going from A> I to A0;;;; I in the anomaly saturation has little meaning. As long as m =F 0 a free qq system is baSically different from a meson bound state, but for m -+ 0 they become indistinguishable. This is obviously a peculiarity of two dimensions. We see therefore that SU(Nh, which has a rich light spectrum for In 0, has a trivial chiral (m '" 0) limit. There, the left and right moving worlds are separated and each includes free mesons and baryons which are trivial composites of massless quarks and antiquarks moving parallely. As it is easy to visualize from this trivial picture, anomalies are saturated by mesons (collinear massless qq states!) while baryon-antibaryons

*"

I April 1982

do not contribute because they cannot be created by the quark currents from the vacuum. This trivial way of realizing chirality is clearly a characterization of two dimensions and is therefore of little help in understand· ing how chirality is realized or broken in four·dimensional gauge theories.

References II] G. 't Hooft. in: Proc. Cargcsc School (1979); sce also: Y. Frishman, A. Schwimmer. T. Banks and S. Yankiclowicz, Nucl. i'hys. BI77 (1981) 157. [2\ S. Colcman and E. Witten, Phys. Rev. Lett. 45 (1980) 100; G. VeneZiano, Phys. Lett. 95B (1980) 90. 13] G. 't Hooft, Nucl. Phys. B75 (1974) 461. 14] W. Biichmiiller, S.T. Love and R.D. Peccei, MPI·PAE/PTh. 70/81 (1981).

(5 J V. Baluni, Phys. Lett. 90B (1980) 407. [6] P.J. Steinhart. Nucl. Phys. BI76 (1980) 100; D. Amali and E. Rabinovici. Phys. Lett. 1018 (1981) 407. [7J S. Elitzur, Y. I'rishman and E. Rabinovici, Phys. Lett. 106B (1981) 403. 18J M. Kurowski and P. Weisz, NucI. Phys. B139 (1978) 455. (9J T. Banks. D. Horn and H. Neuberger, Nucl. Phys. BI08 (1976) 119. [ 10] S. Coleman, Erice lecture notes (1975).

320 Volume 114B, number 2,3

22 July 1982

PHYSICS LETfERS

ON THE DETERMINATION OF EFFECTIVE POTENTIALS IN SUPERSYMMETRIC THEORIES D. AMATI and Kuang-chao CHOU

I

C.ERN. Geneva. Switzerland

Received 8 April 1982

We propose·a renormalization procedure for dynamically generated effective actions. We show that. as expected, it leads to no spontaneous supersymmetIy breaking if this is Iinbroken at the tree level. We also understand why the usually adopted renormalization prescription has led in some models to an apparent supersymmetry breaking for an unacceptable negativeenergy vacuum state.

Application of the well-established l/N expansion to some supersymmetric models seemed to indicate the possibility of spontaneous dynamical breaking of supersymmetry violating general properties such as positivity of the ground-state energy or the index theorem tJ. This would shed negative light on l/N techniques which. nevertheless. seem applicable to supersymmetric theories. This puzzling result is well illustrated by Zanon's model {2l where a non-supersymmetric minimum was found for a negative value of the effective potential. In this note we wish to show that this apparent contradiction stems from an un appropriate renormalization procedure [3] adopted in the evaluation of effective potentials. Moreover. if the renormalization is cor· rectly performed. the effective potential. in terms of the dynamical fields. vanishes at the origin and is otherwise positive. thus confirming that supersymmetry will be preserved by radiative corrections if it is not broken at the tree level [4]. We shall use Zanon's model to illustrate the correct renonnalization procedure and fmd the loophole in that used in ref. [2]. The model consists of N + I chiral

supermultiplets tP and tPi' j = 1 • ...• N. described by the action *2

S=Jd4X d4 6 (ii>itPi+ii>tP)

-Jd x 2

d2 6 {[~mtP2 +~motPt + (g/..jiii)tPtPr] + h.c.}

The tPi fields appearing only bilinearIy may be integrated over. thus leading to an effective action depending only on tP. The well-known fact that these theories need only a wave function renormalization suggests a rescaling ifJ-+(NZ) 1/2 ifJ.

g-+Z- 1/2g •

m -+Z-I m .

(1)

Fields. coupling constants and masses will now represent renormalized quantities in terms of which the integration of S over tPi leads to S eff =N(ZStf> - ~Iog det 9f).

(2)

where

(3) and det 9f

On leave from Institute of Theoretical Physics. Academia Sinica. Beijing. People's Republic of China. *! The index theorem states that no spontaneous breaking of supersymmetry could happen if the numbers of the zero energy fermionic and bosonic statcs are unequal. see ref. [I J.

= det CU det- I en

I

o 031-9163/82/0000-0000/$02.75 © 1982 North-Holland

t2

We have introduced a mass m for the fields tf>j. It avoids infrared problems. The final potential is infrared free so that m may be set to zero if one wishes to make a comparison with ref. [2).

129

321 Volume 114B, number 2,3

r~F C)L-0

.-~-mo

PHYSICS LEITERS

-0

-·2gA- mo

-2g°Fo

0

0

0

-2g°Ao- m

o

-]

-:~.---j -]

0

A (x) and F(x) being the lowest componen t and the auxiliary field of the supermultiplet 4>, respectively. The usual way to evaluate the exact formal expression of eq. (2) is to develop it around constant fields_ In so dOing, one has to regularize ultraviolet divergences. We adopt a Pauli-Villars prescription which suffices for the case under analysis, and call M the regulator mass. We now evaluate the log det term in eq. (2) through its liN expansion around constant fields A = a, F = f, X = O. We shall see a singular behaviour in M for M -+ 00 not only in the potential (Le., -Eerr for constant fields) but also in the coefficients of the A and X kinetic terms. We thus write

Eerr =(Z +CA)A*OA + (Z + Cx) ~ixO""a"x - V +R ,

+ VI'

(6)

where CA , Cx and VI are explicit functions of M,a and/. VI is given by

VI

=(1/641T2)[(a2 + fj)21n(a 2 + (3) + (a 2 - fj)2ln(a 2 - (3) - 2a4 ln a 2 - 2{321n(M2 + ( 2) - 3(32) ,

(7)

with a 2 = 12ga + mol2,

{3 =21gfl .

(8)

CA has the form

CA = (lgI2/81T 2 ) In [(M2 +( 2)/Q2) + terms regular in M . 130

(9)

(10)

Surely enough this is a legitimate renormalization condition in the sense that it eliminates all M2 singularities in eq. (5) and in particular in the potential V. But if the auxiliary field f is eliminated in terms of a by

av/af=o,

(11)

~hen

V= V(a,f(a)) acquires negative values, in particular for sufficiently large a. Moreover, if there is a stationary point,

dV(a,f(a»/da = 0,

(12)

away from the origin, it happens for negative values of V. This was the stationary point detected in ref. [2] and to which was assigned the responsibility of spontaneous symmetry breaking. In order to understand the reason for this apparent inconsistency with positivity of the ground-state energy implied by supersymmetry, let us analyze the kinetic terms of eq. (5). Using (9) and (10) we fmd, after settingM = 00 Z + CA = 1 + (lgI2/81T 2 ) In iJ. 2/a 2 + ... ,

(5)

where R is regular in M and contains derivatives in the fields A, F and X higher than those explicitly written in eq. (5). V will have an expression of the form V= -Zrf+maf+m*a*f*

The usual renormalization prescription [3] consists of determining Z by imposing a condition on Vat an arbitrary specific value of f and a that defines a renormalization scale iJ. 2 . This implies

Z = I - (lgI2/81T2) InM2/iJ. 2 . (4)

22 July 1982

(13)

an expression which becomes negative for large values of Q. The same happens for Z + Cx and it is possible to see that the negativity of both kinetic terms is correlated with the negativity of the potential. Thus the stationary point of ref. (2) is a property of a ghost potenti.al and cannot be identified with a stationary ground state. This rather unpleasant description may be circumvented by an alternative renormalization prescription showing clearly that supersymmetry is unbroken in this theory. A wave function renormalization controls the normalization of the corresponding kinetic term. We are thus naturally led to normalize it at the background field values (a andfin our notation) that we wish to consider. The minimum conditions on the potential thus obtained determine those values for which linear terms in the fluctuations are absent, thus allowing the identification of Q andfat the minimum with (A) and (F), respectively. We could determine Z from Z + CA = 1, but to avoid useless complications with the regular terms in

322 Volume 114B, number 2,3

PHYSICS LEITERS

m5

eq. (9), let us choose Z= 1 -(lgI2/87T2)lnM2/0i 2 ,

(14)

which eliminates from Z + CA' and therefore from the A kinetic term, the logarithmic term which was at the origin of its negative values for the choice of Z in eq. (10). Eq. (1) shows that if Z depends on a, the renormal· ized parameters g and m depend on a through Z. This dependence is the usual one, Le., g-2(0i) =g-2(Oio) + (1/87T 2) In 0i5/0i 2 .

(IS)

Eqs. (6), (7) and (14) imply

V= -lfI 2 +maf+m*a*r + O/647T2) [(0i2 + (j)2In(0i2 + (j) + (0i 2 - (j)2 In(0i 2 _ (j) _ 2a41n 0i2 _ 2(j21n 0i 2 _ 3(j2] . (16) It is easy to see that

v - f iW/af - r av/ar = Ifl2 + O/647T2){(0i4 _ (j2) In [(0i4 _ ;;;;. 0,

(j2)/a4] + (j2}

(17)

for all a and (j '" (X2 , the equality sign in (17) holding only for (j = O. It is then clear that on the line

av/af= av/ar =0 ,

(18)

which expresses f in terms of a, the potential V = V(a, f(a» is a positive function of a with its absolute min· imum at a = 0 where V vanishes. Therefore (,4) = {F} = 0 and supersymmetry is unbroken. We see therefore that with our renormalization prescription, we succeeded in describing the theory for all choices of the background field a with a posi. tive potential and in terms of fluctuations which have positive kinetic terms. On the other hand, in order to describe the same theory at different values of a, we found that the coupling constant depends on a, as described in eq. (I 5). If we call go the coupling constant at a =0, we find

g2(a)=g5/[l + (g5/ 87T2 )In m5/0i 2 ] ,

22 July 1982

(19)

which shows a Landau-type pole at 0i 2 = exp(87T 2/ ga) characteristic of a non.asymptotic free theory like the one analyzed here. To summarize, we have shown that the renormalization prescription of refs. [2] and [3] generates nega· tive potentials together with negative kinetic terms. Ghosts try to increase their potential energy so that a negative stationary point is energetically unfavourable as compared with a zero potential configuration. Thus, even in this language, we understand why supersymmetry is not broken in the model of ref. [2]. Moreover, we have shown how to avoid this pathological ghost interpretation through an alternative renormalization prescription that leads to bona fide fields with positive kinetic energy and with non·negative potentials as reo quired by supersymmetry. We may wonder why the generally adopted prescrip· tion of renormalizing the interaction at a fixed scale does not work for supersymmetric theories while it is perfectly applicable to usual theories as "Np4 [5]. In conventional theories the coupling constant is renor· malized independently of the wave function and thus any preSCription for the first one cannot influence the kinetic terms which are controlled by the second. In supersymmetry the only renormalization is the wave function one and therefore is determined by the kinetic terms. Or, at least if a definition of the interaction is introduced which leads to negative kinetic energy, it is impossible to appeal to another independent renor· malization to correct that sign. We wish to acknowledge fruitful discussions with L. Girardello, 1. Iliopoulos, R. Cahn, G. Veneziano and S. Yankielowicz.

References 11) E. Witten, Lecture notes at Trieste (1981); S. Ceeotti and L. GirardeUo, Phys. Lett. llOB (1982) 39. [2) D. Zanan, Phys. Lett. 100B (1981) 127. [3) M. Hug, Phys. Rev. D14 (1976) 3548; D16 (1977) 1733. [4) L. O'Raifearlaigh and G. Parravieini, Nue!. Phys. Bll1 (1976) 516; W. Lang, Nue!. Phys. B114 (1976) 123. [5) S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888.

131

323 PHYSICAL REVIEW D

I SEPTEMBER 1983

VOLUME 28, NUMBER 5

Koba-Nielsen-Olesen scaling and production mechanism in high-energy collisions Chou Kuang-chao Institute of Theoretical Physics, Academia Sinica, Beijing, China

Liu Lian-sou' and Meng Ta-chung Institutfiir Theorelische Physik der Freie Universitiit Berlin, Berlin, Gemany (Received 14 March 1983)

An analysis of the existing data on photoproduction and electroproduction of protons is made. Koba-Nielsen-Olesen (KNO) scaling is observed in both cases. The scaling function of the nondiffractive rp processes tums out to be the same as that for nondiffractive hadron-hadron collisions, but the scaling function for deep-inelastic e -p collisions is very much different from that for e -e + annihilation processes. Taken together with the observed difference in KNO scaling functions in e -e + annihilation and nondiffractive hadron-hadron processes these empirical facts provide further evidence for the conjecture: The KNO scaling function of a given collision process reflects its reaction mechanism. Arguments for this conjecture are given in terms of a semiclassical picture. It is shown that, in the framework of the proposed picture, explicit expressions for the above-mentioned KNO scaling functions can be derived from rather general assumptions.

I. INTRODUCTION

The recent CERN pp collider eXr!?ments,' in which Koba-Nielsen-Olesen (KNO) scaling has been observed, have initiated considerable interest 3,4 in studying the implications of this remarkable property. A physical picture has been proposed in an earlier paper4 to understand the KNO scaling in the above-mentioned experiments,' and in pp~ and e+e- reactions. 6 It is suggested in particular that the qualitative difference between the KNO scaling function in e + e - annihilation and those in nondiffractive hadron-hadron collisions is due to the difference in reaction mechanisms. In this paper we report on the result of a systematic analysis of high-energy 'YP and e -P data,,8 as well as that of a theoretical study of the possible reaction mechanisms of these and other related processes. We show the following. (A) KNO scaling is valid also in high-energy 'YP and e - P processes. The scaling functions for nondiffractive 'YP and low_Q2 (invariant momentum-transfer squared) e-p processes are the same as for nondiffractive hadronhadron collisions, but the scaling function for deepinelastic e-p collisions is very much different from that for e - e + annihilation processes. (B) The KNO scaling function for e - e + annihilation, 1/J(z) = 6z 2exp( _az 3), a'/3= T) , (I)

n

and that for nondiffractive hadron-hadron collisions, 1/J(z)= l6/5(3dexp( -6z)

KNO scaling functions mentioned in (A) can be understood in the framework of the proposed picture. II. KNO SCALING IN 'YP AND e -p PROCESSES

We studied photoproduction and electroproduction of protons at incident energies above the resonance region. We made a systematic analysis of the existing data,,8 and found that: there is KNO scaling in e -pas well as in 'YP processes. (See Figs. I and 2.) The KNO scaling function for nondiffractive 'YP processes and that for e-p at low momentum transfer are the same as that for nondiffraclive hadron-hadron collisions. (See Fig. I.) The KNO scaling function for deep-inelastic e -P collisions is very much different from that for e + e - annihilation processes. (See Fig. 2.) The similarity between the KNO scaling function in nondiffractive 'YP (and low-momentum-transfer e - p) and that in nondiffractive hadron-hadron processes is not very surprising. In fact, it shows nothing else but the wellknown fact 9 that real (or almost real) photons at high energies behave like hadrons. But does the difference in KNO scaling functions in e-e+ and deep-inelastic e-p processes indicate that the reaction mechanisms of these two kinds of processes are qualitatively different from each other? Before we try to answer this question, let us first examine in more detail the relatiOriship between KNO scaling functions and reaction mechanisms in e - e + annihilation and in nondiffractive hadron-hadron collisions.

(2)

(here z =n I( n ), n is the charged multiplicity and (n) is its average value), can be obtained from the basic assumptions of the proposed physical picture using statistical methods. (C) The similarities and differences between observed

III. e-e+ ANNIHILATION: FORMATION AND BREAKUP OF ELONGATED BAG

The KNO scaling function in e - e + annihilation processes is shown in Fig. 3. It is sharply peaked at n I (n ) = I (n is the mUltiplicity of the charged hadrons 1080

® 1983 The American Physical Society

324 1081

KOBA-NIELSEN-OLESEN SCALING AND PRODUCTION ...

0"2.8-8.0 (GeV/d 2

2 .S

.4

\ +

.3

\

2

\ \

N

Elf(GeV)

~

1.4-16 2.8-3.6 4.S-S.7 7.4-9.4

\

C

a: €

\ \

.05 .04 .03

\ \

.02

~

.01

0.3-0.S 0.5-0.7 1.8-2.2 2.2-2.8. •

.01

•

•

WinGeV 02,n (GeV/c)2

0

z",n/
FIG. I. The scaled multiplicity distribution for nondiffractive yp and low-momentum-transfer e -p reactions. The experimental data are taken from Refs. 7 and 8, respectively. The curve is obtained from Eq. (2).

\

\ \

.005 .004 .003 .002 .001

.0010~---;;D.5'::-----!1.0;;--""'1.S=--""'2"'.0=---=2'=.5--'-3"'.OO-

\ \

W(GeV) 2.2-2.8 • 2.8-3.8 ..

\ \ \

,

o.s

1.5

2

2.5

3

z·n/
e-e+ and (n) is its mean value) and can be approximated by a Gaussian. 4,lo It can be qualitatively understood4 as follows: In e - e + annihilation processes, the electron and the positron can be considered as pointlike particles. Hadronization takes place when the colliding e- and e+ hit each other so Violently that the entire amount of the initial energy and momentum is deposited into a single system which subsequently breaks up into pieces. It is clear that, without further specifications, this picture would be too general and too crude to account for all the characteristic features of the e - e + annihilation processes. In fact, the observed two-jet structure ll and the central rapidity
ljI{z)' 6 a z2exp(-a Z3)

a '/3. r(4/3)

3 2

N .5

~

oS

a. c: "v

.1 .05

vs(GeV) 9.4 -35

D1U---L-U~5----~-----1~.5----~2~~-

z·n/cn> FIG. 3. The scaled multiplicity distribution for e -e + annihilation processes. The experimental data are taken from Ref. 6. The curve is the scaling function given by Eq. (I).

325 CHOU KUANG-CHAO, LIU LIAN-SOU, AND MENG TA-CHUNG

1082

colilSlon

a typical violent collision

e-et

example

annihilation

coUis,on

-------

first stage after the collision

~q

before the

,/

/'

a typical gentle colllSiQtl non·d,ffracl'\le pp collision

p

----0 0--

-0

(0

G-

is~~.~:~~ collision

FIG. 4. Two qualitatively different types of high-energy collisions are illustrated; and one characteristic example in each case is given.

not seen in e - e + annihilation processes, where the decay of the virtual photon into a quark-antiquark (qq) pair is generally accepted to be true. In fact, it is envisaged that as q and ij of the original qij pair move apart (at almost the velocity of light) a color-electric field is developed, and a number of polarized pairs (secondary qij pairs) are formed between them. Now, since the gluon exchanges allow q (ij) of arbitrarily high subenergy to interact with finite probability, an "inside-outside" cascade l6 takes place and as a consequence only color-singlet hadrons are produced. Quantitative comparisons between experiments ll ,I2 and the Lund model,l7 which is a semiclassical model that incorporates all the relevant features of the Schwinger model,I4 have been made." The agreement seems to be very impressive. Is it possible to understand the observed KNO scaling behavior in e-e+ annihilation processes in models based on the Schwinger mechanism? We now show that this question-which does not seem to have been asked before--can be answered in the affirmative. In order to study mUltiplicity distributions in this framework, we need to know the relationship between the observed multiplicity of charged hadrons and the properties of the elongated bag. The following points are of particular importance in establishing this relationship: (a) Since the number of sub-bags at the final stage of a given event is nothing else but the total mUltiplicity of hadrons in that event, it seems plausible to assume that the total multiplicity of charged hadrons (n) is proportional to the final length (1) of the elongated bag in every

approximately the same transversed momentum with respect to the jet axis; (ii) The multiplicity of charged hadrons is distributed mainly around its average value (n) which is rather high at incident energies where KNO scaling has been observed (e.g., (n) "" 7 and 14 at Vs = 10 and 30 GeV, respectively). We note that the final length I is determined by the first breakup of the elongated bag. This is essentially a kinematical effect which can be readily demonstrated in terms of the one-dimensional Lund model 17 as shown in Fig. 5. The generalization from the one-dimensional string to a three-dimensional elongated bag does not influence the arguments used to reach this conclusion. The reason is: In the present model, the existence of a qq pair is not a sufficient, but a necessary condition for the breakup. (fJ) As the original quark-antiquark pair fly apart, their kinetic energy is converted into volume and surface energies. Secondary qq pairs are produced and the elongated bag begins to split when the bag reaches a certain length such that it is energetically more favorable to do so. Note that the collective effect due to color interaction is a substantial part of the bag concept. Hence, it is expected that the probability of bag splitting should depend on the global rather than the loeapo properties of the entire system. We shall assume, for the sake of simplicity, that the elongated bag is uniform in the longitudinal direction, and that the probability df Idl l for a bag of length I to break somewhere (at II, say, where 0 < II < /) is proportional to I, that is approximately proportional to the total energy U of the elongated bag. 21 This means [(1)=

f: ;(

df ="A.l dl l

dl l

(4) (5)

'

where A is a constant. It should be mentioned that we do

event,

n =("111 0 )1.

(3)

Here, 10 is the average length of the elongated hadron-bags in their "rest frames" and "1 is the inverse of the average Lorentz contraction factors of the hadrons along the jet axis. This means, we have assumed that "1110 depends only on the total c.m. energy Vs, provided that n is not too small compared to (n). Obviously, Eq. (3) is in accordance with the following empirical facts 6• 19 : (i) The overwhelming part of the produced hadrons are pions of

FIG. 5. The one-dimensional Lund model (see Ref. 17) is used to demonstrate that in e - e + annihilation processes the fi·

nal length of the elongated bag is determined by the first breakup. Here, t and x denote the time and space coordinates,

respec~

tively. Note that Ihe generalization from the one-dimensional string to a three-dimensional elongated bag does not influence the arguments used to reach this conclusion.

326 KOBA-NIELSEN-OLESEN SCALING AND PRODUCTION ... not know why the above-mentioned I dependence [Eq. (5)] should be linear. What we know for the moment is: By assuming a power behavior Ik for df Idll> the experimental data require k = I. (y) Having obtained the probability f(l) for an elongated bag of length I to break up, the density function for the I-distribution P (l) can be calculated in the following way: Consider N events, among which in N (l) of them the bag has reached the length without breaking and dN of them will break up in the interval (1,1 +dn, then

dN = - f(llN(l)dl .

(6)

It follows from Eqs. (4), (5), and (6)

dN P(l)=di =CI2exp( _Al 3 13) ,

(7)

where C is a normalization constant. The corresponding density function for multiplicity distribution Pin) is therefore [see Eq. (3)] (8)

The constants A and B are determined by the usual normalization conditions22 :

fo"'P(n)dn =2,

(9)

fo'" nP(n)dn =2(n) . From Eqs. (8), (9), and

(10),

(10)

we have

(n )P(n)=t/J(n/(n» ,

(II)

where t/J(z) is given by Eq. (1). Comparison with experiments 6 is shown in Fig. 3. The following should be pointed out: (a) The KNO scaling behavior is obtained as a direct consequence of Eqs. (8), (9), and (10). (b) The elongated-bag model, which is obviously consistent with the physical picture discussed in Ref. 4, is more specific and gives a better description (than the Gaussian approximation) of the existing data. (c) There is a discrepancy between model and data for z < o. 3. This is due to the fact that Eq. (3) is only a poor apProximation for n « (n ). (d) since the final length is determined by the first breakup of the elongated bag, the existence of intermediate states does not influence the observed multiplicity of charged hadrons. IV. NONDIFFRACTIVE HADRON-HADRON COLLISIONS: FORMATION AND DECAY OF THREE-FIREBALLS We now tum to Eq. (2) and show that it can be derived in the framework of the proposed picture under more general conditions than those mentioned previously. We recall that, according to this picture,' the dominating part of the high-energy inelastic hadron-hadron collision events are nondiffractive. The reaction mechanism of such processes can be summarized as follows: Both the projectile hadron (P) and the target hadron (n are spatially extended objects with many degrees of freedom. They go through each other during the interaction and distribute their energies in three distinct kinematical regions in phase space:

1083

the projectile fragmentation region R (P*), the target fragmentation region R (T*), and the central rapidity region R (e*). Part of these energies materialize and become hadrons. We denote these parts by Epo, E p , and Eco, respectively. They are the internal (or excitation) energies of the respective systems. The difference in reaction mechanisms of e - e + annihilations and nondiffractive hadron-hadron collisions is illustrated in Fig. 4. Let us consider the internal energy E j of the system i (i =p*,T*,e*) in a large number of collision events. Viewed from the rest frame of the system i, both the projectile (P) and the target (n before the collision are moving with a considerable amount of kinetic energy. The interaction between P and T causes them to convert part of their kinetic energies into internal energies of the systems P*, T*, and e*. Hence, each system i has two energy sources so that E j can be expressed as (12)

where E jp and E jT are the contributions from the source P and that from the source T, respectively. Note that the two sources are independent of each other, and that among the nine variables in Eq. (12) six of them are completely random. Let Fp(Ejp ) be the probability for the system i to receive the amount E jp from P, and FT(EjT ) is that for the system i to receive E jT from T, then the probability for the system i to obtain E 1P from P and E jT from T is the product Fp(Ejp)FT(EjT). Physically, it is very likely that the system i completely forgets its history as soon as the system is formed. This means, the probability for the system i to obtain E;p from P and E jT from T depends only on the sum Ejp+E,T . That is (13)

where E; and E jp and E jT are related to one another by Eq. (12). Hence

d

d

dE [lnFp(E,P )]= - dE [lnFT(Ej-E jp )] , jp 1P

(14)

that is Fp(E1P)=Apexp( -BE,p) ,

(15)

FT(Ej -Ejp)=ATexp[ -B(Ej -E,p)] ,

(16)

where A p, AT' and B are constants. In order to obtain the total probability P(Ej ) for the system i to be in a state characterized by a given energy Ej, without asking the question "How much of E j is contributed from P and how much of it from T/" we have to integrate over all the possible values of E jp and E'T under the condition given in Eq. (12). That is P(Ej =

)

f dEjpdEjT8(Ej-Ejp-EjT)Fp(Ejp)F.r(EiT)·

(17)

It follows from Eqs. (15), (16), and (17) P(Ej)=CEiexp(-BEj ) ,

(18)

where the constants Band C are determined by the normalization conditions

327 1084

CHOU KUANG-CHAO, LIU LIAN-SOU, AND MENG TA-CHUNG !P(Ej)dEj=I,

(19)

! EjP(Ej)dEj=(Ej }

(20)

Hence (EI }P(Ej )=4E;I(Ej )exp( -2E;I(Ej »

,

(2 I)

which is Eq. (11) of Ref. 4. The mUltiplicity (nND) distribution for nondiffractive hadron-hadron collisions given by Eq. (2) is obtained by taking into account (for details, see Ref. 4) n;l(nj})=E,I(Ej } , i=P*, T*, and C*"

(22)

and (23)

Note that z in Eq. (2) stands for nNo/( nNO)' It should be emphasized that the simple relationship, Ejlnj=constant (i=C*,P*,T*), is an idealization. In reality, fluctuation in nj for a given E j is expected. Such effects have been taken into account by assuming that the KNO distribution for e-e+ annihilation (which can be approximated by a Gaussian; see Ref. 4) is due to the fluctuation of n.. I (n .. ) about I, and that the fluctuations of nj about (nj) is of the same magnitude. These fluctuations are folded into the distributions obtained from the three-fireball model for hadron-hadron processes (a detailed discussion on this point is given in the preliminary version of our paper; see Ref. 20 of Ref. 4). Comparison between data and results of that calculation shows, however, that the effect is negligible in first-order approximation.

freedom (possibly a large number of colored gluons and sea quarks in addition to the colored valence quarks) such that various colorless objects can be formed in an excited proton, it seems natural to conjecture that deep-inelastic e --p processes take place as follows: The virtual photon in such collision processes interacts with a part of the proton gently in the sense that it "picks up" a certain amount of colorless matter in order to fragment. 2s Note that by picking up a certain amount of colorless matter from the proton, the virtual photon becomes a real physical object. The fragmentation products of this object are nothing else but the "current fragments" observed in lepton-nucleon reactions. 26 This conjecture can be readily tested experimentally. Because, if it is correct, we should see: First, the average multiplicity (n) does not depend on Q2 (the invariant momentum transfer). Second, (n) depends on W (the total energy of the hadronic system) in the same way as the average multiplicity in hadron-hadron collisions depends on v'S (the total c.m. energy). Third, the rapidity distribution in single-particle inclusive reactions shows a dip in the central rapidity region (near Yc.m. =0) at sufficiently high incident energies. This is because the center of the current fragments (formed by the virtual photon and the colorless matter it picked up from the photon) and that of the residue target (the rest of the target proton) move away from the central region in opposite directions. Fourth, the KNO scaling function is !/I(z)=4/3(4z>3exp( -4z).

(24)

This is because, according to the proposed picture4 the two fragmenting systems mentioned above act ind~dently, and the KNO scaling function of each system is [See Eq. (21)1

V. A POSSIBLE REACTION MECHANISM FOR DEEP-INELASTIC e -p PROCESSES

We now come back to the question raised at the end of Sec. II. According to the conventional picture23 for deepinelastic e - p collisions, one of the colored quarks inside the proton is hit so violently that it tends to flyaway from the rest (to which it is bounded by the confining forces). As a consequence quark-diquark jet structure is expected. 23 Hence, it is natural to believe that also in this case elongated bags lS or strings l7 are formed which hadronize. In fact, compared with the above-mentioned model for e - e + annihilation, the only difference would be that the bags, tubes, or strings end with quark and diquark, instead of quark and anti quark. If this were true, the KNO scaling function for deep-inelastic e -p processes would be the same as that for e - e + annihilation. The qUalitative difference in KNO scaling functions of e - e + and deep-inelastic e - p collisions is probably because the virtual photon in e -p processes behaves differently as that mentioned in the conventional picture. Once we accept that (a) the virtual photon in deepinelastic e - p collisions cannot fragment like a hadron in hadron-hadron collisions because it has an energy deficiency compared with its momentum,24 and (b) the proton is a spatially extended object with many internal degrees of

!/I(z)=4zexp(-2z) .

(25)

That is, the mechanism of e - p deep-inelastic scattering can be described as the formation and decay of two fireballs. Here we have assumed, by analogy with the nondiffractive hadron-hadron collision, that the average multiplicities of the two fireballs are equal. The first and the second points are well-known experimental facts. 27 In connection with the third point, we see that the rapidity distribution in neutrino-proton reactions at W > 8 GeV clearly shows the expected dip. (See, e.g., Fig. 10 of Ref. 26.) Corresponding data for electronproton reactions at comparable energies is expected to exhibit the same characteristic feature. La~t but not least, Fig. 2 shows that Eq. (24) (the fourth point mentioned above) is indeed in agreement with the data. The conclusion that there should be two fireballs in the intermediate stage of deep-inelastic e - p collisions can also be reached without referring to the properties of the virtual photons, provided that such collisions take place as follows: The pointlike electron goes through the spatially extended proton, gives part of its energy and momentum to a colorless subsystem of the proton al.d separates this subsystem from the rest. While the incident electron is only deflected due to the interaction, the two separated subsys-

328 KOBA-NIELSEN-OLESEN SCALING AND PRODUCTION ... terns of the proton become excited and subsequently decay. It should also be pointed out that, if this conjecture is correct, we expect to see only one central fireball in the e - e + ->e - e + X processes at sufficiently large momentum transfer. In that case the corresponding KNO scaling function should be the same as that given in Eq. (25). It would be very interesting to see whether this and other

'On leave from Hua-Zhong Teachers' College, Wuhan, and Peking University, Beijing, China. 'K. Alpgard el al., Phys. Lett. !!lTIl, 315 (1981); G. Amison el al., ibid. I07B, 320 (1981); 123B, 108 (1983); K. Alpgard ., al., ibid. illB. 209 (1983); and the papers cited therein. 2Z. Koba, H. B. Nielsen, and P. Olesen, NucL Phys. B4O,317 (1972). 3See, e.g., S. Barshay, Phys. Lett . .!l6ll, 197 (1982); T. T. Chou and C. N. Yang, ibid. llill, 301 (1982); Y. K. Lim and K. K. Phua, Phys. Rev. D 2.6, 1785 (1982); C. S. Lam and P. S. Yeung, Phys. Lett. l!2, 445 (1982); F. W. Bopp, Report No. SI-82-14, 1982 (unpublished). 4Liu Lian-sou and Meng Ta-chung, Phys. Rev. D n, 2640 (1983). 5See P. Slattery, Phys. Rev. D I, 2073 (1973); C. Bromberg el al., Phys. Rev. Lett. 11, 1563 (1973); D. Bogert el al., ibid. 11, 1271 (1973); S. Barish el al., Phys. Rev. D 2, 268 (1974); J. Whitmore, Phys. Rep. lOC, 273 (1974); A. Firestone el al., Phys. Rev. D !ll, 2080 (\974); W. Thome el al., Nucl. Phys. Bll2. 365 (1977); W. M. Morse el al., Phys. Rev. D 12, 66 (1977); R. L. Cool el al., Phys. Rev. Lett. ~, 1451 (1982); and the papers cited therein. liSee, e.g., R. FeIst, in Proceedings of Ihe 1981 Internalional Symposium on Leplon and Pholon Interaclions al High Energies, Bonn, edited by W. Pfeil (Physikalisches Institut, Universitiit Bonn, Bonn, 1981), p. 52 and the papers cited therein. 7R. Erbe el al., Phys. Rev. 175, 1669 (1968); R. Schiffer el al., Nucl. Phys. IDl!, 628 (1972); J. Ballam el al., Phys. Rev. D ~, 545 (1972); 1,3150 (1973); H. H. Bingham el al., ibid. .!!.,1277 (1973). See also H. Meyer, in Proceedings of 'he Sixth Internalional Symposium on Eleclron and Pholon Inleraclions al High Energy, Bonn, Germany, 1973, edited by H. Rollnik and W. Pfeil (North-Holland, Amsterdam, 1974), p. 175. By. Eckardt el al., Nucl. Phys. ~,45 (1973); J. T. Dakin el al., Phys. Rev. Lett . .N, 142 (1973); Phys. Rev. D .!!., 687 (1973); L. Ahrens el al., Phys. Rev. Lett. n, 131 (1973); Phys. Rev. D 2, 1894 (1974); P. H. Carbincius al., Phys. Rev. Lett. 32, 328 (1974); C. K. Chen el al., Nuel. Phys. I!.!ll, 13 (1978). 9J. D. Bjorken, in Proceedings oflhe Third Internalional Symposium on Eleclron and Pholon Inleraclions al High Energies (SLAC, Stanford, 1967), p. 109; H. T. Nieh. Phys. Rev. D 1. 3161 (1970). '0K. Goulianos el al., Phys. Rev. Lett. ~. 1454 (1982). liSee, e.g., D. Haidt, in Proceedings of Ihe 1981 Inlernalional Symposium on Leplon and Pholon Inleraclions al High Energies, Bonn, (Ref. 6), p. 558 and references given therein.

e'

1085

consequences of the proposed reaction mechanism will agree with future experiments. ACKNOWLEDGMENT This work was supported in part by Deutsche Forschungsgemeinschaft Grant No. Me-470-4/1.

I2See, e.g., W. Hofmann, Jets of Hadron •• Yol. 90 of Springer Tracls in Modern Physics. (Springer, Berlin, 1981) and papers cited therein. I3See, e.g., W. Hofmann (Ref. 12), pp. 25 and 47 and the papers cited therein. 14J. Schwinger, Phys. Rev. ill, 397 (1962); ill, 2425 (1962). ISA. Casher et al., Phys. Rev. D Ill. 732 (1974). 16J. D. Bjorken, in Curren I Induced Reactions, proceedings of the International Summer Institute on Theoretical Particle Physics, Hamburg, 1975, edited by J. G. KOrner, G. K. Kramer, and D. Schildknecht (Springer, Berlin, 1976). 178. Andersson et al., Z. Phys. C 1, 105 (1979). 18See, e.g., D. Fournier, in Proceedings of 'he 1981 Inle,nalional Symposium on Lepton and Pholon Interactions al High Energies, Bonn (Ref. 6) p. 91 and the papers cited therein. 19gee, e.g., Refs. 6, II, and 12 and the papers cited therein. 2OSee, e.g., Ref. 17. 21 In first-order approximation the energy of the elongated bag at a given instant is proportional to its length at that moment. This energy can be considered as the total potential energy of the qq system. We recall: Hadron-spectroscopy strongly suggests that the interaction inside a hadron can be described by a linear potential (the potential energy is directly proportional to the distance) between q and q, provided that they can be considered as a static source and sink of color flux. This is, e.g., the case when the quark-antiquark pairs are heavy (ci:. bli, etc.). Now, in the case of e -e + annihilation processes, since the primary q and q are always on the two ends of the elongated bag while they separate, it is always possible to envisage the existence of an instantaneous static source and a corresponding sink in each bag. Hence we can assume the existence of such linear potentials for all kinds of primary qqpairs. 22S ee, e.g., Refs. 1,5,6, and 10 and the papers cited therein. 23S ee, e.g., Hofmann (Ref. 12), p. 71 and papers cited therein. 24T. T. Chou and C. N. Yang, Phys. Rev. D~. 200S (1971). 25This colorless matter is not chargeless. It consists probably of a large number of sea quarks. Note also that the existence of such processes does not necessarily contradict the underlying picture of the quark-parton model which may be true to the impulse approximation. 26See, e.g., N. Schmitz. in Proceedings of Ihe 1979 Inlernational Symposium on Lepton and Photon Interactions al High Energies, Fermi/ab, edited by T. B. W. Kirk, and H. D. I. Abarbanel (Fermilah, Batavia, Illinois, 1980), p. 259. 27See, e.g., Chen el al. (the last paper of Ref. 8).

329 Co"",,,,". in Theor. Phys. (Beijillg, China)

Vol. ;:, No. 2 (1983)

97l-982

NONLINEAR a-MODEL ON MULTIDIMENSICNAL CURVED SPACE WITH CERTAIN CYLINDRICAL SYMMETRY CHOU Kuang-chao ( f.l1t~ Institute of

Tt.·~oretical

and

)

Physics, Academia Siflica, Beijing, China

SONG Xing-chang (

*tH: )

Institute of Theoretical Physics, Physics Department, Pekinq University Beijinq, China

Received September 20, 1£F2

Abstract It

dual transformation is fount! for a class of nonlinear

l'-

model definlJd on a multidimens':'or.a:' curt-"ad space k!ith cylind~ical symmetry.

T."'le syst-e:r. is ':'nllariar:t :.znc.er a proper

the dual transrormation and the ~n

~~neral

combination

of

coordinate transformation.

infinite number of nonlocal conser"-rltion laws as well as the

Kac-Mood'l alge!>ra follow directly from tile dual transformatior:. A Backlund transformation that cenerates new solutions from a siven one can also be constructed.

I.

Introduction

In recent years considerable progress has been madt' in th" i n\"est igtl t iC'n of the two-dimensional a-model or chiral model[l], which possesses a lot of rather interesting and mutually connected properties like the soliton solutions[2 1• the Backlund transformation[3], the infinite number of conservation laws ,fiJ associated with a hidden symmetry,G-8] and the close similaritr to thE' sE'H-dun] Yan~-Mills theory in four dimensions. Besides, it has also been shown that th~ chiral field equation in three-dimensional cylindrical symmetric case benrs ~ resemblance to the Ernst equation in yeneral relativity. Moreover, in the 50called super-unification theory an important role has been played by the nonlinl;;ur a-model though the four-dimension'al versio.m of which has not been fully investigated yet. Therefore it is meaninll'ful to extend our earlier work on two-dimensional nonlinear a-model to the case of higher dimensions with certain cylindrical symmetry. First let us recall some of the important results on the ordinary twodimensional nonlinear sigma model. These results are formulated in the K-form which has the advantage of uniformity and simplicity and will be used throu~hDut this paper. The Lagrangian for the ordinary two-dimensional non-linear sigma mode1 can be written in one of the following forms[Sl.

,4

~ =-fTr I<,. kJ& = ~ TT Op. cp-l DlLcp

=t Tr

;))l i

ct'J..

(1.ln) (LIb) (J,. .1e)

330 972

CHOU Kuang-chao and SONG Xing-chang

where the physical field ." takes values in the symmetric coset space G/H with its covariant differentiation defined as (1.2)

rent"

In Eq.C1.2) the "composite" gauge potential l\J and the "covariant curKIJ are defined through the following equations:

n)l= H,.+K"=~\L~ H" == Ln,. +If" n". IT")

KJl = :=

(1.3)

t

(1.3a)

(1.3b)

~ (!l)&-f".!l./Lf")

cp"'D....

,=-( D}&cp-I) IJI

(1. 3c) J

whereas 0 is the involution operator with respect to which the symmetric space cola is defined l7 ] (i.e., oho=h, 1fhf H; 0"'0=.,,-1, 1f." E G/H; 0 2 =1). The gauge invariant field q is defined byl7] (1.4 )

q2 = 1 .

The Euler-Lagrange equation of motion reads or or

D"l<,.. =- ;))101(" + [H''",K,,] = D" ~ '-lD"qI) CJf\D)lcp) = if a}L q.-( a"'~> ~ (~<J,) = 0

The compatibility condition for

C)l

nlJ

(1. Sa)

(l.Sb) (1. Sc)

gives for symmetric coset space

H" - 0, H, + U·t. · H,J ... -

~ K~-D,KJL=o

0

\)

[k",~] •

(1.6a) ( 1.6b)

.

These two equations are known as the Gauss-Coddazi equations. In light cone coordinates

.[l~=t -x

(1. 7)

Eqs.(l.Sa) and (1.6) can be written in the fOrm (1.

Sa)

(1.

8b)

Explicitly, the dual transformation

K(-I\i =(1(1

K~ -

K~ = r-IK~

H~-H~=H1

H~-H~=H~

(1.9a) (1.9b)

with y as an arbitrary constant. leaves the Lagrangian and the equations (1.8) invariant. From this transforma~ion an infinite nUlllber of conservation laws, both local and nonlocal. can be deduced[S]. Under the transformation (1.9), the physical field ." and/or q transforms as

91'%) -", 'T. [ep]) ,IX) ii."\ r, [f]) = ep'l r.:x) q.IXl ,,-1, r. (cp) = IJ.', T. Xl where the transformation functional (amplitude) satisfies the linearized

(1.10a) (1.10b)

331 Nonlineat a-MOdel on Multidimensional Curved Space with Certain Cylindrical Symmetry

973

equations.

and

Ju

O~'I'(t.(P]'=+(l"-11'1'(t, (P)'J1 ('1'J

(l.11a)

O~'I'(t, (lP])=+o':!.J)ljIlt, (~)'J~ (f)

(1.11b)

is the Noether current fbr

~,.lobal

G

svmll1E't ry:

J;.[IP] = 2 cP K)I (q>] 'P"=- 2CPD)( cP" = - 'l.~p. 'l.

(1.12)

.

In a recent note[S] we have sho~~ that, by choosing the essential parameter a and the boundary condition for the transfor~ation amplitude ~(y(a), l;(x']) properly, the one-parameter family of the dual transformation (1.£) forms an Abelian group A(l) (1.13)

and

CIjI}

Fives a nonlinear realization of this

~roup

(1.14 ,

..

It can be further shown that[8] when corr.tinin~ the element ot A(1), a(a)/: ,1.(1). with that of the internal syrr.r.letl'Y gEG to forl:l tt.e prodl!ct a-1(a)gafcr.), i t remai~s tc be an element in G. This fact im~lies that the p~ysical system is invariant under the semi-direct product group GIlA(l) with G as the normal subgroup, and from this important feature follows the Kac-A!oody algebra lS ]. In this paper we will discuss the possibility to define a dual-like transformation for the non-linear sigma model in curved space and give a brief account of the results associated with this transformation.

II. Dual-like tranSTormation Consider an N+2-dimensional ClJrvt!d bl-oaCe, brought to the form

th"

m"Lr.C of which can be (2.1)

where r(t,r) Dnd ll(t,r) are functions of the time t and the first space coordinate r, while gaB depends only on xa, a,S=3,4 ••• N+2. The action of the cylindrically ~ymmetric non-linear si~a model can be written as

Cti d"U S= Jr~ p.~ Tr ( ~Jl ~ a~ '). >,..rB X OC fTy(K11<'l)6(i;~>d~d~

(2.2 )

where 121;=t+r and 1211=t-r. The phYSical field ¢(x) is assun;ed to be independent of the last N coordinates xa, a=3,4 ..• :<+2, f".nd KI; and Kn are defined as in Eq.(1.3). The main probleu. we will discuss is: does there exist a hidden symmetry for the model (2.2) in the curved space similar to the one in two-dimensional flat space? The anSW6r is yes if the reduced metric 1l(1;,11) takes a special form, namely, ll(I;,I1)=f(I;)-g(I1). The argument goes as follows.

332 974

CHOU Kuang-chao and SONG XJng-chang

The equatjon of motion given by the action (16) reads

OJ'( K" A) Combined with the

== D1 lK, Ll HoD,' I<~A ) =

inte~rability

0

(2.3)

condition as in Eq.(1.6) it gives

*"K

~ Kt =D~1<1 i (~K, T 1) H1'~ - HH -(H S ,H,J = (1(1 ' K~] zo_

(2.4a) (2.4b)

where we have used the abbreviation A£=dr.'l etc •• Consider a transformation of the for~

H~-H;= H\

H~=Hi K; = )"(1, ~)1<1 fI. ( ~ , ~) il'( 1 , ~ ) Hj K1 -

with

y(;,n)

and

A'(;,n)

1<'1-I<~ = Y-'l~,~) k~

to be determined later.

(2.5)

Under this transformation

Eq.(2.4b) is invariant and Eq.(2.4a) becomes

Substituting Eq.(2.5) into Eq.(2.4a') and comparing the coefficients of and

Kr,

on both sides, we have

2. r~-{ Y~I)

~~ =0= ZY"'lj-'-O"-.!.1J ~1

with the solution (2.6a) (2.6b) and then

~

(:'..7)

= ~t-LL-f(~)-3l~) ,

where HE:> and g(n) are arbitrary functions of £ and n respectively. From Eq.(2.7) we see that the transformation (2.5) can be well defined if and only if

I\«("n)

n, i.e.,

A=f(E)-g(~).

can be decomposed into a sum of separative functions of

and

E:

Moreover it is evident that such a kind of decomposition

is not unique. If, 'for instance·, A-I'=fC nand A_=G(n) is a reasonable decomposition, then A+=f(;)+k and A_=g(n)+k (k const.) is also a possible choice.

Then Eqs.(2.6a,b) give

the function of the parameter

y

as the functional of

feE)

and

g(n)

and

k (2.8)

The relation between

A(F.,n)

and

A'(E,n)

is also determined by Eq.(2.4a') (2.9a) (2.9b)

Substituting the expressions for

y2

and

A

given in Eqs.(2.7) and (2.8) into

333 Nonljnear a-Hodel on HUltjdjmensional Curved Space with Certajn Cyljndrical

Symmetr~

975

(2.9), we obtain by integration (2.10) The integration constant

C

can be determined by the boundary condition

Y-J,

Therefore

t:.'-D.,

c-f··

(2.11)

(2.10' ) Therefore we see that, under the condition (2.7). the dual-like tran~forma tion for the cylindrically symmetric non-linear si~ma mode) is specified by Eqs.(2.S), (2.8) and (2.10') with k-1=tl as the group parameter. tl=O (k"''''') corresponds to the identical transformation y=1, and two transformations with parameters CI. and tl' give another transformation with tl"=tl+tl':

A("~.~l

'"

=f'1'-St~"'-fi~i'=

AC1,')

0('

=f'Ci'-9<" -

A'(i,~)

=f(1)-Si.~1

(2.12)

fiif'+ c(' = fr(,-'+ (ct. +",,',

This gives an Abelian group A(1) with ':l=k- 1 as the group parameter. Besides the group parameter tl, the transformEtion parameter y also depends on the reducec:\ metric l\( f;,,,), as given in Eq. (2. S) • Later on we wi 11 denote it as y_ rlDl ,Ll) (2.13) For the succeeding ope rat ion of

tl

and

tl', we have

(2.14) Then we see from Eq.(2.S) that

K; ==

d) YCd.,Il) K~ = net", Ll) J<.~ , K~ = ,....\0{; A'l K~ = Y'ld.', Lrl r. ...(oI.AII<,= r-'(d.",A1K'l Y"ICl', Ll')

Kj =

n()(~

(2.1Sa) (2.1Sbl

This means that the equation of motion takes a similar form under the operation of the Abelian group A(1). It must be pointed out that the Lagrangian (2.2) is not invariant under the dual-like transformation we have considered above. So the dual-like transforma't:ion itself is r:ot a symmetry of our system. But this shortcoming can. be cured by an appropriate general coordinate transformation which we will discuss in Sec. IV. The Lagrangian (2.2) and the equation of motion remain invariant under the combined operation of these two transformations which form a symmetry group of the physical system under consideration.

334 976

I I I.

CHOU Kuang-chao and SONG Xing-chang

Linearized equation and conserved currents

From Eq.(2.5) we can see irrur.ediately that, just as the case in ordinl'.ry model, the quantity

is a pure gauge, so that it can be expressed as (3.11

with

g

being an element in lji- 1

Therefore the field

G.

Then by setting

g

= lji~, we have

a1J 1jI = ~(K~-KIJH-l.

or

q(x)

undergoes the transformation under Eq.(2.5)

'I' (r, ["IXJ)l 'iIX) ,,,.'( r. ('1'1%)))= ~(r;X)

CPI".() _

and the transformation amplitude

~(Y,r~(x)l)

•

(3.2a) (3.2b)

satisfies the equation

0i1f'lf. ('P IX1)= lIY-I) ",(r, ('Pm) M-s (1jlIX)) , ()~ o/(r, (Cl'IX»))=t(Y":"'1) 'I'(r. (IPIX))) Mit ('PIXl]

(3.3a) (3.3b)

Bqs. (3.2) and (3.3)· have almost the same forms as Eqs. (1.10) and (1.11) except that Y is space dependent as given in Eq.(2.l3). The quantity MIJ in Eq.(3.3) is defined as (3.-1)

which satisfies the curvatureless condition (3.5) The Noether current for the global

r,

symmetry now becomes (3.6)

which is conserved

-a1'J; (
(3.71

as can be verified directly from the equation of motion (2.3). It can be shown by a straightforward deduction that the compatibility condition for Eqs.(3.3) is just the curvature less condition (3.5) for MlJr~l and the conservation equation (3.7) for JIJ[~l. So Eqs.(3.3) are nothing but the linearization system of equations in the inverse scattering problem, with a being the spectral parameter. Similar to Eq.(1.14), from Eqs.(2.l4) and (3.3) we can deduce the relation (3.8)

Again, {1jI} as a whole forms a nonlinear realization of the Abelian group A(l). It has been noticed in Sec. II that after applying the dual-like transformat ion ( 2 • 5 ) , ' -I .1.('~. "-6(1. =YKi ' Kt-KIt ..... Y 1<,

n' Ks-K;

335 Nonliner a-Hodel on Multidimensional Curved Space with Certain Cylindrical Symmetry

the transformed quantities h' and just like Eq.(2.4a) is ~atisfied by rent can be defined by

K~

h

977

satisfy an equation of motion (2.4a') and K~: The transformed Noether cur-

;-; (
=2.A9 K~a"'

(3.9)

where h', K~ and $' are given by Eqs.(2.5),(2.10') and (3.2a). K~r¢'J is the same functional as K~[~l in Eq.(1.3c) with Q rp.placed by $' and is equivalent to K'IJ up to a gauge transformation in the subgroup H, i.e.,

K,('l'l= i I<~ K-' with h taking value in the subgroup H as defined in Eq.(3.2). straightforward to show that J~[$'l is conserved.

(/' J"~

r,,'J=

d1 J"~ ('P']

+ d~Ji ['P'] = 0

Then it is

(3.10)

::ow since J~ro'1 contains ex as an arbitrarv parameter Eq.(3.10) yields an infinite number of conservation laws when it is expanded in powers of the spectral parameter ex.

IV, General coordinate transformation and standard fanTIs Thus far we have only considered the dual-like transformation which is coordinate dependent. It is evident that the sigma ~ode1 in curved space can also undergo general coordinate transformations. It has been pointed out in Sec. II that the dual-like transformation C[1I' be defined if and only if the reduced metric ~(~,n) takes the particular form

or equivalently satisfies the Laplacian equation (4.1 ) It is clear that by applying an appropriate general coordinate transformation, various models can be cast into four standard forms in accordance with the form of A(l;.r.): (i) A=const. We obtain the ordinary two-dimensional sigma model; (ii) h=f(t)pconst. By applyin~ a transformation ~'=f(~) the equation of motion (2.4a) turns out to be

Therefore we obtain a model corresponding to '~'=~'; (iii) A=-g(n)pconst. By using a similar transformation ~'=g(n), a model corresponding to h'=-n' is achieved; (iv) A=f(~)-g(n) with neither f(~) nor g(n) being a constant. In this case, by making use of a transformation ~'-f(~), n'=g(n), we obtain (4.2) This is the cylindrically symmetric sigma model in 2+1--dimensiona1 spacetime.

336 978

CHOU Kuang-chao and SONG Xing-chang

As an application of the above ~eneral coordinate transformation, let us reformulate the inverse scatterin~ equation discussed in t~e last section. Hereafter we assume that the model has been brought to the standard form so that ~(~,n) takes the form

The correspond in. Lax pair has the form (4.3) Introducjn~

a function

as

~

>(~,';l)=JzlJi+i +j~+i

)2

(4.4)

,

then it follows by a simple calculation that

/- t r -r-=T+Y

I

I-tY -2.-=--:s=r

(4.5)

Therefore Eqs,(4.3) become (4.3' ) This is just the pair of equations ~iven by Mikhailov and Yarimecruk l9 ] wrich has been used to construct the soliton-like solutions. It is well known that, besides beinF consIdered as the li~ht cone coordinates of the variables t and r as ~iven in Eq.(1.7), (~,n) can also be used as the complex combination of two Euclidean coordin::.tes, say. z and r

.fl1 =t:c-Y

,

(4.6)

Therefore, the sallie equations in case (iv), can be used equally well to descritle the static axially symmetric model in 3-dimensional space. In this case our form of the linear scatterinF: problelll associated with t.he sirma model, Eq. (4.3), coinsides with that given by Bais and Sasaki l101 . The spectral parameter s given in Ref.IIO) is related to u we used above by the formuh\

cc=iZ.[2S .

(4.7)

For the static axially symmetric model, by 'changing the variables from to (r,z), and introducin~ w=i~ instead of c into Eq.(4.4) we can show that w=c.ocr, ~ ;J1.)=l\.-~ -rj T"- (7I.-rt (4.8) (~,n)

with

(4.9)

Kow it iti easy to find that (4.10)

Passing to the variables

(r,z), Eq.(4.3) can be rewrjtten in the form (4.11)

where

w given by Eq.(4.8) 1s a function of the coordinates

rand

z

and of

337 Nonlinear a-MOdel on Multidimensional Curved Space with Certain Cylindrical Symmetry

979

the spectral parameter ).. The solut ion of Eq. (4.11) may be considered as a functional of M as well as a function of (r,z), i.e., (4.12)

From this point of view the differentiation with respect to the a compound one:

(r,z)

must be (4.13a)

(4.13b) These two differentiation operators have been denoted as Dr (D 2 ) and Dz(D 1 ) respectively by Belinsky and Zakharov in their paper on the gra\'ity field .. equaticn[II]. From our formulation it is straightforward to show the commutath'ity of these two operators:

(Dr' DiI ]

== [d,. , d.} =0

(4.14)

,

and the equivJleRce of the pair of equations[ll]

Dr 'IIr'= T

y J,

+ U) .T.

y'+w'

(4.15)

to the linearzation system of equations we have obtained in the last section. \Veemphasizehere once again that in our formulation, " is a transformation functional which carries one field into another satisfyin~ a similar equation of motion. As another example of utilizat.ion of the general coordinate transformation we give here the Backlund transformation of our sigma model. Suppose we start with a model in the standarc form

and cp(I;,n) is a solution of this model. By using- the dual-like transformation, ~'(I;,n) as defined in Eq.(3.2a) is a solution to the model witr tpe reduced metric ,

~1;(Il

LlrV\) = ~-t

(4.16)

- T-r

It is possible to carry the reduced metric back into dinate transformation (c.t)

Ll'-~=i-~

6

by means of a coor(4.17)

Therefore we obtain another solution (4.18) corresponding to the oriyinal 6=I;-n. This is the Backlund tranbformation we have obtained~ Evidently, it is just the combined transformation

338 980

CHOU Kuang-chao and SONG Xing-chang

which leaves the equation of motion (2.4) (and hence the invariant.

La~rancian

(2.2»

V. Kac-Moody algebra In Sectiol,;; II and III we have shown that by taking- cx=k- l as the parameter, the dual-like transformations as given by Eqs.(2.5), (2.12) and (2.14) form an one parameter Abelian yroup A(l): (5.1) and that by choosin~ the boundary condition properly the set of the transformation functiona1s {w} becomes a nonlinear realization for this ~oup (5.2) From Eq.(5.2) we have VI-OI; CP'IOI.~

.,1) =-V-l(

01;

Under the ac~ion of the internal symmetric group, underyoes transformation

(5.3)

ge

G, the field

¢(~,~)

'.

q>(~,~)Lcp(1,~Jecp(S. (tpl1·\») = 3 "(1, \) lil{S. [IP])

(5.4)

Here ag-ain h(g, reP) is an element taking value in the subgroup H. and depends on both g and eP itself. Then following the same procedure as given in Ref.fS], we have

and we can prove that the product (5.5) is another global element taking value in the group G. This implies that the system we discussed has the symmetry group GJ\A( 1) with the internal €'roup G as the normal subgroup. Then it follOws that if the generators of G (~j being the parameters of G)

I, =-il! J

I

(5.6)

"31j 1=0

satisfy the commutation relation (5.7)

then the operators defined by

. d,9tCl;81 Ij(OI)=-l~ satisfy the Similar relation

J

I 1-0

(5.S)

(5.9)

from which th~ comp3ct form ot a Kac-Moody algebra can bp deduc~d ~J.

339 Nonl~near

VI,

a-Hodel on MUltidimensional Curved Space with Certain Cylindrical symmetry

981

Conclusion

1. For chiral field defined on curved space with a special form of reduced metric 4 as given in Eq.(2.7), the dual-like transformatiop can be defined whict yives rise to a pair of equations known as the linearization system in the inverse scattering problem and reduces to the ordinary dual transformation in the case of flat space. The physical system is invariant under the proper combination of the dual-like transformation and the general coordinate transformation. 2. From this symmetry, the infinite number of nonlocal conservation laws as well as Kac-Moody algebra can be deduced. 3. In the standard form, the mode] describes the cylindrically symmetric or the static axially symmetric c!:iral field. The equation of motion for the present model reveals great similarity to the axially symmetric Yang-Mills theory in four dimenSions and to the Ernst equation in general relativity. The equation obtained can be considered as a generalization (with arbitrary group G) of the Ernst eq~atlon, which is completely integrable. The solutions can be found by the standa~d inverse scattering method as discussed in Ref.[9] and by other generating techniques used in the theory of gravity[12J.

References 1.

K. Pohlmeyer, Comm. Math. Phys., H. Eichenherr, Nucl. Phys., ibid,

2.

~

(1976) 207; M. LUscher and K. Pohlmeyer, Nucl. Phys.

(1978) 46; V.E. Zakharov and A.V. Hikhailov, (Sov. Phys.) JETP 47 (1978) 1071;

~

~

~

(1978) 215;

~

(1979) 544; H. Eichenherr and H. Forger,

(1979) 381; V.L. Gala, and A.M. Perelomov, Phys. Lett., B79 (1978) 112.

A.A. Belavin and A.M. Polyakov, (Sov. Phys.) JETP [.ett.,

~

(1975) 245; A. d'Adda,

M. Luscher and P. di Vecchia, Nuc1. Phys., ~ (1978) 63; A. LUscher, Phys. Lett.,

!!!.!

(1978) 465; E. Witten, Nucl. Phys.,

~

(19791 285; D. Gross, Nuc1. Phys.,

(1978) 439; F. Gursey and X.C. Tze, Ann. of Phys., Phys. Lett., 3.

(1980) 415.

~

(1978) 2528; M.K. Prasad et al., Phys. Rev. Lett.,

Math. Phys.,

Z!

~

(1979j 750; K. pohlmeyer, COIIIIII.

(1980) 317.

E. Brezin, et al., Phys. Lett., B82 (1978) 422; H.J. de Vdga, ibi?, B87- (1979) 233; A.T. Ogielski, Phys. Rev.,

~

(1980) 406; I.V. Cherednik, Theor. Math. Phys.,

210; M.K. Prasad, et al., Phys. Lett.,

~

5.

K.C. Chou and X.C. Song, Seientia Siniea, 25A (1982) 716; 825. L. Dolan and A. Roos, Phys. Rev.,

~

(1980); B.Y. Hou et al., Phys. Rev.,

!!!!

(1981) 2238; L. Dolan, Phys. Rev. Lett., £(1981)

~

(1981) 411; C. Devchand and D.B. Fairlie,

(1982), 232; L.L. Chau et a1., Phys. Rev., D25 (1982) 1080; 1086;

K. Ueno and Y. Nakamura RIMS-375; 376 (1981). B.Y. HOU, AS-ITP-81-016.

8_

K.C. Chou and X.C. Song, to be published in Co""",·r:. in Theor. Phys.

9.

A.V. Mikhai10v and A.I. Yarimchuk, Nuel. Phys.

10.

(1979)

(1980), 2018; B.Y. Hou, Yale preprint YRP 80-92

~

1371; M.L. Ge and Y.S. Wu, Phys. Lett., Nuel. Phys.,

2!

(1979) 237.

6.

7.

!!!:E.

(1980) 29; B.Y. Hou, and P. Wang,

E. Corrigan et al., COIIIIII. Math. Phys·., 58 (1978) 223; Y •. Brihaye et al., Jour, Hath. Phys., ~

4.

~

F.A. Bais and R. Sasaki, Nuel. Phys.,

~

~

(1982) 508.

(1982) 522.

340 CHOU Kuang-chao and SONG Xing-cMng

982

11.

V.E. Zahkarovand V.A. Belinskg, (Sov. Phgs.) JETP 50 (1980) 1.

12.'

W. Kinnersleg, J. Hath. Phgs. 18 (1977), 1529; W. Kinnersleg and D.H. Chitre, ibid,

l! ~

(1977) 1538;

~

(1978). 162;

!£.

(1978) 1926;

~

(1978) 2017; I. Hauser and F.J. E!nst, Phgs. Rev.

(1979) 978; J. Math. Phgs.,

E:..

(1980), 1126;

E:..

(1980) 1418; V.E.

Zakharo,v and A.V. Shabat, Funct. Anal. Its Appl., 13 (1979) 13; R. Sasaki, Phgs. Lett., 73A (1979) 77; Proc. Rog. Soc. Lond., ~ (1980) 373. R. Sasaki and R.K. Bullough, ibid., A376 (l981) 401.

~ added ~ proof:

The similarity of the stationary, axially symmetric

Einstein equations to the non~inear sigma models has been discussed by D. liaison (J. Math. Phys., 20 (1979) 871.),

The results are somewhat

similar to parts of our discussion~ ~iven in Sec. IV.

The authors would

like to express their thanks to Prof. B.Y. Fou for calling their artention to this paper.

341 Cammun. in Theor. Phgs. (Beijing, China)

Vol.2, No.5 (1983)

1391-1398

KAC-MOODY ALGEBRA FOR TWO DIMENSIONAL PRINCIPAL CHIRPL MODELS CEOU Kuang-chao (

f,j;lt;il )

Institute of Theoretical Phgsics, Academia Sini.7OJ, Beijing, China

and SONG Xing-chang (

*-ft* )

Institute Of Theoretical Phgsics, Peking University, Beijing, China

Received July 26, lS82 t

Abstract

A Darboux cransformation depending on single continuous parameter

t

is constructed for a principal chiral field.

The

transforuation forms a nonlinear representation of the group for any fixed value of

t.

Part of the kernel in the

~iemann

Hilbert transform is shown to be related to the Darboux transformation with its generators forming a I(aC-Moody·algebra. Conserved currents associated with the Kac-Moodg alqebra (If the linearized equations and the Ncether current for the group transformations with fixed value of

I.

t

are obtained.

Introcuction

The technolol!'y of constructinl!' Kac-):oody all!'ebra in a two dimension:>l principal chiral model [1-8] !Jas developed rapidly ovpr the past few Yl.!~rs. Some beautiful mathematical results and a great deal of insight into the nature of the hidden symmetry inherent in two dimensional integrable systems have been achieved. In spite of all these developments, the conserved currents obtained in Refs. (1-8] consist of only half of the Kac-Moody algebra. T~e older derivation [1-6] uses two different values of the parameters 1 and 1', which makes the whole procedure too cumbersome. In papers [7-8] the regular Riemann-Hilbert (R-H) tran",formaticns were used to "demonstrate the: algebraiC structure of the linearized equations in a much simpler and more elegant way. However, no relation between the transformation of the 1 inenl'izerl equations and that of the basic field was given in Refs. [7-S]. In the present note a Darboux transformation is constructed that forms a nonlinear representati.on of the chiral group containintr .a continuous parall'eter t. The R-H kernel H(t) is assumed to be analytic inside an annular regicn a~ltl~b in the complex plane" and expanded into a Laurent series. That part of the Laurent expansion analytiC at t=±l is shown to be relaT~d to the Darboux transformation whose generators form a I<'ac.~lllody a 1gebra. The transformation of the basic field is obtained with the l'ame function a.t a parti('u1:11' value of t. The conserved current associ9ted with the Kac-Moody algebra of the linearized equations and ti'e ~r5ether current for th.e group transforrr.ations t

Revised Hay 17, 1983.

342 l392

CHOU Kuang-chao and SONG Xing-chang

with a fixed value of

t

are obtained.

II. Principal chiral field and its linearized equations The basic field q(x} of the principal chiral model is a function taking values on a simple compact group G. The current defined as

is both curvatureless and conserved

jll=q-1allq

(2.I)

alljv-avj\l+[jll,jvJ=O,

(2.2)

a;!.:ill=o.

(2.3)

In the light-cone coordinates 12f;=t+x,

I2n=t-x.

Eqs.(2.2} and (2.3) can be written in the form af;jn-~nj~+ [jf;,jn]~O,

a~jn+anj~ ..O.

It is very convenient to work with the dual current (2.4)

and the I-form for bdth

jll

and

.ill

j=jlldxll=j~df;+ jndn,

(2.5)

3=Jlldxll=-j~dE;+ .lndn,

in the following where the curvaturelessness and conservation of rewritten as dj+jAj=O, d3=o. ~he

jll

can be (2.2' ) (2.3' )

Lax pair for the model takes the form dl/l (t , j )...p (t , j )0 (t , j ) ,

(2.6)

where (2.7)

wjth the compactibility condition O=d 21/1=1/1 (d!'l+OAO)

=,,,,. {-

t 2 (d"+" "). t 2 d": } I_t2 J JAJ + I-t2 J ,

(2.8)

which implies both the curvaturelessness and the conservation of the current j as given in Eqs.(2.2') and (2.3').

343 Kac-Moodg Algebra for

~

Dimensional Principal Chiral MOdels

lJ93

III, Darboux transformations Let lP(t,j) ampli tude

be a function valued in the group

G

such that the new

lP(t,jjX):: IjI(t,j)X(t,j) satisfies the same equation as

lP(t,j)

(3.1)

does, i.e.,

dljl ( t ,j jX ) = II' ( t , j j X)G ( t , j j X ). Then the new potential old one

Q(t,jjX)

(3.2)

must be a "gauge transformed form" of the

n(t,j) n (t ,j jX ) = X- 1 (t ,j ) dX (t ,j )+X -1 (t , j )12 (t ,j )X ( t ,j) .

If the

t

dependence of Q

n(t,jjX'>

is of the same form as

(t ,j jX) =-1:t j E; (j jX ) dE; + 1; t,i'1 (j jX

(3.3)

!l(t,j). Le.,

)d~,

(2.4)

the new current

(3.5) The relation (3.1) is often called

will be both curvatureless and conserved. Darboux transformation.

IV. Riemann-Hilbert transformations Solution

lP(t,j)

of the linearized equatjons (2.6) c:ln in genE'ral have

singularities representing solitons in the complex t-plane other than To find the desired transformation function with a fundamental solution analytic in an

annul~r

lP(t,j)

xCt,j)

to the linearized

region in the complex t-plane.

t=±l.

in Eq.C3.1), I.e start s~'stem

(2.6) which if.;

In the following it is

useful to construct a function H(t,j,u(t» =r1(t,j)u(t)W(t,j) with

u(t)

(4.1)

a group element independent of space-time and analytic in the same

annular region in the complex t-plane.

For an infinitl"sirr.al group transforma-

tion u(t) =1+w(t), we have

p. ( t ,,j , u (t ) )

(4.2)

= ~.. ,i , II Ct

))•

h ( t , j , II" ( t ) ) = 1/1-1 ( t , j )11 ( t ) If; ( t ,j ) • By assuming the annular region in which

as I t I~b

h(t.,j,w(t»

is analytic to be

with O
it is possible to expand

(4.4)

h(t,j,w(t»

h(t"i,w(t» = =

as a Laurent series

!. h_n(j,w)t- n + I hnC,i,wlt n n-O

where

(4.3)

n-l

ii_ (t ,j ,w(t) )+h+ (t ,j ,w(t»,

(4.5)

344 1394

CHOU Kuang-chao and SONG Xinq-chang

...

h_(t,j,w) = ! l'_n(j,w)C n

(-4.6)

n=O

is analytic in the region Itl~a, including the pOints t=±l. Both the kernal H(t,j,u) and h(t,j,w) satisfy an equation cf the form (4.7)

dH(t,j,u) = [H(t,j,u), l'2(t,j)]. I f follows that

di(t,j,u) = [H(t,j,u), IHt,j)]_ ,

(4.7')

w~ere

[H,I'2]_ denotes that part of the Laurent expansion with zero or negative powers of t. Since l'2(t,j) given in Eq.(2.7) is analytic in the region Itl
=

r t n +1

~=O

r

(J'

d<"+(_1)n+1 J. dn) E: .. " .,

t n + 1 n (n+1 ~

(4.8)

n-O

In the region where both

h(t,j,w)

and

l'2(t,j)

are analytic we have

dh_(t,j,w)+[I'2(t,j), h_(t,j,w») =- [U(t,j), h(t,j,w)]~+[r.(t,j), ii (t,j,w)]

=; £ t n+1n=O 111=0 =-

i r

III

rl'2(n+l)

'

h

] -III

(

4. 9)

t n + 1 [j dE:+ C_1)n+l-m j d", h_",]

E:

n=O/ll=O

"

=-1~t[jE:dE:, h_C1,j,w(1»] + l!t[j"d", h_C-l,j,w(-l»]. Eq.C4.9) clearly indicates that the

t

dependence of the quantity

is the same as that of l'2(t,j). We can therefore use this fact to build an infinitesimal tr"ansformation X. To f1rst order in wet) we put x(t,j,u(t» = 1+h_Ct,j,w(t». It is easy to Cind from

Eq.(3.~)

(4.10)

that

Q(t,j,u(t» =1'2(t,j)+dh_(t,j,w)+ [1'2(t,j), n_(t,j,w)] (4.11) =- 1:t j E:(j,U(t»dE:+ i;tj"(j,U(t»d,, with jE:<.i,u) =j;+ IjE;' n_(1,.j,w(l»), j,,(j,u) = jrt fj", h_(-1,j,w(-1».

(4.12)

As n(t,j,u(t» depends on u(t), we have modified our notation adopted in Sec.II from Q(t,j,X> to Q(t,j,u) and from j(j;X) to j(j,u). With the help of Eqs. (4.10) and (4.11) i t is s-traitforward to verify that the transformed amplitudes given in Eq.(3.1) is a solution of the linerized equations (3.2) with j(u) in Eq. (4.12) both curvatureless and conserved.

345 Kac-Moody Algebra for

rwo Dimensional Principal Cbjral Models

1395

V. Group properties of the transformations ap~lying

Under successive transformations by

Uo

first and

u l ' next

$ (t ,j) -~1/1 (t, j, Uo ) = $ ( t ,j)X (t ,j ,u o ),

1/1 ( t, j , u.) ~"'1/1 ( t , j ( u 0

U1

) ,

)

(5.1)

= ljI( t , j , u 0 ) X( t , j ( u 0 ) , \: 1 ) =1/I(t,j)X(l,j,1I 0 )X(t"i(u o)'u l ), (5.2)

a nonlinear representation of the group will be formed if the condition X (t,j,u1U O ) =X (t,j,uu)X (t,j(u o ) ,u 1

is satisfied.

x's

(5.3)

)

For infinitesimal transformations

are given by Eq.(4.10) and the condition (5.3) turns out to be a commuta-

tion relation x(t ,j.,u,)X (t ,j (u,) ,u l )-X (t, ,1 ,ul)X (t ,j (u l ), u o )

(5.3' )

=i_(t,j, ~1,wa])' From infinitesimal transformation Eq.(4.3) it is easily found that h (t, j ,wl ) h( t ,j ,wo )

and

= h( t, j ,w l Wo ) ,

(5.4)

h( t ,j (u o ) , W1) = $-1 (t , j (U o »"'1 (t )1/1 (t, j (u o ) ) = X-1 (t ,.1 ,u 0 )$-1 (t ,j )w 1 (t)1/I (t ,j ) X(t ,j , Uu ) = X-I Ct ,j, II. ) h( t,.1 ,WI )X (t ,j

(5.5)

'U o )

=h(t,j,Wl)+Ih(t,j,w l ), h_(t,j,W o )]' After success.ive transformations of and

WOWI

Uo

and

u l ' we obtain to order

w~,

WI

the following XCt,j,UO)XCt,j(u.),u l ) = [1+ ii_ ( t , j , Wo ) ] [1+ ~_ ( t ,j ,W 1

) ]

=1+ ii _( t , j , Wa )+ ii_( t , j , W1)+ IN t , j , WI)'

ii_ (t ,j ,W a ) ]

(5.6a)

+h_(t,j,wo)h_(t,j,Wl)' and

X(t,j,Ul)X(t,j(U 1 ),U O) =1+ h_ (t ,j 'W l )+ii_(t,j ,w,)+ [be t, j ,w.),

ii_ (t, j, WI)]

(5.6b)

+ii_Ct "j,w l »)l_Ct, j,w.). The commutation relation that guarantees the group properties is

ii _ (t ,j ,

[w I

, W

01 ) = [h _(t , j

,w ~ ),

ii_(t "i ,w I ) 1

+ [h(t,j,w l ), ii_(t,j,w o )]_-Ih(t"j,w.), h_Ct,j,wl)l_, which can be easily proved as follows: h_(t,j, [WI'W·,]) = [h(t,j,w l ), h(t,j,w o )]_ - ~_(t,j,W1)' ii_(t,j,w u )] +h+ Ct,j,Wl)' h_Ct,j,wu>C+ [ii_(t,j ,WI

>,

h+(t,j,w o )]_

(5.3")

346 1396

CHOU Kuang-chao and SONU Xing-chang

=--lh_Ct,j,w 1 ), ii_(t,j,w o )] +[h1t,j,w 1 >, h_(t,j,w o )]_+[h_Ct,.i,w 1 >, hCt,j,w c )]_ Therefore the infinitesimal transformations form a Lie algebra. explicit let us take

To be more (5.7)

with

Ia

the generators of the group

G and form the co~mutator (5.8)

where k and 1 are positive or negative integers. With this choice of wet) it is easily seen that our transformation generates an algebra isomor~hic to the Kac-Moody algebra GxC(t- 1 ,t). After this work was finished, we learned that the authors of Refs. [7] and [8] have given similar proof to the same problem. We decide to publish our res~lts because of the siwplicity of our approach. In our proof neither a~xi liary quantities such as G(R.',R.)=

r

G(m,n)R.,mR. n =R.':2.{R.'_2.

(.1/.,)-1

p.)}

m,n=O

nor transformations corresponding to different values of

t

are needed.

VI. Transformation of the basic field Putting

t~m

in Eq.(2.7) we obtain immediately Q(",j)=j.

(6.1)

Substituting Eq.(6.1) into Eq.(2.6) and using Eq.(2.1) we find (6.2) with v a constant element of the group G. Therefore the corresponding transformation for the field q(x) is nothing but x(m,j,u), i.e., q ~ qx(m,j,u):: qX(j,u).

(6.3)

Furthermore, in the limit t+m, the gauge transformation (3.3) for the potential Q(t,j) reduces to the one for the current j j (j

,u)=X- 1 (j ,u)dX (j ,u)+x -1 (j ,U)jX (j, u).

(6.4)

According to Eq.(5.3) the function X(t,j,u) forms a nOlllinear representation of the group G for any fixed value of- t, in particular, for t=m. So the same is true of the transformation on the basic field q(x) (6.5) By virtue

of Eqs. (4.10) and (4.5). only the zero power term in the

347 Kac-Moody Algebra for Two Dimensional Principal Chiral Models

expansion of

b(t,j,w)

appears in

1397

X(j,u)=X(~,j,u)

X (j,u)=t+h. (,j,w).

(6.6)

For particular choice with

w(u)=wala kept fixed and k integers, we change the notation from to h(t,j,k) etc. and denote h(t,j,k=O) by A(t,j)

h(t,j,~)

(6.7)

Expanding

A(t,j)

into a Laurent series +~

A(t,j)=

r

n An(j)t ,

(6.8)

we get h(t,j ,k)=I.

.\n(j )t,,+k.

(Ci. 8' )

n=-cm

from which it follows that

x(j ,0) =1+ h. (j • 0) =1+ A• ( j ) • x(j ,k)=l+h D (j ,k )=l+A_k (j).

(6.9) (6.9' )

Tberefore for any fixed value of integer k the transformation X(j,k) on the field q(x) is generated by only one term A_k(j) in the expansion of A(t,j). Multiplying t-:< on A_k(j) and sun.ming over k, we see that A(t,j) itself generates a t-dependent transformation on the field q(x), i.e., (6.10)

q -... q+qA(t,j).

This is nothing but the transformation considered in Refs. [2], [5] and [7], from which the hidden symmetry is extracted. These authors have c~osen a particular class of fundamental solutions such that W(t,j) is analytic in a circle with the center at the origin t=o. In t~is case the negative powers disappear in the expansion of A(t,j). Fence in their treatmen! the transformation corresponding to k-positive integer is trivial and the nontrivial algebra associated with its transformation consists of only half of the Kac-Moody algebra.

VII. Sorre comments 1.

If the analytic region of l
I t I ~b<~,

h(t,j,w(t»

is (4.4' )

an inverse transformation that changes t to lit can be performed and a similar analysis leading to Kac-Moody algebra can be carried out without any difficul ty • 2. On the classical level no central charges appear in the Kac-Moody algebra obtained so far. It is of great interest to look for them in the quantum version of the chiral model.

348 1398

CHOU Kuang-cbaa and SONG Xing-cbang

3. For given k the infinitesimal variation of the current Eq.(4.12) can be further put into the form OJ F;(k)= [j F;' h_ (1, j ,k)] =aE:A_k(j)+ [jF;' A_ k (j)]=aF;\-k+l(j),

(7.1a)

Ojn(k)= [jn' ii_(-l,j,k)] =anA_k (j)+ b , A_k(j )]=-a A_ k +l (j) n n

(7.1b)

from which the conservation of the transformed current j~+cju(k) is obvious. On the other hand, the N~ether current corresponding to the transformation (6.10) has the form[9] JF;(t) = ~~~ ~'(t,j)jtli'-l(t,j),

(7.2a)

In(t) = i.;~ Ij/(t,j)jnlj/-l (t,j).

(7.2b)

Multplying with the generator wal a and taking trace we obtain from Eqs.(7.2a) and (7 .2b) the k-th order current b~' expanding into power series in t J

J

t) =Tl {jF; (-h.(j ,-k)+2ii_ (1, j ,-k»}, (k)

n

(7.3a)

=Tr{ j (-h 0 (j, -k)+ 2h (-1, j, -k»}. n -

(7.3b)

The relationship between the .two sets of conserved currents, further implications of the transformation acting upon the fields and the intp.gral form of our transformations will be discussed in a seperate article [10].

References ~an

and A. Ross, Pbys. Rev., ~ (1980) 2018.

1.

L.

2.

B.f. HOV, fale preprint 80-29; B.Y. ROU, M.L. GE and Y.S. WU, Phys. Rev., D24 (1981) 2238.

3.

L. Dolan, Pbys. Rev. Lett.,

£.

(1981) 371.

!!.!!

4.

C. Devcband and D.B. Fairlie, Nucl. Phys.,

S.

L.L. CHAU, Y.S. WU, B.Y. HOV and M.L. GE, Scientia Sinica,

6.

K. Uena, RIMS preprint 374 (1981).

7.

K. Uena and f. Nakamura, Pbys. Lett.,

8.

Y.S. WU, CPT-82 p.1423 (1982).

9.

(1~82)

~

(1982) 907.

208.

Set>, for example, K.C. CHOU and X.C. SONG, Sci;mtia Sinica, .A2S (1982) 716; B.f. HOU, AS~ITP-81-016

10.

!!2!

(1982) 232.

(1981).

K.C. CHOU and X.C. SONG. in preparativn.

349 5 January 1984

PHYSICS LETTERS

Volume 134B. number 1,2

ON THE GAUGE lNV ARlANCE AND ANOMALY -FREE CONI)ITlON OF THE WESS-ZUMINO-WITTEN EFFECTIVE ACTION CHOU Kuang-chao, GUO Han-ying, WU Ke Institute of Theoretical Physics, Academia Sinica. PO Box 2735. Beijing, China

and SONG Xing-chang Institute of Theoretical Physics. Peking University. Beijing. China ReceiVed 17 August 1983 Revised manuscript received 3 October 1983

A global anomaly-free condition and a non-abelian gauge invariant Wess-Zumino-Witten effectiv; action with less terms have been found by a systematical method rather than by trial and enOl". The condition requires the difference between the left- and right-handed Chern-Simon five-fonns wrt the gauge group must vanish and it turns out to be the usual condition in the local sense.

A few months ago, Witten (1} has proposed a new framework to show the global aspects of the WessZumino chiral effective action [2]. In addition to Witten's intriguing results, we would like to point out that in order to gauge an arbitrary subgroup H <; SU(3)L X SU(3)R' the global symmetry of the action, a global anomaly-free condition should be satisfied by the gauged group H. This condition requires that the left-handed and right-handed Chern-Simon five-forms associated with the group H must be equal to each other and it turns out in the local sense to be the usual anomaly-free condition presented by the perturbative computations at the quark level. Instead of the t'fial and error Noether method we would also propose a systematical way to gauging the global symmetries of the action. In the case of non-abelian symmetries, a gauge invariant effective action with less terms is found. Although the difference between Witten's action and ours concerns only with some rare processes, it would be quite interesting to find in principle a way to discriminate them. Let us start from the ungauged Wess-ZuminoWitten effective action I

*

r(u):= _ _ 1_ jdT.iiklm 24071 2 Q

X tr(U-la;U u-Iap U-1aku u-IaiU U-1a m U). (2) Under the transfonnation of the gauge group HL X HR

<; SU(3k X SU(3)R. the field U(x), being an element of SU(3). transfonns as

U(x) ..... U'(x) = L(x)U(x)R-I(x). L (R) E HL(R) . (3) Our purpose is to search for an action

F;' I(A/l' U) = -}6

J d x tr D/lU D/lU4

S4

I

+nr(A/l' U). (4)

such that it is gauge invariant under certain physical *1 We take the notations and conventions in ref. [I]. except those explained in the present letter.

0.031-9163/84/$ 03.00 © Elsevier Science Publishers B.v. (North-Holland Physics Publishing Division)

67

350 Volume 134B, number 1,2

PHYSICS LETTERS

conditions. In the expression (4), DV is the covariant derivative of V defined as

5 January 1984

If the condition (6) is not obeyed, the variation of

r under the gauge transformation (3) becomes

DV=dV+ALV-VAR' AL(R}=A~R}TL(R}' (5)

r

and the term consists of £,(V) together with the least amount of other terms defined on the compactified spacetime M - S4. The results obtained are the following: Under such necessary and sufficient condition that the left-handed Chern-Simon five-form nL and the right-handed one OR are equal to each other

X

tr[ApL o.,A",LL --1 0/lL + ~A"LAvLAaLL -13/lL

+ ~ApLA"LL -13aLL--lol+~A"LL-l a"LAaLL -la~

- ~A"LL-la"LL-laaLL-la~-(L-+R)].

(10)

where the coefficient of the Chern-Simon five-form is defined by

where r(L) and r(R) are the same as r(U) defined by (2) as long as V is replaced by Land R respectively. In the case of the gauge transformation being infinitesimal

lIjjklm ,. -(247T 2)-1

L = 1 + €L' R == 1 + fR'

(6)

X tr(FjjFklAm - ~FjjAkAIAm + hAjAjAkAIAm). (7)

fL(R} = fL(R}TL(R) ,(11)

the variation ~r becomes an infinitesimal one as well

the expression

rCA

"

,V) =£,(V) + _1_ Jd 4x €"va/lw 487T 2 "va/l

(8)

is gauge invariant under the transformation (3) and then 1 defined by (4) with expression (8) is the gauge invariant effective action, where

+A"L3.,AaL V/lL) +(L-+R)] +3pAvLVA aRV-lV/l L+V3pAvRV-I AaL V/lL

-! [A"LVvLA",LV/lL -(L-+R)] +A"L VAvR V-1VaL VIlL -VA"R V-1 AvL VaL VIlL -A"L 3ALVA/lRV-1-3"A vL AaL VA/l R V-I

X tr{€L [opA"L a..AilL + ~oiA"LAaLAIlL)]-(L-+R)}. (12) which is just Witten's result in agreement with the perturbative computations at the quark level of the anomalous variation of the effective action under an infinitesimal gauge transformation. This correspondence shows that the condition (6) is in fact a global extension of the usual perturbative anomaly-free condition. Furthermore, in comparison with Witten's gauge invariant effective action in the non-abelian case, the action 1 presented here contains less terms than Witten's. It is easily shown that the difference between the two actions is invariant under the gauge transformation (3). In the case of the gauge group H = U(l), the electromagnetic group, the action] is the same as Witten's, i.e. we have

+ApR O.,AaR V-IAIlLV+OpA"RAaR V-1A IlL V +A"LVAvRV-1AaLVIlL +A"RV-IA"LVA",RV/lR

X tr{-ApQ(VvLVatU/lL +V"RU",RV/lR)

+ [A"LA"LA",LV/lLt{L-+R)J -A "LA "LA OIL VA IlR V-I

+ 2F""A a (Q2U/l L +Q2VIlR +QVQV-1U13d}. (13)

+A"RA"RA",R V-IA/lLU- ~A"L VA"RV-lA",LUA/lRV-l}. (9) 6~

We now illustrate how to reach the results. At first, along the standard road from global to local symmetries, one can easily write down a gauge in-

351 Volume 134B, number 1,2

r

variant generalization of the effective action (I) F2

j

= - I~

f d x tr DjJV DjJ V-·l + nr(A i , V). 4

(14)

r

X tr(V-l DiV V-lDP V-IDkV V-I DzU V-I DmV),

(15) which follows straightforwardly from replacing derivatives of V by covariant derivatives of V, dV ~ DV on both the spacetime M - S4 and the disc Q. Although I is obviously gauge invariant, it cannot be used as a physical gauge invariant effective action, because it can not be reduced to one defined on the physical spacetime M- S4 = aQ. A deeper observation, however, shows that the I" term can be decomposed into two parts r(A j, V) = r(AjJ' V)

Ilklm

Q

-n~

Ilklm

The first part of consists of the expression of I" and the integral of the difference between the left- and the right-handed Chern-Simon five-forms (in the abelian case, the Chern-Simon five-form vanishes automatical· Iy) ami the second part itself is gauge invariant on Q. Thus, one could pick up the first part of to define a physical gauge invariant effective action as is shown in the expressions (4) and (8) if and only if the difference between the two Chern-Simon five-forms vanishes which is just the necessary and sufficient global anomaly-free condition (6). The form ofr is, in general, not unique. Arbitrary addition of gauge invariant terms dermed in M - S4 will change r. OUf construction ofr is unique in the sense that after introducing the minimum coupling on r we have made a maximum subtraction of gauge invariant terms on Q. The resulting I" thus found contains less terms than that of Witten's.

r

r(A., V):= - _1- J dl: ijklm I 2407r2 Q

+ Jd"T.jjklm(n~

5 January 1984

PHYSICS LETTERS

) __1- jd"T.jjklm 48 2 W

We thank the referee for drawing our attention to a preprint by B. Zumino. Y.S. Wu and A. Zee. They independently discovered the relation between the Chern-Simon secondary class and the gauge anomaly.

Q

X tr(Fl:D vcrlnucrlD Vcrl_2F~.F.LD ucr 1

References

-F~D UF.R crl+F~U-lD UU- 1D UU-1D U 1/ k 1m 1/ kim

[1] E. Witten, Global aspects of current algebra, Princeton preprint (1983). [2] J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95.

// k

-1

m

1/ kl m

-2F~F%iU-lDmU-F~crIDkucrlF[~U). (16) A similar expression holds in the abelian case, fem(Aj,u)=rem(AjJ'U) -

~ J d"T. i/ klm 48w Q

X tr [FjjQ(DkUU-lDIUU-l DmUU-l

+ U-l DkUU- 1DI UU-l Dm U) -2FjjFk/(Q2DmUU-l +Q2U-lDm V +QUQU-IDmUU-l)! .

(\7)

352 COlllllun. in

Theo~

Phys. (Beijing, China)

Vol.3, No.1 {1984}

73-80

ON WITTEN'S EFFECTIVE LAGRANGIAN FOR CHIRAL FIELD CHOU Kuang-chao( JH';1i ), GUO Han-ying( ~iX.~ and WU KeC:~ if) Institute of Theoretical Physics, Academia Sinica P. O. Bar 2735, Beijing, China

and SONG Xing-chang(

*fH: )

Institute of Theoretical Physics, Peking university, Beijing, China

Received October 4, 1983

Abstract Wess and Zumino's effective Lagrangian for the chiral field is derived by a new method whiCh shows the direct relationship between the Lagrangian and the difference of the left handed and right handed Chern-Simons topological invariants. The resulting Lagrangian is explicitly left and right anti symmetric and equal to the one obtained in a previous letter by a different method.

In a previous letter[l] the effective Lagrangian for the chiral fields proposed f,irstly by Wess and Zumino[2] and then by Witten[3] was shown to be connected with the Chern-Simons topological invariants in higher dimensions. The effective Lagrangian constructed in Ref.[l] has less terms than that of Witten's after correcting some misprints and errors in Ref.[3]. In the present work we shall derive the same effective Lagrangian by a new method which shows direct connection with the Chern-Simons topological invariants. The result obtained coincides with the one· given in Ref.[l]. The advantage of the present method consists in that the Lagrangian is written explicitly in a left-right antisymmetric form through the introduction of a matrix Ys. This is precisely why our Lagrangian contains less terms than that of Witten's because in the latter there are left-right symmetrical terms. Since Witten's formula in its original form is not correct, we shall give a corrected one in the Appendix. Before constrUcting our Lagrangian we shall briefly review some known facts about the principal chiral model and the Chern-Simons topological invariants. The element of the principal chiral group G ® G will be denoted by

353 CHOU Kuang-chao,GrJO Han-ying, WU Ke and SONG Xing-chang

74

(1)

The principal chiral field U(x) is an element belonging to G, which transforms under g in the following torm (2)

It is more convenient to define the chiral field as

Q(x)"

Uo(X») 0 ( U-1(x)

(3)

which forms a regular representation of the chiral group Q(X)---+Q'(X)=gQ(X)g-l

)

(4)

Moreover, it is idempotent. i.e ••

or ( 5)

Q-l(X)=Q(X).

Let

be the gauge potential of the chiral group where ~a a=1, •.•• 2N are generators of the group. The gauge potential Au transforms under a local transformation gex) as (6)

the

Using the chir'al field Q(x) it is possible to define a new gauge field in fo~lowlng way AO(x)=Q(x)(A (x)+a,,)Q(x) 1I

•

\.I

..

354 On Witten's Effecti¥e Lagrangian for Chiral Field

75

(7)

which is the local involution or the local space reflection of the. original gauge field A~, and transforms under local gauge transformation just as A~ does in Eq. (6). It is easily verified that the field strength of the involutionary gauge field (8)

Now we define a 5-form (9)

where

o -1

),

(10)

and (11)

(12)

The 5-form ITs(A) is 2~ times the difference of the left handed and right handed Chern-Simons 5-forms. The exterior differential of ~5(A) is 2~ times the difference of the left handed and right handed third Chern class (13) O

A similar 5-form ITs(Ao~ for the involutionary gauge field A can also be defined as (14)

With

~he

help of the relation

355 CHOU Kuang-chao,GVO Han-ying. WU Ke and SONG Xing-~hang

76

Q(x)YsQ(x)=-ys and Eq.(8) we obtain

(15)

(16)

for arbitrary gauge field A defined on the chiral group. Therefore rrs(A)+us(AO) is a closed form and in the homology trivial region it can be locally written as an exact form (17)

Next we study the transformation properties of the Chern-Simons invariants. Under a gauge transformation(6) we can write

topologic~l

(18)

where (19)

By a straightforward

calculation we get (20)

where (21)

and

(22)

Therefore O(ITS(A)-ITs(AO)=or_ s =d(t~(A,X)-~~(Aa,X).

It is possible to find a 4-form

t~

(23)

(A ,AO) such that unde.r gauge transformation

(18) (24)

356 On Witten's Effective Lagrangian for Chiral Field

77

If such a ~~(A,Aa) exists,

r_s=rrs(A)-rrs(Aa)+d~~

(25)

will be gauge invariant up to a to~al divergence. The form ~~(A,Aa) can be easily found with the result

(26)

where (27) (28)

Now we define an effective Lagrangian

rs

on the 5-dimensional disc Q with

the 4-dimensional space-time as its boundary

1

a

~[~~(A,A )-w~(A,A

a

)] (29)

rs is a closed form and so the effective Lagrangian in the orrtinary space-time is an integer times (30)

where "." is the dual operator. As noted independently by both the authors[l] and Zumino et al.,[4] under an infinitesimal gauge transformation (31) where G is the standard anomaly given by Bardeen ["5] and others [6] (32)

Using Eqs.(29) and (31) we find

~-Tr{Ys(GAX)}+total

divergence.

(33)

357

78

CHOU Kuang-chao,GVO Han-ying, NU Ke and SONG Xing-chang

It is now clear that fs multiplied by some integer can be chosen as the effective Lagrangian of the chiral field interacting with the external gauge field because it satisfies the anomalous Ward identity in the form of Eq.(33). It will be consistent only when the holonomy group is anomaly free. Inserting the explicit form of Aa Eq.(7) into ITs(A a ) and ;~(A,Aa) we obtain after a lengthy but straightforward calculation the effective Lagrangian f s •

(34')

in accordance with the result given in Ref.[l] by a different method. Lt is seen step by step in our derivation that the effective Lagrangian is left and right antisymmetric. This is not so in Witten's trial and error method. Besides our Lagrangian contains less terms than that of Witten's (See Appendix). Appendix In this Appendix we will e~lain that Witten's effective Lagrangian[3] (24) in its original form is, however, not correct. Apart from the misprint some terms that are necessary for the gauge invariance of the effective potential are missing. We have found that the following two additional terms have to be added to the Witten's formula (A.I) After adding these two terms and correcting the misprint Witten's formula becomes

358 on Witten's Effective Lagrangian for Chiral Field

79

-(J.,_R)]+Tr[A ~L UA VR U- 1UaL Us L -A ~R U- 1AVL UUaRU aR ]

+Tr[(a~AVRAaR+A~RaVAaR)U-1AaLU -(a~AvLAaL+A~LavAaL)UAaRU]

(A.2)

To see the necessity for the addition of these terms, the easie~t way is to check the gauge invariance in the special case where AR(or AL ) ie zero. In this case only five terms (with the last one added by us) remain. They are

(A.3)

Under the infinitesimal transformat·ion

A_ ~

A~ =A ~ +E:A ~ -A ~ E:-a ~ E:

,

(A.4)

the change of the effective Lagrangian is just the anomaly t All the fozmulae in this Appendix are written in the notation of our previous paper{l] BlI changing A to iA one obtains the Witten's formulae.

359 80

CHOU Kuang-chao,GUO Han-ying, WU Ke and SONG Xing-chang

(A.S)

as claimed by Witten. Without the last term added the change of the effective Lagrangian is

(A.6)

which obviously is not gauge invariant under the anomaly free condition. Now it is clear that Witten's formula contains four more terms than ours

(A.7)

The difference between Witten's and ours can be combined in a gauge invariant form modula an exact differential as mentioned by us in a previous letter.[l] The result is (A.s)

References 1. K.C.CHOU, H.Y.GUO, K.WU and X.C.SONG "On the Gauge Invariance and Anomaly-free Condition of Wess-Zumino-witten Effective Action", Phys.Lett., ~(1984)67. 2. J.Wess and B.Zumino, Phys. Lett., 37B(1971)95. 3. E.Witten, Nucl. Phys., ~(1983)422. 4. B.Zumino. Y.S.WU and A.Zee, "Chiral Anomalies, Higher Dimensions, and Differential Geometry", preprint (Revised) 10048-18(1983)P3. 5. W.A.Bardeen, Phys. Rev., !!!(1969)1848. 6. D.J.Gross and R.Jackiw, Phys.Rev., ~(1972)477.

360 Commun. in

Theo~Phys.

(Beijing China)

Vol.3; No.1 (1984)

THE TOPOLOG I CAL OR I GINS OF GAUGE CHOU Kuang-chao( liJ,til and WU Ke(*

) ,GUO iif)

J 25-129

ANO~1ALI ES

Han-ying(~~;XJt)

Institute of Theoretical Physics, Academia Sinica, P.O. Box 2i35, Beijing, China

ar.d SONG Xing-chang(

*fH: )

Institute of Theoretical Physics, Peking University, Beijing, China

Received October 4, 1983

Abstract In addition to the recent disccvery that the Chern-Simons secondary cl~sses are the topological ori~ins generating special unitary foauqe group anomalies, it is shown that same special orthogonal gauge grou~ anomalies ~an also be gEnerated, fram the Pontrjagin and the Euler

second~ry cl.:lsses. The :::ommoll differential geometrical and topologic:al background are explai·ned as well.

Recently, it has been discovered independently both by the

pr~sent

authors

rl] and b)' Zumino et al. [2] that The gauge anomalies associated wi th the spe-

cial unitary groups h3vP profound topological origin, the Chern-Simons secondary classes. In audition to thi" discovery, we would like to show in 1:;le present note that gau~e anomalies of the special or1:ho~onal groups have also similar topological origins, the Pontrjagin secondary classes and the Euler secondary classes. We will first in this n01:e explain the differential geometrical and 1:0pological background of the relation between the anomalies and the ChernSimons secondary classes and then extend this consideration to 1:he cases of the gauge fields of the. special ortho~jonal groups. Let us start from the SU(3) gauge field dnfined on a 6-d.imensional sphere S6. Similar to the well known monopole and instanton analysis, we introduce two neighborhoods H+~S6_{S}. H_;S6_{N}to cover the S': where the {s} and {N} is the south and the north pole respectively, and the equator of S6, the unit five sphere S~H+nH_. According to the classification theorem of bundles over the sphere,[3] the characteristic map, which is defined ~s the trans1tion function
361 CHOU Kuang-chao,GVO Han-ying and WU Ke and SONG Xing-chang

l;;6

(1)

On the other hand, the bundl"e can also be classified by means of the third Chern number C3 [3 1, the integral of the third Chern class over the 5' n=dA+A~.A,

( 2·)

where A is the SU(3) gauge field l-form and n the field strength 2-form. And this classification is equivalent to the previous one. This equivalence can easily be seen in the following way. By definition, the third Chern class can locally be written as an exterior differential of the Chern-Simons secondary class on H+ and H_ respectively, so we have C3 =

f

drr;(A+,n+)+ H+

1 d!l~(L,n_),

(~)

H_

where rr~ is the Chern-Simons 5-form ( 0.1)

By taking the equator S~ to be the common boundary of the H+ and H_ and using the Stokes formula, we have (5)

Making a gauge transformation with respect to the U(x) A_~ A+=U-1A_U+U-1dU=:U-1DU, 1

SL..E... 11+=U- I1U

(6)

and substituting A+.I1+ by the transformed ones, we can prove that (7)

where

(8)

Again by the Stokes formula and as!=~, we obtain

c,=r(sg,u).

(9)

362 The Tbpological Origins of Gauge Anomalies

127

This result shows that the two classifications are equivalent to each other. Instead of the unit five sphere S~, the equator of S6, let us now consider a 5-dimensional disc QS whose boundary is taken to be the compactified spacetime M"-S". If the disc is embedded in an SS whose volume is, by normaliz~ tion as.Witten has done,[4] 2n times the volume of the equator it is easy to obtain the following expression

S;,

(10)

where (11)

which is the same as the one introduced by Witten[4] to show the global properties of the effective action theory. The expression (10) shows that the i(Qs,U) term is essentially linked to the Chern-Simons secondary topological invariants as long as the gauge field defined on M~S" can be continued onto the disc QS without any topological obstruction.From the point of view of gauge transformation, the expression (10) shows the gauge variation of the ChernSimons 5-form rr~(A Po). If the gauge transformation function Vex) is infinitesimal (12)

i3"'df:(x) ,

U(x)=l+f:(x),

the variation becomes an infinitesimal one as well (13)

Eqs.(lO) and (13) are in fact the gauge anomalies with a minus sign in finite and infinitesimal forms respectively. [1,2] This fact means that the Chern-Simons secondary class is the topological origin of usual purturbative chiral anomalies with respect to the SU(3) gauge field defined on 4-dimentional spacetime M'l..S'. [5] ·~rOm the pOint of view of differential geometry and top610gy, the abo~e considerations can easily be extended to the SUCk) gauge field defined on an arbitrary even dimensional compactified spacetime M2n~ S2n-2 and a 2n-ldimensional oriented disc Q2n-l, aQ 2n-l=S2n-2. In fact, we can write 2n

1Q

C

2n_lrIT2n_l(U

-1

DU,U

-1

c

QU)-IT 2n _ l

CA,Q) ] =rCQ 2n-l ,U)+ 2 n

J

GC 2n-2 2n-2

s (14)

which is the general form of the finite chiral anomaly of SUCk) gauge field, c 2n-2 2n-2 where G _ is a certain (2n-2)-form defined on the spacetime M -- S 2n 2 We now turn to the ch~ral anomalies associated with the SO(m) gauge fields. As is well known, the SOem) gauge fields on the 4-dimensional spacetime do not have anomalies except for ms 6.[6] This fact can be understood

363 CHOU Kuang-c:hao GUO Han-ying and W Ke and SOWG Xing-chang

128

with the help of differential geometry and topology. It is well known that there are two characteristic classes related to the SO(m) gauge fields, the Pontrjagin class P and the Euler class x.[3] The first can only be defined over 4n-dimensional compact manifolds and the second can be defined over 2n-dimensional ones as the Chern class does. By difinition, each of them can locally be written as an exterior differential of corresponding secondary characteristic classes. And under the SO(m) gauge transformations, the variations of those secondary classes are also divided into a surface term plus a term corresponding to the winding number. If we consider an odd-dimensional disc Q with boundary taken to be the even-dimensional compactified spacetime, we can obtain the expressions similar to the expression (14) for the gauge variations of both Pontrjagin secondary class IT~n_1 and Euler secondary class ITX as follows: 2n-1

2~

and

fo4n-1 (IT P4n _1 (R

-1

DR,R

-1

P

OR)-IT 4n _ 1 (A,O)=r(Q

4n-1

,R)+2~

Jr

P 4n_2 G 4n_2

S

(15)

(16) where R is an SO(m) gauge transformation function. Corresponding infinitesimal variations can also be obtained if the gauge function is taken to be an infinitesimal one. Based upon similar considerations in the case of SUCk) gauge anomalies generated by the Chern-Simons secondary classes, it is reasonable to expect that Eqs.(15) and (16) give rise to the SO(m) gauge anomalies generated by the Pontrjagin secondary classes and the Euler secondary classes respectively. In order to verify this pOint, let us explore the SO(m) gauge anomalies on the 4-dimensional compactified spacetime M~S~. From the expression (15), it follows that there do not exist SO(m) gauge anomalies generated by the Pontrjagin secondary classes, since the Pontrjagin secondary classes cannot be defined on a 5-dimensional disc Q' at all. On the other band, SO(6) anomaly can be generated by the Euler secondary invariant 5-form defined on QS because this 5-form corresponds to third Euler class on a S' whose vector bundle takes SO(6) as the structure group. Furthermore, explicit calculation shows that in the case of infinitesimal gauge transformation the expression (16) gives rise to

(17)

which is just the infinitesimal SO(6) gauge anomaly in agreement with the SU(4) gauge anomaly. Thus, we conclude that Eqs •. (15) and (16) can be regarded as the general expressions for the SO(m) gauge nnomalies on the compactified spacetimes of

364 The Topological Origins of Gauge Anomalies

129

certain dimensions. In the case of 4n-2 dimensional spacetimes (n=1.2 •... ). there exist certain special orthogonal gague group anomalies generated by the. correspond'i.ng Pontrj agin secondary classes. I n the case of 2n-2 dimensional spacetimes (n=1.2.3 •.•. ). the SO(2n) gauge anomaly can be generated by the Euler secondary invariant (2n-l)-forms defined on the 2n-1 dimension:.1 disc q2n-l.

References 1. CHOU Kuang-chao, GVO Han-ying, WU Ke and SONG Xing-chang, Phys. Lett.,

~(:9S4)67;

Cammun. in Theor.Phys. (Beijing, China), this issue. 2. B. Zumino, Y.-S. WU and A. Zee, Preprint, (Revised}40018-18 (198J}p3. J. N. Steenrod, The Topology of Fibre Bundles, princeton univ. Press, S. Kobayashi and K. Nomizu, Pub., 1969. 4. E. Witten, Nucl.

Phys.,

19~1;

Foundations of Differential Geometry , '·ol.II,

~(1983}422.

5. W.rl. Bardeen, Phys. Rev., !!i(1969}1848; D.J. Gross and R. Jackiw, Phys. Rev.,

~(19i2}47i.

6. H.Georgi and S. GJ.show, Phys. Rev., ~(1972}429. 7. 5.5. Chern, Ann. Math., 46(194S}674.

".~eersc.

365 COIIIlIun. in 2'heor. Phys. (Beijing)

Vol.3, No.2 (1984) 221-229

ON THE DYMAMICAL SYMMETRY BREAKING FOR THE N=l PURE SUPERSYMMETRIC YANG~'ILLS MODEL CHOU Kuang-chao( Jil1f31 ), DAI Yuan-bene and CHANG Chao-hsiC

**iJ )

ti* )

Institute of Theoretical Physics, Academia Sinica, P.O.Box 2735, Beijing, China

Received November 15, 1983

AbstraCfl An approx:i1llate effective Lagrangian of composite operators in superfield formalism for the N=l pure supersymmetric Yang-Hills model is ob,tained with the help of renormalization group equations for the generating functional. While the supersymmetrll is alwalls kept unbroken by this effective Lagrangian we find that the chiral symmetry mag and mall not be unbroken.

The potential applir.ation of SUSY to GUTS and composite model building makes it more and more stimulating to explore and to understand the properties of SUSY pne of the important problems is about the possible dynamical symmetry breaking e.g.whether the super-symmetry and the chiral symmetry, both or either,.are broken dynamically or not,and if they are indeed broken then how and where to break.One approach to these problems is. to analyse the effective Lagrangian for those composite fields which may condense. This approach has already been adopted recently to treat supersymmetric Yang-Mills(SSYM) mOdels with or without matter fields by many authors[l,2,8J. The work l1 ] by Veneziano and Yankielowicz is based on the anomalous Ward identities for scale, chiral and superconformal transformation.[6 J Their conclusion is that the chiral symmetry is broken dynamically but the supersymmetry is not for N=l pure supersymmetric Yang-Mills model. I.n this article, the N=l pure supersymmetric Yang-Mills model is to be examined, using renormalization group analysis i.e. an equation for the effective action is obtained, which can be interpreted as the anomalous Ward identity for certain scale transformations. And we shall see that the anomalous dimensions of the composite fields enter into this anomalous Ward identity. This is in contrast with the approach adopted in Ref.[lJ, where the composite fields are scaled according to their naive dimensions. Our approach bears some resemLlance to that used in Refs.[5] and [1] for QCD and SSYM. In addition, under certain approximation the equation can be solved and the solutions can be determined up to some free constant coefficients, which

366 222

CHOU lCUanq-chao, DAI Yuan-ben and CHANG Chao-hsi

do not affect the final qualitative conclusion, so that some interesting results, different from those in Ref.[l], can be drawn. Now let us look at the N=l pure supersymmetric Yang-Mills model with the non-Abelian group BU(N c )(Nc =3 for BUSY QCD). We will start with the gauge covariant chiral superfield Wa ' which is related to the vector superfield V by (1)

and V=vBT~ Ta are the adjoint representation matrices of BU(N c ) generators, and Tr(Ta Tb )=6 ab , g is the coupling constant. Let 1

a

(2)

B2'gTTrW Wa In W-Z gauge

then (3)

where Da is the auxiliary field, the Yang-Mills f'ield strength is (4)

and ~aa(Xba) is the Weyl spinor component. DaD is the covariant derivative m

(5) Cabc is the structure constant of the group BU(N c ).

In order to investigate the problem of condensation for the component fields of the superfield B, chiral superfields J and J+ are introduced as extecnal sources of Band S+. The Lagrangian for the pure Yang-Mills theory including the gauge fixing term, ghost fields C, C and the external source terms is

~=~ Jd2es~ +

Jd 26S+

l~a

Jd-eTrD vD 2V 2

Jd-eTr(C'-C' )Lqv[ (C+C)+(coth L.i!)(C-C)] T

+

2

i d 6JB+ I d 6J+B++counter terms 2

=~Jc+counter

Z

terms,

(6)

367 On the Dynamical Symmetry Breaking for theN=l pyre supersurnmetric Yana-Mills Model

223

where (7)

L2:,!x=ig[ V,Xl • 2

The generating functional is defined by (8)

Let us assume that the theory is made finite by the following renormalization

(9)

Since S is a hard operator and the chiral superfield J is dimensionless, Z(J) is in general not a constant but may depend on J due to multiple insertions of the operator S. From Eq.(9) we can derive the following renormalization group equation for the generating functional

Z:

( 1 () )

where

~

is the scale parameter introduced by renormalizat.ion,

Y=lIda~lnZ(J)

(11 )

:lnd J(YJ)c5~T=J(YJ)i c5~. wit.h J i and (yJ)i denoting components of (~f)rr""I'(Jn ~

ding chiral superfields. Since the external source J is gauge invariant, it is not difficult to see adaZ=o.

(1~

)

The equation of motion of the V superfield reads (13)

368 CHOU Kuang-chao, DM Yuan-ben and CHANG Chao-hsi

224

Combining Eqs.(12), (13) with Eq.(10) we obtain -lla
[.2'}C]~( ga
2at -

fd ~XV-fv)( 1 Jd ~x y.j)e ild ' K~ J~ (14)

Writing

~

in the normal product form (15)

we obtain the equation

-~Z=J£I,!IV]r~][1IC] ( ~iN Jd-XSF)eiJd-JC~

f

+ d - X [ (Y

~)Jir+h.C. lZ

(16)

from Eq.(14) ,here and later the subscript F of SF means to take F-component of S. Let us introduce r[S c , S~l , c through the Legendre transformation (17)

with (18)

Since

(19)

From Eq'. (16) we find that

(20)

In the neighbourhood of the minima

369 Dr;lJlI/IJical S~trr; BreiJcing for the N-1 PUre Supersymmetric Yang-Nillb MOdel

Qn the

225

we can make the approximation (21) Nothing is lost in this approximation if one uses it to find the minima of r only. Under this approximation, Eq.(19) is reduced to the form:

where

N_

2B

Y=Y.~

+ • At J=J -0, we have

cSr or cssc CSs+

0,

C

the first term on the right-hand side of Eq.(22) becomes the trace anomaly of the J-O theory.[7] f[Sc'S;] consists of 'two pieces, the F part and the D part as follows: (23)

With the help of the dimension counting rule,e~uation (22) can be rewritten into the following two equations (24)

and

2ra-[(3+Y)Id-XSccS~ ra+b.c.]-O •

(25)

c

Eqs.(24) and (25) have the structure of anomalous Ward idehtities for some scale transformations. If we make the following infinitesimal scale transformations in r (26 ) Eqs.(25) and (26) imply that the change of r will be (27) Therefore. if we consider (unjustifiably) the ~fd-XSCF term as the trace

370 CHOU Kuang-chao, DAr Yuan-ben and

226

C~NG

Chao-hsi

anomaly of present J=O theory as done in Ref.[ll, the superfield Sc in the effective action r should be scaled as if it has the djmension 3+Y. As we are interested only in slowly varying field we can confine ourselves to the solution of Ens.(24) and (25) local in x. A class of solution in this case is (28)

and 1

r =C d

(S S+)3+"7

(29)

dec

where C f and Cd are constants with an appropriate dimension. In the theory there is only one dimensional parameter 1.1 (the renormalization 3i 27 3+<[ ];7 parameter), so C f and Cd can be represented as Cf =C;l.I and ~d=Cd1J '

Cd

respectively where C and are dimensionless.constants. Now, in this f approximation, r can be considei'ed as the effective Lagrangian ';t~ff of this SUSY Yang-Mills model, h~nce we are in a pOSition to analyse the symmetries of vacuum determined by the effective Lagrangian, what we are interested in mainly, so as to examine if the supersymmetry and/or the chiral symmetry arr.' broken. We'll consider two different cases separately, case Il II)

Cf=O

and case

CiFO. In case I) the effective .Lagrangian 'is J ';teff=-B( gy) -1 (SCF +S;;)+C [( Sc S ; d

)M1D

(30)

where the foot indices F and D imply taking F-term and D-term respectively. It is easy to read out the scalar potential 4+2:;

V,$,L)=2S(g'Y)-I L- Cd $-3+'Y L2 ,

(31 )

(32)

In Eq.(31) we have assumed that the parity is not broken spontaneously, i.e., $=';'* and L=L*. From ilV_O , we obtain ilL

(33)

and

371 On the Dynamical Symmetry Breaking for the N-l Pure Supersymmetric Yang-Mills Model

227

(34)

From the requirement of positive de'finiteness of the kinetic energy terms of the effective Lagrangian we have Cd>O.Assuming ~:~Y>o (This is certainly true for sufficiently large ~ at which the renormalized coupling constant g(~) is small). it is easy to see that the minimum of Eq.(34) is (35)

at L=O

and

(36 )

.p=0.

That isin this case the supersymmetry and chiral symmetry are both unbroken. In case II), ;tlle' effective Lagrangian is 3

Yeff=-S(gy)-1(SCF+s;F)+Cf(Sc3~~)F 3

1

+Cf[(S+)3+~lF+C [(s S+)mlD CdC c and

th~

(37)

scalar potential is (38)

wherE' (39) ~v

From aL-O, we obtain (40)

and (41 )

There is one minimum for the V(¢,L)I

L=L

of Eq.(41) (ymin(¢,L)=O) at 0.:

(and L. 2 =0).

(42 )

372 CHOU Kuang-ehao, DAI Yuan-ben and CHANG Chao-hsi

228

For ;:Cil~O ,there is another minimum (Vmin(~,L)=O) at 28

3+'1

-1

~=~I=(yg C f

)

-cy-

(and L02 =0) •

(43)

Both the degenerate minima are reached at L Q 2=0, with one of them at ~=O and the other at ~~O. This means the supersymmetry is unbroken no matter which minimum is taken and the chiral symmetry may be broken (when ~-~1) and may not be broken (when ~=~o). If the non-renormalization theorem[3] valid for arbi trary order in perturbation theory can be used here we should expect Cf=O.i.e •• case I), the chiral symmetry as well as supersymmetry is unbroken as shown above. More generally, the solutions of Eqs.(24) and (25) may have terms containing D acting on Se or Se+ but we would not consider them at this moment in this paper as is done in Ref.[2]. In addition, the solution of a' +b' Eq.(25) may also have terms of the form Cdi(Se~Se ~+h·c.) provided ai+b i =

~ , but the, can only exchange Cd (See Eq.(3l»

in, the scalar potential

sothey too _ill not affect the conclusion. Therefore the scalar potential Eqs.(3l) and (38) are quite general. Finally, results obtained in Ref.[9] indicate that in SSYM theory, apart from the renormalization made in Ref.[9], the superfield V may need additional non-linear renormalization. As this additional renormalization is equivalent to a gauge transfQrma~ion we can expect that it d~~s not affect the gauge invariant quantity discussed here. In summary. we conclude that in consistency with Witten I s index theorem the supersymmetry is unbroken for the N-l pure Yang-Mills supersymmetric model but the equation obtained here admits of both, chiral symmetry preserving and chiral symmetry breaking, solutions.

References J. G.veneziano and S. rankielowiez, Phgs.LBtt., 1l1!J19tf2)23J. 2.r.R.!aglor, G.P8neziano and S.rankielowicz, BU~.Phys.,~(J983)493; N.B.Pesldn, ,preprint SLAC-Pub.-306J; A.Davis, N.Dine and N.Se1berg, Phgs.Lett., ~(J983)487; H.P.Nilles, Phgs.Lett., ~(J983)J03. 3.B.Zum:tno, BUcl.Phys., !!!!,(J975)535;

P.flest, BUel.Phgs .., !!.!E2JJ976)2J9;

".Lang,

MICl.l'hys., BU4(J976)J23;

N.'.t.Grium, II.Rocek and ".Siegel, NUcl.PhYs., ~(t979)429.

4. J.fless and J.Bagger, ·supers!PJlllMltry and Supergravlty'!, Princeton Lectrzre Notes(t98t). 5.R.l'Ukada and Phgs. Rev.,

r.JraZlllllol,

Phgs.Rev.Lett., 1l.,(J980}J142;

~(J980}485.

6. S.Ferrara and B.Zlllllino, BUcl.Phgs., !!l(J975}207;

373 an the Dynamical Symmetry Breaking fOr the

H.'

_ _ _~229

PYre supersymmetric Yang-Mills MOdel

~.CUrtright,

L.F.Abbott,

Phys.Lett., M.~.Grisaru

~(1977)185;

and H.J.Schnitzer, Phys.Rev.,

~(1977)2995;

Phys.Lett.,

~(977)161;

M.Grisaru, in "Recent Developments in Gravitation" rCarge se 1980) eds.M.Levy and S.Deser. 7. J.C.Collins, A.Duncan and S.D.Joglekar, Phys. Rev., 8. Y.Kazama, Preprint KIlNS 683 HE 9. O.Piguet

~d

('1'H)

~(1977)438.

83/09.

K. Sibold, Nucl.Phys., BI97(1982)257, 272.

374 COJ/IDW2 •. in Theoz. Ph?s.

(Beijing, China)

Vol.3, NO.4 (1984)

491-498

THE UNIFIED SCHEME OF THE EFFECTIVE ACTION AND CHlRAL ANOMALIES IK ANY EVEN DIMENSIONS Kuang-chao CHOU{ JIlti ), Han-ying GUO( tlSi(~ Xiao-yuan LI( f/}' ) and Ke WO( ~ if) Institute of Theoretical Phgsics, Academia Sinica, P.O.Box 2735, Beijing, China

and Xing-chang SONG t ( ;IHf* ) Institute for fheoretical Physics, State university of Ne., York at Stong BrooJc, Stony BrooJc, Ne., York 11794, USA

Received April 29, 1984

Abstract

_thad,.

Based on the flei1 homomorplliSlll a unified scheme in which all the important topological properties of the pseudoscalar GOldstone boson and gauge fields in even dimensional space are described in one remarkablg compact fpzm is given. fhase properties include the effective action, the skgZl/lion anomalous current, Abelian anaIIIalll, spIIIIIetric and as!IIIINtric non-Abelian c:hiral anomalies, and anomalll free conditions.

A year ago E. Witten discussed the global aspec~s of current algebra[1] and proposed that the Wess-Zumino chiral effective action[2] can be described in a new mathematical framework. He pointed out that this action obeys a priori quantization law. analogous to Dirac's quantization of magnetic charge, and incoporates in current algebra both perturbative[3] and non-perturbative anomalies[4]. Applications to the standard weak interaction model require an arbitrary subgroup of global flavorsymm.t~ieato be gauged. However. the standard road to gauging global symmetry of the Wess-Zumino action is not available since its topological nature is unknown. In Witten's original work the author did this by resorting to the trial and error Noether method. In their series of works[5-8] CHOU-GUO-WO-SONG suggested that one can as usual introduce the minimal gauge coupling in 5-dimensional space first. then the gauge invariant Wess-Zumino action can be obtained by a series of nontrivial but systematic mathematical ma~ipulations.[5] In the processes a deep connection among the manifestly gauge invariant 5-forms. the Chern-Simons secondary topological invariants. anomaly free condition and the gauge invariant Wess-Zumino action has beeu found. Remarkably. the structure of symmetriC chiral anomalies[9] can be determined by studying the gauge transformation properties of the Chern-Simons. secondary topological invariants without having to evaluate any Feynman diagram[5,10]. Furthermore it has been shown[6] that th~ Bardeen's t Pezmanent address: Depar1:lllent of Phgsics, Peking university, Beijing, ChiDa.

375 492

Kuanrr-chao CHOU, Han-ying GUO, Xiao-yuan !.I ,Ke WU and Xinq-chanq SONR

asymmetric non-Abelian anomalies[ll] and the corresponding counterterms can be obtained if one does Goldstone boson expansions in the gauge invariant effective Wt.'ss-Zumino action functional.The relation between the symmetric and asymmetrlc non-Abelian anomalies has also been found[7,l2).Recently,it has been !'!uggested[l3) that the gauge invariant Wess-Zumino actior. functional in 2ndimensional space can b" obtain.,.,:l by properly constructing the m:.nifestly gauge invaraint (2n+l)-forms which sa.i3.t'y tht: Abelian anomalous Ward identity in (2n+2)-di~ensional space. The problem about the uniqueness of the gauge-invariant Wess-Zumino effective action has also been discussed. [l41 I~ this note, we would like to suggest a unified scheme of the gauge invariant Wess-Zumino effective action functional, Abelian and non-Abelian chiral anomalies, the anomaly free condition and the r.lanifestly gauge invariant generalized skyrmion anomalous current[15] in any even dimensional space. The description is based on the Weil homomorphism method in differential geometry[lG). It turns out that the Abelian anomaly in (2n+2)-dimensional space, the gauge invariant Wess-Zumino effective action, the symmetric and asymmetric non-Abelian anomalies and anomaly free conditions in 2n-dimensional space can all be given in a remarkably compact form. The formalism further reveals the intrinsic connection among these objects. We start by considering the theory with a chiral SU(N)!.xSU(N)R symmetry which is spontaneously broken down to the diagonal group SU(N). Under an SU(N)!. xSU(N)R transformation by unitary matrices (gL,gR)' U(x) transforms as (1)

In order to obtain a gauge invariant action functional under the gauge group the'rauge covariant derivative

H~SUCR)LXSU(N)R'

(2)

should be introduced, where AL(RJ the field strength 2-form is

is~tbe

gauge connection 1-form of RL(RJ' and

(3)

respectively. Under the gauge transformation

(4)

We have

376 The unified Sche1/le' of the Effective Action ad Cbiral Anomalies in 1In!l Even Dimensions

493

(5)

In particular, if we define n=u-1Du=u-l(d+AL)U-~R=UAL-AR

where

U

(6 )

AL(x) is the "gauge transformed" gauge field l-Iorm (7)

then, under the gauge transformation (Eq.(4»

we have

( 8')

where the "gauge transformed" field strength 2-form UFL(X) is defined as (9)

Eq.(8) implies that the gauge transformation property of UAL is exactly the same as those of AR' and n is the gauge covariant I-form. u Let us consider the interpolation between AR and AL '

o
(10)

then the field strength 2-form (11)

(12 )

where

Q

+

n 1

is the normalization constant (13)

Then using the Chern-Weil formula we iDlllediately have (14)

377 Kuang-chao CHOU; Han-lJing GUO, xiao-!luan LI , Ke IiU and xtna-chana SONG

494

where Cn+1(F) is the (n+l)-th Chern class of A. Q(UAL , AR) defined by Eq.(12) is constructed out of the gauge covariant I-form and 2-forms UFLand FR , and thus is manifestly gauge invariant. In the reference [10] the Chern-Simons secondary topological invariant is defined as the object which consists of only gauge fields and thus is not gauge invariant. It should cause no contusion. Using the Chern-Weil identity Eq. (14) can be written as

C15)

where Rzn+1(A,F) is the Chern-Simons secondary topological invariant This means

of A.

(16)

should be a closea form. In what follows we would like to give its expression in a compact and explicit form. Let us define a connection Aab depending upon two parameters a and b with O"a.b~l

(17)

Denote the corresponding field strength as (18)

and consider the integral (19) over a one dimensional path which is clockwise triangle in the (a,b) plane going from the origin to the point (0,1) to (1.0) and back to the origin. The strai"ghtforward calculation gives (20) On the other hand, one can apply Stokes' theorem to the integral Eq.(19) and transform it into an integral over tbe inside of the triangle. It follows that I-d{} o

0

Using the same trick. one can prove that

(21)

378 Th. unifitld Sche_ of the Effecti tie Action and Cbiral Anomalies in }J.lJY Ewn Dimensions

495

(22) +d{(n+l)nan+lf10VJ1-VoWSTr(UdU-1ALF:;1)} , o

0

where the field strength 2-form F

-dA

vw

vw

(23)

+Az

vw

and (24) is the connection I-form depending upon two parameters v and w with 0.;;; v, w .;; 1. From Eqs.(20), (21) and (22) we come to our final result U

Q2~+1( AL,AR)-IT2n+l(AL,FL)-IT2n+l(AR,FR)+IT2n+l(U

+d {(n+l )nan+1

r

ov fl-Vc5wSTr(UdU-1ALF:;1) }

o

dU,O) (25)

0

+d{(n+l)nan+1J10aJl-aObSTr(ARUALF:~1)} o

-1

•

0

Eq.(25) summarizes all the important topological properties of the pseudoscalar Goldstone boson and gauge fields in (2n+2)- and 2n- dimensional spaces·, as will be shown later. 1) Consider (26) where "." is the Hodge star operator and 12n+l is an arbitrary manifestly gauge invariant and exact (2n+l)-forms which conSists of nand FR.UF L , Substituting Eq.(26) into Eq.(25) and exterior differentiating it. we have (27) A straightforward calculation indicates in the 4-dimensional space (n=l) J defined by Eq.(26) is none other than the gauge invariant anomalous skyrmion current form[17]. Therefore (28 )

can be considered as the generalized gauge invariant skyrmion anomalous current I-form in (2n+2)-dimensional space. Obviously. this current is not conserved. It suflers from the Abelian anomaly which is simply given by Eq.(27). Note

379 Kuang-chao CHOU, Han-!ling GUO, Xiao-!luan LI,KB MJ and Xing-chang SONG

496

that J is constructed out of only boson fields without referring to underlying ferm10ns. However, the subtle pOint is that the sxyrmion, the soliton of the non-linear sigma model, can be quantized as a fermion when the number of colors of the undvrlying fermion theory is odd[15]. 2} Define

J

rCu,Q 2n+l }=2wi 02n+1IT2n+1CU -1 dU,O},

C29}

where Q2n+l is a (2n+l)-d1mensional manifold~ 3Q2n+l=y2n is a 2n-dimensional compactified space. For the case of n-2, rCU,g5) is just the Wess-Zumino term[2] proposed by Witten[l]. Thus r(u,Q2n+l} can be considered as the generalized Wess-Zumino term in y2n. Now the left-hand side of Eq.C25} is manifestly gauge invariant while the Chern-Simons secondary to~ological invariants are not. A necessary and sufficent condition for getting the gauge invariant WessZumino term in y2n is the gauge variations satisfy (30) If this condition can be satisfied, the generalized gauge invariant Wess-Zumino term in y2n can be defined as C31 } where the·2n-form W2n is given by Eq.(25} W2n -Cn+1 }nan+1

J c5a 1

a

Jl-a

a

c5b STrCAR U ALFn-l ab }

+(n+1 }nan+1 ( c5v Jl-V c5w STrCUdU- ALF~:l) 1

a

~2}

a

For the case of n-2,3, the gauge invariaut Wess-Zumino terms[5,12,13] in Y-and y' can be given by the strai~htforward calculations from r(U,05) and r(U,Q7} respectively. Furthermore, the zero order term and first order term ia Goldstone boson field give the Bardeen's counterterm R3 and asymmetric non-Abelian anomalies[6]. This indicates that r(U,Q5} satisfies the non-Abelian anomalous Ward indentity and the Wess-Zumino consistency condition[2]. The same conclusions are also true fqr rCU,Q2n+l} in M2n. In particular, it can be proved that the Bardeen's counterterm R3Cy2n} and asymmetric anomalies GA Cy2n} in y2n are[ 18 1

1

G:CM2n}C2WiCn+l)nan+lf c5VC1-V}STrAad[ALFn-1CVAL)] a

I

(33)

380 'l'htl Unified Scheme of the Effective Action and Chiral Anomalies in Ang Even Dimensions

497

STrAa{[d(En-1An)+A En~lA +En-1A A ] n

L

R

R L

+(n-l) [En-2ARALF(VAL)-F(VAL)En-2ARAL] +(n-I)(v-v 2 )d[A En - 2A A _E n- 2A A2] L

R L

R L

+(n-l)vw[d(E n - 2A A A )+A En - 2A A A R L R

L

R L R

respectively, where

3) If Eq.(30) for cancellation of anomalies is not obeyed, then -r(U,Q 2n+l· ) is not gauge invariant. From Eq.(25) the variation of r(U,Q2n+l) under a gauge transformation Eq.(4) does not vanish but is (35) Using the formula analogous to Eq.(22), it is not difficult to find that

(36)

The general expression of the infinitesUnal variatiol: 6r(U ,Q 2n +1 ) can alai be given. For the case of n=2, 6r(U,Q5) is just the Gross-Jackiw's symmetric anoma. [4] lies[9], and Ar(U,Qs) is likely the Witten's 8U(2) non-perturbative anomaly • Thus, Eq.(22) can be cOnsidered as the global version of the anomaly free condition in yZn. It is not only the generalization of the anomaly free condition in the perturbative theory, but also includes the condition of free of nonperturbative anomaly. On the other hand, the anomaly-tree condition Eq.(20) is equivalent to the condition (37)

which is nothing but the Abelian anomaly free condition in M2n+2. In summary, we have presented in this note a uni·fled scheme in which all the important topological properties of the pseudoscalar Goldstone boson and gauge fields in (2n+2)-and 2n-dimensional spaces are described in a remarkably compact form. These properties include the gauge invariant skyrmion anomalous current, the Abelian anomaly in (2n+2) dimensional space, the gauge invariant

381 Kuang-chao CHOCl, Han-!ling GUO,Xiao-!luan LX, and Xing-chang SONG

498

ICe W

Wass-Zumino action, symmetric and asymmetric non-Abelian anomalies and anomaly free conditions in 2n dimensional space. Eq.(25) clearly embodies the deep connection among them. Finally, we would like to mention that the Abelian anomaly and the non-per~ turbative SU(2) anomaly [4] are both related to the famous Atiyah-Singer theorem[16], as is well known. Thus there should be some direct or indirect relation between the non-Abelian anomalies and xhe Atiyah-Singer index theorem. We will discuss this point elsewhere.

References and Footnotes [1]

E. witten, Nucl. Ph!ls. B223 (1983)422.

[2J J.Ness and B.Zumino, Phys. Lett. {3J S.Adler, Phys. Rev.

~(1971)95.

~(1969)2426;

J.Bell and R.Jac1ciw, {4J E.Nitten, Phys. Lett.

NuOVO

Cim.

~(1969)'47.

~(1982)324.

{5J K.C.CHOU, H.Y.GVQ, K.fIfU and X.C.SONG,Phys. Lett. 134B(1984)67. {6J K.C.CHOU, H.Y.GUO, K.fIfU and X.C.SONG,to be published in Physica Energiae Fortis et Physica Nuclearis (1984) (in Chinese). {7J K.C.CHOU, H.Y.GUO, K.fIfU -'nd X.C.SONG, COI/IIIIIln. in Theor. Phys. 1}1984)73. {8J K.C.CHOU, H.Y.GUO, K.fIfU and X.C.SONG, Commun. in Theor. phys.

!

(1984}125;

Stony Brook preprint AS-SB-84-18. {9J D.Gross and R.Jac1ciw, Phys. Rev. E!(1972)477. {10J B.Zumino, LeS BOuenes lectures 1983, LBL-16747 (1983); B.Zlllllioo. Y.S.

JfU and A.Zee,

to be published in HUcl. Phys. B;

L.Bonora and P.Pasti, Phys. Lett. 132B(1983)75. {11J N.A.Bard.en, Phys. Rev.

~(1969)1848.

(12J K.Kawai and S.-H.H. 2'ye, Cornell preprint also,

o.

CLNS~/S95;

Kaymakcalan, S. Rajeev and J. Seneenter, Syracuse PZ'fIpZint SU-4222-278(Dec. 1983).

03J Y.P./CUANG, X.LX, K.fIfU and Z.Y.ZHAO, COIDIIIun. in Theor. Phys. ,th1:tl$sue. {U]

C.H.CHANG, H.Y.GUO and K.fIfU, Institute of Theoretical Physics

Acad51ia

Sinica preprint

AS-Ifp-84-016 (April, 1984). [15J T.H.R. Skyrme, Proc. ROy. Soc. (London)

~(1961)1271

J.Goldstone and F.Nilczek, Phys. Rev. Lett. E.Nitten, Nucl. Phys. A.Zee, Phys. Lett.

~(1981)986;

~(1983)433;

~(J984)307;

F.Wilczek and A.Zee, Phys. Rev. Lett.

~(1983)2250.

{16] For example, T. Eguchi, P.B. Gilkey and A.J.Hanson, Phys. Rep.

66(1980)21~.

{17J For example, A Zee, reference {15J. [18J A.Andrianov, L. Bonora and P. Pasti, Padova University preprint PD 30/83 (October, 1983).

382 Vol.3, NO.5 (1984)

Commun. in Theor. Ph!ls. (Beijing, China)

593-603

SYMMETRIC AND ASVr1METRIC ANOMALIES AND EFFECTIVE LAGRANGIAN Kuang-chao CHOU( JiJ!?l Han-ying GUO( ~iX.~ ) and Ke WU( ~

iif )

Institute of Theoretical Physics, P.o. Box 2735 Beijing, China

and Xing-Chang SONGt(

*.fj"* )

Institute for Theoretical Physics State Uni"ersity of New York at Stony Brook Stony Brook, NY 11794, U.S.A.

Received June 7, 1984

Abstract The properties of the effective Lagrangian for the low-energg Goldstone-gauge field system constructed from the Chern-S:imons topological illvariant are further discussed. Both the s!l1Dmetric anomaly and tne asymmetric anomaly are connected with this Lagrangian.

In a recent paper[l], Witten has reformulated the non-iinear effective Lagrangian of QCD, which was first considered by Wess and" zumino[2], by adding to the ordinary chiral field action r D a Wess-Zumino like term r, a five d~en sional integral over a volumn D whose surface is the image of the physical fourdimensional space M on the group manifold. This Lagrangian, when gauged by introducing flavor gauge fields, may preCisely describe all effects of anomalies in low-energy processes con'taining Goldstone bosons. r D term is gauged as usual by minimal coupling while the gauged form of viZ term r is obtained by tria-land-error method[l] -r=f-

i

481T f

Iz.

.

(1)

StartinF from a different pOint of view and making use of the properties of the Chern-Simons topological invariant, the present authors have proposed a systematical method of gauging the WZ term[3,4]. The result is similar to but different from that of Witten's

;·fW•

r=r- 48

M

Under the infinitesimal gauge transformation, both rand f[5] give correct t On leave frOlll the Institute of Theoretical Physics, Peking University, Beijing, China.

(2)

383 594

Kuang-cbao CHOU, Han-ying GUO, Ke flU and Xing-Cbang SONG

anomalies G, the anomalies in the left-right (anti-) symmetric form, which ap'peared at an intermediate state in Bardeen's work[6] and is in agreement with the computation of the anomalies at the quark level by Gross and Jackiw[7]. As is well known, in order ~o obtain the normal Ward identity for the vector current, Bardeen has further introduced several counter terms[6] so that the anomaly only appears in axial-vector part. We shall call this form of the anomalies the asymmetric form Ga for convenience. The main purpose of the present cote is to discuss the relation of the anomalies, both in the symmetric form G and the asymmetric form Ga to our effective Lagrangian. To begin with, we give our notations. The canonical form of the principal chiral fields is used by introducing the double matrix[4]

U),

Qa( U-

U=exp {

1

r

2· lr

L8

Aa lr

a}

(3)

a=J

And the gauge potentials ,1-form) and strengths (2-form) are also grouped into

(4)

where

and the same is true of right-handed quantities. The gauge transformation takes the form

(5)

A _

g(A+d)g_l ,

The covariant derivative of Q fields is defined as (6)

DQ=dQ+[A, Q] And for infinitesimal gauge transformation EL(X)

E(x)=

we have

(

384 59_eric and As~tric Anomalies and Effective Lagrangian

6Q=[E.Q] •

6DQ= [E .DQ]

6A=-dE+[ E .A] •

6F=[E,F]

S9S

,

( 7)·

Now the gauged form of the chiral field action ro can be written as

F;f

.r~ o- 16

~

1

d~x tr(D~U)(D U- )

F2

3iJTr(DQ)~(.DQ) •

(8)

where tr denqtes the trace for single matrix and Tr for double matrix, and "." is the Hodge star (dual form) in four dimensions. The WZ term[l]

(9) 1

is indeed an integral of Chern-Simons 5-form of the pure gauge fields U- dU. 1 i.e., Ws (U- dU). over the region D. and[4]

(10)

where UdU- 1

-1

K=QdQ- (

Ys= (

(11)

It can be easily seen that both r and W~ are odd under the left-right exchange U_

u- 1

•

(12a)

A -+ oAo •

C12b)

which can be expressed as the involution Q-+OQO • ~

Meanwhile, ro in Eq.(8) is even under the same transformation. This fact indicates that, just like the ungauged action r -ro+nr considered by Witten[l]. the eff gauged action reff=ro+nr with r given in Eq.(2) has no superfluous symmetries other than the ones given by QCD[4]. A

..........

A

385 596

KU.ul!1-chao CHOU, Han-ying GUO, Ke WU and Xin!1-Chang SONG

On the other hand, Witten's :esult reS] contains more terms than those in our

r. The difference between rand r can be seen from the relation (13) which contains no Vs so that it is even under transformation (12b). Therefore, even has no definite parity under space reflection. Since Eq.(13) is gauge invariant, the variation of or gives the same anomalies under infinitesimal gauge transformation[8J

r

r

6r=6r= f TrEG= f tr(ERGR_ELGL) M

r

,

M

(14) which is the leit,right antisymmetric form given in Refs. [6] and [7], as 1s mentioned above, wbereas Bardeen's asymmetric anomaly, which only couples to the divergence of the axial-vector current, is

where the 2-forms FV and FA are defined as (16)

and related to FL and

pR through the relations (17)

It seems that the effective Lagrangians obtained from topological consideration[1,3,4,8J always yield the symmetric anomalies under infinitesimal gauge transformation. Then, bow about the asymmetric one? Does it bave anytbing to do witb our effective Lagrangian? Indeed, as an effective Lagrangian, r must contain all anomalous interactions among gauge fields and Goldstone fields. In particular it must be the asymmetric anomaly which couples to the single Goldstone vertex. This important feature can be revealed through an appropriate chiral transformation applied to the effective Lagrangian. As a matter of fact, in the early paper on the consistency condition for cbiral anomalies [2] , Wess and Zumino already obtained their effective Lagrangian by such a consideration. The result is nt,A]'"

l-expt.Y _ Af;G [AJ , f;Y

A

(18)

a

- are tbe local chiral operators acting upon the gauge fields and where f; a "F'I21''I1' a , Y A

386 symmetric and Asymmetric Anomalies and Effective Lagrangian

597

From (18), it follows immediately that (19 ) i.e., the terms linear in ~ fields are just the asymmetric form of the chiral anomaly. Now we would like to see whether the effective Lagrangian r or same property. Expanding U and integrating by parts, we have

r

has the

(20) with

r[O,A]

48;2

Jtr{(AR~dAR+dAR~AR)~A~-(AL~dAL+dAL~AL)~AR

+(AR~AR~AR~AR)_{AL~AL~AL~AL) ~AL~AR~AL"AR

J

(20a)

and

Here we have introduced the abbreviation symbols

to simplify the lengthy expression. By inserting relations (16) and (17) into Eq.(15) and completing a lengthy but straightforward calculation, we can show that the integrand on the right-hand side of Eq.(20b) is nothing but the product ~a.Ga. ~ a'

• l.e., ~

a.

rl=*G a

(21)

In comparison with Wess-Zumino's effective Lagrangian (18), our r[~,A] has the same first order piece ~.GG but is different from Eq.(19) by a zeroth order part r[O,A], which has the following interesting properties (22a)

(22b)

Indeed, r[O,A] is nothing but the counter term which is used in Bardeen's paper[6] to remove the anomaly from Ward identity of vector current, i.e., (2e:

A

_e:R_e: L )

387 Kuang-chao CHOU, Han""ging GUO, Ke WU and Xing·-au!'.ng SONG

598

(23 )

OrU;,A]-or[O,A]= Jtre: AGiI [A] M

As for Witten's Lagrangian

r,

it is not difficult to show from relation (12)

that

with i r[O,A]=r[O,A]-48rr 2

-

A

Jtr(F AF L

R)

,

r~[A]=r~[A]148;2tr{Aa(FP.AFL_FLAFR)}

(24)

Therefore,. neither Eq. (21) nor Eqs. (22) and (23) are satisfied for r. It is worth notice from the expressions in Eqs.(20) that Bardeen's anomaly G , although asymmetric in V and A, is symmetric in AL and AR, whereas the suba traction term r[O,A] is antisymmetric. This shows that the subtracted effective Lagrangian r[E;,A]-r[O,A] producing Bardeen's "asymmetric" anomaly, as well as the unsubtracted one r[E;,A] producing the "symmetric" form of anomalies, is odd under the left-right exchange (12) and then even under the real parity transformaA

A

tion. The above results can be further understood through the following analysis. Let us first assume· that the whole system of the Goldstone and gauge fields can be well described by a Lagrangian density in four dimensions as usual. ~ can be considered as functional of Q,DQ,A and F, as well as functional of Q,aQt aA and A. We use the notation (25a) to denote the functional derivative of ~ with respect to A keeping Q, aQ and aA fixed, while (25b)

to denote the functional derivative of keeping Q,DQ and F fixed. Then we have (26a)

(26b)

388 SlJIII1III!tric and Asg_tric Anomalies and Effective Lagrangian

599

(26c)

( 26d·)

The assumption made h'ere is a reasonable one since, as is pOinted out in Ref. [4]: although r= fD P is defined at the beginning as an integral over a five ~ dimensio~al hyperspace,S itself can be locally expressed as an exact form S=d*~, where *~ is the dual form of an appropriate 4-dimensional Lagrangian density. The part of S containing gauge fields has alreaQY been expressed as an exact form as in (10). The other part ws(U-1dU) is a closed 5-form so it can also be put into w5 (u-1dTJ )=d*!t'. locally, therefore, the effective Lagrangian corre~ cs sponding to r

~=~ -~*W cs 48""

(27) •

exists in principle. And the total Lagrangian is (28)

57

where ~o is the term corresponding to f 0 as in Eq. (8), the free Lagrangian of the gauge fields, and n an integer[l] equal to the number of colors in QCD which will be set to one for simplicity in the following discussions. Now, just like the case for odd dimensions[9], since ~ contains all kinds of interaction, including the anomalies, it should give the correct equations of motion, i.e., (29a)

for chiral fields (with constraints Q2=1), and (29b)

for gauge field. Secondly, since the infinitesimal variation of f gives correct anomalies G, and G itself can bp. expressed as a total divergence of an appropriate anomalous current J a [7 1, we see or=id(EG)=

f EG= -f Ed*Ja

II

,

II

from which we have (30)

389 Kuang-chao CHOU, Han-ying GUO, Ke WU and Xing-chang SONG

600

Tbirdly, as ,indicated in Ref.[4], for Lagrangian wbich bas been put into the global invariant form instead of considering tbe infinitesimal gauge transformation Q _gQg-l

,

G can also be generated by taking the variation A- A+oA, Q, DQ, and F fixed,

i.e. ,

6p (SP) G=o;rcre=6A ' from which J

a

=(o$!') oA

(3 ... )

Tben, from Eqs.(30) and (31), we have (32)

Now, consider tbe variation of the total tesimal gauge transformation (7)

La~rangian

(28) under the infini-

(33a)

Taking account of the equations of motion (29) and using the relations (26), we see (33b)

(33c)

In particular, tbe part concerning the cbiral fields is

390 Symmetric and Asymmetric Anomalies and Effective Lagrangian

=[Q,

6~

O~]

[

601

6!£]

6QI-a~ 6a~Q +a~ Q, 6a~Q

(34a)

or

+Q( 6D~~

6ft'. =£Q(cS~+6D ( Tt)AF-6E

~J

6!£)

=[Q, (~~ )-D~(6g~)]+D\.[Q'.(6g~)]

(34b)

Therefore, because of the equations of motion, the meson current

J~[Q]=[ Q, 6~~]

(35)

~

is conserved if a~only if 6~/6EIA,F=O, or is covariantly conserved if and only if (6~/6E)A,F=0. For ungau~ed Wess-Zumino-Witten Lagrangian, since ~= ~o+ ~cs is invariant under the global guage transformation, we obtain the conservation equation

F2

ai~ {+(Qa~Q)+16;1 E~"(1a (Qa"Q)(o.a(1Q)(QCl aQ)Y 5

t=0

(36a)

To the first order in w fields, this gives Fw

-ilra~(ysCl~w)=O .

(36b)

For gauged chiral fields without WZ term, ~=~o being invariant under local gauge transfornations, it yields the covariant conservation law

F2

IiD~[Q, D~Q]=O •

(37)

And for the total Lagrangian (29), the current (35) is neither conserved nor covariantly conserved, and the Euler-Lagrange equation can be written in the form (38) w~ere

we have collected all the terms from the normal ~art of the Lagrangian, into current and terms from the anomalous part, ~, into source. The current appear.ing in Eq.(38) is just the current in Eq.(37) and the one in Eq.(36b) to the "first order. It can be shown by straightforward calculation that the leading ~D'

order of the source S[Q.A] (terms containing no w's) is just Bardeen's asymmetric anomaly (apart from an overall constant coming from the coefficient in the expression of U). he.,

391 Kuang-chao CHOU, .Han-'ling GUO, Ke WU and Xing-Chang SONG

602

(39)

S [0, AJ=Ga[AJ • This is just what we found in Eq.(21). Similarly, Eq.(33b) implies that the total current defined as

(40 )

will be conserved if and only i f OseIOE=O. Now, from relation (32) and the fact thllt the Dormal part of the total Lagrangian, ~ + se F,is gauge invariant, we see J p is not conserved in general. (41) And the conserved current is (42) Eq.(33c) gives nothing new but the covariant conservation law (34b) as discussed above.

Notice that when a term ~ depending only on the gauge fields is subtracted from the Lagrangian se, both the definition of J [AJ and the form of the anomalies II .~-aIlJ~ will be changed simultaneously. However, the covariant divergence of the meson current, D"J\I[QJ, remains the same since 5e is irrelevant to the S Goldstone fields. For Bardeen's counter-term r[O,AJ, this is checked by a direct calculation. As is well known, as a consistent gauge theory, the anomalies G must be set to zero. This gives the anomaly free condition as discussed by Gross and Jackiw[7]. Alternatively, Lagrangian (29) may not be complete and there may exist some new sector (e.g. lepton sector) which also couples to the gauge fields and gives other anomalies to cancel the contribution of G. However, in this case, the role of the asymmetric anomaly Ga does not change. It still appears as part of the source coupled to the Goldstone fields, provided the Goldstone fields have nothing to do with the fields in the new sector. In conclusion, the effective Lagrangian introduced in our previous paper[3,4 J , i.e., in Eqs.(2),(9) and (10), is a correct one for describing the low-energy ..

A

r

interaction among Goldstone mesons and gauge fields. It has the correct symmetries of QCD and contains the information of both symmetric and asymmetric anomalies. The anomalies coupled to the whole system of Goldstone and gauge fields, as shown in Eq.(41), may be the symmetric ones or the asymmetriC one. It depends on the scheme of renormalization or, in other words, on the subtraction made

392 symmetric and Asymmetric Anomalies and Effective Lagrangian

603

or not, and gives rise to the anomaly-free condition. It must be the asymmetric anomaly which is coupled to the Goldstone fields just like the Abelian chiral artomaly ~oF*F in any consistent theory. This is not affected by the subtraction scheme or by the existence of the new sector. Obviously, the results of this note can be generalized to the case of any even dimensions without difficulty.

Note 4Qded in Proof After completing this work, we received two papers from O. Kaymakcalan, S. Rajeev and J. Schechter[10] and H. Kawai and S.H.H. Tye[ll]. in which the same effective action rU;,A] (first presented in Ref.[3J) are obtained hy different methods and the fact that r[O,A] is just Bardeen's counter-term is also noticed. We thank Professor H.T. Nieh for calling our attention to these papers.

Acknowledgements One of the authors (X-C.S) deeply thanks Professor C.N. YANG for his concern and inspiration. He'also greatly acknowledges the financial help obtained from a Fung King-Hey fellowship through the Committee for Educational Exchange with China. Thanks are also rll:e to Professor H. T. Nieh for his helpful discussions and careful reading of the manuscript. This work was supported in part by NSF Grant #PRY 81-09110 A-Ol.

Footnotes and References [1J E. liitten, HUcl. Phys. !E1..(I983}422.

{2J J. liess and B. Zumino, Phys.

Lett.~(1971}95.

{3J K.C. CHOU, H.Y. GUO. K. WU and X.C. SONG, Phys. Lett. 134B(1984}67. {4J K.C. CHOU, H.Y. GUO, K. IiU and X.C. SONG, Commun. in Theor. Phys.

~(1984}7].

{5J Here we mean the result presented in the Appendix of Ref. {4J, which is obtained by correcting some misprints and adding some missed terms. It produces correct anomalies and talees the form nearest to the original one in Eq. (24) of Ref. {IJ.

{6J Ii.A. Bardeen, Phys. Rep.

~(I969}1848;

R.Ii. Brown, C.C. Shih, and B.L. Young, Phys. Rev.

,!!!(1969} 1491.

{7J D.J. Gross and R. Jackiw, Phys. Rev. £!(1972}477. {8J The fact that the infinitesimal variation of the Chern-Simons 5-form gives anomalies has been independently noticed by B. Zumino, Y.S. liashington preprint 40048-17, to be published in Nucl. Phys.

IiU

correct

and A. Zee, University of

See also B. Zumino,

lectures given at Les Houches, LBL-16746, Aug.198]. {9J See R. Jackiw, Lecture given at Les Houches, MIT preprint CTP #1108, Aug. 1983. FOr original discussion about equation of motion and conservation law in anomalous case, see S. Adler in Lectures on Elementazy Particles and QUantum Field Theory, edited by

S. Deser, II. Grisaru and H. Pendleton (IIIT Press, Cambridge, IIA I970). {10J

o. KaljllJalecalan, S. ROljeevand

{IIJ H. Kawai and S.H.H.

~e,

J. Schechter, Syracuse preprint SU-4222-278, Dec. 1983.

Cornell preprint CLNS-84/595, Jan. 1984.

393 Vol.3, No.6 (1984)

Commun. in Theor. Phys. (Beijing, China)

767-710

ON THE lWG~IMENSIONAL NON-LINEAR cr MODEL WITH WESS-ZUMI NO TERM I. Classical Theory CHOU Kuang-chao( J'aJ*~ ) and DAI Yuan-bene ~i* InStitute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing, China

Received July

9, 1984

Abstract The classical solutions of the two-dimensional non-linear cr model wit.'! Wess-Zumino term are Sbolom to be equivalent to that without the Wess-Zumino term under a suitable transformation.

It is now well known that two-dimensional non-linear cr models possess a lot of rather interesting and mutually connected properties like the soliton solutions, the Racklund transformation and the infinite number of conservation laws. Thenon-linear cr model is of interest physically because it describes the low energy behavior of Goldstone bosons produced by the spontaneous symmetry breaking of an underlying fermion theory with chiral symmetry. It was pointed out by Witten[1] that the usual effective Lagrangian for the cr model has more discrete symmetries than the underlying fermion theory. Besides, it does not contain the term due to the anomaly caused by the triangular diagram (or bouble diagram in two dimensiQns) of the fermion loop. To cure these defects a second term called Wess-Zumino ·term has to be added to the effective La~ranp;ial1 which has compact form only in n+1 dimensions for a cr model in n dimensions. Then it was shown by Witten and others[2,3] that the two-dimensional SU(N) chiral model with Wess-Zumino term added is equivalent to a free massless fermion theory when the coupling constants satisfy a special condition. In the present note we shall study only the classical behaviors cf a two-dimensional cr model with Wess-Zumino term added. The quantum theory will he dealt in a separate paper. It will be shown in the follcw.ing that the classical theory of two-dimensional cr model with Wess-Zumino term is almost independent of the two coupling constants. Classical solutions from models with different coupling constants are shown tu be eauivalent to each other by a transformation. The special condition that makes the model equivalent to a free massless fermion theory becomes a singular point of the general transformation mentioned above. The action for the two-dimensional cr model with Wess-Zumino term added has the followinp; form

394 CHOU Kuang-chao and DAI Yuan-ben

768

(1)

where the field g(x) is an element of a simple compact group G; n is an integer and A is the coupling constant. D is a domain in three dimensions whose boundary is the two-dimensional space time under consideration. The action is invariant under the chiral transformation

where gL and gR are two independent elements of the group G. The equation of motion for the field g(x) can be written in terms of a pure gauge field (2)

in the following form (3)

The integrability condition for g(x) is (4 ).

when a gauge field

A~

is found to satisfy Eqs.(3) and (4), the field

g(x)

can

be found by integrating Eq.(2). It is more convenient to work in th'=? li.cht

cone coordinates

f;=

~ (t:+-x)

n- let-x)

12

'j

(5 )

'

in which Eqs.(3) and (4) become (3' )

and (4' )

where a= n>.2

F

is the only parameter of the model. Witten studied the case where a=±l

(6)

395 an

the XWO-Dimensianal Nan-Linear a Model with Wess-Zumino Term

769

and showed that the model with coupling constants A and n satisfying this special condition is equivalent to a free massless fermion theory when the grnup G is SU(N)[2], The usual non-linear a model corresponds to the case a=O, where the equation of motion (3) and curvatureless condition have the form

a;An+-'nA;=o , a;An-anA;+[A;, An]=O .

I

(7)

Now we star't to prove the equivalence of models with a different parameter a. For this purpose let

.I

and

(8)

which satisfies a conservation equation from Eq.(3') (9)

One can easily show that

B~

is also curvature less since

(10)

Using the equation of motion for

A~

=(a~A +a A~)-a(a~A -d A~) .. II <; II II ..

n"

,

(11 )

we obtain from Eq.(lO)

(12)

Therefore B is a pure gauge field satisfying the conservation equation just like the A~(x,a) for the parameter a=O. This means that all the classical solutions of the two-dimensional a model with a¢O can be obtained from that with a=O by the transformation. Eq.(8).

396 CHOU Kuang-chao and DAI YUan-ben

770

It is noted that the transformation Eq.(8) is singular at a=±l which is the case considered by Witten and others[2,3]. In that special case the finite solutions at a=±l corresponds only to part of the solutions at a=O that are either left-handed or right-handed moving (i.e. , either Bf;=O or Bn=O). Since we have established the equivalence of the classical solutions of two-dimensional v model with different coupling constants except for the singular pOints.a=±l, we conclude that all the classical behaviors are the same fer these models. It is also easy to prove directly that they all contain hidden symmetries making the infinite number of conserved charges form a Kac-Moody algebra. As pOinted out above, only part of the solutions for a=O correspond to all the finite solutions at a=±l, and it is interesting to speculate that the model with a~±l contains massless fermion in the tree approximation as part of its particle content. We shall return to this question in a paper on the auantum theory of these models.

References II} E. Witten, Nucl. Phys. B223 (1983)442. 12} E. Witten,

Commun.

Hath. Phys. 92(1984)455.

13} P.Di Vecchia, B.Durhuus and J.L.Petersen, "The Wess-Zumino Action in t~ Dimensions and Non-Abelian Bosonization", preprint NBI-HE-84-02(1984); P.Di Vecchia and P.Rossi, "On the Equivalence between the "'ess-Zumino Action and the

Free Fermi TheoZ'll in two Dimensions" preprint TH 3808-CERN (1984).

397 CP VIOLATION FROM THE STANDARD K>DEL

Kuaog-chao Cllou

Institute of Theoretical Physics Academia Sinica Beijing, China IHTRQDUCTION Two decades have passed since the first observation of CP violation in kaon decayl.

The subject is still not well under-

stood and the progress is rather slow compared with what has been achieved in the other branches of weak interactions.

As we all know now, nature has chosen the standard SU(2) x U(l) gauge model to describe physics at an energy scale below 100 GeV.

Both W and ZO bosons have already been seen

within the error predicted by the theory2.

It is therefore of

great interest to accomodate CP violation in gauge theories which seem to be the most promising ways from a theoretical point of view.

As Kobayashi and Maskawa 3 (K-M) first pointed out, CP violation can occur in the standard model through complex phases in mass matrix with more than two generations.

For three

generations favored by the present experiment there is only one phase causing CP violation.

Could this single phase be suffi-

cient to explain all the CP violation effects? welcome if it could.

It is certainly

However, it can not be answered a priori.

The origin of CP violation is closely related to that of the 609

398 masses and the number of generations, which in turn are described by physics at much higher energy scales.

It would not be a

surprise if some new ingredients had to be added to solve the CP problem.

We shall wait and see.

Since there were excellent review papers not long ag0 4 , it is unnecessary for me to repeat all the known results to you. What I would like to report is a recent analysis of CP violation in the K-M model after the measurement of the unexpected long lifetime of the b-quarks. The outline of this talk is as follows: I.

Parametrization of the K-M matrix;

II.

Physics of £ and £' in kaon systems;

III. Neutral particle-antiparticle mixing and CP violation in BO-BO system;

IV.

CP violation in partial decay rates of particles and antiparticles;

V.

Concluding remarks.

I. PARAMETRIZATION OF THE K-M MATRIX

For three generations of quark the K-M matrix containing three angles and one phase is usually expressed in the following form

V

=

=

610

Vud

Vus

Vcd

V cs

V td

V ts

Vub \

V)

V:b

(I.I)

399 where ci(si), i = 1,2,3, are the cosine (sine) of the angle 6i·

The Cabbibo angle 61 is determined to be~ s

1

=

•

227+. 0104 -.0110

0.2)

Recent measurements on b quark lifetime and the branching ratio rb+u/rb+c have put stringent bounds on the matrix elements IVCb/VUbl.

Their values can be found in the talks given by

Lee-Franzini and Kleinknecht in this conference. IVCbl = 0.0435 ± 0.0047 ,

(1. 3)

IVub IV cb I ~ O. 119

0.4)

•

Both IVcbl and IVub/Vcbl have been reduced from the 1983 values 5 and The fact that IVcbl is small and of the order of s1 2 can be used to simplify the K-M matrix.

In a first order approxima-

tion where ReVij are correct to order s1 3 and ImVij to order SIS

we have c1 V

=

s1 c 1 -s2 s3 e

-s1

s2+ s 3e

-s1 s2

i6

i6

0.5) -e

i6

Writing Vts and Vcb in the _following form V ts

=

Vcb

= s3

s2 + s3 e

i6

=

Ivtsle

i6 ts

0.6)

i6

=

I Vcb I e iticb

0.7)

and + s2 e

611

400 it is easily proved that V

.~

ts

.. el. u V

*

(1.8)

cb

Hence we obtain (1.9)

and (1.10)

One can now redefine the phases of the band t quarks by a transformation (l.ll)

(1.12)

+

and get from Eqs. (1.5)-(1.12) a form first suggested by Wolfenstein6

IV cbl

V ..

-io

-s s e t s

1 2

(1.13 )

1

The phases 0ts' Ocb are related to 0 by the following relations: (1.14)

(1.15)

612

401 The advantage of the present form for the K-M matrix is that ImVij is always proportional to a common factor (1.16)

which is the appropriate parameter measuring CP violation effects in various processes.

A similar but rigorous representation of the K-M matrix was obtained recently by Chau and Keung7 •

Since both s2 and s3 are proportional to IVCbl, it is more convenient in numerical calculations to scale it out. We write s3 :: a

Iv cb I =.!.... sl Iv ub I

(l.17)

(1.18)

From the experimental bound Sq. (1.4) and the value of sl we find a

<

(1.19)

.524

(1.20)

With given a and B, s2 can be solved from Eq. (1.9) to be

(I.21)

where we have adopted the convention of positive s2 and s3.

The

solution 82 with positive (negative) sign in the bracket in Eq. (1.21) corresponds to coso

II.

<0

(coso> 0) in Eq. (1.9).

PHYSICS OF E AND E' IN lAOS SYSTEMS So far CP violation has been observed only in neutral kaon

systems.

The ratio of the amplitudes for KL

+

2w and KS

+

2w

613

402 ~s

given in the standard notation as

=e

n+_

where 00

+ e'/(l + 00112)

I

(2.1)

e - 2e ' /(1 - 1200) ,

(2.2)

= ReA2 /ReA O is

known to be approximately .05.

of the amplitudes AI for KO

+

In terms

2w(I), with I the final state

isospin, the mass matrix element M12 , e and e ' can be expressed in the following forms

(2.3) and

eI where

or

_ -

1

72 00

(

~2 - ~O)

(2.4)

are final state interaction phases and (2.5)

The experimental value of e is well established to be Ree = 0.00162 ± .000088 while that of e ' is still uncertain.

As

will be discus&ed later, accurate measurement of lel/el is extremely .important in our understanding of the origin of CP violation. The mass matrix element for KO-iO transition consists of a short distance part usually identified to be the contribution from the box diagram and a long distance part determined by the low energy intermediate states sd soft MI2 = M12 + M12 The mass difference of relation 614

It and KS is related to ReM12 by the

(2.6)

403

2 ReM

12

2(ReMsd 12 + ReMsl2oft)

=

(2.7)

soft has been estimated long ago 8 • The real part of M12

Its value

is very sensitive to the small parameter that breaks the SU(3) Even the sign of ReM~~ft can not be determined sd reliably. The box diagram contribution to 2 ReM 12 is dominated by charm quark exchange and consists only 1/4 - 3/4 of symmetry.

dMK. 5

Since ReM 12 has nothing to do with CP violation, I shall use in the following the experimental value of ~MK in evaluating the parameter E.

However, I would like to remark that

if one finds eventually that AK_

"-1{

*2

soft box (ReM12 + ReM1 2) '

it will indicate the existence of new

=2

~S

short distance

interaction besides the box diagram, and thus, possible new sources of CP violation. soft can be estimated by current The imaginary part of M12 algebra and Penguin diagram dominance. soft ImM 12 -...:;.~-= soft ReM 12

2~O

In this approximation

(2.8)

•

Using Eqs. (2.7) and (2.8) it is possible to eliminate the soft 9 part of M12 and rewrite Eq. (2.3) in the form 1 E '"'

where

{2

b

sd

4" ImM12 e

sd ReM 12

(~ + 2~O -xM")

(2.9)

M~; is calculated by the box diagram 10

615

404

m2 .f.n _t_ }

(2.10)

m 2

c

where L

(2.11) ~

f(x)

(2.12)

and ntt .. 0.6, ncc = 0.7, nct = 0.4 are the QCD correction factors.

The factor BK accounts for the uncertainty in

determining the matrix element (2.13) Current algebraic estimation tells us that BK is around 1/3, while some lattice calculations ll give the value about I, close to the vacuum-insertion value.

We shall keep BK to be a

parameter in the following calculations. Eqs. (2.9)-{2.l0) have been used to predict the minimum top quark mass when the K-M angles are given, or the other way around, to set lower bound on the CP violation parameter Xcp = s2s3sino when the top quark mass is assumed S ,l2-lS.

In

these calculations the second term in Eq. (2.9) proportional to

~o was neglected and the 1983 experimental values of

IVub/Vcb I were used.

Ivcb /

and

The parameter ~o could be estimated by using experimental value of ReAO • AO and Penguin diagram value of ImAO• found 13 - 14 that 616

It is

405

(2.14)

~O

where Q6 is a (V-A)x(V+A) Penguin operator with Wilson coefficient c6 in the effective Hamiltonian for Penguin diagram. c6 = Imc6/s2s2sino is estimated in the leading logarithmic approximation to all orders in the strong interaction to be -D.l and is quite stable against the choice of parameters. For the matrix element <2n(I=O)IQ6IKO) shall follow the analysis of Gilman and Hagelin 13 to use the bag model value for a conservative estimation.

One finds finally that

(2.15)

A similar estimation gives ~2 • O.

The parameter

1£1/£1

can then

be determined to be

(2.16)

Now a combined analysis of

1£1

and

1£1/£/

can be made with

the help of Eqs. (2.9), (2.10), (2.15) and (2.16) when s3(a), s2sino(S), IVcbl and BK are given. Tables 1-7 and Figs. 1-3, where both be functions of a and S.

The results are given in

me

and

1£1/£1

are shown to

The minimum top quark mass occurs at

the point where s3(a) saturates its upper bound and 0 in the second quadrant where s2 is larger.

Its value increases as BK

and IVcbl decrease. For BK = 0.33 and /Vcbl < .059 the minimum top quark mass is already over 60 GeV. It is also noted that

1£1/£1

is large at the point where mt is minimum.

present experimental value of

1£1/£1 = -

The

.003±.015 is barely

consistent with the one calculated at the point of minimum top quark mass.

617

.j:>.

0

TABLE

1.

1£'1£1,

Values of Mt'

0)

IVcbl = 0.0388 and

CD

I XBd and XBsof Eq. (3.10) as functions of s2/1Vcbl for BK ; 0.33,

IVub/Vcb I =

.119

821/Vcbl

.852

1.031

1.110

1.171

1.222

1.266

1.305

1.339

1.395

1.440

M (GeV) t

325.3

254.3

235.8

226.0

221.0

219.3

220.2

223.4

236.9

162.4

1£' 1£1

.0057

.0065

.0066

.0066

.0065

.0063

.0061

.0058

.0051

.0043

XBd

2.10

2.10

2.17

2.26

2.38

2.52

2.70

2.90

3.45

4.28

X Bs

56.1

38.3

34.2

32.0

30.96

30.6

30.8

31.5

34.4

40.07

Same as Table 1

except

! Vcb ! =

.0435

TABLE 2. sz'IVcbl

.852

1.031

1.110

1. 771

1. 222

1.266

1.305

1.339

1. 395

1.440

M (GeV) t

233.6

180.9

167.6

160.7

157.3

156.3

157.3

159.9

170.3

188.9

1£' 1£1

.0071

.0082

.0083

.0083

.0082

.0079

.0076

.0073

.0064

.0054

~d

1.58

1. 57

1.62

1. 69

1. 79

1.90

2.03

2.19

2.63

3.27

X Bs

42.4

28.7

25.5

24.0

23.2

22.96

23.2

23.8

26.2

30.6

m

TABLE

l.

Values of Mt • /£1/£/. XBd and xBsas functions of s2'IVcb and I Vub/Vcb I = .119

0.0482

.852

1.031

1.110

1.171

1.222

1.266

1.305

1.369

1.419

1.458

M (GeV) t

171. 3

132.1

122.4

117.6

115.3

114.8

115.7

121.4

132.5

150.9

1£1/£1

.0087

.0100

.0103

.0102

.0100

.0097

.0094

.0084

.0073

.0060

1. 21

1.19

1. 22

1. 28

1.35

1.44

1.55

1.83

2.26

2.92

32.4

21.7

19.2

18.1

17.5

17.4

17.6

19.0

21.8

42"607

X Bs

TABLE

4.

Same as Tab Ie 3

except

\ \b\

.0588

l

.852

1.031

1.110

1.171

1. 222

1.266

1.339

1. 395

1.440

1.474

M (GeV) t

91.9

70.4

65.3

63.0

62.1

62.3

64.8

70.3

79.1

93.4

\£1/£1

.0130

.0150

.0153

.0152

.0149

.0145

.0133

.0117

.0099

.0079

XBd

.68

.64

.65

.68

.72

. 78

.93

1.16

1. 52

2.08

X Bs

IB.l

11.6

10.2

9.62

9.40

9.43

10.09

11.6

14.2

1B.6

s2/IVcb

...co

BK = 0.33. IVcbl

s2'\V Cb \

XBd

m

l for

I

.j:>.

0

-...j

~

0

00

0'1 N

TABLE

5.

0

Values of Mt • and IVub/Vcbl

j

1£'/£1. = .119

XBd and xBsas functions of s2/IVcb l for BK

= 1.

I IVcbl

0.0388

.852

1.031

1.110

1.171

1.222

1.305

1.369

1.419

1.458

1.499

M (GeV) t

113.4

81.9

76.1

73.6

72.9

75.7

82.0

84.2

111.8

161. 3

1£'/£1

.0057

.0065

.0066

.0066

.0065

.0061

.0055

.0047

.0039

.0025

XBd

.41

.34

.37

.39

.42

.50

.61

.85

1.18

2.21

XB s

11.0

6.3

5.8

5.5

5.43

5.72

6.30

8.20

10.80

24.9

s2/IVcb

TABLE 6.

Same as Tab Ie 5

except

Vcbl

=

.0435

s2/lv cb l

.852

1.031

1.110

1.171

1.222

1.305

1.395

1.440

1.474

1.499

M (GeV) t

71.5

49.5

45,1

43 .• 7

44.0

47.5

58.4

69.8

86.8

114.7

1£' /£1

.0071

.0082

.0083

.0083

.0082

.0076

.0064

.0054

.0043

.0031

XBd

.24

.190

.188

.197

.217

.28

.46

.67

1.01

1.64

X Bs

6.5

3.47

2.95

2.79

2.81

3.24

4.66

6.25

9.02

14.14

TABLE

7.

Values of Mt •

1&'/£1.

and IVub/Vcbl

=

1.

\Vcb\

0.0482

.119

.852

1.031

1.110

1.171

1. 222

1.305

1.369

1.419

1.458

1.487

M (GeV) t

42.9

24.22

20.8

20.5

21.8

26.9

33.9

42.8

54.6

71.7

1£' 1£1

.0087

.0100

.0103

.0102

.0100

.0094

.0084

.0073

.0060

.0046

XBd

.12

.062

.054

.058

.071

.12

.21

.34

.55

.92

X Bs

3.30

1.14

0.85

.82

.93 6

1. 38

2.14

3.29

5.02

8.06

TABLE 8.

Same as Tab 1e 7

except

Vcb I

=

.0435 and Re

sd M12/~ =

0.36

/ 1Vcbl

.852

1.031

1.110

1.171

1. 222

1.266

1.305

1.338

1. 395

1.440.

M (GeV) t

82.8

60.4

55.7

53.8

53.5

54.3

55.9

58.3

65.2

75.7

1£' 1£1

.0071

.0082

.0083

.0083

.0082

.0079

.0076

.0073

.0064

.0054

XBd

.316

.267

.272

.285

.308

.340

.378

.429

.559

.767

X Bs

8.44

4.88

4.29

4.04

4.00

4.11

4.31

4.65

5.58

7.18

82

en

=

/ 1Vcbl

82

!'.)

XBd and xBsas functions of s2/1Vcbl for BK

I

.j:>.

0

co

.j:>.

...... o

0)

N N

TABLE

9.

Values of Mt • le'/EI. XBd and xBsas functions of s2/1Vcbl for BK IVub/VCbl

= .119

and Re

M~~/~ =

1. IVcbl

0.0482.

.36.

v 2 / 1 cbl

.852

1.031

1.110

1.171

1. 222

1.266

1.305

1.339

1. 395

1.440

M (GeV) t

54.7

37.7

34.3

33.3

33.6

34.8

36.6

38.9

45.3

54.2

I E' lEI

.0087

.0100

.0103

.0102

.0100

.0097

.0094

.0089

.0079

.0066

XBd

.189

.143

.139

.147

.163

.186

.217

.256

.365

.530

XB s

5.05

2.61

2.19

2.08

2.111

2.25

2.47

2.77

3.64

4.96

8

411

200

~

2 :f

IVcbl =0.0482 100 50 20 10 00.05 0.20

0.40

0.60

0.80

1.00

sinS

Fig. 1

Top quark mass for BK

623

412

200

..

:; ~

~

100 50

20

10~~~~~____~~~~~-L~~

o

0.10 0.20

0.60

0.80

1.00

sin 8

Fig. 2

624

Top quark mass for BK

=1

and IVub/Vcbl

=

.119.

413

IVcbl'0.0588

0.016

Sin

Fig. 3

1£'1£1

8

as function of sino.

625

414 To get a definite conclusion we need more accurate measurement on bound on mt.

I€'/€/,

lower bound on IVUb/VCbl and upper

More reliable theoretical evaluations of BK'

C6<2w(I=0)IQ6IKO> and ~~ft are also required.

What we could say

at the present is that new sources of CP violation besides the K-M phase 0 might exist if

/€'h/.5.

~

is found to be around 40 GeV,

.005 and BK .5. 0.6.

The second term proportional to ~O in € is also estimated and the result is given in Tables 8-9. For ~ - 1, 2 ReMbox ~ 3/4 AMK, the effect of this term is appreciable at 12 the point where

me

is a minimum.

It raises mt roughly by 30%

owing to the negative sign of ~O.16 III. NEUTRAL PARrICLE-ARrIPAR.TICLE HIXING AND CP VIOLATION Dl BO-BO SYSTEH.

The mass eigenstates of neutral bosons P and Pare (3.1)

where €p is determined by the mass matrix elements relating pO and

pO. (3.2)

where IItj .. ~j - ifij are the mass matrix elements of the neutral pO_pO system. For a state being pO at t = 0, /~(t=O» time t

626

.. /pO> and later at

415

'.(t»

= f+(t) /pO)

1-&p + I+Ep f_(t)

/pO) ,

(3.3)

where

(3.4) Mixing of

pO and pO is necessary for the observation of CP

violation effects.

There are two cases of maximal mixing. In the kaon case ~r/r • 1. either with the KO or i O to begin with,

it will quickly end up as KL' which is almost an equal mixture of KO and KO• The second possibility occurs when ~m/r = 1 » ~r/r In this case, before decaying, the state oscillates quickly between pO and

pO

and appears as an equal mixture of pO and

pO.

Due to the simple fact that the decay width r for the D and T particles are K-M angle nonsuppressed, yet &m and K-M angle suppressed; the values of

~m/r

and

~r/r

~r

are always

are both small

and the observation of CP violation in neutral D and T systems is extremely difficult. The situation is different in B systems.

There one expects

large mixing effects and possibly large CP violation effects S ,17-18. One special feature of the B systems is that the complex parameter £B is almost imaginary and of the order I.' The imaginary nature follows from the fact that the phase of M12 is the as that of r12 and the condition M12 » f12. 1•

,ame

Therefore the observable effects, depending on the Re£B such as the fractional difference of same-sign dilepton production and the asymmetry 1n semileptonic decay. are very small.

As has been emphasized by Bigi, Carter and Sanda17 that nonleptonic on shell transition in the bottom sector might produce CP asymmetries of the order 10- 1-10-2 , whereas the effects due to CP impurities in the mixing is less than 10- 3 • The effects fall into two categories.

The asymmetry for

6V

416 initially pure BO and BO states to decay into the same final state f is found to be A ... -xa sin2f 1+y cos2
(3.5)

where ~M

y = ~r 2r

x =-

r

(3.6)

1_-..2

a =--L

(3.7)

1+x2 The angle ~ ... (

is defined as

1 - &B 1

M

) - ... - e

-2i~ ~

+ ~ M

(3.8)

with M and M the amplitudes

(3.9) The se~ond type of experiment produces a BOBO pair with charge parity C and measures the decays of one B meson into a lepton and the other to a hadronic state f. to be A ... -2xa2 sin2 f

for C

The CP asymmetry was computed

even

l+r+ycos2
0

for C = odd

(3.10)

For both Bs and Bd systems the parameter y is small compared to unity and can be neglected in Eqs. (3.5) and (3.10). The parameters

Xs

and xd for the Bs and Bd systems are

calculated using the box diagram value for ReM12 and the experimental value for the B meson lifetime 7 ,18. Results of the calculated x are shown in Tables 1-9, which depend very much on the parameters used.

628

417

For Bd system the angle

~

depends on the decay channel

=

for b

+

c decays

tg4> = tgll

for b

+

u decays •

tg~

For Bs system the angle

~

tg~

(3.11 )

is very small for the b+c channel and for b

+

(3.12)

u decays •

The asymmetries for Bd system can reach 20-40% in a certain range of parameters.

However, if

IE'fEI

is found to be small in

future experiments, 8in& will become small and the CP violation effects in BD_BD system will also be small. IV.

CP VIOLATION IN PARTIAL DECAY RATES OP PARTICLES AND ANTIPARTICLES

To begin with I shall recall some general remarks made by Pais and Treiman 19 concerning the constraints imposed by CPT invariance.

Let fi be a set of final states connected by

sfrong or electromagnetic interactions into which the particle p can decay and £i the corresponding conjugate states. invariance requires th~ partial width of p and

p to

CPT be equal (3.13 )

provided this set of final states has no strong or electromagnettic interactions with the other decaying channels.

If the set

consists of only a single state no information about CP violation can be found.

For two final states the decaying amplitudes of

the particle have the form (3.14 )

629

418

where f and g are complex coupling constants in the effective weak interaction while Mi' i = 1,2 are different amplitudes leading to the same final state. for the antiparticles are

-a

= f M

1

*

+g~

The corresponding amplitudes

*

(3.15)

we can easily calculate the asymmetry to be

(3.16)

In the case of strange particle decay the asymmetry is found to be very smal1 20 • In this case

M2)

Im(M1*

IMl12 + IM212 are final state interaction phases and Im(f *g) both of which are very

s~all.

«

sl 2s2s3 sin6,

As the real parts of the

corresponding coupling constants f and g in kaon decays are of the order 1, the asymmetry of partial rates for strange decays is very small of the order 10- 5 - 10- 6 • In general the number of the final hadronic channels increases rapidly as one progresses from strange to bottom decays.

The final state interactions are much stronger and

2 Im(Ml~)/(IMI12 + IM212) might reach several percent in B-decays.

For those decaying channels where Ifl2 and

Ig l2

are

comparable with Im(f*g) « s12s2s3 sin6, the asymmetry can be quite big. Examples 21 are given in Table 10. Chau and Cheng proved a theorem using quark diagram techniques to show that the CP violation effects in the asymmetry of partial decay rates of particles and antiparticles are always proportional to s12s2s3 sin6 which is a very small number of the order of

630

419

10- 5 - 10-6 •

The apparent I arge asymmetry in the B-decays is due

to the smallness of the denominator in Eq. (3.16).

Therefore for

chose decay channels where che asymmetry is large the branching ratio is always small.

The number of events required to observe

the CP violation effects in B-decay is still large, of the order of 10 5-10 6 • It is not an easier task than the observation of CP violation in kaon decays.

V.

CONCLUDING REMARKS 1)

1£'1£1

Measurements of

and the lower limit of IVub/Vcbl

as well as the discovery of the top quark are crucial in finding a consistent picture of CP violation within the standard model. More careful calculation of also necessary.

~, C6<2w/2=0IQ6IKO> and M~~ft are

A low value of

me

less than 40 GeV together

with BK around 0.33 might rule out the K-M model with three generations of quarks.

In this case the K-M phases might still

be the only source of CP violation if more than three generations are found.

A reduction of the ~pper limit of

pushes the theoretical sin~.

me

1£'1£1

not only

up but also reduces the value of

thus making the observation of other CP violation effects

more difficult. 2)

CP impurities in BO_BO mass eigenstates are very small.

Therefore observations on the fractional difference of same-sign diplepton production and the asymmetries in the semileptonic decays are very difficult. 3)

The mixing for BsO systems is large, but the asymmetries in the BsO(b+c) channels are small. The mixing for Bd O systems and the asymmetries in the decay channels BdO(b+c) may reach several percent in a certain range of parameters. 4)

CP violation effects in partial decay rates of particles

and antiparticles are small for strange particle decays. much larger for charged B decays.

It is

However the number of events

631

~

a m

Lt)

N

TABLE 10

Asymmetry and number of events needed to observe CP violation in D and B decays. Taken from Chau and Cheng 21 •

I

• • GK-

u

"

A.pliluiea

lIeaelioa

.2. /2

(V b,V* (e) C

o V

• O-oG*

tree

ub

V* (aobodoc)(

~

i-u.c.t

e

rii D

Vcb V~. (d+e) + V

ut.

... -0 0

V* (a+e) U5

VcbVcd(d+e) + VubV*ud(a+e)

+ .OD-

I (V hV*d(doe) /2 c c

-

+ VubV~d(b+e)1 F+ .. K01l.

VudV~d(a+e) + V

_6"10- 2

).8'1O-~

1.]"10'

(0)

(-1.r,-1O- 2 )

1.4"10-'

(7.1·IO b )

-1.r,>10- 2 (-0.81»

VcbV~.(b+e) + Vubv~.(d+e)

+

1- •

No. of Event. Heeded

0

UI

VCbV~d(a+e)

K-J/,

Ie

C8

+ VubV~d(d+e) +

"Penluia

V* (d-e)

us ca

1>10- 1 (7. )"10-")

-1"10-'

]>10- 3

1.]>10' 3

(4.1"10- )

0.»10 2

S"10- 3

2.P10

2 "IO-~

].5-105

( 1.8"10- 2 )

(1.4-10")

)

9

V·bV. (;) I

as 0.19 (6.2-10- 2 )

_1>10- 2 (7.2"10-')

-4 _10- 2

1.7_10- 3

2.1"10-"

2.7>10 6

(-0.86)

(1"10- 3 )

(8.S-IO-')

(1.6-10 3 )

-0.17

-2.2_10- 2

1.1'10-'

2.1"10'

(-0.R6)

(-S.7-10- 3 )

(4.8-10-')

2.8"10'

-1.6'10-"

-l.l-IO-~

1.8"10- 3

(_2>10- 3 )

(-1.7 xIO-',

0.4 x I0- 3 )

4.5-10 9 0.4"\0"

421

needed to observe the CP violation effects is large, of the order 10 5-10 6 •

ACKNOWLEDGMENT This talk is the result of a collaboration with Wu Yue-liang and Xie Yan-bo. Discussions with Profs. Li Xiao-yuan, Chu Chen-yuan and L.-L. Chau have helped enormously in improving my understanding of the problem. I would like to thank Mrs. Isabell for her kindness and support in typing the manuscript. REFERENCES 1.

J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay, Phys. Rev. Lett. 13 (1964) 138.

2.

G. Arnison et ale , Phys. Lett. 126B (1983 ) 398; ibid 129B (1983) 273; P. Bagnaia et ale , Phys. Lett. 129B (1983) 130.

3.

M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

4.

For a recent review, see L.-L. Chau, "Quark Mixing in Weak Interactions", Phys. Rep. 95, No.1 (1983).

5.

L.-L. Chau and W.-Y. Keung, preprint BNL-23811 (1983).

6.

L. Wolfenstein, Phys. Rev. Lett. 51 (1984) 1945.

7.

L.-L. Chau and W.-Y. Keung, BNL pre print (1984).

8.

C. Itzykson, M. Jacob and G. Mahoux, Nouvo Cim. Supple 5 (1967) 978.

9.

J.S. Hagelin, Phys. Lett. 117B (1982) 441.

10.

T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297.

11.

N. Cabibbo and C. Martinelli, TH-3774 CERN (1983); R.C. Brower, G. Maturana, M.B. Gavela and R. Gupta, HUTP-84/A004 NUB # 2625 (1983).

12.

P.H. Ginsparg, S.L. Glashow and M.B. Wise, Phys. Rev. Lett. 50 (1983) 1415.

13.

F.J. Gilman and J.S. Hagelin, preprint SLAC-PUB-3226 (1983).

14.

P.H. Ginsparg and M.B. Wise, Phys. Lett. 127B (1983) 265.

633

422

15.

Pham Xuan-Yen and Vu Xuan-Chi, preprint, PAR LPTHE 82/28 (1983).

16.

K.C. Chou, Y.L. Wu and Y.B. Xie, preprint ASITP-84-005 (1984).

17.

J.S. Hagelin, Nucl. Phys. B193 (1981) 123; B. Carter and A.I. Sanda, Phys. Rev. Lett. 45 (1980) 952; Phys. Rev. D23 (1981) 1567; L.I. Bigi and A.I. Sanda, Nucl. Phys. B193 (1981) 85; Ya. I. Azimov and A.A. Iogansen, Yad. Fix. 33 (1981) 383, [Sov. J. of Nucl. Phys. 33 (1981) 205].

18.

L.I. Bigi and A.I. Sanda, preprint NSF-ITP 83-168 (1983); L. Wolfenstein, preprint CMU-HEG 83-9; NSF-ITP-83-146 (1983); E.A. Paschos and U. TGrke, preprint NSF-ITP-83-168 (1983).

19.

A. Pais and S.B. Treiman, Phys. Rev. D12 (1975) 2744.

20.

L.L. Chau and W.Y. Keung, Phys. Rev. D29 (1984) 592.

21.

J. Bernabeu and C. Jarlskog, Z. Phys. C8 (1981) 233; L.L. Chau and H.Y. Cheng, BNL preprint (1984).

634

423

DISCUSSION PAVLOPOULOS: Does a t-quark mass of 60-70 GeV lead to a hopelessly small E '/E? CHOU: No, if ~ is around 0.33, E'/E can reach .01 for rot ~ 60-70 GeV. HITLIN: If IEI/ELis small and sin~ is of the appropriate value, then the t quark ss could be greater than the W or Z mass. Could enough tt pairs be produced at the SPS collider to make the decay t + W + b a plausible explanation of the recently reported events at the SPS Collider? CHOU: We better keep in mind such possibilities.

635

424 CHINESE PHYSICS LETTERS

Nov. 1984

Vol.1. No.2

TOP QUARK MASS AND THE FOURTH GENERATION OF QUARK CHOU Kuang-chao, WU Yue-liang, ~IE ran-bo (Institute of Theoretical Physics, Academia Sinica, Beijing) (Received 9 August 1984) To explain CP violation in Kaon system in the light of the recently measured b-lifetime within the franework of three generations of quarks, the top quark mass has to be greater than 50GeV if the current algebra value of the factor ~ is adopted and [e:'/E\ < 0.01. In this letter a fourth generation of quark is considered which can fit the present experimental data on CP violation, and KL+~+~- decay rate for top quark mass is around40GeV. The mass of the new charge 2/3 quark is predicted to be over 100 GeV.

The unexpected long lifetime of quark in the picosecond rangJlJ has changed the

whole picture on the CP violation and the mass

difference

In the Kobayashi-Maskawa theor~ with three genera-

in the KL-KS system.

tions of quark the mixing angles &J. and 63

determined from the b decays

are found to be very small[3]. Consequently, the contribution due to top quark exchange to the box diagram of the small unless

top quark mass

is

large.

KO-KG

mass matrix element

is

For top quark mass less than 1

TeV there will be no appreciable effect to the Kr. -Ks

mass difference by

top quark exchange. All theoretical caculation of the box diagram contains matrix element relating the KO and i{0 states to the product of quark operators.

Current

algebra evaluation[4\ of this matrix element differs from its vaccuum insertion value by a factor of BK=0.33±O.17. In a

previous worJJ5] we have said the

following:

New sources of CP

violation besides the single K-M phase might exist if IlIt around 40GeV,

[E'/E l:ii

ticle of mass

around 40GeV,

0.005 and BK:ii a

is found to be

0.6. Just a few weeks ago a new par-

candidate of the top quark,

was announced

by UAL group in CERrJ6J • The measurement of E'/e: has also reduced its upper bound[7J.

Now we

are facing the possibility that something new might be

needed in order to explain all the existing experimental data.

The simp-

lest possibility is to add a new generation of quark. Standard model with four generations of Oakes[B] .

been

studied by

He pointed out that a relation Mt' =Mt=40GeV exists if 8uras an-

alysiJ9I on KL -I
quark has

mass difference and an

KL"'IJ+~-

decay rate is used.

Buras

upper bound of about 40GeV on top quark mass.

in Buras calculation the KL-KS mass difference is

How-

c:x:mpletely attri-

buted to the top quark exchange which requires large mixing angles ~ and ~

contradicting the b-lifetime measurement.

Actually, the main

contri-

bution to the ~ -Ks mass difference Comes from the charm quark exchange.

425 48

CHOU Kuang-chao, et al.

Vol. 1, No.2

Therefore the upper bound of top quark mass and the relation Mt'=Mt found in this paper are not

justified.

A more careful estimation can raise

the upper bound of Mt up to at least 140GeV. We have redone the calculation of CP violation parameters and in Kaon decay and the

~"'lJ.+\l-

decay rate based on four generations of quark.

re are now six mixing angles and three complex phases. Kobayashi-Maskawa parameters a

The-

By varying these

satisfactory explanation of all existing

data can be found with the top quark mass chosen to be 40GeV.

The mass

of the new charge 2/3 quark is found to be around or even greater than 100GeV, for B\C< 0.6, 1£'/£\=0.008, 'b=1.S, the matrix elements V~are

Vcd

and

taken to be approximately equal to that in the case of three gen-

erations,

other

parameters Can vary freely.

The KL -Ks mass

difference

still can not be accounted for only by the short distance contribution from the quarks.

A

detail account of the calculation will be presented

elsewhere. We would like to thank Profs.

L~L.

Chan, G. Kane, J.T. He for inform-

ing us the experimental results on I~Y£I and top quark mass. REFERENCES [1] Preprint, SLAC-PUB-3323 (1984). [2] M. Kobayashi and T. Maskawa, Prog. Theor. Phys., 49(1973), 652. [3] Chou Kuang-chao, Wu Yue-liang and Xie Yan-bo Preprint, AS-ITP-84-005 (1984) [4] J. Donoghue, E. Colowich, and B. Holstein, Phys. Lett., 119B(1982), 412. [5] K.C. Chou, Invite Talk at Europhysics conference held at Erice,Sicily, March 1984. [6] Private Communication. [7) Private Communication. [8] Andrzej J. Buras, Phys. Rev. Lett., 46(1981), 1354. [9] R.J. Oakes, Phys. Rev., D26(1982), 1128

426 COlIDIJun. in Theor. Ph!ls. (Beijinq, China)

Vol.4, No.1 (1985)

91-10!

ANOi1ALIES OF ARBITRARY GAUGE GROUP AND ITS REDUCTION GROUP, EINSTEIN AND LORENTZ ANOMALIES Kuang-chao CHOU( Ji):Jt;i ), Han-ying GUO( and Ke WU( ~ iJ

jit~

Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing, China

Received July 4, 1984

Abstract Based upon the properties of the characteristic classes and their CheroD-Simans secondary characteristic classes, the -Abelian- anomalies in H2n+2, the Euler-Heisenberg effective actions in ~+!, as well as the non-Abelian anomalies in ~n for arbitrary gauge group and its reduction subgroup have been investigated thoroughly and the application to the gravitational anOll/alies is made. It is shoIm that the -Abeliananomalies of such groups are equal to each other, their EulerHeisenberg actions are also closely related to each other, and their· non-Abelian anomalies are also equivalent if their common generating functional can be taken as a counter-tezm. For ebe gravitational anOll/alies we present the common generating functional for both non-Abelian Einstein and Lorentz anOlllalies in s4n +2 and show the relationship between them.

I. Introduction In a previous note rl ] we have generalized the investigations on some aspects of' gauge anomalies[2,3] and shown that the dual of Abelian or singlet anomalous current l-form in M2n+2 with respect to the gauge fields of leftand right-handed special unitary groups is ac~ually a peculiar Chern-Simons 2n+l_form[4,S] which is gauge invariant. We have also shown that from the decomposition formula ot thiE Chern-SiMons secondary characteristic class in y2n+J follow the anomaly free condition, the generating functional, the general expressions for the non-Abelian anomalies, and the gauge invariant Wess-ZuminoWitten type ef~ect1ve Lagrangian[6,7] in M~n taken to be the boundary of M2n+l. In addition to these results, it is shown[8,9] that the anomalous current in M2n + J is the functional derivative of the Chern-Simons invariant, i.e., the Euler-Heisenberg effecti~e action, with respect to the gauge potential. Therefore, it i~ found that there exist very profound connections among thase subjects in different diMensional spacetimes and they share a common topological hackground--theChern-Simons secondary ~haracteristic classes. We will point out in the present paper that these results can be extended to gauge theories with arbitrary gauge groups, especially, the theories with special orthogonal gauge groups which not only will deepen our investigation on the topological origin of the anomaly associated with the special orthogonal

427 92

Kuang-chao CHOU, )Ian-yinV GlJ() and Ke WU

gauge group[~], but also will relate to the gravitational anomalies[10-12]. It is well known that in the case of quantum spinor field in external gravitational field in M4a + 2 one should take into account both general coordinate transformatinns( i. e., the Einstein transformations) and the local Lorentz transformations. As far as the tangent space at a fixed point is concerned, the Einstein transformations are actually the gauge transformations of the group GL(4n+2, R) and the Lorentz transformations are those of the group SO(4n+2, R). Because the local orthogonal frame can be regarded as an element of the coset space GL(4n+2, R)/SO(4n+2, R) and its (dbuble) covariant derivative is equal to zero, the gauge potential of GL(4n+2, R), i.e., the Riemann-Christoffel connection for torsion free gravitational field, can always be reduced to the Lorentz gauge potential, i.e., the Ricci rotating coefficients. Corresponding to these two sorts of transformations, one should deal with two sorts of anomalies, namely the Einstein anomaly and the Lorentz anomaly. It is clear that there should be some link between them. It will be proved in.the following that, in general, there exists a link between the anomaly of a gauge group G and that of its reduction subgroup H via the coset space G/H. In the present paper, we will first extend the results given in Ref.[l] to the arbitrary gaug~ groups. Secondly we will explore the relation between the anomalies of a gauge group and its reduction group in detail. Then we will analyse Einstein anomaly and Lorentz anomaly and point out that not only they share a common generating functional in the case of non-Abelian anomalies and can be expressed in terms of the local frame and the gauge potential of the counterpart but they are equivalent if the common generating functional can be taken as a counterterm[ f1 1.

II. The

Anomalie~

in Arbitrary Dimensional Spacetimes

Let U(x) be an element of a Lie group G or its non-singular matrix representation. Under the action of tbe gauge group GLxGR , U(x) transforms as (2.1) and its covariant derivative is defined as (2.2 ) where AH(H=L,R) is the gauge potential I-form witb respect to group GH ar.d the corresponding field strength 2-form is

(wedge symbol suppressed). Under the gauge transformations (2.1)

428 .~.r.omalies

of Arbitrary Gauge Group and Its Reduction Group, Einstein and LOrentz Anomalies

(2.3) DU"'"g DUg- 1 L R

If the anomalous current l-fQrm consisting of field U(x) and gauge fields in 2n+2 dimensional Euclidean compactified spacetime M2n + 2 satisfies the Abelian or singlet anomalous dive~gence equation d*J(U A A )=P(~+J)_p(Fn+J) ,

L'

R

L

R'

(2.4)

where * is the Hodge star, P(pn+J) is a homogenous invariant polynomial[4,5] of degree 20+2 consisting of the fielci strength 2-forms, then by means of 0U~ method presented in Ref.[l], one can easily obtain the general solution 01 the singlet anomalous current. Introducing

(2.5)

"nel making use of the relation between the characteristic polynomials anci Chern-Simons secondary characteristic classes[ 4,:jJ

Q(2n+1)

(A' ,A)=(n+1)

Jo cit 1

P(A'-A)[d[A+t(A'-A)] (2.5)

it follows that

.) (2.7)

*jr(U,AL,A ) is an arbitrary gauge invariant exact form. R Following the method of Reis.[1,3], one can prove that t~e Chern-Simons secondary characte.ristiC; classes satisfy

where

Q(A',A)=-Q(A,A') , Q(A' ,A)=Q(A' ,O)-Q(A,O)+dG(A' ,A) ,

(2.8)

429 94

Kuang-chao CHDU, Han-ying GUO and Ke fiU

where (2.9) Then it follows that Q(ij-1ALU+U-ldU,AR)=Q(AL,O)-Q(AR,0)+Q(U-ldU,O)+dW(U,AL' AR ),

I

dW(U,A ,A )"d(G(U- 1 A U+U-1dU,AR)+G(A ,UdU-1).l • L L L R (2.10) In fact, Q(U- 1 A U+U- 1 dU,A ) can be regarded as the gauge invariant EulerL R Heisenberg effective Lagrangian in 2n+1-dimensional Euclidean compactified spacetime 1&2n+1, and the corresponding dynamical current defined by the functional derivative of the Lagrangian with respect to the gauge potential is the anomalous current in 1&2n+1. It should be pointed out that the gauge theory with topological mass term in 3-dimensions proposed by Deser, Jackiw and Templeton[8] and the recent discovery on the anomaly in odd dimenSions[9] can be generalized in terms of such a gauge invariant Euler-Heisenberg effective Lagrangian and corresPQnding anomalous current. Let us now turn to the case in 2n-dimensional Euclidean compactified spacetime 142n which can be dealt :with the boundary of 142n +1 ,o i.e., 0,.2n+1= =142n. From Eq.(2.10) in such an ,.2n+1 it follows that under thl:! necessary and sufficient anomaly free condition tn M2n {2.11) the gauge invariant oLject (2.12) gives rtse to the Wess-Zumino .type effective anomalous action with gauge fields in l4~n. Qtherwise (2.13) defines the generating functional of the non-Abelian anomalies in the left and right antisymmetric form and under the gauge transformation (2.1) they generate the finite form of such anomalies as follows: (2.14)

430 AnOllJalies of Arbitrary Gauge Group and Its Reduction Group, Einstein and Lorencz An:Jl1l4.1ies

95

whose infiniteSimal form is AL_R(£H,AH )=

J £aLG~(AL,T~)-(L+R)

2n H

,

G~(AL,T~)=~nn(n+l) f:dt(l-t)P{T~d(ALFn-l(tAL»}

I

(2.15)

The finite fonn (~.i4) contains the global iniormations about the anomalies such as so-called non-purturbative anomalies[7]. We can also define the genera:ting functional of the non-Abelian anomalies ill the axial current form rather than the CL,R)-antisymmetric form and obtain the finite anomalies of such a kind. In fact, in the case the gauge groups are special unitary groups, W(4) (U=l,AL ,A ) is the counter term R3 introduced by 13ardeen[13] to transfer the anomaly fr~ the (L,R)-ant.isymmetric form into the axial current one in 4-dimensional spacetime and correspondingly W(2n) (U=l,A L ,AR ) should be such anomaly transfer term for the gauge fields of arbitrary group in M2n. We have so far completed the generalization of our analysis on the anomalies with respect to the special unitary gauge group in arbitrary dimensions[l] to the arbitrary gauge groups. We have seen that the Chern-Simons secondary characteristic classes play a very important role in the course of analyses.

III. Anomalies of a Gauge Group and Its Reduction Group ~e now establish some relations between the anomalies of a gauge group G aad its reduction group H as well as the coset space G/H. Let gEG. hEH and a coset field ,(x)EG/H. In general, we have decomposition rule and transformation property

(3.1) (3.2) If G and H are gauged, the covariant derivative of .(x)-field should be defined as

where r anc y is the gauge potential I-form of group G and H respectively. It is well known[14] that the necessary and sufficient condition fur local reduction of r to y is the covariant derivative (3.3) to be vanishing, then we have r=,y,-1+,d.- 1=.y , y=,-l r .+,-ld,=.

-1

r

.}

(3.4)

Furthermore, it can be seen that the transformation (3.2) and covariant deri-

431 96

Kuang-chao CHOU, Han-yint; GUO and ICe IiU

v~tive

(3.3) are similar to Eqs.(2.1) and (2.2) respectively, we are able to establish a relation between the anomalies of G and H, the reduction group of G, with the help of the results obtained in the last section. Let us start with the singlet anomalies in M2a+2. If there does not exist the' reduction relation between. rand y, the singlet anomalies theUlsel ves appear as anomalous divergence of a singlet anomalous current

"J(IP,r,Y)=Q(~

-1

r,y)+dn(~,r,y) •

(3.;)

However, if there does exist the reduction relation (3.4) between rand it follows that Q(~

y,

-1

r,y)=o,

(3.6,

This means that the singlet anomalies of group G and its reduction "roul' Hare cOQpletely equivalent to each other. ~imilar to Eq.(2.10).in general, we have Q(~

-1

r.Y)=Q(r,O)-Q(y,O)+Q(~-ld,.O)+dW(~,r,~) ,

~-l -1 dW(~,r.y)=d(G(Y r,y)+G(r,lPd~

»

(3.8)

Under the reduction relation (3.4), it follows that (3.9) where G(r.~d~-I) is given by Eq.(~.9). If we take the Chern-Simons seconda.ry characteristic classe~ as the Euler-Heisenberg ~ffective Lagrangian in M2n+l, the Eq.(3.9) gives rise to a relation between the two Euler-Heisenberg actions with respect to the gauge fields of G and of its reduction group H. Obviously, when the scalar field ¢(x) as an element of the coset space G/H has nOll-trivia! topology and the buundary of ~2n+l is not empty. i.e.,aM 2n +1 ¢0, the twO Euler-Heisenberg actions are not equivalent to each other. When the boundary of manifold M2n +! is the spa~etime M2n under consideration. according to the analyses in the l'ast section

271J .M2n _1

QU,O)

and

~11

r

Q(y,O)

(3.10)

12n+1

;or

are the generating functionals of the non-Abelian anomalies of G and II in M2n=au2n+l respectively. Because the gauge potential y of the subgroup H can

432 Anomalies of Arbitrary Gauge Group and Its Reduction Group, Einstein and Lorentz Anomalies

97

be kept invariant under the gauge transformation of group G and vice versa, one can also take

=27Tf

(3.11 )

(Q(Y,O'-Q(f,O»

M2n + 1

as the common generating functional for the non-Abelian anomalies of both group G and reduction group H in y2n, i.e., under the gauge transformation of G, it generates the non-Abelian anomaly of G, while under the transformation of H, it generates the anomaly of H. However, the expressions for two nonAbelian anomalies are not the same, because the transformation properties of r(~,r,y) under the two sorts of transformation are not the same. In fact, the finite forms of two anomalies are (3.12)

A(h,y)=27T

J

Q(hdh _1 ,0)+271

M2n+1

J G(y,hdh

M2n

-1

).

(3.13)

In the case of infinitesimal tran'sformations, Le., g=1+e:(g), h=1+e:(h), the two anomalies become the infinitesimal ones a=1, ••• ,dim G;

a=1, .•• ,dim If; a

(3.14)

(3.1!j)

a

where T (g.) ,T (h) are the generators of the Lie algebras of group G and subgroup ' 1 y, G(g)(r,T(g) a a H respect1ve

a a and G(h)(y,T(h)

are given by the secon d equa t·10n

of (2.15), for example, (3.16) It is clear that neither finite form nor infinitesimal one the non-Al)elian anohlalies of group G and of reduction group HC:G are not the same. It should be, pOinted out that making use of the decomposition rule (3.1) of an element g of group G and the reduction relation (3.4) between the gauge potentials r ana y, the non-Abelian anomalies of G (the G-anomaly) can always be reexpressed with the element of coset space G/H, " the element of subgroup H,h, and the gauge potential y; and conversely, the H-anomalies can also be rewritten in terms of ~, g, and r. Does this fact mean that the two anomalies

433 98

Kuang-cnao ClJC}l, Hall-ying GW and Ke ffU

are equivalent to each other? One may suppose whether the common generating func"ionaJ of the anomalies can be redefined so as to eliminate either the G-anomaly or the H-anomaly or transfer one to its counterpart. In order to transfer, for ex~ple, the G-anomaly to the H-anomaly one needs to regard (3.17) as the G-anomaly transfer term other than the genera"ing functional and to add this term to the common generat1ng functional r(~.r,y) so that under the gauge transformation of G the new generating functional is invariant. However, from the definition (3.11) of r(~,r,y), such a new generating functional is nothing but the generating functional of H-anomaly. In other wo~ds, the anomaly transfer of this sort works only in the sense that their comm~n generating functional f(q"r,y) can be taken as counterterm.

IV. Einstein Anomaly and Lorentz Anomaly For the sake of simplicity, we merely analyse in the present paper the Einst.ein anomalies and the Lorentz anomalies associated with the tangent sPace at a given point in gravitational field without torsion, i.e., the anomaly of general linear group and that of its reduction group via local frame" the special orthogonal group. In this case, we can directly make use of the results obtained in last section. And the generalizations to the Einstein transformations involving a Lie rlerivative besides the gnuge transformations and the covariant forms of the anomalies[ 12 1 as well as to the case wi"h torsion can also be don~ without difficulties and obscurities in principle. We start with the singlet anomalies in M4n+4[f31. It is plain that the singlet Einstein anomaly and Lorentz one are determined by the n+lst-Pontrjagin classes P(F 2n +2 (r» and P(F 2n +iy» with respect to the groups GL(4n+4, R) and 80(4n+4, R)in M4n +4 respectively[ f4 1

"0

(4.1) and r is taken to be the Riemann-Christoffel and y the Ricci connection 1-form without torsion in y4n+4, r=r kdx

k

,

rk=(r~k)O J dO~m 1 ~,J", . .. , 0

y =(y. ) k

bk a,b=J, •••

0

,d~m

Moo

M.

Becau~e there exists the reduction relatioD between rand y via the local moving frame

434 Anomalies of Azbitrary Gauge Group and Its Reduction Group, Einstein and Lorentz Anomalies

E=Ce ai ) . . a,Jal, •.• ,d~m we

99

Hi

ha~e

DE=dE+rE-Ey=O ,

C4.2)

so the two singlet anomalies are the same. In y4n+J, we also have the relation between the corresponding Chern-Simons secondary classes C4.3) with r.y and E taken to be the objects in y4n+3, and 1

Qcr,O)a28 n+ l Cn+1 ) J 0 dtTr{rF 2n + 1 Ctr», G( r ,EdE- 1 )=13

n+l

(4.4)

(2n+2)( 2n+l) Jl d"t P -td~ 0

Jo

(4.5) 1

The expressions, similar to Eq.(4.4), for Q(y,O) and Q(E- dE,O) can also be written. As is analysed in the last section, the Eq.(4.3) gives rise to the relation between the Euler-Heisenberg effective action& of gauge groups GL(4n+3,R) in M4n + 3 • Finally, we analyse the non-Abelian Einstein anomaly and Lorentz anom&ly in y4n+2 and assume that M4n +2 can be regarded as the boundary of M4n+J, i.e. y4n+2=3y 4n+3[f2]. According to the analyses in last section, we can define 1

f(E,r,Y)=211)

Q(E- dE,O)+211

~4n+J

J

G(r,EdE-

1 )

(4.6)

M4n+2

as t·he common generating functional of both Einstein and Lorentz anomalies with r, y, and E tak.ell to be the objects in M4n+2. Under the gauge transformations of GL(4n+~,R) and SO(4n+2,R), it gen@rates Einstein anomaly and Lorentz anomaly respectively. Their finite forms are of AE ( g, r )=:;m

J Q(g H 4n +J

-1

dg,O)+21r jr

Gcr,g -. dg),

gEGL(4n+2,R) ;

(4.7)

hESO(4n+2,R);

(4.8)

M4n+2

and G(r,g-l dg ), G(y,hdh~l) have expressions similar to Eq.(4.5). The infinitesimal forms of the anomalies can also be easily written as a=l, ••• , dirnGL( 4n+2, R),

(4.9)

435 100

Kuang-chao CHOU, Han-gina GUO and Ke flU

a=1 ••••• dimSO(4n+2.R).

(4.10)

wllere a Gal )U.T ( ) )=~ •g

g

Il+

1(:J1I+~)(2n+1)21T

l

Jadt(l-t)Tr{Ta(

,

gJ

dUF·n(tr»},

(4.11)

(4.12)

which follow directly from Eq.(3.16) and the definition (4.1) of the Pontrjagin class. Obviously. because of the reduction condition (4.2). the Einstein anomaly either in finite or in infinitesimal form can be rewritten in terms of the local frame and the objects consisting of the Lorentz anomaly and vice versa. Therefore. it is possible to redefine the generating functional so as to eliminate one of the two anomalies. such as the Lorentz anomaly and to tran~ fer it to its counterpart. However. because of the same reason analysed in the last section. this requires that the common generating functional can be taken. as a counterterm and only in this sense the gravitational and Lorentz anomalies are equivalent.

Footnotes [ll] After we finish",d this work.we received an interesting preprir:t by Bardeen

and Zumino[12]. Besides many important points about consistent and covariant anomalies as well as gravitational anomalies they explained. they also claimed that the Einstein and Lorentz anomalies in non'-Abelian case are equivalent. [f2] This means that M2n has no boundary. like.2n-dimensional sphere s2n etG., otherwise K2n cannot be regarded as the boundary of an K2n+1. We will leave the case of y2n with boundary for further investigation. [f3] The reasons why gravitational anomalies appear only in K4n +4 and y4n+2 for Abelian and non-Abelian case respectively are the same as those explained in Ref.[12]. Cf4] The polynomial p. in the general case of Dirac particle 1n gravitational field. should be taken as A(M)ch(F). the product of A roof genus A(M) and the Chern character ch(F). In this paper. for simplicity. we take only the Pontrjagin classes as the polynomial P.

Keferences [lJ X.C. CHOU, H.Y. GUO, X.Y. LI, K.

wu

and X.C. SaNG, Commun. Theor. Phys., 1(1984)491.

[:./} X.C. CHOU, H.Y. GUO, K. WU and X.C. SaNG, Phys. Lett., ~(1984J67; COIIlIIIun. Theor. Phys., 1(1984)73; 12S. [3J B. Zumino, "Les Houches lectures 1983"; to be published

by NOrth-Holland, ed.

R. Stora and B. DeWitt; B. Zumino, Y.S. IfU and A.

Zee, NUcl. Phys., !E2.(1984)477.

436 Anomalies or Arbitrary Gauge Group and Its Reduction 6roup, Einstein and Lorentz Anomalies

101

[41 5.5. Chern, "Complex Manifolds without POtential Theory". Springer-Varlag, (1979). {5] T. Egucbi, P.B. Gilkey and A.J. Hanson, Pbys. Reports,

~(1980)213.

{6] J. Wess and B. ZumillO, Phys. Lett., 37B(1971)95. [7]

E. Witten, Nllcl. Phys.,

~(19/J3)422.

is] s. Deser, R. Jackiw and S. Templeton, Pbys. Rev. Lett., • Ar>.n.

Pbys., (N.Y)

~(1982)9i5;

~(l982)372.

{9] A.J. Niemi and G.W. Semenoff, Pbys. Rev. Lett., .2l...(1983)2077;

A.N. Redlicb, ibid.,

~(I984)18.

(10) L. Alvarez-Gaume and E. Witten, Nucl. Pbys., [II] L.N.

C~NG

~(I984)269.

and H.T. NIEH, Stony Brook preprint ITP-SB-84-25.

(12) W.A. Bardeen and B. Zumino, Nuel. Phys.

~(1984)421.

113] W.A. Bardeen, Phys. Rev., !!!(I969)1848. [14] s. Kobayashi and K. Nomizu, "FOundations of Differential Geometry", Vol.l, Interscience(1963).

437 Commun. in Theor. Phys. (Beijing, China)

NON-LINEAR

Vo1.4, NO.1

(1985)

123-127

ON THE TWO DIMENSIONAL C1 IYiODEL WITH WESS-ZUf1INO TERf1 II QUANTUM THEORY

CHOU Kuang-chao( }l)Jty ) and DAI Yuan-bent Institute of Theoretical Physics, Academia P.O.Box 2735, Beijing, China

lOt*

S~nica,

Received August 30, 1984

Abstract

The canonical quantization is carried out for the two dimensional non-linear a model with Wess-Zumino cerm. Ct is shown tIlat the currents in t1:is model satisfy a Kac-Moodg algebra for arbitrary values of the coupling constant.

Last year the two dimensional non-linear c model with lVess-Zumino term· was invesTigated in several articles. Using the canonical quantization in the light-cone formulation lVitten[l] found that currents in this model at the special value

A;"!! of the coupling constant satisfy a Kac-Moody algebra n

which is of the same form as that for curr~nts in the ~ero-mass free fermion theory. TaAing into consideration the uniqueness of the representation of the Kac-Moody algebra for n=l, this result implies an isomorphis'r.l between trilbert spaces for these two models at n=1. Witten conjectured that at these special values of constants this model is equivalent to the zero-mass free fermion ~ouel. This problem was also investigated in Refs.[2,3] w~th the method of functional integration. It was founu that the physical equivalence between these two theories at n =

#Ao = 1

is true only in some restricted sense. In Ref.

[4J we found some simple isomorphism for classical solutiopsof this model at uifferent values of the constants )., and n. In this art icle canonical quantizat"ion of this mOdel for arbitrary values of constants .\, and n is carried out. We find that the currents in this model at arbitrary values of AO and n~O suitably modified satisfy a Kac-Moody algebra which is of the same form as that for currents in the zero-mass free fermion theory. This result establishes an isomorphism between the quantum theories of these two models at n=l and arbitrary values of Ao. The action of the model is of the ,ollowing form (1)

where

438 124

CHOU Kuang-chao and DAI Yuan-ben u.

1 o

1.1

~1

"O=4)..2Tr(a g

·a g) ,

(2)

1.1

(3 )

where D is a three dimensional disk with t'No dimenSional space-time S2 as its boundary and g is an element of the D(N) matrix group. The equation of motion derived from this Lagrangian is the following (4)

where (5) ~atisfies

the curvatureless condition (6)

For canonical quantization of the theory one needs to know the Poisson brackets of fielg quantities. One can derive from the commutation relations written down below that the constraint gTg=I commutes with the Hamiltonian. Therefore, the constraint is of the first kind and can be imposed on pbyaical states without -the need of introducing Dirac brackets. As a result we can write down the Poisson brackets as if the matrix elements gab are independent generalized coordinates. Let (7)

where (8)

Taking into consideration tbat y is linear in g and changes Sign under the replacement 0xgab,gcd)+(gab' Qxgcd) we can obta.in the following formulas for the Hamiltonian and the momentum of the fields

H-ITr(~Tg-~)dx=fTr(~~g- ~o)dx = -

rtr JdXTr(A~+A~)

,

(9)

P=-fTr(~Ta~g)dx=-JTr(~~axg)dX =2!ifdXTr (AoAd •

(10)

439 ~ith

on the Two Dimensional Nan-Linear a HOdel

From the Poisson brackets between discrete space of lattice spacing E _ aL =

P ab

Elf

agab

Wess-Zumino Term II Quantum Theory

gab

12S

and the conjugate momenta in the

,

ab

we find the POisson brackets between

and

gab

lfo~bas

the following (11)

{gab(~n)' 1T oc d(x m )}=6 ac 0bdo an E-!

(12 )

{Poab(X n ), Pocd(Xm)}={Pocd(Xm)' En

+ {m

a a.gcd (xm ).,'"~ '(

P

0

L

,

)7

3(X

ag ab

Y "}

n

,

,( x ) J

OaD

n

(13)

where

The evaluation of the right-hand side of Eq.(13) is facilitated by noting that the change of r under the variation of g is the following, 3 ~ a ( ag b(x )7dt' at b(x) a nan

)

E~Y1 ...

From Eqs.(13) and (14) we find (15)

Using the formula 6

we obtain that

ar(

gab Xn

1 IlV -1 -1 (a g (x)a g(x)g (X»b 1T Il n v n n a

)--Er£

'

(16)

440 116

CHOU Kuang-cbao and DAI Yuan-ben

(1I

oa

b(X); 11

n

oc

d(X

m

)}=

-...!L(Cl 811

g-lg-l

x ad cb

(17) It

noted tha, using the formulas (9), (10) for Hand P and Poisson brackets (1~) and (17) the effect of the r term in the theory appears explici tely only in the modification of the Poisson brackets between two lI~S in Eq.(17). In the limit of continuous space we find from Eqs.(11), (12) and (20) the following Poisson brackets of A~ i~

(lll,

(18)

- ...!...Tr(~-rB)6 I (x-y) ~1I

(19) '

{Tr~Ao (x). TrTBAo (y) }=Tr[~. TBl (Ao (x)-nAl (x» ,,"here Til are generators of the O(N) group and -

•

1

Ao = m:rAo

(20)

(21 )

U~il1g Eqs.(18)-(21) it can be easily verified that the Poisson equations for Al and Ao are identical with the curvaturelesscondition (6) and the Lagrangian equation of motion (4) respectively. We now introduce the current

(22) and the current J_ conjugate to J+ with respect to the space reflection (23)

For n+O J~s and J:s for all g and x are distinct.The equal time commutation relations between J+ and J_ obtained from the corrdspondence to Poisson brackets (18), (19) and (20) are the following

(24)

(25)

(26)

441 On

·the Two Dimensional Non-Linear a HadEd with Wess-Zumino Term I l Quantum 'l'illiory

127

commutation relations (24), (25) and (26) are identical in form with that for currents in the n-flavor zero-mass free fermion model and form the direct sum of two Kac-Moody algebras. At the special value of the coupling constant, we have from Eqs.(9) and (10) H±P =

f),2jdXTrJ~

(27)

Since operators J+ and J_ comm'lte it follows from Eq.(27) that they depend only on x+ = A(t+X) and x =

A(t-x) respectively.

After the rescaling Jr~.2J;,

the equal time commutation relations (24), (25) and (26) reduce to that obtained by Witten in Ref.r1] using light-cone formali~~ for the special value

A~=!l!. n A~ noted by Witten[l] since there is no operator commuting with all Jr(x)'s, the Hilbert space of the model is an irreducible representation of the direct sum of two Kac-Moody algebras spanned by J+(x) and J_(x). Since the representation of Kac-Moody algebra is essentially unique for n=1 our results imply an isomorphism between the Hilbert space of the non-linear a model at n=l and arbitrary value of AD and that of the one-flavor zero-mass free fermion model. ThiS, however, does not imply the c~mplete physical equivalence of these two models, because formulas for physical quantities like Hand P are not identical in form for these two mOdels. Note that J_ is actually the charge density or tbe current (28)

which is conserved (modula anomalies) by the equation of motion (~). One can derive commutation relations between time and space components of currents (~8) from Eqs.(18), (19) and (20). These commutation relations are similar to those in the current algebra of the fermion theory except that the anomaly appears not only in the commutators of a space component and a time component but also in the commutators of two time components of currents.

References {I} E. Witten. Comm. Math. Phys .•

~(1984)455.

[2/ P.Di Vecchia, B. Durhuus and J.L. Petersen, Preprint of the Niels Sonr Institute NBI-HE-84-02; P. Di Vecchia and P. Rossi, CERN Preprint Ref. TH. 3808-CERlI. {J} A.M. PolIJaKOV

and P.B. Wiegmann, Phys. Lett., ~(19d4)223.

[41 CHOU Kuang-chao and DAI Yuan-ben, CommUD. Theor. PhlJs., !J1984)767.

442 Commun. in Theor. Phgs. (Beijing, China)

Vol.5, No.4(198;6)

359-364

DERIVATION OF THE ANOMALOUS TERM IN VIRASORO ALGEBRA BY TOPOLOGICAL METHOD WU Yue-liang( ~.ffi.l ), XIE Yan-bo( and ZHOU Guang-zhao( Jil1t.g )

iMliIl: )

Insti tute of Theoretical Phgsics, Academia Sinica, P.O. Box 2735, Beijing, China

Received October 21, 1985 Abstract A 2-cocgcle is constructed which gives the anomalous term in the Virasoro algebra.

Recently a unified mathematical scheme in terms of cocycles was developed to describe the anomalies characterizing ch1ral fermions inter·actingwith a Yang-Mills field[1,2]. Specifically, the 2cocycle descending from the Chern density with two higher numbers of dimensions is the anomalous fixed time commutator of gauge group generators [3,4] • It is well known in the string theory that the Virasoro algebra has an anomalous term[5]. In t~is case the generators Ln form a conformal algebra in the classical theory [L n • Lm ]~(m-n)L n+m •

n= ••• -1.0.1 ••.••

(1)

when the string is quantized an additional anomalous term of the form (·2 )

appears in the commutator. Physically. no gauge field has been introduced in the string theory. An interesting question is whether the above anomaly can be described by a 2-cocycle induced by an fictitious gauge field introduced purely as a tool of calculation. ThiS note is addressed to this question. The Virasoro algebra with anomalous term has the form (3)

443 w rue-liang,

360

XIE Yan-be and ZHOU Guang-zhao

where Cm,n are antisymmetric (4)

Cm ,n=-Cn ,.m

and satisfy an equation due to Jacobi indentity

Solutions of Eqs.(4) and (5) can be easily found. Cm,n=(m-n)b(m+n)+dc5 m+n ,O m3

•

where b(m+n) is an arbitrary function of (m+n). relations (3) now become

They are (6)

The commutation

(7)

Making the change ~)

we can rewrite Eq.(7) in the following form (9)

The center in Virasoro algebra (10 )

can be cast into the form of Eq.(9) by a change of the generator Lo -

C Lo + 24

(11)

and d

=..£.. 12

(12)

Our next problem is to find a 2-cocycle which produces this anomalous center term. For this purpose we consider a representation of Ln

444 Derivation of the Anomalous Term in Virasoro Algebra b!l Topological Method

361

(13)

The arbitrary function fez) can be eliminated by a transformation of the wave fUnction ~(z)

~(z)

ff(Z) d

_

e-

- z - Z ~(z)

(14)

Therefore without loss of generality we can take f(z)=O. Under a finite transformation the wave function ~(z) changes into ~'(Z)=e

iEEnLn

n

~(z)=U (z)~(z') E

(15)

•

where En (n= ••• -l,O,l, ••• ) are group parameters and (16)

The coordinate

can be obtained in the following way. such that d'{ =

Let us define a function '{(z)

1

(18)

dz I:£ zn+l ' n n

then (19)

The operator

is a displacement operator of the coordinate '{.

Therefore

a z'=eaTZ (T)=Z('{(Z)+l) •

(21)

445 362

flU rue-liang, XIE Yan-bo and ZHOU Guang-zhao

As a special case we take (22)

It is easily found that R.

n=z

(23)

e:

In this special case the phase factor U(z) can also be found as (24) which is seen

~o

satisfy the group relation (25)

The presence of the phase factor U(z) indicates that besides the coordinate transformation a U(l) gauge field ~(z) is needed to define a covariant derivative. Under the conformal transformation described by the Virasoro algebra the gauge potential transforms as (26)

Here we use the exterior form and differentiation in a complex manifold. To calculate the commutator anomaly we have to find the 2-cocycle descending from the Chern density in four dimensions (27)

where F=dA+A 2

(28)

is the ~ield strength. Using the standard procedure developed in Refs.[1-4] the anomalous term is proportional to

8;liJVdV .

(29)

446 Derivation of the Anomalous Term in Virasoro Algebra by fbpoloqical Method

363

where

where ~(i)(i=l,2) are two parameters varying from 0 to 1 and are infinitesimal group parameters. It is· easily found that

Ein

(31)

Integrating Eq.(31) over ~(i) (i=l,2) and z we obtain for the nontrivial 2-cocycle

where 1 is the number of times that the line integral wraps around z=O. The factor !2a21m3d • n+m, 0 is the anomal~us center term for the Virasoro algebra which has the same form deduced by directly solving the Jacobi indentity. This is not surprising since Eq.(5) is an infinitesimal form of the coclosed condition for a 2-cocycle. What we have done in this paper is to Show that the solution of the coclosed condition can be obtained by constructing a fictitious gauge fiel~ and using the standard procedu~e starting from the Chern density.

Re.ferences {lJ B. Zumino, Y.S. WU and A. Zee, Nucl. Phrjs. ~(1984)477; B. Zumino, Nucl. Ph!1s. B253 (1985)477. {2l H.Y. GUO, B.Y. HOU, S.X. JiANG and X. W, Commun. Theor. Phys. ~.f1985)233-251 (Beijing, China); X.C. Chou) Y.L. WU and Y.B. Xie, Anomalies and the General Chern-Simons Cochain, nProceedings

of the Tenth Hawaii Conference in High

Energy Ph!1sics n (1985). {3l L.D. Faddeev, Ph!1s. Lett. ~(1984)81; J. M1ckelsson, Commun. Math. Phys.

2!..(1985)361; S.Y. Jo, MIT preprint CIP, E,!!(1985). {4l R •.Jackiw, Phlls. Rev. Lett. ~(1985)159; B. Grossman, Phys. Lett. ~(1985)92; X.C. Chou, l':.L WU and Y.B. Xi-e, preprint AS-ITP-85-026;

Y.S. WU and A. Zee, Phys. Lett. lS2B(198S)98; Bo-yu Hou and So-yuan Hou, Chinese Phys. Lett. (in press).

447 364 [5] A. Neveu and J.B. Schwarz, Nuel. Ph!/s. !!l(1971}B6; P.Ramond, Ph!/s. Rev. D3(1971J2415; P. Goddard, J. Goldstone, C. Rebb and C.B. Thorn, Nuel. Ph!/s. !!2,!(1973) 109.

448 Modem Physics Letters A Vol. 1 No.1 (1986) 23-27

© World Scientific Publishing Company

EFFECTIVE ACTION OF SIGMA MODEL ANOMALIES WITH EXTERNAL GAUGE FIELDS YIE-LIANG WU, YAN-80 XIE and GUANG-ZHAO ZHOU Institute of Theoretical Physics, Academill Sinica P. O. Box 2735, Beijing, People's Republic of China Received 28 January 1986

The nonlinear sigma model describes Goldstone bosons originating from spontaneous symmetry breaking. A set of local counterterms is found to shift the anomaly of the nonlinear sigma model to that of the original model with fermions interacting with external gauge fields. The 't Hooft consistency conditions are matched automatically.

Nonlinear sigma models and their effective actions are of special interest in recent years. They are useful not only in the context of chiral dynamics but also in the geometrical interpretation and possible compactification of some superstring theories. We consider in this note those sigma models that arise from spontaneous symmetry breaking of a theory with symmetry group G down to a subgroup H. The original theory has anomalies when its fermions interact with chiral gauge fields. After symmetry breaking the fermions will interact with reduced gauge fields and produce different anomalies. Local counterterms then have to be added to shift the anomalies back to that of the original theory. A simple derivation will be given in this note of these counterterms including the interaction of the Goldstone bosons with the external gauge fields that has been neglected in the Refs. 1 and 2. Consider a theory with symmetry group G spontaneously broken down to a subgroup H. The effective theory then describes Goldstone boson fields in the direction of the coset space G/H. Any group element ge G can be decomposed into the product of an element cp belonging to the coset G/H and an element h of the subgroup H:

9 = cph.

(1)

The Goldstone fields are then described by a function cp(x)eG/H. Under a group transformation the element gcp(x) can again be decomposed in the form of Eq. (1)

gcp(x) = cp(g, cp(x))h(g, cp(x}) , where cp(g, cp(x») E G/H and h(g, cp(x» e H satisfy the group multiplication law 23

(2)

449 24

Y.·L. Wu. Y.·B. Xie & G.·Z. Zhou

cp(g', cp(g, cp(X»)) = cp(g'g, cp(X»), (3)

h(g', cp(g, cp(x)))h(g, cp(X») = h(g'g, cp(X»). Therefore the Goldstone field cp(x) transforms as

cp(x) -+ cp'(x) = g-l cp(x)h(g, cp(x»).

(4)

The I-form cp(x) dcp-1(X) is valued in the Lie algebra of G which can be decomposed into two parts (5)

where H is a I-form valued in the Lie algebra of the subgroup and K that of the coset G/H. They transform as

H -+ H' = h(g, cp(x»)(H

+ d)h- 1(g, cp(x»)

(6)

and

K

-+

K' = h(g, cp(x»)Kh- 1(g, cp(x».

(7)

The kinetic energy of the Goldstone boson fields is constructed to be

(8)

tr(K",K"'),

which is seen to be invariant under the transformation Eq. (7). Gauge fields have to be added if the symmetry group is locaL In this case Eq. (5) changes to the following form

(9) where A is a gauge potential transforming as

A -+ A' = g(d

+ A)g-l.

(10)

When fermions and gauge fields are present, they are described by a Lagrangian (11)

before symmetry breaking, where left-handed t/lL and A are representations of the group G. After symmetry bteaking the fermions and the gauge fields become

450

Effeclive AClion of Sigma Model. . . 25

(12)

and the corresponding Lagrangian fakes the form (13)

The effective action after symmetry breaking has to produce the same anomalies as the original action. This requires adding compensating terms to the effective Lagrangian of the nonlinear sigma model. As in the case of the Wess-Zumino-Witten effective action, a closed form can be found only in the space of one higher dimension. Consider a form P2n-l in (2n - 1) dimensions. What conditions does it have to satisfy in order to describe a Lagrangian in a (2n - 2)-dimensional manifold M which is considered to be the only boundary of the (2n - 1)-dimensional manifold S? There are three of them: (i) Locally P2n-l should be an exact form so that dP2n-1

= o.

(ii) Globally it should not depend on the manifold S chosen so far as it has the same boundary M. This implies tfiat P2n-l integrated over any closed manifold should equal 2n: times an integer:

r

JSI

P2n-l -

r P2n-l =lP2n_l = 2n:m J

Js.

(14)

such that the uniqueness of the quantum amplitude e ir is guaranteed. (iii) Under an infinitesimal group transformation it produces the difference of the current anomalies of the original theory with the reduced sigma model. Let us introduce a double matrix notation and write

0

Q(x) = ( cp-l (x)

A (x) d(x) = ( 0

CP(x)) 0

0 cp-l(d

(IS)

'

+ A)cp

)

(16)

.

We have (17)

dO. = Q(d(x)

+ d)Q(x) =

d(x),

(18)

451 26

Y. .£. Wu, Y.-B. Xie & G.-Z. Zhou

and (19)

The Chern density j" C£

=-" (2tr)"n!'

(20)

vanishes since Q"Q = -".

(21)

The Chern-Simons secondary class descending from the Chern density in Eq. (20) is l22) where

(23)

with

F; = dAr

+ A/

(24)

and A, = tA

+ (1

- t)B.

(25)

It is easily seen that (26)

whose variation under a group transformation produces the difference of the current anomalies before and after symmetry breaking. From Eq. (18) we obtain

Using the identity derived from the generalized Chern-Simons characteristic classes

3

452 Effective Action of Sigma Model. ..

27

where

(29)

Fr'

= dAr + Ar2, IvIr = drAr,

and

we get

The first term in Eq. (30) is a closed form whose integral over the (211 - I)-dimensional manifold is the winding number of the mapping described by the coset function cp(x). Therefore 27t JQ2n-I.1 ~(sJ, 0) satisfies all the conditions required and can be chosen to be the compensating term for the effective action of .the nonlinear sigma model. When the external gauge fields are switched off, we get the same result obtained in Refs. 1 and 2, which is described by the first term of Eq. (30). Our result is more general since it contains interaction of the Goldstone bosons with the gauge fields. described by the second term of Eq. (30). From our derivation it is also apparent that if the original theory is free from anomalies so is the nonlinear sigma model arising from spontaneous symmetry breaking. If some fermions become heavy after symmetry breaking the low energy effective theory will be consistent only when these fermions do not contribute anomalies. One could therefore omit them from the effective action. The compensating term obtained above is still valid and 't Hooft's consistency conditions are automatically incorporated.

References 1. L. Alvarez-Gaumi: and P. Ginsparg, Harvard preprint HUTP 85/A015 (1985). 2. J. Bagger, D. Nemeschansky and S. Yankielowicz. SLAC PUB3698 and Harvard preprint HUTP 85/A042 (1985). 3. H.-Y. Guo, B.-Y. Hou, S.-K. Wang and K. Wu, Commun. Theor. Phys., 4 (1985) 233.

453 Co~un.

vol.7, No.1 (l,98i)

in Theor. Phys. (Beijing, China)

17-37

A SIMPlIFIED DERIVATION OF CHERN-SIMONS COCHAIN Mil) A POSSIBLE ORIGIN OF a-VACUUM TER~l CHOU K:.lang-chao( ,iJ7tE ), WU Yue-liar.g( and XIE Yan-bo ( i# g ~~ ) Insti tuta of The'Ci;etical Physic:;;,

P.o. Bo>.:

2735~

.~cademia.

;'-5- ~

S.:..nica,

3.;ijinq, C!i.i::a

Received June 24, 1985

Abstract In this paper, it is shown that the cohomology of r;:eneralized secondary classes, the F~ddeev type cohcrnolc~y a~= the generalized gaug~ ~~ansfQrmation can be easily obtai~ed by expandi~g the Chern form accordi:2q to t.r,e d,=gre~ of t ..1c :o=ns ~n its submanifolds and usi~g :~~ c1csed ;:roye=~~' of the C..'-:er:-: =C~:::. =-t is a.2so snot"n :::..:It.a -vacuum ~e.,:,,~ ':'.10'1 t.=:e =::ec'tit·-c :~g':a,,'1q~ar: arises when gauge rield ~n c~c grcu~ ~anif=ld it; pre.c:er..t.

I. Introduction Rece~tly, H.Y. Guo et al.(11 have introduce~ higher ordpr charac'teristic classe~ and cocycles y;hich lfeneraliz~ 1:!~" Weyl homodescribed by the coboundary of the Chern class. About the same ti~e Faddeev[2] has constructed another kind of higher order co~rcles based on the group manifold, which ha~e been elaborated in aef.[3]. In chis paper, a simclified deriv~tio~ of Cher~-Sirnons cochain is given. \'ie expand &lie Chern form according to 'the degree of the forms in its submanifolds. Only by usj.ng the closed property of the Chern form, can one obtain a Chern-Simons cochain which represents a descent relation between both submanifolds. In some special cas~s, one .can easily obtain famil.iar results gh·.;!i. in Refs. [1,2,3]. III Sec.II we show the ca.se of the c0ho!'lolcgy of th;: generalized secondary classes. In Sec.III, we ~ive the Factdeev type cohomology. In Sec. IV, the generalized gauge transformaticn and the Chern form gaugG potential are discussed. There the P. po,rameter ill the f.-vacuum is interpreted as a line integral of gauge potential in the group manifold. morp~yism

454 CHOU Kuang-chao, rvu Yue-liang and XIE Yan-bo

28

II. Generalized Secondary Characteristic Class The Chern density and the Chern-Simons secondary class are now familiar to theoretical physicists. In a 2n-dimensional space with gauge potential A=A,,(x)dxi.' ,..

(2.1)

and curvature (2.2)

F=dA+A 2 the Chern density is defined as

(2.3)

which is a 2m-form.

Using Bianchi identity

dF=[F,A]

(2.4)

it is easily verified that

~(F)

is closed (2.5)

d;1 2 ::1 ( F) =0

and can be expressed as (2.6)

where 1

n~m_l(A,F)=mLdtn2m_l(A,F~-1) 2

2

F t =tdA+t A

,

I

(2.7)

is the Chern-Simons secondary class. In a recent paper[l] the authors introduced generalized secondary characteristic classes relating to the higher order cochain :l.rod cohomology. We shall show in this note that their results can be easily dedaced from Eq.(2.5) by the general property that the density is a closed form. Consider a manifold consisting of two submanifolds, one of which is the ordinary space manifold of dimension Nl with coordinates x~, u=1,2 .•. N1 , The other submanifold is one of parameters ~1,i=1,2,.,.Nz, The exterior differential operators are Cher~

455 A Simplified Derivation of Chern-Simons Cochain and a Possible Origin of a-vacuum Term

29

(2.8)

and

The gauge potential ..4 can be decomposed into two parts

(2.9) where

1

and

(2.10)

j

The curvature .7 now becomes .9=(d x +d: )(A+B)+(A+B):=F+G+~I

(2.11)

,

'>

~vhere

I

G=d.B+BZ

and

(2.12)

J

The Chern denisty

(2.13) satisfies Eq.(2.5), i.e.,

By expanding ':2n(:?) according to the degree of the forms in d:'.4, we

have 2n

n2n c,9)= L

m=O

where

n2n _ m ,m(A,B)

m in f,i.

1'2 2n _ m ,m(A,B)

(2.15)

,

is a form of degrees 2n-m in

,.

x~

and of degrees

456 30

CHOU Kuang-chao, WU Yue-liang and XIE :'"a.n-bo

Substituting Eq.(2.15) into Eq.(2.14) and comparing the degrees of the form, we obtain

(2.16)

As a special case, we choose

1

B(x,O=O ,

A(x,O=A{O)(x)+~.;i(A(i)(X)_A{O)(x»

,

f

(2.17)

~

then G=O,

I

\:2.18)

By a change of the notation

M-H.

}

(2.19)

Eq.(2.16) can be easily seen to be just the theorem 1 proved in Ref.[1].

The cohomology and generalized secondary characteristic

classes then follow

by merely integrating over simplexes in the

submanifold .;.

III. Faddeev Type Cohomology In this section we show that the cohomology of gauge groups in Faddeev's approach I2 ] can be deduced as a special case of Eqs.(2.6) and (2.7)

(3.1)

457 A Simplified Derivation of Chern-Simons Cochain and a Possible Origin of 6-vaccum Term

31

Expanding the Chern-Simons secondary class according to the of the form in d~i 2n

;:;

n-

1 (A ,B

)=E n~n_m m-l (A,B). m=l'

degre~~

(3.2)

Substituting Eqs.(2.15) and (3.2) into Eq.(3.1) and comparing the degrees of the forms, we have

n2 n, 0 =d x n~

~n-

I~

2n-m,m

~.

-d

=d

1,0 '

nO

x 2n-m-1,m

(3.3)

+d 11 0 ~

2n-m,m-l ,

,.,0

"0,2n- ('0,2n-1

As a special case, we choose B(X,;)=U-l(X,~)d~U(x,~) , t,

A(x,~)=U-l(X,;)(A(x)+dx)U(X,;)

where

U(X,~)

,

1j

(3.4)

is an element belonging to the gauge group G.

Then

}f=O, G=O,

(::.5)

F=U-l(X,~)F(x)U(x,~)

,

F(x)=d x A(x)+A 2 (x) •

In this special case the sequence Eqs.(3.3) become

(3.6)

d~n~,2n_l=O

•

Eqs.(3.6) are just the results obtained in Ref.[3].

Therefore one

458 32

CHOU Kuang-chao, WU Yue-liang and XIE Yan-bo

can choose (3.7)

where

(3.8)

The Faddeev type cohomology then follows by integrating over the simplexes in the submanifold s consisting of points pi

=~":)~l ,..,.~2 ... ,..,:i-1~! ,...., '-,:i+1 • ... ). =(1,1, ... ,1,0,1, ••. ),

(3.9) i=1,2, .•. , Po=(1,1, ...• 1.1.1 •... ).

IV. Gauge Fields in the Group Space and the 9-vacuum In this section, we consider

anothe~

special case

1

M=d_A+d B+AB+BA=O t, x

(4.1)

J

F=d x A+A 2 ;100 .

In this case the sequence Eqs.(3.3) become

d

d

n°

-

d ~o

f; 2n-2k-1,2k-2-- x"2n-2k ,2k-l

'

n° =-d nO +:] ~ 2n-2k,2k-l x 2n-2k-1,2k 2n-2k,2k

(4.2) J

459 A Simplified Derivation of Chern-Simons Cochain and a Possible origin of a-vacuum Term

where

n2n - 2k ,2k

33

satisfies (4.3)

transfor~5-

This case can also be regarded as a generalized gauge tion.

For example, by choosing B(X,~)=C(;)+U-l(X,;)d>U(x,;)=C(;)+V(x,;), "?

I

(4.4)

we have G=d~B+B:=d~C(;)

...

.'

,

M=d~A+dxV+VA+AV=O

where form

C(~)

1

\ 4.5)

J

,

is Abelian gauge potential only depending on ~, so Chern

~2n-lk,2k

can be written as i.'1

"2n-2k ,2k

0'

P

Tr(F,"1-k Gk)

n!(271)2

=~,,:; n-2:~

2n-2k,2k-1

\

lk-1

• C(" (d,C(

(n_k)!k!(2~)n

: "i . (i

)

n-k. J

<»'-1 TrF

~

Introducing a new Chern density ~Q

"2n-2k,2k-1

=po

+n.

'2n-2k ,2k-1 '"2n-2k ,2k-l

'

(4.7)

one can see that the sequence Eqs.(4.2) become

d

~

nO2n-2k+l,2k-2 =-d x nO2n-2k,2k-l ,

(4.8)

460 34

CHOU Kuanq-chao, WU YUe-liang and XIE Yan-bo

d

nC

E; O,2n-l

=0

We now extend the definition of effective action in xEM2n-2 which should have no boundary.

=

r(A,U)+r'(A,6) •

(4 .• 9)

where (4.10) It is just the Wess-Zumino-Witten effective action in M2n - 2 manifold. r'(A,O)=2n =~

I

I

0'

X£M2n-2 ~£Ml 2n-2,1

.n-l

1

J

(n_1)!(2n)n-l X(M 2n - 2

TrFn-1.

(4.11)

where (4.12) when

n=3

(4.13) It is just the ordinary a-vacuum term in a four-dimensional Euclidean space. From Eq.(4.12), one can see that the parameter a for a-vacuum is the line integration of Abelian gauge potential which corresponds to the phase factor (or 1-cocycle) of the representation of the cranslation group presented by gauge covariant operator in the f,;-space

461 A Simplified Derivation of Chern-Simons Cochain and a POssible Origin of 6-vacuum Term T(f,;)=exp(E;oD) D=d~+B

35

,

, (4.14)

~1(X,E;)=r(E;)~(X,D)=eXp(J:B(X,E;»~(x,E;) ,

=exp(i6)U(x,E;)~(x,E;)

U(x,OEG

We now introduce a new gauge potential and field strength in ~-space through the Chern l-form n 2n - l ,1 or n~n-2,1 and 2-form n2n - 2 ,2 or n~n-3,2 by integrating over x when the ordinary gauge

potential B(x,E;) vanishes. One will see that they are just the e-vacuum and topological mass term comparing with that discussed in Ref.[5]. From Eqs.(3.3) one has

J

J

0. -1 l(A,D)= XfH2n-l 2 n , X£H 2n - l

+

J

dcn~n_l o(A,D) ~ ,

Xf.M2n-l

dxn o (A,D). 2n-2,1

(4.15)

~onsider a boundary1ess manifold M2n - l by Stokes' theorem, it

can be rewritten as (4.16) the Chern form 'gauge potential is defined as (4.17) It is a pure U(l) gauge potential.

f

Consider the 1-cocycle

J

61= .5V1=6 dE; n~ -1 o(A(x,O,O) E;(H' E;f.M' X€H 2 n-l n ,

J

=6 I f 101 2n- l

then

J

Xf.M 2 n-l

[Q;

n

-1

,

o(A(x,E;'),O)-r.; -1 n

,

o(A(X,~),O)]. 2n

can be dealt with as the boundary of an M

61=6

f

°2n,o(A(x,E;»

X(S2n

where when n=2,6 1 is just the ordinary 6-vacuum term.

(4.18)

and

(4.19)

462 36

CHOU Kuang-chao, WU Yue-lianq and

~IE

Yan-bo

If ~2n-l have boundary 3M2n-l=M2n-2 by Stokes' theorem,Eq.(4.13)

becomes

J

X€112n-2=aM2n-1 =

f

Q2 n -1

x€M,2n-1'

~") ~ 11- ..",1·( A ( X , ~ ) , 0 ) 1 ( A , 0 ) -d <"

f

~X(M

2 -1::

n

~ n-1, 0

(4.20)

The Chern form gauge potential is defined as .9?1=aJ

M2n-2

n~ n -2 , I(A(x.O,O) (4.21)

the field strength is (4.22) Consider n=2,

:,i 2 =ClM 3 =S2 x

.9B=ct}52 n2 ,I (A,O),

(4.23)

0

x

~ =-a J\1

5:

2,2

( A ,0 ) •

From the definition fl2 .,(A,B)= - S12Tddt"A+d B+AB+BA)2 ,~ ~ ~ x

(4.24)

,

it is easily checked that

1

(4.25)

Considering the integral of G over s~ in ~-space and supposing that it is toPological nontrivial,we then have

(4.26)

463 A Simplified Derivation of Chern-Simons Cochain and a Possible Origin of a-vacuum Term

37 ~-space,

On the other hand, if there exists a monopole in

in accor-

dance with the monopole quantization condition one obtains (4.27 )

21Tn=3o.w(g)

where W(g)=241 11

2

--7:iA':"24 1 ~

When wind number

~

f f ., f'".r sl

~

S3

w(~)=l,

s~

Tr

S

~Trn

,

-

(4.28)

g - l d~g( g - l d g) 2 ~

X

we have

'}

a =

g - l d .. g(g - l d~f.:)-"

x

.

('L29 )

n=l,2 •...

corresponds to the topological mass.

References [1]

H.r. Guc, ;:. ,.,";,,:, B.!r". Hou, and S.K. r\:'ar:q,

"~ncr.:a':'ies,

C;:t.=-:o~c~.:'q~...

Generalized Secondar~' Classes", prr:pr i:i.c AS-:r~P-8.J-£.i44, f2} ~.:;.

[J]

FaddC'~~",

5. Zunino,

~...:95';)

~:-::a

..

Phys.Lett. 145B(1984)81.

"C::Jhomoloqy of l~auqe Groups: ,~·oc~·c1.es

santa Barbara ,VSF-ITF preprir:t,

.1::;C

S..-:h:'i.nge" T:'",7:"

(1984).

[4] h.C. CHOU, Y.!.. WU and LB. XIE, Commun. Theor. Phys. ~(19S6)95.

[5] LS. rou and A. Zee, "j1.belian Gauge Str:.zcture inside Nc.'n-.::..be2;'an Gallqe 'Z'hecrems",

preprint 40045-35(1:'54).

464 Conunun. in Theo.r. Phys. (Beijing,

Cn~na)

Vol. 8

«

No.3 (1987)

341-367

THE GENERAL CHERN-SIMONS CHARACTERISTIC CLASSES AND THEIR PHYSICAL APPLICATIONS WU Yue-liang( ~.ffi-l ), XIE Yan-bo( and ZHOU Guang-zhao( jiJ;7\;{! )

lit II i)

Institute of Theoretical Physics, Academia Sinica, P.o. Box 2735, Beijing, China

Received January 3D, 1986

Abstract Simple derivation of the general Chern-Simons characteristic classes is presented. Application to construct current conservation anomalies and effective action is discussed in detail. The effective action of nonlinearcr model where the fermions are interacting wi th external gauge field!. is obtained. The central term of Virasoro algebra is also derived by topological method. possible existence of a gauge potential in the group manifold and its relation to g-vac"um is indicated.

I. Introduction Anomalies due to quantum loops of fermions appear in the conservation equations and the commutator equations. They have played a strikingly important role in the development of particle physics and quantum field theory. Just mention one example, the renormalizabili ty of gauge theories enforces the condition of cancellation of chiral anomalies, leading to the fascinating correlation between quarks and leptons and the gauge group in the superstring theory. The fact that the expression of chiral anomalies obtained in the lowest order perturbatio~ is exact must have a deep reason. Indeed it was realized only rd.:;ently that anomalies have topological origin and represent the non-trivial elements of the cohomology of the gauge fields. Now it is possible to calculate anomalies and the effective action by differential geometric method based on the Chern-Simons characteristic classes without ever calculating a Feynman diagram. In this article we shall review briefly some recent developments related to the work done in China[l]. It is in the direction of using cohomology of gauge groups to study the anomalies and the effective action. For alternative views and presentations please read those

465 WU Yue-liang, XIE ran-bo and ZHOU Guang-zhao

342

excellent articles by Faddeev, Jackiw, Witten, Zumino and many others[2-5].

II. Mathematical Preliminaries Points, lines, triungles, tetrahedrons and their. generalization to higher dimensions are called simplices. An n-simplex is constructed from n+l distinct ordered points a.a1, •••• a n in a manifold and will be denoted by the symbol (a.al, ••• ,an ). A zero simplex is a single point; a I-simplex (a.a 1 ) is a line bounded by two zero simplices (a) and (a 1 ) with definite orientation. We define the boundary operator a such that (2.1)

In general. the boundary of an n-simplex consists of n+1 (n-I)-simplices. The boundary operator a is defined to be

(2.2) The orientation of the boundary is taken into account by the permutation factor (_1)2. It is easily verified that the boundary of a boundary vanishes, i.e. (2.3)

A real function of n-simplex f(a.al, ••• ,an ) is called an n-cochain. It is possible to define a coboundary operator dual to the boundary operator in the following way

(2.4)

It is also satisfied the nilpotent condition ~2=O

•

(2.5)

An n-cochain f(a,a1, ••• ,an ) is called a cocycle if its coboundary is zero (2.6)

It is called coexact if it is the coboundary of an (n-l)-cochain

466 The General Chern-Simons Characteristic Classes and Their Physical Applications

343

h(a,al'.·. ,an - 1 ) (2.7)

Two cochains f and g which differ by a coexact cochain are equivalent if

(2.8)

f-g=t.h.

The equivalent classes fo~m a group callp.d cohomology group. We are interested in the manifold composed of gauge fields represented as points die Like in the case of monopole it is not possible in general to describe a topological nontrivial object by one gauge field regular everywhere. We have to divide the space into several covers with one gauge field in each cover, and in their intersections the gauge fields are related by gauge transformations so that the field strength defined by these gauge fields are gauge equivalent. Therefore, the cochain f(A, •.• An ) with Ai equals thp. gauge transforma.tion of A (2.9)

is of particular inter~st where gi(x) are grou~ case the cochain can be written in another form

...

f(A,A1, ..• An)?f(A;gl, .•• gn)'

elemen~s.

In this

C2.10)

Either form \'."ill be used in the following, where the,... will be omitA~ochain

ted for simplicity. the condition

is called invariant if it satisfies

(2.11)

or (2.12)

To illustrate the usefulness of these mathematical concepts in physics, let us consider the effective action of an underlying theory wi th massless fermions interacting- with external vector and axial vector gauge fields defined on a non-Abelian chiral group. The effective action cannot be a group invariant since the conservation law is broken by chiral anomalies. Let the quantum amplitude be

-

Z= .. ifO!J

.

(2.13)

467 344

WU YUe-liang, XIE Yan-bo and ZHOU Guang-zhao

where r is the effective action and A the external gauge fields. Under a group transformation the quantum amplitude will be changed by a phase (2.14) The infinitesimal variation of the phase n1 (A;g) in the group manifold gives the anomaly for current conservations. The group multiplication condition (2.15) implies that

Ql(A;g) is a cocycle

~Ql(A;gl,g2)

=Ql(glA;g2)-Ql(A;glg2)+Ql(A;gl) (2.16)

=0.

I·f Ql(A;g) is the trival coexact such that

ele~ent

of the cocycle condition, it is

(2.17) In this case, we would redefine the quantum amplitude (2.18) to get an invariant effective action

and eliminate the anomalies by adding counter terms. This is not the case if Q1 is a nontrivial element of the cohomology. Therefore, the chiral anomaly appeared in the conservation equation of the current is determined by the cohomology of the group manifold. The coboundary condition Eq.(2.16) is a global form of the Wess-Zumino consitency condition for the anomalies. A second example concerns the 2-cocycles and the Schwinger term in the commutation relation of the currents. Consider a representation of the group on a functional of the gauge fields (2.20)

468 The General Chern-Simons Characteristic Classes and Their Physical Applications

345

where V(A;g) forms a projective representation (2.21) The condition for the group multiplication satisfying the associative law

requires that the phase n2 (A;gl,g2) is a 2-cocycle. In some cases is a matrix instead of a real function. In that case the cocycle condition has to be modified to incorporate the noncommutative nature of the phase n 2 , when r. z is a real function as in the case of Abelian gauge theory, the cocycle condition can be easily worked out to be

n2(A;gl,g2)

=0.

(2.23)

(2.24)

the condition Eq.(2.23) satisfies automatically and n z is a trivial element of the cohomology. In this case a change of the phase V(A;g)--+V'(A;g)=V(A,g)e-iW1(A;g)

(2.25)

will eliminate the phase and V'(A;g) forms an ordinary representation. A nontrivial phase n2(A;gl,g2) contributes an anomalous term ~o the commutation relation of the generators of the group defi~ed by the V(A;g) representation. Higher order cocycles are studied in the literature. The 3-cocyeIe is related to the breakdown of the Jacobi identity. We shall not discuss them further. In differential geometry the exterior differentiation d is a coboundary operator satisfying the nilpotent codition d 2 =O.

The gauge potential A is a one-form

(2.26)

469 WU Yue-liang, XIE Yan-bo and ZHOU Guang-zhao

346

(2.27)

Ia

where A~(x) are real functions and strength is defined to be a two-form

the group generator.

F=dA+A 2 with A2 the exterior product of two gauge potentials. the Bianchi identity

The field (2.2.3)

F satisfies (2.29)

An infinitiesirnal gauge transformation is defined to be OA=-AB-BA-dB,

(2.30)

where B is the exterior differential in the group manifold and B is a one-form usually defined to be a pure gauge B=g

-1

6&.

(2.31)

In this paper we shall keep B to be an arbitrary one-form in the group space and define a dual field strength G=8B+B2

(2.32)

satisfying a dual Bianchi identity oG=[G,B] •

(2.33)

The condition fer the gauge potentials A and B to be single valued requires that of=[F ,B] , dG=[G,A] ,

(2.34) (2.35)

and [F,G]=O.

(2.36)

III. Chern Class and the General Chern-Simons Cochain It is well known in mathematics that the Chern class gives a representation of the cohomology classes by the curvature forms of a connection in the fibre bundle. In this section a simplified derivation of the general Chern-Simons characteristic classes will be brie~ly described. The Chern density is a 2n-form

470 The General Chern-Simons Cbaracteristic Classes and Their Physical Applications

347

(3.1)

where

is a

norm~lization

that

C2~(F)

constant.

is closed for

Using Eq.(2.29), it is easily verified

arbitr~ry

gauge potential

in so far as F is defined by Eq.(2.28). We shall show in "the following that Eq.(3.3) 18 sufticient for tis to derive all the ~esults connec~ed to a whole chain of higher order characte~istic classes from which nontrivial elements of the cohomology can be f()U;:).d. Consider a manifold consisting cf two subma~ifolds, one of whicn is the ordinary space wit~ ~oc~dicates x~ a~d tbR other a manifold of auxiliary para:neters 1;2.. The exte::-ior elf re:::-en-cial operator is (3.4)

where (3.5)

and (3.6)

Any gauge poten"tial Ji can be decomposed into two parts ..s'(x,t)=A(x,t)+B(x,t) • where A(x,t) is a one-form in x~ only and B(x,t) that in t The field streng"th § now becomes

(3.7) i

.

(3.8)

where F=d x A+A 2

,

(3.9)

M=dtA+dxB+AB+BA •

(3.10)

G=dt: B+ B2

(3.11)

and

The Chern density p.12)

471 WU Yue-liang, XIE Yan-bo and ZHOU GUang-zhao

348

satisfies the Eq.(3.3) (3.13) Expanding C2n(F+M+G) according to the degrees of the forms in dt i we have 2n

C2n(F+M+G)=

E

m=O

C2n _ m ,lotA,B),

(3.14)

where C 2n _ m,m(A.B) is a form of degrees 2n-m in dx~ and m in dti. Substituting Eq.(3.14) 'into Eq.(3.13) and comparing the degrees of both sides we obtain d x C2n (F)=0

,

dtC2n(F)=-dxC2~_1,1

d t C2n(G)=0

,

1

(3.15)

J

Now choose the gauge potential to be

,

B(x,t)=O

A(x,t)=E tiA(i)(x)

}

(3.16)

i

The boundary of the t-submanifold is determined by the condition (3.17) In this special case G=O ,

(3.18)

(3.19) and (3.20) It is easily found that (3.21)

472 The General Chern-Simons Characteristic Classes and Their Physical Applications

349

where Str means symmetrization of the operators inside the trace operation. The Chern-Simons secondary characteristic class can be derived by taking only one parameter t=t(l)=l_t(O) and integrating the equation.

along a straight line (a one simplex) from t=O to t=l. of t we have

In terms

{3.22) (3.23)

and (3.24)

After integration we obtain C

2n

(F(l)_C

2n

(F(O)·)=_dJ1na. x

_

0

()

n

Tr«A.(l)_A(O)Fn-l)dt t

--d x .'2n-l (1 (A

(1)

,A

(0)

).

(3.25)

This is the well-known formula of Weyl homomorphism which states that the Chern density of two different gauge field~ are cohomologous in space manifold. Taking A(0) (x)=O and A (1) (.x)=A(x) in Eq. (3.25) i t follows that

=dQ2n-l (A)

,

(3.26)

where (3.27)

with (3.28) Q2n-1 is the Chern-Simons secondary characteristic class. In the general case we choose m+l parameters t i and form an

m-simplex Sm with m+l vert1ces ao=(l,o, ••• ,O) ,

473 fro

"350

a 1 =(0,1, ••• ,0), a~=(o,o

Yue-liang,

XIE

j\

Yan-bo

and ZHOU

Guang-zhao

(3.29)

•...• 1),

whose boundary consis"ting of m+1 (m-1 )-simplices Sl~~~ is specified by the equation I ti=l or i

(0) ( --8 m-I a1

,···

( 1 ) ( a ,:1 ,am ) -,Sm-I o 2

, •••

+ ••• +( -~1)m,,(m)( ) :::i _ m 1 aa,···,a m_ 1 •

,am )

(3.30)

Integrating the equation (3.31)

over 1:he :n-simplex

S!IJ(a~

•••• ,am) we cbt:lin by Stoke'S theorem that

(,(O)

A0

LJ:Ji.2n_m+l l m_l:So

=

Ei

t ••• ,

A(m), J

(_l)iJ

,... (i)v2n-m+l,m-I Sm_"l

=-d () (A(O) A(m) x""2n-lll,m , ••• ,

,_

(3.32)

where t:. is the coboundary operator and (3.33) is the integral of C2n-l,1 over the I-simplex in toe parameter space. 02n-l,1 is called the lth Chern-Simons characteristic class which

satisfies a sequence of descent equations starting from the Chern density. For gauge potential re~~lar everywhere on a compact space manifold M2r.-m+l of dimension 2n-m+l the Eq.(3.32) can be integrated over M2n - m+ 1 • The right-hand side of Eq.(3.32} vanishes after integration and therefore

t:.r~n2n-m+l,m-l=0 or

J r. is

,

(3.34)

a (m-l) -cocycle. In case when nontrivial topological object exists, the gauge potential cannot be regular everywhere. We have to divide the space manifold into several covers M

474 The General Chern-Simons Characteristic Classes and Their Physical Applications

351

(3.35) In each cover Mj a regular gauge field A(j)(x) can be defined.

In

the intersection Min Mj the gauge fields A(i) and A(j) are related by a gauge transformation (3.36) so that the invariants of the gauge fields are uniquely defined over the whole space manifold.

We shal.l show that the cocycle condi-

tion Eq.(3.34) is no longer v~lid when the transition function gij(x), which is a mapping of the boundary aMi=-aM j to the group manifold, has nontrivial homotopy. As an illustration, suppose two covers Ml and M2 with aMl=-aM2 are sufficient to define a gauge potential.

t

C 2n(F)=

~

t

From Eq.(3.26), we have

.C 2n (F)= JaM (rl.2n-l ,1CA(1) ,0) z

~

-rl. 2n - 1 ,l(A

(2)

(3.37)

,0».

It is seen from the definition that rl. 2n _ 1 ,l(A,B) is a gauge-invariant cochain (3.38)

rl.2n_l,l(gA,gB)=rl.2n_l,l(A,B). Therefore from Eqs.(3.36) and (3.38) we get

(3.39)

o2n-l.l (A ( 1 )

'

A. ( 2 ) ) _0

2n-l,1

(A {() )

=-d x"2n-2,2 n (A(O) , A(1) , A(2)

!\. (;' ) ) +0

'"

.

:n'-1,1

(A (0)

'

A (1) )

(3.40)

,

we obtain

(0 ,g12 -ldg12 ) + d xrl.2n-2,~ ( A (0) , A(1) , A(2) --fl 2n - l ,1

•

Integrating Eq.(3.40) over aM 1 it follows from Eq.(3.37) that

JMC2n (F)

(3.41)

475 352

WU Yue-liang, XIE Yan-bo and ZHOU GUang-zhao

=-fdM2n2n-l'1(O.g~~dg12) =_nanlldt.t2n-l(t_1)2n-1J~ • 0

cM2

Tr(g~idg12)2n-l

=-nanB(2n.2n)J~aM2 Tr(g~idg12)2n-l (3.42)

=winding number.

If the transition function g12 is a nontrivial element of the homo-

topy group TI 2n - 1 • the integral of the Chern density will be equal to the winding number of the mapping of a (2n-1)-dimensional sphere to the group manifold.

IV, Group Cohomology of Faddeev In order to get the group cohomology of Faddeev. we start from Eq.(3.26) (4.1) This relation is also true for arbitrary gauge field and manifolds. Let us apply this equation to a manifold consisting of the direct product of a space and a group submanifold with the coordinates x~ and ~i respectively.

The gauge field is (4.2)

where

A(x.~)

is a one-form in the space manifold and

in the group manifold.

B(x,~)

is that

The field strength is (4.3)

where F=d x A+A

2

•

M=d~A+dxB+AB+BA, G=d~B+B2

1 J"

(4".4)

.

Eq.(4.1) then has the form (4.5) Expanding both the Chern density and Chern-Simons secondary class according to the degrees of d~i, we have

476 The General Chern-Simons Characteristic Classes and Their Physical Applications

and

353

C2n (F+M+G)= ~ C2n _ m,m(A,B) , W2n - l

(A,B)=

~

(4.6) (4.7)

W2n - m ,m-l(A,B).

Substituting Eqs.(4.6) and (4.7) into Eq.(4.5) and comparing the degrees of d~i on both sides we obtain

(4.8)

As a special case we choose A(x,()=U-1(x,()(A(x)+dx)U(x,()

(4.9) B(x,~)=u

-1

(x,~)d~U(x,()+C«()

where U(x,() is a group element and tial in the group manifold.

,

C(~)

is an Abelian gauge poten-

Then

F(X,()=U-1(X.~)F(x)U(x,~)

M(x,()=O ,

We see that the conditions Eq.(2.34)-(2.36) are satisfied.

(4.10)

From the

expansion (4.6) we have (4.11)

(4.12) An m-simplex in the group manifold will be defined with m+1 vertices corresponding to m+1 group elements go, gogl' •••• goJ ••• 'gm. to do so we define a function

In order

h(~.g(x»=e(l-~)U(X)

(4.13)

h(l, g(x»=l ,

(4.14)

such that

and

477 354

WU Yue-liang, XIE Yan-bo and ZHOU Guang-zhao

h(O, g(x»=eu(x):g(x) • The function

h(~,g(x»

(4.15)

maps a line segment (1,0) in the

~

axis to

a line running from the identity element to g(x) in the group manifold. Now choose (4.16) The simplex in the

~

_(~o ai.. , •••

space consisting of the following m+1 vertices ,~

i-I

i

,~,~

i+1 , •••

,~

m)

(4.17)

=(0, •.. ,0,1,0 •••• ,0) will map into a simplex in the group manifold with vertices

(4.18) From Eq.(4.12) the term C2n - 2 ,2 can be written as a total dif!erential in E;

where (4.19) The descent Eq.(4.8) can now be written as

C2n(F(x»=dxn2n_I,O(A(x»,

1

d~W2n-l,o=-dxW2n-2,1 ,

r

(4.20)

where (4.21) Integrating Eq.(4.20) over simplices in E; manifold, we get equations similar to Eq.(3.32).

V, The Effective Action of Wess-Zumino-Witten Consider a system of quark field

~

interacting with color gauge

field C corresponding to color group SU(N c ) and some external chi~al flavor gauge fields AL and AR. The Lagrangian has the form

478 The General Chern-Simons Characteristic Classes and Their Physical Applications

355

(5.1)

where we ha"/e used a double matrix notation for cOllvenience (5.2)

The color gauge field c~ is diagonal and C~v is its field strength. The flavor currents are covariantly conserved on the classical level (5.3)

where Ai i=l, ••• ,N are the flavor group generating matrices. Howeyer, the quantum corrections break the conservation equation and produce the current anomaly .

.

",.uJ~-G~ZJ ].,1-

where

Nc

Nc

.

1

•• , ' 24n2 E: lJVC"LT ry 5/\,.l~(A"'A 0],.: Va p 0 + 2"""\i.",:.8.:1 J,

is the number of colors and

1

'(5'::

(~

(5.4)

0

-1).

There are different forms of current anomaly. The one presented in Eq.(5.4) is called the symmetric form which contributes to both vector and axial vector current conservations. It is possible to add local counter term to the effective Lagrangian so that the vector current is conserved and tht: anomalies are shifted completely to the axial vector current. This form of anomaly is called unsymmetrical one. When color degrees of freedom have been integrated, we get an effective theory describing the low energy phenomenology. The effective action r under infinitesimal falvor group transformation should reproduce the anomaly (5.5)

0E;r=Tr(Gv) , where

v=g-l0E;g=Ii(E;)dE;i G=Gir i (E;)

·1

(5.6)

A

with

Ii

the generators of the group.

Under exterior differentiation (5.7)

479 356

wu

Yue-liang,XIE Yan-bo and ZHOU Guang-zhao

where f ijk is the structure constant of the flavor group. Eq.(5.5) we have

From

o~r=O=Tr(o~Gv+Go~v)

=Tr( o~Gli (Od~i + ~fijklk( Od~id~j),

(5.8)

hence (5.9) This is the Wess-Zumino consistency condition that has to be satisfied by the current anomaly • The descent Eq.(4.20) (5.10) implies that d~W2n-2,1 integrated over a 2-simplex in the group manifold is homologous to a cocycle whose space integral can be indentified with the phase Q1 of the effective action r as dicussed in Eq.(2.14). Eq.(5.10), its infinitiesimal form, integrating over a (2n-2)-compact space manifold has the form of Eq.(5.5) for an anomaly in (2n-2)-space dimensions. Therefore W2n-2,l, with suitable field contents, may be identified as the current anomaly which has been verified by theoretical calculations. In order to get an explicit form of the effective action one has to perform an integration in the group manifold for W2n - 2 ,1. ~his h~s been done first by Wess-Zumino[9]. They could not find a close form in 2n-2 dimensions. It has been realized later by Witten that a close form is possible only in one dimension higher[5]. Consider a form P2n-l in 2n-1 dimensions. What conditions it has to satisfy in order to describe a Lagrangian in (2n-2)-dimensional manifold M which is considered to be the'only boundary of a (2n-1)dimensional manifold S? There are three of them: i). Locally, P2n - 1 should be an exact form so that

ii). Globally, it should not depend on the manifold S choosen so far as i.t has the same boundary M. This implies that P2n-l integrated over any closed manifold should be equal to 2n times integer f SlP2n-l- fS2P2n-l=PS1+S2P2n-l=2nn

(5.11)

480 Tbe General Chern-Simons Characteristic Classes and Their Physical Applications

357

such that the uniqueness of the quantum amplitude e ir is guaranteed. iii). Under an infinitesiinal group transformation, i t pr('lduces the correct current anomaly (5.12) The third condition is easily seen to be satisfied by 2nW2n-l,1 which, however, does not satisfy the first two. To find P2n-l satisfying all three conditions, a Goldstone field U(x) transforming as (5.13) has to be used.

In the matrix notation we write

U(X») o

=Q

-l (x)

(5.14)

which transforms as (5.15)

Q(x) -+gQ(x)g-l with g defined to be

(5.16)

Now define a new gauge field

(5.17)

where A is the one defined in Eq.(5.2). the same way A~g(A+d)g - l

,

I

Both A and A

transform in

(5.18)

AO~g(AO+d)g-l •

It is easily found that

o

_1

F =QFQ

•

(5.19)

Chern density (5.20)

481

wv

358

rue-liang, XIE Yan-bo and ZHOU Guang-zhao

which is the difference of the Chern densities for the left-handed and right-handed gauge fields. From this Chern density one can derive the sequence of equations (5.21) (5)

h were 02n-1,1 describes correctly the current anomaly indicated in Eq.(5.4). Since -1

Q y sQ=-y 5

(5.22)

'

it follows that (5.23) Therefore (5)

(5)

Q

(5.24)

n 2n - 1 , 1(A,0)+02 n- 1 , l(A ,0) is a close form.

From gauge invariance and Eq.(5.22), we have Q

( (5)

_

(5)

J 2n _1 ,1(A ,0)--n 2n _1 ,1(A,Q

-1

dQ).

(5.25)

Using Eq.(3.41), we get (5)

(

(5)

Q

n 2n - 1 ,1 A,0)+n 2n _1 ,1(A ,0) =n(5} (A 0)_n(5) (A Q-1 dQ ) 2n-1' 2n-1,1' n ( =n(5} 2n-1,1 (Q-l dQ,O ) -d x "2n_2,2 A,O,Q-1 dQ) ,

(5.26)

where

=a n B(n,n)Tr{ys(Q-1 dQ )2n-1} =2a nB(n,n)Tr{(U- 1dU)2n-1}

(5.27)

is a closed form. Integration of n(5} (Q-1 dQ ,O) is an even integer 2n-1,1 corresponding to the winding number of a mapping from the 2n-1 sphere to the coset manifold GLxGRIG L+R • At the same time we have n(5}

(

(5)

(Q

"2n-1,1 A,O)-n 2n _ 1 ,1 A ,0)

482 The General Chern-Simons Characteristic Classes and Their Physical Applications

359

=Q;!:l,l(A,AQ)-dxQ2n_2,2(A,O,AO) , (5)

l5.28)

0

where Q2n-l,l(A,A ) is a gauge-invariant function. Eqs.(5.26) and (5.28) we have Q(5)

2n-l,1

_1

Combining

(A,O)

(5)

-1

- '2{Q2n-l,1 (Q

0

(5)

dQ,0)+Q2n-l,1 (A,A )

- d X[Q2n_2,2(A,0,Q -1 dQ)+Q2n_2,2(A,O,A 0 )]}.

(5.29)

We now define _ (5) 1 (5) 0 P2n-.1- 2 'IT[Q2n_I,1 (A,O) - '2Q 2n-l,1 (A,A )]

_

(5)

--'IT [Q2n-l,1 (Q

-1

_

(5)

dQ, 0 )-d x [S'2n-2 , /A, O,Q

-1

(5) 0 ] +Q2n_2,2(A,O,A ) ].

dQ) (5.30)

It is easily verified that p thus defined satisfies all three 2n-l conditions stated above. The effective action related to the anomaly can therefore be written in the form

(5.31) The only term in a (2n-2)-dimensional effective Lagrangian that does not have local expression in 2n-2 dimension is the one corresponding to the local expression of a winding number expressed in one dimension higher. When the flavor symmetry SU(Nf)LxSU(Nf)R breaks spontaneously into SU(Nf)£+R in the low energy limit, pion-like Goldstone bosons are created in the direction of the coset generators. These chiral Goldstone boson fields are described by the matrix U{x) introduced above. We write

u(x)=exp{f~~i'ITi(X)} , where ~i' i=1, •• ,N f

,

are Gell-Mann matrices and

'lTi{x)

are the pion-

483 360

WU Yue-liang, .YIE yan-bo and ZHOU Guang-zhao

like Goldstone field. Expanding the effective action Eq.(5.31) in power series of the pion fields, we get the self-interaction between pion fields and their interactions with the gauge fields. It is interesting to note that the term linear in the pion field n.(x) is proportional to the unsymmetrical current anomaly for chiral ~urrents[5] If an Abelian gauge field C(~) is taken into consideration and the effective Lagrangian in constructed from W2n - 2 ,1 defined in Eq.(4.21), we obtain from Eq.(4.19) that

where r9=2nJ x~M JWI2n- 2 , z(A,B)=9JC 2n -

2(F)

with

This is a possible ce~didate for the 9-vacuum term. Since 9 is a line integral of a gauge potential in group space, it is uniquely defined only when its field strength G=d~C(~) vanishes. Therefore C(~) is a pure gauge C(~)=d9(~) and 9=9(1)-9(0). We know that the axial U(l) transformation produces a 9-vacuum term in the effective Lagrangian, which causes the strong CP problem, a serious trouble in particle phYSics. It could be that the axial U(l) transformation is closely related to a gauge transformation of a gauge potential in the group manifoid so that both contribute a 9-vacuum term while the two terms cancel each other.

VI. The Effective Action of a Nonlinear External Gauge Field

cr

Model Interacting wfth

Consider a theory with symmetry group G spontaneously broken down to a subgroup H. The effective theory then describes Goldstone boson fields in the direction of the coset space G/H. Any group element gEG can be decomposed into the product of an element q, belonging to the coset G/H and an element h of the subgroup H. (6.1)

The Goldstone fields are then described by a function ~(x)EG/H. Under a group transformation the element gq,(x) can again decompose

484 The General Chern-Simons Characteristic Classes and Their Physical Applications

361

in the form of Eq.(6.1) g'$(x)=$(g,$(x»h(g,$(x», where $(g,$(x»EG/H and law

h(g.$(x»~H

(6.2) satisfy the group multiplication

$(g',$(g,$(x»)=$(g'g,$(x»,

I

(6.3)

h(g',$(g,$(x»)h(g,$(x»=h(g'g,$(x». Therefore the Goldstone field $(x) transforms as (6.4) The one-form $-l(x)d$(x) is valued in the Lie algebra of G which can be decomposed into two parts ¢

-1

(6.5)

(x)dq,(x)=H+K,

where H is a one-form valued in the Lie algebra of the subgroup H and K is that of the coset G/H. Under a global transformation of g. we have from Eqs.(6.4) and (6.5) that

H~H'=h(g, <1>(x) )(H+d)h- (g,
(6.6)

and K-K'=h(g.,q,(x»Kh

-1

(g,¢(x».

(6.7)

The kinetic energy of the Goldstone boson fields in constructed to be (6.8) which is easily seen to be an invariant with the nonlinear

reali~a

tion of the field q,(x) described by Eq.(6.4). When fermions and external gauge fields are present, they are described by a Lagrangian

(6.9) Before symmetry breaking, left-handed 1/.'L and A are representations of the group G.

After the symmetry breaking the fermions and the

gauge fields become

485 }62~________________________~wu~~~~ue~-~1~~~·a~nzg~,~X~T~E~~~a~n~-bo~~a~n~d~Z~n~O~U~Gu~a~n~g~-~z~h=ao~

(6.10) and the corresponding Lagrangian takes the form (6.11)

The effective action after the symmetry breaking has to produce the same anomaly as the original action. terms depending on

~(x)

nonlinear a model.

This requires additional

and A(x) to the effective Lagrangian for the

These additional terms have to satisfy the same

conditions discussed in the prevous sections. Let us introduce a double matrix notation as before and write

(6.12)

and

o

A(x) .,d(x)= (

0

(6.13)

~-l(x)(A(x)+d)~(x)

we have (6.14)

.JtI Q=Q( fl (x )+d)Q(x )=fl (x)

(6.15)

and (6.16) The Chern density (6.17) vanishes since (6.18) The Chern-Simons secondary class descending from the Chern density

Eq. (6.17) is (6.19)

486 The General Chern-Simons Characteristic Classes and Their Phgsical Applications

363

wnose variation under group transformation produces the difference of the current anomaly before and after the symmetry breaking. From Eq.(6.15) we have

njn-l,1 (..d ,0) =njn-1,1 (.,4Q ,O)=-nin_1,1 ( .,4 ,QdQ).

(6.20)

Hence

n~n_1 . , 1 (.,4 ,0)

(6.21) The first term in Eq.(6.21) is a closed form whose integral over (2n-1)-manifold is a winding number of the mapping described by the coset function ~(x). Therefore, 2~jsn~n-1,1(~'O) satisfies all the conditions required and can be choosen to be the additional ~erm of the effective action for the nonlinear a m.)del. When the external gauge fields are switched off, we get the same result obtained in Ref.[6]. Our result is more general since it contains interaction of the Goldstone boson with the gauge field describe1 by the term IMn~n_2,2(A,O,QdQ~. From our derivation it is also apparent that if the original theory has no anomaly so does the effective theory. The t'Hooft consistency condition is automatically incorporated.

VII. Central Tenm in Virasoro Algebra As discussed in the previous sections, a 2-cocycle can be constructed from the Chern density with two higher numbers of dimens·ions. This 2-cocycle is proportional to the anomalous term in the commutator of gauge group generators[21. It is well known in the string theory that the Virasoro algebra has an anomalous term called the central term[71. We shall show in this section how this central term can be obtained by constructing a 2-cocycle appropriate to this case. The generators Ln of the Virasoro string form a conformal algebra in the classical theory

n,m= •.. ,-l,O,l, . . . •

(7.1)

487 WU Yue-liang, XIE Yan-bo and ZHOU Guang-zhao

364

When the strin·g is quantized, an additional term of the form C 3 ~n,-m 12(n -n)

(7.2)

appears in the commutator. Physically, no gauge field has been introduced in the string theory. An interesting question is whether the above anomaly can be described by a 2-cocycle induced by an fictitious gauge fiel'd introduced purely as a tool of calculations. The Virasoro algebra with anomalous term has the form (7.3) where Cm,n are antisymmetric (7.4)

C

m,n =-C n,m

and satisfy an equation due to Jacobi identity (m-n)C

n+(n-~)C

m+n,~

n

n+~,m

+(~-m)Cn

~+m,n

=0.

(7.5)

Solutions of Eqs.(7.4) and (7.5) can be easily found.

where b(n+m) is an arbitrary function of (m+n). relations (7.3) now become

They are

The commutation

(7.7) Making the change (7.8)

we can rewrite Eq.(7.7) in the following form (7.9) The center in Virasoro algebra

~2~m+n,o(m3_m)

(7.10)

can be cast into the form of Eq.(7.9) by a change of the generator (7.11>

488 The General Chern-Simons Characteristic Classes and Their Physical Applications

365

and

=..£.. 12

d

(7.12)

Our next problem is to find a 2-cocycle which produces this anomalous center term. tion of Ln

For this purpose we consider a representa-

(7.13) The arbitrary function fez) can be eliminated by a transformation of the wave function q,(z)

~(z)--+e-ff(Z)/Z'dZq,(z) •

(7.14)

Therefore without loss of generality we can take f(z)=O. Under a finite transformation the wave function q,(z) changes into (7.15) where E n (n= •••• 1.0.1 ... ) are group parameters and L =zn ( z...L + an)

aZ

n

(7.16)

'

The coordinate

z' =e

~ En Z

n+l

aa Z

(7.17 )

Z

can be obtained in the following way.

Let us define a function

,(z) such that

(7.18) then

(7.19) The operator \

t.En Z

n+l

en

a

a

--

(7.20)

aZ=eC!T

is a displacement operator of the coordinate T.

Therefore

a

zt=eaTz(T)=Z(T(Z)+l)

(7.21)

489 366

Pro Yue-liang, XIE Yan-bo and ZHOU Guang-zhao

As a special case we take (7.22)

It is easily found that n -lin =z z'=z(l-nEz) E

(7.23)

In this special case the phase factor U(z) can also be found as U E (z)

= _-=1=--__

(7.24)

(l_nEz n )a

wbich is seen to satisfy the group relation (7.25)

The presence of the phase factor U(z) indicates that besides the coordinate transformation, a U(l) gauge field A(z) is needed to define a covariant derivative. Under the conformal transformation described by the Virasoro algebra, the gauge potential transforms as -l

A(z)--+A'(z)=A(z')+U E (z)dUE(z) .

(7.26)

To calculate the commutator anomaly we have to find the 2-cocycIe descending from the Chern density in four dimensions. From Eq.(3.34), the anomalous term is proportional to

Jn2,2=8~JVdV

(7.27)

where -l

v=UEdE;Ue: =

n En {dE;(I)(E In +E;,(2)E 2n )+E;,(I)dE;,(2)E 2}anz n·

Here E;,(i) , i=1,2, are two parameters varying from 0 to 1 and infinitesimal group parwmeters. It is eassily found that

(7.28) E ln

are

vdv= Emwmanzn+m-1dzE;,(I} dE;,(l) dE;,(2) n,m (ElnE2m-ElmE2n)·

(7.29)

490 The General Chern-Simons Characterist~c Classes and Their Physical Applications

367

Integrating Eq.(7.29) over ~(i)(i=l,2) and z, we obtain for the nontrivial 2-cocycle (7.30)

where ~ is the number of times which is determined by the line integral wrapping around z=O. The factor. a2~m30n+m,0/2 is the anomalous center term for the Virasoro algebra which has the same form deduced by directly solving the Jacobi identity. Thls is not surprising since Eq.(7.5) is an infinitesimal form of the coclosed condition fca 2-cocycle.

References [1]

X.C. CHOU, H.Y. GUO, K. WU and X.C. SONG, Cammun. Theor. Phys.(Beijing ) 1(1984)125,129; K.C. CHOU, H.Y. GUO, X.Y. LI, K. WU and X.C. SONG, Commun. Theor. Phys. (Beijing

1

1(1984)491,498; K.C. CHOU, H.Y. GUO, K. WU and

X.C. SONG, Commun. Theor. Phys.(Beijing)1(1984)593,603; K.C. CHOU, H.Y. GUO and K. WU, Conrmun. Theor. Phys.

~J1985)91,

101; H.Y. HUO, B.Y. GOU, S.X. WA.VG

and K. W, Commun. Theor. phys. i(1985) 233 ,251. [2)

L.D. Faddeev, Phys. Lett. l45B(1984)81.

{31

R. Jackiw, Phys. Rev. Lett.

[4]

~(1985)159.

B. Zumino, Y.S. WU and A. Zee, Nucl. Phys. B. Zumino, Nuel. Phys.

~(1984)477i

~(1985)477.

[5]

E. Witten, Nuel. Phys. ~(1983)422i L. Alvarez Gaume and E. witten, Nucl.

[6]

Phys. ~~(1983)269. L. Alvarez Gaume and P. Ginsparg, HUTP preprint 85/A015(i985); J. Bagger, D. Nemeschansky and S. YanlcieloftTicz, SLAC PUB3698 HUTP 85/A042

(1985) •

[7]

A. Neveu and J.B. Schwarz, Nuel. Phys. !!.,!(1971)86; P. Ramond, Phys. Rev. ~(197l)2415; P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorr, Nuc.z. Phys. B56(1973)109.

C8]

W.A. Bardeen, Phys. Rev. 184(1969)1848; D.J. Gross and R. JackiftT, Phys. Rev. P2,.(1972) 477 •

C9]

J. Wess and B. Zumino, Phys. Lett. 37B(1971)95.

491 PHYSICAL REVIEW D

15 JANUARY 1991

VOLUME 43, NUMBER 2

Signature for chiral-symmetry breaking at high temperatures Lay-Nam Chang Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

Ngee-Pong Chang Department 0/ Physics, The City Col/ege o/The City University 0/ New York, New York, New York 10031

Kuang-Chao Chou Institute a/Theoretical Physics, Academia Sinica, Beijing, China (Received II July 1990>

In this paper We study the temperature dependence of chiral-symmetry breaking. Our real-time calculation shows that the thermal fermion propagator has a Lorentz-invariant massive particle pole even as (vacl¢¢lvac) 6 vanishes for T> A,e 2lJ • The traditional signature of chiral-symmetry breaking is found only after transforming the Dirac field to the new chiral basis at high temperatures.

I. INTRODUCI'lON Ever since the pioneering study I by Nambu and Iona(NIL) of the dynamical chiral-symmetry breaking In_quantum field theory, it has been recognized that ( ",,,,) gives the signature of chiraI-symmetry breaking in the theory. The question that we shall address in this 'paper is whether this remains the signature for chiralsymmetry breaking at higher temperatures. Naively, we would have thought that all we need do is cO.!ltinue to evaluate the vacuum expectation value of ( "'''') in the new thermal vacuum and if it vanishes we would conclude that chiral symmetry is restored at high temperatures. The problem comes in, however, when we begin to study the thennal fermion propagator and discover the existence of a Lorentz-invariant massive pole that persists at high temperatures even as m" the 'tree Lagrangian fermion mass, approaches zero. In the same m,-+O limit, however, (vacl¢"",lvac)/J vanishes for T> T, where T, =A,e 2 /3. The conflict is resolved when we recognize the need for a transform of '" at high temperatures to correctly describe the Lorentz-invariant massive particle. ~asinio

new ground state is given by ,

(1)

p.'

where 9p is related to the dynamical mass m acquired by the quarks tan29 =.!!!.

(2)

p

p

and p here is the magnitude of the momentum. We wiII show in this section how the NIL ground state may be understood as a rotation in chiral space and demonstrate how it affects the Dirac equation. The original massless Dirac field ",(x,O) is taken here to have the expansion at time 1=0:

",(x,O)=

)-V 1: [ [:L;Pp::P,L 1 P.!

R

p,R

11 e'P" . + [ SR(p)b~p,R t -SL(p)b_p,L I

.

== . ~ 1: "'(p)e'P" v V

,

(3)

(4)

p

where S, are helicity eigenstates satisfying

q·PS,(p)=SS.,(p) .

II. NJL VACUUM To appreciate the problem, we review the picture of the chiral-broken vacuum as first formulated by NIL. (We assume that we have integrated over the gluon degrees of freedom in QCD and are here discussing the effective theory involving only fermions.) In the presence of dynamical interactions, the naive vacuum lo} may no longer be the lowest-energy eigenstate of the Hamiltonian. The new ground state of the Hamiltonian Ivac} is the analog of the BCS ground state with quark and antiquark pairings. If ap" and bp " are the annihilation operators for the massless quarks and antiquarks, respectively, with helicities s =± for the R ,L states, then the

II (cos9p - $ sin9p a)"b ~p., )10)

lvac} =

(5)

Note that the Fourier component fields "'(p) are in general time dependent. For the purposes of our discussion here, we focus on the time slice at 1=0. The NIL chiral-broken ground state can be obtained by an infinite chain of SU(2)p chiral rotations around the (parity-conserving) two-axis by angle 29p : IvaC> = '=

II e2i8px2lPllo}

(6)

II RpU:Jp)IO}

(7)

=010) . 596

(8)

@1991 The American Physical Society

492 597

SIGNATURE FOR CHIRAL-SYMMETRY BREAKING AT HIGH ...

For simplicity, we shall refer to this rotation around the two-axis as the chiral-2 rotation. A. SU(l). allebra

(23)

The Xi(p) are the generators X 3 (p)=-i- l:s(o:',op.,+b:"p.,b_ p,,) ,

,

X 2(p)=f

l:s(o:"b~p"-b_p,,op,,), ,

(9) (0)

j,"4=;"0= [0I 0Ij ' [0 -ia 0

(i2)

And if we further take the sum over the momenta and form the global generators

Xi= l:Xi(P)'

(24)

y= ia

They satisfy the SU(2)p algebra at each momentum p: [X,(p},Xj(p'}j=iEilkXk(P)/iP,P' .

Following NIL, we may treat the massive modes as approximate eigenstates of the total Hamiltonian, and consider the time evolution of the massive modes as if they were free particles. With the explicit representation given in Eq. (23) it is straightforward to derive, using the representation

(13)

the free field equation for the chiral rotated spatially nonlocal equation

v' -V 2 +m 2 a [ 1i=Vi "'V+"Oat

q,.

1_\11=0.

It is the

(25)

p

we see that they form the global SU(2) algebra (14)

Note that we are here dealing with the infinitedimensional representation of the global sum algebra.

B. Transforming the Dirac equation

Under the chiral-2 rotation in the Hilbert space, the annihilation operators and the massless Dirac field transformas Qp,s-.A p ,. ,

(is)

bp,s-...".Bp,s ,

(6)

This nonlocal Dirac equation is strange because it appears to show chiral invariance, and yet we know by construction that the chiral-rotated q, describes the massive particle free field. It is also not Lorentz invariant. But in our context of temperature field theory this latter objection plays no role. It is therefore reassuring to find that there is a similarity transformation acting on the components of ifJ that transforms away the nonlocality in the Dirac equation. _ . If we work with the Fourer components of \II In momentum space, and define the similarity transformation ifJ(p,t):ae -i8p Y'il \ll(p,tl,

(26)

then the nonlocal equation for ifJ implies the usual massive Dirac equation for \II:

[Y'V+"O;t +m j\ll(X,tl=o.

(27)

where A p" =:Rp«()p }Op.,:R;I«()p) =cos()pop" +s sin()pb :"p" , Bp,s =:Rp«()p )bp,,:R; I( ()p)

= cos()p bp,s -s sin6po :"p,s ,

(17)

(18)

From Eq. (26), we can show in our representation of Dirac matrices that \II( x, t) has the usual expansion (po=v'p2+m 2)

(9)

(20)

and

+ V _p,sB t_p,se + iPO') e ip·", ,

Ap"lvaC>=Bk,slvac> =0.

The new operators Ap,soBp" describe the excitons with dynamically generated mass m that propagate in the new medium. Under the Hilbert-space chiral-2 rotation, the massless Dirac field of Eq. (3) transforms in an obvious way into .p(x,O}->ifJ(x,O) , where

(28)

(21) where the massive spinors are given explicitly by

(29)

(22)

(30)

493 LAY -NAM CHANG, NGEE-PONG CHANG, AND KUANG-CHAO CHOU

598

sin8pSL(p) 1 [ -cos8pSL(p) ,

V_ p• L

=

V

= [COS8pSR(P) -p.R

1

-sin8ps R (p).

III. HIGH-TEMPERATURE RESULT (31)

(32)

and satisfy the Dirac equations {iy·p-iYo·po+m )Up., =0 ,

(33)

(iY'p-iYo'po-m lVp., =0.

(34)

Equation (26) is in fact an example of a transformation of the generic Foldy-Wouthuysen type (more precisely, a Cini-Touschek transformation).2 The result of this section may therefore be summarized in the transformation law of the massless Dirac field under the Hilbert-space chiral-2 rotation

n~(x,O)n-l =e -/ly.v/"cv;'I'(x,O) ,

(35)

where 8 is here the differential operator in threedimensional space implied by the momentum-space equation (2). The Cini-Touschek similarity transformation is the analog of the :Dati- A) similarity transform of ~ that results from a Hilbert-space Lorentz transformation. Loosely speaking, a Hilbert-space chiral-2 rotation induces on the ~ a Cini-Touschek similarity transform. The nonlocality of the similarity transform reflects the infinite-dimensional nature of the SU(2) representation involved here. The nonlocality of the ijI field equation is also a warning that it is the wrong basis on which to discuss the chirality of the theory. Indeed, Eq. (25) gives the false indication of chiral conservation. It is only after the nonlocal Dirac equation has been straightened out that one can test for chiral-symmetry breaking under the chiral X J rotations. As we shall see, at high temperatures, the thermal radiative corrections lead to a nonlocal Dirac equation for the massive particle pole in the Green's function. Proper physical interpretation of the signature of chiral breaking requires that we do a Cini-Touschek transformation to get rid of the nonlocality in the Dirac equation of the renormalized particle pole in the Green's function. After this similarity transformation, the inherent chiralsymmetry breaking due to temperature effects becomes evident. Before we close this section, we note the equality ~(x,O)='I'(x,O)

.

(36)

This may be verified by direct substitution of the inverse transformation to Eqs. (18) and (20) into the expansion for ~ as given in Eq. (3). At zero temperature, then, the correct signal for chiral-symmetry breaking is to use either ~ or 'I' and calculate the expectation values of ifl/l or iii'l' with respect to the full vacuum. The equality between the two Heisenberg operators at t =0, Eq. (36), guarantees that the vacuum expectations values obtained by the two different I/I's agree: (vaclifl/llvac) = (vacliii\jllvac) .

(37)

In an earlier work, J we had reported the result of a real-time temperature-dependent field-theory calculation of dynamical chiral-symmetry breaking at high temperatures. We found that, for QCD, dynamical symmetry breaking persists at high temperatures. In this section, we present an analysis of the result and point out the close connection between the zero-temperature chiral rotation of the NJL vacuum and the Cini-Touschek transformation needed in the renormalization of the temperature-dependent Fermion two-point function. We perform our calculation in real time.' Our technique is to introduce into the Lagrangian an explicit mass term for the fermion, put the system in a heat !?ath, use renormalization-group analysis to sum over higher loops, and study the critical limit as m, __ 0. If in this limit, the thermal fermion propagator shows a Lorentz-invariant massive particle pole, then we say that dynamical symmetry breaking has occurred. At zero temperature,S we found that the chiral flip part of S,-l(p), for P,J',. in some finite domain, actually survives the critical m,--+O limit, thus signaling the bifurcation in chiral-symmetry breaking. In this section, we look for the temperature dependence of this chiral-symmetry breaking. The result of the real-time thermal field-theory calculation may be put in the form (our results for A ,B agree with the one-loop calculation of Weldon,6 except that we have also included InT /m terms; Weldon does not introduce an explicit mass term, and thus did not look for a perturbative root around the original m, pole)

Si 1 (p2,P5T)=iY'p(1 + A )-iYoPo(1 +B )+m,(1+C) , (38)

where A, B , C are functions of P, Po, and T. In terms of the parameters Ipl::m,sinhs,po::m,cosh s , we have, using the Feynman gauge, and in the limit of T2 /m;» I,

(39)

(40)

(41)

Here we have dropped terms that are of order I as T 2/m; __ oo. Also we have defined >",=g;/(16"/1"2) and the relation T Q • T Q : : C/1. The Lorentz-invariant massive particle pole in the thermal fermion Green's function occurs at Pn=Vp2+.M\ where perturbatively

494 SIGNATURE FOR CHIRAL-SYMMETRY BREAKING AT HIGH ...

m;

. 4] +--41T2T2 .M. 2 = hm m,2 { I+A,Ct [ -6[ I n - - -

'",-0

m;

3

,..,.2

1

T2 -6In--+const + ...

m;

}.

renormalization-group analysis to sum over higher loops . The existence of this particle mass at high temperature is already a good signal that chiral-symmetry breaking persists at high temperatures. What we want to study is whether the traditional signature of chiral-symmetry breaking is still good at high temperatures, viz.,

(42)

The critical limit m,_O is taken using the fixed-point theorem of bifurcation theory. [See the discussion following Eq. (68) in Ref. 3.] As was shown in Ref. 3, this mass survives the critical limit as m, -->0, so that it is a temperature-dependent dynamical mass: ,M.2 _ T-• ..,

2~ ~ 3 In T2 .

(43)

A~ Donoghue and Holstein' considered the case m,~O and also found in the one-loop perturbative calculation the Lorentz-invariant massive pole at high temperatures. Our technique goes beyond one loop by using the

1

Z2 Sii lchiralftip=

2:~om, {I +A,Ct [-3 [In :: -

= lim m,(A,y) -6Cf

599

lim (vacljf!JIlvac)p*0.

(44)

'",-0

A potential conflict with the usual notion of chiralsymmetry breaking arises when we study the chiral flip part of S i 1 [Eq. (38)] and directly take the limit as m,-->O. This would be consistent with the idea that we simply evaluate the (vacljf!JIlvac)/1 in the thermal vacuum without doing any more renormalization than the minimal ones needed at T=O. In our case, our result in Eq. (38) is in Feynman gauge, which explains why even at T=O the coefficient of 'Yol'o is not unity. So we renormalize S 1 by setting the coefficient of 'Yol'o equal to unity when T=O. We do this by multiplying it by the T=O wave-function renormalization Z2:

ii

411 + ... }

(45) (46)

lb ,

m,-O

where

Z2=I-A,Ct !ln : ; -2]+'"

y=.J...+E. A, 2

=E.2

[In T2 A~

[In T2

,.,.2

-~3

-~3

I

(47)

I

(48)

(49)

.

Here y is a renormalization-group invariant. Each term in the perturbative series is valid so long as T2» Since in the end m,_O, it would appear that the series should be valid for all T. However, the positivity requirement for y due to the representation, Eq. (46), shows that we need to impose the condition T> Ace 213 for the validity of the sum. When we now take the critical limit m,_O, y to oneloop renormalization-group accuracy does not depend on m, and does not approach the y = fixed point [see the discussion following Eq. (68) in Ref. 3], and so the chiral flip part does not survive the critical limit. It vanishes for T> Ace2l3. The only exception is when the temperature T is at the critical temperature8 •9

m;.

°

(50)

At

that

point, our one-loop renormalization-group

analysis fails since it diverges even before the critical limit. We need to go to two-loop renormalization-group analysis to study further the order of the phase transition at Te' Based only on the calculation thus far, we would conclude that chiral-symmetry breaking goes away for high temperatures, viz., for T> Te. And yet the same calculation shows a Lorentz-invariant massive particle pole in the thermal fermion Green's function, which is a signal that chiral-symmetry breaking has occurred. To reconcile between the presence of a massive particle and the vanishing of the traditional signature of chiral-symmetry breaking, we proceed with the analysis of the thermal fermion Green's function. The coefficients A ,B, C in Eq. (38) are nonlinear functions of the momenta. At the particle pole Po = v'P 2+,M. 2 the residue of the fermion thermal Green's function is not like the usual

(-i'Y·p+i'YoI'o+.M.) ,

(51)

but instead has the form [-i'Y'p(1 +.A)+i'YnPn(I+'7l)+m,(l +6')] ,

(52)

where .A,.:B, 6' are non polynomial functions of p2= Ipl2, obtai.,d by evaluating A ,B, C on the particle mass shell, Po= p2+.JIt 2. The Dirac equation for the massive particle is then the peculiar one:

495 LAY-NAM CHANG, NGEE-PONG CHANG, AND KUANG-CHAO CHOU

600

[iy·p( I +.A )-iyo"Vp2+.M. 2(1 + 13Hm,( I +@))u=O . (53)

Ifwe naively form the renormalized field operator by

ifi =_I_l:,(u A R

Vv

p.s

p.s

eip.x-nlp2+JI!2r p•.'

+vP•. Btp •. e -iP'x+nlp2+JI!2r) f

f

,

jifiR =0.

We now study the chiral flip part of the renormalized inverse thermal Green's function, S iii, and take the critical limit as mr -->0. If the chiral flip part survives this limit, we then can properly claim that chiral symmetry persists at high temperatures. Perturbatively, the chiral flip part of S fiRI is given by

(54)

we would find that this renormalzed field does not satisfy the usual Dirac equation, but instead obeys the nonlocal equation [(1+.A)y'V+(I+13)YO;t +m,(I+@)

(65)

-I _ { S(3R Ichiralftip-mr I+ArCf

[[

m; 4]

-3 In7-"3

+

2rrT m,2

2 -

3In

Em;

(55)

1

+const + ... }, (66) Based on our experience in the preceding section, it is clear that it would be dangerous to conclude anything about chiral-symmetry breaking based on this ifi R • Thermal radiative corrections have induced some chiral-2 rotation in the vacuum structure and we must find the generalized Cini-Touschek transformation that can remove the associated non locality. If we introduce (56) then the Dirac equation for U is "straightened out" to the usual massive one (iy·p-iY oV p '+.M. 2 +.M.lU=0, p[m,( I + Cl')-.M.(I +.A))

(58)

p'( I +.A )+m,.M.( 1 +@)

The renormalization of the thermal field-theory propagator should thus involve an extra Cini-Touschek transformation on top of the wave-function renormalization factor (59)

where 9 is the differential operator in three-dimensional space as implied by Eq. (58). Accordingly, we have

-z 2(3e iepY'~S(3Re i9pY'~ S (3-

(60)

and as a result Sii=iY'p(l+A')-iYoPo(I+B'Hm,(1+C') ,

I + A '=(1 +':8)-1

[(I + A )cos29 +

I+B'=(I+13)-I(I+B),

p

(67)

(57)

provided

where

so that to first order in renormalization-group analysis, the chiral ftip part of the renormalized inverse Green's function is simply the Lorentz-invariant physical mass of the particle pole,.M.. As shown in our earlier work, this mass survives the critical limit mr-->O. Chiral-symmetry breaking persists at high temperatures. Our conclusion therefore is that at high temperatures the traditional signature of chiral-symmetry breaking reappears only when one uses the transformed '" R field, and we calculate its vacuum expectation value

:r I sin29p

(61)

(62) (63)

I +C'=(1 +13)-I[m r (l +C)cos29p -p( I + A )sin29pl , (64)

so that, at the massive particle pole, the coefficient of 1'0 is properly normalized. Here

Unfortunately, our results at this stage cannot be used to evaluate this thermal vacuum expectation value. We need a study of the thermal propagator in the full complex Po plane. IV. CONCLUSION

In conclusion, we note once again that the traditional signature for chiral-symmetry breaking (vac I~"'I vac ) (3 is an inadequate indicator of chiral-symmetry breaking at high temperatures. Our calculations with dynamical symmetry breaking in QCD at high temperatures show a persistent Lorentz-invariant massive particle pole in the thermal fermion propagator at high temperatures-and this in spite of a vanishing (vacl~"'lvac)(3 at high temperatures. Thermal radiative corrections induce a further chiral-2 rotation of the vacuum structure, and the resulting renormalized Dirac field undergoes a generalized CiniTouschek similarity transformation. In the transformed basis, the traditional signature for chiral-symmetry breaking reappears. ACKNOWLEDGMENTS This work was written up while one of us (N.P.C.) was visiting KEK, Japan, and he wishes to thank Dr. K. Higashijima for some stimulating conversations and the Theory Group for the warm hospitality. This work was supported in part by the NSF U.S.-China Cooperative Program, by grants from the NSF, the Department of Energy, and from PSC-BHE of the City University of New York.

496 SIGNATURE FOR CHIRAL-SVMMETRV BREAKING AT HIGH ... IV. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961). 2M. Cini and B. Touschek, Nuovo Cimento 7, 422 (\958). The Cini~ Touschek transformation is a special case of a genera] class of Foldy-Wouthuysen transformations; see L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). 3L. N. Chang, N. P. Chang, and K. C. Chou, in Third AsiaPacific Physics Conference, proceedings, Hong Kong, 1988, edited by Y. W. Chan, A. F. Leung, C. N. Yang, and K. Voung (World Scientific, Singapore, 1988). 4For a comprehensive review of the subject, the formalism, as well as citation of earlier work, see Zhou Guang-zhao (K. C. Chou), Su Zhao-bin, Hao Bai-lin, and Vu Lu, Phys. Rep. 118, 1 (1985). For other equivalent approaches, see H. Umezawa, H. Matsumoto, and M. Tachiki, Thermofield Dynamics and Condensed States (North-Holland, Amsterdam, 1982); R. L. Kobes, G. W. Semenoff, and N. Weiss, Z. Phys. C 29, 371 (1985); A. Niemi and G. W. Semenoff, Nucl. Phys. B230, 181 (19841; Ann. Phys. (N.Y.) 152, !O5 (1984).

601

SL. N. Chang and N. P. Chang, Phys. Rev. LeU. 54, 2407 (1985); Phys. Rev. D 29, 312 (1984); see also N. P. Chang and D. X. Li, ibid. 30,790 (1984). 6H. A. Weldon, Phys. Rev. D 26, 2789 (19821. 7J. F. Donoghue and B. R. Holstein, Phys. Rev. D 28, 340 (1983); 29, 3004(El (\984); J. F. Donoghue, B. R. Holstein, and R. W. Robinett, Ann. Phys. (N.V.! 164,233 (19851. See also R. Pisarski, Nucl. Phys. A498, 423C (\ 989). G. Barton, Ann. Phys. (N.V.) 200, 271 (1990), has a very nice discussion of the physical origin of the Lorentz invariance of the massive particle pole. 81t is interesting to note that lattice gauge simulations have found a chiral transition temperature of T, / A.,. of around 2 (for a nucleon mass of 940 MeV) (see Ref. 9), where Ms denotes the modified minimal subtraction scheme. 9For a review, see A. Ukawa, in Lattice '89, proceedings of the International Symposium, Capri, Italy, 1989, edited by R. Petronzio el al. [Nucl. Phys. B (Proc. Suppl.J (in pressl].

497 PHYSICAL REVIEW D

I APRIL 1996

VOLUME 53, NUMBER 7

CP violation, fermion masses and mixings in a predictive 8U8Y 80(10) x .:1(48) x U(I) model with small tanfJ K. C. Chou Chinese Academy of Sciences. Beijing 100864. China

y. L. Wu Department of Physics. Ohio ~iate Unit'"rsity. Colllml>lIs. Ohio 43210 (Received 8 November 1995) Fermion masses and mixing angles are studied in an SUSY SOt 10) x a( 48) X U( I) model with small tan,8. Thirteen parameters involving masses and mixing angles in the quark and charged lepton sector are successfully predicted by a single Yukawa coupling and three ratios of VEV's caused by necessary symmetry breaking. Ten relations among the low energy parameters have been found with four of them free from renormalization modifications. They could be tested directly by low energy experiments. PACS number(s): 12.15.Ff, 11.30.Er, 12.10.Dm, 12.60.Jv

The standard model (SM) is a great success. Eighteen phenomenological paramcters in the SM, which are introduced to describe all the low energy data, have been extracted from various experiments although they are not yet equally well known. Some of them have an accuracy of better than 1%, but some others less than 10%. To improve the accuracy for these parameters and understand them is a hig challenge for particle physics. The mass spectrum and the mixing angles observed remind us that we are in a stage similar to that of atomic spectroscopy before Balmer. Much effort has been made along this direction. The well-known examples are the Fritzsch ansatz [I] and Georgi-Jarlskog texture [2]. A general analysis and review of the previous studies on the texture structure was given by Raby in [3]. Recently, Babu and Barr [4], and Mohapatra [5], and Shafi [6], Hall and Raby [7], Berezhiani [8], Kaplan and Schmaltz [9J, Kusenko and Shrock [I OJ constructed some interesting models with texture zeros based on supersymmetric (SUSY) SO(lO). Anderson, Dimopoulos, Hall, Raby, and Starkman [II J presented a general operator analysis for the quark and charged lepton Yukawa coupling matrices with two zero textures "1\" and "13:' The 13 observablcs in the quark and charged lepton sector were found to be successfully fitted by only six parameters with large tan.8. Along this direction we have shown [12] that the same 13 parameters can be successfully described, in an SUSY SOt 10) X ~(48) X U(I) model with large values oftanf3-m,/m", by only five parameters with three of them determined by the symmetry-breaking scales of U( I), sot I 0), SU(5), and SU(2) L' Ten parameters in the neutrino sector could also be predicted, though not unique, with one additional parameter. In this Rapid Communication wc shall present. based on thc symmctry group SUSY SOt 10) X ~(48) X U( I), an alternative model with small values of tanf3-1 which is of phcnomenological interest in testing the Higgs scctor in the minimum supersymmetric standard model (MSSM) at colliders [13]. The dihedral group ~(48), a subgroup ofSU(3), is taken as the family group. U(I) is family-independent and is introduced to distinguish \'urious fields which belonl! to the same reprcsentations of SOt 10) X M 48). The irr;ducible

representations of ~(48) consisting of five triplets and three singlets are found to be sutlicient to build an interesting texture structure for ferm ion mass matrices. The symmetry ~(48)X U(I) naturally ensures the texture structure with zeros for Yukawa coupling matrices, while the coupling coetlieients of the resulting interaction terms in the superpotential are unconstrained by this symmetry. To reduce the possible free parameters, the universality of coupling constants in the superpotential is assumed; Le., all the coupling coetlicients are assumed to be equal and have the same origins from perhaps a more fundamental theory. We know in general that universality of charges occurs only in gauge interactions due to charge conservation, like the electric charge of different partieles. In the absence of strong interactions, family symmetry could keep the universality of weak interactions in a good approximation after breaking. In our case there are so many heavy fermions above the grand unified theory (GUT) scale and their interactions are taken to be universal in the GUT scale where family symmetries have been broken. It can only be an ansatz at the present moment where we do not know the answer governing the behavior of nature above the GUT scale. As the numerical predictions on the low energy parameters so found are very encouraging and interesting, we believe that there must be a deeper rcason that has to be found in the future. Choosing the structure of the physical vacuum carefully, the Yukawa coupling matrices which determine the masses and mixings of all quarks and Icptons are given by

0556-2821/96/53(7)/3492(4)1$10.00

R3492

o

o (I)

o and '\.'.' 19% The American Physical Society

498 R3493

CP VIOLATION. FERMION MASSES AND MIXINGS IN A ...

- tZiE~

w,,= ~A ){E2 16,(' ~)' "(Au)' ('~) 10 I (~') (A.,) -- (3 N - G - A. u. A. A· v-

3Jjf:~e;d>

..\

".\ I

.\ .

,

forj=d.e, and

_ )20 X

10 , (VIO)(A:)' - ('V - IO)"+I] 16,. (VIO) Ax Ax Us As -

X -

~2

,

7 :! _pcp

(

o

2

- '2 X ,.EC

o

)

(3)

W,.

for Dirac-type neutrino coupling, where the integer II reflects the possible choice of heavy fermion fields above the GUT scale. n = 4 is found to be the best choice in this set of models for a consistent prediction on top and charm quark masses. This is because, for n >4, the resulting value of tanj3 becomes too small, as a consequence, the predicted top quark mass wi II be below the present experimental lower limit. For n<4, the values oftanj3 will become larger, the resulting charm quark mass will be above the present upper bound. AH is a universal coupling constant expected 12, be of order one. EG""uslulO and Ep""u;IMp with M p, U 10, and U; being the vacuum expectation values (VEV's) for U(I)XA(48), S0(10), and SU(5) symmetry breaking, respectively. cfJ is the physical C P phase arising from the VEV's. The assumption of maximum C P violation implies that cfJ=7r/2. xI' YI' zl' and wI (j=u,d,e,v) are the Clebsch factors of SO(IO) determined by the directions of symmetry breaking of the adjoints 45's. The three directions of symmetry breaking have been chosen as (A x)=u lO diag(2,2,2,2,2)®T" (A-)=usdiag(-t-t -i,-2,-2)®T2' (A u )=u s diagO,H,t,n-®T2' The CI~bs~h factors associated with the symmetry-breaking directions can be easily read off from the U(I) hypercharges of the adjoints 45's and the related efTective operators which are obtained when the symmetry SO(IO)XA(48)XU(I) is broken and heavy fermion pairs are integrated out and dccoupled:

The factor IIJI +2(u IO IA x )2 1"+11 arising from the mixing is equal to 1I.j3 for the up-type quark and almost unity for other fermions due to suppression of large Clebsch factors in the second term of the square root. The relative phase (or sign) between the two tcrms in the operator W 12 has been fixed. The resulting Clebsch factors are w"=w,,=w.

=wv=l, x.,="519, x,,=7/27, x.=-1/3, x,.=1/5, y"="O, y,,=y./3=2/27, y.=4/45, z.,=I, z,,=z.=-27, z.=-15 3 =-337S, z~=I-S/9=4/9, z:l=z"+7/729=z,,. z;=z. -1/81=z., z;.=z,.+ I/IS 3 =z, .. An adjoint 45 A x an~ a 16-dimensional (16D) representation Higgs field (
XIOI(~)"+I ---;===== .j3 16,. (U ) 2,,,+ II

Ax

1+2 ~

.

(4)

( III)" + I(A _) )2"' + II ,Ax . ~

.,!3 1+2 (

Ax

X(~)IOI(~)(A :)('~)' ,,+1 162 , A.I

. AI

Vj

AI"

(F=" U.D,E,N)

with

=[I+(A~i)2K,r'12. where K,=3/(M s )/47r 2 with I(Ms)

=

Ax

.V

rrt_l( £I';(MG)/£I';(Ms»,"r/2h;

c!'=(H,3,l;"), c?=({s,3,l;"), cf=(f;.3,0). c>(B-,3,0), b;=('f.I,-3). and R;I=exp[-f:~::;~(A,(t)/47r)2df]

V

I'j.

f::::;;~-i(f)dl. The numerical value for I taken from Ref

[15] is 113.8 for MS=III,= 170 GcY. Other RG scaling factors arc derived by running Yukawa couplings below Ms· 111;(111;)= '1;III;(Ms) for (i=c.b) and 111;(1 GeVl = '1;111;( M.d for (i= I/,d,s). The physical top quark mass is given by M,=III,(III,)[ I + ~".,(III,)I1Tl. Using the wellmeasured charged lepton masses III c= O.SII MeV, III~= 10S.66 MeV. and 1117= 1.777 GeV we obtain four impOitant RG scaling-independent predictions:

499 R3494

K. C. CHOU AND Y. L. WU TABLE I. Output paramcters and their predicted valucs with a..{M z )=O.113 and inplll parametcrs: 111<=0.511 cV, III~= 105.66 MeV, 111,= 1.777 GcV, and 111,,=4.25 Gev.

Output paramctcrs

Output valucs

Data [14]

Output parameters

Output values

182 1.27 4.31 156.5 6.26 0.22 0.083

180± 15 1.27±().()5

Jcp=A 'I>." '1

4.75± 1.65 165±(,5

f3

8.5± 3.0 0.221 ± 0.003 0.08±0.03

Ep=u,IMp

2.68X 10-' 86.28° 21.11 ° 71.61° 2.33 2.987X 10-' LOll x 10-'

0.209

0.24±0.11

,

1.30

0.0393 0.90 207 (1.4± 1.0) X 10- 3

0.039±0.005 [19] 0.82±0.1O [17,16] 200±70 [18,16] (1.5±0.8)X 10-'80

Al, [GcV] 111,(111,) [GcV] III ,,(I GcV) [MeV] 111,(1 GcV) [MeV] 111,,(1 GeV) [MeV]

I v",1 = I>.

~ '1-

IV""I I vchl

--=I>.~p-+

IV"/I J I-p)-+ " '1jVJ=l>.( IV,hl=AI>.'

BK fBJB[MeV] Re(,,'le)

2)

16 m )

\

( 1+675111~ -Vr;;;; -;;;=-

V\=\V\=3 us G

lIS

mp. (

t.r

Y tanf3= u, lv,

EG= Us 1l!._lo

I>.G

1/2

_

- 0.22,

me 1+9mp'

(5) Veh Veh IVUhllVUhl -

= -

G

= (4)2mT~e -15 -mp' = 083 mp' . ,

°

I-V'VIS' I = I-V'VIS' I

G

=3 V;;;-=0.209, 1m:

m,.,.

/II e

)-2

1-m,.,.

=0044 .,

(8)

_ 1513-7 5 15v3 4v3

/lip.

(0.80)

III,

,

\Veh\-\Veh\GR,-~~-R,=0.0391 -R-' , (9)

MeV,

(10)

=4. 25

mIle I GcV)=

C~~)( ;~~:) (~.~~)

5 (' 4 )'

3'. 45

m: 111,

=4.23(:!!!')··

2.2

v2

sinf3) ( 1'/u) (

= 174.9 (0.92

3.33

f8.65) ( ~) -VT, 0.965

GeV,

GcV,

(I I)

_

1'/"R;'m,

(o.SO)' ( R;' 174

1Il,(m,) )

GeV

where the miraculus numbers in the above relations are due to the Clebsch factors. The scaling factor R, or coupling A.~=( II JI- R; 121R;6 is determined by the mass ratio of the bottom quark and T lepton. tanf3 is fixed by the T lepton mass via eosf3=mTfil TJ£1'/TVA.~. The above 10 relations are our main results which contain only low energy observables. As an analogy to the Balmer series formula, these relations may be considered as empirical at the present moment. They have been tested by the existing experimental data to a good approximation and can be tested further directly by more precise experiments in the future. In numerical predictions we take cr-I(M z )= 127.9, .l'2(M.)=0.2319, M l =91.187 GeV, £1','(111,)=58.59, cr;-'(I11,)=30.02, and cr,I(Mc;)=cr;-I(Mc;)=crJ"I(M(;) =24 with M G-2X 10'6 Gev. For a.(M l )=0.113, the RG scaling factors have values (TJ".".s' 711"' TJh, 1'/".P..T' 1'/('. 7JIJ 17J£= 1'/[J!E' TJE' TJs) = (2.20, 2.00. 1.49, 1.02, 3.33, 2.06, 1.58, 1.41). The corresponding predictions on femlion masses and mixings thus obtained are found to be remarkable. Our numerical predictions for ".,( M l) = 0.1 13 are given in Table I with four input parameters: three charged lepton masses and .!?ottom quark mass II1h(II1,,)=4.25 GeV, where Ih and fa \'8 in Table I are two imp0l1ant hadronic

JK.)

and six RG scaling-dependent predictions:

_

vK,

(14)

/lie (

Ins

(7)

_ 1'/u ~I R- 12U . a ~VI-J{, .- r;; Sill,...

/II, ( /II, ) -

1IIp.

m,,( 1 -m,,)-2 =9ms

(6)

MeV.

( 12)

500 CP VIOLATION, FERMION MASSES AND MIXINGS IN A ...

8°_8°

R3495

parameters and extractcd from KO-K() and mixing parameters Eli and x". Re(e'/E) is the direct CP-violating parameter in kaon decays. where large unccrtantics mainly arise from the hadronic matrix elements. a. (3. and yare three angles of the unitarily triangle in the CabibboKobayashi-Maskawa (CKM) matrix. J cp is the rephaseinvariant CP-violating quantity. It is amazing that nature has allowed us to make predictions on fermion masses and mixings in terms of a single Yukawa coupling constant and three ratios of the VEV's determined by the structure of the physical vacuum and understand the low energy physics from the GUT scale physics. It has also suggested that nature favors maximal spontancous C P violation. A detailed analysis including the neutrino sector will be presented in a longer paper [20]. In comparison with the models with large tan{3-m,lm", the present model

has provided a consistent picture on the 13 parameters in the SM with better accuracy. Besides, ten relations involving fermion masses and CKM matrix elements are obtained with four of them independent of the RG scaling effects. The two types of the models corresponding to the large and low tan{3 might be distinguished by testing the MSSM Higgs sector at colliders as well as by precisely measuring the ratio IV"f> I vc,,1 since this ratio docs not receive radiative corrections in both models. It is cxpectcd that more precise measurements from C P violation and various low energy experiments in the ncar future could provide crucial tests on the ten realtions obtained in the present model.

[I] H. Fritzsch, Phys. Lett. 708,436 (1977). [2] H. Georgi and C. Jarlskog, Phys. Lett. 868. 297 (1979). [3] S. Raby, Ohio State University Report No. OHSTPY-HEP-T95-024, 1995 (unpublished). [4] K.S. Babu and S.B. Barr, Phys. Rev. Lett. 75, 2088 (1995). [5] K.S. Babu and R.N. Mohaptra. Phys. Rev. Lett. 74, 2418 (1995). [6] K.S. Babu and Q. ShaH, Phys. Lett. B 357, 365 (1995). [7] L.J. Hall and S. Raby, Phys. Rev. D51, 6524 (1995). [8] Z.G. Berezhiani, Phys. Lett. B 355, 178 (1995). [9] D. Kaplan and M. Schmaltz, Phys. Rev. D 49, 3741 (1994); M. Schmaltz, ibid. 52, 1643 (1995). [10] A. Kusenko and R. Shrock, Phys. Rev. D 49, 4962 (1994). [II] G. Anderson et al., Phys. Rev. D 49, 3660 (1994). [12] K.C. Chou and Y.L. Wu, Science in China (Sci. Sin.) 39A, 65 (1996). [13] See, for example, 1. Ellis. presented at the 17th International Symposium on Lepton-Photon Interactions, Beijing, China, 1995 (unpublished). [14] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 74, 2626 (1995); DO COllaboration. S. Abachi et al., ibid. 74, 2632 (1995); J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982); H. Leutwyler, Nucl. Phys. B337. 108 (1990); Particle Data

Group, L. Montanet et al.. Phys. Rev. D 50, 1173 (1994). [15] V. Barger, M.S. Berger, T. Han, and M. Zralek, Phys. Rev. Lett. 68, 3394 (1992). [16] J. Shigemitsu, in Proceedings of the XXVI/International Conference on High Energy Physics, Glasgow, Scotland, 1994, edited by P.J. Bussey and I.G. Knowles (lOP, Bristol, 1995). [17] c. Bernard and A. Soni, in Lattice '94, Proceedings of the International Symposium, Bielefeld, Germany, 1994 edited by F. Karsch et al. [Nucl. Phys. B (Proc. Suppl.) 42 (1995)]; S.R. Sharpe, in Lattice '93, Proceedings of the International Conference, Dallas, Texas, edited by T. Draper et al. [ibid. 34, 403 (1994)]. [18] C. Allton, Report No. hep-lat/9509084, 1995 (unpublished), and references therein. [19] R. Patterson, in Proceedings of the XXVI/International Conference on f1igh Energy Physics [16], Vol. I, p. 149; M. Neubert, Report No. CERN-TH/95-107, hep-ph/9505238, 1995 (unpublished); A. Ali and D. London, Report No. DESY 95148. UdeM-GPP-TH-95-32, hep-ph/9508272, 1995 (unpublished). [20] K.C. Chou and YL. Wu. Chinese Academy of Sciences and Ohio State University Report No. CAS-HEP-T-96-03/004, OH STPY-HEP-T-96-008, 1996.

YL.W. would like to thank Professor S. Raby and Professor G. Steigman for useful discussions. YL.W. was supported in part by U.S. Department of Energy Grant No. DOE/ERl01545-662.

501 Vol. 39 No.1

SCIENCE IN CHINA (Series A)

January 1996

Low energy phenomena in a model with symmetry group SUSY SO (10) X L1(48) XU(l) ZHOU Guangzhao (K.C. CHOU

}iJJ't~)

(Chinese Academy of Scienres, Beijing 100864, China)

and WU Yueliang (~ffi ~ ) (Department of PhysiCi, Ohio Stilte University, Colwnb~, Ohio 43210, USA) Recehed September 4, 1995 Abstract

Fermion masses and milling angles including that of 1X:utriJlt'!: ~.re studied i;"1 a n-.xic! witb

symmetry group SUSY SO(IO) x .1 (48) x V (I). Universality of Yukawa Q.ll1pling of ~u"erfieJds is alisumcd. The resulting texture of mass matrices in the low energy regio:! ckp'~nd£ only VEVs ca~ed by neressary symmetry breaking.

n

(1)

;).

sinlle lX)upling ~tant and

parameters im~lvi~ :na!'!C~ and mixing angles in the quark

and charged lepton sector are suocessMly d.'!Saibed by mllY Ihe parameters with two of them determined by the scales of V(l), S0(10) lIl":J SIJ(5) &yrrr::r."lt;~1 breiking IX)mpatible with the requirement of grand unifJaltion and proton decay. Tre nt:u1ri,:o masses and mixing angles in the Ieptonic sector are also determined with the addition (,f a Mlij..'lfl:11ll IX)L'.Pi"lng tc:rm. It is found that LSND "Y. -. V. events, atmospheric neutrino defJc::it and the mass lill.'it put by bot dark matter can be naturalIy explained. Solar neutrino puzzle can be solved only by introducing sterile neutrino witb one additional parameter. More precise measurements of a,(Mz ), V.. ,V.. fV.. ,

as well lIS various CP violation and neutrino oscillation experiments will provide aucial tests of the present model.

m6' m"

Keywords: symmetry group, low energy.

The standard model (SM) is a great success. To understand the origin of the 18 free parameters (or 25 if neutrinos are massive) is a big challenge to high energy physics. Many efforts have been made in this direction. It was first observed by Gatto et aI., Cabbibo and Maiani[1J that the Cabbibo angle is close to J md/ m•. This observation initiated the investigation of the texture structure with zero elements l21 in the fermion Yukawa coupling matrices. A general analysis and review of the previous studies on the texture structure was given by Rabyl). In refs. [3,4] Anderson et al. presented an interesting model based on SUSY SO (10) and U (1) family symmetries with two zero textures 'll' and '13' followed naturally. Though the texture '22' and '32' are not unique they could fit successfully the 13 observables in the quark and charged lepton sector with only six parameters[3). In this paper we will follow their general considerations and make the following modifications : • Project supported in part by Department of Energy Grant # DOE/ER/OJ54~S5. J) For a recent review, see Raby, S., Ohio State Univ., Preprint, OHSTPY-HEP-T-95-024.

502 66

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SCIENCE IN CIDNA (Series A)

(i) We will use a discrete dihedral group Ll (3n 2 ) with n=4, a subgroup of SU (3), as our family group instead of U(l) used in ref.[5]. This kind of dihedral group was first used by Kaplan and Schmaltzl61 with n = 5. This group, having only triplet and singlet irreducible representations, suits well our purposes. (ii) We will assume universality of Yukawd coupling before symmetry breaking so as to reduce the possible free parameters. In this kind of theories there are very rich structures above -the GUT scale with many heavy fermions and scalars. All heavy fields must have some reasons to exist and interact with each other which we do not understand at this moment. So we will just take the universality of coupling constants at the GUT scale as a working assumption and not worry about the possible radiative effects. If the phenomenology is all right, one has to be more serious to find a deeper rea:mn for it. (iii) We will choose some symmetry breaking din~(;ti(m~ different from ihose in refs. [3,4] to ensure the needed Oebsch coefficients in l)rdt,r to eliminate further arbitrariness of the parameters. This paper is organized as fe liO\'\s: In sec. ], we will present the results of the Yukawa coupling matrices. lbe resulting masses and CKM quark mixings are also presented. In sec. 2, neutrino masses and CKM-type mixings in the lepton sector are presented. All existing neutrmo experiments are discussed and found to be understandable in the present model. In sec. 3, the model with superfields and superpotential is explicitly presented. Conclusions and remarks are presented in the last section. 1 Yukawa coupling matrices With the above considerations, a model based on group SUSY so (10) x Ll (48) x U (1) with a single coupling constant is constructed. Here U(l) is family-independent, introduced to distinguish various fields which belong to the same representations of SO (10) x Ll (48). Yukawa coupling matrices which determine the masses and mixings of all quarks and charged leptons are obtained by carf"!\I1ly choosing the structure of the physical vacuum. We find

0 G

3

/I

'2 zfe j. 0

for j=u,d,e, and

0

3YJE~ei
.J3 x e 2 J

p

,

3

2"

rJ=-A

2"3 z(e 2

~, 2

xJe,(;

wJ

2

G

(I)

503 ND. 1

LOW ENERGY PHENOMENA

67

3 1 2 Iz.1

o

o

3 1 2 Iz.1

(2)

o for Dirac type neutrino coupling. AH is the universal coupling constant expected to be of order one. F.G=vs/v!O and F.p= vs/Mp with M p, VIO and Vs being the VEVs for U(I), SO(1O) and SU(5) symmetry breaking, respectively. xf'Yf' zf and wf(f=u, d, e, v) are the Oebsch fattoIS of SO(10) determined by the directions of symmetry breaking of the adjoints 45 s. The following three directions have been chosen for symmetry breaking, namciy

=VIO diag (2, 2, 2, 2,

<.4.> = Vs diag (

2)

®T2 ; (Az> =vs diag (

~, ~, ~, ~,

1-) ®

T2 •

~,

;,

+,

-2,

--2) 0T2 and.

The resulting Oebsch facton1 arc: w. =Wd =

w,=w,= 1; x.= -7/9, Xd= -5/27, x.= 1, X.= -1/15; y.=u, Yd=-'y./3=2/27, y.=4/45; z.= 1, -27, z.= -153 = -3375. qJ i!: the physical CI? phase!) arising from the VEVs. The Clebsch factors associated with the ~YTlnnctry breaking directions can be easily read off from etTective operators which err;: obtained when the beavy fermion pairs are integrated out and decoupled z~=z,=

Wn=O AH) ; F.~ 162----r====7==r-6 (~:)(~:) 1O!(~!:)(~; )---;=1+=F.;==:(FA=x-=;:)~6 16

2,

(3)

VIO

where the factor

I!

I +1:~(AX)6 arises from mixing. The

e~

term in the square root is

VIO

negligible for quarks and charged leptons, but it becomes dominant for the neutrinos due to the large Clebsch factor z.. In obtaining the rfG matrices, some small terms arising from 1) We bave rotated away other possible phases by a phase redefinition of the fennion fields.

504 68

SCIENCE IN CHINA (Series A)

Vol. 39

mixings betwc~n the chiral fermion 16i and heavy fermion pairs l/Ij(l/Ij) are neglected. They are expected to change the numerical results no more than a few percent. The factor 1/..[3 associated with the third family is due to the maximum mixing between the third family fermion and heavy fermions. This set of effective operators which lead to the aforementioned YUkaWd coupling matrices rJ is quite unique. Uniqueness of the structure of operator WI2 was first observed by Anderson et al.[4J from the mass ratios of m./m~ and md/m•. The effective operator W33 is also fixed at the GUT scale[7.8.3) in the case of large tanfJ· There is only one candidate for effective operator W22 when the direction of breaking is chosen to be A,. , with Oebsch factors satisfying Y.:Yd:Y. = 0: 1:319] so as to obtain a correct mass ratio m~/m •. The three parameters A.Jl , SG and Sp are determined by the three measUred mass ratios mb/mt' mp/m t and m./mt' Thus, the mass ratio mc/m, and the CKM mixing elements Vcb and V. b put strong constraint to a unique choice of the symmetry bl'eaking direction A. for effective operator W 32 • Unlike many other models in which W 33 is assumed to be a renormalizable interaction before synu'netry breaking, the Yukawa couplings of all the quarks and leptons (both heavy and light) in tbe prese:nt nlQt;ld are generated in the GUT scale after the breakdown of the family gr')up and SO(lO). Therefore, the initial conditions of renormalization group (RG) evo'lIltion will be set in the GUT scale for all the quark and lepton Yukawa .:ouphngs. ConseqLlently, one could avoid the possible Landau pole and flavor ch... mghJg problems encountered in many other models due to RG running of the third famil) Yukawa couplings from the GUT scale to the Planck scale. The hierarchy among the three families is described by the two ratios sG and Sp. Mass splittings between quarks and leptons as well as between the up and down quarks are determined by the Oebsch factors of SO (10). From the GUT scale down to low energies, RG evolution has been taken into account. Top-bottom splitting in the present model is mainly attributed to the hierarchy of the VEVs VI and V2 of the two light Higgs doublets in the weak scale. I) An adjoint 45 Ax and a 16-D representation Higgs field <1>(<1» are needed for breaking SO(1O) down to SU(5). Adjoint 45 A; and A. are needed to break SU(5) further down to the standard model SU(3)cXSULX U(I)(.

The numerical predictions for the quark, lepton masses and quark m1Xlngs are pre· sented in table l(b) while the input parameters and their values are given in table l(a). RG effects have been considered following the standard schemel7,4J by integrating the full two-loop RG equations from the GUT scale down to the weak scale using MsuSy~Mwr-AX1: M,~ 180 GeV. From the weak scale down to the lower energy scale, three loops in QCD and two loops in QED are taken into consideration. SUSY threshold effects are not considered in detail here since the spectrum of particles is not yet determined. The bottom I) In the case of small tan{J, i.e. ~-Vl' the top-bottom splitting rould be 'l!lused by the Yukawa oouplings. shall discuss the alternative interesting ClSe eleswhere.

we

505 No. I

LOW ENERGY PHENOMENA

quark mass may receive corrections as large as 30Ofo[8] due to large tanp. However, it could be reduced by taking a suitable spectrwn of superparticles. Therefore, one should not expect to have precise predictions until the spectrum of the particles is well determined. The strong coupling constant 1X•• (M z ) is taken to be a free parameter with values given by the present experimental bounds IX.• (Mz) =0.117 ± 0.005[IIJ in the following. Table 1(a)

Parameters and their values as a function of the strong ooupling a, (Mz) determined

by m•• m" m•• m. a, (Mz ) 0.110 0.115 0.120

and

IV.. I=A.

rp

£G=v,/v,o

E,=='V,/'M,

tanfJ

73.4 •

2.66 x 10-' 2.51 x 10-' 2.34 x 10-'

0.89 x 10-' 0.83 x 10-' O. 77x 10-'

51 55 58

77.5· 81.5·

Table 1(b) Observables and their predicted values with the val,,'.:,;; of the parameters given in table 1(a). Input m.(m.)/GcV m,/GeV m./MeV m./Mp.V

4.35 1.78 iOS.6 0.5)

IV.I""A

0.22

(J,(M~)

O.!H!

0.115

0.120

165 1.14

185 1.37

6.5 3.3 0.045

176 1.30 172 7.2 4.3 0.045

197 8.0

1~1""4p'+r( V,.

0.053

0.056

0.063

I..!:::!!. V,. I"'A..J (1- p)'+,,'

0.201

0.199

0.198

Output with

m,iGeVI "' •. (m,) [GeVj

m,tl 'JeV) [MeV] m.(1 GeV)tMeV] m.(I GeV) [MeV].

IV..I ""AA'

152

6.1 0.043

From table lea), one knows that the model has large tanp solution with tanp=v2/v J m.lmb· CP violation is near the maximwn with a phase 1'p-80°. The vacuwn structure between the GUT scale and Planck scale has a hierarchic structure BG=Vs/vJO-.A.=0.22 and 3 £p=VS/Mp-.A. • Assuming (Mp/Mp)2~IXG~I/24-.A.2 (here aG is the unified gauge coupling, Mp is the Planck mass), we have

Mp=2.S x 1018 GeV, vlO~(O.86±0.16) x·IO J7 GeV, { vs=M(;~(2.2±O.2) x 1016 GeV,

(4)

where thi: resulting value for the GUT scale agree well with the one obtained from the gauge coupling unification. M p is also very close to the reduced Planck scale Mp = Mp/ .f&t=2.4 X 1018 GeV and may be regarded as the scale for gravity unification.

It is evident in table l(b) that the predictions on fermion masses and Cabbibo-KobayashiMaskawa (CKM) mixing angles fall in the range allowed by the experimental data[JO- J2]:

506 70

Vol. 39

SCIENCE IN ClllNA (Series A)

mr =1777 MeV, m,,= 105.6 MeV, m.=0.51 MeV, mb(mb) =(4.15 -4.35) GeV, m.(1 GeV)=(105-230) MeV, md(l GeV) =(5.5-11.5) MeV, m,(m,)=(157-191) GeV, m.(l GeV) =(3.1-6.4) MeV, me(me) =(1.22-1.32) GeV, (5) and

_( Vud V.. Vuh) _ (0.9747-0.9759 V- Vcd Ves Veb 0.218-0.224 VIII

V"

V,b

0.004-0.015

0.218-0.224 0.002-0.005 ) . 0.9738-0.9752 0.032-0.048 0.03-0.048 0.9988 - 0.9995

(6)

The model also gives a consistent prediction for the If_Bo mixing and CP violation in kaon decays (a detailed analysis will be presented elsewhere).

r;

It is of interest to expand the above fermion Yukawa coupling matrices in terms of the parameter ).=0.22 (the Cabbibo angle), which wm. found in ref. (13] to be very useful for expanding the CKM mixing matrix. With the input ValUl!li given in table 1 (a), we find

r~r::; ~ (O~7J" 0.~7A6 -Q~~9'f); r:r::; ; (-1~27A4 1.3~;;~~~/2 -O~97).l); (7) ).H

).H

\. 0

-·O.89.~?

1

0

-0.97A.l

r:~ ~ .. (- +.. o.~;:~~ 1.~6+ ~~; '+~6A' o.~~~:.

-+. ) 1

for cx.(Mz ) =0.115.

1 Neutrino masses and mixings

To find the neutrino masses and rruxmgs will .be crucial tests of the model. Many unification theories predict a see-saw type masS[I4J my,-m;,/MN with Uj=U, c, t being up-type quarks. For M N r::;(l0-l-1O-4) MGlITr::; 1012_101l GeV, one has my, < 10- 7 eV,

m -lO-leV y,

'

m ,-(3-2l) eV. y

(8)

In this case solar neutrino anomalous could be explained by v. - v" oscillation, and the mass of Vr is in the range relevant to hot dark matter. However, LSND events and atmospheric neutrino deficit cannot be explained in this scenario. By choosing Majorana type Yukawa coupling matrix differently, one can' construct many models of neutrino mass matrix. We shall present one here, which is found to be of interest with the following texture:

507 No. I

71

LOW ENERGY PHENOMENA

li~

G_

M N-

AHV IO - 2 liG

0

0

0

YN

1

2" ZN

°

~:")

(9)

wN

The corresponding effective operators are given by

(10)

where

W N , YN

and

ZN

are Oebsch facturs with wN='~4/3, YN= 16/9, zN=2/3. They are deter-

mined by additional 45s AB .' L and and (A 3R >= Vs diag (0, 0, 0,

AJR

with (A B - L>=vsdiag (

~, ~) ®t"2'

~, ~, ~,

0,

°)®t"2

The 45 A B - L is also necessary for doublet-

triplet mass splitting[l5) in the Higgs lOl' The light neutrino mass matrix is given via see-saw mechanism as follow.;:

1.

ZN

_1_

4

Iz,1

YN

li~ -itp e -3 -ZN 2 -y,- 2

2

Z,

YN

lip

_.:ll. 4

.,

Z.'/

~

Il..

Iz,1

YN

lip

G

0. 73AHe -iO.H6./2

-0.86).4

(11)

1

- -3 2 -E~ Wit. h M04 li p

I - -I -V2 l': • ed to -1-1 V 2 All' Here '1. is the RG evolution Jactor eshmat Z, Z N

'1,

be

VIO

~,~1.35. Diagonalizing the above mass matrix,

we obtain the masses of light Majorana

508 72

SCIENCE IN CHINA (Series .A)

Vol. 39

neutrinos:

m., = 1.- ~ _1_ =0.83 x 10-4, m., 4 Iz.1 YN m. WN - ' =1-3-m., Iz.lzN

X. e~ -J3Iz.1 e

~0.998,

(12)

p

m.. ~Mo~2.U.HeV.

The three heavy Majorana neutrinos have masses

(13)

The CKM-type lepton mixing matrix is predicted to be

v:LEP =vvt= •• (

V•• V•..

V.,P V.,p

CP-violating eiTt'Cts

ht'e

('

Ji~. '

'-0.(>51 \ 0.045

V.~. )~:

V",

O.~'S·76

V•.• \

0.068 0.000 ) 0.748 0.665 . -0.664 0.748

(14)

found to be small in the lepton mixing matrix. As a result we find

(i) a \'~(;p) -il.(V.) short wave-length oscillation with (15)

which is consistent with the LSND experimentl61 (16)

(ii) a vp (vp )

v.(v.) long-wave length oscillation with

-

Am~=m~, -m~. ~(1.6-2.4)

x 1Q-2eV 2,

(17)

which could explain the atmospheric neutrino deficitll7l : &n!.=m~,-m~,~(0.5-2.4) x 1Q-2eV 2,

(18)

with the best fit ll7l (19) However, (vp-v,) oscillation will be beyond the reach of CHORUS/NOMAD and E803. (iii) Two massive neutrinos vp and v, with

.

m.

~m., ~(2.0-2.4)

eV,

which fall in the range required by possible hot dark matte:rl I8J •

(20)

509 No. I

73

LOW ENERGY PHENOMENA

In this case, solar neutrino deficit has to be explained by oscillation between v. and a sterile neutrino[l\lI V,. Since strong bounds on the number of neutrino species both from the invisible ZO-width and from primordial nucleosynthesis[aJ,2JI require the additional neutrino to be sterile (singlet of SU(2) x U(l), or singlet of SO(IO) in the GUT SO (I 0) model). Masses and mixings of the triplet sterile neutrinos can be chosen by introducing an additional singlet scalar with VEV vs~450 GeV. We find

with the mixing angle consistent with the requirement nucleosynthesis[22] given by ref. [20]. The resulting parameters

necessary for

primordial

(22)

are consistent with the values[l9J obtained by fitting the experimental d8.ta: (23)

This scenario can be testl!d by the next generation solar .neutrino experiments in Sudhuray Neutrino Observatc..ry (SNO) and Super-Kamiokanda (Super-K), both planning to start operation in 1996. By measuring neutral current events, one could identify v. - v, or v. - vI' (v since the sterile neutrinos have no weak gauge interactions. By measuring seasonal variation, one can further distinguish the small-angle MSW22] oscillation from vacuum mixing oscillation. t )

3 Superpotential for· fermion Yukawa interactions

Non-Abelian discrete family symmetry ..1 (48) is important in the present model for constructing interesting texture structures of the Yukawa coupling matrices. It originates from the basic considerations that all three families are treated on the same footing at the GUT scale, namely the three families should belong to an irreducible triplet representation of a family symmetry group. Based on the well-known fact that the masses of the three families have a hierarchic structure, the family symmetry group must be a group with at least rank three if the group is a continuous one. However, within the known simple continuous groups, it is difficult to find a rank three group which has irreducible triplet representations. This limitation of the continuous groups is thus avoided by their tinite and discoimected subgroups. A simple example is the finite and disconnected group A (48), a subgroup of SU(3). The generators of the A(3n2 ) group consist of the matrices

510 74

Vol. 39

scmNCE IN ClllNA (Series A)

E(O,O) = (

and

A.(P,q) =

~

c;' ~

o1 o

0 )

(24)

1

0

0

0 . 2<

e'-q •

0

0

e -i1:!!..(p+q) •

) .

(25)

It is clear that there are n2 different elements A.(p,q) since if p is fixed, q can take

n different values. There are three different element types: A.(p,q),

E. (p,q) = A. (p,q)E(O, 0) ,

c. (p,q) = A" (p,q) E2 (0,0)

in the Li (3n2) group; therefore the order of the .1 (31'12) gw\,;p is ~~,2. The irreducible representations of the Li(3n2 ) groups consist of 0) (n 2'-1)/3 tripi~ts and three singlets when n/3 is not an integer and (ii) (n 2 - 3)/3 triplets and rJne singlets when n/3 is an integer. The character of the triplet representations can be expressed as[.lj

r.1;,01,

(A.(p,q»=e i ~

[m,p+m,q]

+e i ~ [m,q-m,(p+q») +ei ~ [-m, (p+q)+m,p]

l Li~,m'(E.(p,q» =Li;,m, (C. (p,q» =0

(26)

with ml, m 2 =0, I, "·,n-1. Note that (-m] +m 2 , -ml) and (-m 2 ,m j -m 2 ) are equivalent to (m l ,m 2)·

One will see that Li(48) (i.e. n=4) is the smallest of the dihedral group Li(3n2 ) with sufficient triplets for constructing interesting texture structures of the Yukawa coupling matrices. The irreducible triplet representations of Li (48) consist of two complex triplets T J (T J) and T3 (T 3) and one real triplet T 2 = T2 as well as three singlet representations. Theil irreducible triplet representations can be expressed in tenns of the matrix representation TJ(ll=diag(i, I, -i), Tpl=diag( -1, 1, -1), { T3(1)=diag(i, -I,i),

T?l=diag(I, -i,i), T?l=diag(I, -I, -I), Tpl=diag( -l,i,i),

T/3l=diag( -i,i, 1); Tp)=diag( -I, -I, I);

(27)

Tp)=diag(i,i, -1).

The matrix representations of Ti i ) and Ttl are the Herrnician conjugates of Tii) and With this representation, we can explicitly construct the invariant tensors.

nil.

All three families with 3 x 16=48 chiral fermions are unified into a triplet 16-dimensional spinor representation of 80(10) x Li(48). Without losing generality, one can assign the three chiral families into the triplet representation T J, which may be simply

511 No. 1

7S

LOW ENERGY PHENOMENA Table 2 Dea>mposition of the product of two triplets, T, ® 1j and T, ®TJ in .d., (SU (3».

I

2

3

"3

112

A33

133

T33 123

133

A23

123 IT3

123

ITI

123 113 233

.II (48)

2

3 Triplets T, and sents a singlet.

1', are simply denoted by i and T, respectively. For example

All T,®1',=A®TJ®"TJ =A33. here A repre-

denoted by f6= 16i T.(i). All the fermions are assumed to obtain their masses through a single 10. of SO(10) into which the needed two Higgs doublets are unified. The model could allow a triplet sterile neutrino with small mixings with the ordinary neutrinos. A singlet scalar near the electro weak scale is necessary for generating small masses of the sterile neutrinos. Superpotentials which lead to the above texture structures (eqs.(l), (2) and (9) 7UOS

with

and effective operators (eqs.(3) and (10» are f(".tmd to be 3

W y= L t/l0.10.t/l02 + t/I"12 Xl/l;3 -:1i2. X2~JI) +t/I 32.l~""L +~3. X3t/123

.-0

+t/lIJi. X~')3 + ~ro. 'XG~(~? -+ ~·33Ax!/J3 + f3 Axt/l 2+ t/I~xt/l. + (liiIlXi+~iI!X+'iiI3Az +t/l23 A• +t/I. y)f6 3

3

2

+G=:O L L SGt/I .{t/lOj+ L (t/I iJAxl/liJ + S,I/Iil/li) + S, 1/133 I/In + Sp 1/1 31/13 j=1 i=l

(28)

for the fermion Yukawa coupling matrices, 3

W R = L (t/I:. 1031/1ii +I/Ii: x;'t/I +liiii x' t/I,~ +I/Ii: Ait/l') +(t/I' X +1/1 A.) f6+tP}Q3tP i=1 3

2

3

LLSGt/li;t/li; +SI'(L t/li;t/li; +liit/l+lii'l/l)

1=1j"'1

i-I

for the right-handed Majorana neutrinos, and

Ws=t/J /101 t/l2' +t/I I' tPv, +t/l2'fP, i6+(v.fP.N. +h.c.) +S/N,N.

(29)

for the sterile neutrino masses and their rnixings with the ordinary neutrinos. In the above superpotentials, each term is ensured by the U(1) symmetry. An appropriate assignment of U (I) charges for the various fields is implied. All t/I fields are triplet 16-D spinor heavy fermions, where the fields "'i3{t/1i3},"'i;{t/I'~}''''i{'''i}(i=I,2,3),

'I' {"ijil'}, 1/12' {I/I2'}, 1/1 {I/I}

and !/I' {I/I'} belong to (16, T I) {(16, T I)} representations of SO(IO)X,148(SU(3»; t/l1I{t/lII} and t/l12{liid belong to (16, TJ {(l6, TJ}; I/Iil{liiil} and t/I;2 (tfa} (i=2,3,O) belong to (16,T3) {(l6,T3)}; t/li;{I/Ii;} and t/I;~{t/Iii}(i=I,2) belong to (16,T3)

((16,T3)}; t/l3:{t/l3:} and t/l3i{t/I~} belong to (l6,T2) {(16,T2)}; X,Y,S/,SI' and fP,aresinglets of SO(IO) X,,148 (SU(3». v, and N, are SO(IO) singlet and A(48) triplet fermions. 103 is an additional SO(lO) lO-representation heavy scalar. All S0(10) singlet X fields are triplets of

512 76

SCIENCE IN ClDNA (Series A)

Vol. 39

T 3, T 3, T I , T 2, T 3, respectively; XI',X2',X3',X' belong to triplet representations T I, T 2, T 3, T 3, respectively. With the above assignment for various fields, one can check that once the triplet field X develops VEV only in the third direction, i.e. (l3» #0, and x' develops VEV only in the second direction, i.e. (t2» #0, the resulting fermion Yukawa coupling matrices at the GUT scale will be automatically forced, due to the special features of ..1(48), into an interesting texture structure with four non-zero textures '33', '32', '22' and '12' which are characterized by

L1(48), where XI' X2' X3' Xo, X belong to triplet representations

XI' X2' X3' and Xo, respectively, and the resulting right-handed Majorana neutrino mass matrix is forced into three non-zero textures '33', '13' and '22' which are characterized by XI', X2', and X3', respectively. It is seen that five triplets are needed. of which one triplet is necessary for unity of the three family fermions, and four triplets are required for obtaining the needed minimal non-zero textures. The symmetry breaking scenario and the structure of th\~ physi(.'al vacuum are considered as follows:

SO(10)

;<. ..1(48)

M

\J

x t'(l) .-.! SO(lO) x L1(48) ~ SU(5) x t\(48)

-~ 8U(3)cxSU(2hX U(l)y ~ SU(3)c X U(l) ...

(30)

and

«I)

(Sp) =M, (X) =vlo = (S/) , «1)(16» = (16» =VIO/J2, (Y) =vs= (SG)' <X (3» = <Xa(i) > = <X' (2» = (X/ (i) =Vs with (i= I, 2, 3; a =0, I, 2, 3; j= 1, 2, 3), <X (1) > = <X (~) = (x'(I»=<x'(3»=0, =vs~450GeV, (H 2 )=v2 =vsinp with v=Jif.+v~=246GeV.

4 Conclusions It is amazing that nature has allowed us to make predictions in terms of a single Yukawa coupling constant and a set of VEVs determined by the structure of the vacuum and to understand the low energy physics from the Planck scale physics. The present model has provided a consistent picture for the 28 parameters in 8M model with massive neutrinos. The neutrino sector is of special interest for further study. Though the recent LSND experiment, atmospheric neutrino deficit, and hot dark matter could be simultaneously explained in the present model, solar neutrino puzzle can only be understood by introducing an SO (10) singlet sterile neutrino. It is expected that more precise measurements from various low energy experiments in the near future could provide crucial tests .on the present model. Aclmowledgement

Wu Yueliang would like to thank Institute for Theoretical Physics, Chinese Academy

of Sciences. for its hospitality and partial support during his visit.

513 No.1

LOW ENERGY PHENOMENA

77

References 1 Gatto, R .. Sartori, G., Tonin, M., Weak self-masses, Cahbibo angle and Breoken SU(2), Phys., Lett., 1968, B28:128. 2 Weinberg, S .. Rabi, I. 1.. Discrete flavor symmetries and a formula for the Cabbibo angle, Phys. Lett., 1977, 840: 418. Raby, S., Ohio State Univ., Preprint, 0IISTPY-HRI'-·1~9>24. 4 Dimopoulos, S. Anderson, G., Raby. S:, et al .. Predictive ansatz for fermion mass matrires in sypersymmetric grand unified theories, Phys. R.~:., 1992, 045:4192. 5 Hall, L.J., Raby, S., On the generality of rertain predictions for quark mixing, I'hys. Lerr., 1993, D135: 164. 6 Fairbairn, W.M., Fulto, T., Klink, W.H., Finite and diswnnected subgroups of SU(3) and their application to the elementary-particle spectrum, J. Math. Phys., 1964, 5: 1038. 7 Kaplan, D., Schmaltz, M., Flavor unification and discrete non-Abelian symmetries, Phys. Rev., 1994, 049:3741. 8 Ananthanarayan, B., La2llrides, G., Shafi, Q., Top-quark-mass prediction from supersymmetric grand unified theories, Phys. Rev. Lett., 1991, 044: 1613. 9 Hall, L., Rattazzi, R" Sarid, U., The top quark mass in supersymmetric SO(IO) unification, Phys. Rev., 1994, D~{):

7048. 10 Geogi, H., Jarlskog, C., A new lepton-quark mass relation in a unified the('ry, Phvs . .rAt., 19 79, Br-6.297. 11

Partide Data Group, Evidence for top quark production in bar {p} P wllisi.')tlf. al s"rr {s} = 1.8 TeV, Phvs. Rev., 1994, DSO:2966.

12 Abachi, S., ObserV'dtion of the top quark, Phys. Rev. Lett., 1<''95, 74'263.

\3 Gasser, J., Leutwyler, H., Quark masses, Phy,;. Rep., 1982, 87:T 14 Wolfenstein, L., Parametrization of

:]y, KGbaYf.shi-Maskll;~rd

mutnx, Phys. Rev. Lerr., 1983, 51:1945.

15 Gell-Mann, M., Ramond. Po, Sli,mkl' R., in Sup«/'yravity (ed. V'dn Nieuwenhui:a:n, F., Freedman, D.), Amsterdam: North Holland, BII;, 315. 16 Dimpopouios, S., W;I~:k, F., i:1 Proceedings of Eric!! Summer School (ed. Zichichi, A.) 1981. 17 Athanassopoulos, C, Candidate ewnts in a search for oscillations, Phys. RI!V. Lett., 1995. 18 Fukuda, Y., Atmospheric ratio in multi-GeV energy range, Phys. Lett., 1994 3358: 237. 19 Primack, J., Holtzman, J., Klypin, A. et al., Cold hot dark matter cosmology. Phys. Rev. Lell., 1995, 74:2160. 20 Caldwell, D.O., Mohapatra, R. N., Cold hot dark matter cosmology in the light of solar and atmospheric neutrino oscillations, Phys. Rev., 1993, 048:3259. 21

Walker, T., Steigman, G., Schramm, D. N. et al., A new look at neutrino limits from big bang nucleosynthesis,

?reprint OSUTA-2/95, hep-ph/9502.IOO. 22 Wolfenslein, L., Neutrino oscillation in malter, Ph}'s. Rev.• 1978, DI7:2369.

514

r'>kICLEAR FHYSICS B

PROCEEDINGS SUPPLEMENTS

Nuclear Physics B (Proc. Suppl.) 52A (1997) 159-162

I-L~EVIr.R

A solution to the puzzles of CP violation, neutrino oscillation, fermion masses and mixings in an SUSY GUT model with small tan,B K.C. Chou

OT1

and Yue-Liang

WU OT1 •

aChinese Academy of Sciences, Beijing 100864, China bDepartment of Physics, Ohio State University, 174 W. 18th Ave., Columbus, OH 43210, USA CP violation, fermion masses and mixing angles including that of neutrinos are studied in an SUSY SO(lO)x.o.(48)x U(l) model with small tanfJ. It is amazing that the model can provide a successful prediction on twenty three observables by only using four parameters. The renormalization group (RG) effects containing those above the GUT scale are considered. Fifteen relations among the low energy parameters are found with nine of them free from RG modifications. They could be tested directly by low energy experiments.

The standard model (SM) is a great success. But it cannot be a fundamental theory. Eighteen phenomenological parameters have been introduced to describe the real world, all of unkown origin. The mass spectrum and the mixing angles observed remind us that we are in a stage similar to that of atomic spectroscopy before Balmer. In this talk, we shall present an interesting model based on the symmetry group SUSY SO(lO) x A(48) x U(l) with small values of tan,B - 1 which is of phenomenological interest in testing the Higgs sector in the minimum supersymmetric standard model (MSSM) at Colliders[l]. For a detailed analysis see ref. [2]. The dihedral group A(48), a subgroup of SU(3), is taken as the family group. U(l) is family-independent and is introduced to distinguish various fields which belong to the same representations of S0(10) x A(48). The irreducible representations of A(4B) consisting of five triplets and three singlets are found to be sufficient to build an interesting texture structure for fermion mass matrices. The symmetry A(48) x U(l) naturally ensures the texture structure with zeros for Yulta.wa coupling matrices, while the coupling coefficients of the resulting interaction terms in the superpotential are unconstrainted by this symme• Supported in part by the US Department of Energy Grant # DOE/ER/0154S-675. Pernu....ent ..ddreso: Institute of Theoretical Phy.ics, Chinese Academy of Sciences, Beijing 100080, China 0920·S632(97)iSI i.OO·n 1997 EIscvi~r Sci~nc" ltV. All righlS reserwd. I'll: SO')20·S632(96)01l553-1

try. To reduce the possible free parameters, the universality of Yukawa coupling constants in the superpotential is assumed, i.e., all the coupling coefficients are assumed to be equal and have the same origins from perhaps a more fundamental theory. Choosing the structure of the physical vacuum carefully, the Yulta.wa coupling matrices which determine the masses and mixings of all quarks and leptons are given at the GUT scale by

r~

= Au

0

~Z~€~

~Zu€~

-3y,,€~ei';

( (

0

0 -~ZIE~ 0

a -

¥-ZJl€~

o

-~:c.. €~ - ~Zl €~

3YJ€~ei'; ~:CJ€~

-

-,{l•.,~ ) w,.

0 -~;Z:f€~ wf

)

¥-Z~€$

15YJlf~ei';

-~;Z:J1f~

with A.. = 2AH /3, AI = A,,( _1)n+1 /3 n (J = d, e) and Av :::: >'J /5 n +1. Hen the integer n reflects the possible choice of heavy fermion fields above the GUT scale. n 4 is found to be the best choice in this set of models for a consistent prediction on top and charm quark masses. This is because for n > 4, the resulting value of tan,B becomes too small, as a consequence, the predicted top quark

=

515 160

KC. Chou. Y.-L. WulNuciear Physics B (Proc. Suppl.j 52A (1997) 159-162

mass will be below the present experimental lower limit. For n < 4, the values of tanf3 will become larger, the resulting charm quark mass will A.~r3, be above the present upper bound. A.H EG

== (;-:;,)

j?; and

=

Ep

== (it;)

JV; are three pa-

rameters_ Where A.~ is a universal coupling constant expected to be of order one, rI, r2 and T3 denote the ratios of the coupling constants of the superpotential at the GUT scale for the textures '12', '22' ('32') and '33' respectively. They represent the possible renormalilllation group (RG) effects running from the scale Mp to the GUT scale. Note that the RG effects for the textures '22' and '32' are considered to be the same since they are generated from a similar superpotentiai structure after integrating out the heavy fermions and concern the fields which bdong to the same repre5~~tations of the symmetry group, this can be expliCItly seen from their effective operators ~22 and W 32 given below. Mp , VIC, and Vs bemg the vacuum expectation values (VEVs) for U~I) x .!l(4~), SO(10) and SU(5) symmetry br~~kmg respectively. rP is the physical CP phase ~I1smg from the VEVs. The assumption of maxImum CP violation implies that rP 11"/2. :c J, YJ, zJ, and wJ (! u, d, e, II) are the Clebsch factors of 50(10) determined by the directions of symmetry breaking of the adjoints 45's. The Clebsch factors associated with the symmetry breaking directions can be easily read off from the U (1) hypercharges of the adjoints 45's and the rdated effective operators which are obtained when the symmetry 50(10) x .!l(48) x U(I) is broken and heavy fermion pairs are integrated out:

=

=

arises from mixing, and provides a factor of 1/..;3 for the up-type quark. It remains almost unity for the down-type quark and charged lepton as well as neutrino due to the suppression of large Clebsch factors in the second term of the square root. The relative phase (or sign) between the two terms in the operator Wn has been fixed. The three directions of symmetry breaking have been chosen as < Ax >= 2vlo diag.(I, I, I, I, 1)®T2' < A. >= 2vs diag.(-~, -~, -~, -I, -1) ® Ta, < Au >= vs/v'3 diag.(2, 2, 2, 1, 1) ® Ta. The resulting Clebsch factors are w.. Wtl W. WI' = 1, :c u = 5/9, :Ctl = 7/27, :c. -1/3, :c" = 1/5 Yu = 0, Ytl = y./3 = 2/27, y" = 4/225, ~ 1, Ztl Z. -27, z" = -11)3 = -3375, zu = 1 - 5/9 4/9, z~ Ztl + 7/729 ~ Ztl, 3 z~ = Z. - 1/81 ::::: Z., z~ z" + 1/15 ~ Z". An adjoint 45 Ax and a I6-dimensional representation Higgs field ~ (~) are needed for breaking SO(IO) down to SU(5). Another two adjoint 45s A. and Au are needed to break SU(5) further down to the standard model SU(3). x SUL(2) x U(I)y. From the Yukawacoupling matrices given above , the 13 parameters in the SM can be determined by only four parameters: a universal coupling constant A.H and three parameters: EG, Ep and tanf3 V2/Vl. The neutrino masses and mixings cannot be uniquely determined as they rely on the choice of the heavy Majorana neutrino mass matrix. The following texture structure with zeros is found to be interesting for the present model

=

=

(1;) (1;) '1A16a (~;) (1;) 71AI62ei~

W 32

=

A. 2 16371X'7A

W 22

=

A.2 16 271A

W I2

=

A.d6 I [

C~·:

+ (~) Mp 2 71A

=

10 1

r

10 1

'1:" 10 1

(

'1~

Au ) ( A. ) Ax 10 1 Ax '7A]16 2

where A.i A.~ri' '7A =-= (vlO/A x ),,+I, '1~ = (vlo/AX),,-3. The factor 'Ix 1/.)1 + 271!

=

=

=

=

=

=

Mf:

= MR (

~ !zNE~ei6"

A.3 16371X'7A 101'1A'7X 163

W33

= =

= =

Y~ ~ZN1eil") 0

wN4

The corresponding effective operators are

Wta" w~ w~ with

MR = A.HV~OE~E~/ Mp, A.f = A.HV104/ Mp, A.i' = A.f E~E~ and A.f = >.i' E~. It is then not difficult to read off the Clebsch factors YN 9/25,

=

516 161

KC Chou. r·L. /YulNuciear Physics B (Proc. Suppl.) 52A (1997) 159-162 ZN = 4 and WN = 256/27. The CP phase 6" is assumed to be maximal 6" = '11"/2. In obtaining physical masses and mixings, renormalization group (RG) effects should be taken into account. The initial conditions of the RG evolution are set at the GUT scale since all the Yukawa couplings of the quarks and leptons are generated at the GUT scale. As most YUDwa couplings in the present model are much smaller than the top quark Yukawa coupling >.f '" 1, in a good approximation, we will only keep top quark Yukawa coupling terms in the RG equations and neglect all other YUDwa coupling terms. The RG evolution will be described by three kinds of scaling factors. 1/F (F = U, D, E, N) and RI arise from running the YUDwa parameters from the GUT scale down to the SUSY breaking scale Ms which is chosen to be close to the top quark mass, i.e., Ms ~ me ~ 170 GeV. They are de-

n:=l (

cr 121>;

=

fined by 1/F(Ms) = ::~::1 (F UJ DE N)withcr,l (~ 3 ~) cP (.1. I , '18' '3 I , - 15 I 3 I ~) 3 , cf 3, 0), cf == (25,3,0), bi (~, 1, -3), 1 and Rezp[- JlnMs r 1nM"(M!l)2dt] [1 + I 4,..

= m,

)

=

=

=

=

31£:'s) with I(Ms) ==

(>.f)2Klt 1/12 , where KI = J,.~:: "Mt)dt with Ms ~ me

= 170GeV. Other RG scaling factors are derived by running YUDwa couplings below Ms· m.;(m.;) = 7Ji m.;(Ms) for (i == c, b) and m.;(lGeV) = 1/i m.;(Ms) for (i u., d, s). The physical top quark mass is given by Me = me(me) (1 + ~a.~",.)). The scal-

=

ing factor RI or coupling >.f = A.- .jl:::p:;rr ~ is v.n.t Rt determined by the mass ratio of the bottom quark and T lepton. tan (3 is fixed by the T lepton mass • ",_.12 I n numenc . al pre d"lctlOns we via cos fIt:l == ~. 1 take a- (Mz) 127.9, s2(Mz) 0.2319, Mz == 91.187 GeV, a;:l(me) 58.59, a;l(me) 30.02 and al1(Mo) = a;l(MG) = ai1(Mo) ~ 24 with Me '" 2 x 10 16 GeV. For a,(Mz) 0.113, the RG scaling factors have values (""'.d ... 1/c, 1/b, 1/•• ,..T, 1/u, 1/D/1/E == 1/DIE, 1/E, 1/N) = (2.20,2.00, 1.49, 1.02, 3.33, 2.06, 1.58, 1.41). The corre' sponding predictions on fermion masses and mixings thus obtained are found to be remarkable. Our numerical predictions for a,(Mz) = 0.113

=

=

=

= =

are given in table 1 with four input parameters. Where BK and fBv7i in table 1 are two important hadronic parameters and extracted from KO - [(0 and BO - iJo mixing parameters eK and Zd. Re(e' /e) is the direct CP-violating parameter in kaon decays, where large uncertanties mainly arise from the hadronic matrix elements. a, (3 and.., are three angles of the unitarity triangle in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Jcp is the rephase-invariant CPviolating quantity. The light neutrino masses and mixings are obtained via see-saw mechanism M" r;(M~)-1(r;)tvV(2Rt61/~). The predicted values for IV..,I, 1V... I/IVc.I, IVcdl/IVc.I, md/ m ., IV"... I, IV".... I, IV" ... I as well as m.,.I m ",. and m.,,./m,,. are RG-independent. From the results in table 1, we observe the following: 1. a 1I,.(ii,.) .... IIT(iiT) long-wave ~. m~,. ~ length oscillation with ~m!T 1.5 x 1O-3 ey2 and sin2 29,... ~ 0.987 could explain the atmospheric neutrino deficit[3]; 2. Two massive neutrinos II,. and II.. with m.,,. ~ m... ~ 2.45 eV fall in the range required by possible hot dark matter[4]; 3 a short wave-length oscillation with ~m~,. = m!,. - m!. ~ 6 eV 2 an sin 2 28.,. ~ 1.0 x 10- 2 is consistent with the LSND experiment[5]. 4. (II,. - II.. ) oscillation will be beyond the reach of CHORUS/NOMAD and E803. However, (II. - II.. ) oscillation may become interesting as a short wave-length oscillation with ~m2 = ~ - ~ ~ 6 ey2 and Majorana neutrino sin 2 28. T ::: 1.0: 10- 2 ; allows neutrinoless double beta decay ({3{3o,,) [6]. The decay rate is found to be rtltl ~ 1.0 x 10- 61 GeV which is below to the present upper limit; 6. solar neutrino deficit has to be explained by oscillation between II. and a sterile neutrino II, [7](singlet of SU(2)x U(l), or singlet of SO(10) in the GUT SO(10) model). Masses and mixings of the triplet sterile neutrinos can be chosen by introducing an additional singlet scalar with VEV v, ~ 336 GeV. They are found to be mil, = >'HV: /VI0 ~ 2.8 x 1O-3 eV and sin 8.. ~ m"t.".Im.,. == V2EP/(2v,E~) ~ 3.8 X 10- 2 • The resulting parameters ~m~. == ~. - m!. ~ 6.2 x 10- 6 eV 2 and sin 2 28.1 ~ 5.8 X 10- 3 ; are consistent with the values [7] obtained from fitting the

=

=

5.

-

517 K.c.

162

ChUlI, Y.-L. IVu/Nllciear Physics B (I'roc. Supp/.) 52A (1997) 159-162

Table 1 Output observables and model parameters and their predicted values with a.(Mz ) == 0.113 and input parameters: m. == 0.511 eV, m,. == 105.66 MeV, m.,. = 1.777 GeV, and mb(fnb) = 4.25 GeV. Output Output Data[8] Output Output Ml [GeV] 182 180 ± 15 Jcp/IO 2.68 m.:(mc ) [GeV] 1.27 1.27 ± 0.05 a 86.28° 4.31 4.75 ± 1.65 /3 22.11° m,.(lGeV) [MeV] m.(lGeV) [MeV] 156.5 165±65 "y 71.61° m/l(lGeV) [MeV] 6.26 8.5 ± 3.0 m.., [eV] 2.4515 IV... I = ~ 0.22 0.221 ± 0.003 m".. reV] 2.4485 3 ~ 0.083 0.08 ± 0.03 m ... [eV]/101.27 v ••

\~:~

IVcbl = ~f tan/3

A~2

= V2/Vl

fG Ep

BK

fBv'B [MeV] Re(t' /£)/10- 3

0.209 0.0393 1.30 2.33 0.2987 0.0101 0.90 207 1.4 ± 1.0

IV..... I IV".... I IV..,. I IV.... I IV.... I T

0.82 ± 0.10 200 ± 70 1.5 ± 0.8

experimental data. It is amazing that nature has allowed us to make predictions on fermion masses and mixings in terms of a single Yukawa coupling constant and three ratios determined by the structure of the physical vacuum and understand the low energy physics from the GUT scale physics. It has also suggested that nature favors maximal spontaneous CP violation. It is expected that more precise measurements from CP violation, neutrino oscillation and various low energy experiments in the near future could provide a good test on the present model and guide us to a more fundamental model. ACKNOWLEDGEMENTS: YLW would like to thank professor R. Mohapatra for a kind invitation to him to present this work at the 4th SUSY96 conference held at University of Maryland, May 29- June 1, 1996.

REFERENCES 1.

m... [eV]/10- 3

0.24 ± 0.11 0.039 ± 0.005

G. Kane, in this proceedings; see also J. Ellis, talk given at 17th Intern. Symposium on Lepton-Photon Interactions, 10-15 August, 1995, Beijing, China.

MN. [GeV] MN. [GeV]/10 6 MNo [GeV)

2.

3. 4. 5. 6.

7.

8.

2.8 -0.049 0.000 -0.049 -0.707 0.038 ~ 333 1.63 333

K.C. Chou and Y.L. Wu, Phys. Rev. D53 (1996) R3492j hep-ph/9511327 and hepph/9603282, 1996. Y. Fukuda et al., Phys. Lett. 335B, 237 (1994). D.O. Caldwell, in this proceedings; J. Primack et al., Phys. Rev. Lett. 74 (1995) 2160. C. Athanassopoulos et al., Phys. Rev. Lett., (1996) nucl-ex/9504002 (1995). For a recent review see, R.N. Mohapatra, Maryland Univ. Report No. UMD-PP-95-147, hep-ph/9507234. D.O. Caldwell and R.N. Mohapatra, Phys. Rev. D 48, 1993) 3259; J. Peltoniemi, D. Tommasini, and J.W.F. Valle, Phys. Lett. 298B (1993) 383. CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 74, 2626 (1995); DO Collaboration, S. Abachi et aI., Phys. Rev. Lett. 74, 2632 (1995); J. Gasser and H. Leutwyler, Phys. Rep. 87, 17 (1982); H. Leutwyler, Nucl. Phys. B337, 108 (1990); Particle Data Group, Phys. Rev. D50, 1173, (1994).

518 Vol. 41 No.3

SCIENCE 1N CHINA (Series A)

March 1998

A possible unification model for all basic forces * WU Yueliang

c'lUli!5U

(Inslitllte of Theoretical Physics. Chinese Academy of Sciences. Beijing 100080. China)

and ZHOU Guangzhao (K. C. Chou, Jj!;]*t~) (Chinese Academy of Sciences. Beijing 100086, China) Received November 10. 1997

Abstract A unification n"Kldel for strong, electromagnetic, weak and gravitati.ll al forces i5 pmol'sed. The tangent space of ordinary coordinate 4-dimensional spacetime is a submanifold of a 14-dirnensiona. i"ternal spacetime spanned by four frame fields. The unification of the standard IT¥ldel with gravity is governed by gauge symmetry in the internal spacetime. KeyWlrds:

unification, internal spI1cetime, SO( I ,13) , gravity, frame fields.

One of the great theoretical endeavours in this century is to unify gravitational force characterized by the general relativity of Einstein[1 .2] with all other elementary particle forces (strong, electromagnetic and weak) described by Yang-Mills gauge theory[3]. One of the difficulties arises from the no-go theorem[4] which was proved based on a local relativistic quantum field theory in 4dimensional spacetime. Most of the attempts to unify all basic forces involve higher-dimensional spacetime, such as Kaluza-Klein Yang-Mills theories[5.6], supergravity theories[7.MJ and superstring theories[9 -12J. In the Kaluza- Klein Yang-Mills theories, in order to have a standard model gauge group as the isometry group of the manifold, the minimal number of total dimensions has to be II [13J. Even&>, the Kaluza- Klein approach is not rich enough to support the fermionic representations of the standard model due to the requirement of the Atiyah- Hirzebruch index theorem. The maxi mum supergravity has SO (8) symmetry, its action is usually al&> formulated as an N = I supergravity theory in II-dimensional spacetime. Unfortunately, the SO(8) symmetry is too small to include the standard model. Consistent superstring theories have al&> been built based on la-dimensional spacetime. In superstring theories, all the known particle interactions can be reproduced, but millions of vacua have been found. The outstanding problem is to find which one is the true vacuum of the theory. In this paper we will consider an alternative scheme. Firstly, we observe that quarks and leptons in the standard modcl[14·· 16 1 can be unified into a single 16-dimensional representation of complex chiral ~;pinors in SO(IO) 117.IM]. Each complex chiral spinor belongs to a single 4-dimensional rcprcsentatlon of SOC I ,3). In a unified theory, it is an attractive idea to treat these 64 real spinor components on the same footing, i. e. they have to be a single representation of a larger group. It is therefore natural to consider SOC 1,13) asour unified group and the gauge potential of SOC 1.13) as the fundamental interaction that unifies the four basic forces (strong. c1eetromag• "mject supported in part by the Outstanding Young Scientist Fund of China.

519 No.3

A POSSIBLE UNIFICATION MODEL FOR ALL BASIC FORCES

325

netic, weak and gravitational) of nature. Secondly, to avoid the restrictions given by no-go theorem and other problems mentioned above, we consider that the ordinary coordinate spacetime remains to be a 4-dimensional manifold 54 wit h metric gil' (x) . f.l, v = 0 ,I .2 ,3. At each point P: XiI,

there is a d-dimensional flat space M" with d > 4 and signature (I , - I, ... , - I). We as-

sume the tangent space T4 of S4 at point P to be a 4-dimensiomil submanifold of M" spanned by fourvectorse;(x)/I=O,I,2,3; A=O,I,"',d-1 such that gil. (x)

where

77AB

= ei; ( x) e; ( x) f)AB ,

(I)

= diag( I , - I , ... , ... I) can be considered as the metric of the flat ~ace Md. We shall

call ej; (x) the generalized vierbein fields or simply the frame fields. Once the frame fields ei; (x) are given, we can always supplement t hem wit h anot her cf-4 vector fields e;;, (x) -e'~, ( ei; (x)) , III = I ,2, ..., cf-4 such that

ei~ ( x) e~, ( x) 77AB = 0, e;~, ( x) e?, (x) 77AB = gil",' (2) where gil", = diag( . . 1 , ... , . . 1). e'!, (x) can be uniquely determined up to an SO( cf-4) rotation. In the flat manifold M" we can use ei; (x) and e;~, (x) to decompose it into two orthogonal manifolds T4 0Cd- 4, where C d _4 will be considered to be the internal space describing SO( cf-4) internal symmetry besides the spin and is spanned by the cf-4 orthonormal vectors e;;, ( x). In the new frame system of M" the metric tenoor is of the form [

J.

gil. ( x)

0

o

gil",

(3)

With ei~ (x) and e'!, (x) , we can now define the covariant vectors as e:'~ (x) and e.~' (x) satisfying ~ ( x) e; (x)

= g~'

,

e:'~(x)e'~,(x) =0,

e.~' ( x) e'! (x)

//I

gil ,

e:~'(x)e:~(x) =0.

(4)

Under general coordinate transformations and the rotations in M", e;: (x) transforms as a covariant vector in ordinary coordinate spacetime and a vector in the M" rotation, e'~, (x) transforms as a covariant vector in the C d - 4 rotation and a vector in the M" rotation. For a theory to be invariant under both general coordinate transformations and local rotations in the flat space Md, it is necessary to introduce affine connection

r;..( x)

for general coordinate transformations and gauge

potential Q,;B (x) = . . 0:,'" (x) for if-dimensional rotation SO (I , cf-I) in Md. These transformations are connected by the requirement that T4 has to be the submanifold of Md spanned by four vectors ei; (x) at point P and e:~ (x) should be a covariantly constant frame and satisfy the condition (5)

It is then easily verified that (6) (7)

With the above considerations, we can now construct an invariant action under general coordinate transformations in t he ordinary coordinate spacetimc and the local SO (I , if-I) group symmetry in M" with eq. (5) as a constraint. In addition, the action is required to have no dimensional parameters and to be renormalizable in the sense of the power counting. The general form of the action which satisfies these requircments is

520 326

Vol. 41

SCIENCE IN CHINA (Series A)

- 2 ~'I-' 1'"Y(/jeB + 2 ~

0,10, c/>:::! Cp -L 1.1..1 + 4 11 '1-'

+ r;F;:~ 1·~,~)g"I}'I.Ht!~e~ +

01

-L

+

-L

1:.12 .,' B , , ;

a~ f

T'8 ·[D "

Y "

.J'Y

IT

1<"" e'Ce8eA eD +

,ly

(8)

I~~ F<;!/e~t!~)e;;e~ ..I8,·.('D "

.,.~

f II' I'~IT e:., e81!cetj

(I]

y

"

where <1>( x) is a scalar field introduced to avoid the dimensional coupling constants. 3) ,

r;. t and

i = I ,2,

A are dimensionless parameters. r1.~ is the field strength defined in a standard way:

F/.! The temor

(Ii (

f7,

is defined as

OyQ,~8

o"Q;8 -

=

+ gu(

~~CQ~8

-

Q;cQ~~

.

(9)

F/. = r'7,! e~.

U sing the frame fields e'~ (x) and e:~1 (e;: (x)) , we can decompose ~~8 (x) into three parts e~ (x) Q:8 (x)

Is (x)

(p,

(1'=

0, 1 ,2.3) which describe the gravity, and e:~1 (x) ~~8 (x) e~ ( x)

which characterize gauge interactions, as well as e:~1 (x) QI~8 (x) e~ (x) which connect gravity with gauge interactions. From the constraints of eq. (5) , we obtain gc;e",j (x) ~~8 ,( x) e~ (x)

=

gc;elllA (x) ~~8 (x) e; (x)

Ij:,r -

= -

(I 0)

e",.,o"d'·4,

(11)

elllA o"e",j .

Similarly, we can reexpress e:~' ( x) ~~8 ( x) e~ (x) as II (

eA

x

)

Q48 ( x ) eBIII (x . ) I'

where A ;:111 (x) = - A ~III (x) ( m , rotation SO( d-4) in Cd - 4.

-.L( eA 0"e

A"11111 (x) - 2

=

II

iliA

gu

III

- eA

0"e) 11041

(12)

,

= 1. "', d - 4) is the gauge potential for ( d-4)-dimensional

11

Note that not all the gauge fields Q,~8 (x) are simply new propagating fields due to the constraints D"e;~

( cr 1) 12

= O.

By counting the constraint equations (4 X4 Xd) , unknown Q~8 (x) (with 4 d

degrees of freedom) and ei~ ( x) (wit h 4 X d degrees offreedom) as well as ~IT( wit h 40

degrees of freedom for the symmetric parts

r; I'ro = [7',,,,,,

and 24 degrees of freedom for antisym-

metric parts J1f"ITI = - J1f""/) ,one sees that besides the antisymmetrie parts rl,"'I, the independent degrees of freedom arc (4£1 +4( d - 4) (d - 5)/2). These independent degrees of freedom coincide with the degrees of freedom of the frame fields ei~ (x) and the gauge fields A .::111 (x) of the group SO ( d - 4). In addition, the gauge conditions in the coset SO (I , d - J) 1 SO ( d - 4) lead to additional constraints (4 d - 10). Thus the independent degrees of freedom are reduced to ( 10 + 4 ( d - 4) ( d - 5) 12) which exactly match with the degrees of freedom of t he metric ten&>r g"y (x) and the gauge fields A::III (x) of the group SO ( d - 4). For d = 14 . the resulting independent degrees of freedo 111 of t he fields are sufficient to describc t hc fo ur basic forccs • where t he general relativity of the Einstein thcory is described by the metric ten!llf. Photon. W-bo&>ns and gluons. that mediate the electromagnetic. weak and strong interactions respectively. are different manifestutions of the guuge potential (.\"1 of the symmetry group SO( 10) [17 .IX]. The cllrvu-

A::

III

ture ten&>r R:,:Y,r and t he Ricci ten!llr R y,T

= R;;YITg~

as well as t he scalar curvat ure R = R ;"g"T are

simply related to the field strength I·~:tl viu R':'Y,r= gur~:tlr!.~euB. R yIT = gl'I<~:~e:~;e'>II and R J

e'.;e~. It is not ditTicult to check that

R;:t

=

gLr;:! .

R:';~J = 1<:~"/<:1I + gi/RI,,}yITRI'I"". and

R '" ,where I~~:I:' (x) is the field strength of the gauge potential A ::'" (x) " I

1<""" l'~'

=

aA ,It

11111

}I

-

a

t ~u'1.,IIII1· ,II

III g L·.(A ,11(1A·,11 If

1.'·IIIA"'~ '1'1 ,II) .

...

(] 3)

521 No.3

327

A POSSIBL E UN IFICA TlON MODEL FOR ALL BASIC FORCES

With these relations, the action SB can be simply reexpressed as S

=

B

j +

r-f

..L

l

<;'"".-11'

.~ 2["[-- b\ ~ -..L1 r " rlll:':",~ gc

4)

al -

..La -ka"'' ' +

R",)y'TR'

I''Y 'Y~ +

2

+

(a:!

..L 4

..L Y u· .Iqi- R

1(1)1

II,::, -

1;) R,,~R'

+

2 a3

\,~ L}

(\4)

RJ

which has the same form as the action of a multiplicatively renormalized unified gauge theory in2 cluding the so-called R -gravity and a renormalizable scalar matter field as well a nonminimal gravitational-scalar coupling, In the real world, there exist three generations of quarks and leptons. Each generation of the quarks and leptons has 64 real degrees of freedom. These degrees of freedom will be represented by the 64 components of a single Weyl fermion IJf + (x) belonging to the fundamental spinor representation of SO(l ,13), The action for fermions is given by SF =

Jx .r--~ -iP e~~r[iall gU~C-i2,"'e~ 1JI++ c.} +

+

h.

(\5)

where ~4B are the generators of the SO (I , d - I) in the spinor representations and given by ~4B =

~ [T.4 ,r8 ].

total action S

r

= SB

are the gamma matrices that Obey{

r ,r)

=

2 TJAB , Note that the resulting

+ SFis simple, but it is nontrival for fermionic interactions since the gauge po-

tentials Q;B (x) are related to the independent degrees of freedom A ;:"' (x) and e;' (x) by some nontrival relations given in eqs. (10) --{ 12), In particular, the supplemented frame fields e'~, (x) will have a highly nonlinear dependence on the frame fields ei~ ( x) , Now let us consider the conservation laws under the general coordinate transformations and .L.

../U ).

local rotation SOU ,d- I). Under the local rotation 1Jf+ (x) -:t- 2"" cult to find the conservation law as D,,(

~S~~B)

~T(AB'

-

~.IIIIJf+ (x)

,it is not diffi(16)

=0

with (I 7) (18)

The gcncral coordinate transformations lead to the wcll-known energy-momentum conservation law as

,r-

D ("- gT,,,)

_ ,,r -gr,,, L.'B ' S. '8

( 19)

=

with T,/Y = gil,!.. - i

-i e;: [p

U sing the l:ovariantly const'lI1t frame fields

S;;rT

=

T,,"T,

=

f

J',., D, 1JI + - (D, IJf +) J',

e;: ( x) , we

S:.~ He) e~:

can project

.

T''''B,e;;e~

IJf

J.

(20)

S';/I and T, . III, into (21 )

=

T,UT - T"",

(22)

The angular momcntum conserv'ltion law becomes

D,'( ~S;:'T) - GT'P'T,

'=0. It is then easy to show that these two conservation laws (eqs. (19) and (23»

(23)

arc esscntially the

522 328

Vol. 41

SCIENCE IN CHINA (Series A)

same as those occurring in special relativity by noticing the following relations:

T {~tTl

=

a."L '; p,TI === a.(t ( Xp T~ -

X

tT

1"'.;) .

(24)

Herc L ;;,r is t he orbital angular mo ment um tensor and J;;,r "'"S;;.r + L ;;,r rep resent s t he total angular moment um tensor. From simple ideas we have provided a unification model for strong. electromagnetic, weak and gravitational forces and constructed the action without dimensional parameters as the basis for quantum theory of all the basic forces of the elementary particles. Such a theory is conjectured to be multiplicatively renormalizable though it may remain an effective theory of a more fundamental theory. One can find a formal proof of the renormalizability of the R 2-gravity with a scalar field in ref. [19]. It was known that in the general relativity only the Einstein equations have been tested to be in good agreement with known experimental data at the cIassicallevel. Thus the general relativityof the Einstein theory may be interpreted as a classical theory in the low energy limit, so that the EinsteirrHilbert and cosmological terms may be induced as a result of the low energy limit[19.2oJ. For instance, these terms may result from a spontaneous symmetry breaking. Finally, we would like to comment on the so-called unitary problem due to the appearance of the higher derivative terms within the framework of perturbation theory. The higher derivative terms become important as the energy scale goes up to near the Planck scale, at that scale gravitational interaction becomes strong so that the treatment by perturbatively expanding the metric fields is no longer suitable. From gauge theories' points of view, the local Lorentz group is not a compact group; not all the components of the gauge fields are physical one; additional conditions have to be introduced to eliminate those unphysical components. This is similar to the case of gauge theories in which gauge conditions have been used to eliminate the umphysical modes (time and longitudinal modes for massless gauge fields). Therefore, to solve the so-called unitary problem in gravitational interactions, a nonperturbative treatment or an alternative approach has to be developed. We shall further study this problem in our future investigations. We hope that the present model has provided us a new insight for unifying all the basic forces within the framework of quantum field thcory. Though both the ideas and the resulting model are simple, more theoretical work and experimental efforts are needed to test whether they are the true choice of nature. References Einstein. fl. .. Sil=. 2

Preu"., . ..Ikuc/.

Wi.I:' .. 1915. 77H. HH4.

Einstein. fl. .. Die Grundlagen der allgemeinen RciativitRt st h<'Orie . ..1,11,.

Pin:>'. '- p=. . 1916. 49: 769.

Yang. C. N .. Mills. R. L. . ('onscrvationllf i,.lIopic spin "nd i..:>tllpic g"uge im'[,ri"nce. Ph.,,,.

4 5

6

R,"· .. 1954.96: 191. Re,· .. 1967. 159: 1251. K:,luzll. Th .. 011 the prohkm "I' unity in physics. Sil=. P,,·us.,· . .Hut!. lI"i.,,, .. 1921. KI: 9Cl11. Klein. O .. Quantent h,'!.,ri,' ullli I' ii,rdimcnsionllie Rcillti\'itRtst h,'!.Hie. 7.. Phpik. 1926. 37: N95.

('olem"n. S .. M:lIldula . J .. Ph.n·,

7

Freedm"n. D.. Fcrrllw . S.. V:lIl Nieuwcnhuizcn . I' .. I''''grcss towllrd II t h''!.HY or supcrgr:,,·it)'. Ph,.s. 3214.

X

l>escr. S.. Zumino. B . . Consistcnt supcrgn,,·ity. Ph,'s.

9

(jrccn. M .. Schll'llrtz..I.II .. OHmillnt description of superstrings. Ph.n'.

10

1.1'11,.

R('\' .. 1976. DI3 :

1'176. 62B: 335. 1.1'11 .•

I<JH-I. 1-I9B: 117.

Canddtls. P.. Ikll"u\\'ilz. G. T .. Slromil1g~r. A. c( ,11. . Vacuum cunfigunttiolls I'(H" sLlp~r~trings . .Vnd.

Phy,\" , . IlJK5.

B25H: -16. II

Gross.D .. II "n'cy,.! . . Martincc. E. etal. .lIcteruticstring. Ph...".

I::!

(jr~cll.

Re,·. Lel/ .. IlJH5.5-1:502.

M.B .. Schwartz.J.n .. Wiuen. E .. SHIH.'",""';fI,!:! rl!f.·fII:\,, Cnmhridg.c: Cnmhl'1dgc Uni\'crsily Prcss. 1')X7.

13

Witten. E.. Scarch ((II''' realistic K:liuza-Kicin th''!.H),. 'vuc/.

14

Gi:lsho",. S. L. . Parlial-symmctries of weak intcnlctiuns. 'vue/.

Ph,.s .. IlJ~l. BIK6: -112. PIli'.",. 1<)61.22: 579.

523 No.3

A POSSIBL E UNIFICA nON MODEL FOR ALL BASIC FORCES

15

Weinberg. S .. A model of leptons. Ph.. '",.

16

Salam. 1\ . . in Pro('eedings

329

ReI". Le/l .. 1967.19: 1264.

(~,. 'he Eight .Vohel Symp(A~·illm. 011 £h'I111'IIIW:\' Particle Tlu..'OI:l". Rl'Ialil'islic Groups. alld ..I'llr

~rlicil. SI'~·"'/cIII/l. SlI"edell. I96X (cd. Svartholm. N. ) . Stockholm: Almq"ist and Wikcll. 196K.

17 IK

Georgi. H .. in P"nicle., "lid Fie"l, 1974 (ed. Curl,.,n. C. ) . New York: Amcr. In~t. or Physics. 1975. Fritzsch. 1·1. . Minkowski. P.. ..J 1111. PI,.. ,,' .. 1975. 93: 193.

19

Stclle. K .. Rcnormu)izalion uf highcl-dcrivaliw quanlum gr,,,·il),. Pin,".

20

Vo ro no\" . H. L .. Tyutin. I. V ..

r"d. Fi=.

IJ.

Nllc!.

ReI".. 1977. D16: 953.

Ph..-",.). 19X4. 39: 99X.

524 Em. Phys. J. C 16, 27!J-28i (2000) Digital Object ident.ifier (001) 1O.1007/s10052000038:3

THE EUROPEAN PHYSICAL JOURNAL C

With kind permission of Springer Science & Business Media.

© c

Societ1t It.alinna di Fisica Springer-Verlag 2000

Searching for rephase-invariant CP- and CPT-violating observables in meson decays K.C. Chou l . W.F. Pa.!mer 2 •a . E.A. Pasthos:l· b , Y.L. 1

3

wuJ.:l,c

institut.e of Theoretical Physics, Chin<'se Academy of Sciences, Beijing 100080, P.R. Chinn Department of Physics, Ohio-State University, Columbus, OH 43210, USA illstitut fUr Physik, UllivCl"sitiit Durtlllllll
© Springer-Verlag 2000

Abstract. \Ve present a general model-independent and rephnse-invariant formalism that cleanly relates Cpo and CPT noninvariant observables to the fundamental parameters. Different types of CP and CPT violat.ions in t.he KO-, BO-, B~- and DO-syst.ems are explicitly defined. Thdr impmtance fOl" interpret.ing experimental measurement.s of CP and CPT violations is emphasized. In particular, we show that the time-dependent measurements allow one to extract a clean signature of CPT violation.

1 Introduction For t.he ciisr.ret.e symmetries of nature, violations have been observed for C, P and the combined CP symmetries[l5]. III fact two t.ypes of CP violA.t.ion have now been established in the K -meson system. It remains an active pwblelll of research to observe CP aSYlUmetries in heavier mesons. In addition there is new interest in investigations of pwperties of the CPT syulllletry[ti]. Up to now, there are ouly bounds on CPT-violating parametcrs[7], whieh are sensitive to the lIIagnitude of alllplitmles, but tests of lhe relaUve phases have nol yel been carried oul. In this article we present tcsts of CPT and ('P,. separately, anci discnss whir.h measurements dist.ingnish between the various symmetry breaking terms. In addition, we cil'riv(' formnlar. which arc manifl'st.ly invariant nnder rephasing of the original mesonic stat;es. The hope is t.o call at.t.cnt.ion 1.0 scveral m~asnrC'ment.s which will be accessible to experiments in the future. Our paper i~ organized as follows: III Sed. 2. we present a complet.e set of parameters charatterizing CPo '1' and CPT lIollcolls~rval.iou arisillg frolll the lllHSS IIlHt.rix, i.e., t,he so-called indirecl CP-, '1'- and CPT-violation. A set of dired CP-, T- and CPT-violat.ing parameters origillat.in/; from t.he dec.A), amplit.ndes are dpfined in Sect.. :l. In Sect. 4, \n' defined all possible independent oiJse1"m.bll's nnd r('lnl.1' 1.lw111 dirC'c·t Iy 1.0 [nncla.nlc'1l1.n.1 p>H·fHnel.c'rs whic·h a.re llUl.nifes1;]y rephasing im·a.rin.n1. a.ml can be a.pplit'cI 1.0 all meson clC'cays. Thl' VA-riOnR typC's of Cl' ilncl CPT viClla.t.ion are dnssilied. indica t.ing how 01ll' cnn ex1.md· purel." CPT 01' CP viula.t.illg dfcl:tS. 111 Sl·d ..j. Wl' illn'Ht.igak in del.nil t.he time evolntion of nwsonic c1("cnys a.nd inl rori 1.1 c:e a

severa.1 time-dependent CP- and CPT-asymmetries which allow one 1.0 meAsure separalely the inciirec:t. c.:PT- finci CP-violating observables as well as direct CPT- and CPviolating observables. In particnlar, we show how one can extract a clean signature of CPT violation from asymmetries in nentral m('son deCAYS. In Ser.t.. G, we apply the general forma.lism to the semileptonic and Ilonleptollic KllIeson decays a.nd show how lIlany repha.~ing im"ctriallt CP and CPT observables can be extracted separately. Our conclusions are presellted in the last section.

2 CP- and CPT-violating parameters in mass matrix Let 111° be the neutral meson (which can be KO or DO or EO or B~) nnd lifO its ant.iparticlC'. The cyolntion of fifO and 11/ 0 stales is ciir.1.at.ed by

"'ith flij = .H,j -irij/2 t.he mat.rix c1emen1-s. and Mi.I, r ij heing I.hC' C\isJlc'rsivC' anel. absorptive par!.s. rl'spC'ctivC'ly. 'fht, dg"l1Y!llnes of tilt' Hamilt.onin.ll a.re

(2)

witll

pnlIIWI"@lllpS.ohio-state.edll

I,

[email protected]

I-d",

l:

,V1Wll.jfi1.p.i:H,:.cll

1 I- d.~1

(/"lid

525 K.c. Chou et ill.: Searching [or r<'ph ..~e-ill\·ariilnt C:P- ilnd C:PT-"iolating obsermhles in meson decays

280

(3)

We not.e alreadv t.hat 6A1 is invarianl, maIer rephasing 01" I.he 51 al·es .\t U ;nd Af u The eigenfunctions of I:he HamilIoniAn drlinp I.he physical 51.>11·1'5. Following Bdl and SI.('inberger[8], .U o tlnd .~fu mix wit.h ench ol.her ilnd I"ol'm two llhysic-al llUl."'iS Pi~enstflt(,R MI

= p.:;I,Uu > +q",IAf >, U

+ Iqsl2

wit.h norma.lization Ipsl2 r.oeIkipnls are given hy

qs

q 1 + LlAl

_

1-

p", = ), 1 - LlAJ = 1

AJ2 = PLIM u > -(ILIAfu >

fS

+ '.:;

!!. _ {Ji;; = 1 - tAl P - VIi;; - 1 + EM

=

qL l'L

IPLI 2 +

(4) IqLI 2 = 1. The

= 9..

1 - LlAJ == 1 - f/, 1'l+LlAJ I+FL

(6)

a:nd is given simply hy -t

,,-iIl 111-II ;

If:";; alld CPT-odd lI~j} paTls. i.e' .. 1+)

1l"fJ=Hr.r.r+

1/1-)

(9)

'If

with (10)

Let f denote the final state of the decay and J its charge ronjngat.e st.AI!', The cle'Cfl), amplitudes of Jl.lo Are defined as

9

== < flHeJJ!lIl o >=

~( ~ Ai

i6, + B) i e

T.. == < IIHofflM u >= L(Ci + D,)"i6,

== L(IC;deiq,~' + lJ)ileiq,~')ej6,

( 11)

with .II; 'tnd C; heinp; CPT-r.onservinp; Amplitudes

(7)

',1/-..:.IA/

= '1-EA/.6

CI'T-en~n

< fIH;;}llIlO >== LAiei6, , < JIH~;)IJIlO >==

1

We discuss next severa.l properties related to the symmetries of the sysl,em. The parameters 6Al And Iq/pl are rephasing inva.riant and so are also other parameters defined in trrms of t.hmu. CPT invarimwe reqnin's 11111 = AJ22 and r l1 = r22 , and implies that 6u = O. Thus the difference het.ween 1/8/r,S and (lI-h'l, represent.s a signal of CPT violation. In other words, .6Al different. from zero indica.tes CPT violat.ion. CP invariance requires the dispersive and absorpth'e parts of HI2 and H21 t.o be, respediveiy, "'1ual allli implies q/p = 1. Also if 'I' invariance holds, then independently of CPT synllnetry, the di~persive alld "h~orpth'e pArl.s of 1112 AUe! H21 mnsl, h(' f''1nnlnp 1,0 A 1.01.al relative common pha.se. implying Iq/pl = 1. Therel"orc a Ref,1/ dil~ fe'rcnl: from Z!'ro e!rs<:rih!'s CP a.ne! T non('onsrlT,).I.ion ancl ca.n be prE'sent. even when CPT is conservecl. Finally, two param!'I.e'rs. (M drsnihing CP violAtion wit.h T non<:onserml:ion and d,1/ chn.mderizil1g CPT violn.1 iOIl wit.h CP 1I011Collst'l"vat.ioI1. a.rc rela.ted t.u (s ClUU (L ,,"ia (8

Let. lief I be th., efi'cdiw Hmnil1.onian which contains

(5)

We' have also introduced the parmutcrs fS,l"Al following [9J. In the CPT conserving case they reduce to the known parameter (AI. Thus we have a complete' des(:ription of the physical states in terms of the mass matrix, and the time evolution i~ deterlllined by the cigellvalue~:

II-h

3 CP- and CPT-violating parameters in decay amplitudes

M

:

(8)

and I'Pcillc'p j.n t.hose..> giv(,11 in [nJ Whf'll llrglt'c'l illg 1 h(' qllrldra.t.i('. I.!mn (,I/LlA/. This is 1). cOlI\plel.!' St'l "f pnTallwl.ers clC.!>f;crihing CPo T (lnd CPT UOl1CCHlSPl"\'ntioll wilil,lt orig;innl,('s ill 1.lw IIlnSS Ilintrix (indirecl). III Ihe lIexl secl ion We' discllss addit jOlln] pnralllC"1 f'rs orig-inatillg; 111 1.111' dl'{"n~' a.mplit.udes (dircn) ns well a.s frolll I'he mixillg b('l.weell IIHlS~ lIIal.rix alld decay alliplihl
L C;e iO ' ,

(12)

And Hi And Vi heing CPT-violAting amplitnnes

< fIH~f)llIlO >== LBi ei6 , , < JIH~f)IJIlO >== LD;e iO •. , (13) Here we have used the notation of [lOJ for J;he amplitude g, and have introduced a new alliplitude h.. The secolld amplit.ude is absent. when one considers only h.-meson de(:IIYS a.nd neglect.s possible viola.tion of d8 = LlQ rule a.s was the CAse in [10J. This is heranse l,h!'. K-m,,~on dl'cAYs obey .68 = .6Q rule via. wea.k interactIons 01 the stan(hu'rl model. The' reason is simple since the' st.rnngr ql1ark can only decay t.o the lip qllark. In the case ~f B.-, B:,and D-nH'son sysl,rms both amplitl1(irs .1/ And II. eXIst. vIa I.he lI'-bosoll e~ehange of weak interactions since both bquark and (~'lnark will haw two dilt'crent trall~it.iol~~ duc \.0 CKI\! '1n;1.rk \Ilixings, i.e., b -t c, 'II. and c -t .s. d (lor explicit d,'en)" nlo
CPIAl"

ndnpt. 1.11(~ phase' {,OIl\T'llt iOll

>= lAIn> ,

( 1·1)

II. is I h"l1 nol dilfil'nlt. 1.0 show I h"l II'IP
H1nJllil.lld('~

of the charge {'olljngalt' 1I1(~SUll J/o hn\'c the f(,lIo,,"ing 1'01'111

-rI >= .Ii == < flHrJIIJI

~(I' ~ .' i -lJ") , (. ;0.

526 ICC. Choll et. al.: Searching for rephase-ill\'ariant. CP- and CPT-violating obsE'n'ablps in meson decays

281

·/'inlalinn.•. Note th~.t. the iA.tter C·.AoSe is more diffic:lIlt t,o est.ablish experimentally since it. requires t.he obsel'Yat,ion of n. rela.tive phas(' he!.w('cll t.wo amplit.nd('s rlist.inp;nislwo "'il,h I.he help of specific quantulll numbers. This was t.he CI:lSt' with the ('If PC.ll'Hllld.el' ill I'-lI1C~OIl decay!;.

In atlAlogy 1.0 Ihe indirp('1 CP- And CP'l'-violaJing parameters ~S.L.M from mass nlat.rix, we define now paramrtrrs (·ont.aiiling oirect. CP ann CPT violAtions I

_

l-h/9

=f

_

I-ii/T..

cM = 1+11/9' cM = 1+,ij/h' II

cM

_

1-9/2

= 1+9/9'

:11

_

=M

=

I-It/I.

_

For final states which are CP conjugate, i.e., If >= CPlf> = If >, the relations h == Ii and h. = .'I hold. and thus the four parameters a.re reduced to two independent ones: c~\1 = c~{1 and ~J = ~.l· The symmetry properties of the amplitudes are as follows. If CP is conserved, independently of CPT symmetry, one has gig = 1 ano h/h:: 1, whic:h implips

= -B;,

¢f

=

in other words:

¢f = 11'/2,

nalllely, Ai and C. are real, while nj ~.nd D j are illlaginary. Similarly T invariance exchanges the iuitial and final states ami implies, independently of CPT symllletry, Hi = Hi,

The E-type parameters defined in (5) and (16) nm not be related to physical ohscrvahlcs sincc thcy are not rcpha.sing invariant. Let us int.roduce CP- and CPT-violating ohsl'rvahles hy considerillg the r~.tio,

(16)

I+h/h

Bj

4 Rephase invariant CPand CPT-violating observables

nj

=

nj

01'

namely, all the amplitudes must be real. Finally, t.Ouserva.tion of CPT requires Bi :: 0 and D, = O. We slllllllutri:w the result.s for the amplitUdes in Table 1. Reading a.cross the tirst row of th", t.able we have the C'Onoil.ions ror C'P ('onsprvat.ion. wit.h T C'OnsPl'\'AJion (firsl, column) and without T-conservation (second column). Th ... relat.ions B j = and D j = -Dr imply T-violation in the presence of CP conservat.ion. The second row of t.he t.ahlr giv('s t h(' ('onoit.ions wh('n T is C'OnsI'rv('(1. wit.h CP ('onservation (First column) 01' VI,it.hout CP ('onservnl:ion (H('COlld COllllIlU). This is a cOIuplctc set of amplitude wit.h t.he C j mlo Di mnplit.l.1cles int.roclu('ed for t.h .. first. time here. As a (,(lIIseqllt'Uce, two luore CP- alld CPT-viola.t.iug parame!.ers E: AI and E.IJ in (16) are needed. III Sllllllllal'Y 01' t.his seet.ioll, we Ita.n' tlte I'ollowillg COIIdllsions. Val lin IIlI' Rp£~1 rl1ld Rpf~{1 d([Tpf'Pllt. fmm Z('I'O dp",c1'ib" CP 1/.onCO'/l.~e7-('llt·i.on independently of T (/nd CPT 8ymmdrip.<. Til" 1'7'F:8eIU'f: nI n;8 (L7/.r/ D:8 iur/ir:a./" .
-n;

Zr.ro <{I and ?:{, '/IIi/.h nnn.z(~m 1111<\1 and JUI~\I iTllpl-i(~,'i T 1I.(mcnn.~enO(!tion. Final/y, ze7'O nnd D,. (wd wm,plc.t II. ".11(/ C, "iY7lul CPT cCJIl .."n:w.;cl7/ willi. CI' lind T

1)J

qs

==

qT,

Tr

< fIHe"I!lf2 > qs PL 1 < fll/,,,IA!I > = cn 1's 1 + rJ

(17)

which enters to the t.ime evolution of the decay amplitudes (see 27 and 28). The parallleters q!;,L and P!;.L were defined in Sect. 2, and we also introduce the notation

1"7 =

(c}s/l's)(h/!I)

7'1.

with a similar definition for Note that the factor qS/qL is necessary for t.lte nOl'1l1aliza.t.ion and also repha.se inYarian!:e, whic:h hA.' not, been alWAYS inc:lnded in I,he literat,ure. In the CPT-conserving case [11] this factor is equal to nnit.y. One ran simply see from the definitions in (:i)-(5) that ijJ is rephasing invariant. The factor qSPL/PSqL = (1 + .t:.Al)2 / (1 - .t:.M)2 is rcphA.,e-invariant sinc.e .t:.Al hAl'; S this property. The rat.ios = (qL.S/PL,S)(h/g) are a.lso rephase-invariant. 'lb see that, let us_make a phase ,!edefiuitiou 11I1 u >~ c:i.plllJ u >, thcn IlIJ u >~ r,-'"'IMu >, HI2 ~ e- 2i "'H I2 and H21 ~ e2i .pH21 , as well as h ~ c,-i.ph aud C) ~ c"
T'Y

1';'S

_ 'IJ

1

== - - 1-

[

+ i .i'L,.<+,,] + a,.< + it" _

T/,j

+a

2 + G.t ...·(lt'

7/.:l

t .•H

,

(18)

\\'here we have nsed the definit.ious (I,s

I = 1+

(1".

=

I*~Y

1.!L::.12 1],0..

2.J.\/

1

_

•

(I, -

- 1-

fLd a,aLl'

1 - 1~12 2Req, _ a., + (/...l I + 1~12 = I + hl~ - ~

'/ILl .'" I

n;

2l?c,~!;

= 1 + If.~12

nil

+

• I I(J Ll

+ .J~I

Till' illgt'hril is desl'ri bed

V

1-

1ill

II II

a]

2 (I.d -

(Hl) ,.)

lI.,;i

527 282

K.C. Chou et. al.: Searching for rephase-im·!triant CP- and CPT-violating obsermbl0s in meson decays

Table 1. CPT-conservation CP-conservation

.eI; '"

T-culIscrvatiUli

A = j

CPT-Violation

Ai c, '" C; Ai r t = (.';"

B; '" -B; f1i

Di '" -Di

= n;

J)j

=

n;

imply'l'-violat.ion CP-,-iulatioll

illlpl~·

CP &; T conserva t.ion

with 1 -lq/pI2

2RefAl

1 + Iq/pl2

(20)

1+ IfAll2 '

2R(-il Al

I

(1.L1 '" 1 + lil M l2'

/1....1 '"

•

2111/.ilAI 1 + lil M I2

'l}j ~

a" •

= 1I +

19/91 19/91 2

=

2Rec~I 1+

IcXd 2

= a." + aELI + a~L1 1 + a~L1 + aLlLl

(21)

where t.he definitionR for a." , aELI. a~L1' a~L1 find aLlLl are presented in the appendix. Analogously, one has

a.nd 1 -lh/hI2 1+

Ih./hI 2

2Ref~I

1+

+ ai',6 + a~,6 (23) 1 + (J.~,6 + (1.,6,6

_ a,"

lEAl 12 -

2'1 [a. + a.' + aLI + a.L1 + aLlLl I

+i(fl c +<' + fI~ + (J.,+<:'", + (1<+-:)]

The dcfinitions of (I.,', (1.,.'+(, and ii.(.«, arc giVCll ill thc appendix. The reader should note that quantities without a hat contain either ollly CP 01' only CPT 1l0nCOlltierVing eITecis, and with a hat contain both CP- and CPTnon conserving effects. AR a" a.,. a.+., ilnd a«, (for t.heir defillit.iol1.~ ReI' appendix) are all rephase-invariant, so are also a. d.' and ii.,,,,,. Note that only three of them are independ~nt. Rim:e (1 - a~)(1 - a~,) = a~+., + {I + a«,)2. Another rephaseinvariant dired CP and CPT 110ninvariRnt obRervRhle is defined as 2

so that olle could ollly kCl'p thl' lillear t.crms of thc I'l'phasl' invariant CPT- and CP-violat.ing observables. With this considera.t.ion, the observable ~f is simplified

wit.h LX, = Dole•. One of the int.eresting cases oc(,urs when the final states are CP eigellst.ates, i.e., f CP = f, ~IIU ill this CH.~l' h = 9 (01' C = A and V = H). AR A ('.onReqlll'n('e, WI' find

(25)

where the definitions for all the rephasE' invaria.nt quantit.ieR are l1;iven in the appendix. ThoRe with index il Rre the CPT-violating observables, the others are CP-violating oncs which havc boon diRC'llsRed in [11]. The formalism so far involves many equations which include CP and CPT violatioll eH'ccts either separately or mixed t.ogether. It has several advantages in comparison with othcr articlcs[12, 10]: 1. Thc formalism iR more general than the oneR reportcd

in the literature and (:an be applied not only to the Kmeson decays but also all othcr heavicr mcson dccays. 2. All observables are manifestly rephasing invariant and well defined by directly relating to the !ladronic mixing matrix elements a.nd decay amplitudes of mesons. 3. All possible indepelldellt observables are c1a.'lsified, whi('h enableR one 1.0 separat.ely measnre diITerenl. l.ypeR of CPT- and CP-violating observables a.nd to ext.ract pnrely CPT or CP violation effectR. 4. The formalism is more elegantly designed for ext.racting variollR rephaRc invariant CPT- and CP- violating observables from time-dependent measurements of meson decays, which will bl' discusscd in dcta.il ill thc Hext section. We have thus defined all possible rephase-invariant CP and CPT noninvarifllJt obsc>rvab!C'R in t.cnnR of eight. parameters related to CP and CPT breaking quantities arising cithcr frolll lllixing or phascs of amplitude. Thc cight paramet.ers (I.·re classified as follows: (AI is 0.1\ indirect. CPviolating paralllet.er anu "l.1I t.he inuired CPT-violat.illg paramet.ers; the pa.rameters
<\I

whf'rf' thC' ('xplidt. dc,nnit.iolls 1'01" H.n~ again givPll in t.he apPf'IHiix. 'I()

ep

see explicit.ly how

1111111)"

tl. f + f ,. 1.I.,+r' .;l

and 1I,.+r' ..l...l

"E'phnse in\'arinnt. CPT and

ohs(lrva.hh~s llllly h(' s('pal'fl.t.C"I,v lll(,fl.SnJ'Pc] frOln f'xpf'ri-

mellt;s, let us consider t.he case for whieh t.he filial st:at·t's are CP dgl'll!";tatt':i aud

SllPPO/';l'

that the violatiolls iln' slllall

1. pll\'I'i:v indiJ'C'ct CP and CPT vinln.l.inl1s whkh arC' ),(ivC'1I by thl' rephase-inmri>lllt CP-,·iolAting observable (1.( muJ CPl'-violating UiJSCITablt.,!':i (l.Ll and 1I~.

528 K.C. Ch01l et a!.: Searching for rCJlhase-inyariant CP- and CPT-dolating observables in meson d,'cays

283

2. pHr(>I~' direc.\. CP and CPT viol'11.ions whic'h art' eharac-

terized by t.he rephase-illvariallt CP-violatillg observf\.hk~ (J.F." and f.I~" and CPT-violating ()hH(~rvn.hl('s tl.!Ll(I.~,.l' (l~~, a~~, a,.1, a~j' ( l j j ami a::;j. 3. I'dlxed-nld\lcl,d CP "ud CPT yiulatiuuH \yhid, arc dl'snibed by CP-violal.ing obsC\"yables (/,+,' and (I.,+(' aud CPT-yiulatillg oL::icrvables (J1+r~~ "(+(~..l~ ll(-I-[~ Cl-ud lll+(~...l'

For th(' ease t.hat. t.he final st.a.t.es flT(' CP eig('nst.at('s. 011<' has a'f.1 = a.f." = a'f' = af Thus, in this case tiE' a.nd a.'E.' also iuuicate purely direct CP aud CPT violatiouH. Wheu the final stat.es are not CP eigenstates, a,' and do not, iu geueral, provide a dear signal of direct. CP violation although they contain direct CP and CPT violations. Their deviatioll from the values a" = ± 1, 0 a.lld = =F1, U ca.lI arise from different CKM angles, final state inl.eractions, or different ha.dronic form factors, but not necessarily from CP and CPT violal.ions. ll

•

a.,

a"

and

r(1I/0(1) --7 J) oc 1.A(t)12 = (191 2 + Ih12) 2 +

a,,,a., + a,.,.' 1 +tLr,o.:

xe- rt { [1 + a•.,. a., + (a,,, + a,' ) Re"1/L!. + a,,,+.,fm"1/Ll 2 + (l.l ....·U('1

+ jj,(s['

-Rer/,l -1"1/..112] cosh(Llrt)

5 Extraction of CPand CPT-violating observables In order 1.0 meMHre I.he rephR.~e-invl\.Iiant, ohservRhles defined a.bove, we consider the proper time evolution [1 3, 141 of the neutral mE'sons 2

IMO(t) > = L~ir:-i(m'-ir'/2)'IMi > 2

11IJO(t) >

=

L(ie-i(m,-ir,/2J'llIIi >

(26)

i=l

wit.h ~I = IJ,)(IJSI'I.+IJI.PS) ami (2 = '18/('181'/.+'/1.1'8) for a pur:,e 111 0 state at t = 0 a.s well as (I = ]JL/(qS]JL + q[,]Js) and {~ = -Ps/('1S1'L +'1L1's) for a pure lI~fo state at t = O. Thus t.he de<:a.y amplitudes of 111° and lI-fo at the time t will be given by

It. follows now that the t.ime-d!'pend('nt. d('cay rn.\".es are

1'(1110(1) --7 f)ocIA(tW=(I!lI~ + 1"12)2! (I,,,,i,, 1;',<" x(~

._f"/ {

- RJ:r/..1

1+0.,,, [1 + (I,,,,i,, + (fI, .< + ,i.,.' ) a,· 'I", + ,i, ,d.,' ["III·'/L!. 2 + Oe,.';-,(I + a.(.<;(,

+ 11/..112]

l"Osh(L1 ri)

where Llr = r 2 - r l and Llm = 1n2 - mI· Here we ha.ve omit,t,en the int.egrRJs from the phase SPfL('.e. Similarly, one can easily write down the decay rates r(1II0(t) --7 J) and r(iii\t) --7 .fl, and t.hen t.he t.imp.-dependent. CP and CPT asymmet.ries are defined by the difference between t.wo decay ratl'S. In addition, in studics of thc time dependen<:e one can isolat.e each of fOllr-terms. One can introduce scveral asymllletries fr01l1 the dccay rates r(1I1°(/) --7 J), relit(t) --7 J) , r(llI°(t) --7 J) and r(lIl(t) --7 J) . Obviously, the tillle dependences contains a. lot of illfol"lllaLion. Therefore studies of time evolution can eliminate Lhe va.rious components (ha.mollics) in cos(dmt), sin(dmt), C"Osh(Llrt) and sinh(Llrt). We now pron'E'd 1.0 apply thE' above geneml analysis to specific processes. As in t.he [111, we may dassify t.h" processes int.o fonr scenaTios: i) 111° --7 .f (MO I> ll, MO --7 J (lIl I> f) , this is the case whell f and J are not. a com ilion final stal.e of lifO and 7ft. Exa.mples a.re: 111° --7 111'- lv, ffO --7 11/'+ Iv; RO --7 n- n~, n- J(+, rr- n~. rr- K+. R°--7 V+/):;.lJ+/\·-, rr+U.:;. rr+/\-: H~ --7 LJ.;·rr', J)~-U+. K-rr+ • J(- IJ+, 13~ --7 IJ;;rr-. D~ IJ-. J(+ rr-, J(+ IJ- . This scenario a Iso applies to ("harged IIleSOIl decays. ii) lifO --7 U = l, j"f'P = J) ;- ill. t.his is I.h(' decc\\' t.o a. ("0111111011 final state \vhi('h is CP eigenstate. SlIch as /30(1]0). IJo(i)O). /\'0(1,0) --7 rr'rr··-. rrorr o..... For I.he filial slat·('s s\lch as rr-p+ ami rr+,,- • althongh ",wh of 1"11<'111 is nol. n. CP C'ig('nR\".I1.1.P of lJ O( jjO) or DO(jjO), one can always decompose l.hE'lIl into CP cigellsl.a.t.es as (rrp)±'" (rr-/'+ J, rrl'p-) with Cl'(7rI')J. = l(rr/')±, This

529 284

K.C. Chou et al.: Searching for rephase-invariant CP- anrl CPT-violating obserntbles in meson decays

rE'col1st.rm·.t.ion is menningfnl sineI' 1[- p+ nnel 1[+ p'" h,wE' the same weak phase as they contain the same qua.rk (:011t·~nt.

iii) MO -7 (f, I It I CP ) t-- .~/. i.e., t.he fina.l sl.ates ")"(' ~UllllIlUU fiual ~t.al·t'~ _[JIlt. are lIot. c!targt:. t·oujl.lgall' slnll's. F~r example, BO(BO) -+ l\·s.J/t", R~(B.~) -7 J\.<;dJ "m[ DU(DU) -7 Ks1[u, KsIP. iv) J\Io -7 (f & J, I CP of. J) +- At , i.e .. bot.h I alld J are the (;()IIIl1IUII final st.ates of A/ u alld At. but they are not CP eigenstates. This is the most general ~:ase. For ~xa.mple, R°(i'l°) -70-1[+, rr-n+; n-p+, p-n+; B~(B2) -7 D; [(+, [(- D;; D°(ijO) -7 [(-1[+, [(+1[-. In this paper, we will only elaborate on the first two s(:ennrios. In s(:enm'io i), t.he nmplitneles Ii. anel J,. nre zero, thus tI f l = -iii' = 1: Q,c.+e' = 0 = a.E+E" a.nd liEf' = -1 = fl.",. For this case, the til1lL~depeudeut rates of (20) aud (30) will become very simple, r(AI°(t)

-7

J) ex [A(t)[2 = [g[2 e-rt . {(l

+ [11..l[2) + (1 -

x cosh ilrt - 2R«"/Ll siuh ilrt X cos ilmt + Im'lLl sin ilmt} r(Jflo(t)

-7

[1/Ll[2)

J) ex [.A(tW = Igl2e-rt . {(l + [11.112) x co.~h ilrt + 2Rel)Ll sinh ilrt x cos ilmt -lml1Ll sin ilmt}

+ (1

-[l1Ll[2)

(31)

It is uot difficult to show that the other two tilllL~depeu dent decay rates which are not allowed at t = 0, can happen at a latcr t, becausc thc Alu dcvelops an A~lo component through mixing. They can be simply expressed as r(J\IU{t)

-7

I) ex

g2

~ Igl2 (1 _

n,,,) (1 -

a,,,) (1 - a-;t

1 + a,,,

2

1 - all

Tlwir ('XflC"t. I'xprf'ssions ("an h" fOllnd in t.hE' appl'ndix. From t.he time-dependent. mf'asurement.s of the above a'~~'mnwtri('s, one' shall h~ ahk to ('xl.m(t. fill obsNvnhks: dill . .;:,r, a.,, a.d, ad and u,,,. FnJlIl t.he above ClSYIIlIJIl't.l'ies. Wl' ca:sily arrive at t.hL' following import.nnt. ObSN\"n.t.iolls: 1. .1 .• /01/..f/ n.. I.he (~:qlP.r;'TTI (m/ (1./ m.(~(I..'!l.rr.1I/.r.n/.....11.01/1 th.at the a.symmetry ACP+CPT(t) 'is not a constant Clnd de1)end., (In time, it pmvidc8 It duun siynul;urc ul indirect CPT "IIiolation Irom mixing.s. 2. For the selllileptonk decays A/u -7 ,~/ -Iv and also for the decay modes in which t.he final state interactions a.re absent, one has a," = 0: a~Ll = 0, a.~Ll = 0 And 0.,,, = a,Ll/(1 + aLlLl), t.hllS nonzero ii,,, will represent direct CPT violation from amplitudes. For this f:flSe, We' come to a strong ronclllsion that once th.e asymmetry ACP+CPT{t) is not zero, then CPT must be ?liolated. 3. By combining measurement of the above asymmetries frolll sClllilcptonic aud uOllleptonic dccays, it allows one, in principle, to sepa.rately measure the indirect CP-violatiug observable a, alld the direct CP-violat.ing observable a." as well as the indirect CPT-violating observables all and a~, a.nd t.he direct CPT-viola.ting observable a,Ll. We 1I0W discuss scellario ii) in which thus af.' = af,"

1-

a",

1-

a1

(,12)

With these four decay rates, we can define three asymllletrie~ which have the following silllple forlll~ wheu Ueglecl;ing t.he quadratic and high order t.erms of the CP find CPT violating para.meters (i.e., a~, a~, a,a~)

.1 C P+CPT{t) = r(JlIO(t)

-7

.f) -

r(JlIO{t) -7 f)

'I'

~ (l.~.11

h = 9 a.lId h = g,

al" ann af,+f' = af,+l'. When

ACP+cPT{t) ~ -(a,

+ aLl)+e-Llrt[{a, + all + ii,,) (36) + (a~ + a,+,,) sill{ilmt)]

2 [-12 + 0.,,,) ( 1 + a.", ) (1 - a-;) 2 (The exact expression is given in the appendix.) I) x 2...2....!L{1 From the above time-dependent. evolution ACP+CPT(t) 2

+ {/.cd +

• (.'P+("PJ"

=

x cos(ilmt)

.(:-I·t(r:osli.ilrl. - cos il"lll.I.)

CO' {/.,"

al'

neglecting the qua.dratic and high order terms and using t.he relat.ions anel definit.ions for t.hl' rephase-invariflllt. observables, the time-dependent asymmetry is simply given hy

·e-rt(coshilrt - cos ilmt) r(AiO{I) -7

=

-all sill!t ilrt

rcxt'cl) -7]')

+ r(Al{l) -7 J)

+ o~ sill ilmt

6 CP and CPT violation in K-meson system

cn.~h.Jrl + co.~.1m.I.

( ) _ r(JlIu(t) -7 f) - r(AloU) -711 I -

+ "~.:J.

(33)

0

r{AI (I) -+ .f) + 2(1.,

+ r(Al°{l)

_

-7

ouc i~ ablc to cxtract t.!trce physica.l tlUautities: onc of thcl/l is the direct CP and CPT Iloninvariant observable 0." and t.he other t.wo are t.he combinatious of CP and CPT 1I01lillvariant observables (a, + all) and (lI~ + ii,+,,). Combining these ll1ea>lurelllents with scella.rio (i), ill whieh t.he illtlirE'C"i. CP nnd CPT noninvarin.nl. ohsl'rvnhll's a" all a.nd a~ are expected to be determined, one will be able 1·0 extract t.hE' mixed-indllced CP And CPT noninva.riAnt. observahle 0,+<" Thus, st.udies of scenarios (i) a.nd (ii) allow liS to s~pm·at.~ t.he t.hr~~ t.yp~s of CP and CPT violat.ions.

(:l4)

f)

..1~:I'I(,l'"r(t) = F(M°{l) -+ f) - r(lIJ'l(t) -7 f) (:l5) r(AlU{I) -7 f) + r(!\/{l) -7 .f) ~ ('os .Jud - 1/., cosh ilrt - (1'11 sinh ilrl. + (J~ sin L11111 - cosh ilrt - a, COS .Jml - ad sinh L1rt + (J.~ sill ilm.f.

The lill"lllalislll all(1 Hlmlysc~ prCSl'lIt('tl above arc gt'llcral and "'til hI' used for fill nellt.ml meson syst.ems. As n sp~~ dhc applkat.ioll, wt' art' guillg to cOllsider tilt' K-IUCSOIl SYSI.Plll. Fro1ll se111ileptoniC" deC'nys or 1\.(1 -+ rr- + /4- + III HlIel gu -+ rr+ + /- + VI, from (:l:l) ~l1d (3~). Ihe till1cd<'pf'ndf'nt. lIH'MlIrf'lllC'lIl·s or t.h~ a._.'·lIlll1pl.ri('s ka.d 1.0 J\.

A(';;+CPT(t) =

r(f(U{I)-+rr-/+III) -

r(Rou) U

-77r+[-v,J

1'(I(O(t) "'>rr-fI"l/,) -I- l'{K (t) .. ) 1[+i,·,7!l

530 ICC. Chou et. al.: Searching for rrphil.se-invariallt CP- and CPT-violating obseryables in meSOll d.-cnys

~ - 1/..,:,

'I",

+

-uLlsillhLlJ'1. -I- U~SillLl·'"KI.

coshLlrt + co.sdmJ(t

(:n)

"

( )._ -~~"""''--'----'''':'':'--'--'--'------'r(l?0(t)--'>7I'-I+vl) - J'(K()(t)-'>7I'+/-VI)

ACP+(.'PT /.....

=::

reg ad

(1)--'>71'-/+'1,)

+ 1'(1\'o(/)--,>;r+/-,-;,)

+ 2".(

2 amplit.ndps. The ",lml' deeomposit.ion holds for B( + -) and B(OO) amplihldes 2 . Considering t.he fad t.ha.t w = 1.121/lil n l =:: 1/22 « 1 (hw t.o t.h!' LJ.! = 1/2 nIle, w(' obtain

iI~." )

(38)

whC'r!' t.IlC' din'(t CP-YioIRt.ing pmmnl'1'/'r II," iR (,X],)(,/'.1 I'el t.o be small as t.he final st.ate int.emetions are ele!'.t.romagnetic. It is then clt-ar t.hat lion-zero a~:q1ml!ctl7l A~~P+CPT(t) ·i•• a clean signature of CPT violation. Its t.ime evolution allows us to extract uirect CPT-violatillg ouservahle a c .:, and indirect CPT-violating observables aLl and a~. The

285

=:: (I,'

+ lI.~Ll + OEL! + (J~Ll'

(i~?O) =:: -2a" - :la.:.Ll - :lil,Ll

+ (f~.:,.

(4~)

and -l+-)

a"E+f' ~

0 0'(+(.1

0 + ae+f~...l + (J.e+<:l + a(+,~~ + (l.e+t:'

combination of the two asymmetries A~~t~C:I-'1'(l) and A~~J'!+cP1'(L) fnrl.her helps lIS 1.0 ext.rad. indired. CP-violating observable a,. In t.he nonlept.onic decays wit.h finRI states being CP eigenstates, the asymmetry ACP+CPT(t) is given ill t.erms of t.he obsC'rvablC's Ii." and iI.,+" which concC'rn bot.h CP and CPT viola.t.ions. In general, it is hard to clearly separatc CP violation frolll CPT violatiull ill the uc<:ay mllplitudes, but it would be of interest to look for possibilities of establishing CPT violatioll arisillg frolll the ue<:ay amplitudes. For the !\:-meson system, there are two unique ue<:ay 1Il0des KU(KU) --'> 71'+71'- a.nd 71'u7I'u which are related via isospin symmet.ry. Their t.imp.-dependent. asymmet.ries are given by

A~~;:;'~'r(L) =::

-(a,

x C'Os(Ll""'J(t)

+ (".~ + ii.:!~») sin(Ll1ll.Kt)]

(r."r.") () A GP+CPT t =:: -

x cus(Llml(t)

+ aLl) + e-Llr'[(a, + a..:, + ii~:-»)

(

.

(39)

_(00») 11., +(J.Ll +" -.<11'1[( II., +1/..:, +1/.,' )

+ (a~ + a:~oJ,)sin(LlmKt)].

(40)

It is seen t.hat sillce the indirect CP-violating observable a, anu illuirect CPT-viulating observaules aLl allu a~ <:all ue extracted from asymmetries in the semileptonic: decays, we then call extra.ct. the uired CP- M.nu CPT-viula.ting observables li.~;-) anel ri.:?O) f\.~ well f\.~ mixC'el-indllCwl CPand CPT-violating observables ii;!;') and a;~U!,. We now uis(;nss how t.o cxtrad purc CPT ur CP violation clIeds by using isospin symmetry. Wheu uegiectillg high order tenlls, we have

ac+ c, ~ ae:+t.' + a.t:+t~..l + a(+E~ (1\1) Not.e t.hat. their dependcl'lC'e on t.he final st.at·es arc underst.ood. Using t.he isuspin s)'lIlllll'1.ry, we lind Al+-) =

.. I (00) =

where we have neglected quadratic terms of", = IA2/Aol. Notc that t.he auoV(' results hulu for allY chuice of phasc conventions. It. is thcn obvious that u (/.e..l

=

2.(+_) :.iU(I

+

1_(00)

(46)

'311c'

ii.::-)

Goo + y;,fi o. , V;, {{ -uo

:3

-

1[23 -U.o

a;?O)

which shows that. once t.he asymmetries and nTp. ml'll.~nrecl, I.hl'ir f'omhination giVl'll abovE' will allow one to extmd a clean signature of CPT violation arising from t.he ueca)' ,unplit ndes. \Vllt're thc ""lncs of ii.~;-) ami ri.;?O) C'a.n be simply cxt.meted from t.he a.symmet.ry .·lcP+cPT(f.) a.t t = 0 in (:3B). It. is nut.i<:ed that WhCll ILlul « 1. i.e., 1"?+'~...I1 « la~+,,1 (whilc d 2 <:ould remain at the ordt'r or one). oue hilS

(-17) (42)

wit.h il l + .. ) alld .1 l0U ) 1'11(' Hmplit.llrks lin' IIJ(' d<'c-a.\' Hlo(ks 1\'°(J~'O) --'> 71'+71'- lUal I\O(F:O) --'> 71'071'0 resped.i\'(~ly. whert' II{) and (/.~ (·olTeS])o'H.\ t.o I hI' isospin! 0 ",,,I ! ~

whkh indic-nl.('R t.hnt. h.\· lIl('n.'1u·illg it.~!;') n.nd ri~~~), on<' milY ext mt't t.h.' direc-t.-illdil'ed. lIlixl'd-induced CP \'iolil.!.iUII. 2 ~ot.e that llol"maJizat.ion of i\(UU) is smaller l.lli.lll t.iu.! usual U\\(' m:.\\\.\1'\lIg 11\ IH,4..~\'i.\t.\\rl·.

by a faet.or

12

531 286

K.C. Chou e\. nl.: S~arching for rephase-im·arinnt. C;P- and CPT-violating obselTables in meson de,·ay"

7 Conclusions

Appendix

In Slll11mAT~'. WI' IlOW!' (I<'V('\OI10(\ 1"11<' p;{']1{'rHI 1TI0cld-incl!'pendent nnd replJase-invarinnt. t"ormalism [or I.esl.ing CPalld Cf'T-lIouiuvnriallt. obs"ly,,]'ks ill UI<'",'II d<,,,,,ys. The rornmlisrn pres(,lIt.ed ill prcviollS art-ides [or CPT is bascd Oll the dellsity l11atrix avproach[lG]. III our Hl't.irIe. we present a complete I.imc-dependenl. and rephase-in\'ariant fonlllllatilm ill tenlls or amplitudes. Tht' rephH.Sl' invariance o[ all CP and CPT noninvarianL observables is mainta.ined throughout the calcula.tion. All possible independent. obsprvahles have been classified syst.emat.icall.v. which is more general and complete than the published results and can he uSNI for all meson dcc.ays. This ('nnhlc.s one to separately measure different types of CPT- and CPviolatiug observables aud to lIeatly distinguish effects of CPT from CP violation. The formalism which involves lllany allll elaborate ddinitiolls is directly related to fUllllamental parameters and call pl'Ove advantageous in establishing CPT-viola.ting para.lneters from tillle-depelldellt measuremenLs o[ meson decays. Several Lime-dependenL CPT- and CP- asymmetries have been introduced, which led t.o some int.erest.ing observat.ions:

Here we wllc('l· sOllie usefnl fOl'l1mli.

II.jyl'

J-

211";;:\1

+ I"~/I' ' -.lim(qsh/psg) 11,.,._." = (I + Ills/fJsl')(1 + 1"/091') 1

+ I"/yl'

= __1 __ 1-

Uta...!

I

[ii.+., }I - a~ - aj - 0.:"(1 + fl.",)]

4Re.(q.~h/Jl"g)

fl.,,,,, =

(1

+ IQs/psI2)(1 + Ih/gI2) J

= 1_

Ucad

+(V1

[_. / nu:, y 1 -

2

_ 1 '2

n.od - a.;:l

,+ a..lO-t:+t:'

-IL~ -1L'1- 1) + lL,a",] .

(A.J)

wit.h

ue I ('

=

i). As long as measurements of the asymmetry ACP+(:p'r(t) i.n I.he ncutml me,~on decay.• (cln .•.•ificrl in thc scenario i) in Sect. 5 ) is not a constant but depend.s on time, one can condude that CPT i1lval'ia1lce 'is b1'Oke1l due to mixing: ii). For the semileptunic decays 111° ~ lit-Iv, Olle may (:ome to a. strong statement that once the asymmetry AUP+CPT(t) is not Z~7V, th~n (:PT -must be v·iolat~d. Among t.he de(:A.Ys t.he semilept.onk dec:A.Ys A.re t.he more representative a nd perhaps the easiest to measure. iii). A comhinen meaSlU'ement. of severn.! time-dependent CPT- and CP- asymmetries from semileptonic a.nd nonlept.onic. decAYs is ncc:eSSAl'Y in order to isolntc separately the indirect. and direct CP'l'- and CP-\'iolating el~ fects. Extradion of a. dean signature on CPT, CP and T violation will play an important role ill test.ing the titandard model aJld local quantum field theory and in addit ion pro\'ide~ all illterestillg willdow 1'01' pl'Obillg lIew phy~ic~. FOI' all these reasons, Lhis topic atLrads a lot of aUent.ion[Hi]. We hope t.ha.t the general repha.se-inva.riant forma.lism present.pd in t.his paper will he useful [or fnrther st ndies of CPT, CP a.lld T in the neutral meson systems prodm:ecl at. l3-factorks. t.he
The definitions for the

l"r.pllHSC'-illvarialll ol)sC'rval,i(,R:

-4Jm.(Qh!1,g) ~(-1-+"""'lq"':/"'p7:12,c)~(1,.-!-'+:";ICfh"'.f"'g"'12~)

2ImfAl(1 -IE'MI") + 2lmE'A/(1-lf,\II') (1 + IfMI')(1 + le~/12) 4Re(qh/pg)

_1

(1 + Iq/pI2)(1 + Ih/gI 2 ) 2 4ImfAf Ime~1 - 2(leAl1 + le~III') (J + leM 12)(1 + le~,\2)

(A.2)

Rcph.....e invariant. ohscrvahloR for purely CP and CPT viulatioll

I L •.4.e'6; 12 - I L. Ai eiJ ;12 I L. A.ei6 , 12 + I L. AieiJ, 12 2 Lij .4.A; sin (a. - aj)

, a • ..!

=

I L; .I\,r.iJ , 12 + I L; .'I;riJ, 12 ' 2L"j ..J..A;(d i + .1;),,08(6; - aj) I L; A;e'J , I' + I L; A;e,J, I' 2i Li.j A,A;(Lli + .1;) sin(o, - OJ) I L. o4.e,6; I' + I L, A:ei6 ; I' ' 2

L,,] A.AjL1.dj COS(di -

o.~", = I L, A.e""I'

(A.3)

0)

+ I L . .-lie'·' I' ,

, 2·;L,.j..1 i J1;L1,djsill(d. -d) a"'Ll = I L •.'I.e"', I' + I L; .-I;e;·" 12

Ackno7l,ierige.me.n/s. W.P was supported in part. by the US d('part.mf'llt of Enc'I"J.;Y: Division of I lip;h EnC'l'P"Y Physic-s . nnci('f

wit.h d

Grant DOE/ER/O 154.5-778. Two of us (B.A.P a.nd Y.L.W)

"hamct.erize dire"t CPT violat.ion in th .. d.."uy amplit.udes.

t.hallks BlIlldesllIinist.('rilllll fijI' Bildllng, Wissl'nschafi., Forsdmll!,; uud 'H'dlllol"p,if' (HMBF), ()!')7l)0!1~P(j), Honn, FH(;, "lid ])1"(; Allt.rag 1'1\-10-1 [ur till' fiwllIcial "Ijljlurt.. Y.L.\V a(:kllowl"dg~s t.iIP slIpport. by the NSF of Chilli! IIlIcler Grant. l!)62551'1.

i

=

(lc--('

Bi! Ai. Here d

=

1

iUE~ \"l~phase-i1\\"ari(mt. qn?lnti1.ics and

211",,,,(1 _1(~1I12) + UI1If:\I(J _1<.1/1') (1 +- 101\2)(1 f 1<:\11')

21""

AI

(I (1

21"".1/(1

1,:"1') +- ~111":"(1 - I, All") . (A..I) + ".1/1")(1 + 1<:"1') .

-I,:"..!I") + 2111).(:"..1(1 -Ind") (1 + I<MI")(I + 1<.1..,1')

532 ICC, Chou et. ,,1.: S.. arching fol' I'ephnse-inyal'iant CP- and C:PT-";olat.ing obs~rvabl ..s in meson decnys wit,h

Nol.~

2L"j Re(A.;Aj)cos(o, -

1-1<\11'

+ 1':\1 I"

I

2L"J Im[A;Aj{.J; -

1(~12

1+

I - 1(~-,12

2 Li.j Re[A;Aj(diLl J )] (:os(o, - 0;)

1 + If~-,I'

I L; A. i e i6 '1" + I L; A;e"'I" 2 Li.j Im(A,Aj) <:OS(Oi - OJ)

2Im(;"

1+1'~'II'

IL;Aiei~'I2+IL,A;e'6;I'

=

+ 1'~12

I L, A;ei6 , I' + I L, Aie"'; I'

= -

2111/,:"-,

2

L,,; /",\.4,A;(.d;.1;)] cos(o, -

+ Ir~-,12

I

(A.5)

2 Li,j Re[AiAj{Ll; - .dj)] Sill(O, - OJ)

2fm(:" I

References

Ll))]sin(,j, - .I,)

IL,.4;«;;'I' + IL, .. 1;"i·'1 2

= -

0.;)

IL,A i e"q2 + IL,Ai e";1 2

The exact cxpn,...,siolls for the tilJlL~dcpclldcllt CP alld CPT asymmetries in the sc..nario i): lI .-le/'j (;/''retl= qAl"{/.)-+J)-r(iV- p.)-+n J'( "/lIe, )-/)+I'{"M"" (1.)_ f)

(A.6)

_ ,i," +'lA(!I"I'(L11I{1+1".o :2) 1"0,,,11 ..:lr/+(1- .'I,Ci 12) '""" .:1 ... ( -

I : 2,;, ".A.('/'T{I)! (I I

A'~'I'

(" •.,.(t.)=

+.

(0.,11

.-l~"I' .

I

tr'...lr") ... ".11 LlrL

I (1-lrl..lI:.1j, " .. .do", 'I

l"liIT::CL1_n_ nMII(t)_i) r(fll

(A.7)

(L)-+/Hr(I\.1"(f.)_f)

~)/{11 ~a,,,) •

C"'T(Ll= rfAlIICt.l-n-r(iT1cl.}-+n

J.

(U)

l"(I\IU(l)-+n-l-I"(A"u(I.)_n

wit.h n..:l

AcP'J' = --_.-,-, sinhdrt

u'"

1-

vI - n

2 -

a'

+ "';'

1- uJ

(1'2

'"

sill dmt .

.

(A.9) a.nd i n

t.llf~

Rf'Ponnrio ii):

ACf'H'I"r(1.) =

~"'(t) + ,1c/"r(t) -

,t.,<

(~~(t) + .1e'I'.,.(t)) + .1CI''J'(t))

.1,,(t) + .1~f''J'(t) - u'sP ... (t)

(A.IO) with Ll",(t) = (a,.< Ll,(t) = (I

+ a,.) cos(dmt) + a,.d"

sin(,1mt)

= (2 + n•.Ji,. + u's")[ 1 '/.-1 '2 -

x (cos Llml, - c..>.rI) I

Llcl'r(t.)

(A.11)

+ ",,
and

LlcP1'(t)

(}....1

= - [-tl...l -I '2 ((l,.<

+

.

U-'

U
+a,.)+~ci,,_.,]

x (("Os ,11111. - e-",r,)

I,hal. wh~n CPT is ('onsel'vPci, ,1cp.,.{l) = ,1ep .,. = 0,

OJ)

I L; /1;,,;;; 12 + I L, ;Ii r ;;; I'

1 _1<'.1'

287

1-(,...1

,

.

+ [~(,.",_ + ii,.) I"'''j

1. '1'.0. I."". C.N. Yang. Phys. Re\". 104.251 (I!l5G) 2. C.S. \-\'n. K 1\lI1bl"r, H.W. Ha.\·warrl, D. HOPI"'S. H..l'. Hu(boll, Phys. Rc\". lOS, 141:.l (195;) 3. R.L. Garwin. L.M. Lederman, M. Weinrich. Phys. R"v. 105, 1415 (185i) .J. J.l. Pricdlll>tll, V.L. Tekgdi, Phys. Rev. 105, [(j1:$1 (1!J57) 5. R. Christenson, J. Cronin, V.L. Fitch, R. Turlay, Phys. Rev. Lett.. 13, 138 (1964) 6. ,J. Schwinger, PhYR. Rev. 82, 914 (1%1); G. Lueders. Dall~k. Mat. Fy~. Meuu. 28, 17 (H)54); W. Pauli, Niel~ Bohr and the Development of Physics (Pergamon, New York, 1955) 7. For re"ent analyseR soo for example: GO. Dib, R.D. Pe,,cei, Phys. Rev. 0 46, 2265 (1992); C.D. Buchanan, et aI., Phys. Rev. 0 45, 4088 (1992) 8. ,J.S. Hell •. J. Sl.einher!1;~r, Proc. Oxford Inl.. Conf. on ele1I1clItary particles, 1965, p. 195 9. ,l.W. Cronin, Rev. Mod. Phys. 53, 373 (1981) 10. V.V. Barmin et ai., Nud. Phys. B 247, 293 (1984) 11. W.F. Palmer, Y.L. Wu, Phys. Lett.. I3 350, 2,1:' (Hl!lr.) 12. see for example, L. Lavoura, Ann. Phys. 207428 (1991), and references therein 13. '1'.0. Lee, C.S. Wn. Annn. Rev. Nlld. Sci. 16, 4il (1966) 14. E.A. Pa;;chos, R. Zacher, Z. Phys. C 28 521 (1!J!!5); For a review see for example, E.A. Paschos, U. Ti.irke, Phys. Rep. 178 147 (1989) Vi .. J. ElliR, .J.S. Hagelin, D.V. NanopouloR, M. Srednic-hi, :'-iuc!. Phys. B 241, 381 (1984); P. Huet, :\I. E. Peskin, :'-iucl. Phys. B 434, 3 (1995) J. Ellis, J.L. Lopez, N.E. i\lavromatos, D.V. Nanopoulos, Phys. Rev. 0 53, 3846 (IUU(j)

16. L, Wolf..nsteill, Phys.Rev.Lett. 83 911 (1999); L. Lavoura, hep-ph/9911209: A. I. Sanda, hep-ph/9902353: L. Lavonra, .J. 1'. Silva, hep-ph/!l!l0~:l4R: 1'. Hue!., hepph/96074:.15 ; J. Ellis, N.E. I\lavrolllatos, D.V. NauopolIlos, hep-ph/9607434; R. Adler, et al (CPLEAR Collaboration), J. Ellis, J. Lopez, :\:. Mavromatos, D. Nanopoulos. hl'I>-"x/!l5 11001; N. J\.IaVl'OlImtos, T. Rul' (for the eollaboration: J. Ellis, J. Lopez, N. J\.Iavromatos, D. Nanopoulos, and t.he CPLEAR Collabom.t.ion), hep-ph/9506395 17. See for example. L. :\Iaiani, ill 'The Second llA
Part II Statistical physics and condensed matter physics

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535

Renormallzation of the closed-time-path Green's functions in nonequilibrium statistical field theory ZHOU Guang-zhao and SU Zhao-bing Institute o/Theoretical Physics, Academia Sinica. Beijing

(Received January 241979) Phys. Energ. Fortis Phys. Nucl. 3 (3),304-313 (May-June 1979) The problem of renormalization of the closed-time-path Green's function in nonequilibrium statistical field theory is studied. Under some reasonable assumptions on the high-energy behavior of the initial correlation functions, it is found that the same counterterms which eliminate the ultraviolet divergences in the usual field theory can also make the closed-time-path Green's functions free of ultraviolet divergences. The renormalization-group equation satisfied by the closed-time-path vertex functions is obtained and the Callan-Symanzik coefficient functions are shown to be the same as in the usual field theory. PACS numbers: 11.1O.Gh, 11.1O.Ef

I. INTRODUCTION

In recent years, with the development of high-energy particle physics and high-energy astrophysics, the general interest in finite-temperature field theory has been aroused. I Many articles have already discussed the renormalization of finite-temperature field theory, and the renormalization-group equation satisfied by the thermal Green's function has already been deduced. 2 In this paper, we shall discuss the problems of ultraviolet divergence and renormalization of the closed-time-path Green's function, which is a method introduced by Schwinger and Keldyshl in studying nonequilibrium statistical processes of a system. With this method, we can discuss not only the properties of physical variables in the ground state (vacuum state) and thermal eqUilibrium state, but also the properties of stationary states far from thermal equilibrium and transport processes while approaching equilibrium. We expect this method to be further developed in high-energy particle physics and high-energy astrophysics. In Sec. II, we shall review briefly the perturbation theory4 of the closed-time-path Green's function and introduce equations and symbols necessary in later proofs. In Sec. III, we shall prove that, under suitable assumptions on the ultraviolet behavior of the initial correlation function, the closed-time-path Green's function is renormalizable. As the usual ultraviolet divergences of the vacuum expectation of the Green's function are canceled, so are the ultraviolet divergences of the closed-time-path Green's function. In Sec. IV, the renormalization-group equation satisfied by the closed-time-path Green's function is discussed. In this paper, we adopt the system of units with Ii = c = I and take the space-time metric as goo

= -

gil

= -

gzz

= -

g33

=

I.

II. PERTURBATION THEORY

Let lPs(x) represent the bare field of the system. tps(x) can have many components: scalar fields, spinors, gauge fields, the ghost fields introduced by gauge-fixing conditions, etc. In general, the various components of tp s (x) are not explicitly indicated; therefore one must take heed that many equations are cast in abbreviated forms: For example, tpB(X}JB(X) represents in fact the sum of the products of the various components. The Lagrangian of the system is (2.1\ where m s is the bare mass of the particle and A. s the bare interaction constant. We assume that 635

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® 1981 American Institute of Physics

635

536 the Lagrangian is renormalizable in the usual sense for ordinary field theory. Introducing the renormalized field tp (x), mass m, and interaction constant A. in the usual sense, we let

=

=

Z~2
Z ..m,

(2.2)

,1,B'= Zl,1"

where Z"., Z ... , and Z" are the renormalization factors which contain all the ultraviolet divergences of the field theory. The Lagrangian may be rewritten as

£,t'(
=

£,t'o(
(2.3)

where .Yo is the free Lagrangian with physical mass m, V (tp (x), A., 6m, 6A. ) is the interaction term which contains the divergent counterterms (represented by 6m and 6A. ). If gauge field are present in the field theory, V contains also the gauge-fixing term and the unitarity-compensating ghost terms. J (x) is the external source. The closed-time-path Green's function of tp (x) is defined as

Gp(xu .... x,)

=

-

jl-lt r

{Tr«iJ(x,)" • .p(x,))p}

(2.4)

where $(x) is the field operator in the Heisenberg picture and p the density matrix of the system. The SUbscript p denotes the closed path along the time axis, running from - 00 to + 00 (positive branch) and then from + 00 to - 00 (negative branch). The time ti in the coordinates Xi (i = 1,2, ... ,1) traces out the closed path, and Tp represents the operator along the closed path p. In Eq. (2.4), if we replace $(x) with $sIx), we obtain the closed-time-path Green's function of the bare field $B(X). The generating functional of the c1osed-time-path Green's functions is introduced as (2.5) the integration being carried out along the closed path p. In order to avoid complete cancellation, the external sources J (x +) and J (x _) on the positive and negative branches, respectively, are taken to be different. Taking the functional derivative with respect to J(x), we obtain, from Eq. (2.5),

Gp(xu ..• ,x, ) -_ I. _~81-:-Z--=-[",-J(,,-x~)"-]-=-1' 8J(x,)· • '8J(X1) }e,,)=O

(2.6)

In taking the functional derivative, we adopt the convention that, when J (x) is an anticommuting c-number, the derivative operation acts from the right. Transforming into the interaction picture, the generating functional can be written as Z [l(x)]

== tr { T ,(exp {- j 1,. [V(rp[(x)) +

.p[(x)J(x)] })p}

(2.7)

where $r(x) is the field operator in the interaction picture, satisfying the equation of motion for a free field,

o

8£,t' n

_

,. 80,.
I

1J£,t' 0 1J
== 0 ,

(2.8)

Equation (2.S) is a linear homogeneous equation in $r(x) and can be written as A-1(o,.)rp[(x) where il -1(0,.) is a derivative operator,

=

0

(2.9)

a,. =salax"; for examples, when $r!,%) is a scalar field, (2.10)

636

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537 where

a2 = apil' is the d'Alembertian operator.

With the interaction term taken out of the trace, Eq. (2.7) becomes Z [J(x)]

=

exp {- i

Lv (i 6J~:I:))

d4 x} t, { T p (exp { - i

L

4

rp,(x) J(:I:) d x }) pl. (2.11)

If Wick's Theorem in field theory is generalized slightly, it can be easily proved that Tp(exp{-i

Ip tP,(x)J(x)d x}) = 4

Zo[J(x)]:exp{-i

L

tP,(x)J(x)d4x}:

(2.12)

where Wick's normal-order notation (: :) implies that all operators enclosed within the double dots are so arranged that all the creation operators are to the left of the annihilation operators. The generating functional Zo[J (x)] of the free field is ZO[](:I:)]

=

exp {-

~

H

J(:I:)Ap(x - ,,)J(y)d4xd4,,}

(2.13)

p

where .dp(x - y) is the vacuum expectation of the closed-time-path Green's function of the free field, (2.14) satisfying the equation

=

A-'(fJzp.)Aix - ,,)

Here

,s; (x -

8~(x -

,,).

(2.1S)

y) is the 6 function in the closed path integral, which satisfies the relation

L/(,,)8~(x

-

y)d'"

=

/(:1:)

(2.16)

for any arbitrary function fIx) defined on the closed path, regardless of whether x is on the positive or negative branch. Substituting Eq. (2.12) into Eq. (2.11), we get 8 Z[J(:I:)] = exp {- if V ( i __) d'x} Zo[J(x)]N[J(:I:)], Jp 8J(x)

(2.17)

where (2.18) is known as the initial correlation functional, which is related to the density matrix describing the initial state of the system. When the system is at the vacuum state, N[J(:I:)]

= 1.

For any initial state, we expand N [J Ix)] by iteration, getting

W N[J(X)] =

t

/-1

N[J(x)] = exp{ - iWN[J(x)]},

(2.19)

~ J... J N(x"

(2.20)

I.

... , x,)J(x,)·· ·J(x,)d4x,· . 'd'x"

p

where N(x" ... ,x,)

= (- i)'-'t,{:tp,(x,) .. ·tP,(x/):p}.;

(2.21)

the subscript c on the right-hand side of Eq. (2.21) indicates that this is an Ith-order iterated term with all the lower·order correlation terms deleted. If the system is in thermal equilibrium, then its statistical distribution is Gaussian and all iterated terms for I> 2 vanish. When Eqs. (2.19) and (2.20) are substituted into Eq. (2.17) and the interaction term V is 637

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637

538 expanded, the perturbation-theory expression is obtained. There are only two differences between the perturbation expansion of the closed-time-path Green's function and that of ordinary field theory. Firstly, the time integration of the closed-time-path Green's function is along a closed path. Thus when resolution into positive and negative branches is made, each Feynman diagram resolves into 2ft diagrams according to whether each of the n vertices are on the positive or negative branch. At the same time, we multiply each vertex on the negative branch by a factor - 1, so that all the time integration is from - 00 to + 00. The other difference is that the closed-time-path Green's function can be used to describe a system in any initial state and is not restricted only to the vacuum state. The effect of the initial state is expressed by the correlation function N(xl> ... ,x,). One can readily prove that the correlation function obeys

,6.-1(8xnN(x" ... , Xl) = 0

j

=

1, ..• , t.

(2.22)

Eq. (2.22) implies that the correlation function contributes only when it is on the mass shell. Moreover, since Eq. (2.18) does not contain the time-ordered-product operator Tp, the values assumed by the correlation function on different branches of the closed path are the same. III. CANCELLATION OF ULTRAVIOLET DIVERGENCES OF THE CLOSED-TIME-PATH GREEN'S FUNCTION AND RENORMALIZATION

Besides the assumption made in Sec. II that the Lagrangian is renormalizable in the usual sense, we introduce the following restriction to the correlation function describing the initial state of the system: In Hilbert space, under the Fock representation of free particles, the matrix elements of the density matrix can be represented by

(P.', •. ·,p~,lplp.. ··· ,p~..),

(3.1)

where Pi is the momentum of the ith particle. For simplicity, the internal degrees of freedom of the particles have been suppressed. Because of the restriction on the physical states, there will naturally not be any particle with infinite momentum. As a result, we assume that, when the momentum Ip;j (or \p;\) approaches infinity, the matrix element (3.1) should approach zero extremely rapidly; for example, when the system is in thermal equilibrium, the density-matrix element approaches zero according to the Gaussian distribution. To be more accurate. we assume that, for any positive integer I.

lim Ip;/l(p;, .•• ,p~,lplpU" .. ,PN>=O.

(3.2)

I"jl~.

Equation (3.2) indicates that the Fourier component N(p ••... ,p,) of the correlation function N (xl, ... ,x,) with X, on either of the branches approaches zero faster than IPi I-I, as IPi 1-00. We have pointed out in Sec. II that N(p ••... ,p,) is not vanishing only on the mass shell; thus the product of N(pl •... ,p,) and any finite polynomial of Pi integrated over d~, converges. From the above discussion. we know that. in the perturbation expansion. for those Feynman diagrams containing N(x ••... ,x,). if the integrand is convergent before integration over X ••••• ,x,. no new divergences can emerge from the integration over X, just because of the presence of N(xl •... ,x,). But this does not mean that a single Feynman diagram containingN(x ••... ,x,) is not divergent. because it is quite possible that, before integration over x ••... ,x,. the integrand is already divergent because of other closed loops which it contains. What we want to prove for the renormalizability of the closed-time-path Green's function is that. when all the diagrams containing N(x., ... ,x,) are summed. their divergences cancel one another. In order to prove this. let us first analyze the structure of the Feynman diagrams representing the vacuum state. With use of the commutation relation 638

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638

539 _8_exp {_ i.. 8J(IIl) 2

=

~

exp { -

rr J(x)l::.p(x -

J)

y)J(y)J4XJ4y}

H

J(x)I::.,(x - y)J(y)J4XJ4y}

,

(8J~IIl)

- i tJ(U)l::.p(U - Z)J4 U), ~~

the generating functional (2.17) can be written as

Z[J(x)]

=

Zo[J(x)]exp {-

8 _ + ( }(y)I::.,(y iJ,r V (i_ 11}(x) J,

x)d4y) d4X} N[J(x)].

(3.4) In order to clarify Eq. (3.4). we first discuss the vacuum state. There N[J(x)]

= 1,

and

Z[J(x)] Z v(j]

=

Zo[J(x) ]exp {- i

=

Zv[J(x)],

+ J,r V (;_8_ 8J(x)

J, J(y)I::.,(y - x)

x}.

d 4y) d 4

(3.5)

Expanding the last term in Eq. (3.5) in terms of J(x). and letting

Zv(jj = Zo[J]

±Jr. .. J,

d 4x,·· .d4x l d4y,.' ·d4YI

1=0

Xl!1 J(y,)"

·J(YI)I::.,(y, - %,) .• ·I::./YI - XI)/V(X", .• , XI),

(3.6)

we can then readily prove that

J... Ld x,·· ·d xll::.,(y, 4

4

x,)·· ·I::.,(YI - XI)/V(X,,", "XI)

= (OIT,(1J(YI)··.cp(YI»lo)v.

(3.7)

The subscript V denotes that the I vertices atYI •... ,y/ are interaction connected; i.e .• the diagrams representing the contribution by the free-field generating functional Zo[J(xll are to be deleted from the vacuum expectation of the Green's function. It is necessary to point out that Eq. (3.7) includes all V-connected Feynman diagrams with I external lines. Using Eq. (2.15) satisfied by .::l,(Y-x). we get, from Eq. (3.7),

11'(x", .. , XI)

=

1::.-1 (os,) 1::.-1 (ox) .• '1::.- 1 (0%1)(0 IT ,(1J(XI)· . 'q,(x,»

IO)v.

(3.8)

Before proceeding further, we need to differentiate the situation on the positive branch from that on the negative. We start our discussion by first putting all the Xi on the positive branch; then

fv(x,+, .•• , XI+)

= A-l(ox) .. 'A-l(ox)(O I T(q,(x/)' . 'q,(XI» IO)v.

(3.9)

On the right-hand side of Eq. (3.9). we have no need to identify Xi and T is just the usual timeordering operator. Transforming into the interaction picture. we have

(0 I T(<jl(x,)· . 'q,(Xl»

10), = <0 IS+T(q,,(x,)· • °q,,(XI)S) Io)v

(3.10)

where § is the S matrix of the field theory. Since the vacuum state is stable. acting on the vacuum. § gives only a phase factor,

(3.11) where L comes from the contribution of closed-loop diagrams. Although it is possibly a diver639

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639

540 gent number. it can be removed through renonnalization. giving no physical effect at all. Thus Iv (XI+>" •• ,x/+)

=

t::.-I(a z)· . • t::.-I(az)e-iL(O I T(rfJ/(xl)' •• «p/(xI)5) IO)v;

(3.12)

the last term on the right is the Green's function ofthe ordinary field theory. which we denote by G(xl •. ·.,x/).The closed-loop diagrams of pure vacuum in G contribute a factor ~ canceling the factor e -;L above. This demonstrates that all contribution of closed-loop diagrams in Iy have been deleted. In momentum space. Eq. (3.12) can be written as Iv+(Pu • •• ,PI) = t::.-I ( -ipI)t::.-I ( -ipz)' •• t::.-I ( -ipl)e-iLG(pu' •. , PI) /

c=

11 e

•••

1

~ Pj"Zj I v+ ( XI'"' =1 "

',X, "

)d4XI"" d 4x/

(3.13)

where the positive sign on the SUbscript of I denotes that all X; are on the positive branch. We know that G (pl ... ·,p') represents the Green's function with I external lines carrying momenta P •• ·.. ,p/. with every external line connected with the interaction term. It has been proved in field theory that. if the system contains scalar fields. spinors. and gauge fields. and the dimension of each terin in the Lagrangian is less than or equal to 4. then when the Green's function G (pl •... ,p/) is expanded in terms of closed·loop diagrams. the ultraviolet divergences in each order of the closed loop diagrams are canceled by countertenns. and also. when Ip; I increases. the contribu· tion from each order of the closed·loop diagrams increases only according to some finite power of Ipd. However. when all orders of the diagrams are summed. is it possible that the series does not converge and instead leads to new divergence? Since the answer to this question is stilI unclear at the present. we limit ourselves merely to the proof that w~en the closed-loop Green's function is expanded in terms of the number of loops. each term is convergent. When all the

Xi

are on the negative branch. similarly we have (3.14)

where G(Xu"" XI)

= (0 IT(rfJ/(xI)"

'/pl(xl)5+) I0),

and f is the anti·time-ordering operator (operator of an earlier time stands to the left). It can be easily proved that (3.15) where G(x I •••• ,x,) is the complex conjugate of the Green's function of the ordinary field theory. In momentum space. for each order of the closed-loop diagrams. iJ has no divergences. and when the momenta increase. iJ increases at the most according to some finite power of the momenta. Thus Iv-(pu •••, PI)

=

t::.-1(-ipl)·· .t::.-1(-ipI)eILG(pu.' ',PI)

(3.16)

is not divergent. Finally we discuss the situation when some x, are on the positive branch and some on the negative branch. It can be easily proved that !V(XI+"'" XjH Xj+I_, ••• , XI_)

=

t::..-1(a z)·· .t::..-I(aXI)

x(O IT(<
(3.17)

With use of the method of expanding Feynman diagrams in the discussion of the unitarity condition. it can easily be proved that

..

(;)~

r r

Iv(xI+," .,Xj+; Xi+l-" • '2 XI_)= ~ ~ \ ••• \ d'Zl' . • d4z~d'"I· • •d'"~ ~,.o

k!·

.

X Iv-(z~, •• ,,·ZI; Xi+U' • ".XI)l::.-(Z~ - "~) •• ·t::.._(ZI - "I)fn(xu" "Xi' 640

Chin. Phys., Vol. 1, No.3, July-Sepl1981

"10' ••

,"t) (3.18j

ZHOU Guang-zhao and SU Zhao-bing 640

541 where (3.19) satisfying the equation (3.20) Going to the momentum representation. from Eq. (3.18). we get

IV+-(Pu ... , Pi; Pi+U""

PI)

~~J '" J~ dq = L..J I .. '--~-Iv-(4

~=o

k!

(2n-)4

(2,,-)4

qlc'" . , - qu Pi+U" ',PI)

(3.21) where ~ _(q) contributes only on the mass shell as a result of Eq. (3.20). and it can be readily verified that ~ _(q) is nonvanishing only when qo> O. From the conservation of energy-momentum (the vacuum expectation of the Green's function is translationally invariant). the integrand of Eq. (3.21) is nonvanishing only when q,

+ ... + qlc =

Pi+1

+ ... + PI = -

(PI

+ ... + PI) •

(3.22)

Thus when integration over d 4q; is performed. there is the restriction (3.23) where Po = Pj+ '.0 + ... + P,O is a constant which should be greater than zero as a result of the positivity of q.o; otherwise Eq. (3.22) vanishes. In the above. we have proved that bothfy_ and /v+ are free of divergences and. because of Eq. (3.23). the integral over qj is a finite one. Thus each term in the expansion of Eq. (3.21) converges. Although we are not sure whether the sum of the series is convergent. yet up to a finite number of loops the series in Eq. (3.21) consists of a finite number of terms and is therefore finite. We have shown in the above discussion of the vacuum expectation of closed-time-loop Green's function that no new ultraviolet divergences emerge. In the following. we want to c;onsider the general case. For any initial condition. the generating functional (3.4) can be written as

Z[J(x)] X:

=

ZdJ(x)]

±r... ,

I=OJ. P

(i 8J~x,) + )p J(YI)~p(YI -

x,)d

d4XI .. ·d4xI1..Iv(x" ... ,xl) II

y,) ... (i 8J~XI) + JPJ(YI)~p(YI -

4

4

':r 1)d YI) :

(3.24)

X N[J(x)]

where. when inside the symbol: :. the operator ~ /SJ(x;) is on the right of J(yj)' In other words. within the symbol: :. ,s/SJ (x;) operates only on N [J (x)] which follows it. At the same time. it can be easily shown that. the function/y(x ..... ,x/) in Eq. (3.24) is the same function given by Eq. (3.8). which depends only on the vacuum expectation of the Green's function. In Eq. (3.24). when i~/SJ(xj) operates on N[J(x)]. some correlation functions N(x ..... ,x,,) appear. Thus when the generating functional Z[J(x)]/Zo[J(x)] is expanded in terms of J(x). its coefficients are the product of/v(x ..... ,x/) and many correlation functions N(xw",x,,) integrated over some coordinates. We have already proved that. for some fixed number of closed loops. no matter whether Xj are on the positive branch or the negative branch. the Fourier components fylp .... ·tP/) offy are free of divergences and increase as some power of Ip;1 as Ipil increases. At the same time, the Fourier components NIp ..... ,p,,) of the initial correlation functions approach 641

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641

542 zero faster than any power of Ipd- I as Ipd increases. Thus, whenfy if multiplied by many correlation fun~tions and then integrated over some coordinates, na new ultraviolet divergences will be produced. Therefore, we can make the conclusion that, as long as the number of closed loops is fixed, the number of Feynman diagrams is finite, and the sum of these diagrams will not lead to ultraviolet divergence. Whether the sum of the whole series is convergent or not is a problem that should be attacked separately. From the above discussion, we see that, if the Green's function of an ordinary field theory is free of divergences, the closed-time-path Green's function in nonequilibrium statistical mechanics is also free of divergences, and also the counterterms in the Lagrangian and the renormalization factors are the same as those of an ordinary field theory. We can make use of these properties to deduce the renormalization-group equationS satisfied by the closed-time-path Green's function. IV. THE RENORMALIZATION-GROUP EQUATION SATISFIED BY THE CLOSED-TIME-PATH GREEN'S FUNCTION

In the same way as in ordinary field theory, we can introduce connected closed-time-path Green's function and closed-time>-path vertex function. Their generating functionals are denoted by W[J(x)] and r[qlc(x)] respectively, which obey W [J(~)]

T[rpc(~)]

W

=

= iln Z [J(~)],

[J(~)]

LfPc(~)J(~)d4X,

-

(4.1)

where qlc(x) is the vacuum expectation of the field $(x), fPc

( ~)

=

BW[J(~)]

(4.2)

BJ(~)'

The closed-time-path vertex function can be written as T,(xu" .,XI; p., m, }., {i)

BIT[pc(~)]

=

•

BfPC(Xl)' •• Brpc(xl)

(4.3)

In the vertex function (4.3), we have included its dependent physical variables explicitly. Besides the mass m and the interaction constant -t, it depends on the choice of the renormalization point J.l and the physical variable denoting the initial correlation function. If we let r PB denotes the bare vertex function, then there exists the relation

ti

r,(xu ••• , XI;P., m, 1, {i)

=

z~1/2rpB(XU""

Xl,

mB' lB' {i),

(4.4)

where Z" is the wave-function renormalization factor. When qI (x) has many components, with thejth component corresponding to xj ' we have

Z~1/2 =

(

U Z"';

+Ih )

•

(4.5)

I

From Eq. (2.2), we know that mB

=

Z",m,

lB

=

Z11

where Z", Zm, and Z.. are functions ofJ.l and the interaction constant -t. (Here we have adopted the method of dimensional regularization developed by 't Hooft to cancel ultraviolet divergences. This method leads to a mass-independent renormalization and therefore the Z's do not depend6 on m.) Differentiating Eq. (4.4) with respect to J.l (with m B and -t B fixed), we arrive at the renormaljzation-group equation 642

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642

543

[ ,.,.~ a,.,. + P(1)~ al +

-2.... - nO)] r<,ll =

r ... Cl)m

am

0

(4.6)

where we have used r~' to denote rp(Xl, ... ,x,,",A.,m,~tl, and

P(1)

= p.~ll

a,.,.

r .. Cl)=""aap. Inm rrCl)

=

= -l,.,.~ln

ap'

lB. mB

11

B. mB

Zl\

'

lB. mB

=-""aaP. 1oZ"'\l

B. mB

(4.7)

'

a a,.,.

-l -,.,.-In Z'I' 11 2 B....B

are the usual Callan-Symanzik coefficients, which are the same as those given by the ordinary field theory. When we fix Xj on the positive or the negative branch and transform to momentum space, we get altogether 2' different vertex functions, everyone of which obeys Eq. (4.6). Thus if we let r~IICpl>"

.,p" I', 1, m,

~,),

i

rO.)]

r." = 0,

=

1,2, ... ,2',

be these vertex functions, they obey

[ ,.,. aa,.,. +

pO.)

~ al + r ...O.)m ~am

We separate the physical variable

'f

s = 1, 2, ... ,2'.

(4.8)

t; denoting the initial correlation function into two parts:

T/;, which is dimensionless; and So which carries the dimension of a mass. The latter can be the

temperature, chemical potential, etc. For those physical variables with still higher dimensions, we can raise them to a certain power and thus reduce them to physical variables of the type S;. According to dimensional analysis, we have r~"[Kp" ..• ,Kp" K.I&, 1, Km,

7];,

K;;]

=

KDrr~')CPI'" .,p",.,., 1, m,

7];,

SI), (4.9)

where Dr is the canonical dimension of the vertex function r~l. From Eq. (4.9), it can be easily shown that

[K

aa

K

+.It

aa + m ~ + ;; ~ I'

am

a;;

Dr]

r~')CKpl>"

.,Kp" ,.,., 1, m, 1/;, S;) = O. (4.10)

Subtracting Eq. (4.8) from Eq. (4.10) and removing pa lap, we arrive at

rlK ~ + ;i ~ - P(1) ~ +

aK a;; x r~"CKpl'" .,Kp"

al

I',

1,

m,

(1

~ r ... O.))m ~ + rr am

7];, ;;) =

Dr]

o.

(4.11)

Eq. (4.11) is the Callan-Symanzik equation satisfied by the closed-time-path Green's function.

When Eq. (4.11) is solved, it leads to the same results4 obtained by Kislinger and Morley using a finite-temperature Green's function. For example, a non-Abelian gauge field possesses the property of asymptotic freedom not only when the momenta are large but also when S; is large (high temperature or high chemical potential). ID. A. Kirzhnits and A. D. Linde, Pbys. Lett. 41 B, 47111971); D. A. Kirzhnits, JETP Lett. 15, 52911972); S. Weinberg. Phys. Rev. D g, 335711974); L. Dolan and R. Jackiw, Phys. Rev. D 9,3320119741; M. B. Kislinger and P. D. Morley, Phys. Rev. D 13, 2765 (19761. 2M. B. Kislinger and P. D. Morley, Phys. Rev. D 13, 2771 (19761; S. Weinberg, Phys. Rev. D 9.3357 (19741; L. Dolan and R. Jackiw, Phys. Rev. D 9,3320 (19741. 643

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

643

544 'J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V. Ke1dysh JETP 20, 1018 (196S); R. A. Craig, J. Math. Phys. 9, 60S (1968); R. Mills, Propagators/or mtlny panicle systems (Gordon and Breach, New York,1969); V. Korenman, Ann. Phys. (N.Y.) 39, 72 (1966); V. L. Berezinskii, JETP 26,137 (1968); O. Niklasson and A. Sjolander, Ann. Phys. (N. Y.) 49,249 (1968); C. P. Enz, The many body problem (Plenum, New York, 1969); R. Sandstrom, Phys. Status Solidi 38,683 (1970); C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, 916 (1971). 4A. O. Hall, Mol. Phys. 28, 1 (1974); J. Phys. A 8, 214 (1974). 'K. Symanzik, Commun. Math. Phys. 13, 49 (1971); C. O. Callan, Phys. Rev. D 5,3202 (1972). 6Q. 't Hooft, Nue!. Phys. B61, 4SS (1973); J. C. Collins and A. J. Macfarlane, Phys. Rev. D 10, 1210 (1974); S. Weinberg, Phys. Rev. D 8 3497 (1973). Translated by King Yuen Ng Edited by Stanley Wu-Wei Liu

644

Chin. Phys., Vol. 1, No.3, Juiy-Sept. 1981

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64

4

545

Dyson equation and Ward-Takahashi identities of the closedtime-path Green's function ZHOU Guang-zhao and SU Zhao-bing Institute o/Theoretical Physics, Academia Sinica. Beijing

(Received 24 January 1979) Phys. Bnerg. Fortis Pbys. Nucl. 3 (3),314-326 (May-June 1979) The Dyson equation satisfied by the closed-time-path Green's function of the order parameters is considered. The transport equation for the number density of the quasiparticles is written down in a general but simple form. With use of the path-integral. formulation for the generating functional of these Green's function, the Ward-Takahashl identites are deduced. PACS numbers: 1l.10.Lm, 1l.10.Bf, 11.10.Np

I. INTRODUCTION In recent years, in the study of high-energy particle physics, many problems concerning collective cooperative phenomena have been raised, e.g., the problem of the phase change of the vacuum state, the problem about the confinement of quarks, the problem of solitons, etc. All of these are not single-particle phenomena, but rather phenomena of collective motion due to the interactions of an infinite number of degrees of freedom. It is expected that, under the interactions of high-energy particles, many degrees of freedom can be excited. When one deals with the motion of these degrees of freedom, the method of nonequilibrium statistical field theory should be used. In our view, the closed-time-path Green's-function formulism developed by Schwinger· and Keldysh· provides an effective method to study nonequilibrium statistical field theory. This method is very similar to the method of Green's function in field theory and, with only small alteration, nearly all methods in field theory can be applied. At the same time, the closed-timepath Green's function contains the statistical correlation for any initial condition; therefore, it can also be used to study the properties of the ground state, the thermal equilibrium state, transport processes, and the properties of stationary states far from equilibrium. In our opinion. the study of nonequilibrium statistical field theory has come under higher and higher demand in the development of high-energy particle physics and astrophysics, and this is a direction that deserves emphasis. Taking solitons as an example, we note that in field theory all solitons are classical solutions of the Euler equation of the primitive Lagrangian. But the solitons observed in solid states physics, like vortex lines in superconductivity, are not classical solutions of the primitive Lagrangian. They are the classical solutions of the equation of an order parameter statistically averaged over all other degrees of freedom. In future particle physics, solitons described by order parameters and not by fundamental fields may also be possible. The main purpose of writing this paper is to arouse the interest of our colleagues in high-energy particle physics in this direction. In Sec. II, the Dyson equation satisfied by the closed-time-path Green's function is studied. This probeIm has already been studied in Ref. 1 and therefore not all of our results are new. However, it is worthwhile writing them out here for our readers because our equations are more general and simpler than those in other references; our transport equation which is in a rather simple and general form deserves special mention. The second-order Green's function in field theory is the Feynman propagator GF ; all the rest are unimportant. Of the second-order closed-time-path Green's functions, there are three independent ones that are very important: They are the second-order retarded Green's function G" the second-order advanced Green's function Ga , and the number density, n, of quasiparticles. In field theory, there is no attenuation when a particle propagates in vacUU!Jl; however, 645

0273-429X/81/010645-14$05.00

@) 1981 American Institute of Physics

645

546 there is in general attenuation in the propagation of quasiparticles. This is why Gr and G are considered as two independent quantities. which contain not only the spectrum of propagation (dispersion) but also attenuation. The Dyson equation of the second-order closed-time-path Green's function can determine not only the propagation and attenuation of quasiparticles. but also" the transportation of the number density. n. of quasiparticles. Q

In Sec. III. the Ward-Takahashi (W-T) identities of the closed-time-path Green's function are discussed. We use the Feynman path-integral method, when the Lagrangian possesses the global symmetry of a Lie group G, .to derive the W -T identities· satisfied by the closed-time-path Green's function. The situation when the symmetry is broken spontaneously2 is also discussed. We have proved that. in general. the equation

lJTjlJQc(x)

= o.

satisfied by the vacuum expectation Q< (x) of the order parameter Q(x) determined by the generating functional of the vertex functions. does not possess stable solutions of the form

Qc(x)

=

Qo(X),,-i"",

W

~

o.

But this does not mean that soliton solution or laser-type solutions do not occur in the system; it only says that, because of quantum effects. the wave packet formed by solitons cannot be stable and it must spread and attenuate. Thus in order to search for soliton solutions. we must at first determine the classical equation satisfied by the order parameter Q (x). In the last section. we discuss the W-T identities satisfied by the closed-time-path Green's function when the Lagrangian possesses local gauge in variance. II. CLOSED-TIME-PATH GREEN'S FUNCTION Let Q(x) and p represent. respectively. in the Heisenberg picture. the physical variable and the density matrix designating the"initial conditions of the system. Here Q(x) c~n be a fundamental field; it can also be a composite operator made up of fundamental fields. If Q(x) contains more than one component. we interpret x as {x!, .il for J1. = 0.1.2.3 and i = 1.2 ..... n. with x!' denoting space-time coordinates and i the various components. The closed-time-path Green's function of the physical variable Q(x) is defined as

G,(XI' ''', xz)

=

(-i)l-ltp{T ,((>(XI)" ·Q(x/))p}.

(2.1)

where the subscript p denotes the closed path varying on the time axis from t = - co to t = + co (called the t + branch) and then from t = + co back to - co (called the t _ branch); Tp is the order operator on the closed path p; the time coordinates of xl .... ,xz can be any point on the closed path. When all x 1.... ,xZ are on the t + branch. Tp is the same as the usual time-ordering operator in field theory. and

G,(XI+' "', x/+)

=

(-i)l-ltp{T«(>(x l )" • (>(xz))p}.

(2.2)

The Green's function Gp(XI+ .... ,xZ+ ) is the vacuum expectation of the operator T(Q(xl) .. ·Q(xzl). It differs from the Green's function of ordinary field theory in that the former is an average over any initial condition (described by the density matrix p) while the latter is an average over the vacuum state. When x I .... ,xj lie on the t _ branch and Xj + 1 .... ,xZ on the t + branch. we have

G,(xl_, .••

,Xj_, Xj+I+' ••• ,

x/+)

= (- ;)1-1 tr{r«(>(x

l

) ' ••

(>(xj))T(Q(xj+,) • •• (>(xz))p}

(2.3)

where T is the anti-time-ordering operator (operator of a later time stands to the right).

Let us introduce the generating functional of the closed-time~path GreeQ's functions Z[J,(x)] =tr{Tp(exp{-iLhCx)(>(X)d4x})p}. 646

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guan9·zhao and SU Zhao-bing

(2.4) 646

547 where Q(x) is the operator in the Heisenberg picture without taking into account of the external source h (x). The integration in Eq. (2.4) is made along the closed path p, the external source h (x) being different when situated on the t + branch and the t _ branch. Differentiating Z [h (x)) with respect to h (x), we get i 8Z [h.(~)] 8h(~)

=

t. {T p(~(~)exp {- i

f

Jp

hey) ~(y)d4y ))p}.

(2.S)

When h (x+) = h (x_~ = h (x),

L

Tp(~(~)exp {- i h(y)~(y)d4y}) = 1J+(t)~(~)U(t)

;:

~A(~)

(2.6)

where

and ~(x) is the operator in the Heisenberg picture when the external source term Jh (y)Q(y)d3y is included in the Hamiltonian. We thus have

(2.7)

Those readers who are not familiar with the closed-time-path Green's functions can learn from Eq. (2.6) why we have to introduce a closed path. Only in this way can we guarantee the operator to be always in the Heisenberg picture; otherwise, on the right-hand side of Eq. (2.7), after application of the trace operator t, there will be an additional factor U (t;), t; being the latest time for x" ...•x,. This factor makes the ordinary Green's functions in any initial state unrelated to the averages of physical variables. Let us introduce the generating functionals of the c1osed-time-path connected Green's functions and vertex functions. W[h(~)l = iln Z[h(~)],

r[Q(~)]

=

Lh(X)Q(~)d4X'

W [hex)] -

(2.8)

where the average of Qh (x) is denoted by Q(~)

= 8W[h(x)] • 8h(x)

Here. we do not require h (x +)

(2.9)

= h (x _), but directly define

QA(~) = T,(Q(~)exp{-i Lh(x)Q(~)d4X}).

(2.10)

When we solve for the observables, we need to take h (x +) = h (x _); then Q (x +) = Q (x _) will be satisfied automatically. Similar to field theory, we have 8T[Q]

8Q(~)

=

=+=

hex).

(2.11)

In the 101l0wing, whenever an equation contains upper and lo~er signs. the upper one applies when Q (x) is a boson operator while the lower one applies when Q (x) is a fermiolN>perator. When 647

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

647

548 Q(x) is a fermion operator, both h (x) and Q (xl are anticommuting c-numbers and all derivatives

operate from the left. Differentiating Eq. (2.9) with respect to Q(Yl and Eq. (2.11) with respect to h (Y), we arrive at two equations,

L

8~(x

G,..(z., y)tl4yTp(Y' z) = -

LTp(X, y)tl4yG pc (Y, z)

= -8~(x

-

z),

(2.12) - z)

where the second-order connected Green's function is Gp«x, y)

=

8h(:;~(Y) == -

i
(2.13)
= Z~h]

tr{T(Qh(X)Qh(Y))p} - Q(x)Q(y)

and the second-order vertex function is (2.14) If Q(x) consists of more than one component, then the integral over d"y implies integration over space-time points and sum over the indices of the components. Also, ~(x - y) will be a fJ function involving the other components. It is defined on the closed path p with the property that

Lf(y)8~(y

- x)tl4y

=

(2.15)

I(x)

where/Ix) is any arbitrary continuous function defined on the positive and negative branches. Because x and y can be on the positive branch or the negative branch, when Gpc(x,y) is expressed as a function of a single time, it becomes four functions which can be written as a matrix

G= where Ga , with a = F,

+, -, and F,

(GG_ F

G+) G,

(2.16)

are themselves also matrices,

= GI'(x, y) = Gp(x+, y+) = - i
(G F ) ... ,

(G .... ) .. ,y

(G,) .. ,y

=

G,(x, y)

=

Gp(x_, y_)

=-

(2.17)

i
If Q(x) is a self-conjugate operator, it can be easily proved that

G+ where

iii

=

-iilGih

(2.18)

are the Pauli matrices,

iil=(~ ~), iiz=(~ ~i).

ii 3

=C

_°1)'

Equation (2.18) can also be written as

648

Gt = -

Gi,

Gt = -

G_.

Chin. Phys., Vol. 1, No.3, Ju1y-Sept.1981

ZHOU Guang-zhao and SU Zhao-bing

(2.18') 648

549 Using the definition of Ga , we can also readily prove Gp

+

G,

=

G+

+

G_.

(2.19)

From Eqs. (2.18) and (2.19), we know that, among the Ga's, only three Hermitian matrices are linearly independent. From Ga , we can also introduce the retarded Green's function Gr and the advanced Green's function Ga :

= G. = Gr

Gp

-

G+ = G_ -

G"

(2.20)

= G+ - G,. Similarly, six vertex functions Fa' with a = F, + , - .p.r, and a, can be defined. In matrix GF

-

G_

representation, let (2.21) Equation (2.12) can be written as

tfJ3 G = - fJ3'

ra

Gti3t = - fJ3.

(2.22)

From Eq. (2.22), it can be easily proved that, when Ga satisfies relations (2.18) and (2.19), will also satisfy similar relations, (2.23)

and Gr

With use of Eq. (2.23), so that

= - f;l,

G.

= - fil.

(2.24)

r can be represented by the three linearly independent Hermitian matrices (2.25)

where 1

ii =

(

1·

1 1) =

B=i.(fF+f,) 2

A

1

+- 7il>

= i.(f++L). 2

(2.26)

1 1 D=-Z(f,.-fF) = --Z(fr+f.), A

= i. 2 (L -

f +) = - i. 2 (fr - f .),

and the matrices B, D, and A are all Hermitian. From Eqs. (2.26) and (2.24), we get Gr

= -

f;1

G. = - fil

=

-=-.....;1,,--_ D + iA'

(2.27)

1

= -:D=---~iA-:-

designating D as the dispersive part and A the attenuation part. If the self-conjugate operator Q(x) consists of only one component, in a uniform system, its Green's function is only a function of the relative coordinate x - y and, in momentum representation, we have 649

Chin. Phys., Vol. 1. No.3. July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

649

550 (2.28) When attenuation A (k) is small, the poles of the retarded Green's function are determined by

o.

D(.n -=

(2.29)

If the energy is determined as ko = (II(k) > 0, then a quasiparticle with energy (II(k) propagates in the system. The amplitude of the quasiparticle is attenuated' exponentially in propagation, with an attenuation constant

r=

A(k)

(2.29')

aD

ako

.\.=.. (k)

From Eq. (2.22), we can also solve for {j and obtain

~ = -..!.. (T;'*. - *.T;') =..!.. (G.*. - *.G.) 2

where

*. = *. =

(2.30)

2

NfJ - ifJ2

+

fJ3

NfJ - i7]2 - fJ3

= =

7j(N

+

fJ3)'

(2.31)

(N - fJ3)fJ,

and N is a matrix determined by the following equation NT• ...,... TrN

=

2iB,

(2.32)

AN) - 2iB.

(2.32')

or ND - DN

=

i(NA

+

At the poles of Gr and G., there exists in the system a quasiparticle carrying energy (II(k). Its Green's function GF can be expressed in terms of the quasiparticle number density matrix n, (2.33) Comparison of Eqs. (2.33) and (2.30) shows the relation between N and particle density n, N latthe poles of Gr = 1 In terms of n, Eq. (2.32') can be written as

nD - Dn

= i(nA +

An) ± i(A - B)

±

2n,

=

;(nA

(2.34)

+

An) ± T+,

(2.35)

which is in fact the transport equation satisfied by the particle number density n. After neglecting the noncommutative parts of n and A, the right-hand side of Eq. (2.35) can be written as

± (1 ±

n)T+ - nT_

which is just the collision term on the right-hand side of the transport equation, r + directly proportional to the rate of emission (absorption) of quasiparticles.

(r _) being

In the following, we consider the case when Qis of single component. In an approximately uniform system, the matrices D (x,)'), etc., can be written as X

=

"21 (x + y),

III

=

(x - y),

which are slowly varying functions of X. In the momentum representation of z, expanding in terms of X,. up to the lowest order of iJ liJX,., we have, at the poles of Gr , 650

Chin. Phys•• Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

650

551

(aD(k, X) anCk, X) _ aD e.! , X) anCk, X)) ak" ax" ax". ak" aD (..E!!.- + v • 'ilx n + aw an). aka aXa ax;. ak"

(nD - Dn)(k, = -

j

X) = _

j

(2.36)

In writing down Eq. (2.36), we have used the pole condition [Eq. (2.29)] of Gr to obtain the velocity of the quasiparticle, v

= 'il.w == _ 'il!D ~ aD' aka

(2.37)

and

aD aw ax" ax" = - aD' aka From Eqs. (2.36) and (2.35), we can obtain the transport equation satisfied by the quasiparticle number density in an approximately uniform system:

~n +

uXa

v.

'ilxn + aw ~ =....:.L {± iT+(l ± n) - if-n}. ax" ak" aD aka

In a uniform system, both nand becomes zero so that

Q)

(2.~8)

do not change with X,. and the right-hand side of Eq. (2.38)

-l±n - = -T_ -,

(2.39)

± T+

n

which is just the Einstein relation of detailed balance. In order to understand the physical meaning of r + and r _, let us take a scalar field as an example. Let Q(x) be the scalar field $(x), which satisfies the equation of motion, (a Z+ mZ)cpC,,)

= jC,,).

(2.40)

It can be easily proved that the second-order vertex function is

TpCx, y) = ca! +

mZ)8~Cx

y) +

-

~pCx,

y),

(2.41)

where l:p(x,y) is the self-energy part, which can be represented as ~pCx, y)

with (

II.p.1

= -

(2.42)

il r {T p C1C:t)jCY))P} •. P.l.,

denoting the one-particle-irreducible part.

Let In) be a complete set of orthogonal eigenstates of the energy-momentum operators and other operators commuting with them. In a uniform system, because of translational invariance, l:p(x - y) is a function of x - y only, and, in momentum space, it becomes

j~-(k) =

JeiH

r- Y ),.{1(:t)jCy)p} •. p.l.d4(x

....

- y)

= ~1ltp.l.Pftft(2 ...)464Ck - P, +

n)_

(2.43)

When ko> 0, the right-hand side of Eq. (2.43) is directly proportional to the quasiparticles 651

Chin. Phys., Vol. 1, No.3, Ju1y-Sept.1981

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651

552 absorption cross section at energy-momentum k. To be more precise,

i.E_(k)

2I k lu •boorp'ion(k)

=

= (;~;3 W.bsorption(k),

(2.44)

where W.bsorp'ion (k ) is the probability of quasiparticles absorption per unit time. In the same way it can be shown that, when k o > 0,

..E+ (I.) '\ -

I

where

Wemission (k

Jf

t!

i~·(%-y) tr {~( ) ~( ) 1 Y1 "

-}

P

-

1.1'.1. -

2k W (I.) (2,..)3 emission '\

(2.44')

) is the probability of quasiparticles emission per unit time.

From Eq. (2.41), one readily verifies that T±(r, y)

=

.E±(r, y).

Thus ir ± (k) has the same physical meaning as il: ± (k), and Eq. (2.39) can be written in the form of the ordinary relation of detailed balance:

+

(I

n) ~mission = n Wabsorption •

(2.39')

In the above, we have made rather detailF discussion of the situation of a single-component operator. But we believe that, in the multicomponent case, n still possesses the physical meaning of quasiparticle number density and Eq. (2.35) can be considered as some generalization of the transport equation. Before closing this section, we want to point out that, because of causality, the retarded Green's function G,(x,y), in the momentum representation corresponding to the relative coordinate x - y, should be analytic in the upper half of the ko complex plane. If r> in Eq. (2.29'), the amplitude of the quasiparticle attenuates when moving in a dissipative system, and the demand of analyticity is fulfilled automatically. If r < 0, the quasiparticle propagates in a proliferous system where its amplitude increase, and we have to take a path of integration to pass over the poles of G, in the upper half plane from above in order to guarantee the preservation of causality.

°

III. THE W-T IDENTITIES AND SPONTANEOUS SYMMETRY BREAKING

Suppose that the Lagrangian of the system is globally invariant under a Lie group G; the symmetry leads to a set ofW-T identities satisfied by the c1osed-time-path Green's function. The group G may include the space-time symmetry· as a subgroup. Let fJ (x) be a fundamental field and Q(x), a function of fJ(x), be the order parameter of interest. There are many components on fJ (x) and Q (x), which form a unitary representation of the basis of G. Under an infinitesimal transformation of G, fJ (x) and Q(x) transform as

11"(,,)

11'(") 611'(")

=

=

11'(")

+ 611'(")'

".(;t~O) - r:(,,)8,,)qJ(")

(3.1)

= itrp("n",

and Q(,,) 6Q(,,)

Q'(,,)

=

=

',,(ii.~O)

Q(,,)

-

+

6Q(,,),

r:8,,)Q(")

(3.2)

= iL"QC"n"

ta

where are the no infinitesimal parameters of the group G; j~1 and f~1 are the matrix representation of the generators acting on ({J (x) and Q(x) and are Hermitian. Under the above transformation of G, ~(x) and the space-time point x,. are related by (3.3)

In the following, we let 652

ta be infinitesimal functions of x. It can then be easily proved that

Chin. Phys., Vol. 1, No.3, July-8ept. 1981

ZHOU Guang-zhao and SU Zhao-bing

652

553 the Lagrangian transforms as

where (3.S)

is the current in direction a. If the Lagrangian is globally invariant under the group G, then (3.6) Equation (3.6) indicates that the current.l.!(x) is conserved when rp (x) is a solution of the Euler equation. With use of Eq. (3.6), Eq. (3.4) can be written as st'(cp'(,,))

:i~'

=

q(cp(x-'))

+ i:c,,)all'aC,,).

(3.7)

Equation (3.7) gives the transformation of the Lagrangian under a local transformation when the Lagrangian is only globally invariant under G. We now proceed to derive the W-T identities satisfied by the closed-time-path Green's function, making use of Eq. (3.7). Employing the method used in field theory, one can readily prove that the generating functional of the closed-time-path Green's functions can be w.-i.tten in the form of a Feynman path integral. Introducing the external sources J(x) and h (x) of rp (x) and Q (x), respectively, the generating functional can be represented by Z[J(,,), he,,)]

=

N

1

[dcp(,,)]exp{i

L

[!.f(cpC,,)) -

- h(")Q(,,)]d4x-}
JC")cpC,,)

=-

00),

(3.8)

where N is the normalization constant. The only qifference from ordinary field theory is that the Feynman path integral has to be integrated along the closed path p, with the boundary conditions at two ends determined by matrix elements of the density matrix p. The variable of integration ofEq. (3.8) is then changed from rp (x) to rp '(x) which is obtained through a local transformation of the group G, the parameters t" (x) being infinitesimal functions satisfying the boundary conditions

'a(x, t± = - 00) = 0, lim 'a(x, t) = o.

(3.9)

hEi-GO

Under such a unitary transformation, the measure [drp (x)] does not change, and, through Eq. (3.9), neither do the matrix elements of the density. As a result, we get

alii: (cp(,,) =

i

lJJ~")) Z [j(,,), h C,,)] = + hC,,)i. a lJh~")]

[J(,,)ta

lJJ~")

Z[J(,,), he,,)].

(3.10)

Let us introduce the generating functionals of connected Green's functions and vertex functions: 653

Chin. Phys., VOl. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

653

554 W[J(~),

T[ep.(so), Q.(so)]

=

(3.11)

h(so)] = iln Z[J(so), h(so)],

and

W[J(so), h(so)] -

L

(J(so)ep.(so)

t

h(so)Q;(so)d4z)

(3.12)

where

ep.(so)

=

6W 6J(so) ,

Q.(so)

=

6W 6h (so)'

(3.13)

With use of the commutation relation

6 _Z i_ 6J(so)

=

6 _], Z rep. (so) + i _ 6J(so)

(3.14)

Eq. (3.10) can be written as

fJ,j: (ep.(so)

+ i 6J~SO)) = -

i[J(:t)la ep.(so)

+ h(so)LaQ.(:r)].

(3.1S)

Equation (3.1S) is then the desired set ofW-T identities. It has the same form as that in ordinary field theory, but here, x can be any point on the closed path p. With use of the representation of the generating functional of connected Green's functions, Eq. (3.1S) can be written as

fJ,j:

C~~) + i 6J~SO)) =

-

i

[J(~)/a 6~~) + h(so) La 6:~)]'

(3.16)

Ifwe differentiate Eq. (3-.16) with respect to J (x) and h (x) and then let J (x) and h (x) approach zero, the W-T identities corresponding to the closed-time-path Green's functions of all orders can be obtained. With use of the representation of the generating functional of vertex functions, Eq. (3.1S) can be Written as

a..1',. (ep. () so + a

'1

I

+

I>

(6J(SO) )-' - - ( 6 ) -() -) 6ep. Y 6ep. Y 6T

j -( )

6P.

~

liT ] = ±,. [ --/aep.(so) 6ep.(~)

' ( LaQ. so).

(3.17)

In Eq. (3.17), we have assumed that the order parameter Q (x) is a boson operator, although the fundamental fields may also include fermion operators. By differentiating Eq. (3.17) with respect to tp.(x) and Q.(x), the W-T identities corresponding to vertex functions of all orders can be derived. Making use of Eq. (3.17), we now discuss the spontaneous breaking of the symmetry, to see the difference between the W-T identities of the closed-time-path Green's functions and those identities of the Green's functions in ordinary field theory. Let us assume that, when the external sources J(x) and h (x) are zero, the equation 6T

(3.18)

--=0, lIep.(~)

=

has the constant solution 'Pc (x) = 0 but with Qc (x+) = Qc (x_) Qco =#= O. This corresponds to a state with symmetry spontaneously broken. Differentiating Eq. (3.17) with respect to Q(y), letting I{Jc =0 and Qc (x+) = Qc (x_) = Qco (x), and then integrating over x along the closed path p, we arrive at 654

Chin. Phys., Vol. 1, No.3, July-Sept 1981

ZHOU Guang-zhao and SU Zhao-bing

654

555 (3.19) In the single-time representation, it can be written as

I

=

Tr(y, x)1,.Q
(3.20)

0,

where

Tr(y, x)

= T,(y+, x+) - T,(YH x_) = Tp(y, x) - T+(y, x)

is the second~rder retarded vertex function. Writing Eq. (3.20) as matrices, we get Tr

l,.Q
•

(3.21)

with iaQcO considered as a vector. Equation (3.21) shows that iaQcO is the eigenvector of the corresponding to eigenvalue zero. matrix

r,

If H is the subgroup which leaves QcO invariant under the operations of the group G and if H has n H generators ia' with a = 1, ... ,nH , then we have

I,.Q
=

,,=

0,

(3.22)

1, .. • ,nH.

In the coset G IH generated by the subgroup H, we have ia QcO .,,0, for a = n H + 1 , ••• ,nG. Thus, in Eq. (3.21), there are only nG - n H nonzero eigenvectors. Taking the complex conjugate of Eq. (3.21), we have

Q:'o/,. • T.

=

(3.23)

0,

r.

where is the advanced Green's function. In term of the dispersive part D and the attenuation part A, we get

Q:'olp • D . l,.Q
=

Q:'ol, • A • l,.QcIl

r,

= o.

(3.24)

r,

From the discussion in Sec. II, we know that G, = -I; the zeros of are therefore the poles of G,. Thus, the second-order retarded Green's function G, has nG - n H lossless collective excitations (poles). When QcO(x) = QcO is a constant solution, in the momentum representation corresponding to the relative coordinate x - y, these collective excitations OCf:ur at energymomentum PI' = 0 and are therefore the ordinary Goldstone excitations. Another important question is whether Eq. (3.18) possesses a stable solution for the order parameter, (3.2S)

When Qo(x) is nonvanishing only in a given region of space, this solution is called a soliton-type solution. If Qo(X)

=

exp {ik • x},

then it is called a laser-type solution. Since these solutions lead to the spontaneous breaking of translational -invariance (soliton) and phase-shift invariance (laser), G, possesses poles in the direction of V .. Qc(x) as- a result of the W-T identities. Physically, these poles indicate that, because of the specification of the coordinates of the soliton, the fluctuations of its momentum will diverge; and, in the same way, because of the specification of the phase of the laser, the fluctuations of the particle number will diverge. But we know that momentum and particle number should be conserved and even if a wave packet is formed at a certain time, it cannot remain stable. A stable wave packet leads to poles in G" which will in turn result in the divergence of many physical variables (e.g., energy). Since this is not allowed. Eq. (3.18) cannot possess any stabie solution. From the above discussion, we learn that it is the quantum effect that 655

Chin. Phys., VOl. 1, No.3, July-Sepl 1981

ZHOU Guang·zhao and SU Zhao-bing

855

556 leads to the spreading of the wave packet. But this does not rule out the existence of soliton-type and laser-type solutions of the classical equation of the order parameter. Thus we should first solve for these solutions from the classical equation, introduce the corresponding collective coordinates, quantize these coordinates, and then study the related quantum effect. The effect of the W-T identities in laser-type solutions will be discussed in another paper.

IV. THE W-T IDENTITIES IN A GAUGE THEORya Let G be a compact Lie group, with nG generators i; which obey the commutation relation (4.1)

where.fjk' are the structure constants of G, which can be made real and totally antisymmetric by a suitable choice of ~. We assume that G is an internal symmetry group which does not affect the space-time coordinates. If the system is gauge invariant under group G, its fields can be divided into gauge fields A ~(x) (with 1= 1, ... ,n G andJL = 0,1,2,3) and matter field 9'a(x) (with a = l, ... n). Here, A ~(x) is a vector field which forms a canonical representation of G while 9'a(x) consists of scalar fields and spinors which form unitary representations of G. Let (j), be the infinitesimal parameters of the . group G. Under an infinitesimal transformation (4.2)

9'a(X) and A ~(x) transform as

ep.(:r) - ep:(s) = ep.(s) + 6ep.(:r), 6ep.(:r) = it!bCO,(:r) epb (s) , A~(s)

At(:r) 6At(s)

=

=

At(:r)

fi/t,A!(s)co,(s)

+ 6At(:r) , + O"COi(S) ,

(4.3) (4.4)

where t ~b is the representation matrix of i, on 9'a. If the system is gauge invariant under the group G, its Lagrangian Lin. (9'a (X), A ~ (X)) transforms according to Eqs. (4.3) and (4.4) as Lin.(ep~(:r), At(s)) = Lin.(ep.(s), At(s)).

(4.5)

where (j)Ax) is any infinitesimal function. In field theory. we have to fix the gauge condition and quantize4 the system in that definite gauge. We write the gauge condition as F/(ep.(s), A~(s)) .... 0,

I = I, ... , nG'

(4.6)

and perform the quantization under this condition. This is equivalent to starting off from the follOwing effective Lagrangian:

L.ff(<
~

F/(ep., ADF,(ep., AD

+

=

I

LI •• (I('., AD c/(s)Mll'(z, y),"/,(y)d4y

(4.7)

where Mll'(x, y)

+

=

6F/(cp:(s), A~(s)) =; ~ 6co/,(y) oep.(:r)

of

OAte:r) [filt/,A!(s)tJ4(z - y)

+

t~epb(s)64(Z _ y)

0,,64(z - y)6i1] ,

(4.8)

and c/(x) and.c/(x) are the Paddeev-Popov ghost fields which are anticommuting scalars. 656

Chin. Phys., Vol. 1, No.3, July-Sept. 1981

ZHOU Guang-zhao and SU Zhao-bing

656

557 As in field theory, in gauge theory we represent the generating functional of the closedtime-path Green's functions in the form of a path integral Z[Jpi(X) , J,(x) , 7J/(x) , ii/ex)]

Xexp {i

L

A~" Ch

[L.ff(ep"

=

J [dep.(x)][dA~(x)][dc/(x)][dc/(x)]

N

c/) -

JpiAI'i -

= - oo)lplc/1(x,

X (c/1(x, t+

].ep. t_

ii/c/ - 7J/cdd4X}

= - 00)

(4.9)

where J,.;(x), J.(X).l1,(x), and i'i,(x) are external sources, 11, and i'i/ being anticommuting cnumbers, ¢Ix) represents the rest of the fields, and N is the normalization constant. For the fields of all nonphysical particles, p corresponds to their vacuum state. Becci, Ronet, and Stora5 have proved that the etrective Lagrangian is invariant under the following global supertransformation (BRS transformation):

= it!bepb(X)C/(x)6l E D!(/Pb)c/(x)61, 6A~(x) = (fj."A!(x) + 6j/Op)c/(x)61 == D~j(A~)c/6}.,

6ep.(x)

6c/(x)

= - -1

IJc/(x)

=

2

(4.10)

fli.,Cj(X) C.,(x) 61 ,

-F/[ep., A~]61,

where S,t is an anticommuting c-number. Equation (4.10) is the global supertransformation which leaves the etrective Lagrangian L. invariant. We can employ the method in Sec. III to deduce the W-T identities for the corresponding closed-time-path Green's function. The W-T identities are of the same form as those in ordinary field theory except that the space-time coordinates can be any point on the closed path. In the following, we shall only present the results without making any further detailed discussion. Let us introduce the following new external source term -

K.(x)D!(epb(X))C/(x) -

Kj(x)D~j(A~(x))c/(x)

_1.. L/(x)f/i.,Cj(X)C.,(x) 2

(4.11)

into the Lagrangian, denoting the generating functional of the closed-time-path Green's functions by

= -

W [J., Jj, '1]1> ii/, K., Kj, Ld

i In Z [l., Jj, 7J/, iiI> K., Kj, Ld

(4.12)

and introduce the generating functional of the vertex functions. r[ep., A~, C/, Cl> K., K I ,

-L

(l.ep.

+ JjA~ +

Ld

ti/ C /

+

=

W[J., J I , 1J/, 7;1> K., Kj,

Ld

1J/ C,)d4x,

(4.13)

where ep.(x) c, (x)

=

6W 6J.(x) '

"( ) 6W A~ x = - ( ) ' 6Jj x

=

6W 67),(x) ,

6W 61J,(*)'

C, (x)

=

(4.14)

In the above ditrerentiation, K., Kj, and L/ remain invariant. Under the BRS transformation, the W-T identites can be written as

~~+

6K.(*) 6ep. (*) 657

6r

IJr

6KI (*) 6A~(*)

Chin. Phys., VOl. 1, No.3, July-Sept. 1981

+'~~+F,~=O. BL,(*) 6c,(*)

(4.15)

6c,(*)

ZHOU Guang-zhao and SU Zhao-bing

657

558 From Eq. (4.15), the W-T identities corresponding to closed-time-path vertex functions of any order can be easily derived. Their applications will not be discussed here. 'I. Schwinger, 1. Math. Phys. %,407 (1961); L. V. K!e1dysh,1ETP ZO, 1018 (1965); R. A. Craig, 1. Math. Phys. !I, 605 (1968); R. Mills, Propagators/or many particle systems (Gordon and Breach, New York, 1969); V. Koren_ man, Ann. Phys. (N.Y.)·B, 72 (1966); V. L. Berezinskii, lETP, 16, 137 (1968); G. Ni~n and A. Sjolander, Ann. Pbys. (N.Y.) 411, 249 (1968); C. P. Enz, The many body problem (Plenum, New York, 1969); R. SandstrOm, Pbys. Status Solidi 38, 683 (1970~ C. Caroli, R. Combescot, P. Nozieres, and D. Saint-lames, J. Pbys. C 4,916 (1971); A. G. Ha, 1. Phys. A 8, 214 (1974); Mol. Phys. 28, I (1974). 2J. C. Ward, Phys. Rev. 78, 1824 (1950); Y. Takahashi, Nuovo Cimento 6, 370 (1957). 3A. A. Slavnov, Tear. Math. Fiz. 10, 153 (1912); J. C. Taylor, Nucl. Pbys. B 33, 436 (1971). 4L. D. Faddeev and V. N. Popov, Pbys. Lett. %5B, 30 (1970); G. 't Roaft, Nucl. Pbys. 833, 173 (1971); B. W. Lee and 1. Zinn-Justin, Pbys. Rev. D 5, 3121 (1972); D 5, 3137 (1972). 5C. Becchi, A. Rouet, and R. Stora, Phys. Lett. 52B, 344 (1974); Ann. Pbys. (N.Y.) !IS, 287 (1976).

Translated by King Yuen Ng Edited by Stanley Wu-Wei Liu

559 PHYSICAL REVIEW B

VOLUME 22, NUMBER 7

I OCTOBER 1980

Closed time path Green's functions and critical dynamics Guang-zhao Zhou, Zhao-bin Su, and Bai-lin Hao Illstitute of TheoreticQI Physics. Academia SillicQ. Beijillll. C/tiIlQ

Lu YU' LYII/QII LQborQto/:~·. Physics Departmellt. Harvard Ulliversi(Y. CQII/bridlle. MQSSQchusl'IIS 02138

(Received 30 October 1979) The closed time path Green's function (CTPGP) formalism is applied to the critical dynamics. The related results for the CTPGF approach are brieny reviewed. Three different forms of CTPGP's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and .conserved var.iables are derived from the vertex functional on the Closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter nuctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGP's are discussed. The problem of renormali.zation for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to syst.ems near the stationary states far from equilibrium, whenever there exists an analog of the potential function ("free energy").

I. INTRODUCTION

The closed time path Green's function (CTPGF) formalism, developed by Schwinger) and Keldysh,2 has been applied to a number of problems.) As pointed out by Zhou and 8U,4-6 this technique is quite effective in investigating the nonequilibrium statistical field theory. They used thi·s method to analyze the Goldstone mode in the steady state for nonequilibrium dissipative systems such as unimode lasers in the saturation region. 7 In this article we apply Ihe CTPGF formalism io study systems near equilibrium phase transition point. The complete system of equations for critical dynamics, including automaticaUy the mode coupling terms and the Lagrangian formulation of the field theory are derived in a unified way. This provides a microscopic justification for the semiphenomenological models in critical dynamics and indicates various possibilities for improving the existing theory. In the vicinity of the phase transition point the long-wave fluctuations dominate. Since the corresponding correlation length is much greater than the thermal wavelength, the quantum effect is irrelevant. However, in the quasiparticle description,

such ·purely· classical field theory does not correspond to the Boltzmann limit, but approaches the "super-Bose" case, i.e., the quasiparticle distribution n ex: TIE where Tis the absolute temperature (with Ir - C - kB = 1) and E is the energy for the elementary excitation. Such a statistical field theory (or fluctuation field theory) has very close analogy with the usual quantum field theory . .In our viewpoint, the CTPGF formalism is a natural theoretical framework for studying such statistical field theory. Assuming the equilibrium density matrix for CTPGF we obtain automaticaUy the ordinary Quantum field theory for the low-temperature limit (T « E) and the existing static criticat phenomena theory for the high-temperature case (T »E) (see Appendix A). If the high-temperature limit near equilibrium state at the critical point is taken, the complete system of equations to describe the critical dynamics foUows naturaUy, as will be shown later in this paper. By introducing the "response fields," noncommutative with the basic fields, Martin, Siggia, and Rosel constructed a classical statistical field theory (the MSR field theory) in close analogy with quantum field theory. As will be shown below the structure of the MSR field theory becomes clearer in 3385

560 3386

ZHOU, SU, HAO, AND YU

the framework of CTPGF's. In Sec. II we briefly summarize the related results for the CTPGF's, some of which are believed to be new, while others are known or have been published elsewhere. 4- 6 A more or less complete list of formulas is given for reference convenience and to make up for the deficiency that papers 4- 6 were published only in Chinese. The perturbation theory and the generating functional formalism for CTPGF's are outlined. Three different forms of general multipoint CTPGF's are defined and the trllnsformations from one to another are described. Some computation rules which greatly simplify the usually complicated calculations involved when using the CTPGF are derived. These seem to be highly desirable especially in view of the fact that the technical complexity is one of the causes why the CTPGF approach has not found applications as wide as it deserves. These algebraic identities are shown to be the consequences of some basic properties of the CTPGF, opening new perspectives not inherent in the ordinary Green'sfunction formalism. The related properties of the two-point functions are outlined. The Feynman path integral presentation for the generating functional of CTPGF's is used to deduce the Ward-Takahashi identities under both linear and nonlinear transformations of the fields. In Sec. III a short account of the existing theory of critical dynamics is given. The generalized Langevin equation, the mode coupling, and the Lagrangian formulation of the classical field theory are briefly reviewed to facilitate the comparison with the results in subsequent sections. In Sec. IV the generalized Langevin equation for macrovariables is derived from the equation satisfied by the generating functional for vertex functions in the CTPGF formalism by differentiating the microand macro-time scales of variation and averaging over the micro-time scale. In generarrorm this is true for both order parameters and conserved variables. The essential point is to determine the transport coefficient matrix y-I (t) connecting these quantities. The proper form of the equation for conserved variables can be deduced from the WardTakahashi identities and the linear-response theory. Comparing this form of equation with the general one yields two blocks of the ')I-I(t) matrix, one of which couples the conserved variables together and the other couples the conserved variables with the order parameters. The other two blocks of ')I-I(t), one of which connects the order parameters and the other one connects the order parameters with the conserved variables are determined through symmetry considerations. It is important to emphasize that mode coupling terms appear naturally in these equations. They are not "introduced from the outside," as in the existing theory. Applications of the general theory to particular dynam ic models are outlined.

In Sec. V the path integral formulation for the CTPGF's is used to derive the effective action for order parameters. Through Fourier transformation of the path integral the generating functional in the random external fields is introduced. Averaging over random fields yields the effe,tive adion, the general properties of which are also discussed. The most plausible trajectories are described by the time dependent Ginzburg-Landau equations, i.e., generalized Langevin equations without random forces. Fluctuations around the most plausible trajectories are considered. There is a possible new way of describing fluctuations in the CTPGF approach arising from the fact that field operators may take different values on positive and negative time branches. It turns out that with the one loop approximation of random fields which is equivalent to the Gaussian averaging, and with second-order fluctuations on different time branches the existing Lagrangian formulation of the classical statistical Jield theory, i.e, the MSR theory reappears. In Sec. VI we summarize the main results obtained and discuss the potential possibilities of the CTPGF approach with regard. to improving the existing theory of critical dynamics. In Appendix A the problem of renormalization in the finite-temperature field theory is discussed. It is emphasized that near the phase transition point the leading infrared divergence has to be separated before the ultraviolet renormalization may be carried out. The necessity of using noncommutative operators to describe the time evolution of classical field theory is also discussed. In Appendix B a proof is given for two theorems of Sec. II dealing with transformations among different forms of CTPGF's. Further useful examples and some technical details are described. Throughout this paper we deal mainly with the applications of the CTPGF approach to dynamic critical phenomena, but it is clear from the presentation that the whole theoretical framework is also applicable to systems near stationary states far from equilibrium, provided the long-wave fluctuations are dominant.

II. SUMMARY OF THE CTPGF FORMALISM A. Definitions and generating functionals

For simplicity we shall consider only multicomponent Hermitian Bose fields ((I(x). Extension to more general cases is obvious. The Lagrangian density can be written as .e=.co(CfJ) - V«({I) -((I(x)J(x) where J(x) is the external field.

(2.1)

561 3387

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

CTPGF for CP(x) is defined as

Ap being a free propagator. Substituting Eq. (2.7) into Eq. (2.6), we obtain

GpO··· n}=(-;)"-ltr[T,(cP(J}·· ·,p(n};» (2.2)

ZU(x» =exp[-i

L

+8J~X}

11ZoU(X})NU(X»

where ifJ (;) and;> are the field operators and density matrix in the Heisenberg representation, index p indicates a closed time path consisting of positive (-00. +oo) and negative (+00. -oo) branches. The time variable I can take values on either branch. T, is the time-ordering operator along the closed time path. The generating functional for the CTPGF's is defined as

as the correlation functional for the initial state. N(J(x» can be expanded into a series of successive cumulants N(J(x» =exp[-iWN(J(x») (2.11)

Z(J(x»=trk,[expl-i LifJ(x}J(X»)]p} ,

W",(J(x»

(2.3)

.h

(2.10)

=

where the integral is taken over the closed time path and the integration variable d 4x is omitted. In Eq. (2.3) the external fields on the positive and negative branches J(x+} and J(x-) are assumed to be different. Taking functional derivatives with respect to J(x} we obtain from Eq. (2.3)

GO· .. n} = ,

i

8"Z (I (x» 8J(I) ... 8J(n)

I

(2.4)

J-O

.

In the interaction representation the generating functional (2.3) can be rewritten as ZU(x»

=trkp[ exp(-i L[v(~,(x» +~ '(X)J(X)))IP} (2.5)

where ~ ,(x) satisfies the Euler equation for the free fields. The interaction term can be taken f.-om behind the trace operator to obtain Z (J(x» =exp[-i

f

p

-I]

vIi _8 8J(x}

x tr{T,[expl-iLcp,(X}J(x)ll;>}

(2.6)

It is easy to show by generalization of the Wick theorem that

T,!expl-i J,
J,
(2.7)

where: : means the normal product and ZoU(x» is the generating functional for the free field ZoU) =exp[-f

f LJ(X}Ap(x -y}J(y}

I.

(2.8)

(2.9)

wit

i

-L

n-I Il!

f ... f,NO

... II}J(I) ... J(n} •

P

(2.12)

where NO ... 1I}=(_;)"-l t r[:rp,(J) .. ,
(2.14)

WU(X» = i InZ U( x» .

G:O . . . II}

= - - - - " '8"W --'-'---

IiJ(J} ... IiJ(II}

I1-0

= (_;)_-1 (Tp[rp(J) ... 'P (n»)), (2.15)

562 3388

ZHOU, SU, HAO, AND YU

where ( )c stands for tr( ... p), with the connected parts to be taken only. The normalization condition for the generating functional is Z(J(x» IJ+(x)-J_(x)-J(x) -1 ,

(2.16)

W(J(x»IJ+(x)-J_(x)-J(xl=O .

(2.17)

It is essential to point out that this condition does not require J(x) itself to vanish in contrast to the ordinary Green's function formalism. We shall frequently make use of ihis basic property below. As in the usual field theory, we perform the Lllgendre transformation for the generating functional r«({',(x» - W(J(x» - LJ(x)({',(x)

(2.18)

where ((,,(x) = Ii W(J(x»/IiJ(x)

(2.19)

As a consequence of Eq. (2.17), it follows from J+(x) -L(x) that ({' ,+(x) - ((',_(x) .

B. Transformations of three sets of CTPGF

In the CTPGF approach we have to deal with three different forms of functions: (a) Functions on the closed time path G,(l2 ... n) with subscript p, which appear under the integrals over the closed time path and are used for a concise writing of formulas. (b) The tensor functions G(12 ... n) with time arguments on positive or negative time branches which appear under the integrals over the single time axis (-00, +00) and are used for constructing the perturbation theory lin what follows we shall specify them by the Greek subscripts G ../I ... p02 ... n) with a, fJ ... = ±, etc.1. (c) The retarded, advanced, and correlation functions, representing the physical quantities G(12 . . . n) which will be denoted by the Latin subscripts Gi} ... (12 .' .. ) with IJ ••• -1,2. Either of tensors Gand G has 2' components. Some of the relationships among these functions were given before,2-4 but our main point is to generalize them to the multipoint function case. To start with the transformations we specify first our notation. The Pauli matrices are written as

(2.20)

u. (~ ~].

From the definition (2.18) we have lir«({',)/Ii({',(x) =-J(x) .

=

(2.21)

This is the basic equation for the vertex functional, from which the generalized Langevin equation will be derived. Taking the functional derivative of Eq. (2.19) with respect to ((',(x) and that of Eq. (2.2l) with respect to J(x), we obtain

U3

G.

U2 =

(~ ~l

U3 =

(~ ~I)

will appear frequently together with The real orthogonal matrix

-II 1

Gand

. u. with

. (2.25)

L G:(x,y) r,( y,z) = -li:(x - z) , L r,(x,y)G:(y,z) = -li:(x - z)

(2.22)

where the two-point vertex function

r

(12)!!! ,

2

1i r«({',(x» 1i({',(l)1i({',(2)'

(2.23)

is used for the transformations between Gand The multipoint step function 9 is defined as

G.

I, ifl.>12 ... >1. , 8(1,2, ... 11) - {O. otherwise . (2.26) It is the product of the two-point step functions

Actually Eq. (2.22) is the Dyson equation for the CTPGF's, from which the kinetic equation for the distribution N and the energy spectrum and dissipation for quasi particles can be deduced. 3- 5 Here Ii,(x - y) is the Ii function on the closed time path. It is defined for arbitrary functions along the closed time path that (2.24) where x can take values on either branch of time. Up to now we have considered CTPGF's for the basic fields ({'(x), but all the above statements about ({' can be repeated word for word for all the composite operators Q«({'(x)).

9(1,2, , .. n)=8(1.2)9(2.3)··· 9(n-l,n) (2.27)

and can be used to define the T product T«({'(I)({'(2) ... (('(n» = !9(PI,P2 ... p,)({'(P.)CP(Pl)

({'(P.)

.

"'. (2.28) where summation goes over all possible permutations. The step functions satisfy some relations such as

563 CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

the normalization condition

3389

with

I9(PI.P2 ... P.) = I

(2.29)

G·(J2) =-i(T(lP(J)lPW)} =i

11'.

f

and the sum rule (n

> m)

G (l2) - -i (lP (2)lP (J ) ) +

I

9(J,2, "'m)~

S(PI,P2···P.) ,

=

i

2

li Z(J) liJ+(J)liJ+(2)

S2 Z (J) IlJ+(J)liJ_(2)

. . . S2 Z (J) G_(J2) = -, (lP(J )"'(2» = I SJ_(J)llh(2)

p.(I.2 ... ",)

(2.30) Gt( l2)

where Po(J, 2 ... m) means Po with 1 preceding 2, 2 preceding 3, etc. Going from Gp to Gwe have only to assign separately the "+" or "-" subscript in accordance to the value of the time argument. Take two-point function for example we have

I

/-0

I I

/-0

•

(2.32)

/-0

= -i (T(lP(l)lP(2») = i SJ}~~!~~(2)

•

1/-0

where t is the inverse time ordering operator. The transformation proposed by Keldysh 2 for twopoint functions (2.33) if written in components (2.34)

(2.31)

can be generalized directly to the multipoint case

G'I'2···'o02 ... n)=2 012 - IQ'\"IQ'Z"2 ... Q,.... G .. I · ..... (I2 ... ,,) .

(2.35)

The inverse transformation is given by

G" I " 2 ... "0 02 ... n)-21-0/2Q~I'IQ~2'2 .,. Q~o'oG'I'2 ... '0(12 ... n) .

(2.36)

We shall see below that Eq. (2.35) contains all possible retarded, advanced, and correlation functions, associated directly with the physical quantities. The expediency of such choice of numerical coefficient becomes clearer somewhat later. The specific features of the CTPGF's in the form of tensors can be characterized by two theorems, proof of which and more involved examples will be postponed to the Appendix B. a. Theorem I. The component of G with all subscripts equal to 1 vanishes, i.e.,

GII

... I (l2

(2.37)

... n)=O .

As consequences of this theorem for one-point and two-point functions we have G\(x) =0,

G+(x) =G_(x) , GII(xy) =0,

G++(xy) +G __(xy) =G+_(xy) +G_+(xy)

(2.38)

or (2.39)

GF+Gt- G ++ G _ .

b. Theorem 2. The other components of

Gcan

be expressed as

I'

(;2 ... 21k).I···1l0-k)(J2 ... n)-(-;)·-I

pi

12···.

("1'2'" P,.

I

9(PIP2 ... P.)«(··· ((1'1),(1'2», ... ).(1'.») •

where ( ...• lP(p,» =

I {I

,lP(p,)!. if k + I =S:p, =s: n .lP(p,) I. if 1 =S:p, =s: k .

(2.40)

the prime over summation indicating that the cases k + 1 =s: 1', =s: n are excluded from the possible permutations. The component (;2 ... 2( 1 ... n) corresponds to the case n = k. All the other components of G are defined as the results of the symmetry properties of the CTPGF's G ...

I ·•• 2 ••. ( " · i " ' j

,,·)=G ...

2 ..• I •..

("·j·" i ".).

(2.41)

564 3390

ZHOU, SU, HAO, AND YU

For the two-point functions we obtain

If we take J+(x) =J_(x) =J(x), then

G 21 (xy) "" G,(xy) = -ie( Ix,ly) ([
(2.48) G 12 (xy) .. G.(xy) = -i8( ly,lx ) ([

(2.42)

or in the matrix form

where the retarded function does appear naturally. For the product of two-point functions we obtain Dp (l2) =A,(13)Bp (32) ,

- [0 G.le G(xy) - G, G

(2.43)

D(l2)

15(12). =

Making use of Eq. ( 2.39), we obtain

(2.49)

A (13) 0"18(32)

or in components

G,=GF-G+-G_-Gt •

(2.44)

G.=GF-G_=G+-Gt Ge~GF+Gt=G++G_

The inverse transformation is given by Eq. (2.36) and can be written as G(xY )=+Ge(xy )llll++G,(xy ++G.(xy)L:

=A (13)0"38(32)

~II

.

)1l

(2.50)

This rule can be generalized to the multiple product with Zp = ApO) Apm ... Apl n )

-II

i

Am 0"3 Al2l t = A(I)O"IA l2l

-I

=

,

0"3 ACn)

(2.51)

... O"IA Cn )

(2.45) The last equation can be written in components as

The first theorem is valid for both connected and disconnected CTPGF's while the second one in general form is applicable only to the disconnected functions. Further details will be given in Appendix B. These relationships among different forms of the CTPGF's will be found quite useful in the applications. Here we shall illustrate them by several simple examples and derive some additional computational rules. The simplest case for the connection of the CTPGF's "in sequence" is the integral of two single point functions

(2.46)

Z, = A,m A,l2l ..• A,cn)

Z. - A.m A.l2l Ze -

i

(2.52)

... A.Cn)

A/I) ... A,Ck-1) AeCkl A.CHI)

k-I

Similarly, by use of the inverse transformation (2.36) and (2.45) we obtain

i

Z,. -

A/I) ... A/ k - Il A ~k) AP+Il ... A.Cn )

k-I

(2.53) where p. = +- or -+. If the multipoint functions stand under the integration, attention has to be paid to the order of the variables. For example the three-point vertex function f,O 23) = if p( 14) f p(25) f p(36) G,{ 456)

where j = (J+,J-) , cP T = (cp 1,
becomes r( 123) = i (r 0"3)( 14) (r0"3) (25) (r0"3) (36) G(456)

(2.54)

and then

r( 123)= i (fO"I) (14) (fO"I)(25)( to"I) (36) G(456)

Rp( I) = Gp ( 12)Jp(2)

or

or in components (2.47)

or

fill

=0 .

f211 =if,f.f.G 211 f221 = i (fef ,f .G I2I + f,fef .G 211

+ f ,f,f .G 22I )

565 3391

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

No additional numerical coefficient appears after transformation in any inherent to the theory relations among the multipoint CTPGF's. Por example, the 4 ) is related to the amfour-point v!rtex function putated Green's functions W. as 9

It is easy to show from the symmetry properties that for the Hermitian Bose field

n

r ;4) = -

+3 Wp(J)Gp!2) Wpm

WpW

(2.63a) which after the Pourier transformation becomes

G(k) =

[6 (-0 JT =

-IT\6"(

t

-0 0"\ = _0"\6 (k) 0"\

It can be transformed into

r(4) =

_

(2.63b)

W(4 ) +3 W(J) ITJG(2) IT] W(J) or

and f(4) = _ W(4)

G(k) = GT( -k) = -O"JG*( -k) O"J = -ITJG t (k) O"J

+ 3 W(J) 0"\ G m IT\ WOl

(2.63c) Contrary to this, a numerical factor may appear in some relations obtained by an artificial contraction. Por example, the relation

becomes

+

A(12) = B(134)( IT\) ll'( IT\) 44'(' (3 '4'2)

+.

with the coefficient These examples justify the choice of the numerical constant 2"/2-\ in the transformation formula from G to G [Eq. (2.35)l. The Ii function on the closed time path 8, (x -y) can be written in the matrix form as S(x - y)

=

/l(x - y)

(2.55)

O"l

where T means transposition, • complex conjugation. and t Hermitian conjugation. All these properties are transmitted to rand through Eq. (2.59). Similarly, the specific features of multipoint Green's functions are conveyed to the corresponding vertex functions through relations like Eq. (2.54). The transitivity of the CTPGP's also holds for some connection "in parallel," i.e., the product of several CTPGP's connecting two points, which itself is the constituent of a Green's function. Consider for example

r

(2.64)

It may be a self-energy part of G,. In fact by use of Eq. (2.39) and G,(l2)G.(l2) =0 we obtain G!+ (12) + G~_ (12) = G!_ (I 2) + G~+ (J2) . (2.65)

after transformation it becomes

Moreover, the matrix They are the time derivatives for the step functions on the closed time path, which in the matrix form are

•

e(I2) =

[e(l2)

1

0

e(21)

)

1

G.(G. +3G/) _ I ( 0 S(l2) = 4" G,( G,2 +3G/l G,[G/+3(G, +Ga)l)

I

(2.66)

(2.57) behaves much like a simple G.

and

_

e(2)

=

(0 e(12)

-e(21)

1

1.

(2.58) C. Further properties of two-point functions

The Dyson equation for the CTPGP's Eq. (2.22) can be rewritten as

GITl=rITJG=-s, GO"l=rIT\G=-8.

By use of Eqs. (2.59)-(2.61) and (2.36) the two point vertex function can be presented as

r

(2.59) It is interesting to note that all characteristic features of Gand Gare "transmitted" automatically to rand f. In particular, we have (2.60)

f= [0 raj

r, r,

(2.6 I)

(2.62)

(2.67) with B

=+iCrF+r t ) =+;(r++L)

(2.68a)

D

= (r F - r F) =

+ r a)

(2.68b)

A

=+;(L-r+)=+iCr,-l'a)

(2.68c)

+

-+ (r,

where 8, D, and A, are Hermitian matrices in the multicomponent fields or in the coordinate presenta-

566 3392

ZHOU, SU, HAO, AND YU

D. Path integral presentation and Ward-Takahashi identities

tion. Equations (2.68b) and (2.68c) can be rewritten as r,=-G,-I =-D -iA (2.69a)

r. =-G.- 1 =-D +iA

(2.69b)

We shall call D the dispersion part, which determines the energy spectrum of the quasiparticle, and call A the dissipative part, which describes the decay rate of the elementary excitation. The solution of the Dyson equation (2.59) can be presented as

Suppose the Lagrangian of the system is globally invariant under the Lie group G which may contain the space-time symmetry group as its subgroup. Let IP (x) be the basic fields, Q (x) order parameters, which are functions of IP(x). Both IP(x) and Q(x) have several components forming the bases of the unitary representation for G. Under th.e infinitesimal transformations of G IP (x) -IP '(x) -IP (x)

-I 1 G- =-T1 ( r,_I N,-N.r. ) =T(G,N,-N.G.) , (2.70)

llIP(x)

with

N,=(I+UI)(N+U3), N.=(N-U3)(I+UI) , (2.7\)

= , ..[ ii~ -

+ 8IP(x)

,

(2.79)

X: (x) B"lIP (x) = ii..IP(x) , ..

,

and Q(x) - Q'(x) = Q(x) +8Q(x) ,

(2.80)

where N is a matrix satisfying the equation

Nr.-r,N=2iB ,

(2.72a)

ND -DN -i(NA +AN) -2iB

(2.72b)

or

The causal propagator of the quasiparticle with energy E(i(), corresponding to the pole of G, and G., can be presented through the density operator n as

G1"=G,O+n)-nG..

(2.73)

By comparison with Eq. (2.70) we obtain (2.74)

Nlpnle=I +2n .

where , .. are a total of nG infinitesimal parameters for group G and i~. i~ are Hermitian representation matrices for the generators of G. (x) are associated with the transformations of coordinates

X:

(2.81)

It can be shown easily that the Lagrangian function transforms as .£(CP'(x»

d44~

dx

=,c(IP(x'»

In terms of n Eq. (2.72b) becomes

nD -Dn -i(nA +An) +i(A - B) =i(nA +An) +r + (2.75) which is the kinetic equation for the quasiparticle density n. S The right-hand side of Eq. (2.75) can be presented as (2.76) if the noncommutativity of n and A is ignored. It corresponds to the collision term in the kinetic equation which vanishes in the thermal equilibrium; i.e.,

r_

r+

=

1 +n _e ll • C• n

1

.

(2.77)

It can be shown s that ir ± - iI± where I± is the proper self-energy part, which itself is proportional to the probability of emission (+) or absorption (-) of the quasiparticles per unit time so

ir± >0 .

(2.78)

Equation (2.77) shows that in thermal equilibrium the probability of absorption is greater than that of emission as expected.

+

8.c

18IP(;) -

8.£

I

a"8a"IP(x) 8IP(x)

(2.82) where g(x) - i

a8~

8 "IP x

) ilP(x) +u:(x)

(2.83)

is the current in direction Q. If the Lagrangian is invariant under the global transformation of G it follows that

" .,,( ) - ·1"

8.£ o"J .. x -/ u"8B"IP(x)

-~I/m() IIIP(x) x. g'l'

(284) .

j:

The Eq. (2.84) shows that the currents (x) are conserved if IP(x) is the solution of the EulerLagrangian equation. By use of Eq. (2.84), Eq. (2.82) can be rewritten as .c(IP'(x»

~4X, (I·X

= .c(IP(x'»

+g

(x)B,,'''(x)

(2.85)

This is the transformation of the Lagrangian under

567 CLOSED TIME PATH GREEN'S fUNCTIONS AND CRITICAL ...

3393

the local action of G, if it is invariant under the global action of the same group. The generating functional for the CTPGF's can be presented in the form of a Feynman path integral by the well-known procedure in the field theory

Z (h (x).J(x»

=

f

N

[d f/I(x)] exp

[i 1: [..c(fP(x)) - hex) CPLr) - J (x) Q (x)] } (CP(x-,+ -00 /ii/cp(x.l- -00» =

=

(2.86) N being the normalization constant. What is different from the path integral in the ordinary field theory is that the integration is carried out over the closed time path and that the boundary conditions are determined by the density matrix p. Transforming integration variable in Eq. (2.86) from 'P (x) to CP'(x) under the local action of group G with infinitesimal parameters Ca(x), satisfying boundary conditions

(2.87) taking into account that the measure [d'P(x)] does not change under the unitary transformation and that the matrix element of p remains the same as a result of Eq. (2.87), we obtain (2.88)

By use of the commutation relation i 8h

~x) Z

= Z!IPc(X) = i 8h

~x)

I

and (2.89)

a,..i:[fPc(X) +i

Eq. (2.88) can be rewritten as

a,.(j:(x»

= a,.j:[fPc(x) +i8/llh(x)]

i

- -;[ hex) afPc(x) +J(x) LaQc(x)]

(2.90) This is the required Ward-Takahashi identity which has the same form as in the usual field theory, but here x can take arbitrary value on the closed time path. Introducing the generating functional for the connected CTPGF's W(h (x).J(x» - i InZ(h (x).J(x»

rIItpc~Y) I

iIIlIP~I;X) i acpc(x) + 1l~I;x) LaQc(x) 1 .

r('Pc(x).Qc(x» = W(h (x) .lex»~ [h(x)'Pc(x) +J(x)Qc(x)]

(2.96) can be used to discuss the spontaneous symmetry breaking and the Goldstone mode. s It is worthwhile to note that with the fluctuation effects being taken into account, the Eq. (2.96) does not have stable solitonlike solution

(2.92)

(2.97)

where rp c(x) = II W /Ilh (x),

Qc(x) = II W /IlJ(x)

(2.93)

we obtain from Eq. (2.90)

a,..Ia.,.!IlW M(x)

+.'8h(x) 8

I

(2.95)

Taking derivatives with respect to h (x), J(x) in Eq. (2.94) and then pUlling them to zero, we obtain successive WT identities for all orders of CTPGF's. The similar procedure in Eq. (2.95) with respect to fPc(x), Qc(x) will yield WT identities for the vertex functions. The equations for the vertex functional r in the vanishing external field

(2.91)

and the vertex function

-1:

1: !1l8;ci~)

where Qu(x) is different from zero in a limited domain of space, or the laser type solution with (2.98)

(2.94)

Up to now we have considered only the linear transformations of fields under the action of symmetry group. In critical dynamics the nonlinear transformations are also nceded. Suppose fPj(x) are basic fields, transforming under

568 3394

ZHOU, SU, HAO, AND YU

the action of an internal symmetry group G (j.e., the space-time coordinates are not involved) like (2.99)

transformation of G, we have ,c-,c'=,c+.it:a"ta(X) where

i" =~A('") . Ba,,'P; a,""

.a where ~a remain infinitesimal group parameters, but contrary to the previous case, here A/a('P) may be arbitrary function of '1'. If the Lagrangian is invariant under the global

(2.100) (2.101)

The Ward-Takahashi identity in this case can be derived also from the path integral presentation of the generating functional, but an additional term comes from the Jacobian of transformation. We have

f (d'P;lexpli J, (,c -)'P»)
xexpli

f (,c -N»)
(2.102)

where (ipl) '" ('P(x.f+=-oo)lpl'P(x,,_=-oo» It follows from Eq. (2.1 02) that

a"it:['Pk(X)+;IIJ;~X) 1= aa~~; ['P.I<+;IlJj~X) I-J;A ..;['Pjc+;IIJj~X)] If the loop correction terms are neglected, Eq. (2.103) turns out to be (2.104) This equation will be used to obtain the nonlinear mode coupling term in the generalized Langevin equation. III. SUMMARY OF CRITICAL DYNAMICS

There was a recent comprehensive review on the critical dynamics. IO We give here a brief summary of the basic results to specify the notations and to facilitate the comparison with our results. The properties of the system near critical point are described in terms of order parameters and conserved variables forming a set of macrovariables Q = (Q;.; -1.2 ... n}. The time evolution of these stochastic variables obeys the generalized Langevin equation (3.1 ) where the random force ~;(,), reflecting the effects of all degrees of freedom, not included in (Q/}, is assumed to be Gaussian distributed, i.e.,

The right-hand side function K;( Q) of Eq. (3.1) con-

(2.103)

sists of two parts (3.3) where the free energy F '" F( Q) as a functional of Q is dependent on concrete models. The static equilibrium condition BF/BQ;-O appears to be the Ginzburg-Landau equation. Therefore Eq. (3. I) without random force ~; is called sometimes the time-dependent Ginzburg-Landau equation (TOGL for short). In principle, the coefficient matrix uiJ may have both symmetric and antisymmetric parts. The symmetric part describes the relaxation. while the antisymmetric one describes the canonical motion. If only relaxation effects are considered, U {j may be taken symmetric. According to the fluctuation dissipation theorem, the same matrix uij does appear both in Eqs. (3.2) and (3.3). In diagonalized form U; = constant (dissipative relaxation) for the nonconserved Q;, and U; = -D; \1 2 (diffusion relaxation, D; being the diffusion constant) for the conserved Q;. The dissipative coupling of different modes can be described by means of the interaction terms in the free-energy functional, but the reversible mode coupling appears as stream term V;( Q) in Eq. (3.3). Usually it takes the form ll

569 3395

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

where the antisymmetric tensor Au is formed from the commutators or the Poisson brackets. As a rule, the linear approximation is accepted, i.e.,

derive the recurrent formulas and to calculate the critical exponents. For the last several years the critical dynamics has been reformulated using the field-theoretical approach. I ).14 The Gaussian stochastic process ei{t) can be presented by a stochastic functionalY Equation D.l) can be considered as a mapping of the Gaussian process ei(l) onto a more complicated process Q,(I). Performing such nonlinear transformation of the Gaussian stochastic functional yields the functional description for process Qi(I).16 A more direct way is to start from the normalization condition for /I functions under the path integral

0.5) where .IijJc are strLicture constants for the underlying symmetry group. The form of the expression (3.4) itself makes the conservation equation for probability to be satisfied, i.e., 0.6)

which means that the IQil space is divergence free, ensuring e- F to be the stationary distribution with detailed balance. It can be seen that the Langevin equation (3.!) is flexible enough to embody all possible factors. To our knowledge Eq. (3.1) is "assembled" by different reasonable arguments, so that it remains a kind of phenomenological model. The widely accepted approach in the theory of dynamical critical phenomena is to construct the perturbation theory by iterating Eq. (3.1),12 Since there are two different kinds of "constituent parts"response and correlation functions, the structure of the perturbation theory becomes quite complicated: The compact presentation of such perturbation procedure is given by the MSR field theory 8 mentioned before. In analogy with the static theory of original K. G. Wilson's formulation, Eq. (3.1) can be used to carry out the renormalization transformation to

I

[dQ)/lI~ -K(Q) -+~(Q) =1.

0.7)

Since the argument of /I function is not Q, but the whole expression O. I) it is necessary to insert the Jacobian I1(Q) for the nonlinear transformation from {; to Qi' Neglecting multiplicative constant I1(Q) turns out to be l6 11( Q) = ex p

(-+ I /I~~Q) tiXl .

D.8)

where dx - d'Xdl is the four-dimensional integration element. In what follows we shall omit dx for short. Presenting the /I functions in Eq. 0.7) in terms of the continuous integral, we obtain

0.9)

The insertion of factor exp{-i IIJ(x)Q(x) +J(X)O(X))} into the integral 0.9) yields the generating functional for the averages of all possible products of the field operators (in theory of probability it is called characteristic or moment-generating functional):

Zf(J)

=

I [dQ)

(~] ex p{ I

(iO(¥, -K(Q)

-eJ- t ~~

0.10)

-i(JQ +jO)]}

with the obvious normalization condition (3.1 J) The random force e( I) obeys the Gaussian distribution W({) cxexp(-+e

.

(3.12)

where 0'-1 is the inverse of the correlation matrix
211' exp {III -'2 QO'Q a,-- K(Q) I-'218K IiQ - iJQ - iJQ - I [dQ) [i!.L1 - -+iQ-Ian --I}

Z(J,J)

=

.

e, we obtain the Lagran(3.13)

570 3396

ZHOU, SU, HAO, AND YU

In their original paper MSR introduced the "response fields," Q in our notation, noncommutative with the basic fields, to simplify the structure of the perturbation theory and the renormalization procedure. As in the ordinary field theory, the noncommutativity of the variables is not evident under the path integration. The introduction of the Q fields doubles the number or operators. In the CTPGF approach the time path is divided into positive and negative

branches, so the number of operators is also doubled. As will be shown in Sec. V, the Lagrangian formulation of the MSR field theory follows naturally from the CTPGF formalism. In can be seen also that the noncommutativity of operators is not an artificial formal trick, but a necessity to describe properly the statistical fluctuations. The Gaussian integration over Qin Eq. (3.13) can be carried ou t to yield

(3.14)

This expression was first obtained as a stochastic functional in Rer. 16, but it turns out that Eq. (3.13) is the more convenient starting point ror the critical dynamics. I). 14

IV. FUNDAMENTAL SYSTEM OF EQUATIONS IN CRITICAL DYNAMICS

Generally speaking, both order parameter and conserved variables when regarded as macro variables are composite operators or the basic fields. We shall specify them by somewhat difrerent notation. The nonconserved order parameters may be written as Qr,(X) , ;=1,2, ... ,n ,

whereas the conserved variables Qr.• +.. (x)-q .. (x). Q=1.2 . .... m .

where q .. corresponds to the average of the zeroth component for the conserved current (4.0

Without sacrifice of generality both Qri and q .. can be taken to be Hermitian. Introducing the generating functional of the CTPGF vertex functions for the composite operators r(Qr), we obtain the equations which require to be satisfied by Qr [Eq. (2.21)); i.e., Ilr/IlQ,,(x) = -Ji(x),

; = 1. 2.... . n

+m

.

(4.2)

After taking the variational derivatives one puts Ji(x+) =Ji(x-) -J,('1,t>, from which it rollows [from Eq. (2.19)] that Qri(X+) = Qr'(x-) = Qi('X,t), where Ji('1.t>, Qi('1.t> are functions defined on the usual time axis (-00. +(0). We next show that Eqs. (4.2) lead to the generalized TOGL equations under the assumption that Qi are smoothly varying runctions or time. Suppose the macrovariables Qi('1. T) to be known at the moment T. At the moment I rollowing closely after T the left-hand side or Eqs. (4.2) can be expanded. If x sits on the positive time branch, we have

(4.3)

where r"ix.y) are two-point retarded vertex functions arter taking Q,.+=Qr-=Q. If xsits on the negative branch or time, the same is true due to Eq. (2.60) r, = r ++ - r +_ = L+ - r __. Since Qj( y.t) in Eq. (4.3) varies smoothly with time, to the first order or (Iy - T) we have (4.4) Substituting Eq. (4.4) back into Eq. (4.3) and taking into account that in the limit I"" Ix

-

T

(4.5)

571 3397

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

where Cij(X'. f.k o. d are Fourier transforms with respect to (Ix - ty) taken at the average time T = (Ix + ty) == T, we obtain

and .iY. For the moment let

+

I;(X'. T) == BBQr

,,+

(4.6) Here the matrix notation is used and Y ij ( X. Y. T ) are considered to be matrix elements with subscripts i X'

I c+ Q

_Q

c-

(4.7)

_QI,)

and we calculate the functional derivative of I;, considering it as a functional of functions Q (X'. T) with three-dimensional argument X'

where

Thus we obtain BI,(X'. d/BQJ( f. -r) =

r ++ij(X'. Y.ko =0. T) - r +_ij(x. f.ko =O.-r)

where the ko-O components of Fourier transforms appear as in Eq. (4.5). It can be shown in the same way that

where the symmetry properties of

r

following from Eqs. (2.59) and (2.63) are used. The difference

Bli(x. T) BQj(

(4.8)

f. -r)

vanishes due to Eq. (2.77), i.e.,

pative part of the vertex function A = Eq. (2.68c) satisfies the condition

r +_ = exp( -,Bko) r _+ near thermal equilibrium, so that there exists a functional F(Q/(x, with

T»

(4.9) Equation (4.6) can be rewritten as y(-r)aQ(T)/aT=-8F/8Q(-r) +J(-r)

(4.10)

If the macrovariables Q (T) do not change with time in the external field J, i.e, in the stationary state, then BF/BQ =J .

(4.1 \)

Hence F is the effective free energy of the system and Eq. (4.11) is the Ginzburg-Landau equation, determining the stationary distribution of macrovariabies. For systems in stationary states far from equilibrium expression (4.8) is equal to zero only if the dissi-

lim Aij(x. f.ko,t} =0 .

+;(

r _+ - r+_)

(4.12)

to-O

I n this case Ii can also be written as a variational derivative of the free energy or effective potential. Some of the stationary states satisfying the so-called "potential conditions," provided by the detailed balance, as discussed by Graham and Haken,11 must belong to this category. In the vicinity of all stationary states with the potential functions F Eq. (4.10) constitutes the system of time-dependent GL equations, but they are much more general than the TDGL equations in the usual sense since the mode coupling terms are also included. It is usually customary to multiply Eq. (4.10) by the inverse matrix .,,-1 (T) to obtain (4.13)

572 3398

ZHOU, SU, HAO, AND YU

Using the symmetry properties of the vertex functions, following from Eqs. (2.59) and (2.63) t

f(k) = fT( -k) = -uJt*( -k) UJ =-ud· (k)uJ

(4.14) it can be shown that the real part of r r is an even function of ko, while the imaginary part is an odd one, so ')I ( t) is a real matrix according to the definition (4.5l. In accordance with the numeration of the subscripts given at the beginning of this section, the matrix y( T) can be divided into four blocks. Two of them, corresponding to the conserved variables YaT./I'yand ')IaTJV' can be fixed completely by comparison with the WT identities. For the general case, the proper form of the two blocks associated with the order parameters can be determined only by the symmetry considerations. This will be discussed below. It is worthwhile to point out that Eq. (4.4) is equivalent to the Markovian approximation. In principle, the original Eq. (4.2) contains in itself the possibility of considering the memory effects. Under the action of the symmetry group G of the system the conserved variables transform as the generators [a of the group, i.e., (4.\ 5)

where 1 all' are the structure constants of the group and CIl are the infinitesimal parameters of transformation. The order parameters Q, transform as some representation i of group G (4.16) As shown in Sec. II D, if the Lagrangian of the system is invariant under the global symmetry transformations, the WT identities (2.95) are valid on the closed time path. In the present case, Eq. (2.95) can be wrillen as

(iJ"jf{( rp»

-

i(

/SQ~j~X) L;j'Qcj(x) + fa/l, /Sq~~x) q,(x) 1 (4.17)

where, as before, r 5 r(Q.,.qa) is the generating functional of the vertex CTPGF's for the composite operators. PUlling Q<+ = Q<_ - Q (T) and let J: '" (J:(rp» in Eq. (4.17), we obtain

aq"/iJT = 'l7Ta- i[Jj(x. T) LjjQj(x. T) +/a/l,J,kx. Tlq,(x. T»)

, (4.18)

where J, Q, q, etc., are functions defined on the usualtime axis (-00. +(0). To determine the conserved currents T.. we perform the following manipulations in analogy with the procedure used to deal with the thermal perturbations in the linear-response theory. By introducing an ad-

ditional artificial external source IlJ, superposed on the original source J, the system is forced to come into the stationary state iJ!!.,/iJl =0. In this case, Ta in Eq. (4.18) changes to j a, so that 'l7T~ -;( (1j + IlJ) LjjQj +/a/l,(1 /I + flJll)q,) =0

(4. I 9) Since the system is in the stationary state, we can use Eq. (4.1 I), i.e., /sF/SQj=Jj+IlJ, . /SF//Sqa=Ja+IlJa •

(4.20)

to replace the source terms Jj , J a by the functional derivatives of the free energy F. According to the linear-response theory the difference between ~ and the conserved current Ta without an artificial source IlJ can be written as (4.2 I) where 1all are the linear transport coefficients. Substituting Eq. (4.21) into Eq. (4.19) yields

I

.,. 21 SF ·I/SF L aQ + f /SF '17 J a = I all '17 \ /Sq /I - J /I + '\ /SQ, ~ J . a/l, /Sq Il q,

I

(4.22) Inserting the expression (4.22) for 'l7Ta back into Eq. (4.18) leads to the equation of motion of the conserved variables

I

~ _I '17 (.H.... - J iJT "Il/Sq/l Il 2

+il:~ -JjILjjQj+fa/l'(:~ -J+,] (4.23) Comparing Eq. (4.23) with the general equations for macrovariables (4.13) in the case of conserved variables, one determines two blocks of y- I matrix [y-I( Tl )aT.IlV= -[la/l'l7; + !fa/l,q,( X.

Tl )/S(J)(x - y)

[y-I(T»)aT.jV=-iLjjQj(x. Tl/Slll(x-y).

(4.24) (4.25)

Now we turn to the equations for the order parameters. If the order parameters Qj form the irreducible representation of the group G and take small values near the critical point, ')I-I can be expanded into a power series of Qj. It follows from the symmetry property tha t (4.26) where IT,. y' not depending on Qj, are determined by the kinetic and dissipative characteristics of the system. Similarly, by symmetry consideration another expansion can be written (4.27)

573 3399

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

where the ellipsis represents higher terms of Qi and f is invariant under group transformations and can only be a numerical constant in the lowest order of Qi' To determine the value of fwe consider the limit of vanishing dissipation. In this case the antisymmetric part of the matrix ')I ( T) in Eq. (4. I 0) is dominant. The same is true for the inverse matrix in Eq. (4.13). Comparison of Eqs. (4.27) and (4.25) gives

f

=

I .

(4.28)

Since the first term of Eq. (4.25) is independent of dissipation, Eq. (4.28) remains valid in its presence. Generally speaking the expansions (4.26) and (4.27) may contain other terms, including the crossover interaction of the dissipative and canonical motion. We shall not touch this problem here. In the approximation discussed above we obtain the following equations for the order parameters:

~~' =CTI:~ -J'I-iLu"Q)I:~ -J ..I ·

so in principle we can go further. Another point is that the derivative terms, coming from the Jacobian, appear only for the basic fields, but not for the composite operators which also transform the derivative terms for the composite operators. FirstlY, they may come from the loop corrections, secondly, and what is more likely in our opinion, they appear as a result of changes of measure in the path integral of the effective action (see the next section). As a concrete example consider the simplest model of the isotropic antiferromagnet, i.e., model G in Ref. 10. This system consist:; of two densities, a nonconserved order parameter Q which is a threecomponent vector representing the staggered magnetization and a conserved density q, also a threecomponent vector representing the total magnetization of the system. From the commutation relations (4.30)

(4.29)

Equations (4.29) and (4.23) form the fundamental system of equations for the critical dynamics. In the CTPGF approach J, coming from J+=L, is the real physical external field. It may contain the additional random fields, representing the effects of degrees of freedom, not included in the macrovar;ables Q,. Therefore we call this system of equations the generalized Langevin equations. The essential point of the above given derivation is that the mode coupling terms naturally appear in the generalized Langevin equations. Moreover, they actually have the form of Eqs. (3.4) and 0.5). To be more exact, the representation· matrix L,y appears in the coupling terms between the order parameter and the conserved variable, while the structure constants f ..fJ~ appears only in coupling terms among conserved variables. In the linear approximation of Qi' the first term of Eq. (3.4) gives no contribution. To be concrete, we divide the matrix A into four blocks. The reversible coupling among order parameters may be ignored, so that Au =0. The coupling terms with conserved variables in the eq uations for the order parameters Ai .. = iL,'jQ) are independent of q .., so that aAi ../aq .. =o. The two other blocks, A ..i=-iLijQj and A ..fJ=-!f"fJyqy appearing in the equations for the conserved variables also give zero contribution due to the antisymmetric character of the representation matrix i and the structure constants. To get the term with derivatives in Eq. (3.4) we have to start from the nonlinear WT identities derived also in Sec. II D. I n fact, by use of Eq. (2.104) we can repeat the derivation for the linear case and obtain the first term of Eq. 0.4). What we want to emphasize is that Kawasaki's formula (3.4) corresponds to the tree approximation in Eq. (2.103),

the structure constants and the representation matrix can be determined immediatelY (4.31) where E ..fJyand E, ..j are fully antisymmetric unit tensors. Substituting Eq. (4.31) into Eqs. (4.23) and (4.29), and bringing together the external field term and the derivative of the free energy, i.e., changing F-F-JiQi-J .. Q .., we obtain

(4.32)

By taking CT = r 0, I afJ = >'08 afJ and changing to the vector notation, we retrieve the system of equations ror model G.IO iii)

~

aT

8F 8F =-r o--::;+KoO x 80

8lj

(4.33)

an

8F /IF _ 8F -"-"- ='>'0'\72- +1:00 x --::;- +Koq x--=aT 8q 80 8q

The models A, B, and C are much simpler due to the absence of the reversible mode coupling. The other models such as E, F, H, and J models 1o and the SSS model lK can be treated in the same way. We shall not repeat these simple calculations here.

574 3400

ZHOU, SU, HAO, AND YU V. LAGRANGIAN FORMULATION OF STATISTICAL FIELD THEORY

Suppose cp" i = 1. 2.... I, are the basic fields of the system, 6i( CP),i = 1. 2, ... n + In, are composite operators representing the order parameters and conserved variables. Some of the basic fields may be order parameters also (as in the case of lasers). For simplicity we take all of them to be Hermitian Bose operators. In what follows operators will not be distinguished by special notations since their meaning is clear from the context. Assuming the randomness of the initial phase, the density matrix is diagonal at the moment l' = TO: (5.1)

The initial distribution of the macrovariables Q,(X) is given by P(Q;("x),

TO) =

tr! II(Qi (x) - Qi( qI (x))) pI

= I

I dql(x) Ia(Q;(x) - Q,( CP(x»)P (CP( x),

TO)

(5.2)

The generating functional for Q;("'(x» IEq. (2.86)1 under the assumption Eq. (5.1) can be written as Z(J(x» =exp[ -iW(J(x» I

= trk,[exp!-i

I J(x) Q (cp(x» IH= N I IdCP(x) I eXP!-i I Lc(cp(x» -JQ (CP(x» 111I(cp+-cp-) (5.3)

where lI(cp+-cP_) "" I dCP'(x)lI(cp(x, 1'+=1'0) -CP'(x»a(CP(x, 1'_=1'0) -cp'(x»P(cp'(x),1"ol

(5.4)

Multiplying the right-hand side of Eq. (5.3) by the normalization factor of the II function on the closed time path IldQIII(Q+-Q_)II(Q(X)-Q('P(X») = I ,

(5.5)

changing the order of integration to replace Q (CP(x» by Q (x) and using the formula II(Q(x)-Q(VI(x»)= I ( : ! }exp!i ilQ(x)-Q(CP(x»]J(x)1

(5.6)

we can rewrite Eq. (5.3) as (5.7)

where e'SefT(Q) ...

I [:!

I

exp

(i II' QI -

iW{/)

I

(5.8)

Here we are performing the direct and inverse Fourier transformations of the path integral. Since the continuous integration is taken over /(x), W(f) can be considered as the generating functional in the random external fields. Calculating the integral by Wentzel-Kramers-Brillouin (WKB) procedure in the one-loop approximation which is equivalent to the Gaussian averaging over the random fields, we obtair.. the effective action Serr(Q> for macrovariables. This is for the case when macrovariables are composite operators. The same is true, if all or part of macrovariables are basic fields themselves. A new field can also be introduced by using the II function. Even if the initial distribution is multiplicative for different components P(CP',

.

TO)

= IIPi(VI:, I-I

the Fourier transformations for the path integral have to be carried out simultaneously for all fields, since for the general case the Lagrangian of the system cannot be presented as a superposition of contributions from different components. We now turn to discuss the general properties of the effective action SefT(Q). The generating functional for CTPGF's in the case of Hermitian Bose fields satisfies the relations W(J+(X),J_(X»IJ+(.d_JJd- O , W*U+(x)'L(x» = - WU_(x),J+(x»

(5.9) (5.10)

Taking successive functional derivatives of Eq. (5.9) and putting J +(x) - J _(x) we get a number of relations between CTPGF's. It is easy to show by use of Eqs. (5.8) and (5.10) that S:rr(Q+(x),Q_(x»=-SefT(Q_(x),Q+(x».

(5.11)

TO)

so SefT is purely imaginary for Q+(x) = Q_(x). Put-

575 CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

ling Q ±(x) = Q + flQ ± and laking functional expansion of Eq. (5.11) around Q, we obtain relations among functional derivatives of different order at the point Q:

~=I~)' 8Q+(x) 8Q_(x) SFij(X,y)

(5.12)

,

= SFj;( y,x) = -Shit y.x)

S ±ij(x,y) = S+J;( y,x) = -S~j;( y,x)

(5.13)

where (5.14) etc. If the system is invariant under the symmetry group G, i.e., the Lagrangian and the initial distribution do not change under q.>1(X) -q.>f(x) - V u ( g)q.>J(x) ,

3401

Up to now we have discussed only the general properties of the effective action S.rr(Q). In principle this can be derived from the microscopic generating functional W by averaging over the random external fields; it can also be constructed phenomenologically in accordance with the required symmetry properties. We shall now show that in the one-loop approximation in the path integral over dl and to the second order macrovariable fluctuations on positive and negative time branches, the current formulation of MSR field theory Il. 14 is retrieved. To calculate the integral (5.8) we expand the exponential factor around the saddle point, given by Eq. (5.15) E=

r

r

J P 01_ w--r-.!.2 J P AIW(2)A/+··· . (5.23)

According to the computation rule described in Sec. "B, E can be rewritten as

£=-r-'2I

f AI
oil)

(5.24)

0

QI(q.»-Qf(f/I) = Vij(g)QJ(q.»

where

then W(J'(x»=W(J(x»,

J!=J;(x)V}(g) .

Serr(Q '(x» -Serr(Q(x», Q f(q.»

=

(5.25)

Vi JdQJ(q.»

This is true if Serr is calculated exactly. In fact, the symmetry properties of SeIT, although related 10 Ihal of the original Lagrangian, may be different from the latter due to the averaging procedure. If the lowest order of WKB, i.e., the tree approximation is taken in Eq. (5.8), it follows that

Q =8W/81 ,

(5.15)

S.rr(Q) =-r(o)

(5.16)

In this case S.rr inherits all the properties of the generating functional r( QJ for the vertex CTPG F's, i.e., S.n·(Q,Q) -0 ,

(5.17)

8S. rr /8Q+IQ+_Q __ Q = 8S.rr/8Q_IQ+_Q __Q

(5.18)

SF+S,=S++S_ ,

8/S. rr

(5.19)

The result of the Gaussian integration, accurate to a constant multiplier is IQJ /S.rr = e-lrlQJI det(
• flJ

(5.27)

is.rrIQ] =-if(O) +ltrlnf

is the two-point vertex function. By use of the transformation formula (2.33) we have Idett(2J I = I detrll) I = I delLl1 detf .1

-I dell', 12 (5.28)

where r,(x,y) =1I21~/8Q(x)IIA(y)

il-I(TpIO/I(I) ... Qi/W]IPI.

(5.20) where I PI means one particle irreducible. According to Eqs. (5.16) and (2.78), -is±(k) >0 (5.21) after the Fourier transforma tion. Near thermal equilibrium we have from Eq. (2.77) (5.22)

(5.29)

Q(x) =+IO+(x) +Q_(x)]

A(x)

=

O+(x) - O-(x)

(5.30)

As shown in Sec. IV

~I 1I1l( y)

-.!. 1_11_1'_ + _1I_r_] 2 111 0+( y) 110_( y) 6-0

6-0 -

(5.3 J)

576 3402

ZHOU, SU, HAO, AND YU

It can be seen from comparing Eq. (5.3 I) with Eqs. (3.1) and (3.7) that 82r/IlQaA is just the transformation matrix, accurate to the coefficient matrix y, from fi to Qi, so the Jacobian can be calculated in the same way. Taking into account that the square power in Eq. (5.28) exactly cancels out the coeFFicient ~ in Eq. (5.27) we have finally

is.rr(Q)=-ir(Q)-~

J ~~dX

Taking J + = J _ = J, in the tree approximation of the path integral over [(x), we obtain

8S.rr (Q) =J(x) =y_U_ "'Q 8Q 01

_~=-

(5.32)

(5.33)

In the path integral (5.7) the most plausible path is determined by the equations (5.34)

Q(x, T+=TO) =Q(X,T_=TO)

(5.35)

8Q

(5.36)

which follows from Eqs. (4.6) and (5.16). This is just the TDGL equation. We now consider the fluctuations around the most plausible trajectories. In the CTPGF approach, in addition to the fluctuations in the usual sense, field variables are permitted to take different values in positive and negative time branches. Changing variables in the path integral (5.7) to the usual time axis by introducing Q(x) and Mx) according to Eq. (5.30) the effective action Serr can be expanded as

where

8S. rr(Q)/8Q±=J±(x) ,

OF +_u_

Denoting +i(S++ +S+_ +S_+ + S __ )(x,y) .. -y(x) oo(x,y)y( y)

(5.37)

and using Eqs. (5.17), (5.32), and (5.36) we obtain e-,W(J(x))

=

J

[dQ(x) )[da(x)]

exp[-~ JMx)y(x)oo(x,y)y( y)M y) + i Jy(x) [~ + :~ ja(x)

- ~ J :~ - J(J~Q +Jo

a)8(L\(x, TO»

i

1

(5.38)

where J~=J+(x) -J_(x),

Jo=~[J+(x) +J_(x)] .

If we take J ~ = J and change variables y(x) Mx) -

Q(x), JOy-I - ithe generating functional for the MSR field theory [Eq. (3.13)] is retrieved. The Gaussian integration over /lex) gives e-iW(J(x)) =

N

J

[dQ (x)] ex p{-

~J 2

[WQ + 1..1---H....- OT

x [OQ( y)

OT

y

8Q(x)

lex)

1100-1 (X,y)

+l..[~ -l(y)l] +~ J ~[l..l£l-i JJQ} y 8Q(y) 8Q y 8Q 1

, (5.39)

which is the generating fu nctional (3.14). It is interesting to point out that 1 = ~(J + +.1_) corresponds to the physical external field, while J - J + - J _ corresponds to the formal source field used for construction of generating functional. It can be seen by comparison of Eqs. (5.39) and 0.14) that
577 CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

By use of Eq. (5.19), valid in the tree approximation, Eq. (5.40) can be rewritten as

(5.41) According to the definition of y [Eq. (4.5»), i.e., 1'= lim i(8/8k olr, kO-O

.

taking into account that only the dissipative part gives any contribution, and using Eqs. (5.16) and (5.22) which are valid near the thermal equilibrium state, we obtain

(5.42) Comparing Eq. (5.42) with Eq. (5.4\) yields the nuctuation dissipation theorem 0" =

2/p",! '

which in the ordinary notation is given by

For _simplicity we derive here the generating functional for the single component Q. Extension to the multicomponent case is obvious.

VI. DISCUSSIONS

Summarizing the main results of this paper, we come to the following conclusions: (j) The CTPGF approach is a natural theoretical framework for statistical field theory to describe systems with dominating long-wavelength nuctuations such as dynamical critical phenomena. By use of the CTPGF's the generalized Langevin equations for order parameters and conserved variables with mode coupling terms included in a natural way and the Lagrangian formulation of the classical field theory are deduced from a unified point of view. The perturbation theory of CTPGF in terms of {; functions has the same structure as that for the ordinary field theory so it is simpler to deal with. In the current theory of critical dynamics and MSR field theory the perturbation expansion is constructed in terms of G functions with two different types of propagators, i.e., the retarded and correlation functions. Therefore, the structure of such perturba tion theory is more complicated. Another advantage of the CTPGF formalism is that the causality is guaranteed automatically. It does not need to be verified order by order,

3403

as in the existing theory.19 (ij) The noncommutativity of field operators, not obvious in the path integral formulation, is not a mathematical trick, but a necessity to describe the time evolution of the statistical field theory. Even if the infrared divergence of the terms, coming from the noncommutativity of operators, is lower than that for other functions, these terms are still needed when considering the time-dependent phenomena, since the infrared divergence of the response function is weaker than that for the correlation function (see Appendix A). (iii) It can be seen from the calculations in this paper what kind of approximations are assumed in the existing theory of critical dynamics and what possible ways may be used to improve the current theory. (a) In the existing theory the transport coefficient matrix for the coupling terms with conserved variables is assumed to be antisymmetric; i.e., only canonical motion is considered. It is possible to analyze the crossover effects of dissipative and canonical motions which may occur, in principle, in the framework of CTPGF's. (b) The one-loop approximation in the path integral over the random fields corresponds to the Gaussian averaging. It is possible to go beyond the Gaussian approximation by calculating higher-loop corrections in the framework of CTPGF's. (c) The current theory of critical dynamics corresponds to the second-order approximation of Mx), the nuctuations on positive and negative time branches. In principle higher-order corrections can be calculated. It may be more convenient to calculate directly the path integral for Q + and Q_, not introducing .1(x) explicitly. (iv) The renormalization of the existing Lagrangian field theory i.s quite complicated. IJ • 14 One of the causes of such complexity lies in the fact that the number of vertices and primitive divergences is much greater than the number of coupling constants and also that Q and Qhave different dimensions. It seems that the renormalization procedure will be simpler in terms of (; functions, since different components of Green's function matrix have the same infrared divergence.

ACKNOWLEDGMENTS

One of the authors (L.Y.) would like to express his sincere gratitude to Professor H. Ehrenreich, Professor B. I. Halperin, Professor P. C. Martin, Professor D. Nelson, and Professor T. T. Wu for the kind hospitality they have extended to him at Harvard and for helpful discussions with them. He is also indebted for Professor A. Aharony, Dr. R. Bruinsma, Dr. B. L. Hu, Professor A. Jaffe, Dr. R. Morf, Dr. H. Sam-

578 3404

ZHOU, SU, HAO, AND YU

polinsky, Professor M. Stephen, and Dr. A. Zippelius for interesting discussions. He is supported in part by the NSF through Grant No. DMR-77-10210.

APPENDIX A: RENORMALIZATION OF THE FINITE-TEMPERATURE FIELD THEORY

As shown by Zhou and Su 6 for the general case, the counter terms introduced in quantum field theory for T =0 K are enough to remove all ultraviolet divergences for the CTPGF's at any temperature. Other authors (see references cited in Ref. 6) come to the same conclusion for finite temperature field theory without resorting to the CTPGF formalism. This result is reasonable from the physical point of view since the statistical average does not change the properties of systems at very short distances and therefore does not contribute new ultraviolet divergences. What we should like to point out is that in considering the phase transitionlike phenomena it is necessary to separate first the leading infrared divergent term and then to carry out the ultraviolet renormalization which is different from that for the usual quantum field theory. To be concrete, consider the relativistic scalar Bose field, the CTPGF propagators for which can be written as·

meant by saying "quantum system in d dimensions corresponds to the classical system in d + 1 dimensions." What has been said above can be verified explicitly by calculating the primitive divergent diagrams for mass, vertex, and wave function renormalization, carrying out the frequency integration, and taking the high-temperature limit T» E(k) to retrieve the results which are identical with that of the current theory of critical phenomena. 2o It is much easier to verify this by use of Matsubara Green's functions, retaining only terms
where n(i()-{exp[E(kl/T)-I}-I, dkl=(kl+ml)1/2 . (A2) Near the phase transition point m dk)/T«l, n(k)

== O.

== T/E(k»> 1

(A3)

for the long-wavelength excitations. Since the n (kl terms appear together with the II function, i.e., on the mass shell, the integration over frequencies can be carried out automatically, so the infrared divergence of these terms is higher than that for other terms by one order of magnitude. Therefore the marginal space dimension for renormalizability for finite temperature cp4 theory is dc - 4, not dc = 4 -I as in the case of the ordinary field theory. This is what is

Therefore the retarded Green's functions are less infrared divergent than the correlation functions. To treat them properly, the noncommutativity of operators has to be taken into account, even though this is a "purely" classical field theory. It is easy to show that all these properties illustrated with the free propagators remain true for the renormalized propagators. As mentioned in the Introduction, the hightemperature limit of statistical field theory corresponds to the "super Bose" limit, but not the Boltzmann limit Usually we consider the classical fields to be commutative, since unity can be ignored in comparison with 11 which is large. If unity cannot be neglected in the phenomena under study we have to start with the noncommutative operators. This is one of the reasons why statistical field theory and quantum field theory have so close an analogy. The physical implications of this analogy are discussed elsewhere. 11

579 3405

CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ... i.e., QI=.,/=(I/.J2)(\,-ll,

APPENDIX B: FURTHER RESULTS ON TRANSFORMATIONS OF CTPGF

QT="I=~I~\I· Q[={=~ln

The main results concerning the transformations of different forms for CTPGF's are described in Sec. II B. Here we shall prove the theorems (2.37) and (2.40), illustrate them by more complicated examples, and discuss the transformations for connected Green's functions. It is more convenient for some cases to introduce the spinor notation. Let

oo.

I(I ...

o z Z(k),100.Ho-k)(J2 •.. n)=2 / -

while the inverse transformation from (; to

G[Eq.

(82)

The normalization condition QQT=I

is expressed as "Ia"la={a{a=\ , "Ia{a={a"la=O,

(B3)

where "la, {a are components of "IT and {T, etc~ In spin or notation the transformation from G to (; [Eq. (2.35)] is

(B1)

GZ

Q2={T=(I/.J2)(I.1l ,

{ak"laH1 ... "IaoGaloo'a.(I2 ... n) ,

(B4)

(2.36)] is

Galazoo.ao02 ... n)-21-'/2(~al .. , {a.GzZoo.z+"Ial~az··· {a.GI2OO' 2 + {al"laZ ... {a. G 2I2 OO 'Z+··· {a._I"Ia. G Zoo .21+··· +"'al ... "'a._I{a. G I00.12

+{al

(B5)

The generating functional Z (J(x)) can be expanded as Z(J(x» -

i...!... J. ... J. 8JO) ..8'Z, 8J(n) I

._on!

I'

-\-; ,_Ii-\n."J. .,. J., -\ f- ... f- G I n! --

GpO'"

=\-;

II_I

J (I) ... Jp(n)

1-01'

I'

-00

n)JpO) ... Jp(n)

(B6)

OO'a 0

a J

"

If we take J+(x) -J_(x) =J(x), Eq. (B6) becomes Z(J(x» - \ - ;

I-If'"

II_I

n!

-00

f- Ga OO'a (I .,. n)."a -00

I

l

.. ,

."a'J(I) '"

J(n) .

(B7)

"

According to the normalization condition (2.16)

Z (J(x» IJ +(%)-J_(%)-J(%) = \ and considering the arbitrariness of J(x) we obtain (;1100.1(12."

n)=2·/2-I"Ial"la2 ... "Ia'G aa OO'a (12··· /1)=0 I 2

(88)

0

thus the first theorem (2.37) has been proved. It follows from Eq. (88) for three-point functions that G++++ G+ __ + G_+_+ G __ += G ___ + G++_ + G+_++ G_++

(89)

Similarly, for four-point functions we have G(+)""T

I

call,.,

(1 +apy8)G a /l ye =G(-)""T

I

(J -apy8)G a/l y &

aJtyI

i.e, the sums of terms with the same signature are equal.

(BI0)

580 3406

ZHOU, SU, HAO, AND YU

To prove the second theorem we first multiply Eq. (84) by the normalization condition of step functions (2.29) to obtain

(BI I) Let

(BI2)

we have

because of the definition of the 9 function (2.26) and the symmetry of CTPGF. Since the lime ordering in the usual sense has already been fixed by the 9 function the action of the Tp operator is reduced to

where (BI4) Such a process of getting rid of Tp can be continued up to the last step to get zero, if 2-·/2+1« ... (lfJ(p,).lfJ(Pl» ."

(I = .,,'1

or

).'P(P.» •

if (I = E'I. We see that the factor 2-·/2+1 exactly cancels Ol!t the numerical coefficient 2"12-1, so proving the theorem (2.40). In Sec. II B we have considered the two-point functions. As a further example we have for n -3

:I

G 2II (123)_(_;)1

p E

8(Iij)(Ill.il.jI) •

11JI

:I (8(ij3)(l!i..i}.3])+9Ci3j)«(Ii.3l.j}» PJI l: 9(ijk) «( (i,j l.k I) =(-i)l«(I1.2l.3})

Gm (I23) =(_;)1

(BI5)

pE

G m (I23) =(_;)1

pEP.)

and for n -4 G11l1 (I234)

=

(_;)l

I P E

9(Iijk)(IIII.iJ..il.k])

11J:1

(BI6) Gml (l234)=(-i)l p

I (9(;jk4)(il\i..i}.k\,41) +9(U4k)(II(i,j \'4!.k I)H}(;4jk) «(\[i.4},.i l.kl» EPJ11

581 CLOSED TIME PATH GREEN'S FUNCTIONS AND CRITICAL ...

3407

where for short we write i 5'1'(;), etc. It is interesting to note that although all the possible combinations of retarded, advanced, and correlation functions are realized in real time, they are defined by the CTPGF approach in a quite natural way. The first theorem (2.37) is valid also for the connected CTPGF, since we can repeat word for word the proof starting from Eq. (2.17). This is not the case for the second theorem, where some complications appear. It follows from the definitions of disconnected [Eq. (2.4») and connected [Eq. (2.15») CTPG F's that G%(J)=GpO) . (BI7a) G%(J. 2) = GpO. 2) + iGp(J )Gp(2)

(BI7b)

G%O. 2. 3) = GpO. 2. 3) +;( GpO )Gp(2. 3) + Gp(2)GpO. 3) + Gp(3) GpO. 2)]- 2Gp(J )Gp(2) GpO) ...

(BI7c)

It can be shown by use of Eq. (BI7) that the formulas for all "purely" retarded functions remain true, as for example

G5 I O. 2 )=G2I O.2)=-i90.2)(l1.2)

, G~II(1.2.3)=GlII(J.2.3) .

G~III(J.2.3.4)=G2II1{\,·2.3.4) ...

(BIS) These functions are similar to r functions used to construct the Lehmann-Symanzik-Zimmermann (LSZ)22 axiomatic field theory, which are the same for both connected and disconnected functions. All other functions are modified, for example, (BI9)

Gi(I2)=-i«!I.2})-!(I).(2»). GV'22 (123) = (_i)2

I

!9(P2.PJ. J)(([ !P2.pJI. II) - ([ (P2)P3. I]) - (lP2 (P3). II»

P2

H)(P2.I.PJ)( (!.[p2. 11.PJ}) - ([P2. I] HpJ»} .

G~22 (23) = (-i)2

I

9(PIP2Pl)(

«( (PI.P21.pd) -

({P!.P2) )(Pl)+2 II (PI). (P2) \. (PJ) P

(B20) (821)

PJ

G~w (1234) = Gml + 2;( GO) G lII (234) + G (2)G lII (134) ]+2;( G,(3)G,(24) + G,(14)G,(23)]

(B22)

GS2lI (1234) - G ml (234) + 2;[ G (J ) G m (234) + G (2) G 22I (134) + G (3) G 22I 0 24)] +2;(Gc 02)G,(34) +Gc (J3)G,(24) +Gc (23)G,04)]

- S(G(J)G (2)G,(34) + GO)G (3)G,(24) + G(2) G (3 )G,{\4)]

·On leave from Ihe Inslitule of Theorelical Physics, Academia Sinica. Beijing, China. IJ. Schwinger, J. Malh. Phys. (N.V'> 2, 407 (1960. 2L. V. Keldysh. Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, lOIS (1965)]. JS ee , e.g., D. Dubois, in LeL'IUtes iI/ Theorelicall'hvsic." ediled by W. E. Brillin (Gordon and Breach, New York, 1967), Vol. [)i:'c; V. Korenman, Ann. Phys. (N.Y'> 39, 72 (I 966); D. Langrelh, in Lillear alld NOlllilleor EleC'II'O/-;;; Trallsporl ill Solids, edited by J. Devreese and V. Van Doren (Plenum, New York, 1976); A.-M. Tremblay, B. Pallon, P. C. Marlin, and P. Maldaque, Phys. Rev. A 19, 1721 (1979). 4Zhou Guang-zhao and Su Zhao-bin, Progress in Statistical Physics (Kexue, Beijing, 10 be published in Chinese), Chap. 5. 5Zhou Guang-zaho and Su ZhaO-bin, Physica Energiae FOrlis el Physica Nuclearis (Beijing) I, 314 ([979). 6Zhou Guang-zhao and Su ZhaO-bin, Physica Energiae Fnrlis el Physica Nuclearis (Beijing) I, 304 ([ 979). 7Zhnu Guang-zhao and Su ZhaO-bin, ACla Phys. Sin. (in press). sp. C. Marlin, E. D. Siggia, and II. A. Rose, Phys. Rev. A S, 423 ([ 973). 911ao Bai-lin, in I'.rogress in Statistical Physics (Kexue,

(823)

Beijing, to be published in Chinese), Chap. I. lOp. C. 1I0henberg and B. I. Halperin, Rev. Mod. Phys. ~, 435 (1977). 11K. Kawasaki, in Crili",tI Ph,·l/ul/wllt•. ediled by M. S. Green (Academic, New York, 1971). USee for example, B. I. Halperin, P. C. Hohenberg, and S. K. Ma, Phys. Rev. B lQ, 139 ([974); S. K. Ma and G. F. Malenko, ibid. 11,4077 (1975), HH. K. Janssen, Z.Phys. B ll, 377 (976); R. Bausch, H. K. Janssen, and H. Wagner, ibid. .?~, 113 (1976). 14C. De Dominicis and L. Peliti, Phys. Rev. B ]!, 353 (l97S). IlL. Onsager and S. Machlup, Phys. Rev. 'l!.. 1505, 1512 ([953). 16R. Graham, Springer Tracts Mod. Phys. ~, I (]973). 17R. Graham and H. Haken, Z. Phys. 243, 2S9 (1971). IlL. Sasvari, F. Schwabl, and P. Szepfalusy, Physic. (Ulrechl) SI, lOS (1975). 19J. Deker.;.J F. Haake, Phys. Rev. A 11, 2043 (1975). 20E. Brclin, J. C. Le Guillou, and J. Zinn-Juslin, in Phase Trumiiliol1s and C,.ili('ul Phenomena, edited by C. Dumb und M. S. Green (Academic, New York, 1976), Vol. VI. 21Yu Lu and Hao Bai-lin, Wu\i (in press). 2211. Lehmann, K. Symanzik, and W. Zimmermann, Nuovo Cimenlo §" 3 19 (1957>'

582 Commun • .in Theor. Phgs. (Beijing), China)

Vol. 1, No.3 (1982)

29S-30f

ON THEORY OF THE STATISTICAL GENERATING FUNCTIONAL FOR THE ORDER PARAMETER(I) - GENERAL FORMALISM

16:**

ZHOU Guang-zhao ( Jil](;?i ) SU Zhao-bin ( HAO Bai-lin ( ]/!~ff; ) YU Lu ( f ~ ) (Institute of Theoretical Physics. Academia Sinica, Beijing, China) Received December 28, 1981.

Abstract A theoretical scheme using closed

time-pathGreen~

func-

tions is proposed to describe the quantum statistical proper.ties of the order parlllllf!ter .in terms of a generating function-

al.

The dgnamic evolution is generated bg a

while the statistical correlation bg a

source,

driving

fluctuation

source.

The statisdcal causalitg is shown to hold ezplicitlg and to give

rise to a number of i.JlIportant consequences.

The pm.bl.em

of determining the quantum statistical properties for the order parameter is reduced to finding a solution of the

func-

tional equation for it.

I.

Introduct1on

It is well known that special tricks have to be used in the s1:andard Green's function techniques[l] to describe systems with broken symmetry such !'.s superconductivity or Bose-Einstein condensation. On the other hand, a systematic theoretical scheme has been developed in the quantum theory of gauge fields to treat the order parameter. [2] The generating functional formalism developed there has been applied successfully to the classical theory Df static critical phenomena. [3] To treat the time-dependent phenomena Martin,Siggia and Rose(MSR) have constructed a noncommutat1ve classical field theory closely analogous to the quantum field theory. The MSR theory has been reformulated into a Lagrangian formalism[5,6] and has been used extensively in studying dynamiC critical phenomena. [7] But, the physical meaning of the "response field"· introduced in that theory is not clear. There are also some attempts of quantum generalization of Langevin and Fokker-Planck equations for describing order parameter. [8] It seems difficult to use those kinds of semi-phenomenological theories to study nonuniform system~ and composite order parameter. The necessity of constructing a unified quantum theory to describe nonuniform, nonequilibrium systems with broken symmetry is obvious from the current research on condensed matterandlsser phenomena. In some cases (e.g. nonequilibrium superconductivity) the dynamic coupling of the order parameter with the elementary excitations is essential. It is almost impossible to incorporate this kind of coupling into the theoretical schemes mentioned above.

583 296

ZHOU Guangzhao, SU Zhaobi.n, HAC Bailj,n, YU Lu

We have shown previously[9] that a unified microscopic description of both equilibrium and nonequilibrium systems is possible by combining the closed timepath Green's functions (CTPGF) with the techniques of generating functional. The transformation propert ies of CTPGF are studied in Refs. [10,11] and the MSR theory appears to be a "physical" representation of CTPGF in the classical(Rayleigh-Jeans) limit. We prove[12] the existence of a generalized potential functional for the order parameter in a nonequilibrium stationary state for systems with time reversal symmetry. The time-dependent Ginsburg-Landau (TDGL) equations for order parameters and conserved densities are derived[11-13] with both time reversible and irreversible parts, also the Lagrangian formulation in critical dynamics is obtained in a low order approximation of CTPGF. It seems to us that CTPGF formalism is a good candidate for the general quantum theory of nonequilibrium systems discussed above. In the present series of papers we propose to construct a quantum theory of the statistical generating functional for order parameter using CTPGF which is capable of describing both aspects of the Liouville problem: dynamic evolution and statistical correlation. The theoretical scheme is general enough to incorporate nonuniform, nonequilibrium systems with either simple or composite order parameter. We will give a practical prescription for constructing the generating functional being applied to concrete sys~ems. The dynamic coupling of the ord~r parameter with quasi-particles will be ~reated in fu~ure publications. The rest of the first paper in the present series is organized as follows. In Sec. II the generating functional for the order parameter is constructed explicitly. The dynamic evolution is generated by a driving source which might be the.actual external field, while the statistical correlation is generated by a fluctuation source which vanishes on tbe completion of calculation. Thestatistical causality and its consequences are studied in Sec. III. In Sec. IVa statistical functional equat.ion for the order parameter on a generalized manifold which is equivalent to the Langevin equation to certain extent is derived and the problem of dete~ining the statistical properties of the order parameter is reduced therefore to finding a solution of this functional equation. The final section contai·ns a few concluding remarks. Throughout this series of papers we will use the system of units with 11 = C = ka= I, but we might come back to the ordinary units if necessary.

II, The statistical generating functional for order parameter Suppose Qa(X), a = l,2 ••. N, are a set of Hermitian order parameters in Heisenberg picture which might be basic field variables or composite operators and ha(X) are corresponding C-number real external sources. Both sets are defined on a closed t~e-path consisting of a positive branch (-~, +~) and a negative branch (+ .. , - .. ). The CTPGF generating functional for order parameters can be written as[9,10]: (2.1)

584 On Theory or the Statistical Generating FUnctional for the Order Parameter (I) --General Formalism

297

where ~ is the density matrix in Heisenberg picture, Tp is time-ordering operator along the closed time-path. Tr has the usual meaning of trace operation in Bllbert space. Jp d4 X... means 4-dimensional integration along the closed t:imepath. We note in passing that the letter "p" will often indicate quantities defined on closed path or operations along it. We have also omitted the index a for ~a(X) and ha(X) and a summation over a is implicitly understood. Introduce index a for the time branch as: (2.2)

"'r-

Q

(X)

~

==

[(Xv)

I

(2.3)

with 0=+ or -, indicating whether the space-time point is at the positive or at the negative branch. Eq.(2.2) defines two independent C-number functions from a single function defined on the closed path. However, Eq.(2.3) defines two operators QO(X) in the ordinary space-time with different time-ordering properties, but representing the same quantity. Under the action of Tp the operator Q-(X) should always precede the operator Q+(X), the ~+ operators are time-ordered in the usual way among themselves. while the Q- operators are antitime-ordered (denoted by T). After time-ordering operations Q+ and Q- operators are set to be identical. Introducing. furthermore.

~~ ::

and defining

Jr-

{

if r=T if

-1

=I

f"7

r At' Q.c(t)=TS .. 1I. lxl.

A

kc(x)

we have

=+'$.

hlr(x)

•

a-~-

(2.4)

a-=+ au -

'"

h ""0"'

[0-11. (X),

(3.5a)

hAlX)='l.,.hr- cx ) ,

(2.5b)

Q.o.(lC)=

ar-(XJ=J,.Qc ex) + f~/I't(X), (2.6a) I{ClCl =

5,,"hc exl + -}~r- k.o. (X).

(2.6b)

The 4-dimensional integration over the closed path is transformed into the ordinary integration as (2.7) According to Eqs. (2.4)-(2.7) and taking into account that

S h (I) th

(2)

SIt""C/) IIII'C2) -

t; (

1- 2) ,

r

or,

&4

<1-2).,

(2.8) (2.9)

where 1.2 are short notations for Xl and X2 • we find that (2.10)

585 298

ZHOU Guangzhao,

sr; Zhaobin, 1I1lO Bailin, ru Lu

(2.11)

(2.12)

The closed time-path 6 - function in Eq. (2.8) is defined as (2.13)

Hereafter a summation over repeated index is always understood except for special reservation like in Eq. (2.10) where no summation over a is assumed. Using Eqs. (2.2), (2.3), (2.6), and (2.7) the generating functional defined by Eq. (2.1) can be rewritten as

:[h4(X),Ir C (xJ]= :,.(hoo) (2.H)

Expanding Z over he and

h~

in the usual space-time we find

(2.15)

(2.16) It can be shown[lQ] using Eqs. (2.2), (2.3) and (2.5) that Eq.(2.16) can be rewritten as

, -

L.

' ... -n)

p( -I

'"

(2.17)

n

where

I,

if

oJ

otherwise,

tr>h ···>t;;:,

e(f.2, ... n)= { (2.18)

586 Theory of the Statistical Generating FUnctional for the Order Parameter (I) --General FOrmalism

On

299

and satisfies the normalization condition

p(

L 1,2,' .•

It)

B ( T,

2, ... il)=

I

(2.19)

T. 2, ... ;[

1:' means summation over all permut'atwns P(!,!,,· .~) 1,2,·· ·n We use here also a simplified notation given by

(... , a(TlJ={

except for cases m+ 1S Isn.

{ ... , Q(T,}+. (2.20)

( .... , Qir,L,

Now we have obtained a new expression for the generating functional given by Eqs. (2.15)-(2.17) with explanation following them. Although this definition is fully equivalent to the original one given by Eq. (2.1), it allows of an entirely new interpretation. The generating functional is now defined in the ordinary space-time, while the closed time-path as an auxiliary tool disappears altogether. To associate with our previous papers we indicate that different representations of CTPGF and transformations from one to the other have been studied in Refs. [10] and [1:1,]. The "natural" representation of CTPGF is defined as .. ._, {~ ~oI~! ~l)} ~., ... ¥ (f 2 ... It)= (-i)

Tr ,Tp(Q

(IJ Q

(2) . . .

Q.

(n)

(2.21)

= i

(-f)"

I"i!(h+(x),

A-ex,)

-I

lh"lIl SIa'cz) ... GIl ('PI.,

'

Itlll'= D which is very convenient for constructing the perturbation theory. By means of an orthogonal transformation£lO, 11) we obtain the "physical" representation Gi j •• •• (1,2 ••• n) with i,j ••• =1,2. In the present notation these G functions can be written as (2.22) which ought to be compared with Eq.(2.i6).The components of a--retarded, advanced, correlation functions and so on---are closely related to physical~y observable quantities. As is seen a~Qve, these functions are functional derivatives of the same genera.ting functional Z[h(X)] with respect to the driving source hc{X) and the fluctuat~on source hb(X). These two independent functions will generate, correspondingly, the statistical average of the commutator--dynamic response, and the average of anticommutator--statistical correlation. We will further discuss the physical consequences of this important fact and compare our formalism with the previous theories in the next two sections. Now we define the generating functional for the connected part as (2.23) and introduce the vertex functional by a Legendre transformation (2.24)

587 300

ZBOU Guangzhao, SU Zhaobin, BAO Bailin,

where

ru

Lu

(2.25) , he lXl) == .r w (LalX) one (XI

~(XJ

(2.26)

It follows from Eqs. (2.24)-(2.26) that (2.27) Gr (" .. lXI, lIe(XI) GII.. (XI

-

he (xl.

(2.28)

It is easy to check from Eqs. (2.1)-(2.7) and Eqs. (2.10)-(2.12) that

where

with

W[h ... eXI. he (0) = Wp (hexI),

(2.29)

r(GaexI, Ilc (XI]=rp (lllX)),

(2.30)

Wp (hexI)=-l I" rp (h(xI) ,

(2.31)

r,. (lIex))= Wp (hexI)- Jl·,d(XIQU'

(2.32)

IIlx)

= SWp(ho(l] •

(2.33)

ah(xl

s r, (II on]

_ hr.,,,).

(2.34)

all"'l

It is worthwhile to emphasize that Eqs. (2.14), (2.29) and (2.30) tell us that the generating functionals originally defi.ned on the closed time-path are generating functionals for Green's functions closely related to the observable quantities, if they are expressed in appropriate functional arguments. This is the starting point of the present formalism.

III. Basic properties of the generating functional In this section ating functional for First of all we r.onventionfollowing

we will discuss some fundamental properties of the generthe order parameters and their physical implications. indicate some basic relations implied by Eq. (2.17) and the it as given by (3.1)

or

SllrCh ... l 1.l.h c tXI]\ She (I)

, , ,

W[h... lXI,

&beenl

=0

(3.2)

=0,

(3.3)

= 0

(3.4)

h... e)()::o

I hC
h... (XI=O

or

c!'''w(hAlXl. ah.ll) .•

,g~c(1I1

h... (X)=o

588 On Theory of the Statistical Generating Functional for the Order Pa:ameter (I)

--General For.maliam

301

and

I(Q,,(X). Q,lX»)

I

=0.

(3.5)

I

=0

(3.6)

QAllll=a

or s"r(Il.AlX).

QclXl)

"Q.,lll ' "EGcln)

Q... l~J= 0

In deriving Eqs. (3.1), (3.3) and (3.5) we have used the

ilOl'IDallza t

ion

cC:Jtlj t

i Of!

(3.7)

T.{f}=I. Also, we have used a special case of Eqs. (2.26) and (3.4), implying

(3.B)

In our previous papers[10,11] Eqs. (3.2), (3.3) have been na~ed nor~aliza~ion conditions for the generating tunctionals and have bee~ shl~n to give r~6e to a number of imp.:lrtant consequences, in part icular, Eqs. (3.:';), (3.4) an ri (:!,'~ which in the previous notations look like

-

4/1 ... , (Q ... n )=0,

(3.9)

rll ... 1(l2 ... n )=O.

(3.10)

It is worthwhile to note that Eqs. (3.1)-(3.10) are valid even in the of an external field hc(X)';'O. Furthermore, three causality relation& equivalent to each other from Eqs. (2.17), (2.20),and (2.23)-(2.28) as given by

S" r [hAlx). dh .. (J)

, •

pl'e:5.:n,'e: 101]0 ..

~c()Cll

'.I' h.. ('" J .I' he (m. I , .. , ; he (n 1

n.1 < . =

( 3, 11)

:

hcC')=~

rW(".. ("I.~cC)(l)

I

rhAw,·· .l'hA(mlrhc.(mtJ) .. ·E".(nl h... (.1=0

=il

.

(3.12)

he (xl=O

r"r

and

(R"OO.

QelXI)

=0 (3.13)

Q"l~)=i)

Qc1x1=alxj

provided any of t j , m + 1 S j S n is greater than all of ti' function Q(X) in Eq. (3.13) is defined as QlX)

== ,-

=TrU QlX)}

G.clx)

h.. tx)=o

1

~

i:ii m.

Tt'e

(3.14)

he ()()= 0

Examining relation (3.11) we notice that in summation over (2.17) the cases of the leading time (t in Eq. (2.18» being tj' m+1:oj::in are excluded. Eq.(~.12) 1 is a direct consequence of Eqs. (2.23) and (3.1l.), wt:ile Eq. (.3.13) follows trom the relation between vertex functions and the amputated Green's functions~3,14J It is important to emphasize that a causality sequence of the statistical Green's functions 1s fixed by Eqs. (3.11)-(3.13), namely, th~ space-time point

589 302

~HOU

Guangzr.ao, SU Zhaobin, BAa Bailin, YU Lu

associate~with he(y), ~(y) cannot precede the space-time pOint associated with ha(X) and Qa(X), since the former is the ~ause while the latter is the consequence. The usual retarded Green's function or retarded product is a ~pecial case of Eqs. (3.11) and (3.12).

We indicate here also some useful product relations implied by Eq. (2.17) and the definition of a-function as given by Eq. (2.18). For example, we have ~

~'2 (I 2)

Gz I

(I 2)

=

~/l2= ~IIG'21= GIIZ 'HI .... GIIZ GI21 =

Z"ZII

0,

(3.15)

4Z11 ',22=

0,

=

0

Gill

'21Z

(3.16)

It is easy to see that the general rule is as follows: (3.17)

provided

i m1 = im;.> = ••• = j",r = 2

and the rest of i are equal to 1 while

.•• = jmr = 1 and the rest of j are equal to either 1 or 2. Now we discuss the physical meaning of ha (X), he(X), Qi. (X) Apart from Eq. (3.8) we obtain

and

jml = jm2=

~

(X).

(3.15)

from Eqs. (2.27) and (3.6), and also

(3.19)

from Eqs. (2.14) and (2.25). Eqs. (3.8) and (3.18) tell us that the relations ha (X) .. 0 and Qa (X) = 0 are equivalent to each other, while Eqs. (3.14) and (3.19) tell us that Qe(X) under the condition ha (X) = 0 is the statistical average of order parameter in the driving field hc(X) and that Q(X) is the expectation value without an external field. As mentioned in Sec. II the functional derivative /i lohe (X) generates the commutac .. !" of 'the order parameter describing the dynamic evolution in the quantur. m~~h3nical sense, while the functional derivative /i/cha(X) generates the ar.ticommutator describing the correlation in the statistical mechanical ::;en",,",. Although the physically obsel;"vable quantities are defined at the fur.ctional manifold h6 (X) = G or Qa (X) .. 0, these functional arguments are needed in addition to he(X) and Qc(X) for a complete description of the statistical systems. These two complementary aspects of the Liouville pl·oblem--dynamic evolution and statistical correlation _forming the essential elements O~ the semi-phenomenologiGal Langevin-Fokker-Planck theory have been grasped in the CTPGF formalism in a natural way as specified above. As an illustrative example we give here the response of statistical correlations to external ~ield up to an arbitrary order as given by

590 OD Theory of the Statistical Generating FunctioDal for the Order Parameter (I) --GeDeral FOrmalism

303

.(3.20)

" +:f.

(3.21)

i=~

The details of the nonlinear response theory and a pos~ible generalization of the fluctuat ion - dissipat ion theorem are discussed elsewhere [15] • Some of the relations discussed in this section have been obtained previously by other authors in the classical limit[5,6,16], but we believe that the present derivation is much more straightforward and complete.

IV. The statistical tunctional equation It is interesting to compare the generating functional for the order parameter constructed from CTPGF as expressed by Eqs. (2.14) and (2.16) with the classical stochastic functional, the so-called Onsager-Machlup functional {17, 4-6] quite extensively used in critical dyn;tmics·. .o\s common features and interrelations between these two formalisms we find the following: (i) Two external sources are needed for describing the statistical properties of each order parameter in both formalisms--one fer dynamic evolution, the other for statistical fluctuation, although the retarded response and the statistical correlatioD are defined differently in the quantum and the classical cases. (ii) The retarded response and the statistical correlation are related by the fluctuation - dissipation theory (FUi') in both formalisms and we have shown t12 ] the FDT in the classical stochastic system [6,.16,18] to be the Rayleigh - Jeans limit of its quantum counterpart. these (iii) The expansion coefficients of the generating functionals in two formalisms, i.e •• the expectation values of different combinations for oreer· parameter products. satisfy the same causality relations as given by Eqs.(3.11~ (3.13). [5.16]

591 304

ZHOU Guangehao, SU Zhaobin, 1IJIO Bailin,

ru

LIZ

(iv) We have shown[10,13] that,for-order parameters slowly varying with time, the Lagrangian formulation of the classical stochastic functional[5,6] is reobtained from its quantum cou~terpart in the one loop approximation with the se~ond order correlation retained if the potential condition for CTPGF is satisfied. [12] In spite of the overall similarity of these two formalisms there ~ one important difference: In the CTPGF formalism the physical field he(X) is coupled with Qa(X) and the fluctuation source ha(X) is coupled with Qc(X) while both sources in the classical stochastic formalism (i. (t) and let) in the notations of Ref. [6] are artificial and an additional driving term has to be introduced in the Fokker-Planck equation to incorporate the coupling with the external physical field. As is seen previously the physical interpretation of he(X) and ha(X) as driving and fluctuation sources correspondingly is possible just because of this apparently "t'IVisted" coupling he(X) ~a(X) + ha(X) Qe(X) which is derived straightforwardly in CTPGF formalism and is not conceivable in the original stochastic formalism. In support of the correctness of this coupling form we notice that. if the projection operator cnto tbe ground state is chosen as the density matrix. then the functional d~rivativ~ H(h4(~)' hc(X») &h/l 1x ,

h,,(X'=O

is just the generating functional for the vacuum expectations of LSZL1S] retarded products in quantum field theory. In summary we can say that the generating functional for the order parameter constructed from CTPGF is general and effective enough to contain physi~ally meaningful information. Both the purely dynamic evolution described by retarded products In LSZ field theory and the classical stochastic process appear to be special cases of this more comprehensive formalism. Therefore, the problem of determining the quantum statistical properties of the order parameter is reduced to finding the CTPGF generating functional for it. In principle, if the generating functional for the vertex functions r[QA(X),Qe(X)] is known, one can find the formal solutions from Eqs. (.2.27) and (2.28) as (4.1) (4.2) and then take the fUnctional derivatives according to Eqs. (2.25) and (2.26)to obtain all Green's fUnctions in the limit ha(X) =he(X) =0. But practically, the formal solutions as given by Eqs. (4.1) and (4.2) are difficult to find. One prefers instead to take consecutive functional derivatives of Eqs.(2.27) and (2.28) to obtain an infinite hierarchy of coupled equations Sr(G.. ,QJ &Q.. I:'O

=0,

(4.3)

592 On Tbeory of the Statistical Generating FUnctional for the Order Parameter (I) --General Formalism

305

(4.4)

~'w [~4)<) Gh4(jJS4.(ZI

+ £'f(G.".Q<)

.n14111~Q"(JI

=0

r'w[h. ,he] ')\ Snc (.YIS;',(2) •

Q,,(.,= 0

(4.5)

Q,(.,=Qc.,

and so on for different Green's functions of order parameter

By solving these equations consecutively one can find all Green's functions needed for the order parameter. Analogous to the Langevin equation in the semi-phenomenological theory of stochastic processes, the functional equations (Z.27) and (2.28') con~ain all statistical information. Usually in phenomenological approach, a random force term is introduced into the dynamic equation for the order parameter to make it a random variable, While in CTPGF formalism a new degree of freedom Q~(X) is introduced for each order parameter to describe its statistical behavior. Although the complete statistical information is contained in Eqs. (2.27) and (2.28) defined on the manifold with both Q~(X) and Qc(X) different from zero, the observable qUantities are defined on the submanifold with QlX)=O like the physically observable quantities which are given by s~atistical averages of the random variable in the Langevin approach. Therefore, in a certain sense, tQe functional equations (2.27) and (2.28) are equivalent to the Langevinequation. However, tbere is one important difference. The random force term in the Langevin equation is postulated independently while the high order st~tis tical correlations in CTPGF formal.ism are determined self-consistently from Eqs. (2.27) and (2.28), which are complete in principle.

V, ConclusIon To sum up, we have established in this paper a theoretical fr:tmework for describing all physically meaningful statistical information for the order parameter' by comb~ning CTPGF formalism with the generatin~ functional techniques. This framework is general enough to include both equilibrium and nonequilibrium, both uniform and non-uniform systems. It i.3 a quantulil theory in nature but incorporates the classical limit as a special case, if the low frequency, long wave-length excitations dominate as in phase-tran~ition-like phenomena. The practica~ value of such a theoretical scheme depends to a great extent upon whether a concrete and effective way can be found to construct the generating functional of the vertex functions for the order parameter r [Qll (X),Qc(X)]. This is the main point we are concerned with in this series of papers. Since a generalized Wick theorem has been establisHed[9,20] for CTPGF, the generating

593 306

ZHOU Guangzhao, SU Zhaob.i.n, 1l1ID Bai.li.n, YU

Lu

functionals defined at the closed time-path as given by Eqs. (2.1), (2.31)(2.34) are very similar to those defined in the ordinary quantum field theory[2], the main difference being the presence of the density matrix~. In the second paper of this series we will analyze in some detail the role of density matrix to adapt the field-theoretical techniques to the statistical systems.

References 1.

See, for example, A.L. Fetter, 3.D.

~lecka,"Ouantum

Theory of Many Particle Systems;

McGraw-Hill, New York, (1971). 2.

B.S. Abers, B.W. Lee, Phys. Rep. 9C (1973), 1.

J.

E. 8rezin, J.C. Le Guil10u and 3. Zinn-Justin, i.n"Phase Transitions and Critical

4.

P.C. Martin, E.D. Siggia, and H.A. Rose, Phys. Rev.

!!

5.

R. Bausch, H.K. Janssen, and H. Wagner, Zei.t. Phys.

~«1976),

6.

C. De Domini.cis and L. Peli.ti, Phys. Rev.

Phenomena~

Vol VI, eds, C. Damb, N.S. Green, Academic (1976), p. 127.

!!!

(1973), 423. 116.

(1978), 353.

7.

P.c. Hohenberg and B.I. Halperi.n, Rev. Mbd. Phys. 49 (1977), 435.

8.

See, for example:

R. Zwanzig, i.nHLectures i.n Theoretical

Physics~

Vol. III, eds. W. Britten et al., Wi.ley, N.Y. (1961); H. Mori, Progr. Theor. Phys.

!!

(1965), 423;

W. Louisell,"ouantum Statistical Properties of

Radi.ation~

Wiley, N.Y. (1973);

H. Haken, Rev. Nod. Phys. 47 (1975), 67. 9.

Z!!DU Guang-zhao, SU Zhao-bin, Ch. 5 in "Progress i.n Stati.sti.cal

Physics~

eds.

HAD Bai-lin et al., KEXUE (Science Press), Bei.jing, 1981.

!!l.!

10.

ZIIDU Guang-zhao, SU Zhao-bin, HAO Bai.-Hn and YU Lu, Phys. Rev.

11.

ZHDU Guang-zhao, YU Lu, HAO Bai-lin, Acta Physi.ca Sini.ca, 29 (1980), 878.

lZ.

ZHDU Guang-zhao, SU Zhao-bin, Acta Physi.ca Si.ni.ca,

1'~

ZHOU Guang-zhao, SU Zhao-bin, HAO Bai-lin and YU Lu, Acta Physica Si.nica, 29 (1980) 961; ZIlD(r

Guang-zhao, HAO Bai.-lin,

and

YU Lu, ibid,

~

~

(1980), 3385.

(1981), 164, 401

(1980), 969.

Physi.cs~

14.

HAD Bai-lin, Ch. 1. "in Progress in Stati.stical

15.

HAD Bai-lin, Physi.ca, A(i.n press);

16.

J. Deker and G. Haake, Phys. Rev. All (1975), 2043.

17.

R. Graham,"Springer Tracts Nod.

18.

G. Agarwal, Zeit. Phys. 252 (1972), 25;

See Ref. [9J

HAO Bai-lin et al., to be publi.shed. Phys~

66 (1973), 1.

S.K. Ma, G. Nazenko, Phys. Rev. Bll (1975), 4077. 19.

H. Lehmann, K. Symanzi.k, W. Zi.mmermann, Nuovo Cimento, 6 (1957), 319.

20.

A. Hall, J. Phys.

!!

(1975), 214.

594 COlIIIIIUn. in Thear. Phg•• (Beiji.ng, China)

Vol. l, No. 3 (l982)

307-3l8

ON THEORY OF THE STATISTICAL GENERATING FUNCTIONAL FOR THE ORDER PARAMETER (II) --DENSITY MATRIX AND THE FIELD-THEORETICAL STRUCTURE OF THE GENERATING FUNCTIONAL ZHOU Guang-zhao ( liJ;;l::i) SU Zhao-bin (?t.¥~j(. HAO Bai-lin ( .~m.ft.) YU Lu ( f ~ ) (Institute of Theoretical Physics, Academia Sinica) Received December 28, 1981.

Abstract Several equivalent expressions for the generating func:tional of the order pariUDflter are derived in this paper to elucidate how the statistical properties are incorporated in its field-theoret.1cal struct:::re. cess of

det.r~ning

the

It is sr.c,,"n that in the pro-

gener3ti~g :~~c:~ona:

the dgnamic ev-

olution and the statistical ir':ormar:ior. can r.e !;''':;:.Jratsd to a

certain extent. which facilitates sclving the problem. whole procedure is greatlg simplified if relat~n

is Gaussian or a generalized

theorem (FD7') holds.

~~e

The

statistical cor-

:l~cr:uation-dissipaticn

As an example, the Gaussian character of

the thermal equilibriuz:J distribution is justified and the .Ya-

tsubara technique along the imaginarg time axis

is

extended

to the real. ads to provide a complete perturbation scheme for the closed time-path Green's functions (CTPGF).

I. Introduction In the first paper of the present series[l] (as (I) hereafter) we havepnr posed a general formalism of the generating functional for the Qrder parameter using CTPGF. In this second paper we will concentrate on the field-theoretical structure of the generating functional and the role of the density matrix to explore the possibility of using field-theoretical techniques in determining the generating functional for the order parameter. In Sec. II We derive two general expressions for the generating functional which serve as a starting pOint for further discussion. It is clear from the structure of these expressions that the effects of the density matrix or the statistical properties of the system can be reduced to adding a new term to the effective action. In Sec. III the thermal -;quililorium situ'l.t!.cn a.s an important special ClXample is ~onsi1ered and an explicit expression for the CTPGF generating functional with Gaussian distribution is derived. Furthermore, the Feynman rules of tbe M&tsubar~ technique developed originally for Green's functions at the imag-

595 308

ZHOU Guangzhao, SU Zhaobin, HAD Bailin,

ru

Lu

inary time axis are extended to the real time axis to give perturbation expansion rules for the real time Green's functions. In the final Sec. IV we show that in principle the process of determining the generating functional for the order parameter can be divided into two steps: To find first a formal generating functional defined on the closed time - path without statistical information and then to include the correct statistical properties in the functional obtained. Such separation of dynamics from statistics to a certain extent would facilitate the transplantation of the field-theoretical techniques and provide a basis for a unified approach to the equilibrium and nonequilibrium phenomena. If the statistical correlation is Gaussian or a generalized FDT holds (as in a nonequilibrium stationary state), the proper generating functional for the order parameter is obtained from the functional without stat·istical information i f the CTPGF vacuum propagators are replaced by the C!l'PGF propagators complying with the density matrix. Throughoulj this paper we will use notations adopted in (I) and send our reader there for detailed explanation.

II. Two equivalent expressions for the CTPGF generating functional In this section we will derive two equivalent expressions for the CTPGF generating functional to analyze the role of the density matrix and to prepare a.basis for the actual determination of the generating functional. For clarity of presentation we will consider a. multi~component nonrelativistic complex field. either Bose or Fermi. As a convention the lower sign will always correspond to the Fermi case. The n-component field operator is represented by ~t. ~b' b .. l.2 •• ··.n while the action of the system is given by

A I. (At") I ( '/'At ,'f'J= 'f','f' tlint (~A) t+, 'f ,

(2.1)

where 10 is the free part. lint -nonlinear interaction. defined on the closed time-path. Suppose

r.[~+, H= Jdtthf+(I) 5;' (I. 2)~12)= q.tS::;'

All these quantities are

,

(2.2)

p

where[2]

(2.3) -I

-

I

Sar (1,2)=So~

iJ'

(l,2J=(i-n+2"...).l"

of.

(2.4)

([-2),

(2.5) 01' 02 being indices of the time branch for arguments 1 and 2 correspondingly. Furthermore. assume the order parameters Qa(x) to be Hermitian composite operators

(2.6)

Introducing external sources

Jb(x), Jb(x)

for

~x).

~x)

and

ha(x)

for

596

On

~heory of the Statistical Generating FUnctional for the Order Parameter (II) --Density Matrix and the Field-~heoretical Structure of the Generating FUnctional

309

Qa(x), the CTPGF generating functional can be written as[1] (2.7) (2.8) with (2.9) fi~J

as the generating functional for the basic

We denote, as before,

i.

tities defined on t~e closed time-path or operations along it by Letters with nAn will mean operators in Heisenberg picture except reservation. Usually we drop the subscripts b, them. The short notations

r+f= ~+J= ~Q

JcI
Jd 4 x p

~

quan-

"1'''. index for special

and imply a summation

over

r:(J()t"C)(" +:O
= !d+" ~,,(.) Q,,[~+(l<). t(XI) p

and so on are always understood. Take tional

J(x) =J+(x) =0

Zp[h;~, J]

in Eqs.(2.7) and (2.8), the CTPGF generating

for the order parameter

Q~(x)

as

Z:p [h(X))=i!f[h(~).

r\:().1("))\

(2.10)

J+(X)=J("'= 0

Eqs.(2.7)-(2.10)

func-

will degenerate into the statistical generating functional

~educe

the problem of determining the role of the

density ma-

trix in the generating functional for the order parameter to the problem of investigating its role in the CTPGF generating functional for

the

constituent

fi-eld. Rewriting Eq.(2.9) in the incoming interaction picture and using the Wick theorem generalized to the case of CTPGF[3,4], one can readily show[2,3] that (2.11)

where

rts.r ==

JdulzJ?(I) S. (1,2 }J(2),

(2.12)

r

86(1,2) being the bare CTPGF satisfying (2.13) and

e,..p{ilofr,. (Jt.r] } =Tr {Af: expli

(J

.'"'1'1 t'f'/J) '" ]: }

(2.14)

i.e. , .. wN(rt. r]=E -I- f P

'".no,

7Il!"!

P

el,···

I

1>0111

J.,-

,-

, ... "'"

(2.15) with

597 310

ZHOU Guangzhao, SU Zr.aobin, HAO Bailin, YU Lu

(2.16) In Eqs.(2.14) and (2.16), $t(x) and $I (x) are operators in the incoming interaction picture, : ... : means normal product. Since the distinction of time ordering along the positive and negative time branches does not make sense under the normal product,one can easily show that (2.17) in notations of (I), i.e., ""pH (J+, :r)= 'WN

witb

'WNrr.+ r. )=i l' 4 , . . ",,JIll

T [JA , JA]

(2.18)

-'-JJ.r", c:/.md.r .. , dn l7l! n! (2.19)

where (2.20) It is important to note that the functional at the left-band side of Eqs.(2.17) and (2.18) is defined at the closed time-path, while that at the right-hand side is defined in the ordinary space-time. Similarly, Eq.(2.16) is an expression' at the closed time-path and, therefore, each argument can take value of either positive or negative time branch, while Eq. (2 .20) is an expression defined in the usual space-time. Furthermore, taking into account that in the incoming picture the field operators satisfy the free field equation, one can easily find that' • I, I ' " P .. " J.p QiJ'-S-,(",)",Nrll,7II'CI"

.. , III , -n. '"

-,)-0 ,

(2.21) (2.22)

or, equivalently,

(' ,,)-N(II ..... ' j '"J,'---' SOY' 1,1 IN Jeli

' - N ( I I mJ

IV

'

"

-

0

-,

(2.23)

{f"'I"·>t,,n"·T,=,

-..

(J '" >n, 1'1 '"

-

-_.r

..• -:-

i ... I ) 5" ... lL ,

I }

=

a,

(2.24)

Since the moment t = -co is chosen as the starti!!g point of the closed timepath, the initial condi'l:ion for the statistical system is fixed at this moment. Therefore, we are not allowed to integrate by parts in respect to the operator a/at arbitrarily. Tbe correct direction of action is indicated by an arrow in Eqs.(2.2l)-(2.24) to incorporate appropriately the initial condition. Substituting Eq. (2.18) into Eq.(2.11) and taking into account Eq.(2.8), we obtain .l,.['; J+l]=eXr{i (laa(-t i,~, - i.~Jtli .. t(ti ,~, - i1j.J)}

xexp{i. (- J+,s.1+W N (J: .I.d)},

(2.25)

598 on

Theorg or the Statistical Generating Functional ror the Order Parameter (II) --Densitg Hatriz and the Field-Theoretical Structure of ~~e Generating Functional

311

which is the first expression for CTPGF generating functional we derive in this section. Eq.(2.25) specifies the generalized Feynman rules forCTPGF and shows clearly how the density matrix contributes ~o CTPGF in terms of WN[J!.J61 in view of the global structure of the perturbation theory. It tells us that the density matrix affects directly only the correlation functions of the constituent field variables describing the statistical fluctuations (corresponding to oJ6T. 6 ). So far as WN(m,n} satisfy Eqs.(2.23) and (2.24). the contribution of the density matrix can be expressed in terms of the initial conditions(sometimes called boundary conditions) for the statistical Green's functions. Now we derive another expression for the CTPGF generating functional. USing the following equality (up to an unimportant constant factor)

oJ

(2.26)

it is easy to show that

(2.27)

if the path integration is taken by parts.

Taking into accounr. that (2.28)

s.,.-

•&. .,._expCi,/,+S •- '.") =e"p (.''1' +5•-, .,. ) ( St'I -S•-, '"T ) •

(2.29)

Eq.(2.27) can be transformed into

e~p{i (-ItS.rt W'rN[.r+. r)}

= Jp (dt') (df) &JCr{L(J+,/,t"'+J+'I'+S.-',/,)}

(2.30)

Using Eq.(2.3) and the convention of (I) we find that (2.31)

t

S

(2.32)

11".I tl.+,f;C1J?, So,,,. (I· X ) = Jd+; t;(I}S~: (1·X.J,

(2.33)

?cr rt+(xtr )

ft~oc) •

(2.34)

and obtain from Eq.(2.18)

599 312

ZHOU Guangzhao, SU Zha"bin, HAO Ba:i.1in, YU Lu

exp{ i =

w; (: i S~ -

,/,+"$.-' ,

'j

S~. - S.-'t)}

(2.3S)

e"p{iwN[-'I';So;', -S.~ 'l'4J}

and

elC?{ i "'; (-,/,"5;' . -5;''I']} = exr{iwN[-'I';S;;, - 5;~ t~J}

(2.36)

considering 1/1;, 1/1&, as well as lIi • '~t as independent funct iona~ arguments. c Substituting Eqs.(2.3S), (2.36) into Eq.(2.30) and putting the result obtained into Eq.(2.11) we get finally

xeJeP { iW,.N [-'/' +--, S. ,

--, 'I' J}

- S.

(2.37)

as the second expression for the CTPGF generating functional for the order ?arameter-a path integral present:ation. It is easy to rederive Eq, (2.25) from Eq.(2.37), so these two expressions are equivalent to each other. This path integral representation is different from what we obtained previously tor the CTPGF generating functional[3 J , in so far as the contribution of the densit:y y + ,,-1 matrix is expressed bere as an additional term in the action given by W'p [-IV ~o, -S~11/lJ. According to Eq.(2.36) tbis term does depend only upon the field variables 1/1~, 1/11 describing the statistical fluctuat:ion, but not upon the field variables describing the dynamic evolution. On the other hand, it is clear that [-1/I+S'~-!., _~11/l J has nonvanishing contribution to the generating functional only at the initial moment t = -.... Expanding w~ in accord with Eq, (2 .1S) and integrating ~1, S;l by parts we find that only terms corresponding to the complete aifferential contribute, because the expansion coefficients satisfy Eqs.(2.21) and (2.22). Since the functional integral is taken over a closed time-path, starting and ending. at t=_CD, ~[_1jI+S;l, _S~11jl] has nonvanishing contribution only at these end points. Before further analyzing Eqs.(2.2S) and (2.37),the derivation of which is the main subject in this section,we apply first these equations to an important special example: the contribution of the density matrix to the CTPGF generating functional in thermal equilibrium,

W;

III. CTPGF generating functional in thenmal equilibrium As an important special case we will derive an explicit expression for the CTPGF generating functional in thermal equilibrium, i.e., for the density matrix given by (3.1)

600 On Theory of the Statistical Generating Funcr:.i.onal .for the Order Parameter (!::) --Density Matrix and the Field-Theoretical Struc~ure of the Generating Functional

31J

(:1.2)

where H is the total Hamiltonian 0: the system, !Ii-operator of particle number, ).I-chemical potential, exp(-n)-normalization consta'nt, or the inverse oJ the partition function. Substituting Eqs.(2.l7) and (2.18) into Eq.(2.14) we find that

."::;. ~;

where $I'~! at the right-ha~d side are operators in the inco~ir.g pict~re. ~: 1s known for the operator AI(t) in the incoming picture that[5]

It is essential to note that

a in

Eq.(3.4) is the total Ham~ltoni~~.

If an an-

alytic continuation -

i 6

is carried out we find that (3.6)

Taking into account that for complex fields the operator of particle number (S.i)

is a conserved quantity. it is easy to prove that (3.8) where ~

~+

A = + 1,

if

AI(t)

=1jII(x)

,

A = -1,

if

AI(t) = $I (x)

,

A =0 ,

if

AI(t)

(3.9)

is Hermitian.

With Eq.(3.8) we can apply, as done by GaUdin[6 1, the following identity

Tr{ (r A(II t

(±)II

= Tr 0[A (I), A(2) J,AU)" t ...

AlIly JAm.·. A(nl}

'Atlll} ±Tr(f A(2) [A (J),AW)",A (f)'"

Aell)}

A .. ·AOI-I) A ("AII).Alnl:; A ]} (±)11-2 Tr fA fAI21

(3.10)

601 314

ZHOU GUangzhao, SU Zhaobin, HAD Bailin, YU Lu

to the right-hand side of Eq.(3.3) to obtain

Tr{ft~: e xp[Hr;tx +t;J41):}

= Tr {frk} <0

f:

exp

(F Je"r [i (J;,f, t t/ J4 J): 10> ,

(3.11~

1

where (3.12 ) ""]

;110

A

A

A

(3.13)

[A • B :r = A BtBA •

In deriving Eq.(3.11) the properties of the normal product and the requirement of particle-number conservation are taken into account properly. Note that for nonrelativistic complex fields the operator ~I(X) contains only the positive frequency part, while $t(x) the negative frequency part, so we find [

where find that

'"1'1 (; ).

"1 'fIr ( 2 .' J'!'

-t

= i S.

is defined by Eq.(2.3).

(3.14)

(I. 2) )

Substituting Eq.(3.14) into Eq.(3.12).

we

with (3.16) Now operating eXP(FI ) upon exp{i(Jb,$I +$iib,)} the result obtained into Eq.(3.3),we find that

in Eq.(3.11) ;,::d

substituting

(3.17) According to Eq. (2.1.8) and ·the convention of

(I)

we Lave (3.18)

where the closed time-path matrix (3.19) Eqs.(3.17) and (3.18) constitute the contrihution of the d~!:.sitr matrix to the CTPGF generating functional in thermal equil!b~ium we consider in this section. As shown by these explicit expressions for the functionals ~h,P [.r+ ,J] and W~[J!,Jb,]. the equilibrium state is a Gaussian pr~cess. the statistical properties of which are described completely by two-point ~orrelation function N

So,p (x,y). Substituting Eq.(3.18) into Eqs.(2.11) and (2.37) we obtain the following expreSSions for the CTPGF generating functional in thermal equilibrium:

602 On Theory of the Statistical Generating FUnctional for the Order Parameter (II) --Density Matrix and the Field-Theoretical Structure of the Generating Functional

]1S

(3.20) and

tp (Jr.;

J'JJ

= Jp[dcy') (clt)exr{~('I'ta.~'t t Ijnt[t~t)+J+'f'+'f'+JthQ(tt,t)}

,

(3.21)

where

Go (r.

N

2) = 50 (/. 2) t So,p (I, 2 ),

(3.22 ) (3.:3)

It is easy to check that

and

Ga(1.2)

are reciprocal to each other.

G~1(1.2)

defined by Eqs.(3.22)and(3.23)

It is also easy to show from.Eqs.(3.16),(3.19) :Lnd

(3.22) that in Fourier representation

-+ (pJ= ±elCp[pCfl.-)o'J JG.t-

~o

(::.24)

(P)

and

Goc (P) =

cth (fJ (P.-)' l/zl th(,6(?-.,"'/z)

(Go~ (Pl- G.cdP)

(3.25)

,

which are the well_known[i,8] equilibrium FDT

satisfied by statistical Green's

fllnctions. Since the equilibrium state is a stationary state, the final

st:atistical

Green's functions obtained after taking functional derivatives should be translationally invariant in time.

In this procedure the only role of the second S~,p(1.2)

term in Eq.(3.23) is to produce

independently in the final result.

5;1

term in

00(1,2)

and cannot appeal"

Therefore, we might not distinguish

~land

from the beginning and take

4-0-r instead of Eq.(3.23).

-I

(I, 2) _. 'S.

(3.26)

(I, 2 1

Irt that case the first term of Eq.(3.22) can be

ered as an inhomogeneous.solution of the

s~ 1

Cro =

eq~ation

I

to describe the dynamics, while the second term of Eq.(3.22) can be

( 3 •2i )

considered

as a homogeneous solution of Eq.(3.27) satisfying the equilibrium. FDT as by Eqs.(3.24) and (3.25).

consid-

given

We can thus rewrite Eq.(3.21) as

t: p [ll; JtJ] =

J [&'1'+) (elt) exp{ i (I.(t+.'I']tlint('f';t )tJ+t+'I'+J+hll.('I':t ))}. P

(3.28)

603 316

ZHOU Guangzhao, SU Zhaobin, HAO Bailin,

ru

Lu

In fact, not differentiating &;1 and S~l, Eq. (3.26) can be obtained directly from Eq. (3.20) by Fourier transformation of' the path integral as given by Eq. (2 .26). An interest ing result follows from Eqs .(3.20) and (3.28) as expressions for the CTPGF generating functional in thermal equilibrium: If all space-time arguments of functions and 4-dimensional integrations are extended to the closed time-path, then the CTPGF generating functional has the same structure as that for the ordinary quantum field theory. Formally, the density matrix disappears, the statistical information being contalned entirely in the propagator Go (1,2), defined at the closed time-path and satisfying FDT, as given by Eq.(3.24). Eqs.(3.20) and(3.26) extend directly the Feynman rules for Matsubara Green's functions at the imaginary time axis to the real time axis. Several authors have. discussed the Possible generalization of the Feynman-Wick expansion for the Matsubara functions. Some of them[9] have analytically continued the integration path from (0, -is) to complex time plane to derive the expansion rules for the real time retarried Green's functions which are too involved to be of practical value. The others[lO] have tried to derive perturbation expansion for the closed time-path si~ilar to ours, but the end-points of the generalized contour they have used differ by i6 which gives rise to some difficulty in justification. This difficulty is avoided by using the incoming picture in our derivation. Eqs.(3.20) and (3.26) show the Feynman rules we established for CTPGF are formally identical to those in the ordinary quantum field theory. Using the transformations derived by us previously[2 j which connect the CTPGF with the real time statistical Green's functions, the perturbation expansion for the re~ time functions can be easily derived.

IV. The field-theoretical structure of the statistical generating functional for the order parameter To "transplant" the techniques of effective action for determining the generating functional in quantum field theory to solving the statistical generating functional for the order parameter, we need to clarify the difference of the field-theoretical structure for these two cases. In .Sec. II we have obtained two equivalent expressions for the statistical generating functional showing that the difference consists in an additional term expressing the contribution of the density matrix -- W(N) -functional for the statistical case. In Sec. III we have shown further that these two generating functionals have identical structure, the contribution of the density matrix being incorporated in the CTPGF bare propagator satisfying FDT. In this section we will analyze the results obtained in Sec. II to discuss the structure of CTPGF generating functional in the general case. For a general nonequilibrium process, the CTPGF generating.functional given by Eq.(2.37) can be rewritten as (4.1) where

604 an

Theory of the Statistical Generating FUnctional IUL the Order Parameter (IIi ---De1l6it!l Matrix and the Field-Theoretical Structu.ce of the Generating Functi·;r:al

317

r PCol (h', J,J t) (4.2)

is

the CTPGF generating functional for the ground state. SinCe Z~O] [hjJ+ ,J] has exactly the same structure a"t "the closed "time-pat:h as r.'1", ,)1"dinary field theory, we can first determine z~O] [hjJ+,J] by a 6"tandard fi~:d theoretical technique and then find Zp[hjJ+ ,J] from ZfO] [hjJ+ ,J] in accord with Eq.(4.1). Such a procedure is similar to what is used for solving liauv!!Ie problem in classical statistical mechanics. The dynamics of the systc:m 1;: O] [hjr ,J], while the initial statistical distribution is g:'V2.il described by N[ + . cS "'-1 . ;t-1 cS ]} b y exp {iW p _l. cSJ"O , l.::)o cJ+ • Many interesting nonequilibrium phenornen:l. c:tr. !::e oeser-ibee. by a ,:.,,~.:;"'.:."._. process, i.e.,

Zb

wN(m.n)

p

(I.2···m;n···T)=O

(4.3)

except for N (/,1 )

Wr

ex,

y)"*

(-1.4 )

D.

In .such a cas.e. we obtain an explicit formula of the CTPGF genera"ting f·<.:ncti.)l::J..L for the order parameter, if the equilibrium e:q:ression for SN ill E4:;'. (3.2G;'o,p a ,l)(X,y) i.e., to let (3.23), given by Eq.(3.19t is replaced by

W:

N

.v(I,I).

S•• p (X,Vl=-W p

l.X,:!!.

(4 ")

..,

It is obvious that the CTPGF generating functional in sllcn a ease has ,!>? "':."~,,. structure at the closed time-path as for the ordinal'Y field theory. On the other hand, i t is clear from Eq.(2.8) that the CTPGF generating functional for the order parameter can be determined as some combination of the generating functional for constituent field in a way independent of the stat isf"J.nct irJnal tical properties. This means that if we write down the generating for the order parameter (4.6)

with (4.7)

and (4.8'

and also the analogous expres·sions for the case without the statistical information as given by Z(O] W(O] r(O] ar..d Z(pO] [hjJ+,J] we find aceordi!l~ to p' p' p Eq. (2.8) that the structure of the generating functional for the order p .... r:urJ€:ti:lI· is identical to each otber in these two cases. Furtbermore, Eq.(2.25) tells us how the density matrix contributes to the Green's functions of tbe constituent

605 318

ZHOU Guangzhao, SU Zhaobin, BAD Bailin, YU Lu

field in view of the homogeneous ~quations for free field (See Eqs.(2.21) and (2.22». These w~(m,r.) functions can be specified as initial conditions for s"tatistical correlations. It is dif'ficult to formula"te these initial conditions in the general case, but many systems we are interested in are existent in stationary or quasi-stationary state (I.e., stationary at a micro-long,macro-short tim~ scale), in which case the effect of the density matrix is realized through the generalized FDT. [8] In the nex"t paper we ~'ill illustrate this point by an example 01' open sys"tems. To summarize WE:' come to the conclusion that the determination of the CTPGF generating functional for the order parameter can ,be divided into two steps: to "forget" first about the density matrix and to find the generating functional ..... i"thout sta"tistical information and then to include the latter in the second step. In general cases, this can l:.e done by using Eq.(4.1). The whole procedure is greatly simplif1ed if the s"ts"tistical correlation is Gaussian or a generalized FDT can be formulated. This division into two steps facilitates the practical calcula"tions and provides a unified approach to both equilibrium and nonequilibri~m problems. preIn tne next paper we will discuss a practical scrip"tion for aete rrni nin;; t!:e g-enf'ra t iag f"nct :'ona1 and ill ustrate it by simpl e examples.

References 1.

ZHOU Guang-zhao, SU Zhao-bi.n, HitO Bai-lin, !'U Lu, t:Je precedi.ng paper.

2.

ZHOU Guang-zhao, SU Zhao-bin, HAO Bai-;'i::, YU J:.L:, Phys. Re,'. ~ (1980), 3385.

3.

ZHOU Guang-zhao, SU Z.'lao-bin, eh. 5 in "pro~ress in Statistical Pl"lysics~ eds. HAO Bai-lin et a:l., KEX::r: (Science Press), Seiji::g, 19a1. ~

4.

A. Hall, J. P.'lys.

5.

See, "for example, P. Roman,"Advanced Quantum Theorr,;~ Addison wesley, 1965.

6.

M. Guadi~, Hucl. Phys. 15 (1960), 89.

7.

(1975), 214.

ZHOU Guang-zhao, St' Zhao-bin, Physica

!.

Energiee Fortis et Nuclearis (Beijing),

(1979), J14.

8.

ZHOU Guang-zhao, sa Zhao-bin, Acta Physica

9.

See, for example, I.i:. Dzyloshinski, Zh. Eks. Teo. Fis.

(JETP) 42 (1962). 1126;

G. Baym, A. Sessle"', Ph,.;;, )lev. 10 ~

See~ for exaziople, R.

Gordon and

i!reacj~,

Sinica, 30 (1981), 164,401.

X.i..l..:..s I t;.

I,

~

(l96ij, ~·345.

Pr~pa.gators ter £'tany-Particle Syste:ns~

r. 1 ~58.

606 COIIIIIIZn • .in 2'beor. Phys. (Beijing, China)

Vol. 1, No. 4 (1982)

389-403

ON THEORY OF THE STATISTICAL GENERATING FUNCTIONAL FOR THE ORDER PARAMETER (III) --EFFECTIVE ACTION FORMALISM FOR THE ORDER PARAMETER ZHOU Guang-zhao ( Ji1!i ) HAO Bai-lin ( ~ffi'# )

SU Zhao-bin ( YU Lu

(f

n** )

~)

Institute o£ 2'beoretical Physics, Academia Sinica , Beijing, China

Received December 28. 1981

Abstra.ct A practical scheme is proposed in tl1i.s paper to determi._ the quantum statistical properties o£ the order parameter in the framework of the closed time-path Green's functions (C2'PGF). crosco~c

As a md-

quantum theory, this formalism is applicable in principle

to both equilibrium and nonequilibrium phenomena and is capable of

dealing with statistical systems both above and below their phase transition point.

'rhis £or1llll.lism can be used to discuss the sta-

tistical properties of simple as well as composite order parameters in either uni£orm or nonuni£orm systems.

As simple illustrations

and check for the theory, the proposed prescription is applied to the contact-interaction model for superconductivity and a unimode laser system coupled with two-energy-level bound electrons.

I. IntroductIon This is the third paper in the present series on the theory of the stat istica1 generating functional for the order parameter in the framework of CTPGF. In the first paper [1] (as [1] hereafter) a general formalism is proposed to construct the generating functional for the order parameter which embodies the statistical properties of the system. In the second paper[2] (as [II] hereafter) it is shown that the determination of the generating functional for the order parameter can be divided into two steps: first to find the CTPGF generating functional formally without statistical information and then to incorporate the appropriate statistical information at the second stage. In this paper we propose a practical prescription to determine the quantum statistical properties of the order parameter. In Sec. II we describe the general procedure of the effective action method for the order parameter, the key steps of which being the construction of the effective action for the composite operators and a loop expansion for the vertex functions. As seen from the presentation in the next section, this technique is applicable to a large class of systems with either simple or composite order parameter expressed as a quadratic form of the constituent field. Within· this framework. the equilibrium and the nonequilibrium phenomena are treated OD an equal footinS. the .ditferencebeing embodied in some cases in the fluctuation-

607 390

ZHOU Guang-zhao, SU Zhao-bin, HAO Bai-lin, YU Lu

dissipation theo~em (FDT). Moreover, this formalism can be used to discuss the space and time variatiCllI!'S of the order parameter and related quantities. To illustrat,~ the specific features of tbe present formalism we apply it to some known examples. As an application to equilibrium system with composite order parameter we discuss the four-Fermion contact interaction model for supercoRductivity and reobtain the BeS gap equation. As an example of nonuniform case we consider superconducting system close to the transition point with weak nonuniformity and rederive the Ginzburg-Landau equation without external magnetic field. Both these two cases are considered in Sec. III. Finally, as an example of far from equilibrium phenomena we discuss in Sec. IV the coupled two-level electron-unimnde laser system and reobtain the result of semiclassical Lamb theory. As illustrated by these examples, the statistical information is incorporated into our formalism as initial values for correlations in some analogy with the ordinary Liouville problem in quantum statis'tics. However, as a general scheme, o~r formalism is applicable to systems near thermal equilibrium (See Sec. III of [II] [2]) as well as open systems far from equilibrium. It is worthwhile to mention that the treatment of equilibrium superconductivity and nonequilibrium laser system in the present framework is analogous to each other to a g~eat exter.t. Although some authors have emphasized the similarity between these two systems [31 , as far as we know, there is no other microscopic theoretical scheme to consider these two systems from a unified point (If view. Throughout this paper we will use notations adopted in (I) and (II) and ask our reader to refer to tbose papers for details.

II.

Effective action method

In this section we describe the effective action method for practical determination of the statistical properties for the order parameter. As in (II), we consider an n-component nl)nr(~lati~'istic comple~ field 1iJb' l/It, b = 1,2 ••• n, for clarity of presentation, but the result obtained is completely general. According toCIIl-(2.37), the CTPGF generating functional of the system can be written as

~r [h ; 1:

n= L[d.yt] [cii' ]e,.-p{ i CI.[-r+, t J+l;nt ['t'~'f'])} x exd; (J+t t

't't]+ltg ['t'~t] +W',.;" [-'f+~'J

(2.1)

-s."'t'JJ}

where 10 [1/1 ,ill] is the free par'[ of the 1j;-field action, nonlinear interaction. The order parameters

lint [~,+ ,1/1]

the

(2.2)

are compos1te operators of the constituent field ha(x) are external sources for ~:(x), ~b(x) and

Jb (x), Jt(x)

and correspondingly,

608 On Theory of el1c Statistical Genera t illg Flulctiollal t"or t.l.u:- Orrier Parar.:et:cr (:!:!I) Effecti ve Actioll Formalism for the Order i'dral1leter

w~ (-1/1+801 ,-SOllji] being the contribution from the density matrix. ating functional fer the order parameter is given by

391

The gener-

(2.3)

with (2.4)

and (~.5)

r,.(Q.l =W,..[h ) -ho. with

fr.. [Il] GQ (;<)

-

(2.6)

h(~) ~enermting

Similarly, we can define CTPGF formation as

functionals without

stati~tical

in-

(2.7)

(2.8)

(!:!.9)

(2.10)

where

hex)

and

Q(x)

0" w~·J (hJ S hex)

satisfy equations

=

.r r:oJ [II.] [Qexl

lI.ex).

(2.11)

-hex) .

(2.12 )

Using the results obtained in (I), (II) and ~eneralizing the steepest ~e scent technique of evaluating the generating functional for the irreduciblp vertex functions in the usual quantum field theOry(~j to the closed time-patt. we can formulate the following practical rules for determining all Green's functions for order parameter. (i) To determine the effective action. for th", order parameter at tile closed time-path, which is formallr defineu as

1[dw]exrti(1

(eft .

rp [1.1")]= (0)

\\·ith

"

La.HnQ.)~

(2.13)

p • " [ ' 'J (. ( III,.CD] [h'] -nL'", t e"'p(d l'[Q.J)=J dn expp "'J

• e.lC

(2.14)

p

In practice, the effective action is found by changing variables in tha pat~ integration and comparing the result obtained with !q.(2.13). In the next section we will give a concrete example. Alternatively, the effective a..:tio:: .::::-: be defined as

609 392

ZHOU Guang-zhao, SU Zhao-bin, HAO Bai-lin, YU Lu

exp(i rei! (Q))=

1[d,/,+][d,!,] i

(Q(X)_ Q [,/,;,/,))

p

xexp{ i (1. (t;'r) t

I;nt ['1'+, 't'] +Jtt t

'I't.n}

I .

(2.15)

J~J=O

(ii) To generalize the steepest descent method of evaluating the functional integr:ll[ 4] to the closed time-path to obtain the following expansion for the vertex functional (without statistical information) (2.16)

lfJ

where r [Q] is the sum of one-particle irreducible (lPI) vacuum diagrams for an equivalent system with action given by (2.17-) where q(x) defined as

is field variable for the equivalent system with free propagator

(2.18) We will call the first term of Eq.(2.16) the tree approximation, the second term one loop contribution and the rest high loop correction. It is worthwhile to note that unlike the quantum field theory, where the field. variable itself is the order parameter in most cases, the effective action Ieff [Q] and, therefore, Ieqv [q], might have an infinite number of interaction vertices. but, only a finite number of vertices contributes to r l: 1 up to finite loops of IPI diagrams. (iii) To include the correct statistical informatio& into the vertex fUDctional given by Eq.(2.16). As discussed in (II), this can be done in most of practically interesting cases by using generalized FDT. (iv) To determine the statistical Green's functions for the 'order parameter from the vertex functional rp[Q] according to the statistical functional equations(D-(2.27), (2.Z8), or the equivalent to the hierarchy of equations tU-(4.3)-(4.5) •

(v) For some systems the functional integration (2.15) is difficult to carry out explicitly, but the effective action Ieff[Q] can be found by transformatioD of path integral for Zljl [hi J+, J] as (2.19) t tp,(oJ (h;J.JJ!

==1

{,e ff [IlJthll)}

+ [da]exp i(l . 3'-J-O l'

.

(~.:.aO)

610 On Theory of the Statistical Generating Functional for the Order P.lr.Jr.:ecer (r.r.r)

-- Effective Action Formalism for the Order Parameter

393

It is important to note that Mo in Eq.(2.19) is a quantity independent of 1 h,J+ and J (in the example of Sec.III, Mo =_2g- ). Let ~lS define

(2.21) It is obvious from Eq.(2.19) that (2.22) Le. WJOl and W,£O] will generate exactly the same connected Green's functions except for the second order CTPGF which differ from e?ch other by Mo. It is easy to show that a relation similar to Eq.(2.22), (2.23)

is also true for the connected function~l with statistical information, if Eq.(11)-(4.1) is taken into account. Since WI' and ?ip have identical statistical properties, we will not distinguish quantities with and without "," in subsequent discussions. The abo,?e-said 5 pOints summarize what we call the effective action method for the order parameter. As we see, this is a pract:c:tJ. p=escription, if we know the explicit expressions for the ori~lnal action I [v~, ~l and the order parameter Q [1/1+, 1/1 ]. Usually, the elementary interaction vertices in the original action have 3 or 4 external lines, while the order parameters are polynomials of the constituent field variables not higher than the setond power. It is easy to see that no difficulty arises in applying tbe described sche~e to such systems. Some of the technical details will be explained in the illustrative examples given in the next two sections, where fer short we will no longer distinguish by notation the CTPGF generating functionals with and without statistical information.

III. Application one - Contact interaction model fer super-conductivity In this section we apply the effective action method to the four-Fermion contact interaction model of superconductivity as an example of equilibrium system with complex composite order parameter. We also consider the superconductor close to the critical temperature with we3k spatial nonunifor~ity to illustrate how proposed formalism is applied to inhomol!l'eneous. syste:ns. The basic field variable is a complex Fermi operator with spin index 1!.b(x) , b = t or oj.. The original action is given by (3.1)

1.['1': 't'J

ip cl.r ilz If

-1 (1,2) tez], 50

I int ['i'~ '1']= ~ 1, dx.'I':llC)

+:(X) +b Cx)+Q.l)()~

(3.2)

(3.3)

611 394

ZIIOU Guang-zhao, SU Zbao-bin, H"O Bai-lin, YU Lu

where

S·'CI, •

2)=(i..l..+...!....V'.j.IJ.I~[~1I-2)(· 10) at

2m

(3.4)

0 I

/,

is diagonal in the spin space, m-E'l ectron mass. lJ -chemical potent ial, gcoupling constant. The order parameter of the system is given by Gorkov composite operators X(x) and X*(x) defined as: (3.5) (3.6)

Introducing the external sources CTPGF generating functional for w(x)

(x), r.;(x), b(x) and h*(x), the and X(x) can be wri"tten as [2}

1'10

1:7]= f [d'f'+] [.1,/,] exr{i(I.['i't.,.]+I;nt['i'~'I'J)}

-rp[lr~ h;

p

x e .. r{i('t'l'.,.t+lth~+.x·htW/[

(3.7)

l5";', ~'r])}"

According to rules given in Sec.II, we "forget" first \'i~ in Eq.(3.7) and carry out the integration o"l.-er 'l' (x) to obtain the effective action. Note that 1jJ(x) is anti-commuting GrasHmann number under the path integration. Generalizing the Gaussian integration formula to the closed time-path, we obtain

" Jf'd. 4;( 'f...+(X)"'" + 'It,(l(Jf.,YJ} e"p{-f-i (>()

=

C~n$f f [df"] [elf] exr{ i (- ; ~. f +f' f, +, +~T 'ij + f

(3.8)

J} .

f'

Substituting Eq.(3.8) into Eq.(3.7), we find up to an unimportant constant factor that

:er [h", II;

I

j

~t, '1]= [d!*J [dt) exr(- f·!) I'

)(J[d't'] [.It] exr{i [tt'tT(~*+h*)'II '/'t +f(f( T+h lT1+t~'/'+f]} .

(3.9)

f'

To carry out the integration over Ferroion field in Eq.(3.9) we use the Nambu spillor presentation as (3.10)

and the corresponding source

(3.11)

to transform EQ.(3.9) as

612 On Theory of the Statistical Generating Functional for the Order Paramet.,,· (III) -- Effective Action Formalism for the Order Parameter

395

rp[h", hj .(, ~J=~[J,x"](d,X]e""di(-3J~'x+(Xt,.X~h-f/j·h)}

xl [cl'ft+J [cl'fJ exp{~( 'i'+cr-I.p + tp+~

t

~t'£t l}

(3.12)

p

where -I

Cr

r'.-I

(I, 2)

,

= G.

-I

__ (.

"". ( r •• ) -

(I, 2) t

3k

( I, Z)

a

(1

z,

~at t C13 2 m Ii'

1".... "

(3.12)

I

1)"d I

r

.

(3.14)

! ( - ," ),

(3.15) °3,°+,°- being Pauli matrices in spin space. tion over d'1'+ and d'¥ in Eq.(3.12) yields

lp [n "'. h: .,~, -;] = ex p(-

can'yin", ou,; the path integra-

;'i' h ) £[d}·] [ d, q

x exp{i (l.rlx]f Id,t[X. *,tJ Th",x t,x*h - S+(f., J}

,

(3 .1S)

where

I. [,x"',;'(]=-J

Jd"'.:rc;X,"'(lC)Y,(X) ,

(3.17)

p

Idet

*

,

[~ ,~] =- I T~J".

~+c;.'5=fdldz

(3.18)

4- -I ,

(3.19)

5+w (fU,2J$C2.J,

p Tr means trace operation in both spin and coordinate space. propagator G satisfies condition

Jd+x4-U, xJ Cf-'(X, 2 )=j d 4 xt;-'l" p

p

>c)

t:;.(x, 2 J = Jr

(1-2

The electron

J.

(3.20)

Comparing Eq.(3.1S) with Eqs.(2.19) and (2.20) we find that Zp has the expected form. According to discussion on point (v) in the preceding section, the effective action for X(x) at the closed time-path is given by (3.21) up to a correction to second order CTPGF given by Eq.(2.23). To illustrate the basic idea of the effective action methal we will confine ourselves to the mean field approximat ion for thl:1 gt::lel'at L1s iu;;;::t i<"llal. According to Eqs.(2.16), (3.21), (3.17), (3.18) (1)-(2.30) !I.nc 't;)e :'::l:!"t~r.cion about the notation, we find the vertex generating functional for tbe ordar parnmeter X(x) in the mean field approximation as

(3.22)

613 3%

ZHOU Guar.g-zhao, SU Zhao-bin, HAO

Bai-li~,

YU Lu

(3.23) (3.24)

(3.25) a,S

being closed time-path inuices. As seen from Eq.(3.16) the G(1,2) function appearing in the mean field approximation of CTPGF ge!'lerating functional :for the order parameter as given by Eq.(3.22) is exactly the second '.1rder CTPGF for electron fields '!I (x), '!'+(x') in the mean field approximacion. According to discussion of Sec.II, the contribution of the equilibrium density matrix can be embodied in FDT satisfied by propa~ator G. In the Fourie~ presentati0n of relative coordinates, FDT is given as [6,7] (3.26 )

where

X = t(:'l + x2)

while

Gc ' G", (;r

a.re defined as (3.27)

In Eq.(3.26) we assume G as well as the order parameter to be slowly varying functions of X. The statistical vertex generating functional :for the order parameter defined by Eqs.(3.22)-(3.27) contains in princi~le all stacistical information for the order paramet.ar in the mean field appro.~imation. Here we will only discuss its statistical expectation value. In accord with Eqs. a~(4.3) and (3.22), the general equ:ltion for the expectation ~'a::'ue of the order parameter is given by r

r pfn ....

'If

•

y' ."

1

I_

J -+! ~O:....:.__....:L:':".\."":"=-7'.._"4_':...'-"c=-'-._"'... c.:..

LX; (>C)

I

I);:

0

(X)=,t"tlC)=O

(3.28)

=:/

. X:... (;< j X" (~) :<:c (.>C)=~(.~) and its complex conjugation. We first consider stationary, homogeneous syscem where X<x), X*(x) are independent of x and Green's functions :lre translational1y invariant. Substituting Eqs.(3.22)-(3.25) into Eq.(3.2d) '.we obtair. the order parameter equation for uniform superconductor in the mean field approximation as

614 On Theory of the Statistical Generating Functional for the Order Parameter (ITI) -- Effective Action Formalism for the Order Parameter

397

(3.29) where

Tr

means trace operation only in the spin space. t.. ()()

Introducing notation

== 3J. ell) •

(3.30) (3.31)

and solving Ga , Gr from Eq.(3.20) in momentum representation, we obtian Gc from FDT given by Eq.(3.26). Substituting Gc thus obtained into Eq.(3.29) yields the famous BCS gap equation as (3.32)

Hereafter in this section we switch onto the usual units. Now we discuss the stationary system with weak inhomogeneity. the scale of space variation obeys the inequality

Assume that

(3.33) where

PF is Fermi momentum of electron. are satisfied by the system

Moreover, the following conditions

(3.34) (3.35) where T = 6-1 definedby[8]

is the temperature of the system, Tc - critical temperature

(3.36) with N(o)=

7IIPF

(3.37)

21C"1; ~

as the density of states at the Fermi surface, wD being Debye frequency of phonon, y- Euler constant. Since Gal is translationally invariant according to Eq.(3.13), the nonuniformity of the system comes from a slow space variation of the order parameter X as follows from Eqs.(3.22)-(3.25). For such a system we can expand the functional arguments X(y), X*(y) in Eq.(3.28) around x(i), x*(i) using O(hl j.tlil IAPI) as small parameter as we did before [7,9]. Taking into account Eqs.(3.33) and (3.35) to neglect quadratic term in 3A(i)/3i, we find that

615 398

ZHOU Guang-zhao, SU Zhao-bin, HAD Bai-lin, YU Lu

(3.38)

. ;;:1, *(-;) . ~iax

o

where (3.39)

(3.40)

Obviously, the first term in Eq.(3.38) is proportional to the left-hand side of Eq.(3.32) with

8

replaced by

8(X).

Considering

6(;) as a small quantity

and keeping its third power we obtain by the well-known technique[8]

& I ..if

&,,:Cxl ;t..•=}:.. =O J&:=~: :X.=J',

(3.41)

substituting Eqs.(3.39) and (3.40) into the four coefficents of Eq.(3.38) with 8

oz

0

in Eq. (3.13) yields (3.42) <1=0

(3.43)

(3.44 )

ar,~

(x. ~)

ifaf

(3.45) =0

616 On Theory of the Statistical Generating Functional for the Order Parameter (III) -- Effective Action Formalism for the Order Parameter

399

Finally, we obtain by putting Eqs.(3.41)-(3.45) into Eq.(3.38)

(3.46) which is the microscopic form of the Ginzburg-Landau equation in the zero external ma&netic field.

IV. Application two: Two-energy-Ievel electron - unimode laser system In this section we apply the effective action method for the order parameter to the two-energy-level electron - unimode laser system as an example of far from equilibrium systems. We describe the positive and negative frequency parts of the vector poten~f

tial

~l (x)'

the unimqde laser field by and

a(x)

and

a*(x)

correspondingly, while

$2 (x)

represent bound electrons at upper energy level lower energy level E~. The action of the system is given as CIO ]

E~

and

(4.1)

I

L. [a~I1)= d.rdz

({"(f)

r

I.[,/,~t]=Jdrd2 p lillf

'f'+( 11 5.-

I

2) (1(2)

(4.2)

J

(4.3)

(I,2)t(Z)

Jp

[a", 0.; 'i'~ '/' ]=-~3 d 4x thJ (l1+aoeJ- rr:o.
where -I

cr,

Llo

KO

4~' (I,

_ . i3

(zat

2)-

-J<.) df

(4.4) (4.5)

((-2).

is the photon energy, g-coupling constant of unimode electro-magnetic field

with bound electron.

The two-level electron system is represented by a spinor

f()(J=('f',(X))

(4.6)

'1'. ()()

with free propagator given by -I

5. where

0.;.,

0_

and

(f,

03

2)=

('

iJ

'aT,

I

t 2M

z

E."+E," V, - - 2 - - - 0;

.o,") ~f (/-2)

E'·-2 E

(4.7)

are Pauli matrices in the space of energy levels, M-

atomic mass. In this example, the basic fields themselves the order parameters we are interested in.

a(x)

and

a*(x)

constitute

Introducing external sources

j*(x), n(x) and n+(x) corresponding to a*(x), a(x), W+(x) CTPGF generating functional is written asCI] ,

and

W(x)

j(x), the

:I'(i*.i; ~t.1J= f[dQ.-) [c:lll.] (d'!'+("JJ[d,/,(~JJ p

)(ex'P{i(l.[~II.]+I.[,/,;'f']tl."t [a~ o.i""~'f']}

xe"r{<j"'a.+ o.*j +~+t of- tty tW; [_0.'" 4;' -A:'aj -'f'+S;'.-~'+J} .

(4.8)

617 400

ZHOU Guang-zhao, SU Zhao-bin, HAO Bai-lin, YU Lu

According to the discussion in Sec.II, we can neglect for a moment the

W~

functional

in Eq.(4.8) to carry out first the integration over

[dlji+] [dlji]

which yields

~r [t,j ; ~t, 7]

=

fCd".·] (da.] p

xexp{i(l.[o.*, a.]+Id..f[/l~a.J+j"CI.+ o.*i- (57 l} ,

(4.9)

where (4.10) , ( S·f (I,2}=S.-, ((,2)-13 cr,.aUi-
Jd+)(

S" (I, x) 5

(x,

2) =

Jd,+)C

x)

(4.11)

•

(4.12)

S-'(X,2J=!r (/-2)

P

p Tr

5 (I,

r ( ) 01' f-2

means here trace operation in both closed path coordinate space and the

space of energy levels. Comparing Eq.(4.9) with Eqs.(2.8) and (2.13) we find the effective action for the order parameters

a(x)

and

a*(x)

to be (4.13 )

Like the preceding example we will discuss only the mean field approximation for the statistical generating functional to illustrate the basic idea of the formalism.

According to Eqs.(2.16), (4.13), (4.2), (4.10), (I)-(2.30) and

the notation adopted, the CTPGF generating functional for

a(x)

and

a*(x)

in

mean field approximation is given by

(4.14) where

(4.15)

I clet [ a..4 , a.. ; ac.. , ac 1= -

. T'r"" 0 -f 5

I

(4.16)

)

2)] =[5"(01 f.!)] _i9[cr.,.(~ accrJt-21t.a..w) [ s·r er ''''"13 • , r "'13 U ~

- IT.. (~.c 1I~;', A~!

a; tft a: OJ)] ~ d.,.jJ [4 U)

((-2)

(4.17)

being defined by (4.18)

with

a,S

as closed path indices.

618 On Theorg of the Statistical Generating Functional for the Order Parameter (III) - Effective Action Formalism for the Order Parameter

401

Unlike Eq.(4.7) we assume in Eq.(4.17) that -/

5.

(1,2)

=(iL+_I_17~- ~- u:~) ""

at,

2M'

D

2

3

2

D

d,p C/-2J-Ze/, 2),

(4.19)

where (4.20) i.e. the CTPGF renormalization correction E(1,2) is included which is diagonal in energy representation in the lowest approximation. As seen from Eq.(4.9) the function S(1,2) appearing in ~he CTPGF generating functional for the order parameter as given by Eq.(4.l4) is the renormalized CTPGF propagator for bound electron. A macroscopically varying in time laser system is a quasi-station.ary state compared with the laser frequency. In the present case people usually neglect the feedback interaction of the laser field upon the distribution of electrons over energy levels. Therefore, in accord with the argument presented in Sec. II, the statistical information is contained in the generalized FDT l5 ,6] satisfied by So(1,2), the CTPGF electron propagator without laser order parameter being incorporated. In momentum presentation the required FDT can be written as l5 ,6] SDC

[P]=5DT [p) (T-2NfPJ)-(t-2NCpJl5....(PJ,

(4.21)

where Sor' Soa and Soc as retarded, advanced and correlation Green's function correspondingly are defined by (4.22) with

N[P]

=

N'a[P] (

(4.23)

Nl [P], N2 [P] are translational momentum distribution functions of bound electrons. Since they are usually fixed by external parameters in the laser system, we do not indicate explicitly their coordinate dependence in Eqs.(4.21)-(4.23). Expressing the renormalized self-energy part of electron in momentum representation in terms of the energy level shift 6E i

(PJ=t(Z~[P1t L~[P])

(4.24)

and the line-width (4.25) we rewrite Eq.(4.21), the FDT for nonequilibrium stationary state,as (4.26)

619 402

ZHOU Guang-zhao, SU Zhao-bin, HAO Bai-lin, YU Lu

where

r~, r~

and

r~

are defined by

(4.27) i

= 1,2

being index for energy level.

We define moreover for convenience

p2 • E,.[p]55-tE. +.6.E. 2M ~ I

(4.28)

The statistical vertex generating functional for the order parameter defined by Eqs.(4.14)-(4.16), (4.21) and (4.27) contains in principle complete statistical information of order parameter in mean field approximation. Take, for example, the expectation values of the order parameters as given by

= Tt{f ~ (x>} ) a (x) = Tr {Aj a."* (x) } •

(4.29)

aw

(4.30)

It

According to Eqs.IIl-(4.30) and (4.14), they equation

satisfy the following general

=0

0.:= a,,=

(4.31)

0

a.; (x)= a.*(X) a., lX}=a.(X) and its complex conjugation.

Substitution of Eqs.(4.14~-(4.17) into (4.31)

yields (4.32) where I Scex.Y'=-2f"C [sex 0I{)] J.jJ '01 )jI

(4.33)

•

and trace is taken here only in the space of energy levels. in Eq.(4.19) in terms of and (4.28), we can find considered.

Ei[PJ,

Sc(x,x)

Yi[PJ

and

Ni [PJ

Expressing (r[p]~B

by use of Eqs.(4.24)-(4.26)

from Eq.(4.12) with Eqs.(4.17),(4.19) being

Finally, we can transform Eq.(4.32) into

x

(4.34)

620 On Theory of the Statistical Generating FJnctional for the Order Parameter (III) -- Effective Action Formalism for the Order Parameter

where

l

403

is the wave vector for the unimode laser field and

(4.35) Eq.(4.34) is just the result of semi-classical Lamb theory of laser I l l ], if the nondiagonal dissipation is neglected. The unimode laser system coupled with two-level bound electrons discussed in this section is a typical example of open systems. As mentioned in the Introduction, since our formalism is set up in some analogy with the usual Liouville problem in quantum statistics, the statistical information appears in i t as initial conditions. This example shows, however, that this formalism is applicable to nonequilibrium, open systems as well. The field variables of a statistical system can be divided into two groups, one of which, the internal parameters, is associated with the subsystem of interest itself (such as the ,'ector potential field in our example), the other one, the external parameters, being connected with field variables outside the subsystem (e.g. the electron variables in our case). The procedure of eliminating the external parameters by means of path integration to obtain the effective action leff(a.:,I1.,,; (1c) embodies itself the characteristic feature of open system - the statistical information has to be "injected" constantly into the subsystem considered by external parameters. In the example discussed here this injection is realized through the contribution of electron propagator 50(1,2) to the effective action of the laser field. Therefore, the effective action Ieff[a.: ,0.,,; Q.cl contains not only the statistical information of the laser field itself but also, and more importantly, the statistical information about the inverted distribution of electrons.

a.:.

0.;,

References Z.~ao-bin,HAO

1.

ZHOU Guang-zhao,su

2.

ZHOU Guang-zhao,SU Zhao-bin,HAO Bai-lin,YU Lu, Commun.in Theor.Phys. (Beijing).l(1982),307.

3.

See, for example, H. Haken, in iaser

4.

R. Jackiw, Phys. Rev.

S.

ZHOU Guang-zhao,

.~msterdam,

!..

Bai-liII,i'U Lu, Ccnunun.ill Theor.Phys. (Be:i.jing),l:J19B2),29S. Handbook, Vol. 1, eds. F. Arrechi et al., North-Hollana

(1972).

su

~

(1974),1686.

Zhao-bin, Physica Energiae Fortis et Physica Nuclearis (Beijing)

(1979), 314. 16~,

6.

ZHOU Guang-zhao, SU Zhao-bin, Acta Physica Sinica, 30(19B1),

7.

ZHOU Guang-zhao, SU Zhao-bin, Ch. 5 in Progress in Statistical Physics, eds.

401.

S.

See, for example, A.L. Fetter, J.D. Walecka, Quantum Theory of Many-particle Systems,

HAO Bai-lin et al., KEXUE (Science Press), Deijing, 1981. MCGraW-Hill, N.Y. 1971.

9.

Guang-zhao ZHOU, Zhao-bin SU, Sai-lin HAO, Lu W, Phys. Rev., !E.1J19S0), 3385.

10.

ZHOU Guang-zhao, SU Zhao-bin, Acta Physica Sir.ica, 29(1980), 618.

11.

R. Sargent III, H. Scully and W. Lamb, Laser Physics, Addison-Wesley, Reading, ~SS.(1974).

621 COIIIIIIIII2. in 2'heor. Ph!;s.

(Beijing, China)

Vol. 1, No.6 (1982)

669-679

ON AN APPROXIMATE FORM OF THE COUPLED EQUATIONS OF THE ORDER PARAMETER WITH THE WEAK ELECTROMAGNETIC FIELD FOR THE IDEAL SUPERCONDUCTOR SU Zhao-bin (

;)j:.~

)

CHOU Kua.ng-chao (

IN:J'C~

Institute of 2'heoretica1 Ph!;sics, Academia Sinica, Beijing, China

Received May 10, 1982

Abstract; A silllplified derivation of the IlliJcroscopic electrod!;nuric

equations of UlIIezawa, llancini et 41. for superconductors is given in the f r . - r k of the closed time path Greell's functions (C2'PGF) using gueralized WaZ'd-2'akalIashi identities.

It is shown that

the for1llS ot the equations obtaiJled. are the same for both thermoequi1ibriwa and nonequi1ibriwa stationar!; states provided the e1e.:-tromagnetic field is weak and its effect on the IIIOdulus of the order parameter can be neglected.

2'he statistical behavior

of the sutes is complete1!; specified in the equations by para-

meters which can be calculated b!; the method of C2'PGF.

I. Introduction The interaction between the phase of the order parameter and the macroscopic electromagnetiC field (hereafter E.M.F.) always )lays an important role in superconductor physics. Both th6 Meissner ~ffect and the Josephson effect are brilliant examples of macroscopic quantum phenomena. With the rapid development of experimental techniques, the research relating to these topiCS is entering the realm beyond thermal equilibrium, fo~ which the Ginzburg-Landau theory [1] or the Bogoliubov-de Gennes theory[2] 01 equilibrium superconductor becomes insufficient in some cases. It· is reasonable to restart from :1 microscopiC :;oint of view to explore a new formalism for the i.:lter:1t'tion batwe'~l1 orde:' parameter and E.M.F., which is suitable for both equilibriwn and Donequilibrium phenomena. In the seventies, a set of macroscopic electrodynamic equations for ground state superconductor, different from the Giazburg-Lalldau equaT.i"n which is e Ue::tive only near the critical point, is given by Umezawa, Mar.,~illi et a1. 13 ], usingthe mapping and transformation technique on the Hilbert spac'i! within a fieldtbeol'atical formalism of ground state superconductors. Th"s<, equations rl"l>lt.-, thE'! pha.se of the order parameter, especially i ts singula~"i ties, to '!:he r.:~·;t"0-;CO~~::; E.M.F. explicitly and are capable of treating a numr:ilr of D'acroscopic '~un'>tmt phenomena for ideal gouIl[ their formalism, it seems hU.l'" :0 "larify the assumptions and approximatior,3 involved, and therefore, difficult to generalize t~eir formalism to a ur.ified theory valid both for ~inite temperatures and stationary states in nonequL1ibrium.

622 670

SU Zhao-bin and CROU Kuang-chao

As a primary check of the "theory of st!ltistical generating functional for the order parameter" [4] , we have rederived the Ginzburg-Landau equation near the critical point [41. In this paper, inspired by Umezawa, Mancini et al., a ,et of interacting weak E.M.F.-order parameter equations for ideal superconductors is derived in the framework of Ref. [4], with its E.M.F.-phase of order parameter part formally identical to that of Ref. [3}. We show that this set of equations is an approximate form of the statistical functional equations for the order parameter f4J • Moreover, the equation for the modulus of the order parameter 1s incorporated into our formalism naturally and statistical information is specified by the parameters of the equations impliCitly in accord with the CTPGF theory. If the intenSity of thp. E.M.F. is not too strong and the superconductor is far from the critical point, these equations are shown to be valid not only for the case of T=O ground state, but also for the case of T~O thermal equilibrium or some nonequilibrium stationary states. All parameters of the equations corresponding to different statistical situations can be calculated by the method given in lief. [4]. In our derivation, a crucial role is played by the gauge symmetry induced Ward-Takahashi CW-Tl identit1es for the vertex generating functional. The general form of the equations follows almost immediately with a transparent physical interprp.tation of our procedure which 1s in some sense the nonrelativistic version of Higgs mechanism for U(l) gauge symmetry wben the Goldstone field has singularities. In Sec.II thp. formulation of the problem is given with the approximations stated explicitly and the general form of the equations for the E.M.F. and the order parameter are derived. In Sec. III a set of coupled E.M.F.~order parameter equations for the ideal superconductor is derived by virtue of the W-T identities generalized to the closed time-path[S]. Finally in Sec.IV, as an example for comparing our results with the known ones and for illustrating the method of evaluating the parameters developed in Ref. r.], we calculate in the long wave-length limit the temperature dependence of the two basic parameters: density of the superconducting electron pairs and phase velocity of the phase excitation of the order parameter.

II. Formulation of the problem and the approximate fo~ of the statistical functIonal equations for the macroscopIc variables For superconductor interacting with the E.M.F., the interesting macroscopic variables are the vector potential of the E.M.F. )l=O. I. 2.3.

(2.1)

and the order parameter of the superconductor

X~X)==.A- (7', (XJti:X, (x I). S IXI exp [-' ® IXI]="Tr{f "'; 01) ~+ O,,} , where

p

is '1:lJenormalized density matrix in Heisenberg picture, AP(x>

(2.2)

being

623 On an Approximate Form of the Coupled Equations of the Order Parameter with the weak Electro~netic Field for the Ideal Superconductor

the Heisenberg operators of the vector potential and electron field operators with up, down spin indices. ventional symbols yields

671

~t(X), ~~(x) - Heisenberg comr .~rison with the con-

A

(2.3)

where d(x) is the energy gap parameter, g the coupling constant of contact interaction for weak coupling superconductor.. We choose gauge condition .; Ca) A/o" == 0

(2.4)

for "the vector potential. In Eq.(2.4), DII(a) is a linear differential operaThe concrete form of DII (3) tor, and "a" is an abbreviation of all = will be determined later. In our paper, we take the metric tensor as

a!1I

;." =

(.!.j.~.; .........), ;

y.,

v= 0, 1,2,3,

(2.5)

-1

. -l and the transformation properties of related variables are defined according to Ref. [6]. There is another variable B(x), the so-called ghost field, needed in the generating functional formalism, which corresponds to a Lagrange multiplier for the gauge condition. In accordance with Ref. [4], introducing the irreducible vertex functional of the macroscopic variables defined along the closed time-path f p =f (A)I(l'J. :::(l'J,@C:r.),8IlLl), we will have the functional equations r satisfied by the macroscopic variables AII(x), X(x), X*(x), B(x)

&Tp(A~.2.®.8JI GA)l,xl

=0

.rre (A~. s.®. 8) G®lXl

I

.t~ .. t-

Ire(A~ s.®. 81 / Ga IX'

srp (A~ s . ® G::\ ,.,

(2.6)

,tt_t-

=

0

(2.7)

=0

(2.8)

0

(2.9)

:t+-.t-

8)

I

;1:'- 1.-

=

Suppose the intensity of the E.M.F. is not too strong and the superconductor is far from the critical pOint. We can neglect the reaction of the E.M.F. to the modulus of the order parameter owing to the finite nonvanishing gap, ::tncl then, assume that 8(1')=

x,t ,

and the equation Eq.(2.9) degenerates into an equation for E.M.F. and with ~(x) taken to be constant

(2.10)

Idl

without the

(2.11)

624 672

SU Zhao-bin and CHOU Kuang-chao

where

=

~.

® . B]

(2.12)

A~= 8=0

$= consl. NOw, the equations for the order parameter interacting with E.M.F. decouple into two parts: the one Eq.(2.11) being the equation for the modulus of the order parameter and the other, Eqs.(2.6)-(2.8), the equations of the E.M.F. vector potential coupled with the phase of the order parameter. We solve the former equation first and then put its solution into the latter one as an input parameter. As the solution of Eq.(2.11) is a problem for the superconductor itself, we will focus on Eqs.(2.6)-(2.8) in the remaining part of this pp.per. Since we have assumed that the intensity of the E.M.F. is not toO strong, we m~y linearize the functional arguments A~(x),~(x) and B(x) linearly in Eqs.(2.6)-(2.8) according to the CTPGF technique given in Ref. [5] as

J-i4-rY {I;.: (x,~)l(V)+I;.; (':':/) C3H~: (X,j'18(t l} = J441 ire: (X"Jl(I Jtf,: rib (~JtfB: (X,YJB(~I} = (ij)

IlI,li

0 ,

(2.13)

0 •

(2.14)

Jd43{r&~ (lq)lclltfb: (lI"J® (1'+f&: (lI"IB(31} =0, where

(2.15)

s'r;cl, ::: . ® .r~ pc... ) .r~

. BJ (I,. )

(2.16 )

'A)I-B=® ... O

s

=s''''t;,"

~

fro

(2;Il)

.t+=t-

are the 2-point retarded vertex functions of the corresponding variables Cij)(x) , B(x) denoted by
A~(x): ~=O,1.2,3,

~±

=

1

02'

(2.17)

?! = ±1 Obviously, by taking Eq.(2.10) into consideration, all the 2-point retarded vertex functions appearing in Eqs.(2.13)-(2,15) are translationally invariant in the linear approximation, i.e., (2.18) In the following, as in the BeS theory, we restrict ourselves +0 dealing with only ideal superconductors, i.e., we will neglect the dissipative part of the 2-point retarded vertex functions. On account of A~(x) ,0 (x) and B(x) being real, all these functions appearing in Eqs.(2.13)-{2.15) are symmetric[5 1, i.e., (2.19)

625 On an Approximate Form of the Coupled Equations of the Order Parameter with the Heak Electromagnetic Field for the Ideal Superconductor

673

where i, j run over ~=O,1,2,3,~, b independently of each other. Along such a line, the time variation scale of the system should be slc\\' compared with that for exciting tlle ignor-ed relax.ation prOCE'sses. Consequently, expanding r~ (x-y) to the second power of ;t' taking account of Eqs.(2.18), (2.19) and tl".t:! local invariance under 3-dimensional rotation, it ts easy to derive the general form of r~ (x-y) as

r:' (X_Vl= __'_(M.)2Z- , [-V') (a 2_ ....

ze'"

:X

0

=_ ~. (:~)

2

•

2

II"

[-V

'J

l

l~' (- v'l('" (aH m~.c "

1

4

V2+~.-'.r cX-Y)

Cl

1j' "j

).r

4(X_

•

JI ,

(2.20)

where and m4 are the constants to be determined wit~ length and mass scales respectively, Z4 ~V2] - the unknown dimensionless renorrnalizatioD operator, v 2 [_V2] _ the unknown operator with dimension of velocity square, (2.21) being the 3-a LaplaCian, c, 1'1 and e being the velocity of light, planck constant and the algebraic value of electron charge. In Eq.(2.20) we have also introduced a symbol (2.22 ) Now, we will fix the gauge condition (2.4) according to Eq.(2.20) by setting

JJ'''"ca 1= (a'. vZe_ V'J a' C·

)

(2.23)

in Eq.{2.4) with

~

.,e}l.(iJ)

l2.24)

=-.8m .

it.

Along the same line, we can expand r~(.x-y) to the second power of Since the transverse part of vector potent~al may behave differently from the scalar and the loncitudinal part, we can introduce the transverse projection operator L(:~ and longitudinal-scalar projection operator L~v as (TJ

~,J=£..." +~,) ~:) =

...... 0:~.......... ;: ... .) (

.q .. : o~1

+a... -a· v.z ~

(2.25)

':P.,,= ( ...

!j...........) :

I V J

. - a.-,a· :

1

(2.26) ,

where ~,v run over time-space coordinate indices 0,1,2,3, while i,j run over space coordinate indices 1,2,3 only. Moreover, by taking into account the gauge invariance, local invariance of 3D rotation and Eqs.(2.18), (2.19), it is also easy to derive the general form for r~v(X-Y) as

626 674

SU Zbao-bin and CHOU Kuang-cbao

(2.27) where (2.28)

Zr[-V 2 1(ZL[-V2 1) bein~ the dimensionless renormalization operators for the transverse(lonritudir.al-scalar)part of the vector potential, iir [-V21, iiL [-V21 and Hoo[-V21 being the corresponding polarization corrections. Furthermore, for convenience of comparison with Umezawa, Mancini et al. [3], a "curved" metric tensor is introduced into Eq.(2.27) as ~,,= ~,,[-V']

I'D :~------ .......)== ....... (

with

:

~ :I'j

(2.29)

IlC-V') C'

(lJ;', - ~ J,) Lr~ = ~v'~r]f (J)(a)~f- ~ (a)Jl,,(a) )L-:

(2.30)

as an easily verifiable identity. In this section, Eqs.(2.11), (2.13)-(2.15) and expressions (2.20), (2.27) are derived under two main assumptions, i.e., smallness of E.M.F. and ideal ness of superconductor. Before continuing our discussion, we would like to emphasize that we have made no assumption on the stat.istical properties of the system, i.e., on the density matrix involved in Eqs.(2.1) and (2.2); or equivalently, no explicit statistical assumption on the Eq. (2.11) and the coefficients in the Eqs.(2.13)-(2.15). Therefore, there are no limitations for the superconductor to be in the ground state.

III. W-T identities, Goldstone-Higgs mechanism and coupled equations for the E.M.F.-phase of order parameter Two of the generalized W-T identities defined on the closed time-path for the irreducible vertex generating functional can be written as [5]

0'" .tfe [04,1, S. ® . [A"" 'x)

sJ _ 2!.. freCl ,:2. ® , sJ iie ;. .s+(a) 8(x)

.rr,.[i, 8, ® ,SJ s s (x)

i@'~)

(3.1)

=

D

(3.2)

where, according to Eqs.(2.22) and (2.24), t\+

IQ

"..
(3.3)

627 On an Approximate Form of the Coupled Equations of the Order parameter with the Weak Electromagnetic Field for the Ideal Superconductor

675

The factor 2e of the second term in Eq. (3.1) is due to the fact that X (x) is the product of two electron field variables. It is important to note that Eqs.(3.1) and (3.2) follow straightforwardly from the gauge invariance principle without any approximation such as the requirement for sex) being constant. Taking functional derivatives along the closed time-.path - L - , MIl(y) ~ and oBt'y) of Eqs.(3.1) and (3.2) successively, and~ theil, returning to tbl ordinary space-time, we can readily verify that (3.4)

z;.,

l'

,. (

a~

:c,)"

a'" i'I rt' 'PrJ

(x

:1(

~:~: and

2e t' rcf.u

= "' _

"-

!~

(;(.1)-

(X"

",

(3.5)

(~,z [.~ (%,1' , fie 7 ..",

(3.6)

14: (X,N'·+ J)ea,,)S4(-l-,y '=0,

(3.7)

4r,r C:C'lJ=.9 lasH' (:(.-3', ~' ~

(3.8)

r

(3.9)

1".,,(%,)"= D J

r.~"r

(X

II, 'I

Combinin~

=

(3.10)

0 •

~au~e

Eq.(2.8) with (3.2), we reobtain the

condition (3.11)

as required by conSistency. By applying Eqs.(3.4)-(3.11) to expressions (2.20), (2.27) and Eqs.(2.13)(2.15) and taking account of Eqs.(2.18), (2.19), it is easy to prove the following theorems consecutively by simple algebraic calculations. Theorem I. If I'4 (x-y), ,. is expressed by Eq •.(2.20). then

m.,,=D 2

(3.12)

or equivalently

r:"8Y (.%_11)== _J\'_ 1 (J£JzZ:'C-v'lc8Ca,,)S4-(X-IJ 2e7 ., •

(3.13)

1/

Theorem II. If

r

rlJv(x-y)

y

r:')'11 (ll_'I> -_

_

is expressed by Eq.(2.27), then

l_{i('r-V'J (1I'·(_v·J . .-lr(~(a}t"':C-v'lc·)~ -.b,)<m.s,(a})L~ L c. 7 L: 1;' J'f

11t'

+!;t[_V2]([J

+ M;~tJCZ) t.).~

} ,,-f(ll-l) •

I;~ (~1 ) = - (!!) ~z Z!;[_vl]~ (a,,) ;4(z_'I) where

M,f:-r)C z

t*

"':(--"1c& li a

411' 1I'Z[_V'] ~ [_Va] 2(' [-r] A.a

C2

.f.'/t i, [-r]

"L

1Tr [__ VI]

'I

,

•

__ U1IJ.~.nfl;".J •

(3.14) (3.15) (3.16) (3.17)

628 676

SU Zhao-bin and CHOU Kuang-chao

Theorem III. If the coefficients of Eqs.(2.13)-(2.15) satisfy Eqs.(2.18), (2.19), (2.20) and (2.27), then these equat:i.ons can be transformed into

' •Jc • ) L A"'(J(H I( ,[-V'] (0 + ""7• (-V'J' l .,L [-v'] (J)(alt M.I-V C) l (T)" A be) 1\':;<111 T 'lii:,.tll

=

47/"

(K)..!... ze

7\.'

v'C-V']

l~'(-V'] ct

C'..

.r-

(®(Xlt

!e ":z",[_V' ] 8 (J()) • nC

(3.18)

.gea) ® lal = 0

(3.19)

oS (jJ) B lX) = 0

(3.2()

where Eq.(3.18) can also be broken up into two equations:

L l(~)= (.am . . M~(-v'Jcz, 1ji 77111 V '(-V'I"r_V z JI·'r_V']L" a (®("1-~e;>"l [-v')S'''') =f..U:).:t1L ~2e}o.' c· z;.~ ;6 ~ i"" nC'" •

(3.21)

(3.22) by using the projection operator technique. Obviously, the difference between the transverse and longitudinal waves will vanish when the wave vector tends to zero. Consequently we have (3.23) (3.24) Up to an unimportant ter",

Sex)

on the right-·hand side of Eqs. (3.18)(or (3.21)

and (3.22», Eqs.(3.18)-(3.20), or equivalently, Eqs.(3.19)-(3.22), with the gauge condition Eq.(3.11),have just the same form as those given by Umezawa, Mancini et ale for the ground state superconductor[3]. Here, these equations are derived straightforwardly by applying the W-T identities to the linear approximation of the statistical functional equatjons for the macroscopic variables under certain explicitly stated assumptions. Moreover, the statistical informations are,implied in Eq.(2.11) and in the coefficients of Eqs.(3.18)(3.20) which are defined according to the corresponding 2-point retarded vertex functions as ~xpressed by Eqs.(2.20), (2.27) and (2.13)-(2.15). Consequently, our formalism is naturally suitable not only for the ground state superconductor, but also for the thermal equilibrium cas~s at finite temperatures, even for certain nonequilibrium stationary states if the approximations stated in Sec. II are satisfied. Furthermore, all parameters corresponding to different statistical sitUations can be systematically calculated according to Ref. [4] in principle. Following the physical interpretation of the 2-point retarded vertex functions, the dispersion relation for the phase excitation of the -order parameter,

629 On ~n Approxi~te Form of the Coupled Equations of the Order Parameter w~th the Weak Electranagnetic Field for the Ideal Superconductor

677

i.e., the so-called Goldstone excitation, is prescribed by the coefficient of the 64 (x-y) term in Eq.(2.20), with v[-v 2 ] being the phase velocities of such excitations. Then, theorem I is just the conventional Goldstone theorem: the rest mass of the Goldstone mode is zero. Theorem II with the related Eqs. (3.14), (3.16) and (3.24) manifests transparently how the E.M.F., primarily the longitudinal part, is acquiring a mass from the phase field of the order parameter. As pointed out in Ref. [3], the right-hand side of Eq.(3.l8) (or (3.19) and (3.20» will give trivial contributions unless the phase of the order parameter behaves singularly. In other words, the singular phase field will give rise to observable effects other than the mass of the E.M.F. Therefore, in a sense, theorems I-III may be interpreted as a generalized form of nonrelativistic Higgs mechanism allowing singular Goldstone field.

IV. Calculation of the long wave-length limits of the density of superconducting electron pairs and the phase velocity of the Goldstone excitation for superconduCtors in thermal equilibrium The long wave-length limit of the transverse mass of the macroscopic E.M.F. is an important observable related to the static penetratin~ depth which is often expressed in terms of the denSity of superconducting electron pairs. For the thermal equilibrium system, one has (4.1) where m is the electron mass, T=a- 1 the temperature and ns[T] the density of superconducting electron pairs as a function of temperature. Besides, the phase velocity of the Goldstone mode is also an interesting quantity djscussed by many workers [7]. These two quantities are the main parameters of Eqs.(3.l9)(3.22) in the long wave-length limit. In order to compare our formalism with the known results of the other theories or experiments on the one hand, and as a simple illustration for deriving Eq.(2.11) and calculating the parameters in Eqs.(3.l9)-(3.22) according to Ref. (4] on the other, in this section we will calcu"late the above-mentioned two parameters for the thermal equilibrium superconductor-E.M.F. system in weak coupling approximation. According to Eqs.(2.11), (2.12) and the relevant discussion, the derivation of Eqs.(2.1l) is a problem for superconductor itself not related to E.M.F .• This problem has already been solved for the thermal equilibrium superconductor with simplified weak contact interaction. Then, the remaining task is to calculate the two parameters mentioned. Transforming ~ (x-y) into its FOUrier representation as (4.2) on account of Eqs.(3.l3) and (2.22), we have immediately

(4.3)

630 678

SU Zhao-bin and CHOU Kuang-chao

Then, with Eqs.(3.24), (4.1) and (4.3), it is easy to verify that

I

l'ls[T]=-!"'![O] 11'[0 ] ...

CJJ] s,. [ oar'; ;T:r il, at

I

,-0

(4.4)

•

-ts,. fa'~ ~J ra"r;; ml aJ a, 1=0 ra,. a,.J,_o '

(4.5)

where Sp mean"s tracing over the second rank tensor in the 3-dimensional space. Since we have made the assumption that the intensity of the E.M.F. is not too strong, we can ignore its nonlinearity and feedback to the modulus of the order parameter. Consistently, the renormalization effect of the E.M.F. in the long wave-length limit should not be important, and we may set (4.6) Besides, in accordance with (2.16), r~ (x-y) is independent of A~(x), B(x) and aex). Consequently, the calculations of ns [T] and v 2 [0] are also problems for the superconductor itself. Therefore, by making use of the standard technique of CTPGF[5] and relevant results of literature [4], after some typical calculations for equilibrium contact coupling superconductor, it is easy to derive (4.7)

(4.8)

where

to'

(4.9)

. } ' = _F_ 2111

ECFJ =

-. -k-

(4.10) (4."11)

-.)1-

ECF' =,./fHp", IAII •

(4.12)

N(OI=~ 1

(4.13)

27t

"-

where P F is the Fermi momentum of electron, lal the energy gap, NCO) the density -of states at the Fermi surface. Substituting Eqs.(4.6), (4.7) and (4.8) into Eqs.(4.4) and (4.5), we have

ll s [T] =- ns• - ;; 11112 N(OljdE(FJ

_

1ts.

_ Lll1laJ~ _1_ 3

(Z1tU'

E(~

(tit .8EC"" -'J AE~

AIt---rl-l~

__,,_

E(1) tl.EiF>

(

EC;h

'/~ i IA\I N(O'jdEt1i-LtIt.IEC1! e'(.,! 2

11""[OJ -rls(r]

For T-O as a special case, we have

7

d:C1) \- ECF)

J

(4.14)

(4.15) (4.16)

631 an an Approximate Form of the Coupled Equations of the Order Parameter ~ith the Meak Electromagnetic Field for the Ideal Superconductor

679

J

f.3 ~ rill' nsa= ,;1;3 = 0 -C-21t"";'1I-)T"j

nS [T= Q]

=

11'2[0]\

• =~=_1_1T; 3m

T=D

(4.17) (4.18)

3

where vI" is the electron velocity at Fermi surface. Eqs.(4.17) and (4.18) are identical with tbe known results in literature[7,8] But, we have not found other calculation for the temperature dependence of the phase velocity for tbe Goldstone mode to compare with. It is interesting to note that Eq. (4.15) coincides with the corresponding result derived from Werthamer's extension[9] of Ginzburg-Landau theory. Making the followinr. approximations (4.19) (4.20)

in the sense of dominant contribution to t~e integration jdE(P) diately reobtain the correspondin~ results from BCS tbeory[8]

noS [T]= nso _

'1t/mfl

3

{O~4cl"(_

we imme-

at (f(1'J») dE(PJ

(4.21)

,

where

f (E ci)) =

I -e-Jl""e"",'F=-)-+--

(4.22)

Acknowledgement The autbors wish to thank Prof. CAl Jian-hua and Prof. WU Hang-sheng for calling this problem to theiF attention and are also grateful to Prof. YU Lu and Prof. CHEN Shi-Fang for helpful discussions and a careful reading of the manuscript.

References 1.

V. Ginzburg, L. LaDdau, Zh. B1csp. Tear. Fi.z. 20(19S0), 1064.

2.

P. de aennes, ·Superconductivity of Metals and Alloys·, Benjamin, N.Y. 1966.

3.

See, for example, L. Leplae, F. mancini, B. Umeza~a, Phys. Rep. ~(1974), lSI. If. IfatsWllOto, B. I1IEza~a, Fortschr. Phys. ~(1976), 3S7.

II. Fusco-Gi.rard, F. Hancini, H. Ifari.nuo, Fortschr. Phys. !!..(1980), 3SS. 4.

ZHOU Guan!1'"zhaO, SU Zhao-bin, ~(1982),

5-.

ZHOU Guan!1'"zbao, SU Zhao-bin,

et 081.,

HAD Bai-lin and YU Lu, COJlllllun. in Theor. Phys-., (Beijing).

295,307,389. Ch. S. in ·Progr. in Statistical Physics", eds. Hao Bai-lin

~XUB (Sci.e~ce)Press , Beijing, 1981.

IIcGra~-Hill,

6.

J. Bjorksn, S. Drell, "Relati.vi.stic Quantum Fields",

7.

J. Schriefer, -Theo.zoy of SUperconductivi.tyN, Benjamin, Reading, Hass. 1964, and references

8.

See,

N.Y. 1981.

tbllrei.n. for ezample, A. Fetter, J. Walacka, NQuantum Theory of Ifilny-Particle Systems",

lie Gr__Bi.ll, N.Y. 1911.

9.

N. JII'erthamu', Ph'JS. Rev • .ill,(1963), 663.

N. Wert~r, i.n ·SUpercollductivit'J- Vol.l

filii. R. Parb, lIarcel Dek1Iar, INC., N.Y. 1969.

632 COIIIIIIW2.

ill Theor. Phys. (Beijing, Chilla)

Vol. 2, No.4 (1983)

1181-ll89

ON ADYNAMIC THEORY OF QUENCHED RANDOM SYSTEM SU Zhao-bin ( $~hlc. ), YU Lu (f and ZHOU G1lang-zhao ( }I]!?l )

*)

Illstitute of Theoretical Physics, Academia Sillica, Beijing, China

Received March 10, 1983

Abstract A dynamical theory for quenched random system is developed in the framework of C'l'PGF.

In steady states the

resu.~ts o~

tailled coincide with those following from the quenched average of the free ener!1!l.

The order parameter q, a matrix in general,

becomes an integral part of Lhe second order connected CTPGF. An equation to determine

q

is derived from the Dyson-Schwinger

equation ill t'l.is formalism.

Some general properties of the

C'l'PGF in a quenched random system are discussed.

I. Introduction In quenched random systems, part of th~ degrees of freedom describing impurities are froze·, =nLO a nonequilibrium but random configuration. This could be accomplished by sudden cooling of a sample in thermal equilibr~um to a state with much lower temperature. The impurities are then frozen into a configuration separated by high potential barriers from an equilibrium one. Diffusion throqgh the potential barriers will cause the nonequilibrium state to vary very slowly in time. As pOinted out by Brout[l], the space average of an observable A in a quenched random system can be replaced by the ensemble average over the impurity degrees of freedom J.

A fA (J) p(J)dJ

(1)

where P(J) is the distribution function. Most of the previous workers~] considered quenched ra:ldom systems as if .• they were static. In this approach one has to evaluate quenched average of the free energy which is proportional to the logarithm of the partition function. It is a formidable task and an enormous machinery of n-replica method is introduced. This method has been applied extensively to systems like spin glass[3-l0J. Recently., several authors [11-16] have proposed dynamiC theories of quenched random systems in the study of spin glass b~sed on the MSR statistical field theory [17]. The advantage of the dynamic theory is that it provides means for averaging out the quenched randomness without using the unphysical replica trick. The results obtained so far can be reproduced. by the replica method with special pattern of replica symmetry breaking, which is itself a s;atic theory [18]. Therefore the full content of the Qynamic theory is still

633 llS:J

SU Zhao-bin, YU Lu and ZHOU Guang-zhao

waiting to be uncOvered. The aim of the present paper is to establish a dynamic theory for the quenched random system using the closed time path Green's function method (CTPGF) [19J. CTPGF is a very general method especially suited to study slowly varying nonequilibrium processes. In it are incorporated automatically causality and fluctuation dissipation theorem (FDT). The order parameter q introduced by Edwards and Anderson ~J appears naturally in the second order Green's function. The new result obtianed in the present paper is a DysonSchwinger equation for the order parameter q. For slowly varying processes it is sufficient to use semiclassical approximation, the one employed in the transport equat·ion. In this way a differential equation that describes the time evolution of the order parameter is obtained. In this paper only the general properties of CTPGF and the Dyson-Schwinger equation are studied. Application to long range quenched ISing model will be presented in a subsequent paper. The paper is organized as follows: In Sec.II we introduce CTPGF for a quenched random system. It is proved that as the system approaches equilibrium, there exists a free energy which is the quenched average of the free energy with fixed random degrees of freedom. In Sec.III a Dyson-Schwinger equation for the order parameter q is deduced and simplified in the semiclassical approximation, Sec.IV contains concluding remarks.

II, CTPGF for a Quenched random system We shall use in the following those symbols and the language adopted in the theory of CTPGF without further explanation. The unfamiliar ~eaders are referred to Ref. [19J. Suppose the dynamical field variable of our system is a(x). The action on a closed time path P has the form

1= ~c&d~O"(X) r;o.DtX:'~llfl,)-~d1V((f"(X),Ji) ,

.

1

+ d.~((l't11;(U.) +(flXljlX»+ Ihe«t

rt,SlI.YVOi,..

(2.1)

'

where hex) is the external field; Jl are random variables with given distribution functions. The a(x)j(x) term represents the interaction of the dynamical field with the reservoir conSisting of a set of harmonic oscillators for instance. If there are more than one dynamical f~elds, a(x) should be considered as a vector with many components. We shall use Path integral to evaluate the generating functional of CTPGF. After integrating over the field variables describing heat reservoir, we get the averaged generating functional

~[~(X)]-fptJJ ~(~(XI,J]dJ' with

=< ~['fi,JJ >J

(2.2)

:i:.[RlXI,JJ=J[dtt]e.iIeft ,

Left ....~ 0"(X)r,'(x.,)1tt3)did~ -~ VCQ"lX7. Ji}d~ +~ltlXl~(Xld~

(2.3)

.

(2.4)

634 on a D!/namic 2'heoZ'll of Quenched Random S!/st:em

ll83

The system is supposed to be prepared at time t=to by suddenly cooling to the temperature of the heat reservoir. In Eq.(2.4) the closed time path starts from t=to to t=+m (positive branch) and runs back from t=+m to t=to (negative branch). r~O)(x-y) is the second order vertex function obtained after integrating over the reservoir degrees of freedom, i.e.,

with the self-energy part r.jO) (x, y) determined by the interaction a (x)j (x) with the reservoir. It is easily proved that r p(o) satisfies the FDT

.......(0'li)=Lcth. ~t. IIII rill r ell T

r,

(2.5)

Here and riO) are the correlation and the retarded vertex function, respectiv.ely. Introducing the generating functional for the connected.CTPGF

and

Z [{Ill] = e.x.p{ i W[ il~lJ}

(2.6)

~(!lXl, j] = up{iW[i(:(I, J]}

(2.7)

it is possible to obtain the connected CTPGF by direct differentiation. have the averaged field the connected CTPGF

~IX)= O/LIX) ~?

-Go

(2.8)

S"ijJ

(2.9)

lX, '" ••• XII ) = -=-~:":""":o-r-P I Sllt,I ..... -S(Xnl

and the corresponding ones from

~ (f(:tl=

We

W[h(x),J].

Eq.(2.2) implies that

<~o-lX:, J) >:r

(2.10)

It is a very important property of Z[h,J} and Zlh(x)} that they are equal to unity in the physical limit when the external field hex) on the positive branch is identified with that on the negative branch. Therefore, the observed field

(j"lX.>=( (JllC.:r):>

(2.11)

J.

satisfying the requirement for a quenched average. Differentiating Eq.(2.10) with respect to hey) in the physical limit, we obtain

Gp (:t. !'+i(fIXlO=(~1 = or

and setting

h(x+)=h(x_)

(GplX.S "T>+i(f(x. J) If(~. :r) \

GplX.,9 1=( G-plX,

~j

J)

>.r+l9.(X:,~)

with the matrix

'l(X, B)=«T"lx.;n(f"(~,J) ~ -

(fIXJ (f'9'

(2.12) (2.13) (2.14)

Edwards and Anderson have defined an order parameter in spin glass

~= Lim «(flO.J"Jlrlt.J"J>

t...,...

(2.15)

635 1184

SU Zhao-bin, YU Lu and ZBOU Guan..,-zi2ao

which is closely related to the matrix q(x,y) deduced in Eq.(2.14). For hermitian field operator o(x) its average a(x,J) is a real function identical on the two tim~ pranches in the physical limit. Hence the matrix q(x,y) is real, symmetrj and equal on the two branches

(2.16) (2.17)

From Eqs.(2.13) and (2.17) we obtain for the retarded, the advanced and the correlation Green's functions the following relations: _GrlX, ~)=

GQCXJ and

(2.18) (2.19)

J

,> == J J)

<

(2.20)

Gc(X,:~fl- GclX'H ; JJ)J+C,!-Cc.,S>.

The appearence of the matrix q(x,y) iE a consequence of the quenched average over the random. variables J i • It characterizes the behavior of a quenched random system. By successive differentiation one ~an easily deduce higher order Green's functions. There one can find new matrices describing quenched random systems. We shall not discuss them in the present paper. However, it is easily proved that for a higher order ratarded (or advanced) Green's function we always have (2.21) GT••••• y (X.······X (X,' ··• __ ·X n ,. J» J n ) = (G Yo •••• - r ' After a suff·icient long time the system is expected to reach a steady but not necessarily equilibrium state, where the field a(x,J) is no longer timedependent. In. a previous work [19] we have shown- that if

ImfGr (i2, - -t2,. J)d.t=O there exists a free energy It{

LJ)=-

F(h,J)

:r .

(2.22)

,

such that (2.23)

For a system in equilibr~um with a heat reservoir condition (2.22) is always satisiied. Hence the free energy F(h,J) exists and is equal to the one derived from the partition functiQn. Eqs.(2.18) and (2.22) imply that (2.24) which guarantees the existence of a free energy

F(h)

such that (2.25)

The equality (2.11) now becomes

aF = <~Ell.J) ) at Soft.:r

(2.26)

For a smooth distribution .function P(J) with finite moments the order of differentiation and averaging can. be changed. Integra"ting Eq. (2.26) we get

636 On A Dynamic Theory of Quenched Random S!/stem

llBS

F.(h) =
(2.27)

As our formalism is quite general it is possible to introduce compound field variables and their corresponding external fields. The temperature can also be treated as an external field coupled to the energy as a compound field variable. In this way we can exhaust all variables in the free energy and obtain finally (2.28) f1. ) = (~ J ))J

F(

,

except for an unimportant constant. Thus the quenched average of the free energy follows from this formalism. This means that results obtained in the dynamical theory will approach that of the static theory when a steady state is reached. Before discussing the time evolution equation for the quenched matrix q(x,y) let us review some general properties of the connected CTPGF. When the random variables Jl are fixed, the Green's functions of a system in equilibrium with a heat reservoir satisfy the FDT. Let (2.29) where a=r, a or c. The metric used is ~xll=kgxg-kiXi' where Xg is the time component. XII II=O,l, ••• d-l are space time variables of macroscopic scale. Similarly, we define (2.30) which could be considered as the classical counterpart of the quantum matrix q(x,y). The FDT ·now reads as follows: (2.31 ) where a-I is the temperature of the heat reservoir. into Eq.(2.20), we have

Substituting Eq.(2.31)

1. FV G (t, X)= icth;-llll Gr( t,X}+1 ''J.(" i, Xl

'==-'

t

This is the FDT satisfied by the quench averaged Green's function. temperature approximation Eq.(2.32) can be rewritten in the form

(2.32) In high

roJ

f'J

Gc(i, x)= -rl:;i 1111 Gr d., XH[~(i, X) .

(2.33)

The retarded Green's function Gr(k,X) is analytic in the upper kg-plane. Its real and imaginary parts obey the dispersion relations. If Gr(k,X) tends to zero as kg+m it is possible to use the unsubtracted dispersion ,-...J relation Re.C:!G (.a X)=J....f 1nGr(~,lX) d-2' (2.34-)

r

11.0"

r

11"

t-la

Using Eq.(2.33) in the high temperature limit we get ,-...J

Re.Gr(i..=o, T,

,...J

X)= Grtla=o,

T, X)

IC.o

637 1186

SU Zhao-bin,

ru

Lu and ZHOU Guang-zhao

(2.35 ) For a long ranged ISing model when the space dependence of the Green's function can be neglected, Eq. (2. 33) becomes Fischer's relation [20]

........

G(t=O t)=~(J-q.(t,t)-if'ltJ) y 0 J

(2.36) •

We see that the validity of Fischer's relation depends crucially on the high frequency behavior of the retarded Green's function such that the unsubtracted dispersion relation holds.

III, Dyson-Schwinger equation for the Quenched matrix Q, We begin with the generating functional for the vertex CTPGF.

f[O"(;o] = w [!mJ-i flXlfmA with

o(x)

determined by Eq.(2.8).

(3.1) p From Eq.(3.l) it is easily found that

(3.2) Higher order vertex function can be defined by successive differentiation

fP lX

Sllf

...... An } = -:-:::--~.:....:~&irlX,l ...••.

I,

(3.3)

Sifo::,,)

The Dyson-Schwinger equation for the second order connected CTPGF follows from Eq.(3.2) by differentiation

(3.4) where lip (x-y) is a Ii-·function defined on the closed time path. The equation satisfied by the physical Green's function can be obtained from Eq.(3.4) by setting the external field hex) to be equal on the two branches of time. Eq.(3.4) then reduces to three equations satisfied by the retarded, the advanced and the correlated Green's f'!nctions in the following matrix form

FrGr=Gyf,=-I, ~ Gil = G-c:~ = -, , and

~ G-c= -

fc G-o.

(3.5) (3.6) (3.7)

In an ordinary system near thermal equilibrium Eqs.(3.5) and (3.6) determine the energy spectrum and the life time (dissipation) of quasi-particle excitation while Eq.(3.7) becomes the transport equation for the quasi-particle distribution in the semiclassical limit. We shall show in a forthcomining paper that in a quenched random system the time evolution of the matrix q follows from Eq.(3_7). From Eq.(3.5) one finds that

638 On a Dgnamic Theozy of QUenched Random Sgstem

tc Gy- GI).)= i GrIm Fr GI).

Gy=

lIm

1187

(3.8)

Now define a new matrix

Q.=-i(Fc - icth1} I~ r;]

(3.9)

which in general is a functional of the matrix q. Then it is easily found from Eqs.(3.7), (3.8), (3.9) and (2.33) the Dyson-Schwinger equation for q (3.10) This is a matrix equation which can be simplified in the semiclassical approximation. The Hermitian conjugate of Eq.(3.10) reads

q,f.=-GCl Y.

(3.11)

It

Separating the Hermitian and anti-Hermitian parts of Eq. (3.10) we get two equations (3.12) F,. q., ~ = -G..Ga. -GrCl (3.13)

+ ia.

Fr 'l--

q.fo. =-G-.GCl+GyG..

To proceed further let us write ......

.-...I

--.J

Gr = Re.G y + i 1m Gy

(3.14)

then from Eqs.(3.5) and (3.6) -

I

,....,

,....,

Ii I" t Re. G-y - j 1m Gy )

r;.

=

Fa.

= 1~I.tRe.~+iIl1I~T)

(3.15)

,

(3.16)

In the semiclassical apprciximation we shall replace the product of two matrices A and B AB = 1/2(AB+-BA)+1I2(AB-BA) by the classical e;Kpression

as

- - i 01 iA as -~ i {--} AB-2:(aioI~-a~K)=AB-2 A,B "11

P.B.

P;

(3.17)

•

The validity of the semiclassical approximation is controlled by the condition

I

where

, ".0' I<< I

(3.18)

'() di)ldIJI.

6 may be either of the functions A and S. In this approximation Eqs.(3.12) and (3.13) become

~-(t I~r 1'"= 21l:rJ ~ {( rl aa )(IIn':::"Gr -af/..G.. oXI'--IGT I" -ax: aA,"jI

T

_

-ReG ~ arl'll~Y )_( at,.. at -IG II dill aG:)(Im r aRI~-R G; ax r

and

~I~fl.(aq:

at,. <Jx!'

r

U'y

to

-IGI1.,'[)= dl~I\a1:_I~I:qG::) r oX!' -a It' ail' r al...

T

(3.19)

gIm~Y)} j) X. (3.20)

639 UBB

SU Zhao-bin, lfU Lu and ZROU Guang-zha9

Eqs.(3.19) and (3.20) are written for systems with only one dynamical field. The generalization to multicomponent system is obvious though tedious. For a homogeneous system in steady state all the functions are independent of the macro space-time variables X~. In this case Eqs.(3.19) and (3.20) reduce to a single equation _ _ ,......J2,

~ .... QIGTI

(3.21)

The functions Q and Or can be calculated by the field theoretical method when the mode! Lagrangian is given. They are functionals of q. Hence Eq. (3.21) could be used to determine the equilibrium value of the matrix q(k). In perturbation theory to first order Q is proportional to q in some cases, say spin glass without external magnetic field (3.22) q-O is now a solution of Eq.(3.21). A nontrivial solution with ting a new quenched pbase might exist if the condition

q~O

exhibi(3.23)

could be satisfied. In a subsequent paper we shall apply the general results obtained to the long range quenched ISing model. We shall show there the condition (3.23) can never be satisfied owing to the stability condition that must be obeyed by the retarded Green's function Gr' Therefore the spin glass is either in a nonsteady state or not characterized by the parameter q.

IV, Conclusions In the present paper a dynamic theory of quenched random system using the method of CTPGF has been developed. In the steady limit the results predicted by the dynamical theory approach those by the quenched average of the free energy. Edwards-Anderson order parameter which is in &eneral a matri~ appears naturally in the second order connected CTPGF. An equation that determines the time evolution of the order parameter q has been derived from the DysonSchwinger equation for the second order connected CTPGF. This equation is in general a matrix equation and is simplified in the semiclassical approximation to be two partial differential equations of the first order. It is certainly true that q(k;X) has a sharp peak at k=O and is a very slowly varying function of X. In a homoJeneous system with no space dependence q will be only a function of the frequency III and the macroscopic time t. In the literature q(III,t) is often approximated by

...,

CJ.(W,t)=q.oltlS(Ul) • In the dynamic theories published so far we have not found any work on how q(t) evolves with time t. We shall try to ftlt this gap by using the equations obtained in this paper. In a subsequent paper we shall try to solve these equations in the long range Ising model and obtain an explicit dependence of q on III and t.

640 1.189

On a Dynamic Theory of Quenched Random S!lstefll

References 1.

R. Brout, Phys. Rev., 115 (1959) 824.

2.

See e.g. P.fi. Anderson and '1'.C. LubensJcy, in Lectures at Ecole de Physics on "Ill Condensed Matter", Les Houches 1978, edited by R. Balian, R. Maynard and G. TOulouse, Holland,

3. 4.

(North

(1979)).

S.F. Edwards and P.W. Anderson, J. Phys., ~ (1975) 965. B. Sllerrington and S. Kirkpatrick, Phys. Rev. Lett., ~ (1975) 1792; Phys. Rev., B17 (1978) 4384.

5.

J.R.L. de Almeida and D.J. '1'houless, J. Phys., All (1978) 9.83.

6.

A. Blandin, II. Gabay and '1'. Garel, J. Phys.,

7.

E. pytte and

8.

A.J. Bray and II.A. NOore., Phys. Rev. Lett., ~ (1978) 1068.

9.

J:

Rudnick., Phys. Rev.,

!!!

m

(1980) 403.

(1979) 3603.

D.J. '1'houiess, P.fi. Anderson and R.G. Palmer., phil. lIag., 35 (1977) 593.

!!

(1979) 1754; J. Phys.,

!!!

10.

G. Parisi, Pllys. Rev. Lett.,

11.

S.K. Ma and J. Rudnick, Phys. Rev. Lett., ~ (1978) 589.

12.

(1980) 1101, L1887, Ll15.

C. De DoIIIinicis, Phys. Rev., Bl8 (1978) 4913; l:BCture notes in Physics, VO.l£!, P.253, edited by C.P. Bnz (Springer Verlag, Berlin, 1979).

13.

J .A. Hertz and R.A. 1CleJJIJJ, Phys. Rev. Lett., ~ (1978) 1397; ~ (1981) 496.

14.

H. SompolinsJcy and A. Zippelius, Phys. Rev. Lett., 47 (1981) 359.

15.

H. S~IinsJcy, Phys. Rev. Lett., ~ (1981) 935.

16.

w.

17.

P.C. lIartin, B.D. Siggia and H.A. Rose, Phys. Rev., ~ (197J) 423.

18.

C. De Dolllinicis, II. Gabay and H. Orland, J. Phys. l:ett.,

19.

ZHOU Guang-zhao, SU Zhao-bin, HAD Bai-l:in and YU Lu, Phys. Rev., B22 (1980) 3"385,

Kinzel and K.H.Fischer, Solid State Coar., ~ (1977) 687.

E..

(1981) L523.

ZHOU Guang-zhao and SU Zhao-bin, Lecture notes on ·progress in statistical Physics·, Ch. V. 20.

(1981, in Chinese).

K.H.FiscbdE, Phys. Rev. Lett., 34 (1975) 1438.

641 CoImauD. in 2'haor. P1P,Js. (Beij.1ng, Ch.1na)

Vol. 2, No. 4 (1983)

1191-1201

A DYNAMICAL THEORY OF THE INFINITE RANGE RANDOM ISING MODEL 50 Zhao-bin ( J¥blc ), YU Lu (f and ZHOU Guang-zhao ( /iJ:fB )

li )

Institute af 2'heoretical Physics, Academia Sinica, Beijing, China

Received March 10, 1983

Abstract

2'he dYnamics of spin glass is studied in the framework of C2'PGF.

A marginal stability line is found on the q-X plane. h
Below 2'c with

follows Fischer's line exponentially to the stability boundary and then decreases in power law along the boundary to its fixed point.

2'be Langevin equation for the spin

valid along the stability boundary.

aft)

is

no

.lcmger

2'be susceptibil1ty is cal-

culated in perturbation theory and found to be in good agreement with those predicted by the projection hypothesis.

2'he

general validity of the projection hypothesi.s is justified

in

the present for1llali_.

I.

Introduction

The simplest model describing spin glass is the long range random Ising model with Hami11:on'ian [1,2]

H.=-..L2 ~ Jii cr; l);-~ ~.If: IlIpJ

where the exchange

J

ij

I,

I

v

I

L

(1.1 )

t

are random variables with a Gaussian distribution (1. 2)

The spin variables a i take the values ±1, and N is the number of nearest neighbors. The quenChed average over J i j is carried out on physical variables like free energy, which is the logarithm of the partition function. The method extenSively used consists of calculating the average. part.ition function of .n replicated systems and taking the lim~t n+O i m *«~n)_I) (1nZ)T=l v n-o 'J

(1. 3)

The model was studied in the mean field limit by Sherrington and Kirkpartick [2]. They found a nontrivial solution for the Edwards-Anderson order parameter q [1]. This solution is however unstable below Tc [3,4] and also yields unphysical negative entropy near T =O~]. Later using a particular scheme of rep1ica symmetry breaking Parasi[S] found a more satisfactory solution, in which the nxn matrix of the order parameter was represented by a continuous function q(x), P~xS1, in the n+O limit. However, in replica

642 1292

SU Zhao-bin, YU Lu and ZBOU Guang-zhllO

theory the physical meaning of the various order parameters remains unclear. An alternative approach based on dynamics has been proposed by Ma and Rudnick~] and developed subsequently by others. The dynamical theory provides not only a means for calculating the quenched average without the unphysical replica trick but also better understanding of the spin glass state, especially so since many of the low temperature properties of real spin glass are dynamic in nature. Earlier dynamic theories ~-8] still failed in describing the spin glass below Tc' The solution exhibits the same instability encountered in the static replica theory. Recently Sompolinsky and ZiPpelius[9] have used a soft spin version of the random Ising model and defined the dynamics of the" random system by a Langevin equation. The formalism adopted was developed by De Dominicis[7J using the functional integral method of Martin, Siggia and Rose~J. In Ref. [9] they introduced a la Sommers [12] two order parameters q and a. Using this approach together with some physically plausible assumptions about the time dependence of q and a, a static solution has been constructed[10] which agrees in general features with Parisi's replica results [S]. This solution has subsequently been derived from the replica theory by a special scheme of replica symmetry breaking[13]. In a certain sense Sompolinsky's solution is an ingeniou~ guess rather than a derivation from a dynamic theory. The physical assumptions involved are very difficult to justify. In a series of papers Parisi and Toulouse n4] proposed a simple projec-tion hypothesis for the mean field theory of the spin glass phase. They showed that a drastically simple extrapolation procedure, projecting physical properties from the instability line onto the spin glass phase, reproduces many of the expected features cominr from the Monte-Carlo simulation and replica symmetry breaking scheme of Parisi. Though the projection hypothesis is very simple and elegant, its theoretical explanation is still lacking. In a previous paper [1S] we have studied the general properties of the Green's function for a random quenched system in the framework of the closed time path Green's function (CTPGF). An equation for the order matrix q(x,y) has been obtained ,and simplified in the semiclassical approximation. The aim of the present paper is to apply the general results obtained therein to the infinite range random ISing model. The main results obtained are: 1. A marginal stability line in the q-X plane is obtained where X is the magnetic susceptibility of the system. 2. Above Tc Fischer's relation, a line in the q-X plane, is inside the stable region. The orde~ parameter q tends exponentially in time to the fixed point q-qo' 3. Below Tc the Fischer'S line intercepts the stabi Uty boundary at a point ql • The order parameter will decay along Fi sC"her' s line quickly (exponentially in time) to ql and then decay further along the boundary slowly (with power law) to qQ' 4. The static fixed point corresponding to the Sherrington and Kirkpatrick solution lies OD the Fis~her'B line in the unstable region. The only stable fixed point is q-O.

643 A Dgnamical Theor!/ of the Infinite Range Random Ising Model

5.

1193

The projection hypothesis can be justified in the present formalism.

The paper is organized as follows: I·n Sec. II the second order connected Green's function and vertex function are studied in a soft spin version of the infinite range randcm Ising model. The stability condition and the low frequency behavior of the Green's function are discussed. The time evolution of the order parameter q(w,t) is studied in Sec.III .• Sec.IV gives the susceptibility in the external magnetic field. Sec.V contains concluding remarks and a brief discussion of the projection hypothesis.

II, The model and Green's functions For the sake of simplicity we consider here a soft spin version of the random ISing model defined by (2.1)

to

The length of the soft spin 0i is allowed to vary continuously from ~. The exchange J ij are random variables with a Gaussian distribution

p( Jij } = (2 1l N/r1 ft exp{-N Ju' /2 J' }

(2.2)

N is the number of the nearest neighbors. Taking into account the interaction with heat reservoir and averaging over the random variables J ij we can write the generating functional of the CTPGF in the following form where

(2.3)

where

Seff= f{ ~ ~ OJ (t>~olt t'>Ojlt')dtdt'~(LlOi"'lt>+ili(tl6"ilt»dt } +{ J :L

4N

(2.4)

~.I6i(tl OJ'(t)dti~rt'i6j(f)dt' ,..) p p L

The notations used here are essentially those in Ref.[IS]. Howeve~, the bar over the quenched average physical variables has been dropped for simplicity. t+t' Any matrix A(t,t') can be represented by its Fourier transform A(w,-:r-) in the relative time t-t' where (2.5)

In this notation system the low frequency approximation for

,..... f';.lUl, t)= (-Yo+iw/ro )

and

'[ tW, t)=i where

S-l

Co

;-r. " •

ro

has the forms e2.6)

(2.7)

is the temperature of the reservoir. In the infinite range limit with N+~ the matrix Gijet,t') could be approximated by 6i j G(t,t·). In this case the second order vertex function can be calculated with the diagram expansion. It is found that

644 ll94

SU Zhao-bin, YU Lu and ZHOU GUang-zhao

(2.8) where ~ time t.

and r are renormalized quantities that could be functions of the To lowest order perturbation in u we get (2.9)

where J is also renormalized and q(t,t) is the order parameter. In obtaining the second term in Eq.(2.9) we make the approximation that q(w,t) has a sharp peak at waO. In Eq.(2.8) tr(w,t) is the self-energy part with the first two terms in the expansion of w and the term proportional to Or is excluded. They are included in the first three terms in Eq.(2.8). Therefore, we have (2.10)

To the same order of approximation we have calculated the vertex function "...." .. 2.."""'" ~ 2 ,...., «.11, t)= L (lUJ ry{W,t) +LJc. l tt,>'l.
rc

I",

+ iJl~lUl,t)- ~c.C.Ul, t) Here

~(w,t)

c~rrelated

(2.11)

is defined to be (2.+2)

where a(t) is the average value of the spin which could be zero if an external magnetic field is applied. It is easily ~(w,t) has a sbarp peak at w=O. Ec(w,t) is the remaining that does· not have a sharp peak at w=o. To lowest order of find

different from proved that self-energy part perturbation we (2.13)

It is to be noted here that

q.{t. t)=

S~~ 'it(uJ, t)

In the low frequency limit the matrix mated by

(2.14)

Q defined in Ref. [15] can be approxi-

Q.(Ul, t)=-i{ Fc.'lW,t)- 'jw 1m rr(W, t>} 2.

-."

= Jc('I.)CJ.(W,t>+:r

2"""" ~lW,t) •

(2.15)

Only the terms tbat have a sharp peak at w=O are retained. Next we shall study the stability of this model. In pertUTbation theory stability should hold order by order. To insure stability in the low frequency limit Imrr , -Im~r and fr (w=O,t) must be positive. ~rom Eq.(2.8) it is easily deduced that (.I)

r _r

I m I r-

,-.J

-lmLr

1- J:l\)

/

li,/

Bence we get the stability condition

(2.16) a

645 A DyniJllli.cal 2'hsory of the InLinit:e Range Random ISing Model

1195

(2.17) In the zero frequency limit Eq. (2.17) becomes (2.18) where X=G (w=O,t) lity line

is the susceptibility of the system.

The marginal stabi(~.19)

in the q-X plane constitutes the boundary of the stability region.· The general analysis given in Ref. [15] tells us teat the nontrivial static fixed point in the absence of a magnetic field is determined by the equation (2.20) From the explicit calculation in perturbation theory Eqs.(2.9) and (2.13) we find that the line representing static fixed point Eq.(2.20) apart from q=O lies outside the stability region in the q-X plane since Jl(q»JJ(q) for all values of q. We conclude therefore that this fixed point cannot be reached and ultimately the order parameter q(t,t) or q(w,t) will tend to zero as the time t+~. Before we go to the next section to study the time evolution of the order parameter, let us study the low frequency behavior of the retarded Green's function Gr(w,t). For this purpose we write (2.21) where 6G r (w,t}+0 as solved in the form

LlGI'=

w+O; Eq.(2.8) regarded as an equation for

2.J;:

X {I-J;(q)l-(~l'+·i~ )~,.... 2. ..-oJ·W 3]'b.}

w is small

6Gr

can be

Ql

((1- J:('i)X~(L.I'+i ~)J. )-4t('l.)(l:r+Y)X When

6G r

(2.22)

can be put into the following form (2.23)

where yet) and v are to be determined. Following an analysis given by Sompolinsky and Zippelis[9] it is easily found that the lowest power of w in the self-energy p-art t'r is w2 \/ for \/lO.i. From Eqs. (2.22) and (2.23) we obtain similar results as in Ref. [9] that \/=1 if the system is inside the stability region while v$i if it is on the boundary of the stability region. We shall use these results to derive t·he time evolution of q(w, t) in the next section.

646 1196

SU Zhao-bin, YU Lu and ZHOU Guang-zhao

III, Time evolution of the order parameter Q(w,t) near critical temperature In Ref. [15] we derived an equation for q(w,t? in the semi-classical approximation. We shall apply it to the model studied in previous sections. Neglecting the external magnetic field the equation for q in the linear approximation reads

(3.1)

In the low frequency limit we take the approximation

dREfiy~LnGr _ ilReGr ~oJ

at

-at

oImG,.-..J ilX ilLnG,. oliJ - -~ ~

(3.2)

Sl.nd (3.3)

Substituting Eqs.(3.2) and (3.3) into Eq.(3.l) it follows that

'dCi: =-2 I-lx"

at

(3.4)

I+JX"

In the stability region we have (3.5)

and

(3.6)

~--.Lx~>o

oW

r

For ising model the susceptibility X is expressed in terms of the stability region by the Fischer's relation

Integrating Eq.(3.4) over w

we get an equation for

q(t,t)

in

q(t,t) (3.8)

is

Above the critical temperature ac=l/J, the only fixed point of Eq.(3.8) q=O. Near q=O the order parameter q will tend exponentially in time

t

to the fixed point

'l. 0:' qin l!.Xp{-

(3.9)

"'Co }

where -'::0=

I + J"~:l 2(1-J,.~a)

~

T=

~: + ~.. ~ 2(~cl_~i) T

(3.10)

Below Tc in the linear approximation Eq.(3.8) has another fixed pOint on the marginal stability line

q,=,-i'- .

(3.11)

The order parameter tends to this point with the time dependence

9.(t',t)=

1_:C +0. exp {- ~, }

(3.12)

647 A Dy1lil1llical TheoZ'!/ of the Infinite Range Random Ising Ifodel

1197

where

(3.13)

It is noted that both 10 and '1 have a simple pole at the critical temperature. In the linear approximation below Tc' the order parameter q will reach a finite value q1 indicating the existence of the spin glass phase. However, this is not true when the higher order effect is taken into consideration. The true fixed point lies on the Fischer's line in the unstable region. After hitting the marginal stability line at q~q1 the order parameter q(t,t) will vary slowly further along the marginal stability line down to q=O. To lowest order of nonlinear correction the two equations for q(~,t) in the low frequency limit have the following forms

d? --2 ,-Jilq)X'

at -

X

oIIIIW

f-1c~lct)X1

and

j)'{ diG;.!

"""IT" --=aur- =

(3.14)

aU)

oS: dl~l aw -=at

Along the marginal stability line

•

q-

X and

(3.15) q

are related by

I

(3.16:

X = J;,2.l il .

Using the explicit form of Jr(q) and Jc(q) given in Eqs.(2.9) and (2.13) i t is easily found from Eqs. (3.14) and (3.16) that aq/at'" q2(t,t)q which is small for small values of q. In the low frequency limit we have from Eq.(2.23) (3.17) fS3nw ) ~r-)(-trb

\w( (cts¥-

with

y(t»O.

Since , 'l2. u"ql

•

( 3.18)

)0

1- Je ('1 >x.= J'+2.S81L1Qt

along the line (3.16), we see from Eq.(3.14) that q will continuously drop to q=O, for however small but finite ~. Neglecting terms like aX/at'" qi! '" q ~ we c·an integrate Eq. (3.• 15) to .get ~

f"tJ

~

,

q«(.O, t)=q( Iwl tVlt»)=qlX)

(3.19)

and (3.20) Using Eqs. (3.19) and (3.20) we obtain from Eq.(3.14)

d~

vt-

f

dx 1

d.t =_ dt

i

142. u

J"lX Il

cx" q

The right-hand side of Eq·. (3.21) is a function of

y1--'

dt =L,3 Q=const.

dt

(3.21) x·

only.

Hence we have (3.22)

648 1198

SU Zhao-bin, YU Lu and ZHOU Guang-zbao

Integrating the above equation we get V

Yet)= ca.t+ b)3 where a and follows that

b

(3.23)

are two constants.

Substituting Eq.(3.23) into Eq.(3.20) it I

9,ct, t)=O( (a.t+bj"-:r

(3.24)

Therefore, q(t,t) decreases according to a power law in time and a>0. Substituting Eq.(3.22) into Eq.(3.21) and inte~ating it, we obtain (3.25) We note that

q(w,t)

has a peak at

w=O

but it is not of the form

qoo(w).

IV. Susceptibility and the order parameter with small external magnetic field to be

In the following we shall use the unit J=Tc=l and take the value of u 1/12 below Tc' The stability condition Eq.(2.1Sj now takes the form (4.1) Inside the stability region the Fischer's relation holds (4.2)

where aCt) is the average value of the spin depending on the external magnetic field. Above Tc the whole line (4.2) lies inside the stability region. The static order parameter qo and the average spin a are related by the equation (4.3) with

X given in Eq.(4.2).

'10=

For small

a

we can solve Eq.(4.3) for

(11 :& I_~" (j +..... .

qo (4.4)

The susceptibility is therefore

dlt

~

A

:1

J<= df= \'-~ 0"+ ".". ~3 flO.

=~-I-f

At the critical point

8=1

+ ..... . Eq.(4.3) is still valid.

(4.5) In this case we

have (4.6)

and

a- 5 .. )'= 1-rff - 2.4- (t +..... =1-..1L+ I -22.+ .ff 2.+ It """

(4.7)

649 A D!1namical 2'heor!1

or

the In:f.tnite Range Random Ising Model

1199

Below Tc there exists a critical 0c corresponding to a critical external field Ile·. Above be the static fixed pOint is still lying in the stable re!!,ion. The critical 0c can be calculated by Eq.(4.3) and the equation

X=~(f-q,a-Irc.2.)=

~' ~

(1+2.~+4-

(4.8)

'Va

ottcl

Near critical temperature it is found that (4.9)

or

(4.10)

where

'[=1-.1 B• For h
and

x.= /- 'l~- ... '" 3 215 = I-(+a-'") _ .... -(4.12)

3 -I.' '13 =1-(4 11 ) - ••••• -

Formulas (4.5), (4.7), (4.10) and (4.12) are in qualitative agreement with the predictions from the projection hypothesis. We have shown in Sec.III that below Tc the order parameter q will fast decay to the intersection point of the Fischer's line with the boundary of the stability region and then decrease slowly along the boundary to q=q •• This point of interesection depends on the external magnetic field and forms a line on the q-h plane, which is just the stability line found by Almeida and Thouless in the static approach~]. The corresponding order parameter ql is determined by th~ equation Q

~

(1-'1- 0-') _ _ --,-:..l_~ \ J - (l+2.q~+4~~)~

Near critical temperature Eq. (4.13) can be solved for n-r2:1.,= .... +""( 2.+"( 3 -(J+ ..... .

(4.13) q

to get (4.14)

which agrees with Parisi's q(x) at x=l. The results obtained so far, though valid only for small q because of the perturbation calculation, indicate strongly that the projection hypothesis is very close to truth. We shall justify this hypothesis theoretically in the following section.

V. Discussions and conclusions Starting from the general formalism of CTPGF we have studied the infinite range random Ising model in some detail. We found in the q-X plane a marginal stability line. Above Tc the Fischer'.s line lies entirely inside

650 1200

SU Zhao-bin, YU Lu and ZHOU Guang-zbao

the stable region. The order parameter q(t,t) tends exponentially in time to its stationary value along the Fischer's line. The low frequency behavior of the vertex function is n9rmal with Imi'r=I~Jr. In this case a Langevin equation for the spin aCt) could be written down from the general formalism. Below Tc the whole Fischer's line is inside the stable region only when the external magnetic field h>h c • In this case the physical picture remains the same as in the case T> Tc. In case h< h c . the Fischer's line will intersect the stability boundary at a point q1. The fixed points on the Fischer'sline are all situated in the unstable region. Hence the time evolution of q is divided into two stages. Firstly q decays exponentially along Fischer's line to point q1. Theil it decreases further with a -1/3 power law to its stationary point qo along the stability boundary. Since the low frequency behavior of the vertex function is abnormal on the boundary with Imrr "'X ZylwlVsgnw, vS1/2 it is not possible to write a Langevin equation for the spin aCt) at the second stage. In our general formalism the stability boundary in the q-X plane is temperature independent. Therefore, the stationary point. (qo'Xo) is temperature independent. As a consequence, the magnetization is also temperature independent and the entropy does not vary with the magnetic field. This is just the content of the projection hypothesis stated in Ref. [14]. It follows naturally from our theory. Parisi and Toulouse have also identified the Edwards-Anderson order parameter with the intersection point q1 which is determined in the whole termperature range by the equation 2

00

i!:z

1=<X2>-=*fd~ iT se.c.h q.~ (~K, +n.) ~. ",2JT

(5.1)

-00

This equation can also be derived from our formalism. Therefore, the thermodynamic properties predicted by our theory will agree with those by projection hypothesis. Comparing our results with those of Sompolinsky ~OJ it 1S natura! to conjecture that the function q(x), OsxSl corresponds to our q(t,t)· varying along the stability boundary from q1 to qo. Not only does the present theory give a clear physical meaning to the function q(t,t) but also an equation that can be solved explicitly for the time evolution of the order parameter q(t,t). The present formalism can be applied to other quenched random systems, which we hope to present in subsequent publications.

651 1201

A Dynamical Theory of the Infinite Range Random Ising MOdel

References ~

1.

S.F. Edwards and P.W. Anderson, J. Phys.,

2.

D. Sherrington and S. Kirkpatrick, Phgs. Rev. Lett., 35 (1975) 1972.

3.

T.R.L De Almeida and D.J. Thouless, J. Phgs.,

4.

E. Pytte and J. Rudnick, Phys. Rev.,

5.

!!2

(1975) 965. ~

(1978) 983.

(1979) 3603.

G. Parisi, Phys. Lett., A73 (1979) 203; Phgs. Rev. Lett., 43 '1979} 1574; J. Phys., Al3 (1980) lBB7. ~

6.

S.K. Ha and J. Rudnick, Phys. Rev. Lett.,

7.

C. De DOminicis, Phys. Rev.,

B.

J.A. Hertz and R.A. Klemm, Phys. Rev. Lett.,

9.

H. Sampolinsky and A. Zippelius, Phys. Rev. Lett., 47 (19Bl) 359.

edited by C.P. Enz

(S~inger

!!!

(197B) 5B9.

(197B) 4913; Lecture notes in Physics, V.I04 p.253,

Verlag, Berlin, 1979).

.£. '(HB1)

~

10.

H. Sqmpolinsky, Phys. Rev. Lett.,

11.

P.C. Hartin, E.D. Siggia and H.A. Rose, Phys. Rev.,

12.

H.J. Sqmmers, Z. Physik,

~

i!

(197B) 1397;

(19B1) 496.

935.

(197B) 301; ibid

~

~

(1973) 423.

(1979) 173.

13.

C. de DOminicis, M. Gabay and H. Orland, J. Phgs. Lett.,

14.

G. Parisi and G. Toulouse, J. Phys. Lett., 41 (19BO) 361; J. Vannimenns, G. Xbulouse and G. Parisi, J. Physique,

15.

~

~

(19Bl) r.-523.

(HB1) 565.

SU Zhao-bin, YU Lu and ZHOU Guang--zhao, HOn a dynamic theory of quenched random system'·, Commun, in Theor. Phys., this issue.

652 474 A Dynamical Theory of Random Quenched System and Its Application to Infinite-Ranged Ising Model* Su Zhao-bin

Zhou Guang-zhao

Yu Lu

(Institute of Theoretical

Physi~s,

Beijing, China)

Abstract A dynamical thoery for quenched random systems is developed in the framework of the closed time-path Green's functions (CTPGF).

The order parameter q, a matrix in general,

appears naturally as an integral part of the second order connected CTPGF.

An equation to determine q is derived from the

Dyson-Schwinger equation.

The formalism developed is applied

to the study of the long-ran!l'ed random Ising model. dary line is found on the

q-I~I

plane.

A boun-

It is argued that the

spin-glass phase is characterized by the fixed point lying on the stability boundary.

The magnetization is calculated in

perturbation and is found to be in good agreement with those predicted by the

projection hypothesis.

The general validi-

ty of the projection hypothesis is discussed.

653 475 I.

Introduction

Much progress has been made in recent years on the understanding of the spin-glass (SG) phase in magnetic systems with infinite-ranged random exchange.

A mean field theory (MFT) with order parameter being a

continuous function q(x)JO~x~l, has been derived by use of a parti' 1 approach 1-10 · cu 1 ar sch erne 0 f rep 1~ca synunetry b reak'~ng or a d ynam~ca The MFT is free of instabilities although the physical meaning of the order parameter q(x) and the origin of the apparent violation of the fluctuation-dissipation theorem (FDT) are still under intensive investiga11-13 ' t ~on .

On the other hand, a drasti,ally simple extrapolation pro-

cedure projecting physical properties from the marginal stability line onto the SG phase has been shown by Parisi and Toulouse all the nice features coming from the MFT.

~onte-Carlo

14

to reproduce

simulation and the

However, a theoretical explanation of this very simple and elegant

projection hypothesis is still lacking. In this note we investigate properties of the infinite-ranged random Ising model in functions (CTPGF) 15.

the framework of the closed time path Green's The order parameter, a matrix q(t,t') in general,

appears naturally as a part of the second order connected CTPGF which satisfies the Dyson-Schwinger equation.

The FDT is assumed to be sa-

tisfied by the CTPGF before averaging over the random exchange. Fischer's relation is found to be violated in the SG phase

16

The The

validity of Fischer's relation depends on the existence of the FDT and the use of a unsubtracted dispersion relat10n for the retarded CTPGF. It is more likely that a subtracted dispersion relation is necessary on the stability boundary and in the unstable

,

I

reg~on.

We believe, this

subst raction rather than the violation of the FDT is the real cause for the breakdown of the Fis cher' s relation in the SG phase. In the present formalism the dynamical behavior and the static properties of the order parameter q(t,t') are determined completely by the Dyson-Schwinger equation which can be solved approximately for small q in the low freq uency limit. played more clearly on the qceptibility of the system.

The physical picture obtained can be dis-

'X'

plane where:t

is the magnetic sus-

& physical boundary is found on this plane.

654 476 Above Tc,the Fischer relation, a line on the q- I{f

plane, lies comple-

tely inside the stable region and the order parameter q tends exponentially in time to its fixed point q=qo.

Below T c , the Fischer line intersects the boundary at the point q=q, , which is a fixed point only when the external magnetic field h=h c . In case h< he' after reaching q=q. along the Fischer line, the order parameter q will decay further along the boundary down to its fixed point q=q..

In a rough apprixima-

tion the decay along the boundary is found to obey a power law.

In this

formalism the physical boundary on the q-I{1 plane is shown to be temperture independent.

Therefore, the fixed pOint on- the boundary characterizing the

SG phase is temperature independent. As a consequence, the magnetization M(h) is also temperature independent and the entropy is independent of the external magnetic field.

This is just the assumption of the projection hy-

pothesis which follows naturally from our theory. The rest of the paper is organized as follows: In Sec. II the general properties of CTPGF for a random quenched system are presented. The stability condition and the physical boundary are discussed in Sec.III for the infinite-ranged random Ising model.

In Sec.IV the susceptibility

is calculated for small external magnetic field and is compared with what follows of the

from the projection hypothesis.

The dynamical evolution

order parameter is also briefly discussed.

The final Sec.V

contains some concluding remarks. II.

CTPGF for a Quenched random sYstem For the sake of simplicity we study the soft-spin version of the

Edwards-Anderson SG model defined by the Hamiltonian

where the interaction Jij are random Gaussian variables with zero average and mean square fluctuation J2 IN, N being th.e number of the neighbors. The generating functional of

C~PGF

with the interaction kept fixed

can be represented by a path integral in the following form

655

where

1\

and

f

is the density matrix.

from t-=to to t=

00

In Eq. (2.3) the closed time-path starts

(positive branch) and runs back from t= oc

t=to (negative branch).

to

rpo (x-y) is the second order vertex func-

tion obtained after integrating over the reservoir degrees of freedom. It satisfies the FDT with the temperture of the reservoir

(2.4) where

reO) (e

and

,(0)

I

r

are the correlation and the retarded vertex functions

respectively. The advantage of using CTPGF for random systems is that the quenched average can be performed directly on the generating functional instead of its logarithm.

This is possible owing to an important property of the

:.!enerating functional Z[h,J ij] , namely, it equals unity in the physical limit when the external magnetic field hex) on the positive branch is identified with that on the negative branch. Introducing the averaged generating functional (2.5) it is possible to calculate the connected CTPGF by a direct differentiation.

Eq.(2.5) then implies that

(2.6) In the physical limit both Z and

Z equal

unity and the observed magneti-

zation satisfies the requirement of a quenched average (2.7)

656 478 Differentiating Eq.(2.6) w.r.t. hex) and setting h(X+)=h(x-) we obtain (2.8) where the order parameter matrix

is real, symmet ric and equal to each other on the two branches, i. e. , (2.10)

From Eqs.(2.8) and (2.11) follow the retarded, the advanced and the correlated Gre.en's fLlnctions

G-y,j(t,t') -

G(l.0 fc,

t')

< GrrL/(t,t;J'j):;-, y

(2.12) (2. 13)

and

The order parameter matrix q(x,y) which characterizes the behavior of a quenched random system, appears in this formalism as an integral part of the second order CTPGF.

By sucessive differentiation one can easily deduce

higher order CTPGFs with additional order parameter matrices which we shall not discuss in this note. Before discussing FDT let us take the Fourier transform w.r.t. the relative coordinates and write the two-point matrices in the form of Wigner's distribution in phase space (2.15)

657 479 After a very short time of microscopic scale the Green's functions of a system with the interaction Jij kept fixed must satisfy the FDT (2. 16) where

~-I is the temperature of the reservoir.

Substituting Eq.(2.16)

into Eq.(2.14) it follows that

~

(W, t) = i et/" f.2"'" Im

~(t.J,t)+ ifrw,tj

(2.17)

which can be regarded as the FDT for the quenched random system. The retarded Green's function Gr(k,X) is analytic in the upper k oplane.

Its real and imaginary parts satisfy a dispersion relation.

In

the stable region it is usually assumed that a unsubtracted dispersion relation holds, Le.,

Re

G-,J~,-t)

(2. 18)

However, this can not be true in the unstable region where either the real part or the imaginary part

changes sign.

Using FDT in the high

temperature approximation we get from Eqs.(2.17) and (2.18) the relation

""

Re6rr (0, t) ==

G- (0, (;) y

=(3 [-i ~c ft, t) - /(t,t)}

(2.19)

For a long-ranged random Ising model where the space dependence can be neglected, the Fischer relation[ Eq. (2.19)J

becomes (2.20)

It should be stressed that the Fischer relation is valid only in the stable region. Now we are going to study the equation governing the dynamic behavior of the order parameter matrix q(x,y).

We begin with the vertex

functional (2.21)

658 480 where

is'

(x) is the averaged magnetization.

The vertex CTPGF can he defined

by successive differentiation of the vertex functional w. r. t. a('X), i. e. ,

\ Lg ()()] =

w [f,.!)tJj - f fIx) (ffXJJ.«X.

(2.22)

p

The Dyson-Schwinger equation for the second order CTPGF has the fo llowing fo rm

5p llidJ 0,tx,j) G-fCl-. 'i) =;f~()('i)fpIJ/~)ddJ where

8i>Cx-y )

== - bl'/><- 'J). is a b -function

(2.23)

defined on the closed time-path.

In

the physical limit Eq.(2.23) reduces to three equations for the retarded, the advanced and the correlated Green's functions in the following mat rix form

= -/

(2.2:' )

-I

(2.15)

.:md

(2 .1h)

Now define a new matrix

(2.27)

which is a functional of the order parameter matrix q and vanishes together with q.

Then the Dyson-Schwinger equation for q is easily found

from Eqs. (2.24), (2.26) and (2.27) to be

rt=-&G-a...

Irl)

(2.28)

This is a matrix equation which can be simplified in the semiclassical approximation where the product of two matrices AB is replaced by the

cl assical expressi~

(2.29)

659 481

,...., Here A and B are the corresponding Wigner's distribution functions. III.

The stability condition and the physical boundary of the model given by the Hamiltonian Eq.(2. 1) After averaging over the random variables J ij the generating func-

tional takes the following form

(3. 1)

where

(fj (f..{fJr.C,'(t-Ou.rt'Jdtdt'-Uu:9-((;)dt 5I2ff = dIfl.;" f d d

+

/l.lt) C!J!t)dt~t (j

u

i h~Z.

(Q".((;}u,{tJdt ((J:{t'J Uj(f)tlt' J j"' " (3.2)

'...J'"TTY .::t-i)'C ~

vf

f

The bar over the quench-averaged quantities ha,s been dropped for simplicity.

In the infin'ite-ranged limit with

can bl:! approximated by

~ijG(t,t').

N~c:>O

the matrix Gij(t,t')

In this case the second order vertex

function can be calcul.ated without difficulty in a diagrammatic pansion.

It

where r and

ex-

is found that

tare

and the time t.

renormalized quantities that could depend on q,

(5"

In the limit where J tends to zero r is the inverse of

the magnetic susceptibility which is proportional to the temperature and increases as the magnetization E) increases.

We shall assume that r will

keep this qualitative behavior even in a random spin system.

In Eq.(3.3)

~

~(oJ,t)

sion of

is the self-energy part with the first two terms in the expan~

and the term proportional to Gr being excluded.

Therefore,

we have (3.4)

66Q 482 To the lowest order perturbation in u we find (3.5) where J is also remorma1ized and q=q(t,t) is the order parameter. obtaining the second

,..,

In

term in Eq.(3.5) we have made the approximation

that q (LJ, t) has a sharp peak at

to =0.

To the same order of approximation we find the correlated vertex function

z."-

(3,6)

"'-

-t iJ L!:.fw,t) - LC(W,"t)J where~(tc..),t)

is defined to be "-

LJ(fAJ,t)

( i c,J 'l:

= Jr,l.7:e

r.-

criti-f-)uft-;)

(3.7)

~

which is also sharply peaked atW =0.

In Eq.(3.6)

.Lc is the remaining

self-energy part that does not have a sharp peak at W

=0.

? J~(q)

can be

easily calculated in perturbation to be (3.8)

In the low frequency limit the matrix Q defined in Eq.(2.27) has therefore the form

(3.9) Here only terms that have a sharp peak at ~ =0 are retained. It is noted 2 2 that Jr(q) is guater than Jc(q) for all values of q. This fact is very important in the following. In the zero frequency limit the Dyson-Schwinger equation for the retarded Green's function[Eq.(2.24Uhas the form (3.10)

£;361 483 where

..t =Gr(o,t)

is the susceptibility of the system.

This equation

can be solved to give

(3.11) The susceptibility increases as r decreases and reaches its maximum at -V-I "y r=2J r (q) where Il. =Jr(q)· Further decrease of r will make '\. complex and the system unstable.

Therefore, the stability region is bounded by

the inequality

(3. 12)

It is easily seen from Eq.(3. 11) that in the unstable region

(3.13)

which is ;] curve on the are

q-Ill

plane.

On this plane all stable points

situated in a region bounded from above by the curve [Eq.(3.l3)]

consisting of marginally stable and unstable points. region q and

.{

In the stable

are related by the Fischer relation.

Hence the physical

state of the random system can only evolve either along the Fischer's line when it is stable, or along the boundary Eq. (3,13) when it is margina] l.y stable or lmstable.

Before turning to the next section let us briefly mention the low frequency behavior of the retarded Green's function.

For this purpose

write

where

ot.. (t)

and

J}

are to be determined.

An analysis similar to that

given in Ref.9 shows that )J~ 1/2 i f the state is marginally stable and )} =1 otherwise.

662 484 IV.

Susceptibility and the order parameter q in small external magnetic field In the fOllowing we shall take the value of u to be 1/12 in the

units J=T c =l. From Eqs.(2.28) and (3.7)-(3.9) the static fixed point for q is determined by the equation (4. 1)

where qo=Lim q(t,t). Above Tc , the whole Fischer line lies inside the stable region and we have

(4.2) For small external magnetic field we can solve Eqs.(4.1) and (4.2) for qo

() 2. + ...

(4.3)

The susceptibility is therefore

(4.4)

==

B

r-

-L -/L2.+. 1-(3'"

At the critical temperature Eq.(4.2) is still valid.

In this case

we have

..90 == l5

!L _ .!.J.. .ft

24-

(J l.

+

(4.5)

and

1

(4.6)

663 485 Below Tc. there exists a critical external field h c • above which the static fixed point in still lying in the stable region. The critical field hc can be calculated and is found to be

n: i =

nearTcwherer=

,£:$

(I-t 3

z: + .'. )

(4.7)

1-1j1·

For T
be reached.

yields negative entropy at low temperatures.

The only possible fixed

point in this case is on the boundary where the solution is" marginally stable.

In these new units the boundary is described by th.e equation (4.8)

Solving Eqs.(4.1) and (4.8) we find (4.9) and

= (4.10)

All the results Obtained in Eqs.(4.3)-(4.6) and (4.9)-(4.10) agree with those predicted by the projection hypothesis. Now we shall give a brief account of the dynamical behavior of the ..... order matrix q(~.t). Details will be presented elsewhere. In the semiclassical approximation stated in the end of Sec.II the Dyson-Schwinger equation for

--

q(~.t)

has the form

664 486 where the external magnetic field is neglected for simplicity.

Above Tc, the only fixed point is q=o.

We can linearize the Eq. (4.

near q=o and find out the relaxation time

L

(4.12)

a

Below Te, the Fischer line intersects the boundary at the point q=q1.

In

the linearized approximation q1 is also a fixed point of the equation (4.11).

The order parameter q(t,t) will tend to q1 with a relaxation tim,

r Note that both

Z"D

/,2.. (4. 13)

I

and

7::,

have simple poles at the critical temperature.

Thuugh in the linear approximation below Tc the order parameter 'I tends to a finite value q1 indicating the existence of a spin-glass phase it is not true when higher order effects are taken intu consideration.

q

is not a tl-ue fixed puint and the order parameter q is still time depende The true fixed point is again q=o on the physical boundary.

After hittin

the boundary ;It q=q\ the order parameter q(t,t) must vary further along t boundary to its fixed puint '1=0.

The time dependence of q(t,t) along the

boundary is nnt clear as it might .pass through some unstable regiuns.

We have shown in the end of Sec. III that the low frequency dependence of the retarded Green's function is drastically different on the marginal stability line.

The time evolution of q(w,t) on the marginal

stability line can be obtained b~substituting Eqs.(3. 14) into Eq.(4.11).

For small q we have

4

- 3where

0(

and (4.8)

p2

-L--/AJ o(J}

/1-)/ "-

g(wr)

~

I

(4.14) •

is the coefficient in the low frequency expansion of the

retarded Green's function which is positive. rv

From Eq.(4. 14) we find ,.....

immediately that q(o,t) does not change with time and q(~,t) for hJ ~o tends to zero as time t goes to infinity.

665 487 If the tAl =0 part of the order parameter has nonzero measure and it constitutes qEA part of ql, then the order parameter q will reach qEA finally along the marginal stability line.

An important question to be

solved is to find the value of qEA which certainly depends on the dynamic processes with infinitely long relaxation time in the N -

v.

00

limit.

Discussions

Starting from the general formalism of CTPGF we have studied the infinite-ranged random Ising model in some detail.

Both the unstable and

the marginally stable states are found to be lying on a boundary line in

I{I plane.

the q-

Above Tc ' the fixed point is on the Fischer line In this case the 10\" frequency behavior of

inside the stable region.

the vertex function is normal and the order parameter q(t,t) tends exponentially in time to its fixed point along the Fischer line. Below Tc and hc there are no fixed point on the Fischer line inside the stable region.

In this case the fixed point is lying on the boundary

in a marginally stable state.

In the presence of the persistent external

magnetic field h the order parameter q

will decay exponentially along

the Fischc r line to the in te rse ct ion poin t q 1 and then decreases further along the boundary down to the fixed point qo.

The magnetization calcu-

lated at the state qo agrees with that follows from the prejection hypothesis.

In the absence of external magnetic field qo=O.

If the

~

=0

part of the order parameter q(t,t) constitutes a finite part of ql, say qEA, the system will finally reach a steady state with q=qEA' It is noted that the boundary line on the qture independent. independent.

l:t/

plane is tempera-

Therefore, the fixed point below Tc is temperature

As a consequence, the magnetization is also temperature

independent and the entropy does not vary with the magnetic field. · state d 1n. . 14 is just the assumption of the projection h.ypot h eS1S

This It

follows naturally from the present formalism. Comparing our results with those of Parisi and Sompolinsky it is natural to conjecture that the function q(x),

O~x~l,

corresponds to

666

our q(t,t) varying along the boundary from ql to qo'

Not only does the

present theory give a clear physical meaning to the order parameter q(t,t) but

it also derives an equation for q that can be solved in principle

to get the time evolution of the system.

667 Commun. in TheeL Fhys. :Beijing, China)

Vol.3, No.2 (1984)

139-148

SYMMETRY AND WARD IDENTITIES FOR DISORDERED ELECTRON SYSTEMS LIN Jian-cheng( #:;t~

)

and ZHOU Guang-zhao (

SHEN Yu(

lit !f:)

fiJ J\'; II )

Institute nf Thecretical Physics, Academia Sinica, P. O. Box 2735, Beijing, China

Received November 20, 1983

Abs&act We study the field theory approach to Anderson localization in the framework of closed time path Green's function (CTPGF). The theory is found to be invariant under an Sp(2) group. Ward identities related to this symmetry are derived. A non-linear a;model arises as a consequence of the dynamical sy~etry breaking caused by the imaginary part ef the retarded Green's function.

I. Introduction Rec~nt

experiment on quantum Hall effect clearly indicates the existence

of extended states for two dimensional disordered electronic system subject to a magnetic field.[1-3] This exciting fact challenges the existing theory of localization which predicts a mobility edge in 2+£ dimensions in the absence of a ma7,netic field.[4-5) Several years ago liegner[6] suggested a ~ield theory model of non-interacting disordered electron gas dealing with

the scaling properties near the mobility edge where the conductance plays the role of the coupling constant of a non-linear a model. This model was further developed by many others[7-9] and derived formally from field theory by Mckane and Stone.[5] Extension to the case with magnetic field in two dimensions is also made recently by Pruisken.[10] To achieve quenched average over the random potential all previous p2pers on the fie'ld theory approach adopt the unphysical replica trick. With n replicated systems an O(n+, n_) or U(2n) symmetry is found. The critical behavior near the mobility edge from the side conSisting of extended states is governed by a Goldstone mode due to the spontaneous breakdown of the replica symmetry. The Goldstone mode is described by an O(n+, n_) or U(2n) non-linear a model where the integer n has to be analytically continued to zero in order to get the physical results from the calculation. Although the replica trick is very successful in many disordered systems, its physical meaning is obscure and mysterious. Sometimes it may lead to ambiguous results

~ossibly

due to the non-uniqueness of the analytic con-

668 LIN Jian-cheng, SHEN Yu and Z110U Guang zhao

140

tinuation of a function fen) to f(n=O) from values defined only on a discrete set of integers n. Even the simplest amorphous system of the Ising spin glass is not well understood in the replica language. The aim of the present series of papers is to reformulate the field theory for the disordered system avoiding the use of the replica trick. This is possible in the framework of CTPGF where a rather successful theory of infinite range Ising spin glass has been developed recently. [II] The present paper is the first one of this series applying CTPGF to the study of localization and discussing mainly the general symmetry properties of the Green's functions.An Sp(2) group symmetry is found and the corresponding Ward identities are derived. Order parameter and symmetry breaking pattern are briefly discussed. Detailed analysis will be given in subsequent papers. In the following notations adopted in CTPGF will be widely used. Readers not familiar with CTPGF can consult papers[12] for further explanations. The paper is organized as follows: In Sec.II the model is formulated in CTPGF. The symmetry and the Ward identities will be given in Sec. III. Finally we discuss the symmetry breaking and the resulting non-linear cr model.

II. Green's functions and their generating functionals We are concerned with the effect of disorder on the Green's functions of a non-interacting electron gas moving in external fields. The Lagrangian of the system in the second quantized language can be written in the following form, (2.1)

where Vex) is a random potential with Gaussian distribution. A(X) -and 4o(x) are the electromagnetic potentials which will be considered as external fields in the following. The energy spectrum specifying the nature of the electronic state either to be extended or local is determined by the imaginary part of the retarded Green's function defined on the vacuum state. In CTPGF the retarded, the advanced and the correlated Green's functions are closely related and form a single Green's function defined on a closed time path p. They can be derived from the generating functional for CTPGF. With fixed potential Vex) the generating functional has the form Z[J ,V]= Jrdl/l] [dl/l+]exp{ i

J[~(x) P

+I(X)+l/I+J(X)+J+l/I{X)]d~X} ,

(2.2}

669 Symmetry and Ward Identities for Disordered Electron Systems 141

where the time integration is along a closed path P running from +~

(positive branch) and turning back from

t_=+~

to

-~

t+=-~

to

(negative branch); J(x)

and J+ (x) are external sources and the integration variables lj!(x) and lj! + (x) are anticommuting Grassmann variables. At the boundary of integration where t±= -~

is the vacuum state whose influence on the path integral is to provide

correct analytic behavior (imaginary parts)

for the various Green's functions.

The effect is summarized in the term I(x) which will be specified later. To proceed further it is more convenient working in the single time formalism and expressing the functions lj!(x), J(x) etc. as column vectors

A

~(x)=

(1/I+(X») J(x)' ( : : : : :

1/1 (x)

where 1/I+(x) (1/I_(x»

1

etc.

(2.3 )

are the function lj!(x) on the positive (negative) branch

of time integration. In single time formalism the term I(x) has the form (2.4)

where n is a positive infinitesimal number and 0i=1,2,3 are the Pauli matrices. This term provides correct analytic behavior for the

retarded, the

advanced and the Feynman Green's functions. Then the generating functional Eq.(2.2) becomes

Z[J,V]=J[d*][d~+]exp{iJ{*+[(i~e~ p

(2.5)

Now the time integration is performed from

t=-~

to

t=+~.

It has been shown in Ref.[ll] that the quenched average over random potential -V can be performed directly on the generating functional Z. Therefore we obtain (2.6)

where P(V) is the distribution functional for the random potential which is in general taken to be Gaussian. The generating functional for the connected Green's functions W[J] is defined to be A

A

W[J]=-ilnZ[J]

(2.7)

670 142

LIN Jian-cheng, SHEN Yu and ZHOU Guang zhao

The vacuum average of the funct ions

ljJ

and

'.[J +

are then (2.8)

where we have adopted the convention that differentiation of the Grassmann variables %J(x) and %~(xl is acting from the right while that of 6/oJ+(xl and 6/o~+(x) is from the left in order to avoid the possible confusion due to the anticommutability of the Grassmann variables. The second connected Green's functions can also be obtained as

G++(x,y)

=(

G+- (x,y)

1

(2.9)

G__ (x,y)

The retarded, the advanced and the correlation Green's function are three independent combinations of the four components defined on the positive and negative

branch~&

of the time axis. They are

(2.10)

Moreover we can construct the generating functional for the vertex (one particle irreducible) function. (2.11)

It is easily found that (2.12)

671 Symmetry and Ward Identities for Disordered Electron· Systems 147

and the second order vertex function (2.13)

which satisfies the Dyson equation with the connected Green's function G(x,y) in the form

(2.14) In the tree level the second order vertex function equals

(2.15) Correspondingly

and (2.16)

It is seen from Eq.(2.16) that terms proportional to ~ give the correct analytic behavior of the vertex functions.

III. Symmetry and Ward identities We shall restrict our discussion to Wegner model where the correlation of the random potential between different energy shells vanishes, i.e. we require

(:;.1 )

Furthermore we assume the external fields to be time independent. Then it is possible to make Fourier transform in the action integral such that

672 144

T,IN Jian-cheng, SHEN Yu and ZHOU Guang zhao

.... + A........ +in~ (x,E)(1-al+ia2)~(x,E) ....

+~

+

A

A

....

A

....

.... + -+ } a 3J(x,E)+J a3~(x,E)

.

(3.2) .

Apart from the source terms and the small terms proportional to n this action has a global Sp(2) symmetry keeping

invariant. The function

~(x)

forms a two dimensional representation of the

Sp(2) group transforming as ~(x)~ ~'(x)=U1jl(x) n

n

,

n

~+(x)~1jI'+(x)=1jI+(x)U+

,

(3.3)

where (3.4) satisfies the condition

(3.5) In Eq.(3.4) A ,i=1,2,3 are group parameters. i

The term proportional to n does not respect this symmetry. Like the small external magnetic field that helps to choose the direction of symmetry breaking in ferromagnets the term (2.4) can be considered as a small external field inducing the breakdown of the Sp(2) symmetry. Actually the Sp(2) symmetry is spontaneously broken by the dynamical generation of the imaginary part of the retarded (advanced)Green's functions. If we make an infinitesimal transformation with group parameters \

(E) of

the integration variables in the path intagral of the generating functional

Z[J)

we obtain three Ward identities corresponding to the three generators

of the Sp(2) group. They are

(3.6)

673 Symmetry and Ward Identities for Disordered. Electron Systems 145

_f

- d

d -1

A

+ -+

A

X[J (X,E)02

oW

oW,'''

X+... I X... o2J(x,E)], oJ (x,E) oJ(x,E)

(3.7)

and

(3.8)

Ward indentities for various Green's functions can be derived from Eqs. (3.6)-(3.8) by differentiating with respect to the external sources J

J+

and

and then putting them to zero. As a special example we shall show how the

dynamical generation of an imaginary part of the retarded Green's function will break the symmetry. It is convenient to introduce two v'ectors and

(3.9)

then A.A+

t;n

A

.....

=a3-~o2'

(3.10)

and

The Green's function can be put into the form

(3.11 )

where Gr =l(G +G -+ -G +- -G 2 ++

),

G =l(G +G -G -G ), a 2 ++ +- -+ -and

(3.12)

674 146

LIN Jian-cheng, SHEN YU and ZHOU Guang zhao

are the retarded, the advanced and the correlated Green's function respectively while Gt =l(G +G -- -G -+ -G +- ) 2 ++

(3.13;

is a fictitious Green's function which always vanishes in the physical limit when the external sources are put equal to zero. In that case Eq.(3.12) reduces to Eq.(Z.10). From Eq.(3.11) we deduce

(3.14)

and (3.15)

Now we are ready to derive a Ward identity for the imaginary part of the retarded Green's function. Taking the derivative 62/6J~(Y.E)6Ja(y,E) on both sides of Eq.(3.7) we obtain

......

As is well known, ImG r (y,y,E) is proportional to the density of the state p(E) at energy E. It is different from zero certainly for extended states and possibly for localized states as n"'O.Therefore the Sp(2) symmetry is spotltaneously broken in these cases. Makane and Stone pointed out in Hef.(5] that there are two ways to satisfy the Ward identity. For extended states dynamic generation of the imaginary part of the retarded Green's function caused by the breakdown of the Sp(2) symmetry leads to the existence of Goldstone modes, which have long range correlation and govern the critical behavior from the side of extended states. For localized states there should be no Goldstone mode described by a pole at the origin in the momentum plane. To satisfy the Ward identity the integrand of the integral on the right hand side of Eq.(3.16) has to be divergent as n"'O before the

675 Symmetry and Ward Identities for Disordered Electron Systems

147

integration. Though the interpretation of Mckane and Stone is given in an entirely different theory based on replica trick we expect it to be true also in our formalism.

IV. Discussions The order parameter to break the Sp(2) symmetry is the imaginary part of the retarded or advanced Green's funciton. A Goldstone mode on the extended state will be generated by the spontaneous breaking of the Sp(2") symmetry, To describe this Goldstone mode it is convenient to introduce a composite matrix field ~

~

A

A+

~

~

(4.1)

q(x,E)=W(x,E)w (x,E)

whose vacuum expectation value is the second order Green's function. Under the Sp(2) group the field q transforms as A

A

q_UqU

+

(4.2)

The vacuum expectation value of q is (4.3)

where toe diagonal part a describes the imaginary part of tne G++ and G__ Green's functions and the part b their real part. Since U satisfies condition Eq.(3.5) we see that the real part b~3 does ~ot break the symmetry while the imaginary part a breaks the Sp(2) symmetry. ThereforeAGoldstone modes will be dynamically generated by the condensation of the q field.We shall show in a subsequent paper that these Goldstone fields satisfy a non-liRear a model equation whose coupling constant is proportional to the conductance of the disordered electron system. We expect that the critical behavior near the mobility edge will be governed by the long range renormalization effect of these Goldstone modes. This problem is now under study and will be reported later.

References 1. K. Kon Kiltzing, G. Dorda and M. Pepper, Phys. Rev. Lett., 2. D. J. Thouless, J. Phys., £!!(1981)3475. 3. Aoki Hand Kamimura, Solid State Commun., 4. F. J. Wegner, Z. Phys.,

~(1977)45.

~(1976}327.

5. A. J. Mckane and M. St:one, Ann. of Phys., y"!'(1981)36.

~(1980)494.

676 148

6. J. Wegner,

LIN Jian-cheng,SHEN Yu and ZHOU Guang zhao

Z. Phys.,

7. E. Brezin, S. Hikami 8. S. Hikami,

Prog.

~(1979)207.

and J.

Zinn-Justin, Nucl. Phys., B165(1980)528.

Theor. Phys., ~(1980)1466.

9. S. Hikami, Phys. Lett., B98(1981)208. 10. A. Pruisken 11. SUzhac-bin, ~(1983)1181,

(Preprint) YU Lu, ZHOU Guang-zhao, Commun. in Theor. Phys.

(Beijing, China)

1191.

12. ZHOU Guang-zhao, SU Zhao-bin,HAO Bai-lin and YU Lu, Phys. Rev., B22(1980)3385.

677 Commun. in TJ\eor. Phgs. (Beijing, China)

Vol.3, No.2 (1984)

.~G3-26·7

DOES PARISI'S SOLUTION OF THE SHERRINGTON-KIRKPATRICK MODEL LOCATE ON THE ABSOLUTE MAXIMUM OF THE FREE ENERGY? SHEN Yu( iii:

1=)

and ZHOU Guang-zhao(

JiJ!fI )

Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beiiing, China

Received November 20, 1983

Abstract In the spin glass phase near the critical temperature we find in a particular replica symmetry breaking pattern an order parameter with free energy greater than that of the Parisi's.

Earlier solutions of the infinite ranged s~in glass model of Sherrington and Kirkpatrick (SK)[l,2] lead to unphysical results such as negative entropy and are indeed unstable. The first satisfactory solution in the replica ap,proach was given by Parisi, who has introduced a replica symmetry breaking scheme involving an order parameter function q(x) with 0<X~l[3]. This solution was found to be locally marginal stable[4] and has remarkable physical properties. In the replica approach the quenched average of the free energy is carried out by first calculating the partition function of n replicated systems, taking average and then letting n analytically continue. to zero. The order parameter is taken to be the correlation of spins in two dif~erent replicated systems (1)

where u.B=l •••• n are replica indices. The limit. for n going to zero. of this matrix QuB is the physical order parameter. which in the Parisi's solution is approximated by a function q(x) with x a parameter defined on the interval to.l). Al though the free energy of n replicated systems wi th n>2, considered as a function of the order parameter QuB' will tend to its minimum as the stable

678 2G4

SHEN Yu and ZHOU Guang-znao

pOint of equilibrium, this is not so when n goes to zero. Strikingly enough, it is the maximum of the free energy considered as a function of Qa8 in the n+O limit that will be the stable point of equilibrium. It is verified that Parisi's q(x) is situated at a iocal maximum of the free energy. Is PariSi's solution an absolute maximum of the free energy? This question will be studied in the present note. The answer is no. We shall show in the following' that there exists particular choice of Q,:xS in the n+O limit whose free energy is greater than that of Parisi's. We shall limit our discussion to the case near the critical temperature Tc where-analytic analysis is possible. Near Tc the matrix Q aS is small (proportional to T8T c -T) so that we can use the power expansion given in Ref.[3] One finds the free energy[2,S]

F(Q)~lim(-TTr(Q2)+lTr{Q3)+YL n+O

aQ" ~)/n, a,p a,p

(2)

1

where Y="4
and

if

(4)

where I(x) denotes the integer part of the number x. In Eq.(4) qi(i=O, ..• K) are real numbers and m1 (i-l •••• K) are integer numbers such that mi _ 1 lm i is an integer (i~l) with mo a l,m k + =n. Parisi assumed that as n+O the order of m, 1 remains unchanged so that we have

It is noted that m -m + changes sign in the process of analytic continuai 1 1 tion of n to zero. By defining the function for it is easily found that 1

K

lim-Tr(Q~)=L(m.-m. )q~ • ~ ~+1 ~

n-+{ln

=-

f

1

dxq2(x)

•

o

It is this negative sign on the right-hand side that changes the free energy

679 Does Parisi's solution of ehe Sherrington-Kirkpatrick Model Locate on ehe Absolute Maxim~~ of ehe Free Energy?

265

from minimum to maximum. In this way Parisi found that ( 5)

By variation of q(x) Parisi found a maximum of F(q,,) near Tc in the spin glass phase where

q(x)=

{!

for

x>3,

for

x<3,

,

(6)

and

(7) In the following we shall calculate the free energy in a different replica breaking s~heme and compare it with that of Parisi's to the first non-trivial order in 1. Since Q

ae

is of the order near Tc and the free energy starts with

,3, it is sufficient to calculate the free energy up to term Tr Q3. Therefore we shall look for the maximum of F(Q,,)=lim(-rTrQ2 +iTr(Q3»/n

(8)

n·O

Let us take n=2np where np is considered to be an integer and write the nAn matrix Q as

where Qp is an n"n matrix which is parametrized according to Parisi and is approximated by a function Q(x) 0<x<1, in the op"'O limit. In Eq.(9) a is an arbitrary real constant.

Fro~Eq.(9)

one easily deduces

Tr(Q2)=Tr(Q~)'2(1+a2)

, (10)

Substituting Eq.(10) into Eq.(8) we get the free energy

_~Xq3(X)_q2(X)

J x q(y)dy]dx 1

(11 )

680 266

SHEN Yu and ZHOU Guang-zhao

where Fp(q,.) is the free energy of Parisi's to the same approximation. The maximum of the free energy F(q,.,a) is therefore (12) It is now easily seen that the free energy obtained in Eq. (12) is greater than that of Eq.(7) when a is chosen to be sufficiently small, say a=I/3. One notices from Eq.(II) that the expression (12) will be correct only when the expansion parameter (13)

is small. Therefore one should not let a+O in Eq. (12) to get the absurd that that F(q,a,.)+'" as a+O. In a reasonable range a.-l/3 the expansion is valid for sufficiently small. and we expect the resulting F(q,.,a) Eq.(12) to be correct. In the Appendix the free energy is analyzed when the number of replica n is a large even integer (n>4). There one has to find the minimum of the free energy. It is shown that Parisi's solution is not an absolute minimum for n large and n. small.

Appendix From Eq.(2) the free energy for n replicated systems near critical temperature has the form (A.I) We shall look for the minimum of this function for n an even integer greater than 4 and. small. In this case the term yEQ~B can be neglected and the Parisi's solution reduces to the symmetric solution of SK in the first nontriviRl approximation so that for for

a=8 , (A.2)

Therefore we get Tr(Q2)=n(n-l)q2 (A.3) Tr(Q3)=n(n-l)(n-2)q3 The minil!lum "f nF(Q,n,r) appears at q _ 2. n-;1

(A.4)

681 Does Parisi's Solution of the Sherrington-Kirkpatrick Model Locate on the Absolute Maximum of the Free Energy?

267

with _ 4 3n(n-1)

nF ( Q.n.T)-~T Tn=2)T

(A.5)

This is Parisi's solution. In the particular case of n=2np Eq.(A.5) becomes

(A.6)

Now consider a particular scheme of replica symmetry breaking by taking for n=2n

p

Q=Q

SK

:)

x(et

(A.7)

,

1

we find the free energy to be

(A.8)

It is easily verified that n (n -1) n (2n -1) 2 p p > p(n _1)2 p (n _2)2 p

for

p

n >2 p

and for all et .

Therefore the free energy F' is smaller than that of Parisi's. Hence we conclude that Parisi's solution

is

not situated at the absolute minimum of the

free energy for n replicated systems with n an even integer greater than 4.

References 1. D.Sherrington and S.Kirkpatrick, Phys. Rev. Lett.,

~(1975)1972;

D.Sherrington and S. Kirkpatrick. Phys. Rev., 817(1978)4385. 2. A.J.Bray and M.A.Moore, Phys. Rev. Lett¥ 3. G.Parisi, J. Phys.,

~(1980)L115;

A13(1980) 1887; Phil os.

~(1978)1068.

J. Phys.,

~(1980)403;

Mag.,~(1980)677;

Phys. ReP., 67(1980) 25. 4. C.De Dominicis and I.Kondor, preprint. 5. F.pytte and T. Rudnik, phys. Rev.,

~(1979)3603.

682 PHYSICS REPORTS (Review Section of Physics Letters) 118, nos. I & 2 (1985) 1-131. North-Holland. Amsterdam

EQUILmRIUM AND NONEQUILmRIUM FORMALISMS MADE UNIFIED Kuang-chao CHOU, Zhao-bin SU, * Bai-lin HAO and Lu YU Institute 0/ Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing, China Received 5 June 1984

COfIIents: 1. Introduction 1.1. Wby closed time-path? 1.2. Few historical remarks 1.3. Outline of the paper 1.4. Notations 2. Basic properties of CfPGF 2.1. Two-point functions 2.2. Generating functionals 2.3. Single time and physical representations 2.4. Normalization and causality 2.5. Lehmann spectral representation 3. Quasiuniform systems 3.1. The Dyson equation 3.2. Systems ncar thermoequilibrium 3.3. Transpon equation 3.4. Muhi-time-scale perturbation 3.5. Time dependent Ginzburg-Landau equation 4. TIme reversal symmetry and nonequilibrium stationary state (NESS) 4.1. TIme inversion and stationarity 4.2. Potential condition and generalized FDT 4.3. Generalized Onsager reciprocity relations 4.4. Symmetry decomposition of the inverse relaxation matrix 5. Theory of nonlinear response 5.1. General expressions for nonlinear response

3

6

7 7 12 18 24 28 31 31

34 39 44 46

48 49 52 53 55 58 58

5.2. General considerations concerning multi-point functions 5.3. Plausible generalization of FDT 6. Path integral representation and symmetry breaking 6.1. Initial correlations 6.2. Order parameter and stability of state 63. Ward-Takahashi identity and Goldstone theorem 6.4. Functional description of fluctuation 7. Practical calculation scheme using CfPGF 7.1. Coupled equations of order parameter and elementary excitations 7.2. Loop expansion for vertex functional 7.3. Generalization of Bogoliubov-de Gennes equation 7.4. Calculation of free energy 8. Quenched random systems 8.1. Dynamic formulation 8.2. Infinite-ranged Ising spin glass 8.3. Disordered electron system 9. Connection with other formalisms 9.1. Imaginary versus real time technique 9.2. Quantum versus fluctuation field theory 9.3. A plausible microscopic derivation of MSR field theory 10. Concluding remarks Note added in proof References

• Current address: Depanment of Physics, City College of New York, New York, NY 10031, U.S.A.

62 67 70 71 76 79 82 Il9

90 92 96 99 103

104 109 114 119 119 123 125 127 128 128

683 KlMlng-chlJO Chou tl aI., Equilibrium and nonequilibrium fonna/isms mcuk unifitd Absll'acl:

In this paper we summarize the work done by our group in developing and applying the closed time-path Green function (CfPGF) formalism, first suggested by J. Schwinger and further elaborated by K.eldysh and others. The generating functional technique and path integral representation are used to discuss the various properties of the CfPGF and to work out a practical calculation scheme. The formalism developed provides a unified framework for describing both equilibrium and nonequilibrium phenomena. It includes the ordinary quantum field theory and the classical tluctuation field theory as its limiting cases. It is well adapted to consider the symmetry breaking with either constituent or composite order parameters. The basic properties of the CfPGF are described, the two-point functions are discussed in some detail with the transport equation and the time dependent Ginzburg-Landau equation derived as illustrations. The implications of the time-reversal symmetry for stationary states are explored to derive the potential condition and to generalize the tluctuation~issipation theorem. A system of coupled equations is derived to determine self-consistently the order parameter as well as the energy spectrum, the dissipation and the particle distribution for elementary excitations. The general formalism and the useful techniques are illustrated by applications to critical dynamics, quenched random systems, theory of nonlinear response, plasma, nuclear many-body problem and so on.

1. Introduction

1.1. Why closed time-path? The field-theoretical technique, introduced into the many-body theory since the late fifties, has proved to be highly successful in studying the ground state, the thermoequilibrium properties and the linear response of the system to the external disturbance [1-3]. However, only limited progress has been made in investigating the non equilibrium properties beyond the linear response by using the fieldtheoretical methods. To appreciate the difficulties encountered here, let us recall some basic ingredients of the field-theoretical approach. The Green function is defined as an average of the time ordered product of Heisenberg field operators over some state which we do not specify for the moment, i.e.,

(1.1) By introducing the interaction picture, (1.1) can be rewritten as

(1.2) where the S matrix is defined as

S == U(C1J, - C1J)::: T exp(-i

J,rt'ln.(t)

dt) ,

(1.3)

with the interacting part of the Hamiltonian 1t'~n.(t) in the interaction picture. If we are interested in the ground-state properties, then

(1.4) where L is a phase factor contributed by the vacuum fluctuations and can be set equal to zero, if the renormalized ground state is considered. Therefore, we can easily get rid of st in (1.2) so that the powerful arsenal of the quantum field theory can be used without major changes in the many-body theory at zero temperature.

684 4

KlUJlIg-chao Owu el al., Equilibrium and lIoM/uilibrium formalisms made unified

For systems in thermoequilibrium at different from zero temperature, we cannot relate observable quantities directly to the elements of the S matrix, but the density matrix in this case take the following form:

p= exp[J3(9" -

(1.5)

k)l,

where fJ is the inverse temperature, ~ the free energy, k the Hamiltonian. If we consider fJ as an imaginary time iI, p behaves like an evolution operator exp(-ikl). The well-known Matsubara technique [4-7] has been successfulIy developed by making use of this property. However, it is not easy to handle the st term in (1.2), if a general nonequilibrium state is considered. An intelligent way out was suggested by J. Schwinger in 1961 [8]. Let us imagine a time-path p which goes from -00 to +00 and then returns back from +00 to -00. We can then define a generalized Sp matrix along this closed time-path (as we call it)

Sp == Tp exp{ -i

I ~inl(l)

(1.6)

dt} ,

p

where Tp is the time-ordering operator along this path p. It is identical to the standard T operator on an anti-time-ordering operator on the negative branch the positive branch (-00, +00) and represents (+00, -00). Also, any point at the negative branch is considered as a later instant than any time at the positive branch. Equipped with such generalized Sp matrix we can define the Green function along the closed time-path p as

t-

Gp(lh (2 ) = -i(Tp(A,(tl)B(12»

= -i(Tp(A,I(t l )B1(12 )Sp».

(1.7)

Although for physical observables the time values II, t2 are on the positive branch, both positive and negative branches will come into play at intermediate steps of calculation if a self-consistent formalism is intended. The introduction of the closed time-path appears at the first glance as a purely formal trick to restore the mathematical analogy with the quantum field theory. Actually, it has deeper motivation. In particle physics, people are mostly interested in scattering processes for which the S matrix providing the probability of transition from the in-states to the out-states, is the most suitable framework. In statistical physics, however, we are mainly concerned with the expectation value of physical quantities at finite time I. It is thus natural to introduce the Sp matrix along the closed time-path p going from the state at -00 along t-axis to the +00 state and returning back to the -00 state (see st in (1.22». This way we can establish a direct connection of Sp with observable quantities. As we will see later, the great merits of the closed time-path Green function (CfPGF) formalism more than justify the technical complications occurring due to the introduction of the additional negative time branch.

1.2. Few historical remarks After Schwinger's initiative in 1961 [8], the closed time-path formalism has been elaborated and

685 5

developed further by Keldysh and many others [9-19]. Some people used to call it Keldysh formalism. For the recent 20 years, this technique has been used to attack a number of interesting problems in statistical physics and condensed matter theory such as spin system [20], superconductivity [21-24], laser [25], tunneling and secondary emission [26-32], plasma [33,34], other transport processes [3~38] and so on. For some of these systems like laser, the application of the CfPGF formalism is essential because the standard technique cannot be used directly for far from eqUilibrium situations, whereas for some of the others the CTPGF approach is used mainly due to its technical convenience. It is our impression, however, that the potential advantages of this formalism have not yet been fully exploited, partly because of its apparent technical complexity. For the last few years we have combined the generating functional technique and the path integral representation, widely used in the quantum field theory [39], with the crPGF approach and have developed a unified framework to describe both equilibrium and nonequilibrium systems with symmetry breaking and dynamical coupling between the order parameter and the elementary excitations [40-49]. To check the formalism developed and to explore its potentiality we have applied it to a number of problems including critical dynamics, quenched random systems, nonlinear response theory, superconductivity, laser, plasma, nuclear matter, quasi-one-dimensional conductor, and so on [~57]. Although most of these problems in principle can be also discussed using other techniques, the logical simplicity and the flexibility, the unified approach to eqUilibrium and nonequilibrium processes as well as the deep insight one can get make the CI'PGF formalism promising and encouraging.

1.3. Outline of the paper In this paper we would like to summarize some of the results obtained by our group in developing and applying the crPGF formalism. Because of the limitation of space we will only outline the main features along with some useful techniques of the CfPGF approach and illustrate them by few examples. Since the major part of our papers was published either in Chinese or in not easily accessible English journals, we will attempt to make this article self-contained as much as possible. Nevertheless, we should warn the reader that some part of this review is still descriptive and sketchy. A brief summary of the CfPGF formalism was given by us earlier [58], but this paper is much more extended and complete. Since we are mainly summarizing our own results, the contributions of other authors in developing and applying the CI'PGF approach may not be emphasized as they should be. We apologize to them for any possible omissions or underestimates. To keep the integrity of presentation we will not distinguish carefully what was known before and what is new. The topics to be covered in this review can be seen from the table of contents. We will not repeat them here. Few remarks, however, are in order. Section 2 is mainly tutorial, but the subsection on normalization and causality is important for further discussion. Section 3 is devoted to a detailed discussion of the two-point functions. The differentiation of the micro- and macro-time scales described there is very useful. In section 4 the potential condition and the fluctuation-dissipation theorem (FDT) are discussed from a microscopic point of view. The theory of nonlinear response which may be important for future applications is outlined in section 5. Section 6 is devoted to the consideration of the symmetry breaking and the Ward-Takahashi (WT) identities. We believe that the CTPGF formalism is advantageous in studying systems with broken symmetry. We also mention there the additional way of describing fluctuations available in the CI'PGF approach. The unified framework of treating the dynamical coupling between the order parameter and the elementary excitations mentioned before, is given in section 7. We could start from this formalism at the very beginning, but the present more

686 6

XIIIJng-chao Chou tt al., Equilibrium and nonequilibrium formalisms made unified

inductive exposition is probably more convenient for the reader_ In section 8 we show that the quenched average may be carried out directly on the generating functional in the CfPGF formalism and the replica trick can be thus avoided. The connections with other formalisms are described in section 9. Readers, familiar with them might have a look at this section before the others. An experienced and busy reader could get a rough idea about the CfPGF approach by a quick scanning of sections 2 and 6-9.

1.4. Notations Throughout this paper we will use the units II = kB = C = 1 except for few paragraphs where the Planck constant II is written out explicitly to emphasize the quasiclassical nature of expansion. The metric tensor we use is given by (1.8)

with the scalar product and the d' Alembertian (1.9)

defined correspondingly. The Fourier transformation with respect to the relative coordinates x-y is defined as ] X+ y G(x,y)=G [-2-'x- y =

J(27rY+l

dd+lp.

• (X + Y

)

exp[-IP'(X- y)]G -2-'P ,

(1.10)

where d is the space dimension and p' X = Pot- p' r. The tilde "." will be omitted wherever no confusion occurs. The formalism presented in this paper can be applied to a broad class of fields including non-Abelian gauge fields, but in most cases we will illustrate it by a real boson field, either relativistic or nonrelativistic (e.g. phonons), and a nonrelativistic complex boson or fermion field. The former will be denoted by iP(x), whereas the latter by ~(x) and ~t(x). Wherever a double sign ± or =+= appears, the upper case will always correspond to the boson field, while the lower one corresponds to the fermion field. As a rule, the field operator is not distinguished by the caret "',, which itself is used in some cases to denote a two-component vector or a 2 x 2 matrix. Also, for simplicity we introduce an abbreviated notation for integration Jq;=

J

J(x)q;(x) =

J

ddxdtJ(x)q;(x).

(1.11)

The form at the right is used only in exceptional cases, while the middle one most frequently. The Pauli matrices are defined as (1.12)

687 Kuang·choo C1IOII et al., Equilibrium and noneqllilibrium formalisms made llllified

7

2. Basic properties of CTPGF As mentioned in the Introduction, this section is mainly tutorial. To get familiar with the concepts and notations used in the crPGF formalism we start from two-point functions (section 2.1) in close contact with the ordinary Green functions. We then define the generating functional and discuss the perturbation theory (section 2.2). The single time representation and the physical representation as well as the transformation from one to another are discussed in section 2.3. Furthermore, the consequences of the normalization and the causality are outlined in section 2.4. Finally, the Lehmann spectral representation is described in section 2.5.

2.1. Two-point functions The two-point Green functions are most useful in practical applications and hence their properties have been most thoroughly investigated. In this section we first define the two-point crPGF and then discuss their connection with the ordinary retarded, advanced and correlation functions along with the causality relations. As we will see later, they are special cases of much more general relations following from the normalization condition of the generating functional and the causality. The explicit expressions will be given for free propagators in thermoequilibrium systems.

2.1.1. Definition The two-point CfPGF for a complex field I/I(x) is defined as G(x, y) l5 -i Tr{Tp(I/I(x)I/It(y»p} l5 -i(Tp(I/I(x)I/It(y») ,

(2.1)

where I/I(x),I/It(y) are Heisenberg operators, p the density matrix, Tp the time ordering operator as discussed in the Introduction. Inasmuch as x, y can assume values on either positive or negative time branches, G(x, y) can be presented as a 2 x 2 matrix (2.2) with

Gp(x, y) == -i(T(I/I(x)I/It(y»).

(2.3a)

G+(x, y); +i(I/It(y)",(x»,

(2.3b)

G_(x, Y)l5 -i(I/I(x)I/It(y»,

(2.3c)

GtI(x, y)= -i(T(I/I(x)I/It(y»).

(2.3d)

Here GF is the uSU'al Feynman causal propagator, whereas the other three are new in the CfPGF formalism. Sometimes G,. defined as expectation value of anti-time-ordering product, is called anti-

688 8

Kuang·chao Chou tt al., Equilibrium and lIonequilibrium formaJums made unified

causal propagator. Using the step function 1, 9(x, y) = {0,

if t" > ty otherwise ,

(2.4)

eqs. (2.3a) and (2.3d) can be rewritten as G~x, y) =

-i9(x, y)(I/I(x)I//(Y») + i9(y, x)(I/It(Y)I/I(x») ,

Gr:(x, y) = -i9(y, x)(I/I(x)I/It(y») + i9(x, yXl/lt(Y)I/I(x»).

(2.5)

These four functions are not independent of each other. There is an algebraic identity or

(2.6) G~x, y)+

Gt;(x, y)= G+(x, y)+ G_(x, y),

folIowing from the normalization of the step function 8(x, y)+ 9(y, x) = 1.

(2.7)

In what folIows we will call CfPGFs defined by relations like (2.3) as "single time" representation and denote them by a tensor G ajl ••. p (12 ... n) with Greek subscripts a, /3, ... ,p = ±. As a whole, the tensor itself is written as G.

2.1.2. Physical representation The CfPGFs defined above are most convenient for calculations, but more direct contact with measurable quantities is established via the "physical" representation defined as Gr(x, y)== -i9(x, y)([f/I(x), f/lt(y)].),

(2.8a)

G.(x, y) == i8(y, x)([I/I(x), I/It(y)]:.),

(2.8b)

Gc(x, y) == -i({I/I(x), I/It(y)}) ,

(2.Be)

where Gro G. and Gc are retarded, advanced and correlation functions, correspondingly. In this definition,

It is straightforward to check that these functions are related to the CfPGF in single time representation as follows:

G.= GF - G_= G+- G F ,

(2.9a) (2.9b)

Gc = GF + Gt;= G++ G_.

(2.9c)

Gr = GF - G+ = G_- G1',

689 KlIIIIIg·chGO Chou eJ a/.• Equilibrium and ~librium lonna/isms made unified

9

The inverse relations are given by (2.10) If we introduce two-component vectors

(2.11) (2.10) can be rewritten as

G= W.e71 t +!G.71t"t +~G.t"t"t ,

(2. 12a)

or in components (2. 12b) Sometimes it is convenient to introduce a matrix form for the physical functions

G= (0

Gr

G.). G.

(2.13)

The transformations (2.10) and (2.12) can then be presented as

G= 0- 1 00,

t; =: 0(;0- 1 ,

(2.14)

using the orthogonal matrix (2.15) which was first introduced by Keldysh [9]. In what follows we will call G the CfPGF in physical representation and denote their components by G 1/"" (12 ... n) with Latin subscripts i, j, ... , n = 1, 2. In the case of two-point functions, G11 = !(Gp+ Gp- G+ - G_) = 0,

(2.16a)

G12 = G.= !(Gp - G_ + G+- G,,),

(2.16b)

G21 = Gr=!(Gp - G+ + G_- G,,),

(2.16c)

G22 =G.=~Gp+ G,,+ G++ G_).

(2.16d)

We see thus Gn is always zero and the other equations of (2.16) are identical to those of (2.9) by virtue of the identity (2.6).

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KUIlllg-chao 0.011 eI al.• Equilibrium and lIonequilibrilim formalisms made IInified

It is obvious from definition (2.8) that 012(X,y)==O.(x,y)= 0, if (,>ry;

021(X, y) == Or(X, y) = 0, if ty > tx ,

(2.17)

and also that (2.18) because 9(x, y)9(y, x) = 0.

(2.19)

As will be shown later, almost all that has been said here, e.g., 0 11 = 0, the transformation of single and physical representations, the causality relations (2.17), (2.18), etc., can be easily generalized to the multi-point functions using the generating functional technique. However, before going on to describe this technique itself we give here the explicit expressions for the free propagators.

2.1.3. Free propagators The Lagrangian of the free fermion field is given by !to =

JI/,t(X)(i ~+ at V2 ) 2m

1/1 (X) ,

(2.20)

where m is the particle mass. The single time CfPGFs are defined by (2.3). To distinguish it from the Bose case we will use the letter S instead of O. H the system is in thermoequilibrium, the free propagator can be evaluated immediately from the definition. In Fourier space the CfPGFs tum out to be

SP(p) =

=

1- nCp) + n(p) po - p2/2m + if Po - p2/2m - if

Po-P

1 212

. . + 27TIn(p)8(po-p2/2m) , m +Ie

(2.21a)

S+(p) = 21Tin(p )8(po - p2/2m) ,

(2.21b)

S_(p) = -21Ti(1- n(p»8(po- p2/2m) ,

(2.21c)

Sr{p) = _

n(p) 1- n(p) po- p2/2m + if po- p2/2m - ie -1

. +21Tin(p)8 (Po- p2/2m). po- pm-Ie 212

(2.21d)

691 Kuang-chao Chou et al., Equilibrium and lIOIItI/uilibrium formalisms made unified

11

where 1

n(p)=------

exp[(po --,u. )IT] + 1

(2.22)

is the Fermi distribution with ,u. as the chemical potential. If n(p) is set equal to zero, we recover the propagator for the "pure" vacuum. It is interesting to note that the additional term 27Tin(p)6(pop2/2m) is the same for all components of G. The reason for this will be clear from the next section. In accord with (2.9) we find the physical functions to be 1

Sr(p) =

Po- p

S.(p) =

2/2 +. ,

(2.23a)

1 2/2

(2.23b)

m

Ie

.,

Po- pm-Ie

S.(p) = -27Ti(l- 2n(p»8(po - p2/2m) .

(2.23c)

We note in passing that the retarded and the advanced Green functions S" Sa do not depend on the particle distribution n(p). Similarly, for the Hermitian boson field described by the Lagrangian (2.24) we have (2.25) In the Fourier space the free boson CTPGFs are (2.26a)

d+(p) = -27Ti(6(po) + f(p»6(p~- W2(p» ,

(2.26b)

.!L(p) = -27Ti(6(-po) + f(p»6(p~- W2(p» ,

(2.26c) (2.26d)

where

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Kuang-chao CIIou el al., Equilibrium and nonequilibrium formalisms made unified

1 f{p) = exp[w(p)/T]-1

(2.27)

is the Bose distribution and w(p) = Vp2+ m2

(2.28)

is the particle energy. If f(p) = 0, we recover the standard boson propagator of the quantum field theory [39]. Also, the additional term proportional to f(p) is the same for all components of J. The corresponding retarded, advanced and correlation functions are given by Ll ( )_ r

1

p - pt.:.·,lif(p) + 2iEpo '

---i

(2.29a)

Ll.(p) = p20-(1) 2() P - 2'IEpo ,

(2.29b)

Llc(p) = - 21Ti(1 + 2f(p »8(pij - w2(p» .

(2.29c)

It can be shown [401 that the expressions for fermion and boson propagators (2.21), (2.26) remain the same for inhomogeneous, nonequilibrium systems provided n(p) and f(p) are replaced by their nonequilibrium counterparts - Wigner distributions n(X, p), !(X, p) in the external field, where

X= (x+ y)/2

(2.30)

is the center of mass coordinates.

2.2. Generating functionals For interacting fields we can construct the perturbation expansion in full analogy with the quantum field theory. The Wick theorem can be generalized to the CTPGF case, most conveniently by using the generating functional. For simplicity, we consider real bosons. The extension to other systems is obvious.

2.2.1. Definition of Z[J] The Lagrangian of the system is given by

J

It = Ito(rp) + (J(x)rp(x) - V(rp» ,

(2.31)

where Ito(tp) is given by (2.24), V(tp) the self-interaction and J(x) the external source. The generating functional for CfPGF is defined as

693 13

KUII/lg-chao elwu tl al.• Equilibrium and /lOMquilibrium formalisms made u/lified

Z[J(x)1 == Tr{ Tp [exp(i

JJ(X)~(X»)]p},

(2.32)

p

where the integration path p and the time ordering product along it Tp have been already defined in the Introduction. In general, the external source on the positive branch J+(x) and the negative branch L(x) are assumed to be different. They will be set equal to each other or both to zero at the end of calculation. The n-point CfPGF is defined as

Gp(l'''n)==(-i)n-lTr[Tp(~(l)'''~(n»pl=i(-l)"

8"Z[J(x)]

SJ(I)"'6J(n)

I .

(2.33)

1-0

2.2.2. Generalized Wick theorem In the incoming interaction picture (2.32) can be rewritten as Z[J(x)] = Tr{Tp[exp(-i

f (V(~I(X»-J(X)~I(X»)]p},

(2.34)

p

where the in-field ~I(X) satisfies the free equation of motion. The interaction term can be then taken from behind the trace operator to obtain

(2.35)

It is easy to show by generalizing the Wick theorem that [40]

r,,[exp(i JJ(X)CPI(X»)]

= Zo[J(x)]: exp[i

p

JJ(X)~I(X)l,

(2.36)

p

where: : means normal product and Zo[J(x)] is the generating functional for the free field Zo[J(x)] =exp{ -

iff

J(x)Gop(X - Y)J(y)} ,

(2.37)

p

Gop being the free propagator given by (2.26) with f(p) = O. Substituting (2.36) into (2.35) we obtain

Z[J(x)] = exp[ -i

Jv(T 8J~X»)

p

]Zo[J(X)]N[J(X)] ,

(2.38)

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Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

with (2.39) as the correlation functional for the initial state. N[J(x)] can be expanded into a series of successive cumulants

N[J(x)] = exp(iW~[J(x)]), ~

1

W~[J(x)] = L, n "=I

I .. ·I W~(1·"

.p

(2.40)

n)J(1)" ·J(n),

(2.41)

p

where (2.42)

It is worthwhile to note that correlation functions contribute to the propagator only on the mass shell because /PI satisfies the homogeneous equation (2.43) where w(p) is the boson energy (2.28). Also, the definition (2.39) is independent of the time branch, i.e., each CfPGF component will get the same additional term as we have seen in the last section on the example of free propagator. Therefore, the perturbation expansion in the CTPGF approach has a structure identical to that of the quantum field theory except that the time integration is carried out over the closed path consisting of positive and negative branches. Rewritten in single time representation, each n-point function (also the corresponding Feynman diagram) is decomposed into 2" functions (diagrams). The presence of initial correlations W~(1··· n) which vanish for the vacuum state, constitutes another difference from the ordinary field theory. In principle, all orders of correlations can be taken into account, but in most cases we will limit ourselves to the second cumulant. It can be shown quite generally that the counter terms of the quantum field theory at zero temperature are enough to remove all ultraviolet divergences for the CfPGFs under reasonable assumption concerning the initial correlations [41]. We will not elaborate further on this point here, but it should be mentioned that near the phase transition point the infrared singularities have to be separated first so that the ultraviolet renormalization for CfPGFs in this case is different from that of the ordinary field theory (see section 9.2).

2.2.3. Connected and vertex generating functionals The generating functional for the connected CfPGF is defined as

W[J(x)] = -i In Z[J(x}] ,

(2.44)

695 Kuang ·chao Chou el al.• Equilibrium and nonequilibrium formalisms made unified

15

GC(l'" n)=(-l)"-1 8"W[J(x)) \ 8J(1)'''8J(n) J=O p

= (-i)"-I(Tp(c,o(I)' .. c,o(n)))c

(2.45)

where ( )c stands for Tr(·· . fJ) with the connected parts taken only. The n = 1 case corresponds to the expectation value rpc(X) = 8W[J(x))/8J(x) = (c,o(x)J

(2.46)

of the field operator in the presence of the external source. Therefore, rpc(x) is also a functional of J(x). Performing the Legendre transformation upon W[J), we obtain the vertex functional r[rpc(x)) = W[J(x)) -

JJ(x)c,oc(x) ,

(2.47)

p

which depends on rpc(x) explicitly as weB as implicitly via J(x) by eq. (2.46). It follows then from (2.46) and (2.47) that (2.48) This is the basic equation of the CTPGF formalism from which we will derive a number of important consequences. The general n-point vertex function, or one particle irreducible (I PI) function, is defined as (2.49)

2.2.4. Dyson equation As an immediate consequence of the definition for the generating functionals W[J(x)) and r[rpc(x)) we derive here the Dyson equation. Taking functional derivative of (2.48) with respect to J(y) and using (2.45) and (2.46)

we obtain the Dyson equation

J

Gp(y, z)rp(z, x) = c\(x - y).

p

(2.50a)

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KUlIng-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

Similarly, by varying (2.46) with respect to IPc(Y) we find

JTp(x, Z)Gp(Z, y) = [jp(X - y) .

(2.50b)

p

Here (2.51) is the two-point connected Green function (the subscript c is suppressed), whereas (2.52) is the two-point vertex function containing only 1PI part. The [j·function on the closed path [jp is defined as

I

[jp(X - y)f(y) = f(x) ,

(2.53)

p

In the single time representation

I j dt =

dt+ -

j

dL,

(2.54)

p

where the minus sign in the second term comes from the definition of the closed time-path. The negative branch goes from +00 to -00. To satisfy eq. (2.53), it should be that [j(x - y), if both x, y on positive branch, [jp(x - y) = -[j(x - y), if both x, y on negative branch, {

(2.55)

0, otherwise.

In matrix notation [jp can be written as S(x - y) = [j(x - y)U3 .

(2.56)

The Dyson equation (2.50) in single time representation is thus (2.57) The transformation properties for the two-point connected Green function are the same as those

697 Kuang-cIuw Owu et aI., Equi/ilJrilllfJ and lIOMIJuilibtium formalisms made unified

17

discussed in section 2.1.2, except that a disconnected part proportional to -ilpe(x)lpe(y) should be subtracted from all Gfunctions, whereas a term -2ilpe(x)lpe(Y) must be subtracted only from Ge with Gr and O. remaining the same. Multiplying (2.57) by matrix 0 from left and 0- 1 (see (2.15» from right we obtain (2.58) where

f

=

OtO-l = O(~: ~:)0-1.

(2.59)

We note in passing that the Pauli matrix U3 always accompanies the CfPGF in single ti~e representation G, whereas 01 appears together with the CI'PGF in physical representation 0, f. As seen from eqs. (2.57) and (2.58) UJ'U3 is the inverse of G, while u1ful is that for G. It is more important to point out that all characteristic features of Green's functions 0 discussed in section 2.1 are transmitted to vertex functions f via the Dyson equations (2.57) and (2.58). In particular, we have

t,

(2.60)

f11 == 0, so that

f=

ra),

(0 \rr fe

(2.61)

with (2.62) The inverse transformation from

f

to

t is given by (2.63)

i.e., exactly the same way as Green's function (2.12). Further discussion on the Dyson equation wiD be postponed to section 3.1. Meanwhile, we would like to emphasize that the "transmissibility" of the CfPGF characteristics is an evidence of the logical consistency of the formalism itself. More examples along with some useful computation rules were given before [40, 43, 44]. One more remark concerning the generating functional technique itself. Up to now we have considered only CfPGFs for the constituent field lp(x), but what has been said for it can be repeated almost word for word for any composite operator O[tp(x)]. In the forthcoming discussion we will use the corresponding formulas without repeating their definitions.

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Kuang-chao OIou el al., Equilibrium and nonequilibrium formalisms made unified

2.3. Single time and physical representations In the CfPGF approach we have to deal with three representations which are equivalent to each other, namely, the closed time-path form Gp used for compact writing of formulas, the single time form G and the physical representation O. In section 2.1 we have already discussed these representations and their mutual transformations on the example of two-point functions. In this section we will use the generating functional to consider general n-point functions. The underlying connection here is that different representations of CfPGF are generated by the same generating functional expressed in different functional arguments. The explicit expressions for n-point functions in physical representation will be obtained along with the transformations from G to 0 and vice versa. As we will see later in section 2.4, these formulas are the starting point for discussing the important normalization and causality relations.

2.3.1. Preliminaries To start with, we need to specify some more notations. The multi-point step function

8(1 2 ... n) = { 1, if 11 >: 12 ••• > t,. , ", 0, otherwise,

(2.64)

is a product of two-point step functions

8(1,2, ... , n) = 8(1,2)8(2,3)··· 8(n -1, n).

(2.65)

It can be used to define the time ordered product

(2.66a)

"" or the anti-time-ordered product T(A 1(1)'" An(n» =

L 8(1, ... , n)A,,(n)··· AI(I). ""

(2.66b)

The summation here is carried out over all permutations of n numbers Pn

(:1 :2·..... Iin). These step functions satisfy the normalization condition

L 8(1, ... , n) = 1 , "" and the summation formula

(2.67)

699 KIUI1IK-c/aao Ow.. tt aI., Equilibrium and IIOMqIIilibrium formalisms made llIIifitd

8(1, 2, ... , m) = ~

19

(2.68)

8(1· .. n) ,

".(1"·m)

where Pn(l'" m) means permutations of n numbers with 1 preceding 2, 2 preceding 3, etc., but the order of the rest is arbitrary. In fact, (2.67) is the special case m = 0 of (2.68). The external source term in the generating functional (2.32) can be presented as

.

1= 1

I I(x)~(x)= I dtd"x(J+(x)~+(x)-L(x)~_(x»== I tU3cP, j

(2.69)

p

where

cP = (9'+(x»),

j

~_(x)

= (J+(x»)

\.'-(x) ,

(2.70)

and also as

(2.71) with

I,. == 1/ tj = I+ - L, '1',. == 1/ tcP = 9'+ - '1'_,

Ie =!~tj =!(I+ + L), '1'. == !~t cP = !('I'+ + '1'_) .

(2.72)

Also, we can express j, cP in terms of Ie, I,., 'l'e and '1',. as (2.73) The functional derivatives are related with each other by the following equations: I)

I)

1

I)

8J.. (x) = 2~.. 8Je(x) + 1/0 8J,.{x) , I)

8Je(X) =

I)

I)

~o 8J.. (X) = TJa 8J(x.. ) ,

& 1 I) 1 & I)I,.(x) = 21/.. 8J.. (X) =2~.. 8J(X.. ) ,

(2.74a) (2.74b)

(2.74c)

with a = ± and summation over repeated indices. Here we have introduced a symbolic notation (2.75)

700 20

Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

which is useful for a compact writing of the definition for Green's function as seen from (2.76). A remark concerning the notation fP.(x) is in order. Previously (see (2.46» we have defined fP.(x) as the expectation value of fP(x) on the closed time-path. Hence it is a two-component vector (fP.+(x), fPc-(x», but we do not make the subscripts +, - explicit. Here (see (2.72» fP.(x) is the linear combination of operators fP+(x) and fP-(x), still, in accord with our convention, no caret is put above it. Later on, the same fPc(x) will denote its expectation value. Hopefully, no confusion will occur, since the meaning of fPc(x) is clear from the context and, moreover, fPc+(x) = fPc-(x) = fPc(x) for l+(x) = L(x) as seen from (2.105). The same remark is effective with respect to other functions like Qc(x), I/Ie(x), I/I~(x) and so on, appearing in the future discussion.

2.3.2. "Physical" representation of the generating functional As we said in the introductory remarks to this section, the same generating functional will generate crPGF in different representation provided the external source term is expressed in the corresponding functional arguments. In particular, the generating functional in the form (2.32) will give rise to CfPGF in the closed time-path representation. If, however, the source is written in single time form as given by (2.69), the same generating functional (2.32) can be then expanded as

(2.76) with G a'''p

( ). 8"Z[I+> '-1 1 .. · n == 1(-1)" 8.T(a)'" 8J(p)'

(2.77)

where

and both the space-time coordinates and the dummy indices a, .. _,p should be summed over. Moreover, if the expression (2.71) for the source is used, the same generating functional (2.32) can be expanded as

where

B" ==

(-i),,8"Z[IA , Ie1

8JA(1) .. , 8JA(m )8Je(m + 1)· .. 8Ie(n) = (Tp (fP.(1)· , '({>e(m )fPA(m

+ 1)' , , fPA(n») == i"-12- m + 1GUU(1' .. n), m

n-m

(2.79)

701 KfI4IIg-clulo Chou et al., EquiUbrium and lIonequilibrium formalisms made unified

21

Now we find another expression of the CfPGF in physical representation, namely, in terms of expectation values of nested commutators and anti-commutators. Using the normalization condition for the step function (2.67), eq. (2.79) can be rewritten as

= 2. 6(1'"

;i)2-"'{"'", {"m1/"'''''' ·""'·(Tp(rp,,,(l)··· rp",.. (m)rp", ..+1(m + 1)'" rp".(n»).

(2.80)

"" For convenience we introduce a unified notation for { and 1J aj _

(

-

{f""

if 1 ::;j :5 m , m + 1 :5 i :5 n .

(2,81)

1J '" , if

Since the order of operators under Tp can be changed arbitrarily, (2.80) can be also presented as BII

= 2. 6(1' .. n)2-",(I .. , (II(Tp(rpj(l) ... rpli(;i») ,

(2.82)

""

where the subscript r == a T. Now let us get rid of Tp in (2.82) step by step for each term of permutation PII' As far as the 6-function ensures n to be the earliest moment on the positive branch and the latest one on the negative branch, we have (II(Tp(rp\(i)· ., rp Ii(n») = ~iI(Tp(rpr(l)" . rp,,(n») = (Tp(rp I(l) ... rp n-I(n - l)}rp(n»

= ({Tp(IPr(l)'"

+ (rp(n)Tp(rp 1(1) ... IP II-I (n - 1»)

IPn-l(n -1», rp(n)}) ,

if

or (II(Tp(rp r(I)· .. rp iI(n») = 1/ iI(Tp(rpi(I)· .. rpll(n») = ([Tp(cPI(i)· .. cp n-I(n

-1», rp ,,(n)]) ,

if

m+l:5n:5n.

Such processes may be continued like "a cicada sloughing its skin" in accord with the Chinese saying, up to the last step to get zero if m + 1:5 I :5 n, or the expectation value of nested commutators (and/or anti-commutators).

702 22

Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

If we introduce a short writing

( ,cp(I»= {{[ ,cp(I)], when m+lslsn ,cp(l)} when Islsm,

(2.83)

we find finally 6~ ~(1' .. n) == (_i)"-1

L' 6(1' .. ii)«' . - ('1'(1), '1'(2»' . " cp(ii») ,

(2.84)

m n-m

where ~' means that permutations m + 1 s As a special case we find for n == 2 that

1s n should be excluded from the summation.

0 21 (12) = -i9(1, 2)([cp(I), cp(2)]),

0 11 (12) == 0,

Od12) = -i({cp(I), cp(2)}) ,

(2.85)

thus recovering 0 2 1> 0 22 as retarded and correlation functions. Using the symmetry

we obtain the advanced Green function as

0 12(12) = -i9(2, 1)(['1'(2), cp(l)]). Furthermore, for n = 3 case we have 0 111 = 0,

0 211 (123) = (-W

L

9(1ij)([[I, i], j]) ,

pea> 0 221 (123) = H)2

L

(9(ij3)([{i,j}, 3]) + 8(i3j)({[i, 3],j}» ,

pe(:~>

0=(123)= (_i)2

L

(2.86)

6(ijk)({{i,j}, k}).

Here for short we write i,j instead of 1P(i), cpU). Therefore, this way we can exhaust all possible n-point functions. Without resorting to the CfPGF formalism this would be a cumbersome task. The functional expansion (2.78) and the explicit expression for physical CfPGF (2.84) are very important equations from which a number of far-going implications will be extracted in the next section. In the meantime we note only that the functional derivative 8/8Jc will generate CPtJ. which in turn yields a commutator in the Green function, whereas the functional derivative 8/8JtJ. gives rise to IPc leading to an anti-commutator. Moreover, in the summation (2.82) none of the time variables m + 1 s s n can take a value larger than all of the time arguments 1 s Js m, because in that case I should be one of (m + 1, ... ,n) excluded from the summation. This fact will give rise to important causality relations.

r

703 23

2.3.3. Transformation formula Using the definitions of CfPGFs in single time and physical representations as expansion coefficients of the generating functional (2.77) and (2.79), respectively, we can readily find the transformation from one to another. In fact, using (2.79) and (2.74) we have

:=:

2- 1 ~'"

•••

~a"7Ja,,,,

... 11'" 0 """'" (1 ... n) .

(2.87)

In a similar way we find the inverse transformation as given by O QI'·'IIII',. (1 · .. n) == 21-"

'~ " !:ten 1'1 '"

110

!tall

0 it"'''' (1 ... n) ,

(2.88)

where i10 ••. , in == 1, 2 ,

l~ == 11",

(2.89)

Using the orthogonal matrix 0 defined by (2.15) these formulas can be rewritten as 0., ...",(1· .. n) == 2n12-10'ICll" • 0""'00 ClI...... (1· .. n),

(2.90a)

o al'·'GI,. (1", n)== 2

(2.90b)

1 nll -

Oalii .. , QTa"l,. 0 •• "." (1 .. ' n) • T

For the case n == 2 eqs. (2.12) and (2.16) obtained in section 2.1 are recovered. For n == 3 we have 0+++(123) == (-iY(T(123»,

0++_(123) == (-iY(3T(12»,

0+ __ (123) == (-iY(T(23)1),

0 ___ (123) == (-iY(T(123» .

(2.91)

The other functions, i.e., G+_+, 0_++, 0_+_, 0 __ + can be obtained by symmetry. For illustration we also write down some of the transformation formulas such as 0 111 = 0(+) - 0(-) == 0,

0Z22 == 0(+) + 0(-) ,

(2.92)

with 0(+)== 0++++ 0+ __ + 0-+-+ 0 __ +, 0 211 = !(O+ .. + G_ .. ),

(2.93)

with 0_ .. == ~ af30-aIl' CI.IJ-:t:

704 24

Kuang·chao Chou er al., Equilibrium arui nonequilibrium formalisms made unified

2.3.4. "Physical" representation for W[J] and r[ipc] We have discussed above different expansions for Z[J} along with some of their consequences. The same thing can be done toward W[J] and f[ipc]. For example, in the physical representation we have (2.94) (2.95) where (2.96) It follows then from (2.95) and (2.96) that

8f[CPA(X), ipe(x)] == -J ( ) 8CPe(X) A X , 8f[ ip A(X), CPe(X )] == -1. ( ) 8cpA(X) eX.

(2.97)

It is obvious that W[JA , Je] and f[CPA' CPe] defined by (2.94) and (2.95) are identical to Wp[J] and fp[CPc]

as given by (2.44) and (2.47) respectively. We should note, however, that the explicit form of the connected Green function is different from that of the "total" (connected + disconnected) Green function obtained as expansion coefficient of Z[J]. For example, Ob(123) ==

(-W L: 8(ijk )[({{i, j}, k}) -

({i,j})(k) + 2{{(i), V)}, (k)}] .

(2.98)

P3

The only exceptions are the "all retarded" Green functions like 0 21 , 02l1, 02lll etc., for which

Therefore, the transformation from (; to (; for the connected Green function and vice versa should also be correspondingly modified. We note in passing that the "all retarded" functions are nothing but the r-functions used to construct the LSZ field theory [59]. Unlike the zero temperature case, these functions alone are not enough to construct the CTPGF formalism, but they still playa very important role here.

2.4. Normalization and causality As we emphasized in the Introduction, the normalization and causality relations are essential for applications. In fact, they are already implied by the expansion of the generating functional discussed in

705 KIIIUI,-c/ulo Chou et aI., Equilibrium "'"' nOlJtllllilibrium formalisms millie wtlfr«J

25

the last section, but we would like to make them explicit here for future reference. We start from the normalization (section 2.4.1), then indicate the consequences of the causality (section 2.4.2) and wind up with few comments on the two aspects of the Liouville problem - dynamical evolution and statistical correlation - naturally embodied in the CfPGF formalism (section 2.4.3).

2.4.1. Normalization If Ill(x) is set equal to zero, i.e., I+(x) =L(x) in the expansion (2.78), we find 3"Z[lr.,I.] 81.(1) ... 81.(n )

I 1,-0

= 0, for

n 2= 1,

(2.99)

because in accord with (2.84) G ll ... 1(1···n)=0.

(2.100)

Using the normalization condition of the density matrix Tr(p) = 1,

(2.101)

we find (2.102) By definition (2.94) we have (2.103)

or, equivalently,

3" W[lr., I.] 81.(1)" . 3I.(n)

I 1,-0

== 0 for any n 2= 0 .

(2.104)

In particular, (2.105) which leads to (2.106) and (2.107)

706 26

KUfUlg-c/uw Chou et aI., Equilibrium and IIonequilibrium formalisms mode unified

or, equivalently

r

(1·,·n)=0.

(2.108)

l1 ... 1

We see thus the algebraic relations obtained before, such as (2.6), (2.16), (2.60) and (2.92) are special cases of these general conditions following from the normalization. We would like to emphasize here that the normalization condition for the CfPGF generating functional is different from that of the quantum field theory or the standard many-body formalism. In this case we require only the equality of the external source on the positive and negative branches I+(x) =L(x) instead of its vanishing. We can thus incorporate the external field Ic(x) =!(J+(x) + I_(x» into the theoretical framework in a natural way. Moreover, this fundamental property will give rise to a number of important consequences which make the CfPGF formalism advantageous in many cases as we will see later. We note also, that eqs. (2.99), (2.104), (2.107) and (2.108) are valid even in the presence of a finite external field Ic(x).

2.4.2. Causality As mentioned in section 2.3.2, in the functional expansion (2.78) none of the time variables with m + 1 :$ r:$ n can take values greater than the time arguments 1:$ J:$ m, because this would contradict the rule established by (2.84) that terms m + 1 :$ I :$ n should be excluded from the summation. Put in another way, S"Z[IA.JJ 81A(1)·· ·SIA(m)8Jc(m + 1)·· ·8Ic(n)

I

-0

(2.109)

Jl=J,-O-

provided any II> with m + 1 :$ i :$ n is greater than all Ij with 1:$ j :$ m. This is one of the causality relations we consider here. It is obvious that the causality relation for the two-point function (2.17) is a special case of (2.109) for m = 1, n =2, i.e.,

In a sense, the algebraic relation (2.99) or (2.100) is also a special case of (2.109) for m = O. Similarly, under the same condition, i.e., the time argument of any Ie, 'Pc is greater than that of all lA, 'PA, we have for the functional derivatives of W[l],

S"W[IA,J.] 8lA(I) .. . 8lA(m)8J.(m + 1) .. · 81.(n)

I

=0

(2.110)

J,-Jl-O

'

I

= o.

and those of the vertex functional 8"r[cpA, CPc] 8'PA(I)· .. 8'PA(m )8'Pc(m + 1)· .. 8cp.(n)

(2.111)

wO, .,,_.,

In deriving (2.111) we have made use of the relations between the vertex functions and the "amputated" connected Green functions [39,60].

707 27

Kuang-chao Chou el 01., Equilibrium and nonequi/ibrium formaUsms made unified

It is worthwhile to emphasize that a causality sequence is established by (2.l09H2.11l), namely, the space-time points associated with Jc(x), fPc(x) should precede those of h(x) and tpA(X), since the former is the cause, whereas the latter is the consequence. We indicate here also some useful product relations as a generalization of (2.18) for a two-point function. For example, we have for three-point functions

(2.112) It is easy to see that the general rule is

(2.113) provided iml = ... = im , :;:: 2 and the rest are equal to 1, whereas jml = ... = jm, = 1 but the rest can be either 1 or 2.

2.4.3. Dynamical evolution and statistical correlation Now we discuss the physical meaning of lA, lc,

tpA

and tpc. In addition to (2.105) we have (2.114)

from (2.97) and (2.107), so that the conditions lA = 0 and follows from (2.91), (2.94) and (2.102) that

tpA =

r

= Tr{(texp(-i =L "=0

f I t

OCI

i"

r

lc(y)tp(y»))tp(X)(Texp(i

'1

dl

-00

I

-00

0 are equivalent to each other. Also, it

I

I

lc(y)tp(y»))p}

"'-1

d2 .. ·

dnlc(I)·"lc(n)Tr{[ .. ·[[tp(x),tp(I)],tp(2)] .. ·],tp(n)]p}.

-00

(2.115) We see thus tpc(x) under lA = 0 is the expectation value of the field operator, i.e., the order parameter in the presence of the external field lc, whereas (2.116) is the expectation value that might cause symmetry breaking in the vanishing field. Equation (2.115) is a nonlinear expansion of the order parameter in the external field. A detailed discussion of the nonlinear response will be given in section 5. As we already mentioned in the last section, in accord with (2.78) and (2.84) the functional derivative a/Blc(x) generates the expectation value of the commutator of the field variables describing the

708 28

Kuang-chao Chou el al., Equilibrium and nOMquilibrium formalisms made unified

dynamical evolution in the quantum mechanical sense, whereas 8/8JA(x) generates the expectation value of anti-commutator describing the statistical correlation in the statistical mechanical sense. Although the physical observables are defined on the manifold h(x) = 0 or 'PA(X) = 0, these functional arguments are needed in addition to lc(x) and 'Pc(x) for a complete description of the statistical system. These two complementary aspects of the Liouville problem - dynamical evolution and statistical correlation - have been embodied in the CfPGF formalism in a natural way. It is worthwhile to note that the response and the correlation functions have found their "proper seats" in the CTPGF formalism just because in the external source term (2.71) 'P and J are "twisted", i.e., 'Pc is coupled with h, while 'PA with Ie as follows directly from the definition of the closed time-path. As we will see later in section 9, this is one of the advantages for the CTPGF formalism compared with the others.

2.5. Lehmann spectral representation In this section we study the analytical properties of the Green functions. As in quantum field theory [39] and in the standard Green function technique [1-3], the Lehmann spectral representation is a powerful tool towards this end. We will discuss in some detail the spectral as well as the symmetry properties of Green functions for a nonrelativistic complex (Bose or Fermi) field defined by eqs. (2.3) and (2.8). The modification needed for a real boson field is also briefly mentioned.

2.5.1. Spectral expansion Assume, the inhomogeneity of the system is caused by the nonuniformity of the state, while the evolution of the Heisenberg operators I/I(x),I/It(x) with x is given by the total energy-momentum operator p as I/I(x) = exp(ip' x)I/I(O) exp(-ip' x),

I/It(x) = exp(ip' x)I/It(O)exp(-ip' x).

(2.117)

Let In} be a complete set determined by p... and other operators commuting with P.... According to (2.117) we have for G_(x, y) defined by (2.3c) (2.118) ,.,m,,,·

Set Z

= x - y, X = (x + y)/2 and take Fourier transform with respect to Z, we obtain

L

iG_(k, X) =

(n!I/I(O)!m) (mW(O)!n'}p,..,. exp[i(p,. - p,..) . X](21T)d+18(k - pm + (p,. + p".)/2).

".m,II'

(2.119)

If

p,..,. == (n'!p!n)

IX

6(p" - p,..) ,

(2.120)

G_(k, X) will not depend on X and the system is homogeneous. In the presence of macroscopic inhomogeneity, p,,',. is different from zero only for p" - p", small compared with k, so that the high orders of a/ax can be neglected.

709 KlUJrlg-chao C1wu el aI., EquilIbrium and IIOIIeqllilibrium formalums made unified

29

2.5.2. Sum rule

For non relativistic fields we have the following equal-time (anti-) commutation relation (2.121) which leads to (2.122) Introducing the spectral function p(k, X) == i(G_(k, X) - G+(k, X»,

(2.123)

we rewrite (2.122) as dk o

f 2'17

p(k, X) = 1 .

(2.124)

For a real boson field we have (2.125) from which one can derive dk o

f 2'17

kopek, X) = 1,

aX4

dko

J2'17 p(k, X) =0 ,

(2.126)

with p(k, X} still defined by (2.123). 2.5.3. Lehmann representation

Presenting the retarded Green function G.(x, y) defined by (2.8a) as

we find G (k X) = r

,

Jo

dk pet, kG. X) 2'17 ko kil or It '

(2.127)

which is analytic in the upper half-plane of ko. Similarly, we have for the advanced function

J

G.(k, X) = ~ p(t. k~, X) 2'17 k(j"=:-tO-=-tr

(2.128)

710 30

Kuang-chao Chou et al., Equilibrium and nonequilibrium lonnalisms made unified

which is analytic in the lower half-plane of ko, Presenting G± as

=

'f dk~ , (1 2fT 0,.,(1, ko, X) ko I

1)

(2.129)

k0 - k'0-1£ , ,

'""--'0'£

lion' 1

we find the spectral form of GF and G F ,

oF\Ik, X) = Gr\, Ik X) G (k X) = 'f dk~ (G_(1, k~, X) + +, I 2 -fe fe';.' fT

0

0

G+(1, k~ X») Ie fe" 1£ ,

(2.130)

1£00

Gr:(k, X) = Gr(k, X) + O_(k, X)

= if dk~ (G+(1, ko, &) G_(1, k~, X») 21T

ko - ko + ie

ko - ko - ie

(2.131)

.

Equations (2.127), (2.128), (2.130) and (2.131) are the Lehmann spectral representations we are looking for.

2.5.4. Symmetry relations It is straightforward to check that for nonrelativistic complex (boson or fermion) field we have

G!(x, y) = -G",(y, x),

G~(x,

y) = -G~, x),

G;(x, y) = G.(y, x),

(2.132)

or in Fourier components

G!(k,X) = -G",(k,X),

G~(k,X)=

GHk,X) = -Gr:(k,X),

G.(k,X) ,

(2.133)

whereas for real boson field we have additionally

(2.134) or in Fourier components (2.135) where T means transposition, * means complex conjugation and t Hermitian conjugation.

2.5.5. Two analytic functions It is obvious from (2.133) that G",(k, X) are purely imaginary on the real axis of ko. If they vanish as Ikol-HO , we can define two analytic functions on the complex plane of ko, namely, G (1 Z X)= i 1

,

,

f dkoG _(1, k~X) 2fT Z - ko '

G (1 Z X)=' fdkoG+(k, ko,x). 2 "

I

2fT

Z - ko

(2.136)

711 Kuallg-chao Chou el al., Equilibrium ami lIollequilibrium formalisms made unified

31

In terms of functions 0 1 and O2 we find Or(k, X) = 0 1(1, ko + ie, X) - O2(1, ko + ie, X), Oa(k, X) = Ol(k, ko - ie, X) - 02(k, ko - ie, X) , OF(k, X) = Ol(le, ko + ie, X) - 02(k, ko - ie, X), O,,(k, X)= O2(1, ko + ie, X) - Ol(k, ko - ie, X), O_(k, X)= Ol(k, ko+ ie, X)- 0 1(1, ko- ie, X), O+(k, X) = 02(k, ko + ie, X) - 02(k, ko - ie, X) ,

(2.137)

i.e., all these functions are superpositions of 0 1 and O2 on approaching the real axis from different sides. It follows from (2.133) and (2.136) that 01.2(k, Z, X)*

= 01,2(k, Z*, X).

(2.138)

We see thus to ensure the causality, the retarded Green function should be analytic on the upper half-plane of ko. If a singularity is found on the upper half of ko during the process of solving On it must be located on the second Riemann sheet. The appropriate analytic continuation is to take the integral along a contour in the complex plane of k~ which circulates the singularity from above. 3. Quasiuniform systems In this section we will discuss in some detail further properties of two-point Green functions, mainly concentrating on quasiuniform systems. The starting point is the Dyson equation formally derived from the generating functional in the last section. The quasiuniformity can be realized only near some stationary state, either thermoequilibrium or nonequiIibrium under steady external conditions. We will derive the stability condition from the analytic properties of Green's functions. In section 3.1 the properties of the Dyson equation are further elaborated, especially for a uniform system. The thermoequilibrium situation is then discussed (section 3.2) mainly for the tutorial purpose. Furthermore, the Dyson equation is used to derive the transport equation (section 3.3). Finally, the multi-time-scale perturbation (section 3.4) and the derivation of the time dependent Ginzburg-Landau (TDGL) equation (section 3.5) are briefly described. The separation of micro- and macro-time scales is the common feature of the last three topics.

3.1. The Dyson equation 3.1.1. An alternative derivation The Dyson equation and its equivalent forms (2.50), (2.57) and (2.58) have been derived from the generating functional. Here we give another derivation which will shed some light on the structure of the vertex function. Consider an Hermitian boson field lp(x) described by the Lagrangian density (3.1)

712 32

Kuang·chao Chou el al.• Equilibrium and nonequilibrium formalisms made unified

For simplicity we assume ~c(x) = Tr(rp(x)p) = 0 in zero external field J(x) = O. The field operator satisfies the equation of motion Or~(X) = j(x);: -& V(~(x»/8rp(x),

(3.2)

where (3.3)

and j(x) is sometimes called the internal source of rp(x). The two-point vertex function defined as (3.4)

can be presented as (3.5) where (3.6)

is the vertex function in the tree approximation and Ip(x, y) the self-energy part due to loop corrections. The inverse of rop is Green's function for free field, satisfying OrGop(x, y) = -8:+ 1 (x - y).

(3.7)

Using (3.2) and the commutation relation (2.125) we find

or OrGp(X, y) =- 8:+ 1(x - y) + i

J[Tr(7;,{j(x)j(z»p) + i8:+ (x _ Z)Tr{ 3cp(x}acp(z) 8V p}] 2

1

p

x Gop(z, y) dd+l Z .

(3.8)

Comparing (3.8) with (2.50) we obtain

I

l'p(x, z)Gp(Z, y) dd+1 z

= -i

I p

[Tr(Tp(j(X)j(Z»p)

+ i8:+1(x - Z)Tr{8CP(!;::(Z)

p} ]Gop(Z, y)

dd+1 Z ,

713 Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made uniMd

33

which yields (3.9)

This expression will be used later to discuss the transition probability. 3.1.2. Matrix representation The matrix representation of the Dyson equation as given by (2.57) and (2.58) are very convenient for practical calculations. For example, we find immediately from (2.58) that

(3.10a) (3. lOb)

Using eqs. (2.9) and (2.62) we find the corresponding relations for

G as (3.11)

The symmetry relations (2.132) and (2.133) valid for G can be also transmitted to r to give n(x, y) = -rf:(Y, x),

n(x, y)= -r",(y, x), r!(k) = -r",(k),

n(k) = -rr{.k),

F;(x, y) = r.(y, x),

F;(k) = r.(k).

(3.12a)

For real field we have from (2.134) and (2.135) t(x, y) = tr(y, x) = -O'1t*(x, Y)O'I = -O'l't(y, x)O't.

(3.12b)

t(k) = F(-k) = -O'1t*(-k)O'I = -O'tf't(k)O'I' 3.1.3. Vertex functions

As seen from (3.12a), only three components of t are independent. They can be set as (3.13a)

where A, Band D are real functions in accord with (3.12a). In terms of unity and Pauli matrices, (3.13a) can be rewritten as

t = iB(I +0'1)+ AU2+ DO'3'

(3.13b)

We then find from (3.13) that rr(k) = D(k)+ iA(k),

r.(k) = D(k) - iA(k),

r P(k) = D(k)+ iB(k),

r~k)=

r.(k) = i(B(k) ± A(k» ,

(3.14) -D(k)+iB(k).

714 34

Kuang·chlUl Chou et al.• Equilibrium and nonequilibrium lonnalisms made unified

In what follows we will call D(k) the dispersive part and A(k) the absorptive part of the self-energy in analogy with the quantum field theory.

3.1.4. Green's functions The expressions for Green's functions follow immediately from (3.10) and (3.11) as G(k)r

-

1

D(k)-f iA(k) ,

G Ik)- Qill-iB(k) F\ - D2(k)+A2(k) ,

1

G.(k) = D(k)'::iA(k) , G (k) = ::.l?(~) - iB(k) r: D2(k) + A2(k) .

(3.15)

It follows also from the matrix equation (2.57) and the symmetry relations (2.133) and (3.12) that (3.16) which can be verified directly from (3.14) and (3.15). By virtue of the definition (3.5) we can express functions A(k), B(k) and D(k) in terms of the self-energy part I as A(k) = ~(I_(k) - I+(k », B(k) = ~i(I+(k) + I-(k », 2 2 D(k) = k - m - !(IP(k)- Ir:(k».

(3.17)

It is unlikely that both A(k) and D(k) have zero on the real axis of ko, so the divergence on the mass shell P = m2 can be removed by the renormalization procedure. If there are no zeroes of D(k) + iA(k) in the upper half-plane of ko, then the causality is guaranteed and the pole of Gr in the lower half-plane of ko will describe a quasiparticle moving in a dissipative medium. On the opposite, if there is a pole a in the upper half-plane, then Gr is analytic only for 1m ko > 1m a. This pole will describe a quasiparticle moving in an amplifying medium with growing amplitude of the wavefunction. In such a case the original state is unstable with respect to a new coherent state of quasiparticles like the laser system beyond the threshold.

3.2. Systems near thermoequilibrium The formal solution of the Dyson equation (3.10) and (3.11) as well as the explicit form of the vertex function (3.14) and Green's function (3.15) are valid for any quasiuniform system near eqUilibrium or nonequilibrium stationary state. In this section we consider the thermoequiIibrium system in more detail. The transition probability is first studied (section 3.2.1), the dispersive part is then discussed (section 3.2.2) to show that the thermoequilibrium system is stable and the detailed balance is ensured (section 3.2.3). Furthermore, formulas for nonrelativistic fields are written out explicitly for future reference (section 3.2.4). Finally, the fluctuation-dissipation theorem is derived for the complex boson and fermion field (section 3.2.5).

3.2.1. Transition probability It follows from (3.9) that

715 Kl/IIIIg-chao Otou et Ill., Equilibrium and IIOIIeqIIilibrillllllormalisms made unified

t-(x, y) = -i TrU(x)j(Y)ph P.I.,

t+(X, y) = -i TrU(y)j(x)ph P.I.·

35

(3.18)

As done in section 2.5, the evolution of j(x) under the space-time translation is given by

j(x) = exp(ip· x)j(O) exp(-ip· x).

Substituting this expression into (3.18) and taking Fourier transformation, we obtain it-(k) = ~ 1(1\j(0)lnhp.I.l2p",,(21T)8d+l (k - PI +P.. ), ~

.

it+(k) = ~ l(nV(0)Il}lP.I.1 2 PII(21T)d+18d+ 1(k - PI +P.. ).

(3. 19a) (3.19b)

I...

Here we neglect the off-diagonal elements of the density matrix because the system is uniform. For ko> 0 each term of (3.19a) corresponds to the probability of transition from the state In) to the state II) by absorbing a quasiparticle of momentum k, i.e.,

(3.20a) while each term of (3.19b) corresponds to the probability of emitting a quasiparticle (3.20b) Since EI > E .. for both cases, P"" > PII in thermoequilibrium, so that (3.21a) for ko > 0, i.e., the probability of absorbing a particle is greater than that of emission. Using the relation

following from (3.12b) we find for ko
(3.2Oc)

it+(k, -Ikol) = 21kolW.(-k) ,

(3.20d)

so that i(t-(k) - t+(k» = 2A(k) < 0 .

(3.21b)

Therefore, we can write A(k) = koy(k) ,

where y(k) is always positive for systems near thermoequilibrium.

(3.22)

716 36

KlIlItIg·chao Chou et al., Equilibrium tI1Id lIOIItquilibrium formalisms made uni/ild

3.2.2. Dispersive part The dispersive part D(k) can be written as (3.23)

where Ilm 2(k) comes from loop correction. Assuming -y(k) and Ilm 2(k) to be small, we find the pole of the retarded Green function located at ko = w(k)- !iZ.. -y(k, w(k» ,

(3.24)

where w(k) = w (k)(1+ Ilm2(k, wo(k») o 2wa(k)

(3.25)

with wo(k) = ±Yk2 + m2,

Z -.. I --

aDI 2

ako

(3.26)

• "O=OI(t)

Here Z;I is the wavefunction renormalization which is close to 1 provided 11m2 is small. However, it can drastically deviate from 1, even become negative for Coulomb field cp(x) in a plasma. As seen from (3.24) the pole is located in the lower half-plane for -y(k) > 0, so that the quasiparticle decays with -y-I as its life time.

3.2.3. Detailed balance The causal Green function OF can be written as (3.27)

where

a =~1 + B(k )/IA(k)1) = W.(k )/[ W.(k) - We(k») ,

(3.28a)

b=~-I+B(k)/IA(k)l)= We(k)/[W.(k)- We(k»).

(3.28b)

For a system in eqUilibrium (3.29)

so that

b = n'b(k) =

1 (QI,.) l' exp "''''0 -

It can be shown also that (3.30) still holds for the case ko < 0 if ko is replaced by

(3.30)

Ikol·

717 KlUUIg·cltao CItou el aI., Equilibrium and lIOMquilibrium formalisms made unijie4

37

Therefore, the detailed balance condition a(k)/b(k) = W.(k)/We(k) =(1 + n(k)]ln(k)

(3.31)

is fulfilled for equilibrium systems. Also, it follows from (3.20), (3.12) and (3.29) that (3.32)

y(k) = W.(k)[I- exp(-J3\ko\)].

One more remark. Green's functions obtained in this section reduce to those for free field discussed in section 2.1, provided 8m 2 and yare ignored. 3.2.4. Complex field Up to now we have discussed mainly the Hermitian field in this section. A complex boson or fermion field with conserved particle number can be treated in a similar way. The Lagrangian is written as

where P(x), J(x) are anti-commuting c numbers for the fermion case, Eo a constant. The connected two-point Green function is defined as Gp(x, y) = - 82 W[P, J]/8P(x)8J(y)

= -i(Tr[Tp{r/I(x)r//(Y»p] - r/I.(x)r/I!(y» , where ",.(x) = 8 W/8P(x),

",!(x) = ±8 W/8J(x).

(3.33)

Here the functional derivative is acting from the left. Introducing the vertex generating functional

we have (3.34)

The two-point vertex function is given by rp(x, y) = 82 r[r/I:, r/I.]l8r/1:(x)8r/1.(Y) =

(iiJ/iJt+ V2/2m -

Eo)8~+1(x -

y)- tp(x, y),

where the self-energy part tp(x, y) is defined as

(3.35)

718 38

Kuang-chao Chou el aI.• Equilibrium and nonequilibrium formalisms made unified

with j(x) =8 V/8,!/(x),

/(x) =8 V/81/1(x).

(3.36)

Expressing t in the form of (3.13b) we have 1 D(k, X) = ko - 2m 12 - Eo - AE(k, X),

i

2i

A(k, X) = (t_(k, X) - t+(k, X» ,

i

B(k, X) = 2(t-(k, X) +t+(k, X» = 2(tp(k, X) + t F(k, X» , AE(k, X) = !(t ,.{k, X) - tF(k, X»

(3.37)

in the Fourier representation of the relative coordinates.

3.2.5. Fluctuation-dissipation theorem (FDT) We should mention that in general the components of G and t are not scalars, especially when we consider the multicomponent field in the coordinate representation. For example, the causal Green function

should be considered as a matrix with subscripts ix and jy, where i, j are indices of internal degrees of freedom. Therefore, different components of G, t as well as A, B, D in (3.13b) do not commute with each other. However, they are commuting scalars for single component field in k space provided these functions do not depend on the center-of-mass coordinates. We have thus (3.38) i.e., (3.39) Using the same technique as in the case of real boson field we can show that in thermoequilibrium (3.40) where p. is the chemical potential. It then follows from (3.39) and (3.40) that the particle distribution n(k)=

1 . exp[.B(ko- ~)] + 1

Using (3.13), (3.38) and (3.40) we find

(3.41)

719 Kuang-c/uw Chou tt aI.• Equilibrium tmd lIOMquilibrium formalisms made unifitd

39

Fe = F+ + F_ = coth{!,B(ko - JL)](F- - F+)

= coth{!,B(ko -

JL)](Fr - Fa),

(3,42a)

for a boson system, and

Fe = tanh{!,B(k o - JL)](L - F+)

= tanh[!,B(ko -

(3.42b)

p. )](Fr - Fa) ,

for a fermion system. This is the well-known fluctuation-dissipation theorem (PDT) for equilibrium systems. Using (3.10), it can be rewritten for the Green functions as Ge =coth[!,B(ko - JL )](Gr - Ga),

(3,43a)

Ge = tanh[!,B(ko - JL )]( Gr - Ga)

(3.43b)

,

for Bose and Fermi cases, respectively. 3.3. Transport equation

We show in this section that the usual transport equation for the quasiparticles can be derived from the Dyson equation for the quasiuniform system (sections 3.3.1 and 3.3.2). As illustrative examples we derive the equation of weak turbulence in a plasma (section 3.3.3) and a generalized Leonard-Balescu equation for charge carriers (section 3.3,4). 3.3.1. Quasiparticle approximation

According to (3.10), (3.11) and (3.13), the correlation function Ge can be presented as

Ge(k, X) == Gp+ Gf: = -r;l(Fp+ Ff:)F;l = -2iF;1(k, X)B(k, X)F;l(k, X),

(3.44)

which can be rewritten as (3.45)

where the matrix N satisfies the equation ND - DN - i(NA + AN) = -2iB .

(3.46)

The energy spectrum of the quasiparticle is determined by putting zero for F" i.e., from the equation D(k,X)=O,

if the dissipation can be neglected. As in the equilibrium case, we assume

(3.47)

720 Kuang·chao Chou el al., Eqwlibrium and nonequilibrium formalisms made unified

40

NlpolC = 1± 2n,

(3.48)

then (3.46) can be rewritten as

nD- Dn - i(nA + An)= +i(B - A)= ±t+.

(3.49a)

or (3.49b)

3.3.2. Quasic/assicai approximation As shown in the preceding sections, t+, t_ are proportional to the probability of emitting and absorbing quasiparticles, respectively, so the right-hand side of (3.49b) is related to the collision tew. whereas the left-hand side to the drift term of the transport equation. It is worthwhile to note that we cannot entirely ignore the noncommutativity of nand D on the left-hand side. To the leading order of gradients (Dn - nD)(k, X) =

I

exp[ik . (x - z)][D(x, y)n(y, z) - n(x, y)D(y, z)] dy d(x - z)

= i{oD(k, X) an(k, X) on(k, X) oD(k, X)} ok,. oX,. Dk,. ~,. ,

(3.50)

where X = !ex + z).

To convince oneself of the validity of (3.50), one can either consider the Poisson bracket as the quasiclassical approximation to the commutator or check it by a straightforward calculation. As we mentioned before, the microscopic scale x - y is a fast variable with respect to which the Fourier transformation is carried out, whereas the center of mass coordinate is a slow variable. Expanding D, n in (3.50) as

F(k, (x + y)/2) = F(k, !ex + z» +!(y - z )oF/ oX + ... with

F=D,n, and integrating over k' by parts, one can easily verify eq. (3.50). Such separation of micro- and macro-time scales and the replacement of the commutator by its quasiclassical counterpart - the Poisson bracket, wiD be used frequently in our future discussion. Assume that the solution of (3.47) is given by

ko=w(k,X),

721 KUIlIIg-CMO

Chou tl aI., Equilibrium and nonequilibrium formalisms made unified

41

then we have aDl -k

a

0

ko~ .. (t, X)

V"w(k,X)+VkD(k,X)=O,

aD 1 aw(k, X) aD(k, X) + O. ako ko ...(t.X) ax,.. ax,..

-

(3.51 )

Using eqs. (3.20), (3.50) and (3.51) we have for ko = w(k, X), an(k, X) + V' Vn, (k X) +-.:'-..:-~ aw(k, X) -~--.!.. an(k, X) at ax,.. ak,.. 2w(k, X) 'I k) {W.(k,X)(I±n(k,X))- W.(k,X)n(k,X)}, (aDI a 0 "o-.. (i, Xl

(3.52)

where v = Vtw(k, X)

(3.53)

is the group velocity. If the renormalization of the wavefunction is neglected, then aD/ak~=

I,

and (3.52) can be simplified as an aw an -+v,Vn+--= W:(I±n)- W:n at aX/A akp.· 8 ,

(3.54)

which is the transport equation for the quasiparticle distribution in the phase space. The last term on the left-hand side of (3.54) comes from the force aw(k, X)/ax,.. due to the variation of the energy w(k, X) with the coordinate. If the space-time dependence of D is unimportant, this term can be neglected and the standard transport equation is recovered.

3.3.3. Plasma equation As an illustration consider the transport equation for plasma. Let I/Ii(x),i = 1,2, ... , be the charged fermion fields (electron and ion). The equation of motion for the Coulomb field is given by (3.55) To the one-loop approximation of free fermions we find the two-point vertex function to be r,,(x, y) = -V2 .s;(x - y)- i ~ eJs,,/(x, y)Spj(y, x),

,

(3.56)

where SpJ(x, y) is the propagator of the 1/1, field (3.57)

722 42

Kuang-chao Chou et al., Equilibrium and nOllelJuilibrium formalisms matk unified

In the Fourier representation of the relative coordinates the retarded vertex function for the Coulomb field can be determined as

.+

2

rr.,(k) = k

= 12 -

I ~ el I (::)4 4

i ~ e:

i

d1 (211"f TrG<1+ u3)Sj(/)U3Sj(/- k)} (nj (I) - nj (/- k »)r;/(l)r;/(/- k) ,

(3.58)

where (3.59) and nj(/) the fermion distribution. Integration of (3.58) over 10 yields (3.60) where (3.61) is the susceptibility. It then follows from the expression

rr., = D., + iA., , that

D., =12 Re(e(k» ,

(3.62) (3.63)

where (3.64)

Equation (3.63) is the well-known Landau formula of dissipation for plasmas in thermoequilibrium. If nj(/) > nj(/- k) for 12> (1- kY, then A., <0, which means the pole of the retarded Green function moves into the upper half-plane of ko and an instability of the plasma occurs. However, near thermoequilibrium we always have A., > 0, so that the plasma oscillation decays in time. Using the expressions for the free fermion propagator (2.21) we can also obtain from (3.56) that

723 43

Kuang·chao CIIou el al., Equilibrium tmd nonequilibrium formalisms made unified

iL,,(k) = -211" ~ e; iF+,,(k) = -211"

~ e;

d31

J(211")3

f

a(k, 1)(1- nJ(I»nj(l- k),

d3 1

(3.65)

(211")3 a(k, l)nj(l)(I- nj(l- k».

J

The Boltzmann equation for the plasmon distribution N(k, X) can then be derived from (3.52) and (3.65). The plasmon energy is determined from Re e(ko, k, X) = 0 , whereas the wavefunction renormalization

aD = k2 a Re(e(k, X» ako

ako

cannot be set equal to 1 in this case. Therefore, the resulting transport equation for the plasmon is

f

3

aN 1 d1 -+v· +aw-aN -211" e 2 --a I I-n l-k at VN ax,.. ak'" - k2(a Re e(k, x)/akoho-.. (Ic) ~, (211")3 (,kin )[ :/()( /( »

x (1 + N(k, X»- (1- n,(l»nj(l- k )N(k, X)Jlko-.. (Ic)

(3.66)

which describes the weak turbulence of the plasma. In eq. (3.66) we take into account only the absorption and emission of plasmons by charge carriers. If the high order self-interaction of the Coulomb field is included, we will have in addition the plasmon-plasmon collision term (wave-wave interaction).

3.3.4. Equation for charge carriers Now we discuss the transport equation for the charge carriers. First we find from (3.11) Green's function

(3.67) We must separate from (3.67) the contribution of plasmons which can be written as . N(k.X) IG+(k)=21Ta(ko-w(k»k 2 R I k . a eeiJ o

To the one-loop approximation the two-point vertex function is given by

(3.68)

724 44

KUIlIlg-clulo Chou el aI.• EquiUbrium and lURll:quilibrium lonna/isms made unified

(3.69) where SpJ(x, y) is the fermion propagator (3.57), while Gp(x, y) is the plasmon propagator. After separating the contribution of the plasmon pole we find

(3.70) where

8(k, I, w) == 8(ko + w(l) - _1_ (1+ k)2) , 2m, 8(k, I, I') == 8(ko+ _1_[1 12 _ (I' -IY] - _1 (I + kY), 2mI' 2m,

r

p and [ means non-pole contribution. Taking into account the relation between r± and We .• , we can readily write down the transport equation for the charge carriers. The first term in (3.70) describes the interaction of the charge carriers with the plasmon, whereas the second term describes the mutual interaction of the charged particles via the screened Coulomb field. If only the second term is retained, the usual Leonard-Balescu equation is recovered. The transport equation for plasmas has been also discussed by DuBois et al. [33, 34] using the CTPGF technique.

3.4. Multi-time-scale perturbation As repeatedly emphasized before, we must distinguish the microscopic (relative) and macroscopic (center-of-mass) space-time scales. However, in the mean field approximation IPc(x) depends only on a single variable x which contains both micro- and macro-scales. To distinguish these two types of change we use the multi-time-scale perturbation theory. The mean field IPc(x) satisfies the following equation:

8r[ CPc]/8cpc(x) = 0 , an expansion of which in powers of CPc(x) can be written as

725 Kuan,-cIuw 0I0Il et III.• Equilibrillm IlII/J IIDMquilibrium formlllisms made Jlllijied

4S

(3.71) p

where E[rpc(x)] contains high-order terms. Separating the diagonal part of rr from (3.71),

we find

I

Do(i a.. )rpc(x ) + borr(x, y)rpc(y) d4y + E[ rpc(X») = 0 .

(3.72)

Assume both borr and E[rpc(x») to be small, then (3.72) has a solution (3.73)

rpc(x) = 'Pc exp(-ik . x),

where k satisfies the dispersion relation (3.74)

Do(k)=O,

from which we find ko = w(k) = a real number.

Now consider the influence of borr. If its imaginary part is less than zero and, in addition, = aDolak~ > 0 provided (3.74) is satisfied, then 'Pc(x) will grow in time to form a laser-type state with its amplitude being limited by nonlinear term E[rp.(x»). Near the critical point when such instability occurs, borr is a small quantity and 'Pc changes with time rather slowly. Assume that the approximate solution of (3.72) can be presented as Z;1

where E is a small parameter which should be set equal to 1 by the end of the calculation, EX describes the slowly varying part. Set x= EX, assume both borr and E to be of order E, the differentiation with respect to x can be written as iJ.. + eiJ~ so that (3.72) becomes

So far as x is a slow variable, we neglect the difference of x and leading orders we have

y in the last two terms. To the first two

Do(iiJ.. )'P~O)(x, i) = 0,

f

Do(iiJ.. )rp~l)(x, i) + iDo/&(iiJ.. )iJ~'P~O)(x, x) + borr(x, Y)'P~)(y, i) d4 y + E['P~)(x, i)] = 0,

(3.75)

726 46

KlUlIIg-chIW Chou el al., Equilibrium and nonequilibrium formalisms made unified

where

Do .. (k) = aDo(k)/ ak" .

(3.76)

As seen from (3.75), the solution is given by "c(x, x) = "k(X) exp(-ik . x),

(3.77)

where k is determined from (3.74). If we require that 1P~1) does not contain a term proportional to lP~o), then the second equation of (3.75) after Fourier transformation becomes (3.78) where we have also replaced the center-of-mass coordinates !(x + y) in rr(x, y) by x. This is an equation satisfied by the oscillating mode of the mean field. We have used this technique to discuss the laser system coupled with two-energy-level electrons [40,41]. We will not reproduce the calculation here, but it should be mentioned that a stable laser state allowed in the classical theory, is unstable in the quantum case. In the quantum theory we must consider the fluctuation of the photon number. Since the laser system is described by a coherent state with fixed phase, the fluctuation of the photon number diverges. This divergence can be removed by a renormalization procedure which leads to the decay of the laser state. Similarly, the soliton solution of IPc(x) is also unstable due to the quantum fluctuation. It is worthwhile to note that such multi-time-scale perturbation technique is quite useful. In fact, we have already made use of its basic idea in deriving the transport equation in the last section. It is also the key point in obtaining the TOOL equation which we are going to discuss now.

3.5. Time dependent Ginzburg-Landau equation The concept of macrovariableness is very useful in critical dynamics, hydrodynamics, and many other fields [61]. Usually, the set of macrovariables includes both order parameters and conserved quantities. As a rule, their microscopic counterparts are composite operators. In this section we use the equation for the vertex functional (2.48) to derive the TOOL equation [61] for their expectation value. As seen from the later discussion, the term TOOL equation is used here in a much more general sense. Let 0, (x ), i = 1,2, ... , be the set of composite operators corresponding to macrovariables. Without loss of generality, we assume them to be Hermitian Bose operators. The order parameter Oc(x) is determined from the equation for the vertex generating functional (3.79a) Suppose, Oc,(x, T) is known for the moment T. At the time t following T, the left-hand side of (3.78) can be expanded as (3.79b)

727 KlUUIg-chao C7Iou el al.• Equilibrium and nOMquilibrium formalisms made unified

47

which is true for t located either on the positive or negative time branches. So far as 0 varies slowly with time, we can write

Substituting this expression back into (3.79) and taking into account that in the limit I == Ix -+ '1",

(3.80) where fr/i(x,y, ko, '1") is the Fourier transform of frij(x, I",y, Iy) with respect to I" - ty, taken at T = ~(I" + Iy) "" '1", we obtain in the matrix form aO(I) 8f '}'(/)-=-

al

I

80c+

(3.81)

+J(I),

0 •• =0.--0

where we change T for t. For the moment let

I

8f I, (x, t)==--

80c,,"

,

0 .. -0.--0

and calculate the functional derivative of h with respect to O(x, t) as a function of three-dimensional argument M(x, t) 80j (y, t)

I

d d { Z .,Tz

=fFjj(x, y, ko = 0, t) 8lj (y, t)

8Q( I X, t)

Z

2

8f 80k +(z) 8f 80d Z)} 801+(x)8Qk+(Z) 80j (y) 801+(x)80k -(z) 80j {J) f+lj(x, y, ko = 0, I),

fFj,(Y,x,ko=O,I)-f+j,(y,x,ko=O,t)

= fF/i(x, y, -k o = 0, I) -

L,j(x, y, -ko = 0, I),

where in the last step we have used the symmetry relation (3.12b). The difference ~) 8I,Cy, I) . l:>0(y ) l:>Q( )= hm (f-,j(x,y,-k o, I)-f+jj(x,y, ko, I» o 1,1

u,X,1

k(t+O

vanishes due to the relation

(3.82)

728 48

Kuang·chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

L

=

L exp(-pko),

(3.83)

following from (3.29) for a system in equilib,ium. Therefore, a free energy functional such that -'O~/'OOj (x,

Ij(x, I) =

~[O(x,

I)] exists (3.84)

I).

Equation (3.81) can then be rewritten as aO(/) 'O~ y(/)-= --+J(/). al '00(/)

(3.85)

If the macrovariables OCt) do not change with time in the external field J, then 'O~/80 =

(3.86)

J.

Hence ~ is actually the free energy of the system and (3.86) is the Ginzburg-Landau equation to determine the stationary distribution of macrovariables. For non equilibrium systems the potential condition, i.e., the vanishing of (3.84) can be realized if lim A (x, y, ko, t) = 0 ,

(3.87)

ko-O

where A is the absorptive part of Of' In the next section we will show that (3.87) is fulfilled for non equilibrium stationary state (NESS) obeying time reversal symmetry. It is usual to multiply eq. (3.85) by y-l(t) to obtain aO(/)

--=

at

}

y-l(t) {'O~ ---+J(t) . 'OO(t)

(3.88)

This is the generalized TDGL equation we would like to derive. If a random source term is added to the right-hand side of (3.88), it will appear like a Langevin equation. However, there is an important difference. Equation (3.88) includes the renormalization effects. Also, the way of describing the fluctuations in CTPGF formalism is very special as we will see in section 6.

4. Time reversal symmetry and nonequilibrlum stationary state (NESS)

It is well known that the principle of local equilibrium and the On sager reciprocity relations are the two underlying principles on which the thermodynamics of irreversible processes is constructed [62]. This is true near thermoequilibrium. Within the framework of statistical mechanics a successful theory of linear response has been developed by Kubo and others [63-65]. The two fluctuation-
729 KlIIIIIg-chtw Owu et aI., Equilibrillm and lIOMJullibrium formalisms made llllijied

49

mobility, follow immediately from the linear response theory [63] along with the Onsager'reciprocal relations. Many attempts have been made to generalize this theory to far from equilibrium systems such as hydrodynamics, laser, chemical reactions and so on (66-731. Recently there has been great interest to study the fluctuation effects in NESS in connection with the light scattering experiments in fluids with temperature gradient [74,75]. However, in most of these treatments a phenomenological approach based on the Fokker-Planck equation or Langevin equation, at most a mesoscopic (semiphenomenological) method using the master equation or the transport equation, were adopted. In particular, the existence condition for a free energy type generalized potential in NESS were discussed in [69-73] using the Fokker-Planck equation. In this section we will explore the consequences of the microscopic time reversal symmetry for NESS, derive the potential condition along with nonequilibrium FDT and Onsager relations, and also decompose the inverse relaxation matrix into symmetrical and antisymmetrical parts within the framework of CfPGF approach [45].

4.1. Time inversion and stationarity The time reversal symmetry is well known and is discussed in detail in text books [76]. Here we recall some basic points to specify our notations.

4.1.1. Time inversion in the Schriidinger picture Suppose the system is conservative, being described by the Hamiltonian ,w[J] = ,w-JQ,

(4.1)

where Q is a multi-component real boson field, either order parameter or conserved quantity, and J the corresponding external source. In general, J is time dependent, but here it is assumed to be constant in time. The wavefunction at moment 1 is given by

(4.2) where SJ (I, to) == exp{-i,w[J](1 - lo)} ,

(4.3)

and I/Io(A)'" 1/1'0(10, 1, A) with A as parameters specifying the initial state. Under time inversion, ~ ~ e,J, ,

A.. ~ e..A.. ,

i = 1, 2 ... n , a = 1,2· •. g ,

where e" ell are ±1 depending on the signature of the quantity considered under time inversion. It is well known that the time inversion in quantum mechanics is implemented by an antiunitary operator R such that

730 50

Kuallg·chao Chou er ai., Equilibrium and nonequi/ibrium formalisms made unified

.rt'[J]---+ R.rt'[J]Rt = .rt'[ eJ] ,

(4.4) (4.5)

If

(4.6) the state is considered to be time reversal invariant, i.e.,

RI/I'o(/, J, A) = 1/1,.(-1, e1, eA).

(4.7)

Analogously, for a statistical ensemble the density matrix

(4.8) transforms as (4.9) It is time reversal invariant if

p'. = po(eA) .

(4.10)

4.1.2. Time inversion in Heisenberg picture Now we turn to the Heisenberg picture. Suppose it coincides with the Schrodinger picture at t = to, then we have (4.11) The density matrix does not change with time in the Heisenberg picture. Set t = to in (4.9) and use (4.10), we find (4.12) The expectation value

O,.(t,1, A) "" Tr{p,. Of.(t)} = eO,.(-t; e1, eA)

(4.13)

by virtue of (4.11), (4.12) and antiunitarity of R. It is important to note that we need the time invariance for both dynamical variable and initial state to get the invariance for the expectation value. The external source is introduced here for mathematical treatment. In the final answer we usually set I = 0, so the stationary state is described by Po(A) which does not depend on I. However, in the process of calculation using the generating functional the dynamics are detennined by ~[l]. To ensure the time translational invariance we have to set to = - 00 and

731 Kuallg-chQlJ C710u et aI., Equilibrium and IIOIIttJIIIlibrium formalisms made uni~d

51

switch on the external source adiabatically. In fact, there are two implicit assumptions. First, the limit (4.14)

lim exp(-i1e[J]('T - to»po(A) exp(i1e[J]('T - to» '0-+- 00

exists in NESS, moreover, it does not depend on with R. It follows then,

'T.

Second, the limiting procedure to-+ -00 commutes

Rp(A, J)Rt =p(eA, eJ) .

(4.15)

4.1.3. Implications for Green's functions The correlation function of Heisenberg operators defined as F 12 .. .,(12, . ·1, J, A):= lim Tr{po(A)Q{'o(1)··· Qf.o(/)} ,

...... -'"

(4.16)

transforms as (4.17) under time inversion as follows from (4. 10}-(4. 12), (4.14) and (4.15). Note that the order of arguments on the RHS of (4.17) is reversed due to the antiunitarity of R. Here 1 stands for t10 etc. It is ready to check that Green's functions for NESS are time translation ally invariant and transform under time reversal as Q,(J) = e,Q,(eJ) ,

(4.18a) (4.18b) (4.1Sc)

in accord with (4.17). Hereafter we drop the parameters for the initial state A for simplicity. We can solve for J from (4.18a) to obtain 1,(Q) = e,J,(eQ).

(4.19)

Using the equation for the vertex functional (3.78) and taking functional derivative of (4.19) in accord with the formula

I)

"Q (E[Q(t)))IOtC'>-Ot = u

,

[f dt,I):~~~~)] ,

Ot~Ot

,

we find that (4.20)

732 52

KlUUIg·chao OIou el al., Equilibrium and noMquilibrium fonna/isms made unified

Also, it follows from the Dyson equation (3.10) and the relation (4.18) that (4.21) Equations (4.18)-(4.21) are the implications of the time reversal symmetry needed for our further discussion.

4.2. Potential condition and generalized FDT 4.2.1. Potential condition We have shown in section 3.5, that the potential condition can be satisfied if the zero frequency limit of the absorptive part A(k, X) vanishes as given by (3.87). Taking the Fourier transform of (4.21) we obtain

Comparing its zero frequency limit with (4.20) we find rrij(k o =0, 0) =r./j(ko =0,0),

(4.22)

which will yield the potential condition (3.87) if combined with the definition of D and A in (3.14). Thus we have shown that one can construct a free-energy-like generalized potential for NESS satisfying the microscopic time reversal symmetry, i.e., one can construct functionals ~, C9 such that

-u" ar[o] I ao(x,,)

=

0.-0--0

_!~ aW[J]1 2

"8J(xa )

a~[O(x)]

(4.23a)

ao(x) ,

_ aC9[J(x)] 1+-1_=1 -

(4.23b)

aJ(x) .

If the macrovariables Oi(l) vary slowly with time as discussed in section 3.5, we should assume the time reversal symmetry in the "local" sense, i.e., the system is invariant for a macroscopically short and microscopically long time scale so that

AiJ(ko = 0, I, 0) =

°

(4.24)

at each moment of t.

4.2.2. Generalized FDT Now we discuss the nonequilibrium FDT. First we split the relaxation matrix defined by (3.81) into symmetrical1'(~J) and antisymmetrical Y[~i1 parts as -!i

-aij=Y(i.i)=2\Yi/+Y/i)=-

iJA/j(w, I, iJ

W

0)1 .. -0

'

(4.25a)

733 53 •

_

ldu =

1

•

Y II. 11 = ~ylj - YII) = 1

aDIj(w, t, aW

Q)\ .. -0

•

(4.25b)

As follows from (3.10) and (3.45), the correlation vertex function f. can be expressed in terms of function N (cf. (3.46» as felj(w, Q) = frik(w, Q)N,'I(W, Q)- Nik(w, Q)fakl(w, Q) = DilcNlci - NikDkJ + i(AikNlcl + N,,,A kJ ) .

(4.26)

In thermal equilibrium NII(w) =N(w)811 = coth(w/2T)8ij = (2T/w)81j,

as follows from (3.38). Hence the FDT (3.42a) can be rewritten as _'r

(

_

0 Q)-I' 2AIj(w, Q)T - 1m ,.,~ w

IJ ell W - ,

4'r' Jail

(4.27)

in the low frequency limit, where aij is the symmetrical part of the relaxation matrix defined by (4.23). If we assume the zero frequency limit of N,I(W, Q) to be finite then we find from (4.26) that i ~ felj (w

= 0, Q) =0

which contradicts the positive definiteness of the quantum fluctuation. It is therefore more natural to assume that lim NII{W, Q) ~ (2/w)Teft811 ,

(4.28)

01-+0

where Teft is the effective temperature. Using once again the potential condition (4.24) we find (4.29) This is the low frequency limit of the FDT for NESS. Substituting (4.29) into (3.10) and carrying out the inverse Fourier transformation, we obtain the FDT for Green's function as iaGJat = 2reft (Gr - Ga)

(4.30)

which has the same form as that used in critical dynamics [77].

4.3. Generalized Onsager reciprocity relations 4.3.1. Kinetic matrix As seen from eq. (3.85) the matrix 'Y1j(t) describes the relaxation of the order parameter. Using the

734 54

KUQIIg·c!uw Chou el al., Equilibrium and nonequilibrium fonnalisms made unified

definition (3.81), the symmetry relation under the time reversal (4.21) and the basic relation (4.31) following from the definition (cf. (3.12b», we can easily find the symmetry relation for Y'j under time inversion (4.32)

4.3.2. Diffusion matrix The standard Langevin equation is written as [61] (4.33) where Y,(Q) is the mode coupling term. The random source (~,)

= 0,

~,

obeys a Gaussian distribution such that

(Mt)Mt'» = 2ul~(t - t') ,

(4.34)

where U,} is the diffusion matrix, appearing in the Fokker-Planck equation. We will show later (section 6.4) that within the CfPGF formalism UiJ can be expressed as (cf. (6.105» (4.35) where T means transposition. Using (3.10), (4.18) and (4.22) we can find (4.36) Substitution of (4.32) and (4.36) into (4.35) will yield (4.37)

U,}(Q) = EIE,UiJ(EQ) ,

provided Yj, is symmetrical. If not, (4.37) can then be easily derived using the PDT (4.29) and the symmetrized YI} as defined by (4.25a). Equation (4.37) is the symmetry relation for the diffusion matrix first obtained by Van Kampen et al. [67-70].

4.3.3. Response matrix Now we turn to the response of the system in NESS to an external disturbance. Consider the density of conserved quantities Q .. (x, t), ex = 1,2, ... , and the corresponding sources i .. (x, t). The rest of the external sources] are stationary in time. The NESS is described by the Hamiltonian (4.1) and the qensity matrix p(A,]) (4.15) invariant under time inversion. The coupling of Q .. (x, t) to the source ] .. (x, t) is treated perturbatively. Let i' (r, t) be the current density satisfying the Heisenberg equation of motion aQ.. (x, t)

at

aj~(x,

+ ax'

t) 0,

(4.38)

735 KU/UIg-chao Owu et al., Equilibrium and lUJMquilibrium lormalums made llllijied

55

where i = 1,2, 3. Using the generating functional (2.32) it is straightforward to find (4.39) to the linear order in

i .. near NESS. The retarded Green function for the current density is defined as (4.40)

whereas

i.. (x) in the interaction Lagrangian varies slowly with coordinates so that the external force

-vi.. (x, t)= X.. =const.

(4.41)

It is then ready to derive from (4.38)-(4.40) by use of the Lehmann representation discussed in section 2.5, that (4.42) where the response matrix

f£~{j

is defined as

f£~{j(J,A)=-i aG~"{j(w'P)1 aw

,-0

.

(4.43)

.. -0

The current operator j~(x) in the SchrOdinger picture transforms under time inversion as j~(x)~ Rj~(x)Rt

= -e.,j~(x) .

(4.44)

So far as j~(x) is similar to Q" whereas the definitions of G~..{j (4.40) and f£~fJ are analogous to their counterparts for the order parameter Q, we can readily find that !t~(J, A) = e.. e~g.,(eJ, eA).

(4.45)

Equations (4.42), (4.43) and (4.45) are the generalization of the Onsager theorem to the NESS case, whereas in the literature eqs. (4.32), (4.37) and (4.45) are known as generalized Onsager relations.

4.4. Symmetry decomposition of the inverse relaxation matrix In this section we will decompose the inverse relaxation matrix 1-\ first introduced in deriving the TDGL equation (3.88), into symmetric and antisymmetric parts and find their explicit expressions, i.e., to complete the derivation of the generalized TDGL equation with both dissipative and mode coupling terms.

4.4.1. Symmetric part Using eq. (4.35) the PDT for NESS (4.29) can be cast into another equivalent form. In fact,

736 56

KUOIIg-chao C1wu el al., Equilibrium and nOMquilibrium lonnalisms made unified

substituting (4.29) into (4.35) yields O'li(Q) = 2Yiklakl(Q}y~-lTeff

=Teff( y-I + yT-l)j),

(4.46)

i.e., the symmetric part of y-l is nothing but the diffusion matrix divided by twice the effective temperature. Equation (4.46) is another form of FDT in close analogy with the Einstein relation.

4.4.2. Antisymmetric part The antisymmetric part of y'j is more complicated. According to the discussion in section 2.5, the Lehmann representation can be written as G (k) = rl)

0

I dk~

CIj(kb) 2' k' k . , 17'1 0- 0 -IE

(4.47)

where (4.48) Integrating (4.48) over ko gives the expectation value of the equal time commutator (4.49) We then obtain from (4.47) and (4.49) that lim koGriJ(ko, Q) = if,J(Q) ,

(4.50)

kcr+'"

i.e., rrlj(ka) increases linearly with ko as k o-+ 00. We can thus write down a subtracted dispersion relation as (4.51)

Differentiating (4.51) with respect to ko and setting ko-+O, we find that (4.52)

where

arrl

aka

(4.53)

=d+ia ko-O

'

737 XlUIIIg-chao OIou el al., Equilibrium and rumequilibrium formalisms mtUk IlIIified

57

(4.54)

In deriving (4.52) we have made use of the consequence of the potential condition lim A/k (k o) = kOa'k (k o) .

(4.55)

11:11""0

Since a,k (k o) is a symmetric matrix, ilk is finite. Solving (d- 1)1I from (4.52) we obtain (4.56) The antisymmetric part of y- 1 can thus be written as -1 Y[i./1

= - (a -)'d)-1 [~/l "" -)'dII 1+ O(a2) = -i{(I + i/J)-lif}1I + O(a 2) = /11

+ O(a 2 , ;:1) .

(4.57)

Usually /11 itself is considered as the antisymmetric part of y-l. However, as seen from our discussion, this approximate result is true if both high order effects of dissipation a2 and the dispersion i are neglected.

4.4.3. Generalized TDGL equation Substituting (4.46) and (4.57) into the TDGL equation (3.88) and setting J =0, we obtain aO/(t)

-1

&[1

-1

&[1

---at = -Y(I.J) &0, - YO.'1 80, 1

=-

&~

&[1

2TcfI 0'11 &0, - III &0, '

(4.58)

where the first term is associated with the irreversible dissipation, whereas the second term is the reversible part due to canonical motion. In many practical problems Q/ are either conserved quantities or a linear representation of some Lie group. In both cases (4.59) where It are either structure constants of the symmetry group or elements of the representation matrix. The TDGL equation (4.58) derived in the CfPGF formalism has the same form as that used in critical

738 58

Kuang-chlW Chou et al., Equilibrium and nonequilibrium lorma/isms made unified

dynamics and other related fields [61,68]. The second term in (4.58) is usually called the mode-coupling term. Previously, we derived the explicit form of (4.58) by comparing the TDGL equation with the Ward-Takahashi identities. Nonetheless, the present derivation is more straightforward. In closing this section we note that as consequences of microscopic reversibility near NESS, many concepts valid for systems in thermoequilibrium can be generalized to these cases. The potential condition, the FDT as well as the reciprocal relations for diffusion and relaxation matrices which constitute the basis for a semiphenomenological consideration of non equilibrium processes, can be justified within the CI'PGF formalism. 5. Theory of nonlinear response As we mentioned in the last section, the linear response theory near thermoequilibrium [63-65], centered on the FDT and the Onsager reciprocity relations, belongs to one of the most successful chapters of nonequilibrium statistical mechanics. For the last twenty years it has been generalized in two directions, namely, to linear response near NESS as we discussed in the last section and to nonlinear response near thermoequilibrium. In spite of few formal developments [78-82], the latter issue has not become an active field of research. In fact there are some reasons for such slow-footed advance in nonlinear response theory. First, the nonlinear response depends not only on the intrinsic properties of the system under consideration, but also on the boundary conditions to remove the heat generated in the nonlinear processes. Second, except for nonlinear optics, there was no urgent need for nonlinear response theory from the experimental point of view. Third, the formulation of the nonlinear theory became tedious due to lack of appropriate framework. However, things are changing. Progress in pico-, even femtosecond pulse technique and multichannel data acquisition and processing make the detection of multi-time response available. Such measurements will certainly yield much more detailed information on the intrinsic properties of the system provided the nonlinear effects are essential. On the other hand, the development of the CI'PGF formalism has furnished a suitable theoretical framework for such nonlinear analysis. This problem has been studied by Hao et aI. [50,51]. Although the discussion for the time being is still rather formal and is restricted to the case of "mechanical" disturbance, i.e., expressible by an additional term in the Hamiltonian, it will serve as a good starting point for further development. As we will summarize in this section, many relations which in principle can be obtained by other, more sophisticated, techniques, appear rather simple and natural in the CI'PGF approach. The general expressions for nonlinear response are presented in section 5.1, whereas different relations among these functions including algebraic, KMS [63,83], time-reversal and spectral, are discussed in section 5.2. A plausible generalization of FDT in nonlinear case is sketched in section 5.3. In this section we do not write out explicitly the factor (_i)"-1 in the definition of Green's functions. 5.1. General expressions for nonlinear response 5.1.1. Model As mentioned in section 2, the CfPGF generating functional can include the physical field lex) = lc(x) by setting J,,(x) = 0, i.e. l+(x) = J-(x). Therefore, the high order response functions are contained in the expansions of the generating functional (2.76) and (2.78). Still, for convenience we will write down here the explicit expressions for nonlinear response.

739 59

Assume the system was in equilibrium state described by the Hamiltonian 'lto and the density matrix po = Z-1 exp(-f3'lto),

Z = Tr(exp(-f3'lto»

(5.1)

in the remote past to = -00 and then a time-dependent external field l(t) has been switched on adiabatically to derive the system out from equilibrium. The field l(t) is coupled to the dynamical variable Q which might be a composite operator, so the total Hamiltonian becomes

(5.2)

'It = 'lto - l(t)Q.

As before, we use in this section the abbreviated notation

, f dx.J;(x, t)Q,(x) ,

1Q == L

(5.3)

sometimes making explicit only the time variable. Here we consider the linear coupling, but, obviously, other cases such as (It· Ef, (II' Hf coupling in the liquid crystal or E,E}PI/ (P,} being the polarization tensor) coupling in second-order light scattering, can be treated as well. We should mention that the subscript "0" for 'lto and Po does not mean free of interaction between particles of the system. In this section we consider the perturbation expansion in powers of the external field, whereas all interaction effects within the system are included in all· Green functions appearing later like .d 2h .d 21l , etc. According to the very spirit of statistical mechanics, an average should be taken over the initial distribution, whereas the evolution of the system is described by the dynamical equation. If we know the various average values (Q,(t}), (Q,(1}Q}(2}) and so on, we would have more and more detailed information of the nonequilibrium properties for the system. So far about the classical system. There is an additional problem for the quantum case. Not every product of operators here corresponds to a physical observable. The question of operator ordering occurs. According to Dirac [84], to present an observable (i) the product must be a real operator (ii) which has a complete set of eigenstates and (iii) satisfies certain supplementary physical conditions. We will consider Hermitian operators and use the Hermicity as a guide line for operator product, but in general it is hard to say anything about the completeness. For example, the combinations AB + BA and i(AB - BA) are both Hermitian, while their expectation values are correspondingly Gc :: G22 and G 21 - G12 = Gr - Ga. Since the problem of operator ordering in the quanto-classical correspondence has not yet been solved, an assumption is made in [50,51] that the quantum counterpart of the average for the product of dynamical variables is just Green's function G with all subscripts equal to 2. In the n =2 case, G22 is the fully symmetrized correlation function, whereas for n > 2 cases they are only partially symmetrized averages.

5.1.2. Analytic expansion of the generating functional In this section (section 5.1) we denote Green's functions with external field by G, whereas those without field by .d. Using the analytic expansion of the generating functional W[l] ,

,.d

.., 1

W[J] =

L

11-1

n.

p

(l· .. n)J(l)·· ·l(n),

(5.4)

740 60

Kuang-chao C1wu et aI., Eqililibrilllll and ROMIlllilibrilllll formalisms mode unified

where

J (1·" n)= p

S"W[J] 81(1)·· ·SJ(n)

I

(5.5)

J-O

are connected propagators without external field. It then immediately follows that

Op(l) == (TpQ(l» = J p (l) + J p (12)1(2) + !J p(123)J(2)J(3) + .. . Op(12) == (Tp(Q(1)Q(2») = SOp«l» = J p (12) + J p (123)1(3)+ .. .

812

Op(l23) ==

(Tp(Q(1)Q(2)Q(3») = 8~g:) = J (123) + J (1234)1(4) + .... p

(5.6)

p

To transform (5.6) into physical representation we need to insert the Pauli matrix obtain

U1

[44] (cf. (2.76» to

1 0(1) = J (1) + J (12)(ud)(2) + 2! J (123)(u 1J)(2)(u1J)(3) + ...• 1 0(12) = J (12) + J (123)(ud)(3) + 2! J (1234)(u1J)(3)(u 1J)(4) + ...• £1(123) = j (123) + j (1234)(u 1J) (4) + ....

(5.7)

5.1.3. Higher order response In accord with the discussion at the beginning of this section we need only retain the "all 2" components of (5.7) and set J+ = '- to get the general expressions for nonlinear response O2(1) = J 2(1) + J 21 (12)J(2) +

2!1 J 211 (123)J(2)1(3) + ...•

1 0 22(12) = J 22(12) + J 221(123)1(3) + 2! J 2211(I234)1(3)1(4) + ...•

0 222(123) =J 222(l23) + J 2221(1234)1(4) + ....

(5.8)

The first expression is a short writing of (2.115). The first two formulas of (5.8) were obtained in [78] by explicit manipulation of integrals. In the CfPGF formalism the structure of high order terms is obvious. It is worthwhile to note that the causality is guaranteed in each step of derivation using CfPGF as emphasized in section 2.4. Terms contradicting the causality drop out in accord with (2.109)-{2.111). Following the convention set in the literature [78-82], J 2h J 211 • J 2111 should be called response functions of the averaged physical observable to external field, while their Fourier transforms are the admittance functions of various order. Accordingly. 0 22• 0 222••••• and J 22 • J 222••••• are called, respectively, nonequilibrium and equilibrium fluctuations of different order, whereas J 22h J 2211 , •••• are response functions of these fluctuations to the external field.

741 Kuang-chlUl Chou et al.• Equilibrium and nonequilibrium formalisms made unified

61

Other components of (5.7) like nonequilibrium retarded function 1 0 21 (12) =.:1 21 (12) + .:1 211 (123)1(3) + 2! 4bu(1234)1(3)1(4) + ... ,

(5.9)

which is the straightforward extension of the usual response function _ 8(0(1»\ .:1 21 (12) - 81(2) J=o'

(5.10)

to (5.11)

by including the higher order effects of the external field, may be named as nonlinear response functions. In fact, by integrating (5.9) over 1(2) we recover the first equation of (5.8). Therefore, no additional information is contained in (5.9) and in terms disappearing in going from (5.7) to (5.8) by setting 1+ = L.

5.1.4. Physically observable functions Independent functions which in principle can be measured are those listed in the following table and their higher order extensions.

Without external field Linear response Second-order response Third-order response

Average

Two-point correlation

Three-point correlation

Four-point correlation

.:12

.:122

.:1222

.:12222

.:121

.:1221

.:12221

.:122221

.:1211

.:12211

.:122211

.:1222211

.:12111

.:122111

.:1222111

.:12222111

In t~is table the function on the oblique line from lower left to upper right are components of the same .:1, so that the relations indicated in [78,80,81] look very natural. To sum up, the observables in nonlinear response theory are partially symmetrized correlation (fluctuation) functions O2 ,022 , 0 222 , • •• ,etc., as functions of the external field, in particular, their zero field derivatives 8' .:1 2 "'21"'1 = - , 0 22"'2(12 . .. k) . 7 , 81

(5.12)

The possibility of detecting them in practice depends on the nonlinearity inherent in the system, i.e., the magnitude of functions (S.12) and the strength of the external field.

742 62

Kuang·chao Cllou et aI., Equilibrium and nonequilibrium formalisms made unified

5.2. Oeneral considerations concerning multi-point functions The n-point (; has 2n components, but not all of them are independent of each other. There are many constraints following from the normalization of the generating functional and the causality, due to the canonical distribution and the time reversal symmetry, etc. Some of these constraints (due to normalization and causality) were described before in section 2.4. Here we will elaborate further on these relations and discuss the others one by one.

5.2.1. Exact algebraic relations First of all, let us recall that the "all one" components such as 0 11 , 0 111 , •• • , vanish in accord with (2.99). Furthermore, eqs. (2.109)-(2.113) are also exact. Using the transformation formulas (2.87), (2.88) and (2.90) for G and (; functions we can easily rewrite (2.99) in different forms. For example, in the single time representation it can be written as (cf. (2.92»

L Cll··· ...... ±

(1 + al ... an)O """,,_ =

L

(1- al ... an)O ",,,,,,_

(S.13)

al'··cz,.-;:t;:

in particular, for three-point functions we have

Equivalently, the same relation for three- and four-point functions can be transformed in the "retarded" combination as (S.14a) ~(02111 + 0 1211 + 0 2211 ) = 0 ++++ + 0

++ __ -

0 +++_ - 0 ++_+,

(S.14b)

or into the "correlation" combination as (5.1Sa) (S.ISb) Equations (S.14a) and (S.ISa) have been derived in [81] by tedious calculations without realizing its connection with (2.99). By careful inspection of (S.14) and (S.IS) one can easily uncover the general rule to write down analogous formulas for higher order functions. We would like to emphasize, however, these formulas do not contain any new information in addition to (2.99). Another set of exact algebraic relations follows from the properties of commutator (anti-commutator) and 8-function [SI] on which we will not elaborate any further here. Nevertheless, we indicate here a symmetry relation following directly from the definition of CfPGF as ... ) O '''1···2·..(... " I J ...0 -- .. ·2"'1 .. · (........ J. .I . ) •

(S.16)

743 KUiIIIg-chao Owu et aI., Equilibrium turd lWMquilibrium formalisms made unified

63

5.2.2. KMS condition [63,83] As mentioned before, in the response theory the system under consideration is assumed to be in thermal equilibrium for to = -00. Introducing a time-dependent external field, the expectation value of the operator product in the interaction picture (with respect to the external field) satisfies the following equation:

(S.17)

Owing to the time translational invariance of equilibrium state, (S.17) can be rewritten as (S.18)

By taking Fourier transformation (S.18) becomes (OJ(W )0,(0»

=rJ

6J

(S.19)

(O,(O)OJ(w»,

This is the so-called KMS [63,83) condition as emphasized in the mathematical treatment of statistical mechanics [8S] , because (S.19) still holds even when the density matrix is ill-defined for systems with infinite degrees of freedom. It follows immediately from (S.19) that (S.20)

which is nothing but the FDT as given by (3.43a). Now consider the multi-point functions. First of all, for any function invariant under time translation (S.21)

we have " 0 '_1 04

~-F=O,

or symbolically after Fourier transformation

"

(S.22)

Next, consider the averaged product of n operators. Transposing the leftmost operator to the rightmost one by one, we get (01(1)02(2)·' . O"(n»

= exp(ipoIX02(2)" . O,,(n)OI(1» = exp[ip(ol + 02)](03(3)· .. 01(1)02(2» = exp(-i po" XO" (n)OI(l) "

. O,,-I(n -1»,

744 64

Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unifi£d

where (5,23) This process can be stopped at any step, say, ith. We introduce the following two functions FH(1", i, i + 1", n)==(Ql(l)'" Qi(i)Qi+I(i + 1)'" Q"(n» , F(+)(l,·· i, i + 1··· n) == (Qi+I(i + 1)··· Q"(n)Ql(1)··· Qi(i» ,

(5.24)

and write FH(l··· n)= exp[i,8(a 1 + ... + a;]F(+)(l'" n)

(5.25)

which yields after Fourier transformation the generalized KMS condition (5.26) Defining two more functions F(C)

== F(-) + F(+) = ({Q\(l)' . , Qj (i), Qi+l(i + I}' , . Q"(n )}),

F(e) == FH - F(+) = ([ Ql(1)' .. Qi(i), Qi+l(i + 1)' .. Q" (n )]),

we find then (5.27)

which looks like a generalization of the FDT but actually it is not, because it connects only the symmetric and antisymmetric parts of the same function rather than different functions as in the two-point function case. Therefore, the KMS condition itself is not enough to give rise to the FDT, For example, the four-point function G++ __ can be represented as G++ __ (1234) = ~(-i~[(T(34)T(12»

+ (T(12)T(34»] + ~(-i)3[(T(34)T(12» - (T(12)T(34»] ,

(5.28)

where the first term corresponds to P(c) whereas the second term to p(e). We can thus derive a relation similar to (5.27) which, as we said, is not the FDT.

5.2.3. Time reversal in variance The implications of the time reversal symmetry for macrosystems have been discussed in section 4. Here we restrict ourselves to the situation when the conditions (4.4), (4.6) and (4.10) are satisfied. It has been shown there also that the average of product from Heisenberg operators transforms under time inversion as (cf. (4.17» F 1... "(1· .. n; J, A)

= El ... EnF n".\(-n·· . -1; el, EA) =

EI'"

EnFr"n(-l'" -n; el, EA),

(5.29)

745 Kuang-chao Chou tl al., Equilibrium and nontquilibrium formalisms madt WlifiM

65

where A is a parameter specifying the initial state. The second line of (5.29) is based on the following equality Tr(AB ... K) = Tr(K ... BA)*

(5.30)

valid for Hermitian operators. It also follows from the Hermicity of Q and po, the density matrix, that (5.31) Since the CfPGF are linear combinations of averaged operator products, so that for systems with time reversal symmetry we can easily write down their transformations under time inversion using (5.28). For example,

Hereafter we do not specify the change of J and A explicitly. We can split three-point function G++- into symmetric and antisymmetric parts under time inversion (do not confuse it with p(c) and p(a) considered in the last subsection) (5.32) where G~+_(123) = G~+_(-1- 2- 3)= !(-if«3T(12»

+ s3(T(12)3»,

G~+_(123) = -G~+_(-1- 2- 3) = !(-if«3T(12»- s3(T(12)3»

(5.33)

with (5.34) Applying the KMS condition to (5.34) leads to (5.35)

5.2.4. Fourier transform and spectral representation It is known that the Fourier transformation itself does not bring about any new information, but it is more convenient to incorporate the KMS condition and the time reversal symmetry in the Fourier space. The Fourier transform of a time translation ally invariant function can be written as F(wl ... w,,) = 21T(~(WI + ... + w,,)F1(Wl .•. , t"

= 0, ... w,,),

(5.36)

where all time arguments in PI are transformed except for t" = 0 with k being any of 1 to n. In other words, the Fourier transformation of the n-point function appears as

746 66

Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

dWI' .. dWk ... dw a • • (21T)a-l exp{-I[wI(11 - Id" . + Wk

r

F(II .. 'Ia ) = •

X

FI(wl ... Ik = O· .. wa )

+ ... Wa (ta - Id]), (5.37)

,

where the W variable with caret is missing. We remind the reader that the Fourier transform in this paper is not distinguished by any special symbols, the meaning being clear from the arguments of functions. As shown in section 2.3, the CTPGF can be presented as a linear combination of products of the 8-function and n-point function. For example, (5.38)

FR(123) = 8(123)F(123) . Factorizing the 8-function as 8(123) = 8(12)8(23) and using the integral representation

i ~ dfJ 8(1) = -2 -fJ . exp(-i{Jt), 1T + IE

J

(5.39)

we can write (5.40) It is easy to read out from (5.40) the general rule to write down the spectral representation for any CTPGF. In particular, for the three-point retarded function

G211 (123) = -8(123)([[1,2],3]) - 8(132)([[1,3],2]),

(5.41)

by use of the KMS condition and the simplified notation d W l dw 2 . . (123) = (1- 2, 2 - 3) = \!"1T"""t- exp[ -IWl(1t - (2) - IW2(12 - 13)](wh W2) ,

J

(321) = (123)* =

dW l dW 2 . . """fJ:1T"""t- exp[-lwl(t 1- (2)- IW2(t2- (3)](-Wh -W2)*,

J

(5.42)

we find that

1 J'" dndfJ? ( fJ +' )( fJ +' ) 1T _'" WI - 1 IE -W2 - 2 IE

+[I-exp(,8fJ2)](-fJh-fJ2+fJl)*}+(2 )2

X ([exp(-{Wl)- exp[~(fJ2 - fJ1)]](n h -n2 ) + [exp(~nl) -

exp[-~(fJ2 -

n1)]](-fJ h n2)*}'

(5.43)

747 KUQlIg-chao Chou et al., Equilibrium and /loMqUilibrium formalisms made u/lified

67

We see thus G211(W1WZ) is analytic in the upper half-plane of W1. There is one pole in the upper half-plane of W2 depending on the position of the pole in WI. As functions of several complex variables, the analytic properties of multi-point functions are much more complicated compared with the two-point case.

5.3. Plausible generalization of FDT Some authors claimed previously [78, 82] that they had already found the nonlinear generalization of the FDT, but indeed this was not the case. As we have seen in the last section, any of the three kinds of relations, namely, the algebraic, the KMS or the time reversal invariance, will not provide by themselves the generalization of the FDT needed. However, their combination might give what we would like to have.

5.3.1. Even and odd combinations The transformation formula from

G a."'a.(I· .. n) = 21- n

L

G to G as given by (2.88) can be rewritten as

(a1)'·· . '(an)"G """.(1· .. n) .

(5.44)

11 ... 1,.-1.2

Under the change a,-+-a, the T-product changes into t-product and a factor (_1)'·+"+1. appears on the right-hand side of (5.44). Therefore, it is natural to distinguish the "even" and "odd" components of G as G no t. and On. '0' where I", u = e, 0 is the number of "1"s among ik • 'l'he even components do not change sign under al-+ -ex, whereas the odd components do. As we will see later in this subsection, there are simple algebraic relations among the same type of 0 n. "', while the connection between these two types of components established by using the KMS condition and the time reversal invariance is the plausible generalization of the FDT. First consider the "all +" and "all -" components of (;

G++ ... +(12"·n)=2 1- n

L

G/t ... ,.(I"· n),

(S.4Sa)

(-1)'.+"'+1'0 ,.... ,.(12 . .. n).

(S.4Sb)

'.'''10=1,2

G --... _{12· .. n) "'" 21- n

L '.",'.-1,2

If we define the symmetrized combination of G n, '" as

G~,,,,=

L'

![1+(-I)"'+i.+ ... +i·]G'."'i.(12"·n),

(5.46)

;1"'1.. -1,2

where' means that the number of "1"s should be equal to I", then

L

G~,10 = 2n - 2 [G++ ... + (12" . n)+ G __ ... _(12·· . n)],

OSlc:!iin

L Is/os"

G~ '0 = 2n - 2 [ G ++ .. +(12, .. n) - G __ ... _(12· .. n)] .

(5.47)

748 68

Kuang·chao Chou el al., Equilibrium and nonequilibrium formalisms mode unifted

Making use of the relation following from the time reversal invariance, we have (S.48) for a = ±. It follows then immediately from (5.47) that

L

(O~.1,,-EI·"f"Ote)=O,

(S.49)

Os1"Sn

where

O<-)S n. '" == OS",1" (-1'" - n) . After Fourier transformation (S.49) becomes

L

(O~.1,,(W)-

fl'"

f"O~,1,,(-W}),

(5.50)

O:S~:S:;n

Since the components of 6 contain the factor (-i)"-I and an average of (n - 1) nested commutator and/or anticommutator, the 0"°. 1• and 0 n•• lo components are real, whereas 0 n •• I. and 0 no. I. are imaginary with their Fourier components related with each other by the following equations O!o.I.(W) = Ono.I.(-W),

O!•. 10(W) = O" •. I.(-W) ,

O!•. I.(W) = -On •. I.(-W},

O~D.ID(W)=- On •. I.(-W),

(5.51)

It is straightforward to see from (S.SO) that the "even" and "odd" components are linearly dependent among themselves but not with each other, as we mentioned at the beginning of this section.

5.3.2. Nonlinear generalization of FDT Now consider the other algebraic relations following from (S.44) as G a\'''a, (12, .. n) ± G -a\"'-a, (12, .. n)

= 2 1- n

L il"';,lJ

1C

[I ± (_I)i'+"'+i'](al)i' ... (a n )i'Oi''''j,(12··· n).

(S.52)

1,2

If {j}, i.e., (h, ... ,jm) from

ak

subscripts are "-" while the rest are "+", (S.52) can be rewritten as

01i-}(12· .. n) ± O{j+}(12' .. n) = 21 - n

2. ;)"'·i,,=1.2

with j; == ijr • Defining

(1 ±(-1)1t+"·+i')(-1}ii+"·+i';'G j ''''i,(12··· n)

(5.53)

749 69

KUlIng-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

G 1o. (j}{l2 ... n) == ~/ W- (-I)iI+"'+")(-I}i i+"'+i:'G il .. · •• (I· .. n),

(5.54)

we rewrite (5.53) as

L

L

G le.lil = 2n - 2( G {j-l + G (j+l),

Osle sn

(5.55)

G lo.{j} = 2n - 2 (G U _1- G{j+l)'

l:S;losn

Consider the combination G li - I (I· .. n) + e n G li _I (-1 ... -n) ± G{j+l(l ... n) ± e n G{j+I(-I' .. -n)

and make use of the following relations: G li-I(1

•..

n) == (-i),,-l(T(h ... jm )T(il ... i n-m» ,

GIi-I(-t··· -n)= en exp [- if3

~ ail] Gu-l(l ... n),

(5.56)

.-1

with

obtained from the KMS and the time reversal invariance conditions (5.25) and (5.29), we find (1 ± 1- e- A + eA) L G ,•. lil + e"(1 ± 1- eA + e- A) ~ G!;{iI ~

~

= (-1 ± 1 +e- A + eA)L G'o.{J) + e"(-l± 1 +e A +e- A) ~

L G!:)!il,

(5.57)

~

where (5.58) G(-)==G(-l···-n).

(5.59)

After Fourier transformation we get finally

L OS/e':slI

[G ,•. (j}(w) + e"G,•. IJl(-W)] = cothW(wiJ + .. + Wi.)

L

[G'o.{j}(W)- enG,o.U)(-w)] ,

ls:/o :S;1I

(5.60a)

750 70

Kuang-chao Chou el al., Equilibrium and nonequilibrium lonnalisms made unified

This is a plausible generalization of FDT to the nonlinear case. For n = 2, (5.60a) is the usual FDT, while (5.60b) is an identity. For arbitrary n, we have 2n - I -1 algebraic relations of type (5.52) (excluding all Irk = + cases) leading to 2n - 1 - 1 pairs of relations given by (5.60) in combination with the KMS and the time reversal invariance conditions. To illustrate (5.60) we write down explicitly the corresponding expressions for n =3 case

= 0 222 , 0 1. {1} = - 0 122 + 0 212 + 0 22 1> 01.{2} = 0 122 - 0 212 + 0 221 , 01.{3} = 0 122 + 0 212 - 0 221 , 02.{2} = -0112 + 0 121 - 0 211 , 02.{3} = 0 112 - 0 121 - 0 211 , O 2.{t} = - 0 112 - 0 121 + 021h OO.{1}

=

OO.{2}

=

OO.13}

(5.61)

O 2.(J}(W) + e302, (J}(-W) + 0222(W) + e30222(-W) =coth(,8w,/2)[ OI.{J}(W) - e30I.{j}(-w») , 02,{j}(W)- e302.{J}(-W) + 0222(W)- e 30222(-W)

= tanh(,8wj/2)[01. (j}(w) + e 0I.{J}(-W»), 3

j

= 1,2,3.

(5.62)

The question whether (5.60) is a correct generalization of FDT should be settled by further studies of nonlinear phenomena. We should mention, however, that Tremblay et al. [38] have considered the heating effects in electric conduction processes using CfPGF formalism. These authors do not discuss the general relations as we do here. In their opinion, the FDT should be model dependent in the nonlinear case. 6. Path integral representation and symmetry breaking The generating functional Z[J(x)] for CfPGF can be presented as a Feynman path integral. In terms of the eigenstate Icp'(x» of the operator cp(x, t = -(0) the density matrix can be written as (6.1) so that the generating functional is given by (6_2) where

U(L

=

-00,

t+

= -oo)=Sp = Tp exp(i

f J(x)cp(x») p

is the evolution operator defined along the closed time-path.

(6.3)

751 KUlIng-chao Chou tl al., Equilibrium and nontquilibrium formalisms made unified

71

It is known in the quantum field theory [39] that

(cpz(x)\ U(tz, t1)\CP1(X») = N

f

'2

[dcp(x)] exp (i

J.P(cp(X» dd+1 X)8(cp(x, tz) - cpz(x»8(cp(x, t1) - CP1(X» , ~

~~

where N is a constant. The path integral representation (6.4) is valid for any t}, t2, so (6.2) can be rewritten as Z[J(x)] = N

f

[dcp(x)]8(cp(x, t+ = -(0)- cp'(x»p",·".8(cp(x, L

x exp[i

J(.P(CP(X» +J(x)cp (X»] ,

= -(0)- cp"(x» (6.5)

p

where the integration in the exponent is carried out over the closed time-path p. Since the functional dependence of p",'",' upon cp'(x) and cp"(x) is rather complicated, in general (6.5) is not very suitable for practical calculation. However, it is useful for discussing the symmetry properties of the generating functional so far as the total Lagrangian of the system appears in the exponent. We will use this representation to discuss the Ward-Takahashi (WT) identities and the Goldstone theorem following from the symmetry (section 6.3). The path integral representation would be well adapted to the practical calculation if the contribution of the density matrix can be expressed in terms of effective Lagrangian in certain simplifying cases. This possibility will be considered in section 6.1. In section 6.2 we briefly discuss the properties of the order parameter and describe two different types of phase transitions. Finally, in section 6.4, the path integral representation is used to consider the fluctuation effects.

6.1. Initial correlations In this section we derive two equivalent expressions for the generating functional to incorporate the effects of the initial correlation in a convenient way [46]. 6.1.1. Model Consider multi-component nonrelativistic field t/lt, t/lb, b = 1,2·· . n which may be either boson or fermion. The action of the system is given by

(6.6) where the free part can be written as

f

1o[t/lt, t/I] = dld2t/1 t (1)Sol(1, 2)t/I(2) == t/l t So 1t/1, p

with

(6.7)

752 72

Kuang-chao Chou el al_. Equilibrium and nonequilibrium formalisms made unified

SOl = !USo/1}t + 1}So.:e + ~Siicln, So/ = So': = [ior + (112m )V2]Sd+I(1- 2),

(6.8)

in accord with (2.23), (2.63) and (3.10)_ 6.1.2. First expression for W ~ The generating functional

(6.9) can be rewritten as (6.10)

in the incoming picture by using the Wick theorem generalized to the CTPGF case as done in section 2.2 for the Hermitian boson field_ Here So is the bare propagator satisfying the equation

(6.11) p

p

and (6.12) Le.,

w~[r, J] =

i +, f dl··· dm d1 -.. dli r(l)' -. r(m) w~m. ")(1· - . m, m.n.

m.n-I

Ii ... I)J(Ii)' .. J(I) ,

p

(6_13)

(6.14)

where 1/1" I/Ir are operators in the incoming picture and: : means normal product. Since the time ordering does not have any effects under normal product, we can rewrite (6.12) as (6.15)

where J~

= J~-J~_

(6_16)

753 KlIIUIg-cIuw 0.011 et III., EquiIibrlIllll tmd 1IOMqllilibrillllllormlllisms millie IUlijied

73

We can then write down an expansion equivalent to (6.13) as

(6.17) with

w(m.n) = i",+n-I Tr{p:IJ1I(m)··· ,J1I(1)t/li(1)'" t/lI(11):}.

(6.18)

We note that W~ and w~m. n) are defined on the closed time-path, whereas WN and w(m. n) are defined on the ordinary time axis. Taking into account that in the incoming picture the field operators satisfy the free field equation, we have

f

di' .5ol (i, i')Wg"·n)(l··· i" .. m, Pi ••• I) = 0,

p

f

di'

w~m.n)(l·· . m, Pi· .• i"" 1)Sol(i', T) = 0,

(6.19)

p

or equivalently,

f f

di' 50/(i, i') w(m. n)(l' .. i' ... m, Pi ••• I) = 0 ,

di' w(m. n)(l' .. m, Pi· .• i" .. 1)So1(i', 1) = O.

(6.20)

As far as the initial condition is fixed at t = -co, we are not allowed to integrate by parts arbitrarily with respect to ill at. The correct direction of acting ill at is indicated by the arrow. Substituting (6.15) into (6.10) we get the first expression for Z[J] we would like to derive as (6.21) from which we can obtain the generalized Feynman rule. We see from (6.21) that the density matrix affects only the correlation generated by 11 and h. So far as W(m. n) satisfy eq. (6.20), the contribution of the density matrix can be expressed in terms of the initial (sometimes called boundary) condition for Green's function.

6.1.3. Second expression fOT W~ Now we derive another expression for the CfPGF generating functional. Using the following identity (up to an unimportant constant)

754 74

Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

f

exp(-irSol) = [dll/)[dl/r] exp{i(I/rtSO'Il/r + P I/r + if/J)},

(6.22)

p

it is easy to show that exp{i(-PSol + W;'[P, J])} =

f

[dl/rt)[dl/r) exp{i(rl/r+ I/rtJ)}exp {iW;' [±i :I/r,i 6~t]}eXp(il/rtSoll/r),

(6.23)

p

if the path integration is taken by parts. Taking into account that

'0: exp(il/rtSillI/r) = exp(il/rtSoll/r) ('O~ ± il/rtSill) , 'O~t exp(il/rtSilll/r) = exp(il/rtSillI/r) ('O~t + iSilll/r ) ,

(6.24)

(6.23) can be transformed into exp{i(-PSol + W;'[P,J])} =

f [dl/rt)[dl/r) exp{iWI/r+

I/rtJ + I/rtSillI/r)}

p

(6.25) Using (6.8) and the convention agreed in (2.69H2.74) we find that (6.26)

f

f f dySiI~(X,Y)I/r4(Y)'

dyI/rt(y)u3Sil 1(y,x),., = dYl/rl(Y)SO'"t(y,x),

,.,t I dySO'l(X,Y)U31/r(Y)=

(6.27)

and obtain from (6.15) that exp{i

w: [± i '0: - I/rtsO'\ i 'O~t - Siill/r ]} = exp{i WN[-I/rlSO'/, - SO'~I/r]} = exp{iW:[-I/rtSO'l, SO'II/r]}.

(6.28)

755 KUlJng·chao Chou et aI.• Equilibrium and nonequilibrium formalisms made unified

75

Substituting (6.28) into (6.25) and the resulting expression into (6.21), we find

J

Zp[Jt, J] == [dq/][dl/l] exp{i(Io[I/It, 1/1] + l in.[ I/It, 1/1] + r 1/1 + I/ItJ)} exp{i W~bl/ SOl, -Soll/l]} p

(6.29) as the second path integral representation for the generating functional. It is easy to rederive (6.21) from (6.29), so these two expressions are equivalent to each other. Note that this expression is different from that given by (6.5) in so far as the contribution of the density matrix appears here as an additional term W;' in the action. According to (6.28), this term does not depend on field variables I/I:,I/Ic describing the dynamical evolution, but does depend on I/Il, 1/111 describing the statistical correlation. It is also obvious that W;'[-I/ItSo\ -So 11/1] has nonvanishing contribution to the path integral only at the end points because of (6.19). 6.1.4. Two-step strategy For a general nonequilibrium process, (6.29) can be rewritten as (6.30) where

z~[r,J]==

J[dl/lt][dl/l] exp{i(Io+l .+rl/l+ I/ItJ)}

(6.31)

in

p

is the generating functional for the ground state. Since z~ has exactly the same structure on the closed time-path as that of the standard quantum field theory, we can first calculate ~ and then "put into" it the statistical information via (6.30). Such "two-step" strategy is well known in solving the Liouville problem in classical statistical mechanics. Many interesting nonequilibrium phenomena can be described by a Gaussian process, i.e., w~m. n)(1

... m, n ... I) == 0 except for

W~· 1)(1, I)

t- 0,

(6.32)

for which the contribution of the density matrix reduces to replacing the bare propagator Sop by Gop (x, y) == Sop(x,y)-

W~I. I)(X,

y).

(6.33)

For the thermoequilibrium case eq. (6.33), after Fourier transformation, is identical to (2.21). A more rigorous derivation of the diagrammatic expansion for thermoequilibrium will be given in section 9.1. Another possibility of simplification comes about when the state is stationary due to the microscopic time reversal invariance, the generalized FDT then holds as shown in section 4.2. As seen from (6.20) and (6.21), w~·n) as solutions of the homogeneous equation can be specified by the FDT.

756 76

Kuang-chao Chou el al.• Equilibrium and nonequilibrium formalisms made unified

To sum up, the determination of the CfPGF generating functional can be divided into two steps: To first "forget" about the density matrix in calculating the generating functional without the statistical information and then "put it into" the generating functional at the second step. In the general case this can be done using (6.30), but a significant simplification results if the initial correlation is Gaussian or a generalized FDT holds. If we are interested in some order parameter Q(x) which is a composite operator of the constituent field, we introduce an additional term h(x)Q(x) 'in the action. The generating functional for the order parameter is given by (6.34) in terms of Zp[Jt, J] for the constituent field. The extension of (6.21), (6.29) and (6.30) to this case is obvious.

6.2. Order parameter and stability of state It is well known that the vertex functional F[ipc] is most suitable for describing the symmetry breaking, inasmuch as it is expressed explicitly in terms of the order parameter rpc. In section 2.2 we have derived an equation (2.48) satisfied by it. Before going on with the discussion of the order parameter, we rewrite this basic equation of the CTPGF formalism in another equivalent form.

6.2.1. Functional form for the vertex equation Let the total action of the system be presented as 1,=1+1.=

I

I

.
(6.35)

The operator equation satisfied by rp(x) is then

81[rp(x)] = -J(x) . 8rp(x)

(6.36)

Here we use the Heisenberg picture including the external source. Multiplying (6.36) by the density matrix and taking trace, we find

-J( )= T {81[rp(x)] -} x r 8ip(x) p . Transforming into the Heisenberg picture without external source, we obtain

757 KUQlIg-chlJQ Chou tt aI., Equilibrium and IIOIItqIIilibrium formalisms made unified

77

Using the commutation relation -i

&J~x) Z[J(x)] :: Z[J(x)] (CPc(x) - i &J~X») ,

(6.38)

(6.37) can be rewritten as

-J(x) =

8:~X) [CPc(x) - i &J~X)] -

(6.39)

Comparing (6.39) with (2_48) we find (6.40) which is a formal relation to derive the generating functional from the action. As in the quantum field theory, the term i(8/8J(x» comes from the loop correction. Hence in the tree approximation

8r[cp.(x)] = 8I[cp(x)] 8cp.(x) 8cp(x)

I.,-.,c'

r[cp.(x)] = f .!t(cp.(x»

dd+l

x_

(6.41) (6.42)

p

We note in passing that we did not emphasize the validity of (6.36) on the closed time-path, but it is true, so that we can put p in eq. (6.42).

6.2.2. Two types of phase transitions Coming back to the order parameter itself, CP.(x) should be a solution of equation 8rt8cp.(x) =0

(6.43)

in the absence of the external field. The CP.(x) = 0 solution corresponds to the "normal" state, whereas the nonzero solution corresponds to the symmetry broken state. Then the question arises which state is more stable. Let us consider the fluctuation around a homogeneous solution 'PcO of (6.43)

CP.(X) = CPcO + CPk exp(-ik . x).

(6.44)

Put (6.44) into (6.43) and obtain the linear equation satisfied by CPk as

f CP. x82rCP. I 8 ()8 (y)

p

"C~"cG

(cp.(y)-cpcO)dy=frr(X,y)CPkexp(-ik,y)dy=rr(k)CPk=O.

(6.45)

758 78

KUQ1Ig-chlW Chou el al., Equilibrium and 1Ionequilibrium formalisms made u1Iified

As follows from (6.45),

q>k

is zero unless k is a solution of the equation

Fr(k) = O.

(6.46)

Assume the solution of (6.46) to be k o =w(l) ,

(6.47)

the fluctuation will decay in time, i.e., the state is stable if 1m w(l) < O. Otherwise, the state is unstable. According to (3.14) Fr(k) = D(k) + iA(k) ,

(3.14')

where A(k)= - ~(L(k)- F+(k».

(3.13')

Near the critical point where 1m w(l) changes sign, 1m w(k), and probably A(k) is a small quantity. We discuss here two possible situations. (1) The equation D(k, ko) = 0 has real solution ko = Re w(k) for all k, it then follows from (6.46) and (3.14') that Imw(k)= _A(k)jaDI ako

.

(6.48)

ko-Re .. (1)

If 1m wei) > 0 for some i, then an instability with this k occurs to form a new space-time structure. However, as discussed in section 3.2, in thermoequilibrium we have A(k) =-!iF_(k)[l- exp(-pko)]

= ko Wa(k)[l- exp(-pko)] >0,

and aD! ako > 0 for k o> 0, whereas both of them change sign for ko < 0, so that such instability cannot occur in an eqUilibrium system. In fact, it usually appears in far-from-equilibrium systems under certain special conditions, for example, in a laser system the q>. = solution is unstable above the threshold of pumping. (2) A(k) is not small compared with D(k) as i ... 0. For an eqUilibrium system we can write

°

A(k) = koY,

y

> 0,

D(k) = Do + ako + ....

Up to the first order of ko, the solution of (6.46) is ko = -D(O)/(a + ioy),

with 'Y

1m ko = -2--2 D(O) . a

+ 'Y

(6.49)

759 Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made uniji£d

79

Hence the phase transition occurs at D(O) = O. This is the ordinary second-order phase transition if the nontrivial solution grows continuously from zero. Otherwise, the point D(O) = 0 will correspond to supercooling or superheating temperature.

6.3. Ward-Takahashi (WT) identity and Goldstone theorem In this section we derive the WT identity satisfied by the CfPGF from the invariance of the Lagrangian of the system with respect to global transformations of a Lie group G.

6.3.1, Group transformations Let ip(x) be the constituent field and O(x) the order parameter. Each of them has several components forming by themselves bases of unitary representations. Under the infinitesimal transformation of G ip(x)~

ip'(x) = ip(x) + &ip(x),

&ip(x) = (,,(iI~L X~(x)a,,)ip(x) = iI"ip(x)(",

(6.50)

= O(x) + &O(x), &O(x) = (.. (iL!!') - X ~ (x )19" )O(x) = iL"O(x )(" ,

(6.51)

O(x)~ O'(x)

where ( .. are a total of nG infinitesimal parameters for group G and I~), L~) representation matrices for the generators of G. X~ are associated with the transformation of coordinates

(6.52) It can be easily shown that the Lagrangian function transforms in this case as

(6.53) where

j:;(x) =i

8O~~X) I,,!p(x)- ..'tX~(x)

(6.54)

is the current in the a direction and (.. (x) is an arbitrary infinitesimal function. If the Lagrangian is invariant under the global transformation of G, it then follows that

or equivalently,

o"],,'''() .( &:t' x = I 0" 8O,.!p(x)

&..'t) ()

&ip(x) I"ip x ,

(6.55)

760 80

Kuang·chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

i.e., the current is conserved provided rp(x) is the solution of the Euler-Lagrangian equation. Substituting (6.55) into (6.53) yields (6.56) which is the change of :£ under local transformation of (, if it is invariant under the global action of the same group.

6.3.2. WT identities The path integral representation for the generating functional (6.5) can be written as

Z[J(x), h(x)] = N

f

[drp(x)] exp

{i f (:£(rp(x» + J(x)rp(x) + h(x)Q(x»} p

x (cp(x,

t+ = -

oo)lplcp(x, L = -00» .

(6.5')

Performing a local transformation of rp(x) in (6.5') with (a(x) satisfying the following boundary conditions (a (x, t± = -00) = lim (x, t) = 0,

(6.57)

Ix 1.... 00

and taking into account that the measure [dcp(x)] does not change under unitary transformations, we obtain from the in variance of the generating functional that (6.58)

(a"f~(x» = -iJ(x)IaCPc(x) - ih(x)LaQc(x).

On the other hand

(a,.j~(x» = Z-la,.j~ (rp(x) = -i 8J~X»)Z[J(x), h(x)],

(6.59)

from which it follows that (6.60) by use of (6.38). Using the generating functional W for the connected Green function, the WT identities (6.60) can be rewritten as

a,.Ja',. (8~';)

-.8J(x) _8_) -__ .IJ ()Ia 8J(x) 8 W _ 'h() 8W X La 8h(x)' J

X

I

(6.61)

761 KuolIg-chao Chou el aI., Equilibrium and lIonequilibrium formalisms made unified

81

Taking functional derivatives of (6.61) with respect to J(y) and then setting J(y) = 0, we obtain WT identities satisfied by CTPGFs of different order. In terms of vertex generating functional r[r,o.] (6.61) can be expressed as

(6.62) Here we allow r,o(x) to be either a boson or fermion field. Taking the functional derivative 8/8r,o.(Y) of (6.62) and putting r,o.(y)::: r,odJ, the symmetry breaking in the absence of J(x), we obtain WT identities for different vertex functions.

6.3.3. Goldstone theorem Now we use the WT identity to discuss the symmetry breaking after phase transition. Suppose the equations

have solutions r,o.(x) = 0, O.+(x) = O.-(x) "I- O. Differentiating (6.62) with respect to O.(y), setting J(x) = h(x) = 0 and integrating over x, we obtain

which can be rewritten in the single time representation as (6.63a) In matrix form (6.63a) appears as (6.63b) i.e., LaO.(x) is the eigenvector of rr with zero eigenvalue. Assume O.(x) is invariant under a subgroup H of G with nH as its dimension. Therefore, a

= 1, ... , nH

if a belongs to the generators of H. On the contrary, if a belongs to the coset G/H, LaO. "I- 0, then (6.63b) shows that rr has no - nH eigenvectors with zero eigenvalue. Suppose the representation to be

762 82

Kuang·chao Chou el aI., Equilibrium and nonequilibrium formalisms made unified

real, then taking complex conjugation of (6.63b) we obtain (6.64)

due to the orthogonality of La. Separating the real and imaginary parts we find OcLa . A = A . LaOc "" 0 ,

(6.65)

i.e., LaOc are zero eigenvalue eigenstates for both D and A. It follows from the Dyson equation (3.10) that the retarded Green function Gr has no - nH non dissipative elementary excitations called Goldstone modes. If Oc does not depend on coordinates, in Fourier representation of x - y such excitation occurs at zero energy and momentum and is called Goldstone particle. 6.3.4. Applications The Goldstone bosons considered above have important consequences in the symmetry broken state. For example, in laser system the U(l) symmetry is broken, so the corresponding Goldstone boson leads to the divergence of the fluctuation which in turn makes the classical solution unstable. This phenomenon in the CTPGF approach was observed by Korenman [25] and was analyzed by us in [40,52]. The WT identity is used to derive a generalized Goldstone theorem in a slowly varying in time system. As its consequence the pole of the Green function splits into two with equal weight, equal energy but different dissipation. Combined with the order parameter (average value of the vector potential) these two quanta (one of which is the Goldstone boson) provide a complete description of the order-disorder transition of the phase symmetry in the saturation state of the laser. We have also used the WT identity in combination with order parameter expansion (cL section 3.5) to derive the generalized TDGL equation [43,47]. We will apply the same identity to discuss the localization problem in section 8.3 [56]. We should mention that the transformations given by (6.50) and (6.51) are linear. We can consider nonlinear transformations under C as we did in [43,47]. In that case we need to take into account the Jacobian of transformation for the path integral. The result thus obtained turns out to be the same as the nonlinear mode-mode coupling introduced phenomenologically by Kawasaki [67]. 6.4. Functional description of fluctuation 6.4.1. Stochastic functional It is known that the Gaussian stochastic process [I (t) appearing iii the Langevin equation (cf. (4.33))

ao;/at = K,(Q)+ [jet)

(6.66)

can be presented by a stochastic integral [86]. Equation (6.66) can then be considered as a nonlinear mapping of the Gaussian process on to a more complicated process O/(t). Realization of such mapping actually results in a functional description of O/(t) [72]. Nevertheless, such functional description can be achieved by a more straightforward way [87-89]. Consider the normalization of the 5-function under path integration (6.67)

763 Kuang·cilao CIIou et aI., Equilibrium and lIonequiJibrium formalisms made llllijied

83

where the Jacobian .1 (0) appears because the argument of the 8-function is not 0 itself but a rather complicated expression. Neglecting multiplicative factors, .1 (0) turns out to be [72] .1(0) = exp

{-! I 8K(0)/80} .

(6.68)

Using the integral representation of 8-function (6.69) (6.67) can be rewritten as (6.70) Inserting the source term

one obtains the generating functional

Zf[J,J] =

I[dO] [~~] exp {I [iO (a~ - K(O)-~) -~:~ + i(JO +JO)]} ,

(6.71)

with the normalization condition

Zf[O,O] = 1. Averaging over the random noise distribution W[~] oc exp(-!t'o'-l~)

(6.72)

with u as the diffusion matrix, one obtains Lagrangian formulation of the generating functional for the statistical fluctuation

Z[J,J] =

I [dO][~~]

exp

The Gaussian integration over Z[J]=

{f [-!OuO+iO (aa~ - K(O») -!:~ +i(JO +JO)]} .

(6.73)

0 can be carried out to yield

18K . ]} . a,-K(O)-JA) u- l(ao a,-K(O)-JA) -28Q+IJO I [dO]exp {I[ -21 (ao

(6.74)

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Kuang·chao Chou et al., Eqwlibrium and noneqwlibrium formalisms made unified

Historically, the theory of noncommutative classical field was first suggested by Martin, Siggia and Rose (MSR hereafter) [90]. This theory has been extensively applied to critical dynamics [61] and has been later reformulated in terms of a Lagrangian field theory [88,89] as presented by (6.73) and (6.74).

6.4.2. Effective action We will show now that such description occurs within the CfPGF formalism in a natural way [43,47], postponing the comparison with the MSR field theory to section 9.3. Let Q/(x) be composite operators of the constituent fields ipJ(x). Both of them are taken to be Hermitian Bose operators. Assuming the randomness of the initial phase, the density matrix is diagonal at moment 1= 10, i.e., (ip'(x, to)lplip"(x, (0

»= P(ip'(x), to)8(ip"(x, to) - ip'(x, to» .

(6.75)

The initial distribution of the macrovariables Q1(x) is then given by P(Q/(x), to) = Tr(8(Q/(x) - Qi(ip(X»P}

f

=

[dip(x)]8(Q/(x)- Q/(cp(x»P(cp(x), to).

(6.76)

The generating functional for Q/ can be written as

Z[J(x)] = exp(i W[J(x)]) =Tr{

Tp[ exp (i f J(x)Q(ip(X»)]p} p

(6.77) where 8(cp+ - 1,0-) ==

f

dcp'(x) 8(ip(x, t+ = to)- ip'(x»8(cp(x, L

= (0)- ip'(x»P(ip'(x), to).

(6.78)

MUltiplying (6.77) by the normalization factor of the 8-function

f [dQ]8(Q+ -

Q_)6(Q(x)- Q(ip(x») = 1,

(6.79)

changing the order of integration to replace Q(ip(x» by Q(x) and using the 8-function representation (6.69) with 6 changed for I, we can rewrite (6.77) as Z[J] =N

f [dQ] exp (i

Sell +

if JQ )8(Q+ - Q-), p

(6.80)

765 K/lQIIg-chao Chou et aI., EquilibriwtJ and nonequilibrium formalisms malk unified

85

where exp(iSefI[O]) =

f

[dl/21T1 exp (i W[I1 - i

f 10) .

(6.81)

p

Here we have performed direct and inverse Fourier transformations of the path integral. So far as a continuous integration is taken over l(x), W[I] can be considered as a generating functional in the random external field. Calculating the functional integral in the one loop approximation which is equivalent to the Gaussian average, we can obtain the effective action Seff[ 0] for O(x). So far we have discussed the case when macrovariables are composite operators. The same is true if part or all of macrovariables are constituent fields themselves. A new "macro" field can be also introduced by use of the 6-function. However, one should carry out the path integration simultaneously in spite of the fact that the initial correlations are multiplicative, because in general the Lagrangian itself is not additive in terms of these variables. Before going on to calculate the integral (6.81) we first discuss the basic properties of the effective action Seff[ 0]. It is ready to check that apart from the normalization condition (2.103) the generating functional for the Hermitian boson field also satisfies the relation (6.82) It then follows from (6.81) that

(6.83) Hence Sell is purely imaginary for O+(x) = O_(x). Setting O,.(x) = 0 + flO± and taking successive functional derivatives of (6.83) near 0, we obtain (6.84) SFi/(X, y) = SFjl(Y, x) = -S;/i(Y, x),

5,,,,/ (x, y) = S"/I (y, x) = -S:/I(Y, X),

(6.85)

where

We see that S,j respect the same symmetry as the two-point Green functions (cf. (2.134» and vertex functions (3.12b). It the system is invariant under a symmetry group G, i.e., both the Lagrangian and the initial distribution do not change under

then

766 86

Kuang-chao Chou et al., Equilibrium and nonequi/ibrium formalisms made unified

WW(x»)

If = ~(X)V;I(g),

= W[I(x»),

Sell[QB(X»)=Sef([Q(x)],

Qf= V;j(g)Qj(q»_

The above said is true if the effective action Sell is calculated exactly. However, the symmetry properties of Sell, being related to those of the Lagrangian, may be different from the latter due to the average procedure. If the lowest order of WKB, i.e., the tree approximation is taken in (6.81) we find that (6.86)

Q = 6W/6I,

(6.87) In this case, Sell inherits all properties of the generating functional r[Q], e.g.,

(6.88) (6.89)

(6.90) (6.91) In accord with (6.87), (3.5) and (3.19) we have (6.92)

-is,,(k) >0

after taking Fourier transformation. Near thermoequilibrium we find from (3.40) S_;j -

S+ij~-

fikoS-I/(k).

(6.93)

kct-+O

6.4.3. Saddle point approximation Up to now we have discussed only the general properties of Sell[ Q). In principle, Sell can be derived from the microscopic generating functional W[I] by averaging over the random external field, it can be also constructed phenomenologically in accord with the symmetry properties required. We now calculate (6.81) in the one-loop approximation. Near the saddle point given by (6.86) we expand the exponential factor in (6.81)

E=W-

I QI=r-~I MG(2)t:.I=r-!I t:.Pui';u~J, p

p

(6.94)

767 Kuallg·chao Chou el al.• Equilibrium and lIonequilibrium fonnalisms made unified

87

where (] is the two-point connected functions, tJ.jT = (tJ.I+, tJ.L). Up to a numerical constant, the result of the Gaussian integration is (6.95) It then follows from the Dyson equation (2.57) that iSefJ[Q] = ir[Q] + !Tr In

t.

(6.96)

By use of the transformation formula (2.59) we have Idet tl = Idet tl = Idet r.lldet f.1 = Idet frl2 , where 82 f fr(x, y) = 8QA(X)8Qc(Y) . As shown in section 3.5, (6.97) Comparing (6.97) with (6.67) we find that 82rt8QA(X)8Qc(Y) is just the transformation matrix up to the numerical factor ,),-1. Therefore, we can calculate the Jacobian in the same way to get

J

iSefJ[ Q] = if[Q]-! 8K/8Q,

(6.98)

with K = -')'-18~/8Q.

In the path integral (6.80) the most plausible path is given by 8SefJ[Q]l8Q(x ±) = -l:t(x) ,

(6.99)

= to) = Q(x, L

(6.100)

Q(x, t+

= to) .

In the tree approximation of (6.81) 8SefJ /8Q

= -J = -,),iJQliJt -

8~/8Q,

which is nothing but the TDGL equation derived in section 3.5 (cf. (3.85».

(6.101)

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KUQIIg-chao Chou et aI., Equilibrium and nonequilibrium formalisms made unified

6.4.4. Role of fluctuations We now discuss fluctuations around the most plausible path. In the CTPGF approach there is an additional way of describing the fluctuation: To aJlow field variables to take different values on the positive and negative time branches. Changing variables in (6.80) to Q = Qe = ~(Q+ + Q_), QtJ. = Q+ - Q-, the effective action can be expanded as

Seff[Q+(X), Q_(x)] = S.f![Q, Q] +!

f

[6S.f!/I)Q(x+)+ 6Se f!/6Q(x- }]QtJ.(x)

f

+ 1 QtJ.(x}(S++ + s__ + S+_ + S_+}(x, y)QtJ.(y) + .... Denoting

!i(S++ + s__ + S+_ + S_+ )(x, y) == - y(x )u(x, y)y(y)

(6.102)

and using (6.88), (6.98), (6.101), we obtain eiW[J(x») =

f

[dQ(x)][dQtJ.(x)] exp [ -!

- i

f

(y(x)

f

QtJ.(x)y(x)u(x, y)y(y)QtJ.(y)

f

aa~ + :~) QtJ.(x)- ~ :~ + i

f

(ftJ.Q + feQtJ.)] 6(QtJ.(x».

(6.103)

If we take ftJ. = f and y(x)QtJ.(x) -+ 6, fey-I -+ j the stochastic generating functional (6.73) is retrieved. Carrying out integration over QtJ.(x) wiIllead to an equation identical to (6.74). It is important to note here, that in the CfPGF approach, fe, the counterpart of j in MSR theory, is the physical external field, whereas ftJ. = f+ - L, the counter part of J in MSR theory is the fictitious field. It is clear by comparing (6.103) and (6.73) that u(x, y) is the correlation matrix for the random force. If Q is a smooth function of x, u can be taken as a constant (6.104) which reduces to (6.105) by virtue of (6.87) and (6.90). This is the expression we have used in section 4 (cf. (4.35}) to discuss the symmetry properties of the kinetic coefficients, if it is generalized to the multi-component case. According to the definition of y given by (3.81) (6.106) Comparison of (6.106) with (6.105) yields the Einstein relation (FDT)

769 KUiIIIg-cl!ao Ow" et al., Equilibrium and nOMqUilibrium formalisms mtule IUliMd

u = 2/Py

89

(6.107)

for the diffusion coefficient in the case of single macrovariable. For simplicity we consider only one component order parameter in this section, but we consciously write some of formulas in such a way, so that the generalization to the mUlti-component case is obvious_ To sum up, the MSR field theory of stochastic functional is retrieved in the CfPGF approach if the one-loop approximation in the random field integration and the second cumulant expansion in Q6(X) are taken. The possibility to go beyond such approximation is apparent. 7. Practical calculation scheme using CTPGF As we have seen, the crPGF provides us with a unified approach to both equilibrium and nonequilibrium systems. However, to make it practically useful we need a unified, flexible enough calculation scheme. Such scheme has been already worked out by us [48,49]. In fact, most of the calculations carried out by us so far using CfPGF [40,46,52-57] can be cast into this framework. Consider a typical situation when fermions t/I(x), t/lt(x) are coupled to the order parameter Q(x) which might be a constituent field like the vector potential A .. (x) in the laser case, or a composite operator like x(x) = t/I dX)t/I ~ (x)

in the theory of superconductivity, or S= t/lt(x)!ut/l(x) in the case of itinerant ferromagnetism, where u are Pauli matrices. The boson field Q(x) via which the fermions interact with each other, may be nonpropagating at the tree level like the Coulomb field. However, the radiative correction will in general make Q(x) a dynamical variable and the fluctuations around the mean field Qc(x) will propagate and form collective excitations. Therefore, the system is characterized by the mean field Qc(x) and the two kinds of quasiparticles - constituent fermions and collective excitations with their own energy spectrum, dissipation and distribution. Such a way of description has been found useful in condensed matter physics [1-3,21-24,91], plasma physics [33,34] as well as in the nuclear many-body theory [92-96]. In this section we first (section 7.1) derive a system of coupled equations satisfied by the order parameter and the two kinds of Green's functions using the generating functional with two-point source terms. Next (section 7.2), the technique of Cornwall, Jackiw and Tomboulius (CIT) [97], developed in the quantum field theory to calculate the effective potential for composite operators is generalized to the CfPGF case and is used as a systematic way of computing the self-energy part by a loop expansion. In thermoequilibrium when the dissipation is negligible, the mean field Qc(x) and the energy spectrum of the fermion field are determined to the first approximation by the Bogoliubov-de Gennes (BdeG) [98] equation, in which the single particle fermion wavefunction satisfies the Hartree type self-consistent equation without Fock exchange term. In section 7.3 we discuss the generalization of the BdeG equation in the four-fermion problem, whereas the free energy in various approximations is calculated explicitly in section 7.4 by directly integrating the functional equation for it. Some problems related to those discussed in this section were considered by Kleinert using the functional integral approach [99].

770 90

Kwollg-chao Chou et al., Equilibrium alld lIonequilibrium formalisms made unified

7.1. Coupled equations of order parameter and elementary excitations 7.1.1. Model Consider a fermion field I/I(x) interacting via a boson field O(x) with the action given by (7.1)

where 10[I/It, 1/1] =

J

I/It(X)SOl(X, y)I/I(Y) ,

(7_2)

p

10 [0]

=!

I

O(X).1ol(X, y)O(Y) ,

(7.3)

P

with So I (x, y), .1 OI(X, y) as inverse fermion and boson propagators respectively, For a system with four-fermion interaction only, we can use the Hubbard-Stratonovich (HS) transformation [100} to introduce the effective fermion-boson interaction. Let the generating functional for the order parameter O(I/It(x), I/I(x» be defined as

Using the Gaussian integral identity, i.e. the HS transformation, eq. (7.4) can be presented as (7.5)

Zp[h] =

I

[dl/lt][dl/l][dO] exp{i[/o[I/It, 1/1] + lo[ OJ + lint[I/It, 1/1, OJ + hOl},

(7.6)

p

where

lint[I/It,I/I,O] being the nonlinear interaction. It is important to note that Mo,.1(j1 are two-point functions, independent of either field variables or external source. Therefore, up to an additive constant Mo the Green functions of the original system are the same as those of the effective system described by Z. The formal ambiguity in defining the [dO(x)] integration [96] can be avoided by imposing the condition

~\ Sh(x)

_ Si,.[h] \ Sh(x) h-O

hzO -

(7.7)

771 KUlJIIg·chao Chou et aI., EAjuilibrium and lIOIIeI/uilibrium formalisms made UIIifitd

91

We see thus the four-fermion interaction can be considered on an equal footing with system of fermions interacting through a constituent boson field Q(x).

7.1.2. Two-point source The generating functional with a two-point source is defined as

Zp[h, r, J, M, K]

J

= [dl//][drfr][dQ] exp{i[Io[t/lt, t/I] + 10[Q] + 1int [t/lt, t/I, Q] p

(7.8)

where W~[t/lt, t/I, Q] takes care of contribution from the density matrix as discussed in section 6.1. Here we adopt the abbreviated notation and M(x, y), K(x, y) are external sources to generate the second-order CfPGFs. Introducing the generating functional for the connected CfPGF as usual W[h, r, J, M, K) = -i In Z[h, r, J, M, K] ,

(7.9)

it then follows that 8Wp/8h (x) = Qe(X) ,

(7.10a)

8 Wp/8r(x) = t/le(X) ,

(7JOb)

8Wp/8J(x) = -t/l~(x),

(7.10c)

y», 8Wp/8K(y, x) = -(t/le(X)t/I~(y)+ iG(x, y».

(7.10d)

8Wp/8M(y, x) = ~Qe(X)Qe(y)+ iJ(x,

(7J0e)

In case of vanishing sources Qe(X), t/le(x) and t/I:(x) become expectation values of the corresponding fields Q(x), t/I(x) and t/lt(x), whereas J(x, y), G(x, y) are the second-order CfPGF for the boson field Q(x) and the fermion field t/I(x), t/lt(x) respectively. 7.1.3. Coupled equations The generating functional for the vertex CfPGF is defined as the Legendre transform of Wp , rp[Qe, t/I:, t/le, J, G) = W[h, r, J, M, K) - hQe- t/I!] - r t/le -! Tr[M(QeQe + iJ)] - Tr[K(t/I.t/I! + iG)] ,

where

(7.11)

772 92

Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

Tr[M(QeQe+ i.1)] ==

f

M(x, y)(Qe(y)Qe(X)+ i.1(y, X»,

p

Tr[K(I/Iel/l~ + iG)] ==

f

K(x, y)(!{Ie(y)I/Ic(x) + iG(y, x».

p

Using (7.10) it is straightforward to deduce from (7.11) that afp = -h(x)aQc(x)

f

M(x, y)Qe(y),

(7.12a)

K(x, Y)I/Ic(y) ,

(7.12b)

p

afp al/l~(x) = -J(x) -

f p

f

afp _ t() t a!{lc(x) - J X + I/Ie(y)K(y, x),

(7.12c)

p

~ 1 a.d(x, y) = 2i M(y, x),

(7.l2d)

8fp 'K(y) 8G(x, y) = I ,x.

(7.l2e)

Equations (7.12) form a set of self-consistent equations to determine the order parameters Qc(x), !{I~(x), I/Ic(x) as well as the second-order CTPGF .1 (x, y) and G(x, y) provided fp is known as a functional of these arguments. In almost all cases of practical interest the condensation of the fermion field is forbidden in the absence of the external source, i.e., I/I! = !{Ie = 0 for = J = O. On the other hand, the condensation of the boson field (elementary or composite) is described by the order parameter Qc(x). Since the energy spectrum, the dissipation and the particle number distribution are determined by the second-order CTPGF, eqs. (7.12a, d, e) are just those equations we are looking for. The only question remaining is how to construct the vertex functional rp. In the next section a systematic loop expansion will be developed for this purpose.

r

7.2_ Loop expansion for vertex functional 7.2.1. CJT rule [97] Without loss of generality in what follows we will set r

= J = 1/1: = !{Ie = O. The vertex functional

773 Kua/lg-chao Chou et al., Equilibrium and /lonequilibrium fomralisms made u/lifted

93

is the generating functional in 0 for the two-particle irreducible (2 PI) Green's functions expressed in terms of propagators .1 and G. To derive a series expansion for rp we note that after absorbing into the effective action, i.e.,

W:

(7.13) the only difference remaining between the CTPGFs and the ordinary Green functions in the quantum field theory is the range of the time axis. For CfPGF the time integration is taken for both positive and negative branches. Hence the loop expansion technique for the vertex functional and its justification developed by eJT [97] in the quantum field theory can be easily extended to the CTPGF formalism provided the difference in the definition of the time axis is properly taken into account. Here we shall simply state the result as

rp[ 0 0 ,.1, G] = I[ 0 0 ] - Wi Tr{ln[.1 ol.1]- L1 01 .1 + I}

+ iii Tr{ln[SoIG] -

GOIG + I} + r 2p [ Oc,.1, G] ,

(7.14)

where

I[Oo] =IetM/, 1/1, O]I",~",t_o '

(7.15a)

Q~Qc

(7.15b)

(7.1Sc)

(7. 15d) Note that G OI is different from SOl and

(7.16)

r

The quantity 2p[ 00'.1, GJ appearing in (7.14) is computed as follows. First shift the field O(x) in the effective action IetI[I/It, 1/1, 0] by Oc(x) and keep only terms cubic and higher in I/It, 1/1 and 0 as interaction vertices which depend on Oc. The 2p is then calculated as a sum of all 2PI vacuum diagrams constructed by vertices described above with ..1 (x, y), G(x, y) as propagators_ In (7.14) the trace, the 1 products..1 0 .1, etc., as well as the logarithm are taken in the functional sense with both internal indices and space-time coordinates summed over.

r

7.2.2. Coupled equations The self-energy parts for the fermion and the boson propagators are defined to be

774 94

K/IIJIIg·chao Chou el 01., Equilibrium and nonequilibrium jormalisms made unified

-i ~ !(X, Y) == h &G(y, X)'

(7.17a)

(7.17b) Hereafter in section 7 we restore the Planck constant h in formulas to show explicitly the order of magnitude. The equations for the order parameter O.(x) and the second-order CTPGF .:1 (x, y) and G(x, y) for a physical system can be obtained from eqs. (7.12) by switching off the external sources. We have thus: l

&fp

&O.(x)

=&I[O.]_iIiTr{&Gii G}+ Sf2p =0

SO.(x)

&O.(x)

&O.(x)

(7.18)

,

2i Sfp _ -I( -I _ Ii M(Y,x)-.:1 x,y)-.:1 o (x,y)+l1(x,y)-O,

(7.19)

(7.20) Rewritten in the ordinary time variable in accord with the rule set in section 2.3, eq. (7.18) becomes the generalized TDGL equation for the order parameter, whereas (7.19) and (7.20) are the Dyson equations for the retarded and advanced Green functions along with the transport equation for the quasiparticle distribution. Equation (7.18) can be rewritten in a symmetric form as

(7.18') The retarded, advanced and correlation Green functions are related to the matrix

A as (7.21)

in accord with (2.12). Therefore, the Dyson equations for the retarded propagators take the form (7.22a)

(7.22b) The corresponding equations for the advanced functions can be obtained by taking the Hermitian conjugation of (7.22), whereas equations for correlation functions appear in the matrix form as

775 KlIIIIIg-chao Chou et aI.• Equilibrium and 1I000uiUbrium formalisms made uni/ild

95

.de =-.dr(.d ilc1-lIe).d a ,

(7.23)

Ge = -G.(GoJ - te)G •.

(7.24)

As shown in section 3.3, the latter equations reduce to transport equations for the quasiparticle distribution. 7.2.3. Summary To sum up, we have derived seven equations to determine seven functions Qc(x), .d" .d., .dc, Gn G. and Ge , from which the order parameter as well as the energy spectrum, the dissipation and the distribution function for the corresponding quasiparticles can be calculated. Up to now we have not yet made any approximations. As is well known [97], the loop expansion is actually a series expansion in fl. Therefore, for systems which can be described by quasi classical approximation one needs only the first few terms of this expansion. In fact, one recovers the mean field result if the contribution from r2p is neglected altogether. In some other cases like in the theory of critical phenomena, one needs to partially resume the most divergent diagrams. For most cases of practical interest including thermoequiIibrium, the initial correlations expressed in terms of W~ are Gaussian. As shown in section 6.1, in such cases the statistical information can be included in the free propagators .do, So by FDT, so that W;-' term drops out from the effective action. Hence the analogy with the quantum field theory can be carried through even further for such systems. As seen from the derivation, this calculation scheme can be applied to both equilibrium and nonequilibrium systems. It is particularly useful when the dynamical coupling between the order parameter and the elementary excitations is essential. We note in passing that the logical simplicity of the present formalism comes partly from introducing the two-point sources M(x, y), K(x, y) and performing Legendre transformation with respect to them. 7.2.4. Comparison with earlier formalism To make contact with the generating functional introduced before (marked by a prime), we note that Z~h]

= Zp[h, M, K]IM_K=O,

(7.25a) (7.25b)

W;[h] = Wp[h, M, KlIM-K-o, r~ Qe]

= rp[ Qe,.d, G]lar"lM-ar"laa=o,

8r~ Qe] 8Qe(x) =

[8rp[ Qe, .1, G] 8Qc(x)

I ]I 4.a

ar"lM-ar"laa-o'

(7.25c) (7.25d)

Previously, an effective action method was introduced by us in the third paper of reference [46] to calculate r; explicitly. The disadvantage of that technique compared with the present formalism lies in tile difficulties connected with fermion renormalization when the fermion degrees of freedom were integrated out at the very beginning. 7.2.5. Applications We have already applied the present formalism to study the weak electromagnetic field in super-

776 96

Kuang-chao Chou et al.• Equilibrium tuuJ nonequilibrium formalisms matk unified

conduction [53] as well as the nonequilibrium superconductivity in general [54], the dynamical behaviour of quenched random systems and a long-ranged spin glass model in particular [55], the quantum fluctuations in the quasi-one-dimensional conductors [57] and the exchange correction in systems with four-fermion interaction [49]. The last topic will be discussed in the following two sections, whereas the random systems are considered in section 8.

7.3. Generalization of Bogoliubov-de Gennes (BdeG) equation As mentioned in the introductory remarks to this section, the self-consistent equations for the order parameter Qc(x) and the complete set of single-particle fermion wavefunctions I/In(x) are known as the BdeG equations which are of Hartree-type without exchange effects being accounted for. There have been some attempts of extending these equations to include the correlation effects with limited success [96]. These authors emphasize the nonuniqueness of the HS [100] transformation and make use of it to derive various approximations. As mentioned in section 7.1, such ambiguity can be avoided by using the generating functional technique with given definition of the order parameter. As we will show in this section, the successive approximations can be derived in a systematic way using the formalism developed in the preceding two sections.

7.3.1. Model The effective action of the system is given by

where i, j are indices of internal degrees of freedom, Oile matrix in this space and .1 o(x - y) = ~(tx - ty)V(x - y).

(7.27)

Using the fermion commutation relation and the matrix notation for Olk, (7.26) can be rewritten as <

I

I[I/It, 1/1] = I/It(x) (ill :t - Ho + !g26 2V(o) )I/I(X)- !g2

J1/1 t(x}61/1(x).1 o(x - Y}I/It(y}Orp(y). (7.26')

The order parameter is defined as Q(x} == g

I

.1 o(X - Y)I/It(y)Orp(y) = g

f

V(x - y)I/It(y, t)61/1(Y, t).

(7.28)

7.3.2. Coupled equations In accord with the result of the last section, up to the two-loop approximation, the coupled equations for the order parameter Qc(x) and the second-order CfPGF are the following:

I)rp[Q] I) x"

-'i.~" -Q() 1

I 0+-0-

=-

f

I" .1 0' I(X - Y)Q(y) - 2111g Sp{ OGc(x, x)} '" 0 ,

(7.29)

777 KlUIlIg-cluw Chou et aI.• Equilibrium and IIOMqIIilibrium formalisms made unified

97

(7.30) with tp(X, y) = ilig20Gp(x, y)O.dp(y, x),

(7.31)

.d ;l(X, y) = .d iJ:(x, y) - IIp (x, y),

(7.32)

with (7.33) where Sp means taking trace over internal indices only, while Tr remains summing over both internal and space-time coordinates. Here (7.34) with (7.35)

7.3.3. Spectral representation For simplicity we consider the stationary states when Q(x) does not depend on time and all Green's functions are time translation ally invariant. Also, we assume the dissipation for both fermions and collective excitations to be small, so that they can be considered as quasiparticles to a good approximation. In particular, the Fourier transformed fermion functions can be expanded in terms of the complete set {",.. (x)} as (7.36a)

(7.36b)

(7.36c) while the spectral functions {",.. (x)} satisfy the following equation (7.37) In eqs. (7.36) and (7.37) E.. , 1.. and N.. are energy spectrum, dissipation and particle distribution respectively.

778 98

Kuang-chao Chou et al.• Equilibrium and nonequilibrium formalisms made unified

It is ready to check that in combination with FDT written as (cf. (4.29»

(7.38) n

Equations (7.36) and (7.37) are just equivalent to the Dyson equation (7.30) in the limit of weak dissipation. The orthonormalization and the completeness of the {"'n(x)} set is then obvious.

7.3.4. Hartree approximation As the first approximation, we neglect the self-energy part altogether, i.e., to set t p (x, y) -+ O. It then follows that 'Yn -+ 0+ and the boson propagator .:1 will not depend on the order parameter. Substituting (7.36c) into (7.29) and setting tr =0 in (7.37), we obtain the well-known BdeG equation

f

V- 1(x- y)Q(y) = g L (Nn -!)"'~(x)O"'n(x) n

(7.39) n

(7.40) An interesting and important feature of these equations is that the energy spectrum {En} should be determined self-consistently with the quasiparticle distribution.

7.3.5. Hartree-Fock (HF) approximation As the second approximation, we keep tp(x, y) but set IIp(x, y) equal to zero. It follows then from

(7.31) and (7.32) that (7.41)

.:1 r (X, y) == .:1 a (x, y) == 8(t" - ty ) ' V(x - y), tc(X, y) == 0,

'Yn ... 0+.

(7.42)

Repeating the procedure carried out in the Hartree case, we find that in the HF approximation the equation for the order parameter Q(x) remains the same as (7.39), but the equation for the spectral functions changes into

(En - Ho- gOQ(X»"'n(X) + g2

f

V(x - y) L OI/Im(x)Nm"':" (y)O"'n (y) = O.

(7.43)

m

Comparing (7.43) with (7.40) we find that the unphysical term !g26 2 V(o) has been cancelled out and a new, Fock exchange term appears here. It is clear from our derivation that the order parameter Q(x) is coupled to the fermion Green functions in the order of " (see (7.29». Hence one should take into account contributions of the same order from tr in (7.37) no matter what kind of interaction one deals

779 Kuang-cMo Chou el m., Equilibrium and nOMquilibrium formalisms made unified

99

with. We see thus how the Fock term has been "lost" in some derivations, but "recovered" in the CfPGF approach.

7.3.6. Higher order corrections We can improve our approximation by keeping also the self-energy part IIp (x, y) in eqs. (7.29}-(7.33). This is the so-calIed random phase approximation (RPA) if only the leading term in II is kept. It is straightforward to check that the equation for the order parameter still remains the same as (7.39) but the equation for the spectral function is now coupled to the collective excitation via the fermion self-energy part !p provided the dissipation is still weak. We come back to this approximation in the next section where the free energy is calculated. As is clear from the presentation, we can in principle continue this systematic process to go on higher order corrections, but we will not elaborate further on them here. 7.3.7. Vacuum fluctuation Before closing this section a remark on the term !g Sp(O)ad(x) in (7.39) is in order. For systems with off-diagonal long-range order like superconductivity this term drops out because Sp(O)=o.

(7.44)

For systems with diagonal long-range order this term will cancel out the divergent contribution of the Fermi sea. By using the particle-hole symmetry the right-hand side of (7.39) can be rewritten as (7.45) n

n

n

where the first sum is carried out over states above the Fermi level, while the second - below it. The rest of the notation is standard. In the next section we will assume for simplicity that eq. (7.44) is fulfilled.

7.4. Calculation of free energy 7.4.1. Functional equation As shown in section 4.2, for systems respecting time reversal symmetry, the potential condition is satisfied and the free-energy functional ~[Q(x)] can be defined in accord with (4.23a). Using (7.29) and (7.36c) this equation can be rewritten as (7.46)

provided the dissipation

')In

is neglected. In the same approximation it follows from (7.37) that

SEn _ t( )A,. () SQ(x) - g"'n X V"'n X +

f "'n
)S!r(En,y,z) 8Q(1)

) "'n(Z.

(7.47)

According to the convention set at the end of the last section, i.e., eq. (7.44) is fulfilled, (7.47) yields

780 100

KUDng-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

(7.48) Insert (7.47) into (7.46), we obtain

(7.49) which can be transformed as

f

&~[ Q(x)] = _ V- 1( - )Q(y) + L (N _ 1) &E" &Q(X)

+ i~ ~

x y

f

dd+l

""

2

&Q(X)

y dd+l z Spe:~~~) Gc(Z,

y)}

(7.50)

by use of (7 .36c) with T/2

T=

f

dt.

-T/2

We would like to emphasize here that the functional equation (7.49) or (7.50) for the free energy is valid for both equilibrium and nonequilibrium systems provided the potential condition is satisfied.

7.4.2. Hartree case In the Hartree approximation when

&~[Ql_ &Q(x) -

& [1 W(;) -2

f

-I

t

p

is neglected altogether, we find immediately

Q(y)V (y - z)Q(z)

]+:-~ N" &Q(x) §g.

(7.51)

with eq. (7.48) being accounted for. For systems in the thermoequilibrium N" is the Fermi distribution (7.52)

the free energy is then given by

f

~[Q] = -~ dd y ddzQ(y)V-1(y- z)Q(z)- ~-1 L In(1+exp[-~(E" - #£)]}. " Taking into account (7.39) and (7.40), the first term in (7.53) can be rewritten as

(7.53)

781 Kuang·chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

101

which obviously is the fermion interaction energy in the Hartree approximation.

7.4.3. Hartree-Fock case In the Hartree-Fock approximation we can proceed exactly the same way as in the previous case and end up with an expression for the free energy containing an extra term (7.54) in addition to that given by (7.53). The physical meaning of (7.54) as exchange interaction between quasiparticIes is obvious.

7.4.4. Random phase approximation (RPA) The RPA case is more complicated. Let us decompose the fermion self-energy as

!p (x, y) = !~l)(X, y) + !~)(x, y) ,

(7.55)

where

t~)(x, y) = ilig20Gp(x, y)O.1 oP (Y' x),

(7.56)

!~)(x, y) = ilig20Gp (x, y)O(.1 p(y, x) - .1 op (y, x».

(7.57)

Correspondingly eqs. (7.48) and (7.50) are rewritten as

8 ( ) 80(x) LE..

1

-2

f L!/I~ 8t~1) f L!/I .. 8!~2) 80 !/I.. 80 !/I.. = 1

....

8[f[0]=_J 80(x)

t

.

-2

V-IO+~(N _!)8E.. +ili

.

£.oJ

..

2

80

(7.58)

.!..JsP (8!~1) G) 80

2T

J(

0,

e

J

1 Sp !(2)_ 8Ge) +--iii 1 8 Sp(!(2)G) -iii -2T r 80 2 T 80 r e,

(7.59)

where the abbreviated notation is used. The second term on the left of (7.58) and the second term on the right of (7.59) can be calculated by the same procedure as previously, whereas the third term of (7.58) and the last term of (7.59) turn out to be of higher order [49]. To calculate the contribution from the fourth term of (7.59) we make use of the identity

f(

2'(2)( r

1

=-2

y, Z

y»)

) 8Ge(z, y) + ~(2)(y )8G.(z, 80(x) 4e ,z 80(x)

f (8I1

r (y, Z) 8I1c(y, Z) ) 80(x) (Je(Z,y)-JOc(Z,y»+ I)O(X) (J.(Z,y)-Jo.(Z,y»,

(7.60)

782 102

KUlUIg·chao Chou et a/., Equilibrium and nOMquilibrium forma/isms made unified

which can be proved using the ~, 1/ vectors and the definition of the self-energy parts. Neglecting terms like Ie and fle which are proportional to the dissipation, taking into account eqs. (7.32) and (7.38) as well as the time translational invariance, the fourth term of (7.59) can be presented as

(7.61) To extract the information about the collective excitations we use the approximate spectral representation for .::l as

(7.62) where w., 1/, Z, are, respectively, the frequency, dissipation and wavefunction renormalization for the quasiparticle. Making use of (7.62), the expression (7.61) can be presented as

M

= ~ 8(licu,) (N 1) £.oJ 80 ' +2 , ,

(7.63)

which is the extra contribution due to the collective excitation. In thermoequilibrium,

N, = [exp(,8liw,) - 1]-1 ,

(7.64)

the additional term in the free energy for the RP A compared with that of HF is given by

PPA -

~HF = f3- 1

L In[l- exp(-f3licu.)) + ~ L liw"

(7.65)

where the last term is the contribution from the zero point motion.

7.4.5. Summary To summarize we note that the calculation scheme presented in this section is rather practical and complete. We cannot only determine the order parameter as well as the energy spectrum, the dissipation and the quasiparticle distribution for both fermion and collective excitation, but also calculate explicitly the free-energy functional in successive approximations without invoking any extra assumptions. It is also interesting to note that for systems with symmetry breaking we can directly integrate out the vertex equation without introducing the symmetry broken ground state in advance. This technique may be helpful for more complicated problems.

783 KlUllIg·chao Chou et aI., EquilIbrium and IIOMquilibrium formalisms made U/lijied

103

8. Quenched random systems In quenched random systems some degrees of freedom associated with impurities are frozen into a nonequilibrium but random configuration. Such situation can be created, say, by sudden cooling of a sample in thermal equilibrium to a state with much lower temperature. The impurities are thus frozen into a configuration separated by high potential barriers from the equilibrium one. Diffusion through these barriers will cause the nonequilibrium state to vary slowly in time. As pointed out by Brout [l01], the space average of an observable A in a quenched random system can be replaced by an ensemble average over the impurity degrees of freedom J.

A=

f

[dJ]A(J)P(J) ,

(8.1)

where P(J) is the distribution function. Most of previous workers considered quenched random systems as if they were static [102}. In this approach one has to average the free energy proportional to the logarithm of the partition function, over the random configuration of impurities. It is a formidable task and a special n-replica trick has been invented. This method is extensively used to study random systems like spin glass [103-110]. Recently, several authors [111-120} have proposed dynamical theories of spin glass based on the MSR [90] statistical field theory. The advantage of the dynamical theory is the possibility of taking quenched average without resorting to the unphysical replica trick. The results obtained so far can be reproduced in most cases by the replica method with special pattern of replica symmetry breaking [121]. Although a kind of plausible physical interpretation of replica has been suggested very recently [122-124}, as far as we understand the whole problem is still controversial. Therefore, a systematic dynamical theory is certainly needed to provide a proper description corresponding to the real experiments. In this section we will show that the CTPGF formalism might be one of the candidates to provide such a dynamical description. As we have seen already, this formalism is suitable for studying slowly varying in time processes, so far as the causality and the FDT are built in the formalism itself. In the CfPGF approach the quenched average can be carried out over the generating functional itself and the counterpart of the Edwards-Anderson(EA) [103} order parameter q appears naturally as an integral part of the second-order CfPGF. As a consequence, a Dyson equation can be derived for q to describe its slow variation using the quasicJassical approximation. In section 8.1 we outline the basic features of the CfPGF dynamic theory for quenched systems, whereas in section 8.2 we apply it to discuss the infinite-ranged Ising spin glass, i.e. the SherringtonKirkpatrick (SK) model [104}. A boundary line of stability is found on the plane q -lxi, where X is the susceptibility. It is argued that the spin glass phase is characterized by the fixed point located at the stability boundary. The magnetization is calculated in perturbation and is found to be in good agreement with those predicted by the projection hypothesis (125, 1261. In section 8.3, we discuss the disordered electron system within the CfPGF framework. The WT identities following from the symplectic symmetry Sp(2) respected by the Lagrangian of the system as well as the localization properties are considered without resorting to the replica trick.

784 104

KlUlIIg-chllQ Chou el al., Equjlibrium and nonequjlibrium formalisms made UIIified

8.1. Dynamic formulation 8.1.1. Model

Suppose the action of the random system is given by

1=

f u(x)r~'O)(x,y)u(y)- f p

V(u(x),J;)+

p

f

(u(x)h(x) + u(x)j(x» + I" ,

(8.2)

p

where h(x) is the external field coupled to the dynamical variable u(x), 1;(x) being random variables with given distribution. The u(x)j(x) term represents the interaction of the dynamical variable with heat bath described by III' u(x) may have one or several components. After integrating over the reservoir degrees of freedom which might be considered as a set of harmonic oscillators, we obtain the effective action Ielf =

f u(x)r~)(x,

y)u(y) -

p

f

V(u(x), 1;)+

p

f

u(x)h(x) ,

(8.3)

p

where (8.4)

with the self-energy part !~) contributed by the interaction term u(x)j(x). Suppose the system is prepared by sudden cooling down to temperature ~-1 at moment to, then r~) satisfies the FDT given by (3.42a).

8.1.2. Averaged generating functional The generating functional averaged over the random distribution of J, is i[h(x)] ==

f

[dJ]P(J)Z[h(x), J] == (Z[h(x),J]h,

(8.5)

f

(8.6)

where Z[h(x), J]

=

[du] exp(iIoIfXt+ = tolplL = to) .

Introducing the generating functional for the connected Green functions as usual W[h(x)] = -i In i[h(x)] ,

(8.7)

W[h(x), J] = -i In Z[h(x), J],

(8.8)

we obtain the average of the field variable

785 KU(/IIg-cllllO Chou et aI., Equilibrium tIIId Itonequilibrium formalisms made Ultified

u(x) = 8W/8h (x) ,

105

(8.9)

and the connected CfPGF as (8.10)

along with the corresponding terms for W[h(x), J]. It follows from (8.5) that i.q(x) = {Zu(x, J)h,

which by virtue of the normalization condition (2.102) reduces to u(x) = (O'(x, J)h

(8.11)

for the physical limit h+(x) = h_(x). Equation (8.11) is what is required for the quenched average. Hereafter we denote all quantities derived from the averaged generating functional by a bar above them, whereas expectation values directly averaged over the ensemble P(J) are presented as (. ·h.

8.1.3. Order parameter Differentiating (8.10) with respect to h(y) and taking the physical limit, we obtain (Gp(x, y; J)h = Gp(x, y)+ iq(x, y),

(8.12)

where q(x, y) =(O'(x, J)O'(y, J)h - u(x )u(y) .

(8.13)

The expectation value O'(x, J) for an Hermitian operator 0' is a real function with equal values on the two-time branches. Hence the matrix is real, symmetric and independent of time branch, i.e. q(x, y)= q(y,x)= q*(x, y),

(8.14)

q(x+, y+) =q(x+, y-)= q(x-, y+) = q(x-, y-).

(8.15)

Edwards and Anderson [103] have introduced the following order parameter qEA = lim (0'(0, J)O'(t, J»

,.....

which looks similar to what is defined by (8.13). It follows from (8.12) that (G.(x, y; J)h = G.(x, y),

(8. 16a)

786 106

Kuang-chao Chou et al., Equilibrium and nonequilibrium formalisms made unified

(G.(x, y; J»)] = G.(x, y),

(8.16b)

(Gc(x, y; J»] = Gc(x, Y) + 2iq(x, y) .

(8.l6c)

It is easy to check that (8.l6a) and (S.l6b) are also true for higher order retarded and advanced functions, i.e.,

etc. The appearance of the matrix q(x, y) is a consequence of quenched average over the random variable Ji • Hence it characterizes the behaviour of the quenched random system.

B.1.4. Free energy After a sufficiently long time the system is expected to reach a steady, yet not necessarily equilibrium state, when the expectation value u(x, J) is no longer time dependent. As follows from the discussion of section 4.2, there exists a free-energy functional such that u(h, J) = -B~[ h(x), J]/Bh (x) ,

(8.17)

if 1m

JGrOt, -~t,

J) dt

=0.

(S.lS)

In accord with (8.l6a), the same is true for the function with bar, i.e., u(h)= -B~/Bh(x).

(S.19)

From (8.17) and (8.19) we find that (S.20) For a smooth function of distribution P(J) with finite moments the order of differentiation and averaging can be changed. Integrating (8.20) yields ~[h 1= (~[ h, J)]

+ terms independent of h .

So far as our formalism is flexible enough to incorporate various composite operators, in particular, the energy density as a conjugate variable of temperature. This way we can exhaust all variables in the free energy and get the equality ~(h] = (.o/'[h, J])],

(8.21)

up to an unimportant constant. We see, therefore, the averaged expectation values of physical variables like magnetic moment, free

787 KUlllIg·chtlO Chou tl al., Equilibrium and lIonequilibrium formalisms made unified

107

energy, etc. can be calculated directly from the averaged generating functional i[h,J). Since the technique of deriving various consequences from a well defined Z[h, J) is highly developed, the predictions of the dynamic theory are unambiguous.

8.1.5. FDT and Fischer law After performing Fourier transformation with respect to the relative coordinates x-y, the FDT for G before average is written as (cf. eq. (3.43a» Gc(k, X, J) = 2i coth(J3k o/2) 1m Gr(k, X, J) ,

(8.22)

whereas for the quench-averaged G we have

Gc(k, X) = 2i coth(.Bk o/2) 1m Or(k, X) - 2iq(k, X) 4i _ "" -k 1m Gr(k, X) - 2iq(k, X) .

(3

(8.23)

0

The retarded function Gr is analytic in the upper half-plane of ko. If Gr(k, X) vanishes as ko-+ 00, the real and imaginary parts of Gr satisfy an unsubtracted dispersion relation Re 0 (k k X) r

0,

,

='!'f 1m Or(k~, k, X) dk' k' _ k o· 1T

0

(8.24)

0

Making use of (8.23) we find in the high temperature limit Re Or(ko =0, k, X) == Or(k o =0, k, X) == - {3

f

dko . 21T (tGc(k o, t, X) - q(ko, t, X» .

(8.25)

For a long-ranged Ising spin model when the space dependence of Green's function can be neglected (8.25) becomes Fischer's relation [127]

x = -Gr(ko = 0, t) = P(l- q(t, t) - u (t». 2

(8.26)

We see thus that the validity of Fischer relation depends crucially on the high-frequency behaviour of the retarded Green function as well as the FDT.

8.1.6. Dynamic equation for q Now we derive the dynamic equation satisfied by the matrix q. The Dyson equation for the quench-averaged function can be written as (cf. (3.10»

tro, = 1,

(8.27a)

taOa= 1,

(8.27b)

788 lOS

KU4IIg-cIuw C'lwu et al., Equilibrium and nonequilibrium formalisms made unijitd

tJjc= -teG..

(8.27c)

Introducing a new matrix

o == iBfc - i coth(.Bkot2) 1m t r] ,

(8.28)

we directly find the Dyson equation for q,

trq = -OG•.

(8.29)

The Hermitian conjugation of eq. (8.29) is given by

qt.=-G.O.

(8.30)

Separating the Hermitian and anti-Hermitian parts of (8.29) and (8.30) we obtain

trq + qt. = - OG. - GrO,

(8.31)

trq - qt. = -OG. + GrO.

(8.32)

In the quasiclassicaI approximation we replace the product of two matrices A and B, AB = (AB + BA)t2+ (AB- BA)t2,

by the classical expression

~~ i ~ ~ ~ ~ i (aA aB aA aB) AB--2 {A , B }P.B. =AB--2 - ak" - , ak" ax -ax

,.

(8.33)

,.

where A, B are Fourier transforms. As seen from the discussion in section 3.3, this approximation is controlled by the inequality 1 - (PO = - I ~1 1 ak"ax,.

o

where

(8.34)

'

0 may be either A or B. In this approximation eqs. (8.31) and (8.32) become (with "~,, dropped) + 010rl2 = ~ {( aq -IOrI2 ao

q

210.12

ax,.

ax,.

)(Im Or aRe G. _ Re G. aim Or) ak"

_(aak'"q -1012 ak" aO)(Im 0 aRe Gr • _ ax,. r

ak"

Re G

•

a~:..Or )} ,

(8.35)

(8.36)

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KlIiJIIg-chao Chou el al., Equilibrium aM nOMquilibrium formalisms made rmijid

Equations (8.35) and (8.36) are transport equations to determine the time evolution of q. They are written here for single dynamic variable. It can be easily, though tediously extended to the multicomponent case. For a homogeneous system in steady state all functions do not depend on the macro-space-time X, then eqs. (8.35) and (8.36) reduce to a single equation (8.37) Using the field theoretical technique Q and They are functions of q. In some cases

IOrl2 can be calculated once the Lagrangian

is specified. (8.38)

Q=Aq

in the first order of perturbation. A nontrivial solution with q -:F 0 exists if the condition (8.39) could be satisfied. For the model to be discussed in the next section (8.39) cannot be fulfilled. Hence either the spin glass is not in a steady state or it cannot be characterized by a nonvanishing q.

B.2. Infinite-ranged Ising spin glass 8.2.1. Model In this section we apply the formalism developed in the previous section to the infinite-ranged Ising spin glass, i.e. the SK [1021 model. For simplicity we consider the soft spin version described by the Hamiltonian '!{ = -~

L l.juiuj + L Grouf + uo-1- u.hi } ,

(8.40)

;'JIj

where the value of spin Gaussian distribution

U'.

is not limited to ±1. The exchange integrals 1., are random variables with

P(liJ) = (21TN/.Pyl/2 exp{-NJ~/2P},

(8.41)

where N is the number of spins interacting with the given one. Taking into account the interaction with the heat bath and averaging over 1.j, we can write the CfPGF generating functional as

I =L f

Z[h.(t)] =

with Sctf

j p

(8.42)

[dud exp(iSctf)(t+ = tolplL = to},

{Uj (t)rop(t, t')Uj (1') - uo-j(t) + Uj (t)hj(t)}+ i :~

L

f

• "" p

U'. (t)Uj (t)

dt

f

U. (t')Uj (t') dt' .

p

(8.43)

790 110

Kuang·chao Chou el al., Equilibrium and nonequi!ibrium formalisms made unified

The notation used here is the same as in the last section, but with the bar over various quantities dropped for simplicity. After Fourier transformation the low-frequency approximation for fo is given by f Or(w, t):; -ro+ iw/fo,

(8.44a)

f Oc(w, t):; 2i/f3fo.

(8.44b)

8.2.2. Self-energy part In the infinite range limit when N ~ 00 the matrix G ii (t, t') can be approximated by liii G(t, t'). In this case the second-order vertex function can be calculated by the diagrammatic expansion. It is found for small q that

f.(w, t) :; - r + iw/ f - J;(q ) Gr(w, t) - Ir(w, t) ,

(8.45)

where r, fare renormalized quantities which might depend on temperature. To the lowest order perturbation in u we find (8.46) where J is also renormalized and q(t, t) is the order parameter. In eq. (8.45) terms proportional to w and Gr are subtracted from the self-energy part, i.e.,

I.{O, t):; O.

(8.47)

In obtaining (8.46) we assumed q(w, t) to be peaked at w :; O. To the same approximation the correlated vertex function is calculated to be 4i

fe(w, t):; -

Bw

1m fr(cu, t}+ 2iJ~(q)q(w, t)+ 2iP.1(w, t)- Ie(w, t),

(8.48)

where

J

.1 (w, t) = dT ei·"u(t+ ~T)U(t -

~T).

(8.49)

The expectation value u(t) might be different from zero when a magnetic field is applied. It can be shown that .1 (w, t) is peaked at w = 0, whereas Ie is the smooth part of the self-energy. To the lowest order in u we find (8.50) In the low-frequency limit the function defined by (8.28) turns out to be

Q(w, t) = -J~(q )q(w, t) - J2.1 (w, t) ,

(8.51)

provided only terms peaked at w = 0 are retained. It is worthwhile noting that J;(q) is greater than q as evident from (8.46) and (8.50). This fact is essential for our later discussion.

J~(g) for all values of

791 KlUIIIg·chao Chou el 0/.• Equilibrium and nonequilibrium formalisms made uni/ild

111

8.2.3. Stability Now we study the stability of the system. As seen from (8.45), in the 1 -+ 0, i.e., the pure limit' is the inverse susceptibility which itself is proportional to the temperature and increases as the magnetization increases. We assume that such qualitative behaviour holds also for random systems. In the zerofrequency limit the Dyson equation for the retarded Green function (8.27a) becomes (8.52) in accord with (8.45), where the magnetic susceptibility

X=-Gr(cu=O).

(8.53)

Equation (8.52) can be solved to yield 1 X = 2J;(q) [, -

v',2 - 41~(q)] .

(8.54)

Thus the magnetic susceptibility increases as , decreases and reaches its maximum at , = 2Jr (q). Further decrease of , will make X complex and the system unstable. Therefore, the stable region is bounded by the inequality

X1r(q) < 1.

(8.55)

It is clear from (8.54) that in the unstable region

(8.56) which is a curve on the plane q -ixi. On this plane all stable points are located in the region bounded from above by the curve (8.56) consisting of marginally stable and unstable points. In the stable region q and X are related by the Fischer relation (8.26). Hence the physical state of the system can evolve either in the stable region or on the boundary (8.56).

8.2.4. Time evolution of q(t, t) Before discussing the time evolution of q(t, t) we first note that in the low-frequency limit G.(cu, t) = - X + a(t)lcul"(coth(m42)- i sgn cu),

(8.57)

where aCt) and II are positive quantities to be determined. An analysis similar to that of ref. [117] using (8.45) shows that II $1/2 if the state is marginally stable and II = 1 otherwise. To study the time evolution of q we start from the Dyson equations in the quasiclassical approximation (8.35) and (8.36). Using (8.51) in the absence of the external field we have

e-

aq = n(q)l~rI2 at 1 + 1~(q)IGrI2

aIO,12/ (a Re Or a 1m Or _ a 1m Or a Re Or)} ~. at

acu

at

acu

at

q

(8.58)

In the low-frequency limit eq. (8.58) can be simplified and after Fourier transformation becomes [55]

792 112

KUIllIg·chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

aq

2

1- 'pfJ2(1- q)2

r

bI = - t--rfflt-7[H1tf-- q) q ,

(8.59)

by use of the Fischer relation valid in the stable region. Above the critical temperature fJ~l = J, the only fixed point of (8.59) is q = O. Near this point the order parameter decays exponentially

q = qin exp(-t/To),

(8.60)

fJ~ + fJ2 fJ To = 2(P~ - fJ2) r .

(8.61)

with

Below Tc there is another fixed point ql on the stability boundary (8.62) The order parameter approaches this point also exponentially with characteristic time

(8.63) It is worthwhile noting that both To and TI have a pole at fJ = fJc, giving rise to a kind of critical slowing down. However, ql is no more a fixed point if higher order terms are taken into account. As seen from (8.58), the nonlinear fixed point is determined by the intersection of Fischer line with

1 = J~(q)lxI2 .

(8.64)

Since J;(q) > J~(q) as we noted before, this curve is outside of the stable region and cannot be reached. In fact, what happens is that after hitting the stability boundary at qh the order parameter further decays along the marginal stability line down to q = O. By inserting (8.57) and (8.64) into (8.58) we obtain

aq(w, - -t)= -4-q2 - 1w11- ~q~(w,t ) . at 3 aJl

(8.65)

Hereafter we use the units J = Tc = 1 and take u = 1/12. It is obvious from (8.65) that q(O, t) does not change with time whereas q(w '# 0, t) tends to zero as t goes to infinity.

8.2.5. Susceptibility and q in a small field It follows from (8.29) and (8.49)-(8.51) that the static fixed point for q is determined from the equation

(8.66) where qo = lim ....... q(t, t).

793 Kuang·chao OIou el aI., Equilibrium and nonequilibrium formalisms made unified

113

For T> Te, the whole Fischer line (8.26) is located inside the stable region, so for small field h we have (8.67)

(8.68) At the critical point the Fischer relation (8.26) still holds to yield (8.69) (8.70) Below Te, there exists a critical magnetic field he above which the static fixed point is still sitting in the stable region. The value of he turns out to be (8.71) near Te with T = 1- p-l. For T < Te and h < he, the intersection of the Fischer line with the fixed point equation (8.66) is outside the physical boundary, so it can never be reached. This false fixed point is just what was found before [104, 105] to yield negative entropy at low temperatures. The only plausible fixed points are those located on the stability boundary (8.56) which in new units appears as (8.72) Solving (8.66) and (8.72) yields (8.73) (8.74) All results obtained in eqs. (8.67)-(8.71) as well as in eqs. (8.73) and (8.74) agree with those predicted by the projection hypothesis [125, 126].

8.2.6. Summary To sum up, we have found from a systematic analysis using the CTPGF formalism that a physical boundary exists on the plane q-Ixl. Above Te, the Fischer line is lying entirely in the stable region and the order parameter approaches the fixed point at this line exponentially in time. Below Te and he there are no fixed points on the Fischer line inside the stable region. In this case the fixed point is located on the stability boundary. In the presence of a persistent magnetic field h the order parameter will first decay exponentially to the intersection point ql and then decreases further along the boundary down to

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KUlJng-chao Chou el ai. Equilibrium and nonequilibn"um formalisms made unified

the fixed point qo. The magnetization calculated for the state qo agrees with what follows from the projection hypothesis. In the absence of magnetic field, qo = O. If the w = 0 component of the order parameter q(t, t) constitutes a finite part of q, say qEA, the system will finally reach a steady state with q = qEA' The value of qEA depends on the dynamic processes with infinite long relaxation time in the N ~oo limit. It is worthwhile to note that the boundary line on the q-IXI plane, and hence the fixed point, is temperature independent. As a consequence, the magnetization is also temperature independent, while the entropy does not vary with magnetic field. This is just the assumption of the projection hypothesis [125,126] which follows from the CTPGF formalism in a natural way. Comparing the results obtained here with those of Parisi [110] as well as Sompolinsky and Zippelius [117-119J, it is natural to conjecture that Parisi's function q(x), D5x 51, corresponds to our q(t, t) varying along the boundary from ql to qo. The present formalism not only elucidates the physical meaning of the order parameter but also provides us with an equation to solve its time evolution.

8.3. Disordered electron system Anderson first showed in 1958 [128] that if the electron site energies in a solid were sufficiently random, some of the energy eigenstates became localized instead of being extended as they would be in a regular crystal. Such localized states will not contribute to the electric conduction. The nature of the state depends on whether the energy value is located on the "localized" or "extended" side of the "mobility edge" (129]. The drastic change in the behaviour of the wavefunction with energy is reminiscent of phase transition. The great success in applying the field-theoretical technique and the renormalization group to critical phenomena [60] encouraged similar attempts in the localization problem. Wegner [130] suggested a nonlinear u-model to study the scaling properties of the noninteracting disordered electron system near the mobility edge [131] with conductance playing the role of coupling constant. This model was later derived from the field theory [132,133] and was further studied by other authors [134-136]. Recently, there is a revival of interest in this problem due to the discovery of the quantized Hall effect in two-dimensional electron system [137,138]. This is a challenge to the theory since according to the existing theory all electron states in disordered two-dimensional system should be localized in the absence of magnetic field [131], whereas extended states are certainly needed for explaining the observed quantized Hall effect. The extension of the field-theoretical approach to include the external magnetic field was made very recently by Pruisken et al. [139,140]. However, almost all studies of field-theoretic approach in the localization problem were based on the replica trick. With n-replicated system all O(n+, n_) or U(2n) symmetry is used to construct the nonlinear u-model. The critical behaviour from the extended state side of the mobility edge is described by a Goldstone mode resulting by virtue of the spontaneous breaking of the replica symmetry. The difficulty of the replica trick is the necessity of continuing n, originally defined for integers, to zero to get the physical result. Such process might not be unique as in the case of spin glass. It turns out that the CfPGF formalism can be also applied to the localization problem without resorting to the replica trick [56]. In this section we describe the symmetry of the model and derive the corresponding WT identities. The order parameter and the symmetry breaking pattern are also briefly sketched.

795 Kuang-choo a.ou el al., Equilibrium and nonequilibrium formalisms made unified

115

8.3.1. Model We are concerned with the effect of disorder on Green's functions of a noninteracting electron gas moving in external fields. The Lagrangian of the system is given by 2=

JI//(X)(i :t-Lo- V) cfr(x) ,

(8.75)

where 1

.

(8.76)

Lo = 2m (-IV - eA(x)f+ erp(x).

and V the random potential. In the CfPGF formalism the generating functional Z[l(x), V] is specified by the effective action which in the single-time representation can be written as (8.77) where cfr(x), lex) are two-component vectors as usual. The vertex function in the tree approximation is given by (8.78) where e is a positive infinitesimal constant. To convinve ourselves in the validity of (8.78) we note that for noninteracting fermions

rOr= E-e(k)+ie,

r

Oa

= E - e(k) - ie,

rOc = 2ie,

(8.79)

as follows from (2.23) and the Dyson equation (3.10). Equation (8.78) is then a direct consequence of (2.63). The generating functional can be thus rewritten as Z[l, V]=

J[dcfr][dl//] exp {i J[cfrt(X)((i :t -Lo-

V)0'3

+ ie(J - 171 + i0'2») cfr(x) + cfrtO'J + ]f0'3cfr ]}.

(8.80)

As shown in section 8.1, the quenched average of random potential can be carried out directly on the generating functional. It is, however, more convenient to work in the energy representation for the localization problem. After Fourier transformation the effective action is given by

f

Soft = d"x dE {cfrt(x, E)[0'3(E - Lo- V) + ie(I - 0'1 + i0'2)]cfr(x, E) + cfrtO'~(x, E) + ]f0'31/1(X, E)}. (8.81)

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KUlmg·chIUJ OIou el al., Equilibrium and nonequilibrium formalisms made unified

We restrict ourselves to the Wegner model [130] where the correlation of the random potential between different energy shells vanishes, i.e., (V(x, E»

= 0,

(V(X, E) V(y, E'» = 18(x - y)8(E - E') .

(8.82)

After averaging over the random potential V the generating functional

I

Z[l] = [dV]Z[l,

V]exp(-2~I V2(X)dx)

(8.83)

is determined by a new effective action S which differs from Self (8.81) by dropping V in the single-particle Lagrangian and a new term (8.84) The generating functionals W[l] and f[ r/lc] are defined in the standard manner so that there is no need to write them down explicitly. We would like to mention, nevertheless, that to avoid the possible confusion with sign of the Grassman variables we adopt the convention that 8/8l(x), 8/8r/1c(x) act from the right, whereas 8/8r(x), 8/8r/1~(x) act from the left.

8.3.2. Symmetry properties Before discussing the specific model under consideration we would like to make a general remark concerning the symmetry properties in the CfPGF formalism. It is well known that the U(l) symmetry of a complex field

corresponds to the charge conservation. In the CfPGF approach we deal with an action defined on the closed time-path which in single time representation appears as

I '"

S=

dt [.
(8.85)

This action, therefore, respects the U(l) x U(l) symmetry, i.e., it is invariant under r/I+(X) -+ exp(ia+ )r/I+(x),

r/lt(x)-+ exp(-ia+)r/lt(x),

r/I-(x)-+ exp(ia_)r/I_(x),

r/I!(x)-+ exp(-ia-)r/I!(x),

(8.86)

where a+, a_ are independent of each other. However, such U(l) x U(l) symmetry is always spontaneously broken, because Fo-+ or Go-+ is different from zero even for vacuum state average as seen from (2.21), so that only U(l) symmetry is

797 KUQIIg-chao CIoou el 01., Equilibrium and nonequilibrium lonnalisms made unified

117

retained. The question whether there are any physical consequences of such spontaneous breaking, has to be studied. Coming back to our model, we note that apart from the source terms and terms proportional to E, the effective action (8.81) (and also the averaged action (8.84)) has a global Sp(2) symmetry keeping '1/ U3r/1 invariant. The function r/I(x) forms a two-dimensional representation of the Sp(2) group transforming as r/I(x)-+ r/I'(x) = Ur/I(x),

r/lt(x)-+ r/lt(x) = r/lt(x)U t ,

(8.87)

where (8.88)

satisfies the condition (8.89) Here AI> A2, A3 are group parameters corresponding to the three generators. The term proportional E does not respect this symmetry. Like a small magnetic field determines the direction of magnetization in an 0(3) ferromagnet, the E term can be considered as a small external field inducing the breakdown of the Sp(2) symmetry. Actually, the Sp(2) symmetry is spontaneously broken by dynamic generation of the imaginary part of the retarded (advanced) Green functions. 8.3.3. WT identities If we make an infinitesimal transformation with group parameters Aj(E) for functions r/lt(x), r/I(x) in the path integral of Z[J] we obtain the following three WT identities corresponding to the three generators of the Sp(2) group:

-2iE =

I I

dd

X

SP{(1 + (71)

ddx [r(x,

(-j 8J\t, ~~(x, E) + 8J~(:'E)8J~:E)]}

E)1718J~(:'E) +8J~:E) utl(x, E)] ,

(8.9Oa)

(8.90b)

(8.9Oc) The WT identities for various Green functions can be obtained by differentiating (8.90) with respect to r(x, E), J(x, E) and then setting r = J = O. As an example, we show how the dynamic generation of the imaginary part breaks the Sp(2) symmetry. Using eq. (2.12) we find that

798 118

Kuang-chQ(} alOu el al.• Equilibrium and nonequilibrium formalisms made unified

(8.91) Taking the functional derivative 'fl/'OP(y, E)'OJ(y, E) of both sides of (8.90b) we obtain

(8.92) As is well known, 1m Or(y, y, E) is proportional to the density of states p(E). It is different from zero certainly for extended states and possibly for localized states as E ~ O. Therefore, the Sp(2) symmetry is broken for both cases. McKane and Stone [133] pointed out that there are two ways to satisfy the WT identities. Although their interpretation is given in an entirely different theory based on the replica trick, we expect it applicable to our case as well. For extended states, the dynamic generation of the imaginary part for the retarded Green function caused by the breakdown of the Sp(2) symmetry leads to Goldstone mode characterized by long-range correlation and governing the critical behaviour from the "extended" side. On the other hand, there is no Goldstone mode with vanishing momentum for localized states, so the integrand on the right-hand side of (8.92) must diverge as E ~ 0 before integration to satisfy the WT identities in this case.

8.3.4. Order parameter and nonlinear u-model As said before, the order parameter breaking the Sp(2) symmetry is proportional to the imaginary part of the retarded Green function. A Goldstone mode is therefore generated. To describe this mode it is convenient to introduce a composite matrix field q(x, E) = ",(x, E)", t(x, E) ,

(8.93)

the vacuum expectation value of which is connected to the second-order CTPGF. Under the Sp(2) group, the field q transforms as (8.94) where U is given by (8.88). The vacuum expectation value of q is (8.95) where the diagonal part of the first term describes the imaginary part of OF and OF, whereas the second term describes their real part. As seen from (8.89) the b term does not break the Sp(2) symmetry, while the a term does. Hence Goldstone modes will be dynamically generated by the condensation of the q field.

799 Kuang-chao Clwu el al.• Eqllilibrilllfl tJNJ nonequilibrilllfl lonnalisms made IDIified

119

In analogy with the earlier work [133], we can derive a nonlinear q-model describing such Goldstone modes and carry out the renormalization procedure to study the scaling behaviour near the mobility edge. However, we will not elaborate further on such discussion here. To summarize section 8, we note that the theoretical framework outlined here for quenched random systems is quite general as well as flexible. It is based on the dynamics of the system itself without resorting to replica trick, so the approximation involved are well under control. Apart from spin glass and disordered electron system discussed in this section, the present formalism can be applied to other quenched random systems as well. In particular, we discussed [141] the controversial problem of the lower critical dimension for an Ising spin system in a random magnetic field [142]. We hope the CTPGF formalism is helpful in solving some of the difficult problems still remaining open in this field. 9. Connection with other formalisms To save space in this paper we attempted to avoid as much as possible digressing from the main subject and comparing the CfPGF approach with other formalisms in passing. We would like such comparison to be concentrated here. Although not so much new information will be presented to experts, hopefully, this section will help the newcomer entering this field to orient himself in the forest of diversified formalisms. We will mainly discuss the connection of the CfPGF approach with the Matsubara technique (section 9.1), quantum and fluctuation theories as low and high temperature limits of the CTPGF formalism (section 9.2) and also the CTPGF formalism as a plausible microscopic justification of the MSR field theory (section 9.3). There are still many related papers not covered in this review for which we apologize to their authors once again.

9.1. Imaginary versus real time technique The Matsubara technique [1-7] using the imaginary time for thermoequilibrium is well developed and highly successful. However, there are two difficulties from the technical point of view. One is associated with the fact that in this technique Green's functions are defined on a finite section of the imaginary time axis (0, -i/3) so the Fourier series expansion in frequency is used instead of Fourier integral. The frequency summation is sometimes quite cumbersome. Another difficulty is connected with the analytiC continuation of the frequency (or time) variable in the final answer. Usually, such a process is rather delicate. We see thus in spite of the great success of the Matsubara technique, a convenient real time formalism would be highly desirable. The CTPGF formalism is one of the possible candidates for this purpose.

9.1.1. Real time diagrammatic technique We have mentioned in section 6.1 that the diagrammatic expansion for thermoequilibrium system at finite temperature is similar to that of the quantum field theory provided the free propagator is given by (6.33). Here we would like to justify this statement using expressions for the effective action derived in section 6.1. The correlation functional W N defined by (6.15) becomes in this case (9.1) where

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Kuang-chao Chou et al., Equilibrium and nOMf/uiiibrium jormaiisms made unified

lith = exp(-n -

(9.2)

/3('Je - J.'N»

where n is the thermodynamic potential, whereas known that for the in-field we have [39]

.pi, .pI are operators in the incoming picture.

It is

(9.3) where '!( is the total Hamiltonian. This point is essential for our derivation. By analytic continuation 7'-+ -i/3 we find (9.4) Taking into account that for a complex field the operator of particle number

is a conserved quantity, it is easy to show that (9.5) where

,\ = 0, if AI(t) is Hermitian. (9.6) Using (9.5) we can apply as done by Gaudin [7] the following identity Tr{(pA(l) + (±)"A(1)p)A(2)-" A(n)} =

Tr{p[A(l), A (2)J,.A(3) - .. A(n )}± {pA(2)[A(1), A(3)J,.A(4)· .. A(n)}

+ _... ,. (±)"-2Tr{pA(2)' .. A(n -l)[A(l), A(n)],.}

(9.7)

to the right-hand side of (9.1) to obtain (9.8) where (9.9)

In deriving (9.8) the properties of the normal product and the particle number conservation are taken

801 KUQIIg-chao Dlou tl aI., Equilibrium and nontquilibrium formalisms made unified

121

into account properly. Note that for nonrelativistic complex fields the operator ",,(x) contains only positive frequencies whereas ",i(x) only negative ones, so that (9.10)

where So-+ is given by (6.8). Substituting (9.10) into (9.9) we find that

F, = ±i

f

8"':(X) Sl;" (x, y)

8"'~(y)

(9.11)

with

a

Sl;"(X,y)= (1+exP

(iP axo -PIL))

-1

So-+(x,y).

(9.12)

It then follows from (9.8) by using (9.11) that

(9.13)

In accord with (6.15) the correlation functional defined on the closed time-path is

(9.14) p

where (9.15)

Equations (9.13) and (9.14) are the contribution of the density matrix to the generating functional. We see thus the initial correlation is actually a Gaussian process described by the two-point correlation function Sl;" (x, y). Substituting (9.14) back into (6.10) and (6.29) we obtain the following expressions for the CfPGF generating functional in thermal equilibrium: (9.16) (9.17) p

where

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Kuang-chao Chou el aI., Equilibrium and nonequilibrium formalisms made unified

(9.18) Gol (l, 2) = Sol(l, 2) -

I

dl d2 501(1, 1)Si;'p(12) So 1(2, 2).

(9.19)

p

It is ready to check that Go and G OI defined by (9.18) and (9.19) are reciprocal to each other and that

the FDT (3.43) is satisfied. Since the Green functions in thermoequilibrium are time translationally invariant, the only role of the second term in (9.19) is to produce Si;'p(l, 2) term in Go(I,2) and cannot appear independently in the final result. Hence we can ignore the difference of SOl and SOl from the very beginning and take (9.20) instead of (9.19). In that case the first and the second terms of (9.18) can be considered as superposition of solutions for inhomogeneous equation SOlp = 1 and homogeneous equation SOlp = O. The numerical coefficient of the latter is determined by the FDT. We see thuG (9.16) and (9.17) are the generalization of the Matsubara technique to real time axis. The advantage of using real time variables in some cases more than justifies the technical complications owing to the matrix representation of the propagator.

9.1.2. Other real time formalisms Several authors previously considered the possible generalization of the Feynman-Wick expansions for the Matsubara functions [143,144]. However, some of these attempts ended up with very involved formalism, whereas the others were difficult to justifiy. We believe the incoming picture adopted here is helpful in avoiding these difficulties. Very recently, Niemi and Semenoff [145] proposed a version of real time technique to study the finite temperature field theory. Their work is close to ours but is still different. The time-path in the complex plane they adopt consists of four pieces (-00, +00), (+00, +00if3/2), (+00 - if3/2, -00 - if312) and (-00 - if3/2, -00 - i(3). Their free boson propagator is given by (9.21) where A

A=

(COSh (J sinh (J ) sinh (J cosh (J

,

cosh 2 (J = exp(f3lkol)/[exp(f3lkol) - 1] .

(9.22)

Obviously, (9.22) is different from that given by (2.26) in our formalism. A detailed comparison of these two versions has to be made by future studies.

9.1.3. Thermo field dynamics For the last ten years Umezawa and coworkers [91] have developed the "thermo field dynamics" and applied it to a number of interesting problems in condensed matter physics. They have adopted a great

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many ideas and techniques from the arsenal of the quantum field theory, especially the operator transformations. So far we have not yet studied all the topics they have covered, and it is hard to make a thorough analysis of the merits as well as the shortcomings of each formalism. It s~ems to us, however, that the extensive use of the generating functional technique, especially the vertex functional in the CfPGF formalism, is advantageous. As we found from the study of the weak electromagnetic field coupled to the superconductor [53] the ambiguity connected with the dynamiC mal!ping and boson transformation occurring in thermo field dynamics [146], can be avoided in the CfPGF formalism. Another merit of the latter is the unified approach to both eqUilibrium and nonequilibrium phenomena, whereas the thermo field dynamics is limited to equilibrium systems up to now.

9.1.4. Kadanoff-Baym formalism We should also mention the Green function formalism developed by Kadanoff and Baym (KB) [147]. These authors do not use the closed time-path, but rather start from the original paper by Martin and Schwinger (83). There are still many common features of these two formalisms. In fact, the G> and G< functions appearing in the KB technique are nothing but G_+ and G+_ in the CTPGF approach. There are many papers applying theKB formalism to different problems in both eqUilibrium and nonequilibrium systems [148]. We will not go on to compare these two formalisms in further detail. The interested readers are referred to their excellent book (147]. 9.2. Quantum versus fluctuation field theory In this section we consider the low and high temperature limits of the CfPGF formalism for thermoequilibrium. It is natural to expect that in the zero temperature limit the standard quantum field theory or its equivalent in the many-body systems should be recovered. In fact, if the boson density is set equal to zero in (2.26), .:h becomes the usual Feynman propagator. Of course, there is no need to duplicate the time axis in this limit. 9.2.1. Critical phenomena Now consider the high temperature limit. As seen from the Bose distribution 1 n(p) = exp[e(p)/T»)-1

for particle number nonconserving system or near the critical point (where the chemical potential II- = 0), the quasiparticle density

n "" T/e(p)~ 1,

(9.23)

e(p)/T~

(9.24)

if 1.

In the ordinary units (9.24) can be rewritten as A ~ k/V2mkBT,

(9.25)

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Kuang·chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

i.e., the characteristic wavelength of elementary excitation is much greater than the thermal wavelength. Hence near the critical point the thermal fluctuations dominate, whereas the quantum fluctuations are irrelevant. This is the basis of the modern theory of critical phenomena. It is worthwhile noting that this classical limit is not described by the Boltzmann distribution which holds when exp[(E(p)- p,(T)/T] ~ 1. Therefore, it is more appropriate to call this limit "super Bose" distribution, as far as the expectation value of at a is of order n ~ 1, so that the noncommutativity of Bose operators can be ignored. We have thus two types of field theory: quantum field theory at zero temperature and classical field theory near the critical temperature. They have many features in common but differ from each other in some essential aspect.

9.2.2. Finite temperature field theory For the recent years many authors study the finite temperature field theory and possible phase transitions in such systems [149]. As mentioned before, we have shown [41] that the counter terms introduced in the quantum field theory for T = 0 K are enough to remove all ultraviolet divergences for CTPGF at any finite temperature, even in nonequilibrium situation. This has been shown also by other authors [149] for thermoequilibrium without resorting to CfPGF. This result is understandable from a physical point of view since the statistical average does not change the properties of the system at very short distance and hence does not contribute new ultraviolet divergences. What we should like to emphasize is that in considering phase transition-like phenomena one must first separate the leading infrared divergent term and then carry out the ultraviolet renormalization which is different from that for the ordinary quantum field theory. 9.2.3. Leading infrared divergence To be specific, let us consider the real relativistic scalar boson theory the free propagator of which is given by (2.26). Nea~ the transition point, the mass m vanishes so the energy spectrum is given by w(k) = \k\ (cf. (2.28». Since terms proportional to the particle density n appear together with the .5-function, i.e., on the mass shell, the integration over frequencies can be carried out immediately to give stronger infrared divergence (k- 2 ) than the other terms (k- 1). Therefore, the marginal dimension of renormalizability for ip4- theory is de = 4, whereas for quantum field theory at T = 0 K the marginal space dimension is de = 4 - 1. This is what is usually meant by saying "quantum system in d dimensions corresponds to classical system in d + 1 dimensions". If we keep only the most infrared divergent terms, then all components of G become (P+ m2t\ i.e., exactly the same as that used in the current theory of critical phenomena [60]. What has been said above can be checked explicitly by calculating the primitive divergent diagrams for mass, vertex and wavefunction renormalization, carrying out the frequency integration and taking the high temperature limit T ~ w(k). The results obtained turned out to be identical to those resulting from the theory of critical phenomena. For example, in 1,04 theory, the primitive mass and wavefunction correction diagrams have quadratic divergence, whereas the vertex correction term diverges logarithmically in four-space dimensions. We know that in the quantum field theory such divergences occur for three-space and one-time dimensions. 9.2.4. High temperature limit in Matsubara formalism The high temperature limit can be easily taken in the Matsubara technique. For example, the free

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125

propagator for non relativistic complex boson field is given by (9.26) where w" := 21fnT, m2- T - Te. Since T ~ k 2+ m2, in the frequency summation to be carried out later we need to keep only the w" := 0 term. Hence the propagator (9.26) reduces to minus the correlation function in the theory of critical phenomena. This fact seemed to be first realized by Landau [150]. Some investigators of finite temperature field theory in the early stage of their work incorrectly used the renormalization constants for T = 0 K to study phase transition related phenomena. As far as the high frequency limit, or, equivalently, the leading infrared divergent terms are picked up, both relativistic and quantum effects are irrelevant. The only exception is the phase transition near T = 0 K when both thermal and quantum fluctuations are essential so a special consideration is needed. Otherwise, the field-theoretic models (including non-Abelian gauge models) cannot provide anything new beyond the current theory of critical phenomena as far as the phase transition is concerned, i.e., they are classified into the same universality classes as their classical counterparts. 9.3. A plaUSible microscopic derivation of MSR field theory

We have mentioned already in section 6.4, that Martin, Siggia and Rose (MSR) [90] proposed a field theory to describe the classical fluctuations. There are several peculiar features of this theory: (i) Being a classical field theory, it deals with noncommutative quantities; (ii) A response field tP is introduced in addition to the ordinary field lP; (iii) Some components of the Green functions should be zero along with their counterparts - vertex functions. Nevertheless, the general structure of this theory is very close to that of the quantum field theory. These authors originally proposed their theory to consider nonequilibrium fluctuations such as those in hydrodynamics, but it has been extensively used in critical dynamics near thermoequilibrium [61,77]. In spite of the great success, its microscopic foundation especially the motivation for using noncommutative variables to describe classical fields, was poorly understood. A few years later, the MSR theory has been reformulated in terms of stochastic functionals as a Lagrangian field theory [87-89]. The noncommutativity of field variables was thus obscured by the continuum integration, whereas the calculation procedure was significantly simplified. Nonetheless, the physical meaning remains not sufficiently clarified. We would like to note that the CTPGF formalism provides us with a plausible microscopic justification for the MSR field theory. In a sense, the MSR theory is nothing but the physical representation, i.e., in terms of retarded, advanced and correlation functions, of the CTPGFs in the quasiclassical (the low frequency) limit. 9.3.1. Noncommutativity First consider the operator nature of the field variable. As we discussed in the last section, in the high temperature limit the noncommutativity of operators can be ignored altogether for static critical phenomena, which implies that all components of the Green functions are replaced by correlation functions. This is no more true for dynamic phenomena. The first term in G++ and G __ (2.26) i.e., (k~ - k 2- m2tl comes from the inhomogeneous term of the Green function equation which in turn is determined by the commutator of operators. If only leading infrared divergent terms are retained, the retarded function

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Kuang-chao Chou el al., Equilibrium and nonequilibrium formalisms made unified

Therefore, to describe the response of the system to the external disturbance we need to keep the next to leading order of infrared divergence. Put another way, the response is less infrared divergent than the correlation function. In the sense of "super Bose" distribution, the commutator of order 1 can be neglected in the leading order of n but not in the next to leading order. This is the physical interpretation for the noncommutativity of a "purely classical" field variable. We would like to emphasize here that the classical field should be considered as condensation of bosons. This is why the "quantum" or wave nature of the classical field comes into play. The deep analogy of the quantum and fluctuation field theories is then understandable. Such parallelism is particularly evident in the functional formulation. The classical path in the quantum field theory corresponds to the mean field, or TDGL orbit in the fluctuation theory.

9.3.2. Doubling of degrees of freedom Next consider the necessity of doubling the degrees of freedom. It has been realized for a long time that to describe the time-dependent phenomena one needs both response and correlation functions. This was, probably, the motivation of introducing the response field Ij; and putting together the response and correlation functions into a matrix function by MSR [90]. In the CfPGF formalism we introduce an extra negative time axis, so also double the degrees of freedom, i.e., to use ifi+, ifi- instead of one ifi. In fact, the MSR response field Ij; = ifi", == ifi+ - ifi- ,

whereas their physical field

The CfPGF formalism is constructed on the functional manifold ifi+ and ifi-, or, equivalently, ifi", and ifie, but in the final answer one should put ifi+ = ifi- to get the physical result. As mentioned before, this is an additional way of describing fluctuations, the physical content of which should be further uncovered. In using the generating functional technique the following external source terms are introduced in the MSR theory r87-891

f

Is = (Iifi + JIj;), where I is the usual source, J the response source. This is rather similar to our generating functional, but with an important difference. As naturally follows from the definition of the closed time path we should set (see eq. (2_71)) Is =

f

(I"'ific +Icifi"').

As seen before, such "twisted" combination is most natural. In fact, the physical source Ie generates the dynamic response in terms of ifi", functional, whereas the fluctuation source I", generates the

807 Kuong-choo Chou el aI•• Equilibrium and nOlU!lJuilibrium fonnalisms made unified

127

statistical correlation in terms of ({Jc functional. Another advantage of such "twisted" combination is that we do not need to introduce any extra physical field in the Hamiltonian like in the MSR theory [87-89). In the CfPGF approach lc is the physical field built in the formalism itself.

9.3.3. Constraints Finally, a remark concerning various constraints imposed on propagators and vertex functions. In the original formulation of MSR [90] such constraints appeared rather difficult to understand. They became more systematic in the latter Lagrangian formulation [89] but remained not so transparent. Within the CfPGF formalism, as shown in section 2.4, they are natural consequences of the normalization for the generating functional and the causality. While in the MSR theory one needs to explore the implications of the causality order by order [151), within the CfPGF framework it is ensured from the right beginning, so that causality violating terms can never occur. As seen before, in the low frequency limit when the MSR theory holds, the CfPGF formalism yields the same results in rather low approximations. We believe, therefore, the CfPGF formalism provides us with a plausible microscopic justification for the MSR theory and indicates how to go beyond it. 10. Concluding remarks It is time now to summarize what has been achieved and what has to be done. (1) The CfPGF formalism is a rather general as well as flexible theoretical framework to study the field theory and many-body systems. It describes the equilibrium and nonequilibrium phenomena on a unified basis. The ordinary quantum field theory and the classical fluctuation field theory are included in this formalism as different limits. The two aspects of the Liouville problem, i.e., the dynamic evolution and the statistical correlation are incorporated into it in a natural way. The formalism is well adapted to consider systems with symmetry breaking described by either constituent or composite order parameters. If different space-time variation scales can be distinguished, a macroscopic or mesoscopic description can be provided for inhomogeneous systems from the first principles. (2) The powerful machinery of the quantum field theory including the generating functional technique and the path integral representation can be transplanted and further developed in the CfPGF formalism to study the general structure of the theory. The implications of the normalization condition for the generating functional and the causality are explored. The consequences of the time reversal symmetry such as the potential condition, the generalized fluctuation-dissipation theorem and reciprocity relations for kinetic coefficients are derived. The role of the initial correlation is clarified. The symmetry properties ,of the system under consideration are studied to derive the Ward-Takahashi identities. Also, a general theory of nonlinear response is worked out. (3) A practical calculation scheme is worked out which derives a system of coupled, self-consistent equations to determine the order parameter along with the energy spectrum, the dissipation and the particle number distribution for both constituent fermions and collective excitations. A systematic loop expansion is developed to calculate the self-energy parts. The Bogoliubov-de Gennes equation is generalized to include the exchange and correlation effects. A way of computing the free energy by a straightforward integration of the functional equation is found. (4) The general formalism is applied to a number of physical problems such as critical dynamics, superconductivity, spin system, plasma, laser, quenched random systems like spin glass and disordered electron system, quasi-one-dimensional conductors and so on. Although most of these problems can be and have been discussed using other formalisms, but, as far as we know, the CTPGF approach is the

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KUIJIIg-chao Chou el al., Equilibrium and IWnequilibrium fonna/isms mcuk unified

only one to consider them within a unified framework. Moreover, new results, new insight or significant simplifications are always found by using the CfPGF approach. (5) In general, systems in stationary state or near it have been studied more thoroughly, whereas the transient processes need further investigation. As far as the formalism itself is concerned, the two-point functions are well under control, but the properties of multi-point functions must be explored further in the future. Our general impression, or our partiality, is that the potentiality of the CfPGF formalism is still great. One has to overcome the "potential barrier" occurring due to its apparent technical complexity to appreciate its logical simplicity and power. It is certainly not a piece of virgin soil, but the efforts of a dedicated explorer will be more than justified. We hope that in applying this formalism to attack more difficult problems in condensed matter, plasma, nuclear physics as well as particle physics and cosmology, its beauty and power will be uncovered to a greater extent. Acknowledgements It is a great pleasure for us to sincerely thank our coworkers, Jiancheng Lin, Zhongheng Lin, Yu Shen, Waiyong Wang, Yaxin Wang and, especially, Prof. Royce K.P. Zia for a fruitful and enjoyable colIaboration. A great many people were helpful to us by their enlightening comments, interesting discussions as welI as sending us preprints prior to publication. Their names are too numerous to enumerate, but we would like to particularly mention Profs. Shi-gang Chen, B.1. Halperin, Tso-hsiu Ho, Yu-ping Huo, P.C. Martin, Huanwu Peng, Chien-hua Tsai and Hang-sheng Wu. To all of them we are deeply grateful.

Note added in proof

After our manuscript was submitted we became aware of some more references [152-158] where the CTPGF formalism was applied to various problems. We would appreciate other colIeagues to inform us of their work using this approach. References [I] A.A. Abrikosov, L.P. Gorkov and I.E. Dzaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, New Jeney, 1963). [2] A.L. Feller and J.D. Walecka, Quantum Theory of Many-Particle Physics (McGraw-Hill, New York, 1971). [3] G.D. Mahan, Many-Particle Physics (Plenum, New York. 1981). [4] T. Matsubara. Progr. Theor. Phys. (Kyoto) 14 (1955) 351. [51 A.A. Abrikosov. L.P. Gorkov and I.E. Dzaloahinskii, Zh. Eks. Teor. Fiz. 36 (1959) 900. [6) E.S. Fradkin. Zh. Eks. Teor. Fiz. 36 (1959) 1286. [7] M. Gaudin. Nucl. Phys. 15 (1960) 89. [8] J. Schwinger. J. Math. Phys. 2 (1961) 407. (9) L.V. Keldysh, JETP 20 (1965) 1018. [101 R. Craig. J. Math. Phys. 9 (1968) 60S. [U) K. Korenman. J. Math. Phys. 10 (1969) 1387. [12] R. MiDs. Propqators for Many-Particle Systems (Gordon and Breach. New York. 1969). [13) A.G. Hall. Molec. Pbys. 28 (1974) 1; J. Pbys. AS (1975) 214; Phys. Let!. 55B (1975) 31.

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Orou et aI., Equilibrium and /lOIIef/uilibrium formalisms made unified

[122) [123) [124) [125) [126) [127) [128) [129) [130) [131) [132) [133) [134) [135) [136) [137)

G. Parisi, Phys. Rev. Lett. SO (1983) 1946. A. Houghton, S. Jain and A.P. Young, J. Phys. C16 (1983) L375. C. de Dominicis and A.P. Young, J. Phys. C16 (1983) L641, A16 (1983) 2063. G. Parisi and G. Toulouse, J. Physique Lett. 41 (1980) 361. J. Vannimenns, G. Toulouse and G. Parisi, J. Physique 42 (1981) 565. K.H. Fischer, Phys. Rev. Lett. 34 (1975) 1438. P.W. Anderson, Phys. Rev. 109 (1958) 1492. See, e.g., D.l. Thouless in the book quoted in ref. [64). F. Wegner, Z. Phys. B 25 (1976) 327, B 35 (1979) '}!l7. E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramankrishnan, Phys. Rev. Lett. 42 (1979) 673. L. Schiifer and F. Wegner, Z. Phys. B 38 (1980) 113. A.l. McKane and M. Stone, Ann. Phys. (N.Y.) 131 (1981) 36. E. Brezin, S.H. Hikami and J. Zinn-Justin, Nuc!. Phys. B 165 (1980) 528. S. Hikami, Progr. Theor. Phys. 64 (1980) 1466; Phys. Lett. B 98 (1981) '}!l8. A.M. Pruisken and L. Schiifer, Phys. Rev. Lett. 46 (1981) 490; Nucl. Phys. B200 [FS4) (1982) '}!l. See, e.g., K. von Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 494; D.C. Tsui, H. Stormer and A. Gossard, Phys. Rev. Lett. 48 (1982) 1559. [138) See, e.g., R. Laughlin, Phys. Rev. B23 (1981) 5652, Phys. Rev. Lett. 50 (1983) 1395. [139) A. Pruisken, Nucl. Phys. B 235 (FS11) (1984) 277. [140) H. Levine, S.B. Libby and A. Pruisken, Phys. Rev. Lett. 51 (1983) 1915, and also Brown University preprints. [141) R.K.P. Zia, G.Z. Zhou, Z.B. Su and L. Yu, unpublished. [142) See, e.g., G. Grinstein and S.K. Ma, Phys. Rev. B 28 (1983) 2588. [143) I.E. Dzyloshinskii, Zh. Eks. Teor. Fiz. 42 (1962) 1126. [144] G. Baym and A. Sessler, Pbys. Rev. 131 (1963) 2345. [145) A.J. Niemi and G.W. Semenoff, Ann. Phys. (N.Y.) 152 (1984) 105; Nucl. Phys. B230 [FSlO) (1984) 181. [146) See, e.g., H. Matsumoto and H. Umezawa, Forts. Phys. 24 (1976) 357; M. Fusco-Girard, F. Mancini and M. Marinaro, Forts. Phys. 28 (1980) 355. [147) L.P. Kadanoff and G. Baym, Quantum Statisitcal Mechanics CN.A. Benjamin, New York, 1962). [148J See, e.g., R.E. Prange and L.P. Kadanoff, Phys. Rev. 134A (1964) 566; D.C. Langreth and J.W. Wilkins, Phys. Rev. B6 (1972) 3189; A.P. Jaucho and J.W. Wilkins, Phys. Rev. Lett. 49 (1982) 762. [149) See, e.g., D.A. Kinhnitz and A.D. Linde, Phys. Lett. 42 B (1971) 471; JETP Lett. 15 (1972) 529; S. Weinberg, Phys. Rev. D 9 (1974) 3357; L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 33'}!l; M.B. Kislinger and P.D. Morley, Phys. Rev. D 13 (1976) 2765, 2771. [ISOJ See, e.g., A,Z"Patashinskii and V.L. Pokrovskii, Fluctuation Theory of Phase Transitions (pergamon, Oxford, 1979). [151) J. Deker and F. Haake, Phys. Rev. A 11 (1975) 2043.

Refermas added j~ proof

[152) 1153] [154J [155) [156] [157) [158]

K.T. Mahanthappa, Phys. Rev. 126 (1962) 329. P.M. Bakshi and K.T. Mahanthappa, J. Math. Phys. 4 (1963) I; 12. T. Ivezic, J. Phys. C8 (1975) 3371. I.B. Levinson, Sov. Phys. JETP 38 (1974) 162. A.L. Ivanov and L.V. Keldysb, Sov. Phys. Dok!. 27 (1982) 482. AL. Ivanov and L.V. Keldysh, Sov. Phys. JETP 57 (1983) 234. Wu Xuan-ru, Liu Chang-fu and Gong Chang-de, Commun. Theor. Phys. (Beijing) 3 (1984) 1.

131

812

34

Progress of Theoretical Physics Supplement No. 86, 1986

Spontaneous Symmetry Breaking and Nambu-Goldstone Mode in a Non-Equilibrium Dissipative System Kuang-Chao CHOU and Zhao-Bin Su

Institute of Theoretical Physics, Academia Sinica p. 0. Box 2735, Beijing (Received December 5, 1985) In a non-equilibrium stationary state with space-time structure a N ambu-Goldstone mode with dissipation arises as a consequence of the spontaneous symmetry breaking. A laser type model is considered as an illustration. From the Ward identity in the scheme of the closed time path Green's function (CTPGF) it is shown that the Nambu-Goldstone mode splits into two waves with different dissipations.

Spontaneous symmetry breaking and the N ambu-Goldstone mode playa very important role in contemporary physics. It is with feeling of high appreciation to write an article to honor one of the founders of these fundamental concepts, Professor N ambu's sixty-fifth birthday. Over the past thirty years our understanding of the micro world is much richer because of the valuable contribution of Professor Nambu who initiated many important research directions in particle and solid state physics. In a stable stationary state of a nonequilibrium open system, variation of the external parameters sometimes causes bifurcation with spontaneous symmetry breaking and the generation of Nambu-Goldstone mode. Typical example of this kind is the laser system, where the vector potential develops a non-vanishing vacuum expectation value when the inversion of electron occupation number exceeds a critical limit. It breaks spontaneously the phase symmetry and generates a N ambu-Goldstone mode. As far as we know most laser theories deal with the coherent wave by semiclassical method without paying attention to the existence of the N ambu-Goldstone mode which in its turn is also a light wave of the same frequency with a tiny width caused by the dissipation of the laser system. It is the aim of the present paper to study the properties of the N ambu-Goldstone mode generated in a nonequilibrium steady state of a laser-like system. The field theory applicable to nonequilibrium system was first developed by Schwinger in the early sixties and by Keldysh and many others later. 1 ),2) A recent review of this method which is called closed time path Green's function (CTPGF) was published in Physics Reports. 3 ) We shall use the formalism developed there without detailed explanation. For further reference the reader should consult the original literature. In the first part a model Lagrangian of a laser-like two-level system is studied and the U(l) phase symmetry is considered to be broken spontaneously by the variation of the occupation number of the two levels. AWard identity is written down which insures the existence of a N ambu-Goldstone mode accompanied by the spontaneous symmetry breaking. In §2 the two-point retarded Green's function is solved and a split of the N ambuGoldstone mode into two waves with equal half probability and different dissipations is found. In §3 it is shown how the broken symmetry is restored and the coherent wave goes

813 Spontaneous Symmetry Breaking and Nambu-Goldstone Mode

35

gradually into one of the N ambu-Goldstone modes with less dissipation. Conclusions and speculations are presented in the last section. Some of the contents of the present paper were published by the authors in Chinese in Ref. 4). § 1.

Model Lagrangian and the symmetry breaking

Consider a two-level system interacting with a scalar field aex). The Lagrangian is

.l=rlh*(i ~ -EI)r/J1+1h*(i ~ -Ez)r/Jz+a*(i ~ -ko)a+g(r/Jz*r/Jla*+r/JI*r/Jza).

(1)

The Lagrangian (1) has a U (1) symmetry

(2)

For simplicity we have neglected the space dependence of the fields involved. In general there are random interactions of the system with the surrounding causing dissipation. To a first approximation we shall include them in the energies E •• i =1.2 and ko. widths rio i=I, 2 and r. respectively. The state of the system is described by a density matrix p and the dynamics can be formulated in terms of a path integral along a closed time path P starting from t = - 00 to t= +00 (positive branch) and back from t= +00 to t= -00 (negative branch). The generating functional is ZU(x}] = eiW[J(ZI) =![dr/J ] [dr/J*][da] eiJp[.£(ZI+JO(zla(zl+aO(z)J(zIWZ

x
where the classical field ac(x) is defined to be

oW

ac(x) = o]*(x)

When ](x+) is put equal to J(x-) the field ac(x+) = ac(x-) is just the average value of the field a(x). The equation of motion for the classical field ac(x) is - ]*(x) .

It is more convenient to write ac(x) as

(5)

814

36

K. C. Chou and Z. B. Su

(6)

ac(.r) =A(t)e- ikot

and assume the amplitude A( t) is a slowly varying function of time, i.e.,

Equation (5) for ac(x) without external source has been calculated in Ref. 5) by summing over all the fermion loops. The result is (8)

where (9)

and

gZIAIZ (rl + rz)Z - rlrZ (ko-LlEo)z+(rl+rz)2

K(A) -

(10)

with LIN = Nl - Nz as the difference of the occupation number and LlEo=E1 - Ez that of the energy. Equation (8) clearly shows that A has a stable point at

IAlz =

rl rz (LIN - LIN ) r(rl +rz) c ,

(11)

provided the inversion of the occupation number LIN exceeds a critical limit LINe

gZ(r1r+rz) [(ko-LlEo)z+(rl+rz)2].

(12)

In this case the field a(x) will develop a nonvanishing average value, thus breaking the U(l) symmetry spontaneously. § 2.

Ward identity and Nambu-Goldstone mode

The Ward indentity for the system has the following form: a,,(p'(x»=i(

8~fx) a(x)-a"(x) 8:'~X»)'

(13)

where the current (14)

The charge j/ is proportional to the difference of the occupation number LIN. We assume that LIN is kept fixed by some external means so that jl is conserved separately. This means that the system is under a stationary environment which makes the inversion number stationary. Then the current ja" will be conserved by itself, whose charge integrated over unit volume consists of two parts (15)

815 37

Spontaneous Symmetry Breaking and Nambu-Goldstone Mode

where

and (16)

Here we have used the capital X or T to denote the center of mass space-time variables which are slowly varying compared with the microscopic processes described by the difference of the two point coordinates in the Green's function. Though the total charge Q is conserved in time there is still transformation of charge from the coherent part IAI2 to that of the N ambu-Goldstone part as time goes on. We shall see this shortly in the following. Doing functional differentiation 8/8ac and 8/8ac* on the Ward identity (13) and putting external source equal to zero in the end we get after integration over a closed time path P that L(rl1(.x, y)ac(Y) - r 12 (.x, y)ac*(y»d 4y=0 , (17)

where

(18)

Equation (17) written in the single time formalism is

firl~(X, y) ac(Y) - rl~(X, y) ac *(y»d 4y =0,

f(r2~(X, y) ac(Y) - g~(x, y}a c*(y» d 4y =0,

(19)

where

and

(20)

are the average value and the retarded vertex function respectively. In order to separate the time dependence of the slowly varying amplitude with the fast moving phase we write r l1 (X' y), r 12 (x, Y»)=(~ll(X' y) e~ikO('X-'Y), !12(X, y) ~-ikO('X+tY») ( r 21 (X, y}, r 22 (X, y) r21(X, y) e,ko('x+'y), r 22 (x, y} e,kO(tx-t y )

(21)

816 38

K. C. Chou and Z. B. Su

and

f (27r)4e d q 4

;; ( X, y) --

1 Ij

-Iq·(x-y);; ( l,j

q, T) .

(22)

The retarded vertex function consists of a free part arid a self-energy part (23)

with r,l/J( lJ

q,

T) =(qo- ko, 0 ) 0 k . , -qo- °

(24)

Now we can expand Eq. (19) to first order of slowly varying amplitude under th assumption that (25) and obtain

rl~(q=O, T)A(T)-rl~(q=O, T)A*(T)=-i~t,

r2~(q=0, T)A(T)-r2~(q=0, T)A*(T)=-i~~

.

(26)

To first order approximation we write (27)

8 is considered to be a small quantity of first order. From Eq. (26) the condition for th, existence of nonvanishing solution A*O is det

-R( _

.8

-

-~l~(q=O, T) =0. -r2~(q=0, T)-i8/2

r l l q-O, T)-z2' r-21 ( q=O, T),

(28)

The retarded Green's function is the inverse of the retarded vertex function. Therefor 15If;(qo, T)rf,.(qo, T) =8 0 . , r.1( qo, T) 15M qo, T)

=8

(29)

ik •

To first order approximation with the assumption (25) under consideration we have r i1(qo, T)

= rl}(qo=O,

0).

T) +(qO' 0, -qo

With the help of Eq. (28), Eq. (29) can be solved to give

(30)

817

39

Spontaneous Symmetry Breaking and Nambu-Goldstone Mode

1 (f2~(O, T) -qo, - fl~(O, T) ) (qo+i8/2)(qo+fl~(0, T)-f2~(0, T)-i8/2) -f2~(0, T), fl~(O, T)+qo . (31)

From the general properties of the Green's function it is easily proved that fl~(O, T)*=f2~(0, T), fl~(O, T)*=f2~(O, T).

(32)

fl~(O, T) - f2~(0, T) = purely imaginary .

(33)

Hence

The Lehmann representation also tells us that the residue of the pole for the Green's function J5~ should be real requiring that fi~ to be purely imaginary. Therefore we have fl~(O, T) = - f2~(0, T) =purely imaginary.

(34)

Finally Eq. (31) can be put into the form -R _ 1 1, e 2i _) 1 Du(qo, T) - 2(qo+ i8 2) - e- 2i.. , -1 + 2(qo+ i(2G+8 2»

(35)

1 -R( ) G=-yr ll 0, T -8/2

(36)

ftz(O, T) = iGe 2i"

(37)

where

and

with both G and P real numbers. Causality requires that

8>0

and

(38)

The retarded Green's function D~(qo, T) can be obtained from J51f;(qo, T) by a shift qo ..... qo-k o ,

(39)

i.e., (40)

In the ordinary case of equilibrium phase transition where the condensed field has no space-time structure, the Nambu-Goldstone mode is a zero energy mode with a pole at qo =0. In this case the dissipation Imrl~ is proportional to qo and becomes very small as qo ..... O. The causality then requires that 8 =O. Therefore the condensed field will stay constant when the system is in equilibrium. However, the Nambu-Goldstone mode excited will still split into two waves of equal weight. One of which propagates without dissipation while the other propagates with dissipation unless Imrl~ is identically vanish-

818 40

K. C. Chou and Z. B. Su

ing as in the case of pure vacuum condensation. § 3.

Transport equation and the restoration of the symmetry

In the model discussed above the dissipation ImTi'f has a finite limiting value as the energy Qo approaches the pole. It is plausible that the dissipation of the second NambuGoldstone mode 2G + 8/ 2 is much larger than that of the first wave. We shall adopt this assumption and justify it later. The occupation numbers of the quanta of the two Nambu-Goldstone waves are defined through correlation Green's function and will be denoted by Nlfl( T) and NJfl( T) respectively. It has been shown in Ref. 5) that a transport equation for Ni;( T) can be deduced from the Schwinger-Dyson equation of the two-point Green's function in the formalism of CTPGF. The result neglecting higher orders of slowly varying function is the following:

a~ (Ng l) = - 8Njfl+ ; nf'( T), a~(NJfl)= -(4G+8)NJf' + ; nf'( T)

(41)

together with the equation for the amplitude A( T) and the conservation equation (42)

a~(IAI2+ ~ Nlf'+ ~ Mfl)=O .

(43)

In (42) rii+(ko, T) + rir,,-(ko, T)

l,-r1(IOI

2i

t

1

=z( Wabs(ko, T) + Wem(ko. T»,

(44)

where Wabs and Wem are respectively the probability of absorbing and emiting one quanta of a(.r) field per unit time. There are two time 'scales in Eqs. (42) - (44) r,=1/8,

rc=l/ (4G+8).

(45)

Since G is of the order of nl~ which is the same time scale for the amplitude A to increase from the unstable point A =0 to the stable point of the condensed phase it should be very fast. Therefore it is plausible to assume that r,~rc and the G wave will reach its saturation point (OI r 11 ( C I -1 _ N 11 i 4G+8

much faster. At the same time by the conservation equation

IA/ 2 +1.. N g) 2

(46)

819

41

Spontaneous Symmetry Breaking and Nambu-Goldstone Mode will also reach its saturation value

nf) IAI 2+lN(I)=l 2 i 20 II

(47)

.

In this time scale it is not possible to excite many quanta for the 0 wave. Therefore Nff) <::IAI2 and

(48) This is just the result for the width obtained by entirely different method in the semiclassical laser theory. 6) In the time scale r, we see from Eqs. (42) and (43) that eventually (49)

A-+O

and

r

(O)

N 11(Il-+_IIio.

(50)

This means that the coherent phase order of the field a(.x) will gradually disappear and the system will restore the original symmetry where the phase is in the disordered state. However, the corresponding Nambu-Goldstone modes are still there. The energy is concentrated in the 0 mode with less dissipation. It is interesting to notice that the coherent wave and the N ambu-Goldstone 0 mode have the same tiny width caused by dissipation. § 4.

Conclusions

We have considered a nonequilibrium open system in its stable stationary state where the symmetry is spontaneously broken by the condensation of an order parameter with space-time structure. In this case the N ambu-Goldstone mode splits into two waves with equal half weight and different dissipation. The coherent wave described by the phase of the order parameter increases very fast to its saturation value and gradually decreases, inducing a tiny but finite width of the coherent wave. The energy eventually goes to one of the N ambu-Goldstone waves, the 0 mode whose width and frequency is the same as that of the coherent wave. The only difference between the coherent wave and the 0 mode is that the former is a classical wave while the latter is quantized. It is interesting to find experiment to distinguish the one from the other. We shall not discuss it further here. Though only a simple model has been analysed in the present paper, we believe that the splitting of the N ambu-Goldstone mode is a quite general phenomenon for systems where dissipation plays an important role~ References 1)

]. Schwinger, J. Math. Phys. 2 (1961), 407.

2)

L. Keldysh, JETP 47 (1964), 1515. See, e.g., D. Dubois, in Lectures· in Theoretical Physics, ed. W. E. Brittin (Gordon and Breach, N. Y., 1967), v. 9C.

820 42

K. C. Chou and Z. B. Su D. Langreth, in Linear and Nonlinear Transport in Solids, ed. J. Devreese and V. Van Doren (Plenum, N. Y., 1976).

Kuang·chao Chou, Zhao·bin Su, Bai·lin Hao and Lu Yu, Phys. Rep. 118 (1985), l. 4) G. Z. Zhou and Z. B. Su, Acta Physica Sinica 29 (1980), 618. S) G. Z. Zhou and Z. B. Su, in Progress in Statistical Mechanics, ed. Bai·lin Hao and Lu Yu (1981) (in Chinese). 6) M. Sargent lIl, M. Scully and W. Lamb, Laser Physics (Addison·Wesley, Reading, Mass., 1974). M. Lax, Phys. Rev. 145 (1966), 110.

3)

821 Volume 123. number 5

PHYSICS LEITERS A

10 August 1987

THE CAiNONICAL DESCRIPTION AND BOHR-SOMMERFELD QUANTIZATION CONDITION FOR THE FRACTIONAL QUANTUM HALL EFFECT SYSTEM Zhao-bin SU, Han-bin PANG, \,an-bo XIE and Kuang-chao CHOU IflStiluleo/Thearelical Physics. Academia Sinica. P.O. Box 2735. Beijing. PR China Received Il May 1987; accepted for publication 1 June 1987 Communicated by D. Bloch

In this note. by introclucins the canonical description which is one dimensional in nature, we establish' the Bohr-Sommerfeld quantizatiOll condition for the fractional quantum Hall efTec:t system. The generic and model independent pbysical implications of this condition are explored. The guidins center representations 'of tbe N-body Scbr6dioger equation for the system is also derived.

Since the fascinating discovery of the integer [1) as well as the fractional [2) quantum Hall effect, the fundamental physiCs of the two-dimensional (2D) electron gas in a strong magnetic field B has attracted -great interest. For th. fractional quantum Hall effect (FQHE) [2), the experiments show that there is a series of plateaus in the Hall conductivity a"y with resistivity p"" minimum at certain "magical" rational values of the dimensionless dt:nsity (fillina factor) /I, where /I is the mean number of electrons in the area 271:.<12 tfJrIB covered by the unit flux quatum tPo=hcle. These phenomena sUge5t [3) that a gap exists in the electron spectrum. The general belief is that the Coulomb interaction between electrons gives rise to this gap when the lowest Landau level is only partially filled. The main problem is then why the gaps only occur for certain rational filling factors. Lauahlin [4] proposed an elegant trial wavefunction mainly for the ground state with filling factor 11m where m is an odd integer. Using this wavefunction, one can explain many aspects of the FQHE in a satisfactory manner. Moreover. extensive numerical studies [5] also show strong evidence that there are cusps in the ground state enefIY E(,,) as a function of II at certain rational values of it. To investigate the excited states, Girvin et al. [6) use an approach similar to that ofFeynman in studying liquid 4He. Recently, Kivelson, K.a11in, Arovas and Schrieffer (KKAS) [7) have proposed an interesting "cooperative ring exchange" mechanism using which these authors try to construct a dynamical theory in the many-body coherent state representation. As fas as we know, the canonical description and the Bohr-Sommerfeld quantization condition can play an essential role for the dynamics of electrons in a magnetic field [8]. Furthermore, the path integral in the c0herent state repieSentation is not as neat as that in the canonical description [9). In this note, we introduce the canonical deSCription and establish the N-fermion Bohr-Sommerfeld quantization condition for the FQHE. Moreover, we also derive the N-electron SchrOdinger equation for a system with Coulomb interaction in a strong magnetic field usiDi the guiding center representations. According to the canonical description, the quantum dynamics of electrons in the FQHE regime is essentially one dimensional, although the physical system itself is two dimensional. In fact, the 20 guiding center space is just the phase space of the 10 canonical coordinates. The Bohr-Sommerfeld (BS) quantization condition for the N-electron (fermion) system in the lowest Landau level (LLL) can also provide us with an interesting intuitive picture. It explains semiclassica1ly why electrons in the LLL have to be distributed in accord with a kind offlux quantization rule. It also explains, to some extent, why the ground states afthe FQHE system will have filling factors 11= 1I(2m+ 1) with even denominators being excluded and the excited states are quantized.

=

0375-9601187/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

249

822 Volume 123, number 5

10 August 1987

PHYSICS LETTERS A

Therefore, the existence of a gap in the excitation spectrum for these ground states with magic filling factors follows from the BS Quantization condition in a natural way. The guiding center representations of the Schr6dinger equation we derive for N interacting electrons in the LLL case are consistent with the general discussion by Girvin and Jach [10). Moreover, our derivation can be easily generalized to higher Landau levels. Consider a 20 electron gas with Coulomb interaction in a strong magnetic field. The N-electron hamiltonian Ii has the following form,

H=

I (I7-+A(r;) 0 )2 +l- L - I,

LN -

;=12

N

(I)

eao;<jlr;-rjl

lor;

where the hamiltonian as well as the energy are scaled by flwc with we = eBlmc being the cyclotron frequency, the 20 electron coordinates are scaled by the magnetic length l with l2=flcleB, the vector potential A(r;) is scaled by flclel with its curl equal to I. ao is the Bohr radius, whereasc, m, e, E are the speed oflight, the effective mass, the electron charge and the dielectric constant, respectively. As usual, we assume that the Landau level spacing as well as the Zeeman energy for the electron spin are much larger than the average Coulomb energy _e 2/E..1.. In the hamiltonian (I) and hereafter, the positive charge background is assumed to be included implicitly. Introduce the guiding center coordinates for the jth electron

(.i;, f;)=(x;-h;y,y;+h;x).

(2)

with ti;= -iolor;+A(r;). These quantities satisfy the following commutation relations [11)

111] =tS;j, (3) and .i;, 1; commute with hi, 111, where h;= (h;x -ih/j)/Ji,. Making use of the 20 Fourier transformation [.i;, 1'j) =itS;j,

[h;,

of the Coulomb interaction and then expressing the electron coordinates in terms of (.i;, Y;) and (h;, fir) in accord with eq. (2), we can derive the following general expression for the hamiltonian (1) as Ii=

N

.L

t= 1

(firh;+~)+ -

l

d k 1 L. f -2 -Ikl exp( -1IkI 7t N

eao

2

2

)

exp[k(fir

t<j

-111)] (4)

where k=(kx -iky )/J2 and \k\2=k~+k~=2kk*. Since the subspace of the LLL is defined as h; I ) :=0, i= 1,2, ... , N, the contributions from the h; degrees of freedom will be projected out neatly. Recalling the commutation relations (3a) we can identify .i; with the generalized coordinates 0; and f; with the generalized momenta Pi;

.i;=0;, Y;=P;;

[O;,Pj]=itS;j

(5)

and

H-H[O, p) =

J:... i eao

;<}

f

2 l d k -k exp( -lkI 2 /2) 27t I I

exp[ikxC~; -aj) +iky(P;-Pj )]

(6)

for the LLL. Here we have combined the constant term with the positive background. Eqs. (5) and (6) provide us with a complete and explicitly defined canonical description for the LLL subspace. As mentioned above, it reduces precisely to a one-dimensional N-particle quantum mechanical problem. For a comparison with the generally interested Laughlin type wavefunctions, express kxQ+k~ in terms of guiding center operator as k2 + k* tt with t = (j{ + i f)1 introduce the guiding center coherent representation, then define from the guiding center representation for the state vector

Ji"

250

823 Volume 123. number S

PHYSICS LETTEItSA

10 AUl!ust 1987

Using a similar alaebraic technique as before, it is easy to derive from eq. (6) that IJf(ZT, ...• ~; t) satisfies the following equation

*f

ift IJf(ZT, ..·, zt; i)= L V ;<j

.. /2

1t6tlo 0

d6 :exp[ -1 cos2 6 (Zr-Z1)(%Z1' -oloZf)]: IJf(ZT, ... ,~; t), (7)

where the normal product: : shifts the %zr to the right of Zr by definition. Eq. (7) is consistent with the general discussion of Girvin and Jach [10]. Since eq. (7) is homogeneous in Zr's of the order zero, its eigenfunction is a homoseneous function of Zr's. In the sense of the WKB approximation, i.e., in the long-wavelength limit, we can show that Laughlin's wavefunction is an approximate eigenfunction of (7). Ifwe start from the hamiltom. in the form of (4). we can surely derive a much more general Schr6dinger equation which can be applied 10 cases beyond the LLL. To establish tibe BS quantization condition we will follow the main lines of Gutzwiller [ 12] and Maslov [13] for the single particle case. trying to generalize them to many-particle systems with Fermi statistics. Introduce the traced propagator ~E)=Tr(E+i,,-b)-1

~E)=-i j dTG[T) exp(iET),

(8a)

o G[T)=

L.,. san ~

f

dQ

<~QI exp( -iRT)IQ) ,

(8b)

where Q(Qh •.•• ~), dQ= (dQh ...• dQN), ~ is the N-particle permutation operator, sgn ~ being the sign of ~, ~ means a permutation operation on Q and ~" means a summation over all the possible N-particle permutations. Clearly, eqs. (8a) and (8b) include all the information needed for the N-particle system with Fermi statistics. Using the canonical path integral representation for eq. (8b). we will make three successive stationary phase approximations (SPA) for each term in the traced propagator ofeq. (8). The first SPA is applied to eq. (8b) for pickinl up the classical path which satisfies the classical canonical equation of motion with Q as its initial coordina~es and ~Q as the final ones. The second SPA is applied to eq. (8a) with respect to the intearation over variable T. The purpose of this step is to pick up the classical characteristic function W. [~Q, Q; E) which is a L¢lendre transformation of the action. The third SPA applied to eq. (13) subsequently conceros the tracina in Q space and its purpose is to pick up the closed smooth orbit in the phase space (Q. P). The corresponding SPA condition is that the arriving (final) canonical momentum equals the departing (initial) momentum at each Q; since we are dealing with the classical orbit with its fmal coordinates being just a permutation of the initial ones. If we forget for the moment about the nondistinguishness of particles, after these successive SPAs, we can actually end up with a smooth periodic orbit in the phase space. This is one of the key procedures to incorporate the principle ofnondistinguishness of identical particles in a semiclassical way. From now on we will mention the periodic orbits being always understood in this sense. The final approximate expression for tbe traced propagator !J(E) becomes, then

~E) - ~ SID ~(

f

d(8QI) 1i5(E) 1112[D)(E)]-112 ) exp( -!irm:+iWc.p[t¥Q. Q; En,

(9)

251

824 Volume 123. number 5

10 August 1987

PHYSICS LETIERS A

where Wc,p[ £;Q. Q; E) is the classical characteristic function for the periodic orbit in phase space and it can be proved that Wc,p[ £;Q. Q; E) =

f

PIC)

'dQ(e) •

(10)

where the integration is taken over the periodic orbit. while Q(e) and P(c) are the conjugate canonical coordinates and momenta. respectively. The phase factor - linx is contributed by the singular part of the quantum fluctuations following from the first and the second SPA producedures where n is the number of times the particles hit the (10) turning points. ID(E) I is the corresponding nonsinguiar part of quantum fluctuations [9.12). D3(E) is also the quantum fluctuations obtained in the third SPA procedures [12]. and fd(~Q,) is due to the existence of a zero mode describing the shift of ~Q, along the periodic orbit. To derive the BS quantization condition for the many fermion system. we will sum up a series of periodic orbits where the final states are just simple cyclic permutations of the initial one involving only N, particles with N, ~N. Since a general permutation can always be decomposed into a direct product of such simple permutations. the following discussion for the cyclic permutation can be easily generalized to the general N-particle cases, (It is interesting to compare this with the "cooperative ring exchange" idea of KKAS [7).) Consider a series ofsimple cyclic permutations {g,~}. s= 1.2... with g,o as a basic simple cycle which transforms Qi ..... Qi+ h i= 1.2•...• N, with QN. =Q,. The contribution of such series of periodic orbits to the traced propagator (8) is given by

+,

- J, sgn £;~( J

d(6Q,) ID(')(E) 1"2[D3 (E)]-"2 ) exp( -lin,x +

iWc.p[£;~Q. Q; ED .

(11)

It can be readily proved that Wc.p[£;~Q. Q; E] =sWc,p[£;oQ, Q; E) and ID(')(E) I D~')(E) is independent of s. Moreover. there are two interesting observations. Firstly. the sth orbit bounces the tuming points two times more than the (s-I)th orbit. i.e.• n,=n._,+2 due to the 10 nature. Secondly. for simple cyc1.es. sgn 9t=1 if N, = odd and sgn ~b = (-1)' if N, = even. Then. eq. (11) can be summed up and is proportional to -lI{l±exp(iWe.p[~oQ.Q;E])}.

(12)

where the .. +" sign corresponds to N, being odd and" -" sign corresponds to N, being even. Since the zero of the denominator of eq. (12) will contribute a pole to the traced propagator ~E). it leads to be BS quantization condition

,p(e) 'dQ(e) = (2m+ I )x. N=odd.

=2mx.

N=even.

(13)

So far we have chosen g, 0 as a simple cycle transforming Qi ..... Qi+' and. then. taking into' account that Q, = X;, Pi = Y1• and also noticing that the (Q, P) phase space is just the 20 guiding center coordinate space (X, Y), the BS quantization condition can be equivalently formulated as: A periodic orbit consisting of N, electrons is possible only if

A/2x.,p =1/J1;u=m+ l. N=odd, =m,

N=even,

(14 )

where A is the orbit closed area and I/J the orbit closed magnetic flux. In eq. (14) we have recovered the dimensions. To study the physical implications of the BS condition we first consider the ground state. It is certainly reasonable to assume that the electrons in the ground state are distributed homogeneously. Then. consider arbitrarily chosen three nearest electrons. Presumably, the successive hopping between themselves can be described 252

825 Volume 123, number 5

PHYSICS LETTERS A

10 Ausust 1987

semiciassically by a periodic orbit connecting them and it is consistent with the motion of the rest N - 3 electrons. Due to the symmetry consideration as weD as the 2D geometry, such an orbit will close an areajust equal to one half of tile average area occupied by each electron. Therefore, the BS condition (21) leads to ll( 2m + 1) with even denominators being prohibited. The excitations are in principle charge density fluctuations. Within the semiclassical picture. the density fluctuation corresponds to the distortion of certain periodic orbits which are homogeneously distributed in the ground state. The BS condition requires that the distortion of periodic orbits inducing inhomogeneity cannot proCeed in a continuous way due to the constraints of its discreteness. This leads to one more interesting implication, namely, the excitations upon ground states with odd " are quantized and the existence of a gap is understandable. As shown above, the BS quantization condition follows mainly from the Fermi statistics and the ID nature of the FQHE system. Although it is established in the semiclassical sense, it is so profound in nature that the above discussions are generally applicable, model independent and even do not depend on the concrete form of the interaction as far as it is repulsive to keep the stability of the period orbits (SPA).

,,=

The authors would like to thank Professor Lu Yu for fruitful discussions.

References (1] K. von Klitziq, G. Corda and M. Pepper, Pbys. Rev. Lett. 45 (1980) 494. (2) D.C. Tlui.H.L StormerandA.G. Oossard, Phys. Rev. Lett. 48 (1982) 1559. (3) B.I. Halperin, Helv. Acta Phys. 56 (1983) 75. [4] R.B. Lalllhlill, Phys. Rev. Lett.. SO (1983) 1395. [5] R. MorfandB.I. HaIperjn, Phys. Rev. B 33 (1986) 2221; W. Lai, K. Yu, Z. Su and 1.. Yu, Solid State Commun. 52 (1984) 339, and references therein. (6) S.M. Girvin, A.H. MacDonald and P.M. P1atzman, Phys. Rev. B 33 (1986) 2481. [1] S. Kivellon, C. KalIin, D. Arovu and J.R. Schrieffer, Phys. Rev. Lett. 56 (1986) 873. (8) LOnIa&cr, Philoe. Mas. 43 (1952) 1006. [9] LS. Schu1mall, T~ques and applications of path integration (Wiley, New York, 1981). (10) S.M. Girvin and T. Jac:h, Phys. Rev. B 29 (1984) 5617. [11) R. Kubo, S. Miyake and N. Hashitsume, in: Solid state physics, Vol. 17, eels. F. Seitz and D. Turnbull (AcademiC Press, New York, 1965). (12) M. Gutzwiller. J. Math. Pbys. 12 (1971) 343. [ 13) V. Maslov and M. Fedoriuk, Semiclassical approximation in quatum mechanics (Reidel, Dotdrecb.t, 1981).

826 1 JUNE 1988

VOLUME 37, NUMBER 16

PHYSICAL REVIEW B

Influence functional and c1osed-time-patb Green's function Zhao-bin Su, Liao-Yuan Chen, Xiao-tong Yu, and Kuang-chao Chou Center of Theoretical Physics. Chinese Center of Advanced Science and Technology (World Laboratory) Beijing. China and Institute of Theoretical Physics. Academia Sinica. P. O. Box 2735. Beijing. China (Received 24 November 1987)

In this Brief Report, we explicitly show the equivalence of the Feynman-Vernon influencefunctional approach and the path-integral formulation of the Schwinger-Keldysh closed-time-path Green's function. The latter simplifies the practical calcuiations considerably with a systematic diagrammatic technique and is expected to have a broader application.

Since the initial work by Feynman and Vernon,l the. idea and technique of the influence functional has been successfully applied to more systems, such as the maser problem, the polaron problem,2.3 the transport problem,4-6 the quantum tunneling in the presence of dissipation, 7 and the semiclassical approach to weak localization. 8 In the Feynman-Vernon approach to quantumstatistical problems, the propagation of both dynamic and statistical information and the quantum interference between different paths are explicitly described by the two different branches of the path integral. 9 On the other hand, the path-integral formulation of the c1osed-timepath Green's function (CTPGF) 10 introduced by Schwinger and Keldysh 11,12 is also a powerful treatment for various kinds of similar problems since it has a systematic diagrammatic-expansion technique. The main purpose of this note is to show the equivalence of the Feynman-Vernon influence functional (IF) and the pathintegral formulation of CTPGF, in particular, to express the IF as a CTPGF-generating functional. The key point is the following. In the original form of the IF, the density matrix p plays the role of an initial condition which makes the path-integral calculation rather difficult, and there-

fore, application of the IF approach is more or iess limited in certain aspects. In the CTPGF approach, however, p is incorporated into an effective action and appears as the thermopropagator of the action. Therefore, it makes practical calculation much more tractable and is expected to have a broader application (e.g., Refs. 5 and 6). After deriving the exact equivalence of the two approaches we will calculate the IF for a harmonic-oscillator system as an explicit illustration. Frequently, situations occur of two systems coupled to each other where one of them is of primary interest, i.e., the measurements are always done on it (test system), while the second system (environmental system) influences the behavior of the first one. The well-known IF (Ref. I) describes all the quantum effects of the environmental system upon the test system. Suppose that the test system and the environmental one are characterized by their general coordinates q,Q, respectively. Their actions are l[qCt>1 and I[Qh)1 while the interaction between them is IIq,QI, which is taken as f dtqh)Qh) for.simplicity. To compute the expectation of observable 0 of the test system, one has I

f Idq +)[dq -l
where the IF is defined as F[q+,q -I -

f [dQ+)[dQ -lc5(Q+(r) -Q -(t»p(Q+(to),Q-(to»exp{j([[Q+1 -/[Q-I +I[q+,Q+I -/[q-,Q-))}

(2)

I

In th~se two equations, q and Q are the eigenvalues of q and Q, respectively. with the index + and - describing the two branches. [dqh)1 means

with to < t I < ... < tN-I < t while [dQI is understood in

F[q+.q -I -

a similar way. Hereafter. we assume the Planck constant h -I and will recover it if necessary. Equation (2) is exactly the original form of the Feynman-Vernon IF (Ref. I) where the effect of the density matrix p is as an initial condition. In terms of the Heisenberg picture of the environmental system. Eq. (2) reads

JdQ_ (to)dQ+(to)dQ- (t)dQ+(t)(Q- (to) I Texp [ -i f,:dt q - (r)Q(r) ) IQ-{t)}(Q- (r) IQ+{t» x(Q+(t} I Texp [if,:dtq+{t-)Q(r») IQ+{to»(Q+(to) IpIQ-(to» .

II

9810

(3)

(5)1988 The American Physical Society

827 BRIEF REPORTS

Following the basic idea of Schwinger and Keldysh,ll.12 we can combine the two paths Q + and Q _ into a closed one Qp which goes along the positive branch Q+ from 10 to 1 and then comes back to 10 along the negative branch Q-. Then the IF Eq. (3) can be written as a CTPGFgenerating functional Flqpl-Tr[pTpexp [ijdlqp(r)Q(t})] , p

(4)

where the trace Tr is taken over the sub-Hilbert space of the environmental system Q, while qp means q + and q_ which is treated as a c number at this stage. The time integral Jpdl is along the closed time path and equal to Jl.dl + minus Jladl -. The time-ordering operator along the closed time path Tp is identical to the usual chronological operator T on the positive branch, identical to the anti-time-ordering operator t on the negative branch, and any operator on the negative time branch should be ordered toward the left, compared to those on the positive

Flqpl-exp [il int [;;q

9811

branch. For further investigation, we assume the environmental system action J[QI-/oIQI+lintlq,QI with its free part

101QIwhere

t

f/'ILdI2Q(II)600;Q(12) ,

(5)

~oop

is the free propagator of the environmental at zero temperature which is defined on the closed time path and has four components. sy~tem

~(rtln ~(ltI2-)l ~P(rI,12)- ~('I-,t) ~(11-'2-)

[

_ [~+(r112) ~~-(r112)l 600+(1112) 600 -(11 12) .

(6)

Introducing the incoming interaction picture for the environmental system we can easily verify that Eq. (4) turns out

1]Tr [pTpexp [iLdlqp(r)Q,(t») 1

Applying Wick's theorem for CTPGF (Ref. 10) to the above equation, the IF becomes exp [ilinl [i;q ] ]ex p [ -

f Ldl,dt2q(rI)~P(tlt2)q{t2)

] Trp:exp [i Ldtq(r)Q/(r»):.

Notice that in the normal ordering : : the difference between the two branches Eq. (8) becomes

(8)

Q+ and Q_ vanishes, the traced part of

withq4-q+-q_. For comparison with Feynman and Vernon, we would also assume a thermal density matrix p-exp( - pHIQ1) for the environmental system in the Q space. Moreover, in order to extract the full information of the density matrix we would use an algebraic identity which was first introduced by Gaudin: 13

TrlpA(J) " . A(n)] -0 -exp-ipa,,) -ITrlpIAO ),A(2)] " . A(n)+ " . +pA(2)' " lAO ),A(n)]} .

(0)

Different from the usual cases, here we have the density matrix with the total Hamiltonian HIQl instead of the free part HolQl of the environmental system, but the identity remains valid since we still have p -IA(r)p -ACt -ip) -exp( -ipa,)ACt) in the incoming picture for an arbitary operator A(r).14 Applying the above identity to Eq. (9), we obtain Trp:exp k(dlq4(r)Q/{t) ): -(0 lexp(A)exp k(dtq,,(r)Q/(r) ) 10) where

(I 2)

in which Q,c+>,Q,c-> mean the positive, negative frequency parts. After some straightforward calculations we arrive at a compact exponential expression

828 9812

BRIEF REPORTS

in which AcINCt,l z) -Iexp( -ipa,.) - I1-I~-Ct,lz)

+ lexp( - ipa,.) -11-IAOi) +(lllz) ,

AGN- [: :)AoN. Here AGN describes

(4)

OS)

the thermostatistical contribution to the CTPGF propagator Aop - Aoop + A&v of the environmental system at temperature lip. Substituting Eq. (3) into Eq. (8), the IF becomes Flqp] - exp

[iii" [ t

xexp ( -

i:q )] J,dl IdtzqpCt I )AopCt,tz)qp Ct z)

1 Cl6)

After a functional Fourier transform we arrive at the final result F[qp] -

f IdQp]expi [t J,dtldlzQCtI)AopIQ(tZ) +IiR,[Q] + J,dtqp(r)Q(t}) • (17)

Compared with the original form of IF Eq. (2), now the density matrix formally disappears and, as a result, the free propagator L\oop is replaced by the thermostatistical propagator Aop • In the above CTPGF form of the Feynman-Vernon IF, the effect of the density matrix p is completely removed into an effective action with the free propagator Aoop be-

Flq+q_] -exp [-

ing replaced by the thermostatistical propagator Aop • With this new approach, we could make practical calculations much easier as complemented by a systematic diagrammatic technique. Moreover, we would like to make tw.o more remarks. First, the thermostatistical propagator satisfies the fluctuation dissipation theorem, i.e., Ao+ - (OJ) -exp( - POJ)Ao- + (OJ) which follows straightforwardly from Eq. (3). Second, the thermo Aop and the free Aoop have the same inverse because the IlGN is the homogeneous solution of the Dyson equation for Aop • Therefore, the form of the CTPGF path integral for the finitetemperature case is formally the same as that of the zerotemperature case. Finally, we would like to calculate the IF for a harmonic oscillator system by using our new form to illustrate its advantages. In this case, the interaction 1..,-0 and the thermostatistical propagator is given as Ao+-(IIIz) - (-i/2mOJ)/N.exp[ -iw(II-Iz)]

+O+N.)expIiOJCt,-tz)]} , (8)

Ao-+Ct,tz)-Ao+-Ctz,II) ,

(19)

AJj+(r,lz) -eUI-tz)Ao-+(tlh) +e(tz-tl)Ao+-Ct,tz) ,

(20)

AOi) -(I,tZ) - eCr z -tl)Ao-+Ct,tz) +eCrI-tz)Ao+-Ct,tz) . When we carry out the Gaussian functional integration in Eq. (17), we get

2~ I.~dlldtz(q+Ao++q++q-Ao- -q- -q+Ao+-q- -q-Ao-+q+»)

(22)

Then we substitute the propagator given in Eqs. Cl6)-(19) into the above equation so we can check that it gives exactly those of Feynman and Vernon both for the zero-temperature case Eq. (4.8) of Ref. 1 and for the finite-temperature case Eq. (4.42) of Ref. I.

'R. P. Feynman and F. L. Vernon, Ann. Phys. (N.Y.) Z4, 118 (1963). zR. P. Feynman, Phys. Rev. "', 660 (\955). 3R. P. Feynman, R. W. Whellwarth, C. K. Iddings, and P. M. Platzman, Phys. Rev. 1Z7, 1004 (1962). 4K. K. Thornber and R. P. Feynman, Phys. Rev. B I, 4099 (1970). sz. B. Su, L. Y. Chen, and J. L. Birman, Phys. Rev. B 35, 9744 (1987). 6Z. B. Su, L. Y. Chen, and C. S. Ting (unpublished). 7A. O. Caldeira and A. J. Legget, Ann. Phys. (N.Y.) 149,374 (1983); S. Chakravarty and A. J. Legget, Phys. Rev. Lett. 51, 5 (1983); R. Bruinsma and Per Bak, ibid. 56,420 (1986).

IS. Chakravarty and A. Schmid, Phys. Rep. 148,193 (1986). 9A. R. Hibbs and R. P. Feynman, Quantum Mechanics and Path integl'tll.s (McGraw-Hili, New York, 1965). 10K. C. Chou, Z. B. Su, B. L. Hao, and L. Yu, Phys. Rep. 118, I (1985). IIJ. Schwinger, J. Math. Pbys. 1, 407 (1961); L. V. Keldysh, Zh. Eksp. Teor. Fiz.47, ISIS (19M) (Soy. Phys. JETP lO, 1018 (1965)1. 12J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). 13M. Gaudin, Nuel. Phys. IS, 89 (1960). 14p. Roman, Aduanced Quantum Theory (Addison-Wesley, Reading, MA, 1965).

829 Thirty Years Since Parity Nonconservation A Symposium for T. D. Lee, ed. Robert Novick, (Birkhlluser), pp. 117-131. Reprinted with pennission.

TIME REVERSAL INVARIANCE AND ITS APPLICATION TO NONEQUILIBRIUM STATIONARY STATES

K. C. Chou

and

Z. B. Su

Center for Theoretical Physics, CCAST (World Laboratory) and Institute of Theoretical Physics, Academia Sinica

Abstract Generalized free energy of order parameters near nonequilibrium stationary states is shown to exist for systems obeying time reversal invariance. A low frequency fluctuation-dissipation theorem similar in form to that in thermoequilibrium is obtained in this case.

I am very pleased and honored to be here to JOln so many eminent physicists to celebrate Professor Lee's sixtieth birthday. Professor Lee has not only made many contributions to world science, but has also done much to promote collaborations between the U.S. and China. So, on behal f of Academia Sinica and all Chinese physicists I would like to extend our warmest congratulations to T. D. and Jeannette, and to wish him a very happy birthday and a long creative life ahead. In 1979, the graduate school in China was reopened a fter the disastrous turmoil which lasted more than a decade, and scientists started to re-educate themselves. That summer T.D. went to China and gave an intensive course on particles and field theory to about 600 graduate students and scientists gathering from allover China. Almost every day for eight weeks, he lectured three hours in the morning and discussed with the students during lunch and sometimes in the afternoon. It was the first time a fter years of ignorance that we were able to touch the frontier of physics through excellent lectures. T.D.'s talent as a great scientist and a great teacher was fully displayed and delighted the audience. 117.

830 Through contact with these brilliant young students, T.D. realized that China, although poor in material production, is rich in human resources. If these young people could be trained at good universities, they certainly would make great contributions not only to the modernization of China, but also to the development of world science. The next year, 198Q, T.D. started the CUSPEA project with great enthusiasm and devotion. Over seventy universities in the United States and all universities in China joined the project. Up to now, 704 Chinese students have been sent to study for their Ph. D. degrees in the United States. I am certain it will have far-reaching impact on the development of U.S.-China friendship and scientific collaboration. attended some of T.D.'s lectures that summer. was also motivated by the stimulating atmosphere and vivid discussions among the audience. The ceaseless endeavor of T. D. to find somethino new hilS encouraged us to CD some work. What I shall tal~ about in the following is research done at that time, after T.D.'s lecture on CPT and spontaneous CP violation. 1 The results hove been published in Chinese. 2 I was interested in laser, a physical system far from thermoequilibrium at that time. We were looking for a generalized free energy to describe processes neor nonequilibrium stationary states from a microscopic rather than phenomenological point of view. The 0 r i e s bas e don the t·, a s t ere qua t ion, the F0 k k e r Planck equation and the Langevin equation were already developed to deal with systems near nonequilibrium stat.ionary states.3-4 These theories are semi phenomenological in nature. In analogy with statistical mechanics, a generalized free energy was introduced. The minimum of the generalized free energy corresponds to the nonequilibrium stationary state and the curvature at the minimum point determines the linear fluctuation of the system near its stationary state. It was shown in the framework of the Fokker-Planck equation that the existence of generalized free energy can be justified under the assumption of detail balancing. 3 In a series of papers,5 we have applied the field theory of closed ti'me path Green's functions (CTPGF) to systems near nonequilibrium stationary states. A time-dependent Ginsburg-Landau equation (TDGL) was derived in the form

(1) where Qa' a = l, .•• n the order parameters are the average values of some composite operators of the underlying fields. J a , a = l, ••• n are the external sources coupled to Qa The generalized free energy F has been shown to exist for systems near thermal equilibrium.

118.

831 For systems far from thermal equilibrium, does the generalized free energy exist? If it does, what is the condition? The purpose of the present talk is to answer these questions. It will be shown that the time reversal invariance of the underlying field theory is the basis for the existence of the generalized free energy. Since we know that CP is not an exact symmetry, neither is time reversal invariance. As was pointed out first by T.D. Lee, CP and T might be broken spontaneously. 1 If this is true, it will be most promising from the theoretical point of view. Fortunately, the argument leading to the existence of generalized free energy is not affected by the spontaneous breakdown of the time reversal symmetry.

1. Time Reversal Invariance Consider a system with the Hamiltonian H

=

o

( 2)

where ~o is the Hamiltonian without the external source term; QCf a = l, •.. n are hermitian composite operators. Under time reversal J a (t) may change sign ( 3)

= In Eq.

(3)

the index

a

+ 1.

(4)

is not summed.

In the following we shall work in the Heisenberg picture and take t = () to be the initial time where the Heisenberg picture coincides with the Schroedinger picture. The physical quantities evaluated at time tare independent of the particular initial time chosen. Time reversal invariance implies the existence of an antiunitary operator R such that .for any hermitian dynamical variables

Q

a

(t,

(5)

J)

The state of a statistical system is described by a speci fied by a set of real paramdensity matrix p (X) eters Xa , a=l, ••• m which may also change sign under time reversal (6)

Xa 119.

832 (7 )

E(0) = ±. 1 •

The state of the system is called time reversal invariant if it satisfies the relation

C (E A )

=

(8 )

If the time reversal invariance is broken spontaneously, there will be some >-0 ~ 0 with E(O) = -1. In this case, we may live in one world, and do not interact wi th the time-reversal counterpart because the barrier between the two worlds is infinitely high. Then we can always assume the density matrix satisfies Eq. (8) in the study of dynamical systems living in our world. The equal to

average

value

of

Do (t; J, A)

=

the

operators

cc

tr(c(A)Q(t,J)~

a

t.

J)

is

(9)

which will be called order parameters in the Following. It is easil y deduced from Eqs. (5) and (8) for a time reversal invariant system that

The correlation functions are

Co 1 .••

=

t Q.

0Q.

tr(p(A)D

;

J, A)

(t 1 ,J)···

01

Q (tn, on x.

J)1.(11)

Time reversal invariance requires that

••• 0Q.

=

t Q. ;

(t 1 ,'"

e (0 1) •• , e (0 Q. )

C

0Q.

J, >.)

... ° 1(- tn, x. ( 12)

From Eqs. (10) and (12) we see that time reversal invariance relates physical quantities in one world to those in the time reversal counterparts if time reversal symmetry is spontaneously broken.

120.

833 2. Time-dependent Ginsburg-Landau Equation (TDGL) In this section we are going to derive TDGL from the field theory of CTPGF and prove the existence of generalized free energy for time reversal invariant systems. Since the relation obtained in the previous section is independent of the particular initial time chosen, we shall take the initial time to minus infinity in the following. Consider the external source term JaOa(JQ) to be, a perturbation adiabatically swi,tched on. Any operator 00 in the Heisenberg picture of HO will change to (13 )

5

where

is the S-matrix co

S

=

T(exp

f

-i

J(-;-) 0(-:- ) d-

~ )

.

(14 )

-co

'+

The S on the left in Eq. (13) is also necessary to guarantee the causality of the interaction. Equation (13) can be expressed as 6 (15 ) \~here

,

Sp

::

Tp f exp f -i .fp J(d Q( .;) dT)}.

(16 )

The path of integration P is a closed path starting from T = -CD to T :: +CD (positive branch) and back from T = +CD to T = -0:) (negative branch). T p is an ordering operator along the whale pa~h P. It is easily.seen that the positive branch of Sp cqrresponds to S in Eq. (13) and the negative branch to S+. In statistical mechanics we are interested in average values of physical variables rather than transition amplitudes. The average value is equal to a trace of a density matrix at an initial time to and some Heisenberg operators at the time t 1, ..• tn which consists of amplitudes propagating from to to the time of the operators and back. This is why we need an S-matrix along a closed path in statistical physics. The generating functional for CTPGF is defined to be Z[J] = exp {-iw[J]l

= trf

P Tp(exp {-i

fp J(T) OCT) dT }) 1.

121.

(17)

834 The external sources Ja(t), a =l, .•• n are taken to be different on positive (J+ a (t» and negative (J -a (t» branches. They will be put equal at the final step when physical results are evaluated. wCJ is the generating functional for the connected Green's functions whose first derivative gives the order parameters

J

(18 )

When the external sources on the two branches equal to the physical external source, we have

=

=

Qa

ph.

(t)

are

(19 )

,

Qa(t~

which is the average value df the physical variable From the definition follows easily that

W [J +

of the generating

J-JIJ:J + -

Let

r

=

1/2(J+ + J

J/:;.

=

J

J

-

+

functional

it

(20)

0

=

set

'}

J

(21)

Equation ( 20) can be rewritten as (22) By differentiation with respect to J I: we obtain a series of equalities among Green's functions from Eq. (22). The first equality is

&W[Jr, J/:;.=O]

Q

& lYa (t) coinciding with Eq.

+0

(t) - Q

-a

(t)

o

(23)

(19).

Second order connected tained by differentiation

Green's

functions

OJa{t) OJ~{tl)

122.

can

be

ob-

(24)

835 In the single time formalism ther!' ,"'" four second order Green's functions for each pair ( 1 : , \,f which only three are independent. They are the re t ,I"" ".1

G

ra~

(t, t' ; J, X)

-i9(t-t') tr {p(X)

[0 ., \ t.

J),

Q

~

(t', J)]l

(25)

the advanced

G

oa~

(t, t'; J, X)

and the correlation Green's functi""

G

ca

B (t, t'; J, X)

=

( ) ~J

SJll.a t

<>

~(~

( ")

A

-i tdp(X) {Qa(t, J), ("r~(t"

The vertex functional is defi.r. f ,.:

where aa(t, J) is determined "/ from Eqs. (18) and (28) that

sr[Q] 6 Q (t) a

123.

=

1,0

:'l.

-'• ' . I

.

J)} -;.1

(27)

be

(18).

It

follows

(29)

836 In terms of the variables ( 30)

and Q we have from Eqs.

l1a

==

Q

+0

- Q

-0

,

(31 )

o

( 32)

(22) and (28)

and

(33 )

Q ==0 11

The vertex functional can be calculated by summing all I-particle irreducible diagrams. Once this is done, the physical order parameter Q fa (t) can be determined from Eq. (33). We shall show in the following that TDGL is an approximation of Eq. (33) for systems near stationary states where the motion is slow. There are four second order vertex functions for each pair a ~ in single time formalism, of which three are independent in the physical region where Q 11 = O. They are the retarded ==

==

- D

a~

- iA

a~

(34 )

the advanced

r aat-'It (t, t'

; Q , ).)

==

==

- D

a~

and the correlation vertex function

124.

+ iA

(.I

a,..

(35)

837

r cal"'R (t,

t' ;

a , X)

5 a ~a (t) 5 a ~ ~ (t')

(36)

In Eqs. (34) and (35) Da~ is called the dispersive part and Aa~ the absorptive part of the vertex. One can easil y deduce the Dyson-Schwinger equations for the second order Green I s functions. In matrix form they are

rr

G

r

G

rr

- 1 ,

(37)

G

r

- 1

( 38)

r

r a Ga

a a

and

rc

r R Gc r a

=

(39)

Now we are prepared to derive TDGL for a system near stationary states. In the following we shall assume the physical external source Jr. to be constant in time and the equation

or[a]

- J

5a~ (t)

a~ = 0

r

(40)

0 [. Then the system is conto have constant solution The question is sidered to be in a stationary state. whether or not there exists a generalized free energy such that 0 r. is the solution of the equation F [0

[J

aF

= - J La

(41)

If this is possible we must have

aF

aa ra

=

125.

cSr (42)

838 Then

J

dt' - - - - - - oar~ (t') OaAO(t) t"

J dt' r

LJ,

~ (t' rI"'o

- t ;

a6.0:: 0, ar(3(t)=aI~

a,).) .

(43)

Here we have used the time displacement invariance when the external sources are time independent. A well-defined funct ion F can be obtained by integrating Eq. (42) if the order of differentiation can be changed in Eq. (43). Therefore the condition for the existence of the generalized free energy is

J dt'

r r~o (t'

-

t .

a ).)

J dt'

I'

r rot"'(.\ (t' - t ; a, ).)

or in Fourier transform

r rof3 (w=o

a,).)

=

r ro~(w=o;

a,).)

r

a,).) .

6(w=0; 00

(44)

The last equality in Eq. (44) follows from the relation

r

~(w; a,).)

ro~.

=

r

~ (-w; a,).) .

(45)

0t" 0

Equation (44) can also be written in the form ..J~

~o~

(w=O; a,).) = o.

(46)

Therefore, vanishing of the zero frequency component of the absorptive part of the vertex function is a sufficient condition for the existence of a generalized free energy.

126.

839 Our next task is to show that Eq. (46) follows from time reversal invariance. For constant external sources, the order parameters are time independent. We have from Eqs. (10) and (12) that Qa(J,~)

=

E (a)

Q a (EJ

, E

(47)

~)

and

From Eq. (47) we obtain by differentiation with respect to constant Jr~ the following relation

Gr~a ( w =

a; J ,

~) =

E

(a)

E

(8) Gr~a (w = 0,

E

J,

E

(49 )

~)

The Fourier transform of Eq. (48) reads (50)

From Eqs. (49) and (50) we get ( 51 )

Since - rr and - ra are the inverse 0 f Grand shown in Eqs. (37) and (38) we obtain finally r~ (w=O;J,~) =

r"a

r

~ (w=O;J,~)

aI-a

Ga

as

( 52 )

•

Equation (52) is just the condition (46) required. Hence we have shown that generalized free energy exists for time reversal invariant sy~tems in stationary states. Near the stationary state the order parameters vary slowly with time. We can expand Eq. (29) around the stationary point at time t

or

or

oQ6.a(t)

+

J dt"

Q6.a=O,

2 6 r

- - - - - - (Q

127.

r~

Qr.~(t)=Qr~

(t')-Q

r~

(t}+ .. ••

(53)

840 To first order approximation in time dependence we neglect higher order terms in the expansion and put (t' - t)

aQL(3

(54)

at

Then Eq. (29) becomes - J

(55)

ta

where !dt'(t'-t)

r r~a (t'-t,

arr~a ( 101 , Q a101

, X)

Q, X)

I

(56)

101=0

Equat ion (55) is the TDGL equation for the order par amet er s Q to Ti mer eve r sal in v a ria n c e imp 1 i e s also a reciprocity relation for the relaxation matrix ~~ ( 57)

This is one kind of Onsager relation generalized to systems near nonequilibrium stationary states. It

was

proved

in

Ref.

2

that

r c yo.,.{L1=0,

Q, X)

both

the

di ffusion

matrix a

a~

-1 = Yoy

r- 1 y 6(3

(58)

and the response matrix

L

a~

=

101=0

(59)

satisfy the reciprocity relations (60)

and

128.

841 Lo ~ (J , >")

=

E (0) E

(~) L~ 0 (e J, E >.. )

(61)

~or"systems near stationary states. We shall not discuss It"l~ detail here. The interested reader can consult the

orlglnal literature.

3. Generalized fluctuation-dissipation Theorem We have shown in Ref. 5 that the correlation function Gc can be written in the form (62)

G = GN-NG era

where ticle state

N is a hermitian matrix related to the quasi pardensity distribution. In the thermal equilbrium

(63)

where

T

is the temperature. lim N (w, Q, >..) w-O a~

Substituting Eq. (39) we obtain

rc

(62)

= 5

into

r r N - N ra

In the low frequency limit 2T a~

the

(64)

w Dyson-Schwinger

= - ON + NO - i (AN + NA)

equation

,

(65)

where D and A are the dispersive part and the absorptive part respectively. Equation (65) is the transport equation for the particle density N in a slightly inhomogeneous system. In a stationary state where the vertex functions depend only on the time difference, Eq. (65) can be written in frequency representation in the following form

-0 (w)N ~(w)+N (w)O ~(w) oy Yl"' oy Yt'

- i(A

or (w)

129.

N A(w)+ N Y\"'

0

(101) A ~(w» • (66)

Y

Y"t'

842 It is easily proved that

2:: r caa (w)

i

a

is the quantum fluctuation which has to be positive definite. Therefore we have

>

tr (A( w) N ( w»

0

( 67)

From time reversal invariance we have already shown in Eq. (46) that A

a~

(w=O)

=

0

In the low frequency limit we may put A

a~

wa

(w)

(68)

a~

where caB is the real part of the relaxation matrix Ya j3 • For a statile stationary state the eigenvalues of the matrix caB have to be positive definite. Since Eq. (67) holds also at zero frequency, the density matrix N must have a pole at w = O. Hence

N

a~

for small frequency. into Eq. (66) we get

o

ay

2 T eff

(w)

w

(69)

a~

Substituting Eqs. (68) and (69) back

(w = 0) T eff = T eff 0 y~

ay

y~

(w = 0)

(70)

and tr

crc (w = 0»

= - 4i

tr(c(w

=0) Teff)

.

(71)

This is the generalized fluctuation-dissipation theorem. In the literature of critical dynamics it is always assumed that T eff =

(72 )

a~

130.

843 In this case we have

r caB (1.1 = 0)

-

. Teff 4I a

(73)

aB

Substituting Eq. (69) into Eq. (62), we obtain form of the fluctuation-dissipation theorem

aGe

at

another

2 (G Teff _ Teff G ) r a (74)

References 1.

T. D. Lee, (1974) .

Phys. Rev. D.!! (1973) 1226; Physics Reports

2.

K.C. Chou and Z.B. 164, 401.

3.

N. Van Kampen, Physica 23 (1957) 707, 816; U. Uhlhorn, Arkiv. foerFysik 17 (1960) 361; R. Graham and H. Haken, Z.PhYS. 243 (1971) (1971) 141. -

Su,

Acta

Physica Sinica

30

(1981)

289;

4.

G. Agarwal, Z.Phys. 252 (1972) 25; J. Deker and F. Haak~Phys.Rev. All (1975) 2043; S. K. Ma and G. Mazenko, Phys. Rev. Bll (1975) 4077.

5.

K.C. Chou, Z.B. Su, B.L. Hao and L. Yu, (1980) 3385; Commun. in Theor. Phys • 307, 389.

6.

j. Schwinger, J.Math.Phys. 2 (1961) 407; L. Ke1dysh, Zh.Eksp.Teor.Fiz. !:2 (1964) 1515.

131.

..!

.2.f.

245

Phys.Rev. 822 (1982) 2'95,'"

This page intentionally left blank

Part III ~ i~, ~ T 49lJ 1Il:3=~ 49lJ 1I

This page intentionally left blank

847

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11

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1955

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aRB

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If!.

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848 300 .10

3r• 10

1r.

= -

53.8

= 1.71

± 0.04

± 0.04 x

x 10- 13 lii.*, lU- 13

mc*,

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flifj~~Il:~1t G.

fij, tIt Wiitt f:lT y,( ~ffl(; :

1f=f~ . :fj1f!~ Cl~:'l[ .L.,~~ ~ rt~, S

1 .pJ=-u(r)X;

(3)

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'it

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u(O)=o, r II{r) = 1 - -

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a

} (5)

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3

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1

21

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2

6

5046 (1+b)'

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864 316

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K 1 (6) = ----'----0-;:-7-'--'--:-:-::--=--=-----'-92406 8 (l+h)8

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It (11

!A(ICH1i.,~:l!ijt, '1'~¥!J:{I.~t-ll. 7 (1950), 339.

(2] [3l

.jf,~l!t~~.

[4J [5]

[6]

Huhhcn, L., KIlI/RI. FYiio. Sill. I. LUlld. Fiirh"lId. Bd. 14, NT. 8, NT. 21 (I9H). I:Id. IS, Nr. 22 (1945). Hulthen, 'L., A"kil'. fiir. Math. Astra. O<'k. Fyi;k. lid. 3SA, Nr. 25 (1948). Chrutian, R. 5., Hart, E. W., Phys. "Rev.77 (1950). HI. ~:m~,Ji!f~*, rplll-1tJJI~# 7 (1950), 309. Pror:. Roy. Soc• •0\. 204 (1951), 176.

*-8#!,;tJllAtJj,':;, [7] [8]

Gerjuoy, E., Schwinger, J., Pk),s. ReI'. 61 (1942), 138. Clark, .4.. C., Pl'ot:. Pkys. 50(,. A.67 (1954), 323.

A DISCUSSION ON TWO-RANGE NUCLEAR FORCE CHOU KUANG-CHAO

AB5TR.oI.CT

The binding energies of nuclei H3, He3, He1, the low energy n-p scattering length and effective range are calculated by using the standard variational methods. A two-range central Yukawa potential is considered in the first two sections. The longer range corresponds to the n meson mass. The smaller range corresponds either to the heavier meson or to the higher order field interactions. No repulsive core appears, when the force parameters are chosen to fit the low energy scattering and deuteron data. The calculated binding energies of nuclei H3, He 3 and He4 are too high. This result is in agreement with most of the previous cakulatioDs. Tensor force of the Schwinger mixed type is considered in the third and fourth sections. The force parameters are chosen to fit the low energy two body data. They are not uniquely determined and are given for a set of possible D-percentage values. The adding of the tensor force redu~es considerably. the calculated binding energies of nuclei H3, He3, He~. But still, the calculated values increase -too fast with the mass number. It does not fit the triton and helium ·binding energies simultaneously. The p
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2. •=*.i[111~. ~1iUf=:9LiE.ftU~~ Ja-liij , a1M;:; HiUJi~r.t% H D Ii H N.1).t 't;iM~.HJIJ f\::1t H

;e:

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m~=~(J9t~-rfJ, H9 ~1J:5Itjq.-$~ H:!N.D.

+ e!~:flFPJ~!?1J&!il1ittlnfi'l'~T.

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..ft

-if"H

0

2N ,D, (z)dz)fl,(u)

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_·ifr

U1 ~-''I' zlE._*o

U, =exp( H 2 l1'.L1.(z)dz) (12) .ii~"i'~ft, iltlb1R.iIi 0 1 (0") ~.r&:T 0 1 (0') ,~(CT) 3Tii~

(J9JJfiit, ~ ~iJHj~JgCliliff.lJi~~ifSl!1U~~.:rj].rm~$~.,~~

iB~~(J") ={H:!.D.+I13+H1+H:+H2+Ht+Hf+H~+If.~ ~(O') ;1t+,

iTii

Il:!.D,(z ,H3 (z). Ut(z)

H~( (I}) =

;

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JJc. H{ z) ~

rll(z')d~'JDr

(10) ~1'JYf~m (JEJ;iftfPJo

[Ii (2:"),

r"

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(13)

869

Il~(:») =

; [[B (:»),

r

H(Z')clz'JN.D.r [H (z"),

1~ [[[H (z),

HU(:») = -

r r

H(z') dZ']D.

r r

r"

[H (fI;") ,

H(z''')dz''']N.D.d~'']

f" H(Z"')d~"']N.D.d:r;"J

J(II" [H (Zip), J/T,U H(z") d:»"]N.D.dz'" ]

H~(:t:) =

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1~' [[[[HCz),

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r

[H (z"') ,

[H Cae).

H(z') dZ']N.D.,

r

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f'u

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-TjJ ((.}jf ilk1lJ k:iW--~OfllE.JlIJ~. 3. '=:~.iE!lIJ;t.

~ U2 :m:~2.lE..r.ro

*_ m*o

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(15)

Ull=exP(-ifH3(~)")

(16)

~~~~~li!.

i2tlllY1liti D.s(0') ~J&7 0 3 (0')" ~.R.t;:iiJ'J gl

ili.iU, _1l:t'~jj1~-'3 ~.1J.H ((.jlfOt: aa( 0') J9i'~~l¥.Jjjfi:itiiJ ~~: . BnaCu) Bo-

'Ii

{H 2.D. + HA,+ HB,

+HO, +HA +HB + HO +HD +HB}O ( U ) 6

6

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;tt'P, H9.D.(z),H1(z),H~(z).Hl(Z),Ht(:»),H:(z),H~(z) ltH~(ic).;fO (10)~

(14)it"PJiJrt:ftti1fJ<J#JlPJo iffi Rf(z) j{lj:m: Hf(z) =

-

fillH (It,,) :.a: (17) ~~, H

r r r'" J

;8 [[[H (x), H(z')dz'] H(x")dz"],

2 •D •

dz''']

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lll ]

r

[H(z''') , (18 )

~-Jl.pe!*1'*=iHltff.l~"-rj] fft.HIJ ;&H~ rf:r~:t"

.$~~~ T jiflZ9 ~:jftfl?l((.j,fJ{-rj].

ffii 1:Cf ,H~.H~ ,Ii~ .& lIf ~JJ{ ((.) )(f1lJ'k*~~.JlIJ

~*T.*'!HJifUMlttTjJo ~r-'liJ:p,~%ll'HI1J~BIJMit (17) :it.p*lJ~~f-.t1 fAjf~o

870 58

Jll1:~1i{(.JlI;.J{IJ~illJ3lT*~&

,m, ;fil ~t}j~;fk 15

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rm,;tt.r:J:r 'IJ ;g~f-itJlo

~~.fP'1~*~(

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2

2

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i: )\ :) ~~tt.~tt.J=:~;Ulfj~ ~~(PJ~~m 0

L(..!.)2}X( g2_)2--.! ~~~ii~ he e he e •

15

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C

0

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0

1tTf.I(J{Jilti?iIt~~tf 'Pc. il!i{)-~ »"Im~

[Pc. (:I:) ,rpJl(a;')] = -iSLMt.(Z- 'J}')

(19)

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[pt(z) ,Q'J:l{(z')] = -'iB1.,Vt.+ (;11- a;')

A+(a:) =

~

(,:).($)

1 A( z ) -- (21T)3 A (l) ( Z )

--

1 3 (21T)

+it.

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JII k;smll'o~o6' die .

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dk i.. " TaCOS liioZofJik.r

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J

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+

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1.

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(1)

d=

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871 59

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f

dz' dz" ~ (z/

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)+ [i]j(a;)OL.p(z)i/i(fll')ON1/1(re').

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872 1955~

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J..;l.;.g [r[H (z), =

r

H(z') d;2l']

r

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-ifdz'[dz"f'dZ"BLp.6.(Z~:lt}{[~(Z)OL1/I(Z) .~(z')O.llI/J(z')]

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(26)

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([qJ.v( z' )tpQ(z'''). Ps( Ziv)tpF(ZV)])o =

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(27)

tJU1S(27)~.#Jr:tt~t i :m~z~.~lH;;£IDEI!JJ

rf r r J r"

f dt' f

d,t"

dt'" dtivf'''dlVA( 1& -:1;" ) ['P,u(z')'PQ(z'''). tpRC Ziv)tp~.(Z") Jp= 0

fiilniiJJa.ruEl!}}

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d.tuA(!l:-:1;" ) [tpQ( z"')I1',,(:1;') • ({Jp:(ZiU)f{JF( ZV) Jo= 0

~~~*. ~'fr'Hl~j1jJ H~=O

1¥1*k=*o

tE~~rrH t tti~~. ~fI"1N.~~JI~-rJ.a

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PI J!imEl!}} H t _D•• H1. H~. H1,

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Hg ,&lIf:;rt=-{;$.iJ&lf'Jt

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m:a:.!J!t1JUJ~

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fllJtr.JJj~, tt.~;a*7b

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(X"»))[ [\p(

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)OQ'" ( ~"') • ifj(Z'O)OEt/J( Xi.,)]

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ittJ(J *ll!Pll.p ~

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:rtJ:fl.m \y(~)OL\fI(Z)

tlz~

!OfPB(i'-i'j) = Ly~j)ofj)a(T-TJ) i ;

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3,i/n:;

:±l*.

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2 i:!l!fj~k

+ 2[QLi) •o.IP]OlP [mf> .o,~) )o]!') F C",j) It ('I"jk)G( 'I"'j)}

(31)

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(32)

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(33)

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'J'lJ>a
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875

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(2) R. S. Christian and N. W. Bart, Physical Review 77, pp.441-453,1950. (3) H. Prima.koff and T. Holstsin, PhysioaJ. Review 55, pp. 1218-1234,1939. L. J'a.nossy, Proceedings of the Cambridge Pltilosophical Society 35, pp. 616-621,1939. (4) G. Wentzel, Physical Revie'l.l! 89! p:p.6&-688,1953. S. D. Drell a.nd Kerson Hua.ng, Physical Review 91, pp.1521-1542,1953. (5) K.A.Bru~kner, Physical Revie~o91, pp.161-762,1953. K. A. Brueckner, J. GeU-Mnn and M. L. Goldberger, Physioa·l Retrie1091, P.• 460, 1953. (6) G. F. Chew, Physical review 95, pp. 1669-1675,1954. (7) K. A. Broeckner and K. M. Watson, Pltysicnlreview 92, pp.l023-1035,1953.

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(20)

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va .fln~1, "i3~~JJl!~)({tfft:~;'11~t~ito 7Iff(20):tt1JoJa.2¥!!:J,

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886 :al8

[1) Lorent:r, II. A. Encyclopiidic (lor nutl/wmatischen Wisscnscfo.a.!len, V. 2. Heft 1. S. 200 [2]

Fokker, A. D . .Re/ati1Jitatstheoritl, GmIlingOll, (1929)

Becker, R. TheO/';'e d8T Eleklrb.itiit Band II. Leipzig, (1933) Van Vleck, J. H The theOl'Y of electric and magnetic susceptibilities, Oxford. (1933)

'1'a)m, H. E. OCKOGtJ/· T(!np1Ll~ 9.le"ml)1I~r.m/1rr, roc-Tc'x., (19,.9) Sauter, F. Zeitschri/t fur phgsik (1949) 126, p. 207. [ 3 ] Rosenfeld, L. TheOl'Y of elect1"01ls Amsterdam, (1951) [-I: j

Mazur, P. and Nijiboer, B. R. A.

Pl!y.~ica

(1953) 19, p. 9il.

[5] Kirkwood, J. G. Journal of chemicrr.l 1>hysics (1946) 14, p, 180.

DERIVATION OF MAXWELL EQUATIONS FROM THE MICROSCOPIC LORENTZ EQUATIONS

AB!I'l'RAC'l'

The space time average method, used ill the derivat;ion of macroscopic ma:\-well equal.ions, is

elaborated ill a systematic way, The technique of Dirac 8-fnl1ctiol1 is used which grcatly simplifies the calculation. The resuLt.s are identiClll with those obtained by MazLU' and Nijiboel'[4], who use the ensemble average method. The idea illYoh'cd ill the present papet' is lUueh simplier than that of Mazur lind Nijiboel' and the calculation is straightforward. The present method

can be extended to the

quantum C:J.se.

der~vation

of hydrodynamical equations and ea5ily generalized to fh3

887

tr. 15 ~

%J Ill! ~ ~J'! ACTA PHYSIC A SlNICA

~U lUI

1959t.f. 5

1'3

Vol. 15, No. 5 :May 1959

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{EI-ft-[5]r-ft ,~flllJttjN-jfT #-~~5(;j-jJ.1t =.f!f.lll.{-r J:.f?~iiltin<J~lIliiJ. ~*,*I"l, -411* Gell-Mann ~U FeYl1lllaIl 6CJll-jlJE:.JitlEr~(J(;. JtIJ (3- i1UftR(J~&~ 20% b=.;(;. rb.:r~-1t--r1t'i!f-r J:·~~~JL~:kJJ,,;{E[5]J:fIglle..~l:iriJ~H;Z, .1!L:~'f.g; 7 f J}t (~>.itbY.. :{£tf1.::: ,,[j l i:) , ~.ltfIDBIt1f.tHI:f'f~ JL~, 1"1'·:r·~·flMt w1t-1-uC] 1fJ ;:X;JDL yJ. & rft-f·:mftf~ ,lIf

~~~. ~.~~~ft~M.rn~~-T~.-rJ:§a~A.,~~G~~.:r-.re~* :1:\. ml!!J 'Vi -lfflM ~ jJ.1l-=f-:(r:1i"'t~ .. I:.(J(lff- ~ , llUi 15" J{IJW-lM~ jJ.1l--=f"(EJjj( -=ft~J::~f1 a.

=.

jJ.

fr·:f-.{f mr-l-t$.J: f:f-t}EB9tfil0i\

••

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(7)

.:j2'(ii,..B(l +'Y )n.. + 2~f .:a,..Bp,u(l +I'G)I~,,). 5

B= J2' [An,./3 P1u(1 +1'r.)U,,+-t;tl-ikilii,./3PI0'(1 +1'5)11,,,-

_.12J1l ~~ki,. (1-1'6)u•• ] •

".K) \

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rE ,u. ~ ~{. &. r.p~.:r ~~ FJ1PiG; r.ff 9='~7f.' lfH*, (8ht liT1if TII((' ,Ilit;;i\ : (9)

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-ik·,;

R=f~/iL'rI-;(j)(.A+B·r7i)C r.P,{LV. (lfl) ; J.t:r+, 'tr-J(j) ;lJiC!'R '~'r~}11(: 'i'={-{)(Jlii](tllPi1l:n:,:t!fM j l-~Ji.:r-;l]jlIT, Jlljf;ftd ..; -' (j) fj: JI)

Jff?1lt'Y::1t.I'..::r-,;&'t;:XJ r ll·:r·.l\IJ 't"")(j)W.jtlJ:~*f1:1l7,J~ilf]M:llL rlli lith :XJ.r'-i-=f.l'~·(I:'!lh:f::i -9 *:t~ Ml il~ f;'~ .~"< • jifi W; 1'1' ~. ,~l1Xltf'( )~ ~j'j~ :ftc jfjj JJlHi:l.Jf;i -=f,j:J.t, fkl JIt;t~ £~ v.t i;~ J.it 'I' i!1 MW:. III r\"IT.llHHiI t~ZiJ.n~.Hl.{. ~-l:'lftM:~I'IG'!iii:ill] l:!.~~'{I~ (4) l:\J:11 11-t.1
898 ~I.;O

15

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~

{(.J*/J-..Jff:X:.

-,~ Lt:jIHTJi L,)~ Jklc·H tJ·{jt ~)(:fl4.l:A{j1"t'Uf

, C.10)A;nJ Ja.~JjICr"·~IJ !~A: R=Aa.+B.b,

(11)

J.l:11·1 (12)

:ttr-t' N

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:Y'I'1<J ~1.~E~m:-;/,J 0',.

,fIUH(ll) ,"f:~Kn;,!~~.Ii!t

'1'1'=

(2;1l;)-~15·'(IJrl-1~-p.-Z),,)R.;f'(O).

r±£~jll cp(x.}hl; IJ.fr-f-(I::Id~l);v}UJ\'IVL3tt.l:n.'Ji#ii;~ ~, I (p(O) 1'.

,

(13)

It;l IIi:

Z3 = ----, ;Il;a6

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(14)

if< ( 14) WY];.JHtt .u--1t-=f-n:)!j(

-{-.t1Ll:f+~(I>J JL~lkjfj ~J!:t. ~Jj -=r.f~:{EMi~**JMt: I'H~l

d1l!= (2~)-&~- .-.._1-.-HIlR(l+u .p )R+dp dk dE , :n:ag 2(2J +1) . "". n·'

(1.1)),

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1 -"P (10,+ = F 1 (EN) . 0._

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('17)

:1..1

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2.T+(~P (~Uln

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(10)

899

2J ~1 Sp b.BIT1b+.B+=iF 6 (EN )Bx B++iF,(EN)pp· (Bx B+),

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;'A(15).& (17)II,·i·f j;.J.:1HiIJ.

On

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(20)

T r~'~tl-l.

~I'fHF; 1 IhR rL"lfI~*fi~H\:.A (1.7), (ttl).&. (15) II',

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mJtI}j:lj ,u.-n-·rn;}f.( '::{-.fhJ:I'f. [X (19

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(£=1,

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b=u1 ,

J-A(21)."U:i·lI~*tq,(17)Jt(18)t\:a<J

F. ~~,~~~l Fl=F~=l, F 3 =O, F,=Fa=l, F~=F7=O.

(22)

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P.. = -k.

.

*(2:)

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rl:1 ("17), (18). (24)&!I(h~ I

1 =1

IIJ j;Jff. H.i

(2-1)

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+3A~+fj(1-A1')+{3.u'(2A+,8.u'i2) + ~ 1'2:

(2(i)

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(27 )

j(X =1-

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{3= .M. f.!1Irr:(:!~)), (26)1*.(27)'1"

,% l' =1 -A, .wWJij f

~- ~ ft-.'J Iii !;{!,BI&~, ;J~frJ ~f1t~f;tlj"[

900 15

382

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[f,Jf-fiif J;.J.;}<W 9-"Jr1IHta<J1h\::, t*flJ (21-i)

;!tIll (l.

= 2 [ .\(.\+1) + ~

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.\+ f-3/k'

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[1' (.\+1) -I-!J.' (.\-/-1) +~~ JL' (I' -/- p.') +-i (j' -/- p.') ] =/3

(I' -I-!J.')

=

2 [ (.\ -1) ( .\ +{ !J.') -

(30)

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(~ f' + ~) (.\ -I- ~ 1'-') ] =

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('\+1+~ +~2 p.),

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c= 2

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(31)

1'I<J-?tf%-'3[5] ~1~~

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901

f

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J ,., =fln*c-ik,f/2CPtflV. r." f.

r~M~~~lli5.~8.~~~.~m*,w~

X~Xd= }(3 +UI'O'~)' X;O',X
1

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SiCu! x
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oJ

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(36)(tlID(17), (:t8)]f(15)'1' ,rtjf.~l E= P~

+ (Pn+k)' +IJ

2111

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kruill

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902 384

15 ·115

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(41)

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a=---}{(A2+,8,u.'A+~~,u.'z )(1;/+2 b=

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(J2)

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X= ~ ~++ ~ A.j- :~Z(i~Jz1~~JL~,JlIJ~~~q, A+-A_

f

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(47)

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1J=2o\ +20~=2( A+~I~' )(1+>"+~.((.='1

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+8 A~ +.8 +lfx. (21-" -

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(50)

l=0,1,2.

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+ hP,.' V +

cPHc·V]V

+

~>p (l

= 2 [J.lo. l +

1)

+

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4

dP"

+ eP Hc ,

(10)

1(ll)

910 16 $

64

b = {31Ct;

~

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c

=

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e

= (1 + ~y + (f3I}.t!; - ~: f? -

-2{31(P;

(J.

I

(12)

+ ~l p; ) [AI -

1 -

(13)

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(14)

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(15)

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(20)

(21)

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(28)

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=

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al(t/),

F2p(l)al(q2).

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(A-I)

p'2

/

1 3

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6 iH=1

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1

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.& (A-4)

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15 %li::b.

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n

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i

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1-1

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+ Pi) ag~tfIlJli",

= ~

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1:1

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i,tJ{[A I -t. 8 1'11;-

/

1 A-I

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3

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.;)

[ 1 J Feynman, R. P., Gell·Mann, M., Ph)'s. Rl!v. 109 (1958), 193. Sudarshan, E. C. O. nnd Uarshak, R. E., Phys. Rev. 109 (1958), 1860. [ 2 J Crawford et a!., PlJyr. Rev. {"#fer! 1 (1958), 377: Nordin ct n\., PI,ys. Rev. {etters 1 (1958), 380. [3] Goldberger, R. L. and Tre;man, S. B., Phys. Rev. 111 (1958), 35~: Wolfenstein, L., NIIOVO Ciml!"1110 8 (1958), 882: Hocjlcjle, B• .'\., }J(3TiP 37 (1959), J59. [4J mJ~g> B. Jli,1lil*l\IT-1£>~i!~1fl, 15 (1959),377. [5 J ~"f" "fl-T.:iI:lIl~:1!iJ:~~if'JIf!:;PJ~ lloJlnHcKnii, 3. f1. It BJlOXmll.\ell, JI. )l(3TrIJ 34 (19;8), 759: 35 (1958), H88; Oberall, H. and Wolfeustein, 1.., Nuovo Cillumto 10 (1958), 136: l'olhock, H. A. and Luyten, L. R., Nile/ear PlJys. 3 (1957), 679: Primakoff, R., Rev. Mod. Phys. 31 (1959), 802. [ 6] Fujii, A. and Prirnakoff, R., NUOVD C;111~tO 10 (1958), 327. [7] 1I!-}'tf:!,!!f;Ji;!~m, 11 (1955), 299. [ 8 J Rosenbluth, M. N., Phys. Reo. 79 (1950), 615.

n.,

MUON CAPTURE IN He3 NUCLEUS

The rate of the muon capture reaction p.- + He3~H3 + v, the angular distribution and the polariz.a.tion of the final nucleus Hl ar
916

;j'} 16~

&f.

1960

2

~

faJ

~2l1Jl

.F.l

:m.

~

Vol. 16, No. 2

ACTA PHYSICA SINICA

.

Feb., 1960

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[1] [2] [3]

[ '" J [ S]

Feynrnan. R. P. and Gell-Mann. M., P/,y~_ RelJ. 109 (1958), 193. Sudarshan, E. C. G_ lind Manhak,.R. B-, Pro£,_ of tIll! Pr.d''''''l!1Iice Confel'/!lII:e. September (1957). Goldberger. M. and Treiman, S. B., Phys. Rev. 111 (1958), 35+. Goldberger, M. and Treiman, S. B.• Phys. Refl_ 110 (1958), 1+78. HoctxiJe, B. Jl., ){C3Tf/J 37 (1959), 1+9; /I.ll't i!3, .mllll7i;lQf 1.;, !/iiJJ!l~ltl, 15 (1959), 377 Ji.

921

2JW

75

ON THE SIGN OF THE EFFECTIVE PSEUDOSCALAR TERM IN ,a-CAPTURE REACTIONS CHOU

KUANG-CHAO

HUANG NIEN-NING

(U"ivmily of Peki"ll) ABSTRACT

Recent experiments seem to show that the sign of the effective pseudoscalar term in p.-cap:ure reactions is different from the theoretical result of Goldberger and Treiman based on V - A theory and dispersion relation technique. A possible explanation of this difference is proposed in the present note. It is shown that a vcr' small p. s. tern ill the HamiHonian (of tbe order 1/100 of the effective p. s. term) can change th~ sign of the effective p. s. tCrm. If this small term is transformed and added as a part of the universal axial vector current, it will produce practically no effect in fJ-decay and .. - lL + v decay. Finally the inclusion of hyperon pairs is can. sidered.

922 ~16~ 1960

Vol. 16, No.2

~2Wl

,:p

2

f:I

Feb., 1960

ACTA PHYSICA SINICA

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+ Voa. p - Z( -fJm + Vo -

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925 2M

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=/12i11

,

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(17)

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m -

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II

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(23)

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1].

= -

~-(l~~-')

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1,2) -

(f~ + g,D (~ £12 + ~ V)] dx. (2-1)

rb (H) ·~JJ:;'{.".ffi·lJ: fh ffi: iJj1 iJ I ~ a{J tH..f~ f(f.lE (to) ~ M ;f& - ·(aZ) (m/ M) (111/ s) lix. (aZ)2(m/ ....t) (m/s). (2-4) rNt1=E~JJi::rr;-J.QJ-'!i r 11 Ft·s'.] (-11) ~\ri<]-!I~B--rJL:k-fb'I<],:hn [11~Jm~H:ltJ<], fli"'f-(18HI1(19hl:.:1r: -F~1j5N~rf*#~~:

.P-~~

••

f.£ffiftJ<Jf!~lE

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~.~M~mfi~,~~

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F'-> C',

C'-,> -F'.

••M.~"w.mmT.

tJ(J~jibWU:~:f:T '-I(m/.I),J)(m/s)"

(25)

~~&~ftmnMM

mt '·2(aZ)(m/M)(m!s). ;ft:r-f-'

1'1_

''If:1lW

~~ift~-=r.

~.5tilfi~@:tJ<]~:5:i1Ht~jf)f ,~{r'tJ{j~~;('I1[ 1] 9~1~J"IJa~f.fi~::&-ffl1~:la~J.

1f:!.f.l3=f[ 11 (JI,J 1tif*JfI a'.JMiiiiN.ilP%c$:(l{J1j~~7t~ rrrtt., fI!!1rJH~~IJtJ<]*~~fX. &Jt~;f[~u<], nn li~ !k%~ JE~{f.J.

[1] Foltly, L. L.. Ford, K. W. 'lI\d Yennie, D. R., Ph,'s R~I'. 113 (1959), 1147. [ 2 1 Schiff, L. I., Quantum Mechanics (1949), p. 322:

THE EFFECT OF RECOIL ON THE SCATTERING OF }.I.-MESONS BY LIGHT NUCLEI CHOU KUANG-CI-IAO

(Peking UlIillcrri:y)

D.u

YUEN-BEN

(/nstifllie of Matl,,·matit:s)

ABSTRACT

The correction of the recoil to the potential in the scattering of p.-mesons (or electrons) uy zero-spin nuclei is derived by means of a simple method. The potential obtained is corrected to first order in the velocity of the nuclei but without restriction on the velocity of the p.-mesons (or electrons). The effect of recoil on the phase-shift and cross section is discussed. The results arc compared with those of Foldy, Ford and Yennie. It is pointed Ollt that the method Ilsed in the laner's work is not perfect, because it makes use of Breit's interaction which is valid only for small velocities.

927 ~"M

l.m ~

1~

Vol. 16. No. 5

ACTA PHYSICA SINICA

*)ct1~ilii7

;r

fl=r-tiFr-iiiifiJ"iiiF~

11

May. 1960

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(1)

(2)

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q

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252

= 0,

928 253

5Wj

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2

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4 -

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(), ---21 1 21-1:,-, w"'\ ~

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(14)

929 25+

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(16)

aq'),

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mZ, ID, I • e

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(L)3J~~p~HiiR'A'

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(22)

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930 5Wj

255

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(26)

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I

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'H;

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= 1,

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qvl(-x),

Dy!n- I = y ..

D = _D T ,

(37)

(38)

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(42)

932 257

(43} (44)

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d:l J:A:&(32) .,m ~W' ~Ii*

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(46)

~~::!i!:~W1~.

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(47)

-I

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e·=-I;

(51)

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01 (

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>

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l

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X p"

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11:

1jSD;( - x)s+,

1r-=fa<J.:fll!l:f'F-m,t:'-YWmE ...c:t'j

mlf1a9~5'i::m: (qlg21 ti.'!::1:1 (glq,l!

~j :1:J;tiIiilt.Cl.¥.J-m~5f:;

+ s. 2

[ 1] JIoryuOB, A. A., 1'O.10POB, H. r., Nuc-l. Ph yr. 10 (l959), 552. [ 2] Screaton. G. R., Nuotlo CinlNlto 11 (l959), 229. f 3 ] Boromo!5oB, H. H., MellBelleB, B. B., nOnllP:lHOB, M. K.. Bonpocl" TeopDil LlllcnepcRoHHblX CoOTlIOWeUHH (l958) . .( 4] Pauli, W., Niels Bohr anu Development of Physics (1''155).

937 262

16 ~

DISPERSION RELATIONS FOR PION PRODUCTION IN PION.NUCLEON COLLISIONS K UAN-CIIAO

DAI YUAN-liEN

(P~kl"l1 U.;/J~rslly)

(At:,tdem;d Slnit:a)

CHOU

AIISTRACf

Dispersion relations for pion production in pion-nucleon collissions nrc discussed. ~cnttering

lae

amplitude is written in a form, in which the operators of nucleon field are ex-

tracted from the state vectors of initial and final stale.

The kinematies and symmetry,

properties are worked out in detail and the contribution of bound slate is evaluated.

938 ~17~ ~3M

1961 •

3

Vol_ 17. No_ 3

J:J

March, 1961

ACT... PIn·SICA SINICA

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A SUGGESTION OF EXPERIMENT TO DETECT THE p-RF.SONANCE IN 'Ir-1r SCATTERING

:\RSTIIACT

An Expeiiment r1e~ignl·,r to tl,!tect the gested.

p-r,;sorl:lllct! in It -

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scattering process is sug-

940 ~ 19 ,'{f;,

1963

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Vol. 19. No. 10

J.I

,\CTA I'HYSICA

1t

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SINICA

October. 1963

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(Z,5a)

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(tJ1'I(E)li) =

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(2.5b)

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=

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()

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(2.13~)

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(2.13c)

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+

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+ O( /),,-"-1), + K.(l),

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+

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;fIJJf:I (2.17a), (2.18)lQ(2.16b),:ff

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=

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+

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.,.,ot,1

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«cde)ltd1,E:dl 1'1CH) IZ(ab»

_..

:J4=~ft tf.H3. 9 Hn (3.12) ,t§-l-:M~rE~ 2 _...

951 660

+

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(3.13) ~,t:r:j-' «cde )lcdl,E:d IJ'I(E)

lie ab»

)1(L,\;/F8.-it- 1'/01

-:x TS4,

~t.EI.{:E;tt.ml1:Hti:~t
t,:" ;lJk.;tt~:H·~lUl-:J~!fz.;HfiJ:>jJ);'(' (3.13)11 1 iJIANJ Kronecker lJ led 0 flJ 61 I, .IX II*' T:JH~~H[[ r~1 ::kJ~ (i'j 1:Fflll /f~ riiJ ilJ1i~ '1~i'?M ,J~ r.p .!Ji,

0(/ 0cd, I.', I)

~,

~ I.+I~J=I;

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~

'I-r j;!:

f.f.~} 81('II~d' 0(1'),,1, I,; l),

(3.14 )

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1r

+

p --.i -I-

+K

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;U:fliW'I*1ft-F,W:ill:

~~ T ~1l!j!5G{t:1ft.J7j-l!lt~7f;f:f

(3.160) Xl '=i(/.bt)/K ('o~na.l; 'oK)=

~ c(jl,elj i=1>1KJ=)X j i= '.Iw t(Q~.a.I)YIK mx(·QX) , i= +'nK.~.I::

:(E(3.16)q-~, 19Jt§~;fI3M~6ll!1l:fJi!, *$fP~"';f§x-tifroili::·IrJ.. ·tEfj~">!:',*

*,15-

(3.16b)

Pi Y:J.fJJ;5 7l" 1t-=t-

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I 2' E -- 'Vp;...,

2

7IJ.

Pi"

, /' 2 , -r,\

2 1111'-

Ei -.

+ E;-p,

(3.16c) (3.16d)

1 2 E he = --fJ/J'If, f ..

2/l-.1n

(3.16e)

P". = o= 1'-.1>

=~

__ . m.

"'.1 +

q,1

tll ,I 11l "

-1- m.

+

+

11l.

(.UnO .

(3.16g)

(3.16h)

- _ .. _ ,

!IIK("'.I

-~-- q" 1/1,1

+ q. + qK.

111.\

_

q.1 _ m.

m.)

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(3.16i)

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10

661

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(3.17 )

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+ P *if1C:!1JIi15JJ 820 lI--IeV,

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if lK

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iii 60 MeV ~llJftfg:buiIJ 75 MeV IIftilI:at ~ m-r1' ltttt-J7fht, :I*~!±lm j[(r.iJ~.~, ::fX1Hi5~ 't:tE,tISJ J: T- ~ 1$ rK ~ J%-'J j'] 5-10 lyt MeV(-I). ~ E.b :iZsitl130 MeV 1ftilI:11t f±l~ Yf ;It m ~. I

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J Ho" ~~~~~~I_~60 90 120 150 EM 60

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m

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+

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=

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iI~:1 J; JlIJ* J = -.l at, P(Y:-I) 2

=

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r

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,

(4;;-)3 - [p(E;)p(EA.)p\EK)]I-

. o.I~B(:\.·)), (4.6)

A(x)

=

2: [T/+t(!ll + l)P;+l(X) -

TI-tUll-l)P;_t(x)],

(4.6a)

I

flex)

=.z= [-TH'~(lll -I- 1) +

TH·Ctll-- J.)]P;C:r);

(4.6b)

I

ni

=

Pi, Pi

n.1.1f

=..:

_PI'". p,,"

Z'i Z·i ' " --- Pi~. ~"

P (I:'~i)

--E-'

p(R,In) "" IL,l.~},b,

pCE}J

(4.7:1)

( 4.7b) =

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( 4.7c)

955 664

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(4.7<1)

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==

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(4.8)

(4.9)

F = o

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( 4.10)

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(

da

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1

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(4.11)

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.

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P~

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2 '

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~ P.I

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r.

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(HI.lla)

I

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+

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(111.13)

(43t)l!l.y' plEA.),o(EK)p(E j )

11 I Hurnblet, 1- ,>nd RuscllfclrJ, L., Nlld. Phys" 26 (! 961), 529. [ 2 J 'OOJm: B33b II OKYllb, )f(3T(/J, 35 (1958), 757. NeWlon, Anllals of Phys" 4 (! 958), 29. [3] B1~tt, J. and \Vcisskopf, V .. Thcon!lical Nuclear Physics (1952). JI~, lbt·J'~J.!l1i;'lil!l' >~ (l%I). [4 J ~Hm: JlorI1lllYc, jJ. yJ. II 'i)I':OY ry::n-Q)K!lO, )K:'JTlP, 39 (1960), 112, 15J W~tson, K, M., Ph"s, Re,-'., 88 (1952), 1163. [6] JhlllllAYC, Jl. I,i. II 'liKOY rY:lII·t.j';{30, JK:'Jr(/J, 39 (1960), 36-1.

[7J [8 J I9 1 [10]

[11]

POllda, L. ~lId Wigner, E. P., AI.tun, 1\'1., ct AlsIOn,;\-I., Ct

Newton, R. G., Phys. R,',-,., 119 (1960), 1394. 1'II),s. 1{~'TJ •., 73 (194Mj, 1002. 01., Ph),,,, 1<e,·. Leiters, 5 (t 960), 520. aL. P/:)'s. R(,,·. Lett ,'r.', (j (J %1l. 300. Erskr. r., C[ "I.. Ret'.<. ,H"d. I'h),.L, 33 (I %1), 43". \'V'olf, S. E., et aI., R",· ..... Mod. l'hy.<., 3:1 (1961), 43<).

963 672 [12] rUJ [14J rIS] r16J

Alston. M. H. anel Ferro-I.uzzi, M., Revs. Mod. Ph)'.r., 33 (1%1), 416. Erwin. A. R., March, R. H. nnd W"ikcI, \'1'. D., lY;{O;!(I Cim"IIIIJ. 2,1 (1962), 237. J\lcxanJcr, G., ct aI., piJys. Reo. l.el/en, 8 (1962),447. Fly, R. P., Cl aI., P/;ys. R"". Letters, 7 (1 %1), 461. Dalitz llnd Tun", Armals of Phy.r., 111 (1960), 307.

INVESTIGATION OF THE CORRELATION BETWEEN RESONANCE EFFECT AND NEAR THRESHOLD EFFECT IN THE 71' + p---+A + 71' + K REACTION Su

ZHAO-DIN

GAO CHONG-SHOU

CHOU KUANG-CH-~O

ABSTRACT

In this paper an experiment for the reaction 'IC -I- p->-ll + 'K + K is proposed. The total energy in the centre of mass system is fixed at 1900 MeV_ The kinetic energy of the final K-meson is then in the range 0-90 MeV. We propose to observe the final state Ib1' resonance and the cusp at the ~'KK threshold_ From the· correlation between the l'esonance and the cusp and from the angular distribution of the resonance, the spin of Y 1* and the ralative parities among Y 1*, A and J: may be determined. Based on the diagonal representation of T-matrix in the channel space, a general phenomenological description of the resonanc.e effect, the near threshold effect and the final state interaction is developed, which takes into account especially the correlation of the cusp with a nearby resonance. The general theory is applied to the present experiment as a special casco

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IS THERE ANY SYMMETRY BETWEEN FLAVOUR AND COLOUR GAUGE INTERACTIONS? ZHOU GUA::-m-ZHAO (lll •• lilllt,·

of TTteoretical Ph.II.·i~8• .:1c:oc/elllia Sillica)

GAO CHONG-SHOU (Peking Unit:el'sit;v)

_\:eSTR.-I.CT

The electro-weak Illodel in SU(3) X [,-(1) is extended to include strong interaction. A possible exchange symmetry between the flayour and the colour gauge interactions is disC'usst'd.

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£~ii!7f:~Ji!.:.

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(2.10)

EI:1 (2.6) ~ (2.10) ~~UE~

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=

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CPA~+J

(2.11)

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(2.14)

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(2.15)

970 612

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= -1- F~.Fi". + Cjjrl'(B,• + A~)'cP + 4

- V(4),

Tr[(D"Cf»+(D"rP)]

4>+) + 1... CjjC/J(I + 'Y,)cP 2

+ ~* Cjj4>+(1

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0.1)

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ttP~*-t-(~-t-)nt S

= 1 I¥.J U~5) ~~,J3i!m~7f'3t, Higgs %.Itt( s =

-2 il~ U~+) ~jijL ~if

~ilElY3tE(3.4)I¥.J~u:~~r, (3.1) j;\;;~7f'5fI¥.J(~T V(4), CPT) a~Ml~~~~rHO.

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TiHi:f.tflJ* Higgs ~}g 4>(6)

*il!:fjlt~,

51-Higgs :tlL-f-. Higgs ~a~~~J!1H~{gIfxf:1 (3.5)

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1::

fiJ I¥.J 1::]?X; 5C?t JJIj}g It! f.lj (3.6)

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~ {I

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.~&~.£~~*&~~*~~.~.~~~~.~£~~~~~~.~~.~~ 1itE~im.g.

~

.

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s"

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R-

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-
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ntU

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EB (3.12);fO (3.13)

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(3.12)

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(3.14)

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f.iL T v± ~ U±±, 1:: ff] 7t JjiJ iW-1' Et! f.ljfO j:i;q l' Et! f.lj , 1:: ff] Et:J ml.i:

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~~

(:'

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T

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{f] :tEJ§Iiiiili~ i!\:-zy it L~.

~T13AAHI3~T!S*13][f'FmES

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fiJm

0, CJ!(1~tH

(3.10) ~ TsL = L, TlR=

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= i

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j!;t:j:r i; N:lm'm'a9 Gdl-Mann i

(4.1 )

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)

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( 4.2)

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+

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+ .jig_ ( 2v I.YI'VI.

2

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-

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+

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X>j~lIf.

i£-r.!:3 Higgs ~at;f§][fFmJ9

1. Cjj41(1 2

;fIlm (3.10) ~

+

Y5)c/J

+ t* Cjj41+(l 2

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(3.15) ~iliJ9

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j*el.eR)XD.

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(4.5) 0

a9

:,<:0 :;:f~l:3~-r1I3i

M~OOa9M~~iiJ~~~,i,t.B.~~at!j£-rl:3.~tt-ra9f§][fFm~&:i!~fiL

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film

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ff::tE;

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(;.1)

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or XJ~lI~Ht

e .•

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(5.5)

T r[(D,lll)+(D,.t1»]

q:r ,:j!;$ t1> E8 (3.15)

:;;.t~tH, D,.t1> EI3(2.11):;;.t~tH.

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W-

ij:;'@;jlj Y-

W+ _1_A __1_Z

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975 617

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idJ

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J\ + J\ +

+

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+

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J

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+

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+

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det ([J

+

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+ <1>11 fl!J1i;~Jm~{t3)\~~:~r~.

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<,"C"), _

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:

:).

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v

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!i

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2

+ :

V'2) Z1 + 4112U+U-- + (v

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J2

(w-V),

l

+

(6.2) ~.

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V'1)(W+W-

+

v' m~

V+V-)

(6.3)

976 618

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4

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=

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2 mU

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'

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2

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~

l.. (_1 2 2 mw

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1

2

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v-

4

(6.6)

+_1)

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+

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4

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v-

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1.

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-+

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#

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4

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(6.9)

977 619

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;i!1It7tff-Ij'

tj)(3),

Higgs

e

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f..lL-J-,

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=

mw; (iii) ~

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f,~iIJ

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~ffj&-&iiJlGJ.iiEiW 6 ~~jf;: !pc') ~lIBUt~~;&~(f.J 6 ~U~jf;:£~ffl"a~.

f.!l!.:tEt-t~ Higgs t1-F1,] !pc') (f.Jfi1~. ~~~~I31tlitaRJS, 1f i 1'~~:I:~~l13ttTt;!i~, iiJlGJ. lmi:t1li!m~m*~,1*~T*(f.Ji2Eg1f 51'~~iI:. :JtffJ~~tE Higgs ~~~JtIj~!l..7'J

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,

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(-S' + J: ),

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mJ..'t:;rIGJ.~~~~r.uSB~~~J!/: 6i ( : ) , ~~~JS(f.J ;, = :, = ;, =

;,

-

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a:JFJUGilll*, ·/liHWT-

*

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~~~(f.J !1t~iiUi~El3 Higgs ~1Y:i~ tFm~m~tI:l. ~~jtliiJl\1Jr£1tl't:J~;jt, @ f'f.m'!i~·gEt ~ 111 rp+ (f.J~SM~~~IY:i~~~IY:i~~m~~tI:l.

:tE*m Higgs:li~ !pc') (f.J~15fF,~IY:i=~~~ • .Rff-1',&P.iE!tT Tr(!Prp+) Jr:lr.Pi;=lX~~lD: 1fdct!P:fIl dct!P+, lX~a9Jlji:!III~:tE V(4', 4'+) tf:1t1:!]Ji!, l(f.:tEX<.tf*~~2t~f!Ratflj*-~"~~, ?tffl ~~vtf:1~#:tE~ff~Jlji,:tE*xm~~tf:1~.M~~#I't:JJljitl:lf.!l!.~I't:JJS*;~~~~.1f=1',~~

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±IN;

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i-I

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i

t

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,~.

±q,icf>Nlq,: = l~ i,A

= 1, ....•. , m ::f'~~:lz:!t:J,UEft.

m.:iIi!lLit~l1I!., 1" I" I" $:m:~9..1f~1'~~:lz:!t:J. ;fIJm4lit:J-~~j;\; (U), :&;r:~~!;] I. = 2Tr(lPlP+) = 1,

= i-/~,

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(U.2) (11.3)

.

L;

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+ ..;T <;b;A<;b:' + rpi;rpt;:'

1. .\=1

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;I!t$ p.', ,.>0,

=

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CJ""'~~iU V:t£ 1. = L2a n;j"~:f&/Ht. ~aI~nfil:J9

13ffm~ml(!;'g

(Il.s)

.pm <rp,,). = ~I -~ ~z ,

~~,~ lP $t1'7tm~ilJ*,~rtilli1'7titJ9~!t:J <;b,,, rp" ;fa~!t:J rp" = J ~; ~!t:J~2tlit~

xo

~~~~JJl::f.Il:

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J9~

rpo fIl

mi = 4/-",

<;b"

$J!f.1ffiQi'F SU(3)

= cpa,

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= cp++

(1l.4)

+ x.

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mw.

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!t:J~f*

X:t~M:. tE""F -:ifiiX. t:p, ~f1J:mtEi!1'~~1lEjilJ-E!m~R:tE~ ,tE)lJ~.!E¥.f~ili~r~-rilJfi!t:Jfl i~.

S. Weinberg. PIIYs. Rev. Lett., 19 (1967). 1264. A. Sal:lm. Elcmentllry Pllrticle Theory. ed. N. SVllrtholm (St~ckhOlm, 1968). [2] .00.1978 ~*~j;j!J~f.j!j~f4l!l!~iSO,(~. [3] Y. Neeman, Phys. Lett., SIB (1979). 190. [4] D. B. Fairlie, PAy,. Lett., 82B (1979), 97; E. J. Squires, Phys. Lett., 82B (19i9). 359. J. G. Taylor. Phys. Lett., 83B (1979). 331; Phys. Lett., 84B (1979), 79; P. H. Doudi aud P. D. Tllrvis, Phys. Lettt., 84B (1979). 75. [1]

980 622

UNIFIED ELECTRO-WEAK MODEL IN SU(3) ZHOU GUANG-ZHAO

(InlJtitute of Theoretical PhyaicB, Academia Sinica)

G-~o CHONG-SHOU

(Pe'kmg 17nit'erBity)

ABSTRACT

In this paper a unified electro-weak model for leptons based on the SU(S) gauge group is suggested by means of four kinds of realization for the generators of the ~oup. For all low energy electro-weak processes, this model predicts the same results as the conventional Weinberg-Salam model does. The Weinberg angle is shown to be sin"e.. = % in a natural way. When the Higgs self potential respeets a discrete symmetry cP --+ - CP, a new conserved quantum number called weak strangeness emerges from the model after spontaneous symmetry breaking. In the present model there exist another four heavy vector gauge bosons V:t and U:t:: together with some heavy fermions and Higgs scalars, which have non vanishing weak strangeness quantu::n numbers. These weak strange particles have no direct couplings with leptons. Their existence will not influence the low energy electro-weak processes. Nevertheless, they can be produced in pairs in high energy collisions and the lightest of them should be stable if the conservation of weak strangeness is exact. The experimental implications and the possibility 'Of violation of the conservation of weak strangeness are also discussed.

981 1980 df.

*~~ml#gft~a~~~&~~~~~~-~*~o~~m~~~~4~ ~~tit~jjtt, 1!~'dijUfffit~IfijlBt&*,

fIJIHa.l:t.911T J(O),

;fila3 J(O)lJt9~,

iE:fi-1-f~ IE J"j, ~

1-

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fttfH~ IJJ!frfti~iIllQ

(-) * :X:Tll&f£;JH}:;;T1ttf:J.:t-~ Dirac

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[Zl.

fE

~ 1933 ~:mtfj~[11, J3,ffi Fcynman:£E 19,18 ~

ffi.;l.Hl!!.fl"JiY-JHii"mJJ:lT~~df~Atr:J45i:~ffi~, ~l:MSIj£ L=}_ql_V(q),

2

. dq q=-"dt

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2

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q.

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*

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at . a ,. 1-.0=0, at "

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oq

j 2~

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= JdqOF (qt ,qot 0) 'f1(qo, to).

~~.~m~~Fre.ffi~~t~~~~~~toH~~ffi~~*7~*.

• 28

1

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984

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• ll . lq" >"'
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dpd

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• 29 •

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2

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)

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)

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)

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f'

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IJ.

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E::

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rkr) r~1

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r' (4.11)

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e,l

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iS o

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:l

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J

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- ;

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[im;~.q)]

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2

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m

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-+

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1/

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e

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n!

t

fZ~,(q,q)'''fZt(q,q)p~I....:''+ln)(O)

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(

.)- aZL(q,g)

mAj! q,q -

rJ.o(l'/J Cq q

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a

12

a,I-'="

>.r)

(4.21)

"'n

(4.22) • ···PT~(.) 1 '1,'/ = - -

II"

n!

fJ"H(p,q)

op .... op~aprOp&

=_1_(~ ~. n!

m Cq

).... (.l.~)~(l.)r' . m Oq

m

(4.23)

• 34 •

990 a~~M~Ma~a~~~~~~~.§~~.~~,.~~~m~A~.~.~~~#~ M:*ft8~:I;~ftil~ 7 ffi~~f£o

[1] [2j

[3] [4] [5]

P.A.~L

Dirac. Physik. Z. Sow-Jetunion, 3 (1933) 64. R. P. Feynman. Rev. Mod. Phys. 20 (1948) 267. Phy.r. Rey. 80 (950) 440. E.S. :\hcrs and 13.W. Lee. "Gauge Theories", Phyr. Report., 9C (1973). T.)). Lee and C.N. Yang. Phys. Rey. 128 (1962) 885. i\UH. Mizrahi. JOrlr. Math. Phys. 16 (1975) 2201,

• 35 •

991

s~

j(&

~

j(& 1 M

1981 ~ 1

J.!

~ ~

1m !It J1I!

!I#o Jm

Vol. S.

PHYSICA ENERGIAE FORTIS ET PHYSICA NUCLEARIS

No.1

Jan •• 1981

~T SU(3) X U( 1) @T~4f51~EI!

~-_m~-~l"tj"~ %J jt ~

...>.-

(lj:Il!iIf4~~llU~!It6l!1!!iJf1EPJT)

~

** "-'

7if

(:tl:~*~)

;f.X#it1 SU(3)XU(I) ~T~*Jt~lt$jt;-~t!~t#g)VHPJft!: (iHt it 1 Higgs :Ii ~ Wi ~~~:7f ME f,if ~ ~ t#g;lf ~ ~.zt ~ Weinberg-Salam ~ 1! ~ :* (ii) ~ ill fA Higgs ~ if Jfl ~H~ ill ~ it f,if ~;jt If.; ~ ;\Jk: ~ iiE!Vi, :#-;t ~iL~::it {r1i.tit; (iii) :llt.1 ~HJ.n:~~A)t.~.Ubl~:T~~~at, ~T~*)t~jJ1:-i.t 1tJJt*'*Jt~r-}( Cabibbo il1-ili ~~*if..

*

*;

tE:>C(l]r:jl ,~{f]tE~m:~~f'F SU(3)

X U(l) (f.J£lilII_t.~ljm~ftl!<:(f.Jggf*~I.m,m

ili 7 --t#f-tli'i~~-f-~~5l~EI!ff3]if'Fm(f.J~~.j!-tm~u£~-t Higgs:IiI.~»H,*-m: VI

~

Vl

zl± == V

I::r ~=.F5E~t-j ,~l:H 71:3 Weinberg-Salam m~[Jl.f§IR.I(f.J~*.:tE:>c

(1]

r:jliE~ili,~ sincp« 1, sin2cp/v« 1 t-j ,j!-t~~~ili(f.J~*tEf~~~m:8ilI*.!~ sinlBw:S

1/4 (f.J

Weinberg-Salam

m~(f.J~*ff3llI,

:tf: .l3-..!:3!.li!.1f~~ffH~.

1R.It-j:tE5Zt-2P"(f.J~iTl~

~~!J!iil!jfi~:i:(f.J~~r:jl, j!-tm~..!:3

Weinberg-Salam ~~(f.J~jJlJ£"ilJlt.l~~(f.J. :tE*:>c r:jl ,~{m~:tE)(( 1 ]~£i.fU:,;;:1 JL-trPJJmf'F5Zt-2P"(f.JW}t.

Y

tE:>c [1] r:jl, ~{f]sIAmm 3 ~~~(f.J Higgs :IiI cPt = 0 fIl Y = -1, 'E:;(f](f.J1{~;Wj*-m:?t-jJlJ;7g

~ cPl ,

(~~ -(n' (~~ ~ (:). ~{f]fJ\j!~~$H~·ftl(f.J.~;7g~~ I.

'E:;{f](f.J Y:I:-f-l!<:?t-jJlJ;7g

(Ll)

1.!l£tEsIAmm 3 ~~~ Higgs :liIa9f'F$r ,iE"ilJ It.l1f ~-f*~Qj.tJi~,f1P cPt ~ cPl (f.J Y :I.:-f-l!<:?t-jJlJ;7g Y = 0 ~ y = + 1, 'E:;{f](f.JA~;Wj* -m:ff3ml:l:&;7g ;;j;.::lt 1979 ~ 9 J.I '4

a 1&111.

992 40

(~,). - ( ~' ), (~~. ~ ( ;, ),

(1.2)

JiJffJJl.ftl;f~Hf!!.ffFXl#Hf!!. II. *lJq:r:f.t{(J#}ffitA±~ta 5I!#~.m~ I ~!:;$X (~~-\%*;gJ11 (1]). ~iJm(1.2)~, "iiJld.M Higgs ±m:bZ~~~i;f]~JPi (Dl'fPl)+(Dl'fPl)

+ (D/Pz)+(Dl'fPz),

(1.3)

1~ ftllvUk:J§J;Jilr'B±m~ ftfl:i;: JPi Xl

~

ilvd z( W+W- +

v+V-

+

~

zz)

+1-ilvzIZ(W+W-+U++U--)+1-lvzll(- g

2

J3

4

Z+Jgl+g'lZ')z.

(1.4)

rtlltl:; ~ fiJiffl' ~ fL T ftflm: f;] myz

= -1g21 vl IZ , 2

~1-ta5l!l:j~m! I q:r~~.

* v>

muz

=

:tEmm! I q:r

21 g-'I l'Z I'-,

m~'= m~

Z = my 2 mv;

+ m~,

+ mu,

( 1.5)

1

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1, !ilUWfLT:m£l1i~I¥J. IJi!.:tEI¥Jm~IIt:jJ,~itv~{iiJ{g,mH~HFfii:TU~Vm

.~~WfLT~.

Z ~ Z' ~ll[~i~~nlG Zl ~ Zz Z

=

COSOIZ\ + sinaZz,

mi = "±"'m~ 3

201

COS l+v

4-

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3

1+v

m~=-m~--

1 2mz ' mu, my, I

-'>- 00,

.IlP%:i:@lfiJT Weinberg-Salam I ~m!ff~~.f.f~" #

tEttm! II t:jJ,

Z'

= -

( 1.6)

sinOlZ\ + COSOlZZ.

[1 + ~(1 + J"3 Sln!p t~ a )2], 4

[ tiOl+v ( tgOl~ 4·

J-3 - 1. -)2] •

(1.7)

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(1.9)

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«

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tI -'>- 00

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m~/m~1

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?§~f1J~~J:B*f*~fiStt mw tmltfA,:iX~>.I<: m~, m~1 -'>- O. ~~~~J:#*~UliH!t

993 41

~~~.~*EI!~:i:*-li:r

y:l:

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liiT

m~

+

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mv

mu

ml::t~*,Mffij~4HIJ!Ji!.1ffltfm~~~*

= m~, ~~Ji:T v ret~~llI sin2cp/v«

0(1). :Jt{f1;!I~~ sincp« 1"

11-

;fI:l

mv

~~I!fj~~!Ji1~it~~ili:i!;pfrt&~&.

Z1,

1

=

•

2

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mu

l+v

00

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a ~m*, &P

1 ~t.*. ytLr~U~rIm:l:79

=

V

•

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--m\V,

l+v

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cos 8 w 1 + v

m;= m~v. 3v cos!8w,4(1+v)sin 2cp

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mtv • 311 ,(L14) cosz8 w 4(I+v)(1-4sin Z 8w )

=

_

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:fif~T Weinberg-Salam ~~tjJfDi§~

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Z1 ~ Z2 f§.@a~tjJtt7nt71JJIJ79 ],v,1'

= P,. _1.. 4

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CPJ~""

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V

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*.

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994 42

B~f&~;95~f'Fmwr1«
Weinberg m~a'g~* B~*a'g m~,

. • ~ II tj:J~#:tE-1-f&~&~~@jfl Weinberg-Salam m~a9~*,

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<

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:::, Higgs ~ S'g §

ik ~tgc

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m~tj:J5IA 7~1- 3 ~t Higgs ~ 4>1 ~ 4>2'

BffJ:tE

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u(1) ~~M:~:;r:;~. -~15l*EB

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(4)t4>I)2, (4)i4>2)2, 4>t4>l4>i4>2, 4>t4>~4>i4>1>

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X

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202

201

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(~,). -(:),

r*Mit.

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(~,). - (~,

., -J:. .

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995 43

SU(3) x U(l) ~~ 91-mm:±1.J, liI!fliJtJ§~:3'tT7~~*;!t* 81-mm:fiLTm!lt~~~ :&:, lZt-f:5it~~M Higgs ±1.J$~¥U 81-?t:i:fH~, ~&PPJ~lmllmm:~~~~ Higgs ±1.J® 81-?t:i:~~f!i!,X14>1 ifO 4>2 ~~jIfF 41-?t:i:. :tE.~WH~HamlLtf'F5E1J'l\~~·,X14>1 ifO 4>2 ?tjJlj~.

4>; = 4>1 + 1. VI 2

;3 + 3 (gl + W g8 ..;

g1

)

, 4>; = 4>2 + 1. Vz

(;4 -

s ig g5 - ig;

) •

2?

-

+ igs

- gs - 2e

..; 3

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f!i!-1- ,afil~i~f!i!fiJ •. lZt-f 4>1 =

VI (

Higgs ±1.J'ilJ~~

+ ..;12 XO) o o

(2.9)

,

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EB'T§1i'.t

Higgs

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:=. ..

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996 44

~='~lF!*-=f=.g~?HJljm 1f!~m Rlo R z, R3 *7F. Higgs

tPh tPZ, tP3 *7F, 3m*JK~!r*-=f£~%, X;j$:ag='1-

±m 1- [fijZ;;iCP(1 +

cP

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'Ys)tPj

2

+ fjZ;;jCP+(1

-

'YS)tPi] (3.1 )

+ 1- [f~i-RiCP+(1 + 'Ys)tPj + f::Z;;jCP(1 - 'YS)RiJ. 2

a#~~~.-ft~.=ft$cP~~*~~1L~.~ag*ffi~~u~ cP

= -

;2 (-:3

~3

cPZ -cPl

sIA1t:5!WH~{R O

=

II, O

=

-:: )~ 0

cP

;= (

(3.2)

:: ),

cP3

0 = 0, ttA(3.1):P:;,#5IA!r*-=f~Itr-F-5 0.3)

mij =- ..;i 2 f ijll

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Rr.lpti:JDiiiJimi:tm1-m:*~15$ mij

i¥Il m~i !€i!:l:L~ mi/bibi

T.

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+ m~iaiaj.

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997 45

1Mt.

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u., c,t~

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@Bffls91Pt

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1±l:

J<.

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• •..t3&it~~~PJla.mj~~~~m@~~~~~m*.

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R,->-R"

!P(I)-+W(I), !P(')-+±;!P{2), ,.·J~-+A~

(!P(') :YIJ~!l!j~;A;F!Hfr..!;;})

;Zr,1lL~fi~~. ;r:::~~ili,1lL~tit[ll$~Jt*T~ Higgs ~a:Jt§][fFmllli;Z~h ;!t"t:;fl:Jlllim~w.~ ~~fI:J • .:tf~~ Higgs :I:iJ~][f'Fm1lL~jfiiJ~}9

+ 1~$;g:;(I)+P_"'i + 1;;R i i:[l(I)+P+",; + l;r$;!p(I)p_Ri + hi;$;g:;(')P+"'1 + h~$;g:;(')+p_",; + h;;l?i!P(')+P+",; + h;r$;(p("P_R; li;$i!P(llP+",;

lI~Iik~FR~Jtlit:J~:itJ:ili~~~~~mit:J~>.It,JilUff 11;t/., I;;"~, h;;""

h;;,,:

m:*

(1.4)

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III

= I.. = I.. = I.. = In = In = 0, I;.

=

I;. = I;, = I;. = I;, = I;. =

(1.3)

0,

998 46

1.11

="" = I." = I." = I." = I. .. = 1.31 = o~ ";, = I.;,

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~,).

(1.6)

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t1:17. :!m*f¥tEl!!Jffl. ~~~it~. :ltell« (I. 2) ~ft:J'-E)(mr-, iiJl~H~i1'-E~ 4 ffl. ~~rt:J R ~~fg:JiQ7'iJ .p.--.p., R._-R•• (I. 7) UI:~R~lY'<'I'~~tI:I~:I:~Il$~

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~(

.;

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[1] [Z]

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°

:).

4 ° h' 0

c· D

m, = 4

;t

0

h'*

I-'

e*

0

0

(I. 8)

1980IF5iJ3Wl~233~.

S. Weinberg, Ph,s. Rev. Lett., 19(1967), 1264; A. Salam, Elemelltary Particle TheOl'~', rd. X. Smr· tholm (Stockholm, 1968). [3] ml:l\';B,a~"" a~~.-"3~!I2., 4(1980), 609. [4] M. :Kobayashi and E:. Maskawa, ProgT. Tlteor. Plli/s., 49 (1973), 65::!; A. Pais and J. Primack, PIIY8. Rev., D8 (1973),3063, L. ~raiani, Plli/s. Lett .• 68B (19i6), 183; S. Pakmsa amI H. Sugn· 'Warn, PhY8. Rev. D14 (1976), 305. [5] H. Fritzsch, Phys. Lett., 70B (1977), 436; 73B (19i8), 31i.

SOME DISCUSSIONS ON THE SU(3)xU(1) ELECTRO-WEAK UNIFIED MODEL OF LEPTONS AND QUARKS ZHOU GUANG-ZliAO (CHOU

KUANG-CHAO)

(Illstitiite of TheoreticaZ PhY8ics, .dcademia Sinica) G..\.o

CHONG-SHOU

(Peking U1Iil'crsity)

ABSTRACT

In this paper we discussed the following problems in the SU(3) xU(l) unified mod· el proposed earlier: (i) Two possible choices of the Higgs fields and their comparison with the Weinberg-Salam model; (ii) The form of the Higgs self potential and the realization of the spontaneous symmetry breaking; (iii) The relation between the mass spectrum and the generalized Cabibbo mixing angles in a model with several generations of fermions.

999 ~8!{& ~2J1JJ

A

1984 ~ 3

~ ~

!IW 3m .!:§

~ ~

3m

Vol. 8.

PHYSICA ENERGIAE FORTIS ET PHYSICA NUCLEARIS

No.2

Mar •• 1984

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:tr:=.f.jfQ;fj=.f.

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F. ~ 190 MeV, U(x) ~ su(3) ®5f;~, ;!t7C~/J\*;r.I:;m3i!S. SU(3)L X SU(3)R ~f* EI.~~~~tl~ft9 su(3) ® Goldstone ~1S~. n ~-:f5:fr:i:Ttt, Q ~ 5 ~m:, ~ifl:{tl¥J ~ ~ 4-1lt~ M - S4 ~ Q ®iiIW. B~:tE1s!iltBM HL X HRc;;;.SU(3)L X SU(3)R

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*;r.I:; n L ~ nil .m~ (6)

(7) ~:tE1s!ilm:se~ (3) r~se®,lZSIifijEl3~;r.I:; (4) ~)Z®

I

~7mm~~®1ff~.yFffl:m:. ~!I,

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(8)

A; ~~)Z:tE 5 ~8€m: Q ..t®1s!ilm:~, Fa ~;f§ElZ®1s!iltB~5j; ifij.§.

Wp.",., =- Tr{ [( -Ap.LUpLU"LU,L

+ 8p.A pLA"LU 8L + Ap.L8.A."LU8L ) + (L - R)]

+ 8p.A.LU.4."R.U- U'L + - 2:.. [Ap.LU. LA"LU 8L 1

2

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(9)

1001 254

Af - T(L) - feR)

-

~ r d4xS.''''''~ Tr {AtU-0.A"LL-I0~L + 1..2 24~-J

A"LApLA,,1.L -lo~L

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R) }

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3lC*

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R

=

1+

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SR,

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+

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+

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1002 255

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(19)

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[ 1] [2]

E. Witten "Global Aspects of Curl'ent Algebra I I, Prineeton Preprint (1983). J. WesB and B. Zumino, Pl''U8. Lett" 37B(1971), 95.

1003 256

ON THE GAUGE INVARIANCE OF WESS-ZUMINOWITTEN EFFECTIVE ACTION CHOU KUANG-CRAO

Guo

HAN-YING

Wu KE

(Inatitll.te of ThBOretielJi Phy8ic8, ..4.cademia Sinioa) SONG XING-ORANG

(Peking 17n'lIerrity)

It is shown that in order to introduce the gauge invariant Wess-Zumino-Witten effective action a global anomalyfree condition should be satisfied by the gauged subgroup of SU(3h xBU(3).. The condition requires that the left handed and the right handed Chern-Simons 5-forms with respect to the gauge group be equal to each other and it turns out in the local sense to be the usual perturbative anomaly-free condition. It is also constructed a gauge invariant effective action under the anomalyfree condition by means of a systematic method rather than the trial and error Noether method. In the non-abelian case, the gauge invariant effective action presented here contains less terms than the one obtained by Witten. The case of pure. gauge is discussed in the present note as well.

1004 ~ ~ ~

$Il4W}

1ff8!{&

1984 ~ 7

J.I

~ ~

:1m Ej

:1m

Vol. 8,

PHYSICA ENERGIAE FORTIS ET PHYSICA NUCLEAlUS

~ i{~

JiJJ't??

~

No."

July, 1984

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( rp~f-1.~WEl1I~!!t6ll1raf~m=)

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JtlT ~ Witten

Chern-Simons

fi:ifr~~:ii:,

Witten

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(4) 1983 ~ 11 J! 19 ElI/U"tJ.

1005 509

= a"uu-1,

U"l.

U"R

u-1a"u

=

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1d4xe,l'·"~ Z ".,,~ -

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[A.u, A.]

(9)

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srr:.MWjjJ~nlG

W

= flQG" + O(fl2)

(12)

G" it~~FX1f*~li1itl9i.

~~~~~~,~fi~~~ •• tt~.tmw~~~~~~.tt~~t~§~A .2:1-~JDt?

:X177.Hfi t

~ f! JDt , ~ 111 jjJ Id. ~ m (13 )

*1il7f t, (1-+)

+ + Id.~

fl"tt

=-

A"RBvA.RA{Jc.

+

B"AvRA.RA~c. -

A~c.A.c.A.c.A8R

1

·4."RA.RA ..RA sc. -

.

2" A,.C.A.RA.C.ASR )

1

4S1IJZ dx4s"""~Tr{nQlQ(2BI'A.c.B.Apc. + + B"Avc.B.AIlR +

(13)

2B,.A.RBaA PR

B,.A.RB.A{Jc. - B,,(A.c.Aac.Apc.)

- B,,(A.RAaRAPR) - B,,(A.c.AtJRAsc.) - B,.(A"RA.C.AJlR) - A"c.B.AaC.AiR - A"c.fJ.A.RAIIR - A"RfJ.A.c.Allc. - A"RfJ.AaRA{Jc. - a,.A.c.AaC.AIIR - fJ ... A rIl A ..RA{Jc. - A"RA"c.fJaA{Jc. - A"C.A.RfJ.A IIR

+ A"c.A"c.A ..C.AIIR + AI'RA"c.Aac.A{Jt + A,.C. A .R/1.C. A IIR + A"RA.c.AtJRAllc. + A"c.A.RAaRAIIR + A"RA.RAaRAllc.J} (16) llmtt.jjJld. iiEOO fr ~~~FX1f*~li m- GQ, tEJtt~1110lJ~·iliA·*H~~)1.1-±:~~~. ~ (9) J!::ftA. (8) J!::iiJ~~ naG"

== ~ s"w'Tr

an -

{nOla

B,.v vVaV{J -

[fJ"V.fJ.V i

+ ..!.. fJ,.A.B.A{J 3

1 1 1. B,.A.V.A/I - - B,.A.A..Vi/ + - fJ,.V.A.AII 3 3 3

-

+.i.. A"B.VaA , +..!.. A"A.BaV{J -..!.. VI'A"B.A, 3

-

VI'V"BaV, -

3

3

1 - AI'A"fJ.A, + VI'V.V"V, 3

+ -31

VI'A"V.A,

1007

1 1 - -AILA,V"VR - - 1 VILV.A .. A~ - -AILA .. A .. A~ ]} 3 3 3

¥f~ AI'- VIL ~ AlLl., AILR ~*~~ (10) lI"G"

ftA

(17) ~,~~*~

= 481TI~ s,,·"/lTr{lI" 1" [Zo!,A,l.O",.-l8l. +

+

oILA.l.o"A8R

+

-

AILl.o.A"'l.Apl.

2oILA.lI.o.. A~1I.

o,.APRo",rl./il.

+

rl.p.c.o • .4."'l.A/IR -

A,ul.o.A~RA;ll.

+

A"c.o.A"RA pR

+ rl.ILRo.A ..l.AdC. - ...J.ILlI.o •. J",l.'.JdR + AILRo.A ..RApl. - AILRo.A ..RA/IR + oILA.l.A ..l.Apl. + oIL •..J.,l.rl. ..C.Adll. + o,,,A.l.tlarRA8l. + o"tl.Rtl ..l.AdR + O,url..il.A",Rrl.pl. + O!,rl..Rrl.",RA iiR + rl.ILC.A.l.O.. Adl. + A"C.A.RO",Adl. + ...J.p.R •.J.c.o",Apl. + A,uc.A.ROa •..J.IIR + Ap.R,rl..l.O",rl.PC. + Ap.R,A.Ro.. rl.HR + AILl.A.l.A ..l.A.ilR + AILRA.l.A",c.A./Il. + A"l.A.Rrl.",r.AiJR + d,uRtl.l.darRrl./il. + rl.ILC.tlPRA"'RA/IR + d"Ra.Rrl.aRrl.~c.J} ~1iX::¥Q,H~~ilJ~

ffiJ

tt

(17)

(18)

~~~F~
Witten ~!i!i *E13 T~ T -@:iji,JiHfFP

f' = fo +

lIaf~

+

(19)

O(TIl)

(zo) (Zl) ~~.~A~~~~m. ~~t~f'~g~.

~7'~,

t ~ W-z (t-]~*W(*)(ll)~);f§~t.,$.?~n~~-IID"J.OO;f§IS1,@ 0 1ID":iji~ (Z2) (23)

= 0 t(ll = 0) = fo W(ll

mz~mlB

fo

=

0)

CP1J.U>~~Jj( {gi2~'fo)~3!rrf1'!mi

I

d Bfo BAt ,/l.=
= =

_1_ Tr1·[2o"A.Ro",A BR 48,..2

1 -Tr1·[ZouA.l.o .. Aill. 48,.-l ,

iW~~~~iE~~Bt 'it~M~~~.

+

+

o,,(A.RA ..RA/lR)]

)] 0"( A.l. A ..l. A 8l.

(24) (25)

:li:Jaa9~~~,*)(f'F.:j!ftE)([2,3]tP~IB~1f~:hi~.,IiP*)((3):t.t t, 'e1!!*7 JiJT1f ~&-.t1f*~~ml,reE~frMf*&1!r, X ~1f~FMf*&1!t, ffij..§. 'eJiJT~~~II~~nn ~~~m1i f'Fm -&~iEiD~.

1008 512

[ 1] E. Witten, Nucl. Phys., B223 (1983), 422.

[2] K. C. Chou, H. Y. Guo, K. Wu, X. C. Song, Ph"s. Lett., 134B (1984), 67; HIIJ'tB ,$II&~,~ iiJ ,*fi*,i'1itm!/!8l1IP3~!/!8111, 8(1984), 2;2. [ 3] K. C. Chou, H. Y. Guo, K. Wu, X. C. Song, "On Witten's effective Lagrangian of chir~l field" Preprint AS-ITP-83-032 to be published in Commun. in Theor. Phys. (Beijing). [4] D. J. Gross, R. ]ackiw, PIlys. Rev., D6 (1972), 477. [ 5] w. A. Bardeen, Phys. Rev., 184 (1969), 1848. [6] J. Wess, B. Zumino, Phys. Lett., 37B "(1971), 95.

SYMMETRICAL ANOMALY, UNSYMMETRICAL ANOMALY AND EFFECTIVE LAGRANGIAN CHOU KUAN'G-CHAO

Guo

HAK-'l"IKG

Wu

KE

(Institute 01 Theoretical Physic-s, Academia Sinica) SONG XIXG-CHANG

(Pe.lidng Un:t·ersity. Beijil1g, Chil/a)

ABSTRACT

B:,th thE' ';YI11lilt'ir!(';:1 anrJmulr and the lUl!;Ylllmetrical anoIllal~'. are deriyed il'CI!TI :,11 effectiw' Lqn',!llgiun recentl J- constructed on the basis of Chern-Simons topological

invariants.

1009 ~9~ 1985

iF

~2J1Jl

~~!lWl!~~!lWl!

J.I

PHYSICA ENERGlAE FORTIS ET PHYSICA NUCLEARlS

3

~v:.~

)i]Jti?

~/J\I

~

Vol. 9.

NO.2

Mar., 1985

-

Pf

(.:p mf-l~~l!UMBJ1Ili/f~.@T)

• •

mT(aJII
Ji'litt19

Ward m~~,~JtJ7'1Alfl3t'J~li'jjftJl[t19-Alt~iZi'A;;:PH'IJJf:l Chern ~~ Chern-SimoDS

~=jtjn:;f:~:i:t19t1:JlIHiEIjij7 Jf:l-1-~'A;iit1M57 2.. ~M~..ttl95fli1!r~-f!t., nnt13:;f:~a!J

M'·+'

Wess-Zumino =A"~f'FJf.I,W.&~tiEli'M't19-Alt*'A;;:l!-i!7.ff.;7

~

M'·

..tl!~rp]Bz

la:JtI9~~.

UI*,~Tlf~f'Fm:i:A9*~~ttmlA9IUf~51j§TjlZa9~., Witten U]

Wess-Zumino &1jHf~f'Fm!lJA9-J!:!t*~rmM:m;

SkyrmetlJ

§"$t1l\5f.;7 *TmTiiI~~~~ f'F m:fl

:tta9:8l1.TA9~lttm:iT~¥tl7m:m[4]; ~lf~f'FmA9~m:~~tt~:&1mm;&-m-a9~;fI-t:g WJi

Jb.\~~jJH~)£ff7~M!5-'J; ~~4o/J:a.l!tj:l ®7.t~ea.1'iIT ~-~*w..:::r~~~~i!! iiI ~~ ~ 'i*ll!i:lf.PiFtEA9I1*~!llJ·lll.

*X~M*~~A9~~~ffg~~~~~~~LA9lf~f'Fm®1m~~~tt~&~~ &'iitA9~;fI-t:gitl1tfj-i1I:-~A9~M.

~{rJ~~.m SU(N)c. X SU(N)R @~i!llUtl~jij

SU(N) A9lf~fFJfH!!l~, ~m Wei! fPl~[U1A915~~f8¥ 211 W &rd

m: ~5:t, ~ ¥tl1m~th-~& 1itjjlt®-Jt;iH~5:t; ~tl m

+

2 ~1M~.t Abel

&*99

Chern ~!3 Chern-Simons ~ =ii5

j'1-~1et®~~;m~ml~lli 211 ~1I1~.tA9~& 'ffl'~¥F .. :Jmffl:~~i¥.J~~fFm~ &~tl3li 1ltA9-i!t~5:t.

211

~¥-f, ~in-t!!)£-;$'mffiT 211 ~~~.tA9~~fFm:&~F Abel

& 'M'~

+ 2 ~1M~.tth-~&1lt~& Abel &1itZIa:Ja9PiJ:tElI*~. -t U(s) !! SU(N) A95G. ,::(£ SU(N)e. X SU(N)R i¥.J5G~ (ge.,gR) A9f'Fffl"'fA91£:Ifc

~ml:X:J

U

-+

ge.U(s)gi'.

(1)

~Y~¥UtE1mm;~ H~SU(N)e. X SU(N)R "'f~~i¥.Jf'FmiZi?EI, &::'~9i5IAUA9~~lM£ ~J!c

(2)

;;$:)( 1984 ~ 4 Jl 29 I3IJltJtJ.

1010 253

;J:tr:j:l AUR) ?HU~ HLCRl 1¥.J;OOi-B~ 1 ~:tt,:mmzl¥.J~5& 2 ~:tt;{)I) F LCRl

dA LCRl

=

+

(3)

AicR).

(4)

ALCRl - hURled + A LCR»)hL"I(R) ' F L(R) - hL(RlF L(RlhLC~l' DU - hL(:r:)DU(:r:)hiil(:r:), U-'DU - hR(:r:)U-1DUhiil(x). U-IDU II

=

U-l(d

+

(5)

(6)

AL)U - AR=:UA L - A R•

AL 1¥.J1£~~llA.!=j A R ~-~I¥.J.

;E: 2n

+2

~~Im$ ,J\!f9;:~1¥.J

&.1it~

Abel

CHI(A~ -

;jI;r:j:l Cn+I(A) ~Al¥.Jm n

.m 1 ~:tt;&jf\tfHIt;{)

11Sfl"~~,~iE!3)J!f~ Goldstone ~u~;OO~~~Jl.liltI¥.J&.1it~

leU, A L , A R ) , jjlU&.1It1tI¥.J~m15~~ d*l(U, A L, A R) = cn+1(AL) - C+I(AR).

X<.tffll. •

;jtr:j:l*~ Hodqe

+

c n+1(AR)

(7)

.:r-.

~{f11igeif~l¥.Jfii]JMI~:fJGJtl-1'f#J.~a<J~3'&Ii 1Ittm l(U ,AL,A R) tA!~15~(7).

A,

=

AR

F, = dA,

+ IU-1DU, O:e;;; I :e;;; 1, + A: = IIiFL + (1 - I)FR

5\ (8)

- 1(1 - 1)(U- 1DU)2

Weil !i5J~~~~ Chern ~I¥.J ~~J:tTI:&P~~Jtl

;ftJJf.I

C+I(UAL) - C n+1(A R)

=

an+l(n

*J(U, A L, AK)=an+l(n

+

1)

~$ *5 0 ~ff~I¥.J1;f}~l¥.JlE~ 2"

Uf#J.~I¥.J,

+

1 ~J:t.

+ l)d!:dITr(U-'DUF~).

1: dITr(U-1DUF~) + *5

0•

(9)

(10)

m(8)~PJ1~1)IUtl U-IDU ~ F, W~~A~

I2SIJtt *J ~1$i!~:;r:~I¥.J. X<.tr 4 ttfl;j~ e" = tm[lll. !Z5IJIt, (lo)J:t~1:I:I7 2,,+2

Jli~P~ Skyrmion

1), H~lt.*J!l3 *50

=

o.

~~lm$I¥.J1;f}~&'1it71ifI¥.J-~

*Jt. ~~,:fln;!!·1!t

2"

tt~lmJ:.1¥.J Wess-Zwnino ~:l&fFJ'IL:fljm Chern ~~ Chern-Simons

~=t:ti;}'~:;r:~.I¥.J~$*~J:t , C+1(A)

kJ..&.

=

dIT<2ft+'l(A) ,

Chern-Simons ::t:~:i:l¥.JttmLm(7) .. (9)PJ~~J~ *J = IICln+l)(U-1dU) + dW Clnl + I1Cln+1l(A L) - I1(ln+1J(A R) ,

(11) (12)

1011 254

(13)

tm

*,

:)ttfJ QlD+! ~ 2n + 1 ~ BQlftH - M" ;;9 2n ~~ ~ ~t IM"~, ~ n = 2ll-J it~ U1 Witten m~a<J Wess-Zumino Jjli[2J. ~~t"tB, f(U, Ql.+I) i:tH::~Jjli1£ M" -.ta<J1'!Ej. N:::JmmYJ~ag, f5IJlttEft:~~flF rr
fJ:lJa1, EBT *J

r, 1lJ~~)(' M'ft -.tflEj!¥.J:tmm:~~ag f(U, Ql.+L)

;;9TJj(tB

Wess-Zumino Jjlj

f(U, QlHl)

=

+

27ti

(14)

L,.

(15)

W ClD ).

Welft) a9-~~:i2s~,~1r15IA

t U A L + uA R 0 ~ t, u ..:;;;; 1, (16) = dA,. + A~., uF L, = tdUA L + fUAi, FRu = ndAR + nlAt ~~ (u,t) 5jZtlii...t~JJ:,¢'C~t(I,O),(O,I);;9JiiIR.\ag,=:jJj~ L,~llI aL = l. IiJl?UlEf!/:] A,. F,.

=

I,

+

duAR)F~.} + rr ClftH)(A R) + *fmlD'

1) LTr{(dtUA L +

=

IJw+I(n

=

_rrClfl+l)(U A L )

(17)

;!!!. *fmlD;;9 *5 0 ~ 0 a1!¥.J *Jl),~1r1j:f:Jfl?ii~ rrCl.Hl(A) ~-15tlii ,~um Green

= IJ.+1(n + 1)

t

dIlTr(AF:), F.

=

IIdA

+

IIlAl,

(18)

0:it

r

+

1)

JL

-= aft+ld {(n

+

l)n

1, - IJftH(n

Tr

{JL au (UALF~,,) - JL at (ARF~,,)}dudt

LStr(ARUALF~,,-I)dudt}.

(19)

*JaIa = rrC2ftH)(UAL) - rrcz"H)(A R ) + d{ (n

+

l)naHl

1R:I f-f , PI ~ iiE f!/:] rrUft+I)(U A L )

=

rrUftH)(AL,) +d{(n

;)ttfJ F,,,,

=

F(IIAz. W

+ WUdU-l) ,

C2 .) ""'"

(n

+

+

+

t 1:-' dt

duStr(A RUALF~u-l)}.

rr
l)naft H ):dll

r·

dwStr(UdU-1ALF:;;;1)},

J)

2,3,

(21)

0":;;;; II,W ~ 1. T~~1r1~?iitl

l)na.+l

1: dt 1:-' duStr{ARUAL,P-l(tUA L + uA R)

+ UdU-1ALP-I(tA L + uUdU-' )}.

!liZ n =

(20)

(22)

H~tt.~a.F.I f(U ,Qlft+!) :fr7JtJ~tB M4 ~ M6 ...t~m~ ~ Wess-Zumino Jjli

/;§;lE}i!J!.ff~tf:J r -fu£tEiXliP5:,J,g),CH,;) nff~tltf:J"'liIIlEi!t [I,,,.

Wess-Zunlino

Jlli. -~~*, EBT •.9'. :i::j£~~~,

r ~;g~

1012

ft-J!i!i*. [5.51ifjjH Goldstonl ~1¥!J~AA"lJii;fO-I!j'I')i1i:$HjlJ~ ill Bardeen I¥!J~ W4 :qn R3 T ~F ;\-j ~15i.~[5J, J!*1lJ3 f(u, Q') tlJE. M4 L~F Abel li',ljt1¥!J Ward m~j.\;~Ut Wess-Zumino !§i~t!E~lHf. ~ffl¥!J~it"EJlcHtj¥tl M~· l:1¥!J f(U, Q2n+l), ~BU~"EJ 1d. iJE a}l M'· L Bardeen a9~MIili R 3 (M l ") .!:3~Fxt~liit G~(M'O) :$}§jU~

R 3 (M z.)

G~(Mzn)

=

(n

+

+

r

C-' auStr{ARAL(h, + FRY)"},

at

.. 0

+

2tri(n

=

l)na.+l

2tri(n

1: at(l -

l)naO+l

+

l)nan+1

~I

l

t)Str{l"a[ALP-I(tA L)]}

dt rl-'duStr(l"{[a(E"-IAR)

Jo

.. 0

+ A LEn- A R +

+

(n - 1)[E"-'A RA LF(tA0 (n - 1)(1 - tl)a[ALEO-lARAL

P-1ARAd

-

F(tAL)Eo-IARAd

-

P-1ARAi]

+

ALEn-lARALAR

-

ALAREn-lARAL -

+

(23)

.0

+

(n - 1)

+

+

u[d(EO-ZARALA R)

EO-IARALARAL - a(AREn-lARA L) AREn-'ARAi]}),

(24)

E = F(tA L) + F(uA R) + tu • (ALAR + ARAL). (25) W.M , :t!1l:~HR¥f(l4 )~fmti\ljJe, ti~z. f(U, QlO+l) &t:if'~:lmm::if'~~, tE:lmm:~~(,,) "f feu , Ql.+1) a9i&~:fi71Je ~f(u ,Ql.+1)

+

2"j~(n(2w+ll(AL)

»=

- U(ln+1l(A R

0,

(26)

;fIJm~MT(21)a90j.\;~~UEllJ.l ~f(u ,Q1n+1)

=

-T(hL,Ql.+1)

- 2tria {(n

+

l)na.+1

+ T(hR ,Q21J+1)

J: J:-. all

awStt(hi:1dhLALF:;I) -

Wi.Jt-f"EJl?t~lliX~Jhi!j(~:i: Of(U, Q2a+1) 1¥!J-~*iZij.\;.

~f(U ,QS)

(L - R)}. (27)

=

21H, .lI:lftt_*llJ.l Gross-Jackiw a9xt~li-m-!l31,ifjj ~f(U, Q') !iltl"EJ~lli Witten I¥!J SU(2) §,.

:5iti:: *m:lEli1it lll • J!*IlA ,~¥f(H)~ MI• ...t:£li1lt~¥fa9*m:mI~j.\;, r~~~:.lf::lijl!Ultt it~.$( "f(E:JCli1it~¥fa9mj, ffiiH -&.:I'i57~F~tt"fl¥!J:JCli 1jta9""¥f. §:}-15iIii, M Ql.+1 L~~¥f(14)"EJ~ C+1(A L) = C O+1(A R).

~tfttft~ M1n+l

LX

,~cz, M20+l

(28)

Abel ~.li~a9~¥f.

LI¥!J

Abel li~ftl MIn La9~F Abel li~, ::'~a9f6tht~~:.m~, ~~

~~Ja9~tElf1C~. :fUl~~, Abel li1it1d.R~F~tt~~F Abel li'Will\5~ Atiyah-Singer ffi ~~J.lI![121a91*~, 12iI!It, im1itI¥!J~F Abel li1it-t!!~M1i:lf-9Ia]:If 1:3 Atiyoh-Singer ~:l!;j§lf1C ··~~J!~~~~n.~~l?tm.-~Mit.

[ 1] E. Witten, Nucl. Phys., B22}(I983), 422.

r 2]

]. Wess and B. Zumino, Phys. Lett., 37B(1971), 95.

1013 256

[3] T. R R. Skyrme, Proc. Roy. Soc. (London), A260(l961), 127. [ 4] E. Witten, Ntid. Phys., B223(l983), 433. G. W. Adkins, C. R. Nappi and E. Witten, ibid., B228(1983), 552. [5 J K. C. Chou, H. Y. Guo, K. Wu and X. C. Song, Phys. Lett., 134B (1984), 67; Institute of Theore. tical Physics, Academia Sinica preprint As..ITP·83~32; 033; Seony Brook preprint ITP SB-84·18. [ 6] [ 7] [ 8] [ 9] [10] [11]

B. Zumino, Les Houches lectures 1983, LBL-16747 (1983). B. Zumino, Y. S. Wu and A. Zee, Nucl. Phys. (to be published). K. Kawai and S. ·H. H. Tye, Cornell preprint CLNS·84/595 .. Y. P. Kuang, X. Y. Li, K. Wu and Z. Y. Zhao, preprint AS-ITP·84-015. c. H. Chang, H. Y. Guo and K. Wu, preprint AS-ITP-84-0l6. ]. Goldstone and F. Wilczek, Phys. Rev. Lett., 47(1981), 986. A. Zee, Phys. Lett., 135B( 1984), 307.

F. Wilczek and A. Zee, Phys. Rev. Lett., 51(1983), 2250. [12J flJtm111 T. Eguchi, P. B. Gilkey and A. ]. Hanson, Phys. Report's, 66(1980), 213,B:~rf!Jjf5Ix1(. [13] D. J. Gross and R. ]ackiw, Phys. Rev., D6(1972), 477.

EFFECTIVE ACTION AND CHIRAL ANOMALIES IN ANY EVEN DIMENSIONAL SPACE Cnou

KUANG-ClJ.AO,

Guo

LI XIAO-YAUN,

HAN-Y1NG

Wu KE

(Institllte of Theoretical Physics, ActJlie1l'lia Sinica) SONG XING-CHANG

(Peking University)

ABSTRACT

General expression for gauge covariant anomalous current is obtained from the difference of left handed and right handed Abelian anomaly in 2n+2 dimensions by the method of Weil homomorphism. The general form of symmetric and asymmetric anomalies, gauge invaria.tllt Wess-Zummo effective action and anoma.}.y free condition in 2n dimensions are summarized in one closed formula, showing the deep connections among all these topological properties of gauge fields and pseudascalar Goldstone fields in 2n and 2n+e dimensions.

1014 ~ 10 *E ~ 2 lUJ 1986 ~ 3 ~

itli

~

!I'm :II 1:3

~

!I'm :II

Vol. 10. No.2

PHYSICA ENER.GIAE FOR.TIS ET PHYSJCA NUCLEARIS

Mar •• 1986

tfii91l Chern-Simons _~-e{n91Jem JiJ!i

*x;jf

~~l

.

it~t

)(

Chern ~ ~ tj i:. -tit. ~ J: fJ!J ~r.?tjh~1 fJ!J PJt- IUl* , ~t-I :!\E~ W

1-~ Chern-Simons

*.

:ll :}i:~U~ ~'t': 1I1fJ!J -

~»I.m.

#"*~l;Vc~~J!(*1fr, -ti1~fjJ Chern ~~fJ!J

- ..

~I

ftl1I,!lW1:!I!*~JIHr6~r'".?t J1faJtEli~:r::iHlHjlt1iTlttE-J!:!;;*tl3m1~ml ,~tI:I7

fi:$1ftltE!a*.

~BgA. -fll~Jj(\Jf.l.tt1:!l!itJiJTft-JdtE~{j£& 1!t1f'i?:a9~~H~jJt,

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*X~~tI:I.il Chern-Simena ~!£~tE-1--fi!iJ."m~.

:tEm=~. ~ Chern ~~

~'ETtJIE~J::a9~~.?t~~rf.JI'!ftIiJiHf, lI};fIJJfl Chern ~~mtE~~, @

11. m=:, gg~, iiM~_-~~,*ml£, m!J::li!llii. ftIl5Mlt:!:-8:-.Q1'f.)1I1£," 8-.~.

Chern-Simons

Faddeev

'i?::i:1m

Chern-Simons ~~~,;f~ JIHl?t J1faJ1'f.)15~

C(,,) = C,(,,)a,,', 1-= 1,2,

w -= Jc + C 2 == W IP d,,' J!:,m ~-fll Jf.l7 ~~ llt?t ~ 53:;. ~Htt?t. T ~: J _ a'l' JL a,,"

'.'J

Ad,,'.

nUIi.'i1(iitft-Jtl-

~tI:I7j )l.tEm=~~~~

N

(2.1)

(2.2)

(2.3)

(2.4)

-------

1015 178

't'::lE 2m-~~,;f1jm Bianchi

tl~~:

Jw-[W, .A.11]m~111~YE Dz.n(W)

(2.5)

C) •.

£f;ijjtlt (2.6) (2.7)

Dz.n(W) "'" JD~"_l(W ,C).

XIliI D2IJI (W) :lE;I;Iii~:;r;:~tf.J. liP: (2.8)

BD:/m(W) ... O.

~mB~~.~~~.T. ~m*.~~~~~: (2.9)

,i ~rf~lt

1J~~>jfHm~f'Fm~~~[ll:

BA -= -dV lJF -

AV - VA

(2.10)

[F, V], V _g-l/ll.

(2.11)

(2.12)

8D~~ .. - JD~_"_l k - 1, 2, ... , 2m

:2:£lI#~mm~!l;ftl~OO~Hl~OO~-m Chern-SiOloos .a~Hl1t~~. :tm:iE~f]~*!!l~~Tt~t:J~~JVttSfJ~,-~:I:.lI~~~I~ X"(,.,.= 1,2,·· ·N,), 9}-~;}9.g:itijT~~~~I~ ~i(i - 1,2, .. ·Nl ) . ~z..A.f]iiJ~:ji;ll~~;fU~~.~-tl!.~ ~~W~:

(2.13) A""" A,.(X, ~)dX", A - Ai(X, ~)d~'

(2.14)

d- dX"~' d-d~;~

ax"

W ....

a~'

Jc + cl ==

F .... dA

+

A

2

F

+ P+

M

,

+:4>, = dA + dA. +

(2.16)

P =dA. M

(2.1 5)

AA

+

A.A.

1016 179

M~~~~.,F.a&~~X~~.~~~.".a~~~g~

•• ~~~.,M~

~flo.Wi.:r-iJlt~~~imf;~:fI. Yltl~ n

IX

Chern !ti¥fiE~:XI:

C. :IJ.i!{E~ D 2"

=

D2" =-

t)

n! 2,.. •

Tr (F

+ F+

M)·

(2.17)

tlt.:r-• • g (f:J5'~mt~IXlt'~::1f,

jt$ D z.-m.... ~--t-m~~~ dX

(f:J 2" -

m

(2.18) .~, ~=-t-m~;&~ d~ (f:J m-~~, 'E

~1tilUtl!J:\;~~ ~:

.o2._(ID+~)'''+~;'''' ~

± (") (m)

STr (F"-""~ M"-~) "'=0 teo m k ~ (2.18) ~ftA (2.6) ~#l±~ d~ (f:J5'Hl7.t.~(f:JYtItAff1~IUU17~¥tl:

(2.19)

d.o,.,o --= -d.o 2. -101 d.o 2"_1 ••

..,.

-dO~._2'Z

(2.20)

~~jj;f§Jjt~7~ • • IW ~ Chern

mil

:j4J. (2.18) ;ftl (2.21)

O~"_I

dOo.z. - 0 Chern .lttl9l8Ht*~.

fut!t

tl~i

Jl3f:

og"_1 """ Dg.-1,o + og"_201 + ... + 03.z,,-. ftA (2.7)~. #l±~ tl~ (f:J1lft1t,A.ff1ilJ~¥tl:

(2.21)

.0 2••0 =- dO~,,_•.o

.oZ"_I •• =- d.o~,,_ ••o + tl.og,,_2 •• (2.22)

(2,23)

:tttfl 03::'~_~,~ re~m~

(m -1)

*~lj1ltmtl3f'FJf.lag5'~mt7.t.;a:;.

reJiHf~ttA (2.12). #lt~.tt5Hl7.t~;a:;(f:JlIftlt,A ff1iJ]"~¥tJ: 803::.1... ,0 - - tlOf._ .. _l.a 603::''..,-1.1 = ilO3:-... -1,O + tlO~"_"_l" (2.24 )

l 6.oo.i.. _...

=

dOo.Z"-"'-l

m = 1, 2"", 2n

1017 180

1*1i!Z ffl. 1£ilf~A1*I1i]~*. ~~L ilJtre~1f-~~*~i4. ~Jlt~{f1*fi-®~,*f~~.

~m1f -~-rtJIE~.l:I¥.J~5!il:X1*, i.i~:

F - dA + :43 - 0, F ~ 0, M ~ 0

(3.1)

E8 Jlt~~ fiJ~ 1f"FJlJ ~;:t: [F, A], dM - [M, A)

dF -

+

tIF

tlM -

[F,

(3.2)

A] + [M, A]

(3.3)

iJz.-..... """ 0 ~ m

>

n

(3.4)

pq-

(3.5)

A(z, ;) = 0 A(z, ;) - A(')($")

+

~ ;i (A(i)($") -

I;i\ M

==

A(O'($"))

(3.6)

~1

~ tl~i(Ai(~) -

A'($")}

(3.7)

i

ttA~ffi§:fjfl (2.20),

#1£ !;i .:r~~tSj(1~LiHt, AfrlitliJ~ft.Jgj )(l¥.Jm=~M:~

[21IiP: ( AW2a_III.III)(A D, AI, A 3 ,

== ;lt$Af!L1ft~ •

=

.:r.

"',

-tlWz._rn_,.m+!(A', Al, A 3 ,

A"+!) Am+!)

••• ,

(3.8)

;It.ffo~:

"'+1

~ (-lYwza_...... (AO, Al, A\ •• '·Ai-l,A", Ai+!, " ' , Alii+!)

(3.9)

i-o

W3fJ_ ....m

-

A Wl n - m .,.

2:m. s., ~ S T~~La9 ml(!~. Jul

f D 311 -",."., Jus.,. =-

f as",+. D JU

m .... 0, 1, 2, 3

(3.10)

lfl - m •m

~.~~illZ:X1B .. g~ .. =1f:l~, ggUii

1018 18-1

lE.:9:u.A.1Tl~~,

WZo->.3

E3

212 - 2 !t~rmag Wess-Zumino-Witten &1itm~~.

:1J r~ ,:6' .A. (fl1fll : A(x, ~)

0,

=

A(x,~) """ U-I(X, ~)(A(:r)

F -= dA

M

=

+

dA s

+

d)U(x,~)

(3.11)

A Z -= U-I(X, ~)F(:r)U(x, ~), -dB -

AB - BA,

= U-I(X, ~)dU(%, dF = [F, B] B

iiD zn •D -

~),

(3.12)

(3.13)

0

(3.14) (3.15)

-

j."

riO U2.-2.1 ... -

()

Tr(BF--I)

(3.16)

"! 2:rr "

(3.17)

~m1i~ BM"-l - M--3 ~ 0, :U~1,.,A.1T1~:Jm~ lE.l±=fjj1ltts~~&;t [31. m1J~m (2.20)

A. {f]1lJ~JU,

'E~~:

(3.18)

~, Faddeev MJ:~

iPJ

lX.m~{f]lI«-~_-1--T-7*~..t.tsiMJj7t.~im-@;:liJ5i7t.~:X1*ts~?£. ti:X1: F- M- 0 (4.1) dF = [FA], iiF -

Qz,,-,..nt =

Chern ~t\;,*Jll" T~*.t. E81J;f1 (2.22), A.1T]~Jll:

A]

(

i" Tr (F-) m .... 0 n! 2:rr)"

1 o

5!fl;t

[F,

m

~o

(4.2)

1019 182

(4.3)

2: f.FA{i1fiUij.jU7 §3-mliff£*~, '81i~ 7tE-riJre~lm Chern ttflla9iiUI~M:J.O'l:.

WJ!;llIr E-r~~~IMjl£~,1f:~1J: A(.r, ;) == U-1(.r, ;)(Aes) + d)U(.r, ;),

(4.4)

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(4.5)

h(ei , h(O,

lieS) lieS»

-

t'(1-Ci)MI(~)3 "ieS) -

-= lieS), h(I,

t'M,(~)

,.ez»

(4.6)

== 1

e

pi=(e', a, ,··ei -" e i , ei +1, ••• ) =(0,0, "',0,1,0, ... ) i=l, 2, .•. Po .... (0, 0, ... , 0, 0, 0, •.• ) m.ll!trf.ll(l~J:~?}.

A {i1ltRJ~jj Paddccv

(4.7)

m[41I¥!JJ:~i.'iJ, pp:

(Aw~._ ....._,) (A. gu la, •. ·k".)

(4.8)

.... -dw:. ___, ... (A, g" g2 • • , .g".)

;)t IP ~:i:J: l1JIUJ 7-, ;tU~f'F ~: (Aw:._ ..... _,)(A, g" g2> .,. g.. ) == _,(A'" 12, •• 'g",) - w~._ ..... _,(A, tala, + (-l) .. _,(A, g" tn .. . g.. -,)

w:._.....

w:._.....

w~.. _.. _, ... A

-

(

Jus",

'50 ...,..)

D~._"_l ... ,

(4.9)

(4.10)

w:._....._,... ( D:._...",_, hEOS",

;)tIP s.. :l:EiI m +

1 ~ R.i.m.ll!trf.lml(l~.

A{i1~li II -

(4.11)

2( 2 W2'.-2.1 ~ Juar.-

it;:J!:t£3Cl!J.1trf.l2 n - 2 ~~Ia.J~ Wess-Zumino-Witten ff~fF:m ••

;film

stoke's ~~. A{i1$~~3Ji!

..... aw:.-301(A,glga)+J .. J~EM.• _.w:._a.l(Aw','a)-(J.. u..---

•• _. EM

w~"_2 •.(A,g,)-O(4.12)

1020 183

li ...

J

5{ Chern-Simons *~ (J-"~

.m~~*~*R~~~~~~~.~~~~~ M .... dA + tlA + AA + AA - 0 tlF -

[F, A], dF -

[F,

(5.1)

A]

(5.2)

iF - [F, A ], (iF .... [F, A]

2:MA-(flPJ ~~J-~~~tt. Chem ~:it~: n

"l.-~.l.

-

j-

nl ( 2.)-

STr (p. ·--P--)

(5.3)

IJ 2. " - tlIJ~._I.a dIJf.-I.D == -tlIJf_-2oI dIJ~._2.1 - -tlIJf"-3.2 dIJ:.._3.2 - -tlIJ:._M .:

0

tlIJ2._2t.2~_1

-

+

IJ2.-2.2

•

-tlIJ2.-~_Io2~

(5.4)

+ IJ2.-2~~

dIJf.-2t-102~ .... -tl.o:'-'~-'.2t+1 -

0

r

.

tl!2,.-zt-2.2t+1 - -tl.o2"-2t-,.~+I

.

-

+

.o2.-2~-2.2o\:+2

tl.o M . -1 A-trJi'±gJtJ2:mil.*~ !3l111ttf:J Chern-Simons iUIUt. .lI.J!M-~~JlM1.JIl- lJi. JX -JjitE tl ~ (i \¥.Jt'Fm""f.~ •• ~bk./jW (2.20) f~~j}:ti"J~-ia. 1;§~~1I1 (5.5) .o'.-2~-lo2t+1 - 0 k - 0, 1, " ' , n - 1 .0 0,28

-

*=f A :IHl1!l:tiiJ.

~1~,A-(f]PJIU1iEJm .of.-Z., (A, A) "2.-41,1

~JlttEl!~.~la.In~lli

Ww-Zumino

liP

Tr(An(A»

nCA.~) _

(5.6)

&')!t~:it

A~PJ~M~~.~\¥.J-~~.~T. ~ A(z, ~) = B(~) + U- 1 (x, e)dU(x, ~), A(z, ~) "'" U-1(z, ~)(A(s) + tl)U(z,~) ~m B(~) :l!.R~#1t S ~*tf:J:t£T~~ s ...t\¥.J Abd ~;l3~. U(z, 5[; ,E~ 1.. Afi1!3:~ liE 00 :

M"" tlA

+ dA +

AA

+

+

A2 ,..., U-I(X,

~) ~JiJT*_Ml'f.Jtlf

AA'" 0,

F = dA + Al = dB, F == tlA

(5.7)

(5.8) ~)F(s)U(z,

e)

1021 ~ 10

184

1&

iSTr (dB)"'p-m 111 (2:rr)i(dB)- Trp-m($) (" - m)!m!(2:rr)" (5.9)

Dlll-lm.zm-l -

517l1:JT~ Chern

i- ( ) B(~)(dB) .. - l TrP($)--" (11 - m)!m! 2:rr •

(5.10)

mit.

-' 0

0

tlD zs - ZIB+1olm-Z dl1z,,-_ol..- l

-tlDI,,-lmolm-l

(5.12)

-tlDf._lIB_1_

-

dDg ol8 - 1 -= 0

5w*AffJ~j~3&f'Fm.a9,;E)(:

r(A, A) - 2:rr IUN'--a J~d1 ~.-Zo1(A, A) .... f(A, U)

+ rCA,

r(A, u) .... 2.. fZEN',,-1 f'ES1

BtH:::ff 2" - 2 ~3G~.im:~ MZ,,-l ..t~

8),

(5.13)

Df.-lol

(5.14)

Wess-Zumino-Witten ff~f'Fm ••

r(A, 8) = 2:rr Jzu,z,,_l

!

1 D;,,-lol

fES

(5.15)

(5.16) ~1\iln

= 311>.t,AffJff:

r'

r Tr pI == 8 64,1 _1_ J r rJ4:rs""ptt P",.. P" .

(8, A) -

-~ 8 8,'?

JzEM'

(5.17)

Ptt.

:l!.!I!:(~Jf.lT P

=! P~.T·d:r"l\d:r·,

TrT"T b

--!

6"b.

~-J.iJi~lliT!:3!m~

Euclidean ~ra]llilJilat 8-.~~~~l1li. ~-J.iJi5m*t£~:Iml:1f~U(, lJ~1,. (5.16);t~

1022

183

ililt:J () mmzE3~?H~H~7C*!£~~-m~-~Ik. (fji!]j:it:m::I:.:r- {t lY:J.

~~~;j( B (~) :Wt::f:m:~:OO~:Wt::f;f§li!Z

.

Loo~~&M~7-@~~M~. *T~~~~~li!Zm~~~-~lY:JM~ .

( 1] B. Zumino Cargese Lectures 1983. { 2] Guo. R Y. Hoo. B. Y.• Wang. S. K. and Wu"Ke (1984) (3] K. Fujikawa Phys. Rev. D21 2848 (1980). (4 J L Faddeev Phys. Lett. 14m 81(1984).

Pteprint

AS-ITP-84-039

and

AS-ITP-84-044

THE GENERAL CHERN·SIMONS COCHAIN AND THEIR APPLICATION CHOU KUAN-OJUO

Wu

YUE-LIANG

XEI

YAN-BO

(Institute of Theoreticttl Physics. At:llliellli4 Sinica)

Some general Chern-Simons cochains are easily obtained by expanding the Chern form a.eeording tG the degree of the form in its submanifold and using the closed property o.f the Chern form. The recurrent relatuns of Chern form between submanifolds are discussed under some constraints. We also consider their application.

1023 ~ 10 !Ii ~ 3 :lUI 1986 ~ 5

A

~ ~ ~ 1,1 ~ ~ ~ 1,1

vol. 10,

PHYSICA ENERGlAE FOR.TIS ET PHYSICA. NUCLEARIS

No.3

May, 1986

-fIPi'T89J:.U!tl_+~~~JJ=J ffiJJt:g

~il

f~1i

(tj:lli!I~4~~l!I! tt~l!I!lVf~3T)

• •

*~.ilil~.~.~~~~~~~.~.~.N~~

r, ~JMlt;f!f.~M~fjJJf~~MJ~-8-.l!frT~*O.

...

#~~.~

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SO(3) ;iJi1.~$.~1:f~;t$;l~:I::r1t*1f.

:iillIA in E. ~refii:l'H~n!l&~7t5tmzm $U~tti~~ $[1.2.31.

f91J~M.,*~~~.:lmt13~

~~$tl(f.)-1SJt ..tPfJ~jn\]=1SJt ..t~HjutliJW.~1±I ~Dt·-m-;m.f.f~-IbT(f.)&1It11·l1.

:l:T:t.J ~ $,

:!lS ~rtE

~ {[lift ~H~ -1'-~t.V! Je.~:fiiR::ii:T~t~ ¥F(f.)i!U(!·a:f4f~j[ 7BlPIr!l&~ 1P13jZ~

~~(f.)~I~d$~tr-131. m~~~1i1~T~Fj!iiJJ)l~m.:&JlI:fl"~N.tt-J"~(f.)iS,

)C.[3JPJT* m(f.)1i~&tfl"~ii7 ,Jt~l25Igt~~Fj!iiJJ)l~~H::~W,it~~4.!ilt1iJ~~(f.)~. i!;f-f)C. [3]m1Jn~Jg.~)((f.)..t3iI!i.Tgt~~m~~Jg.*~ii~)(.

~1i1:f4ftE~=~$~I±I~~~(f.)..t3iI!i.T,~~tE~=~$M~E(f.)mzm.

ti-1'-~Tm~7t.7BlPEi!l&c/J(~) ~IHfi~. c/J(:r:) ~~1'-~G(f.)~~~~;I;~.

~

~~IP1fftE-1'-~m;~

A(:r:) = A~(:r:)tlx"1·,

1. H::~(f.)~.!iltJG,

(1)

A(:r:) ~If G (f.)i$I\jfi~~. ~{fl~ittE:Al/NflI-€iMm. r ,.7BlPIrlttE~rPJ

3j2.~~~ rrt(Wi{r~~~t

(2) U(a)

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Wz(x

+

b, r

+

a

+

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+

b, x

+

a

+

(7) b)

+ a + b, x).

(8)

Wz ~El31-mMl!MJ:17l.~~T 8 f.i¥tl~ 2-~M. ~ (5) ~ftA (8)~, ~!{$~~33i! Wz ~~.=:jti~ (x, x

+ a + b, x +

b) #~ x fF~il§..a~~~1j-.

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iP4(~

(9)

•

+ a)Wii'C~),

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+ II + b + c)Wz(x,x + II,X + a + b + x + II + b, x + a + b + c)c/I(x + a + b + c)

_ Wl(x, x

c)Wz(x

+

a,

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(10)

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r,

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+

a, %+ a

c, x)

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== W3(X +

a

+

b

(11)

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1

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1025 285

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c)

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at § IS] ~ iW$.

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r

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!fag~IiX5t;.

[ 1] K. C. Chou, H. Y. Guo, K. \Vu, X. C. Song Beijing preprint AS-ITP-8.J.018 B. Zumino Seattle preprint LBL 16746 Ucn-PnI-83/IC R. Stora Ahnecy preprint LAAP-Th-94(1983). [ 2] L. D. Fuddeev Phys. Lett~ B.145(19841. 81. [ 3] R. Jackiw MIT preprint CPT 1209(1984) Hou Bo-}u, Hou Bo-}uan r-;orth\\-estern Unh-ersity. China preprint NWU 84-8. Y. s. Wu A. Zee University of Washington 40084-29 P4 (1984). [4] G. t'Hooft Nucl. Phys~ B79(1974), 276. [5] ~f8!¥, .iC;jI;, S;lI;f!I:, $!!iI~l1Il, ZS(l9i6), 514. [ 6]

!i!~it<:t, jr5.it.j", tl::*~, fIl&~, Jj~~l1Il~~~~, 1(19i7), 53.

A NEW CO-BOUNDARY OPERATOR AND ITS APPLICATIONS CHOU KUANG-CHAO

(Institute

0/

Wu YUE-LL~XG

XIE lAX-BO

Theoretical Physics, Academia Sinic.:z)

ABSTR,~CT

In this note new co-boundary operators are defined in the product form. The associative composition law of spatial translation group field is discussed using these new operators. The quantization condition of monQPoles in BU(2) and BU(3) gauge theories follows easily from the new formalism.

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Part IV ~1ftt~Jll:§:)~~~~!I

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1031 ~ 29 ~ .~ 5

1980 ~ 5

WI

Vol. 29,

A

No.5

May, 1980

ACTA PHYSICA SINICA

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= --'---

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=

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+

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]

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==

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1043 630

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~ ("!"...;Y
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(4.33)

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1044 6.H

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1045 632

29 ~

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~u,m.

tE~~~~X1.~_m~~~~IT.-f-~. ~-t-~~M.~~m~T~m(.~T)~

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m, -1. . ~ "liiT~.

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ill.,

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•

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(A.I)

1046

633

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aFR DR + aFR DR aFR DR)

aT

aq.

aq,

aT

.""

= 0

(A.2)

•

p.3)

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21.

T"

~

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0

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(A.6)

I.....

~:ti:~=(A.3) ~:c~.

tEi1M:i';7
+

Ttl rT:O.l -

(A.5)

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f1t (.\.4), ( ..U) {i1 (A.h) ;\;lZ:UpU)'miiiE (A.3)

c =

(A.4)

(0, -1 ).

.Qlt~mefl, ·Zm(.'·16)~~?J~ .t'Jl

f = +rtl [T:k] - f.

;~if1.ri ill:j,"itll ,bA.,g2!l;'jrt:Jt1'tMH~,tEl'!!.fIl~:5f+:tEj.\Ijl'iI1lmIil:WJ: iiE(t~Jt.(~.Z2)~I:;;;:{ll:f:E T. G't~.1:!ilZ!l:,ifii!l.f:E!l!:fu!A~ T,

(B. 1)

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tP i¥1il;ltl'lil'l'm1:JM:ilj

fi "(--'--r ) A Aq.

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:&iH\.

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taiJi:if.i, :I¥tami T ~#c~!J'1t!. T(f.J~Ift~I&m=ltii[.

fAlJl(i.ta;t:86

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:j!f~,tE;I1;~m1Lf\:ffJ!Ifliliil~lltilt{l;(r.·Jr, a(k, T)

fO a*(k,n -jE,~;ij'ffii1i3'<:J{,~i:XjffJljHJl!t£ rMT;k] It:.! 00, ~~FHE!B-1'~'l?:mi{1'Ett~'I'iiJ~ Feynman a(z) a*Cz')_ (T ,d(z) a+(z'»ccnn. ~~iiEM ,i.!iW;§-€;i3l*ff.J:o:iit

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m:

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+ t.A'H'

[T; k]

+

+.

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(B.2)

kJ + 1;l'1'<:.

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~

wr 11 'f' iiJ ~

Fcynman IIIl1\lfm~JlI;1i:a:;1lffm-€;.

( 1]

J. Schwinger, J. Math. P1&ys., 2(1961), 407. n. KenbAbllU, )f(3Tct>, 47 (1964), 1515; JU/l D. Dubois, In "Lectures in Theoretieal Physics," Vol. 90, ed. Brittin, Gordon and Breach, N. Y. (1967); D. Langreth, In "Linear and Nonltinear Transport in Solids". eds J. Devreese and

1047 634

29 ~

V. V'llI Doren, Plenum, N. Y. (1976). Ko~enman. Ann. PI,ys. (N. Y.). 39(1966). 72. [ :1 J :\1. Sm'goot III. :)[. Scully. and W. Lamb, Luscr Physie.s, Addboll·Weslcy. Reading, Mass. (1974); :\1. Lax. Pllgs. Ber., 145(1966), 110; In "Physics of Quantum Electronics". cu •. P. Kelle~' et al.; MrGrnw Hill. X. Y. (1966); H. Hakcn, LaSH" Theory, Va!. XXV/::!e of Elleyclo. lll'um of Physics. Springer, Berlin (19iO). f' 4] )1. S'L1'gcnt III ai" ·ibid., Chnpte. II. Adclison·Wesley. Rradillg. :I!as~. (1974). [5] mJ:l'tB,m:~/Jjc,~itf!4Jl1li1f:.~,~Hi:G:,il'lljtrj:.j(~fJilm,f4!!J!:!l:\JIli:iUP""HIlJl&: ~hiI:l'ttL ;5~i*, ~liri:~JlIl~ laf!4ll1! ,3(1979) ,304: 3(1979) ,314. l 6 J RiJ:l'tB,i1Hii/Jjc,iGlifj'6~llI!E,jtA~llI!,3(I979),314. [i] RiJ:l'tB,;5£/Jjc,~tt!It.JllI!;l£Ii,~lif,'f, § 5.2.4; $I!~**~fJ;,*li,f4!!J!:tllJ\[i:i:t!!P:m!llJl&. [ S J FnLt,§ 5.2.5. l 9] fIltm M. Sargent III, et al., Laser Physics. §20-2, Audison·Wesley. R()3cling. Mass. (1974). [10] NiI:l'tB~,!l!:;'(mU 7 ],§ 5.2. [11] 1i'il..!:.,§ii.1.7. 1.1::!] ~..!:.,§ 5.:!.3fjH 'j.::!.6. r13] 1ll.I..!:.,§ 5.3.2. [14] !;.;j"!:',§ 5.2.2. [15] IfJtIIl H. Hakcn. ibid .. ~rv. i. I Hi] JI L:,x. III "P1'~'si", of (Ju,",tU'" Elccl,·orr;(·s". Ells. P. Kl'II('~·. cl ,,1., MI'Grnw·HiJI, N. Y. (1!J66) .

l 2] Y.

,.t

ON THE GOLDSTONE MODE IN THE STATIONARY STATE OF A NON-EQUILIBRIUM DISSIPATIVE SYSTEM

(11,~tillll,

uJ' Ihl'Orc/iclIl Ph!J.i,·~ • .1 ..ali l 1llia S';/lical

.AnSTR.\CT

In consl:'quencc (If the spontaneous symmetry breaking, non-zero energy Goldstone modes with dissipation art> (>xcited in II non-equilibrium stationary state with spacetime structure. In this paper. as a specific example, the Ward-Takahashi identities formulated in the close time path Green's function method is applied to the saturation state of a single modr h!ser. A generalized Goldstone theorem in a weak inhomogeneous d~ssipatiye system is rstablished and tIle physical interpretation of the Goldstone mode is discussed. As 11 rpsuIt of the Goldstone theorem, the PQle in the Green's fUllction of the laser light splits into two with equal weights, each corresponding' to it quanta with the same frequpncy but diffrrcnt dissipation. Together with the order parameter (the aYel'age value of the vector potential), these two kinds or quanta (one of which is the Goldstone mode) give a complete description of the order-disorder transition of the phas . . symmetry in the saturation state of the laser. A detailed discussion on the restoration of the spontaneously broken symmtry of the phase is given.

1048

3IJ 29 -m 3IJ 7 AlI 1980

q:.

7

Vol. 29,

J'I

ACTA PHYSICA SINICA

~Jt~

~

t

No.7

July, 1980

~#!*

(1P!i!I.f!j.o!j!:~JlI~~l!I!!i}fRPJT )

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~l.!i!m~lIIattt. ;

.~ml~U&~!I?4~*~atmlli,il!!rifrJ;jtI*lI*pjgltat~!S$ G(I,

2,··· ,n), :atl21. r~~ ;lt$ i, i, ... = I, 2. G fIl G~!S$m~ 2·1-5t~. tEff~~ {nZIa]at~~*~L HU , J'[;nR)E;4i:)( ~mat-~i2 ~. i'§~J~l5$i2Jg

)(::f:H}:~ffiJg Gij...(1, 2, ... , n),

(1.1)

0= 1 -

Ji!tl~)E

G1:3 G Zra]at~~. $.¢.til'l!iti?1Siltat)E5(Jg

iuz

-/2

= ._1_ (1 -1)

-/2

1

1·

(1.2)

1049 7

AA

879

B(

.,. • •

1, -,

=

)

, n

't?:£tt~ mR.i. it I!1t i?f,i tta9 ~~: B(1, 2,···, n)

{I

(~"l > Iz > 0 (;!it173t.~).

... >

t.),

== 80,2)8(2,3)·· ·8(n-l,

(1.3)

0.4 )

n),

JI3~:Q::)(lf;jff.-f-T

T(c/J(l)c/J(2)·· ·c/J(n») =

2: 8(Pl, PH···, P.)c/J(PI)c/J(Pl)·· ·rjJ(P.),

(15)

p.

J1t~X1 n

p. ;jt~ (*)(R~~~it~ c/J ~~J).

-ttta9-t)Jji:/!R:

8 ~t!<:iJiif,@;EfT-JN:*

~,tw!J3-~~:

2:

8(Pl, P2, •.• , P.)

=

( 1.6)

I,

p.

~

8(l,2,···,m)=

(1.7)

8(PI,p!,···,P.),

P,.(1.:!.·· ".1,.,

p.(1, 2, ... , m):£ 1 :e 2

ID:£f.E#a9).

n= m

+

8(123)

=

mr,···,

1 :£Em Wi"a9 n .~]i~ (l!!f;j 'I, tl'···'

m -

1m

S9*,j\

la9m~~:£:e~1fa9m-tIJ;f~lj7'~¥HI-1-,tm

8(4123)

+ 8(1423) + 17(1243) + 8(1234).

(1.8)

8 ~ tttf.H~~·ttrilJIaJi* , tw 8(12)8(34)

=

(1.9 ) 1#,,(1

~T!.

3;itT'"

~~~iA1f;jti~IJ;f~glJ;f ,9l'ar*~,@]g~ .~*~:

8(12)8(134)

== 8(1-234) + 8(1324) + 8(1H2).

(LlO)

nR.i.m~~**i?Eitt Gp (1, 2,···, n) ... (-i)·-ITr{T,(rjJ(1)c/J(2)·· ·c/J(n»p}

== (-i)·-I(T,(1,

2, ..• , n»

t/C;!it~~:&:?}.TIE.~fftXa9~f!lt1l~~*,~{~jU

(2.1)

G a9~-t5G~.

Kddysh[5J ~AmT

MmR.i.~tt G jtJ G a9~~*~ G(12)

==

(2.2)

QG(12)Q-l

.-2

G1,1...... (1, 2, ···,n)

2 Qt;•• Qi .......Qi .... G...........(l, == 2-.

2.···,n).

(2.3)

~{m~t:R:~.jtJ, (2.3) J;t$ ~~j;l!Hgm7 -t)J~ R.i.ft;g~mmr~**mtt~*lI*mtta9:Q::

"-2

)(. (2.3) J;t$I!C_f51-f- 22- a9j£iI~:er-ll~~.{fJ!iiUl. j!mR~mili,

173%1

1050 880

29

(

G1) Gz

1 (G+ - G_) G+ + G_

="2

( 0) G G+=G_=G

=

~ • • ®m4m~~~Am~~®~~~.

!ffi

(2.4) •

~®~X~w (2.5)

L/(X)8 p (x - y)dx = I(y) '
I(x - y)

1!!£l 8(x - y) == 8 C41 (X -

e,

=

8(x - y)0"3' H(x - y)

y)

Atfiia9 8-~Jt.

=

8(x - Y)O"\o

(2.6)

iEliJ~~X=:~I;ajls:HtM-~~ fJ p ,

iJ ~

~$ fJ(l2)

_ fJ(l2)

=

fJ(l2) , D) ( 1, fJ( 2!) ,

= •

OfJO- 1 = - -

( D, 0(12),

-fJ(2!)) . 1

(2.7)

:taHI~:~1fm4~!tt®~¥-J,~~~~tI:\ ~

G

=

G ~ G p;;j~~I5$®*~,.!iltl1f 1 (GC+1 - GC-l, G. r + G .• 0) =2 Gr. + Ga., GC+1 + GC-l '

A

Ovo- I

-

-

(2.8)

;U;$sIA 7i2% G++ -

G+_

= ~ ClG+-,

G•. = G_+ -

G __

=

Gr.

=

.

G. r = G++ - G_+

.

G .•

~ ClG_a ,

GC+J = G++ GC-J = G+_

+

G __ ....

+

G_+

..!.. ~ 2

2

== Gr. =- Ga. ==

G.

== G. =

G+_ -

(1

.a

= ..!. ~

Gr

r

=

G __

= ~

.

=

ClGa+,

.

~ ClGa-,

+ crP)G.."

(I - crP)G..,.

(2.9)

a/J

-iO(l2)([l, 2]),

(2.IDa)

G .• =- -ifJ(21)([2, 1]),

(2.10b)

~"X1~1t*~~I!("[6J

G< = GC+1 =- GC-l ... -i({l, 2}).

(2.10) ~$[,]~{, };HjU~~X1~~&X1·SrT.

G=

(2.10c)

rA

(0. Go\. G" G)

(2.11)

1051 881

G imM G a9':::1-~F"j(;~*~1il(JJ

1),

G="!'GJ1 2 \1

l')+ ..!..G.e -1)+1..G.(1 11 2 ,] -1 2 -1-1

(2.12)

~m~m:ie~~IHBUfa9.

~~~.8a~r~~~~B. ftm~M':::~~D,

:II''i' 1m l~ ~ tf{ a9 ~ fi;J , n: ~

•• ~ffi~~~~a9~ •.

~

(2.13)

W~U

G~Wa9 161-j(;~1il G uu

= 1.. (G<+l

-

GC-'), Gzm

2

=1.. (GC+l +

Guu

= -1

G ZZII

= 1.. (G rr .. +

G"" .. )

~ 61-,

1.. (G rrr • +

G.... )

~ 41",

2

(Gr ...

+

2

Gm ! =

GC-,),

2

2

G•... ) ~

41-,

(2.14)

.ill

G ....

=

L: ap(G++a~ + G--a~)

~

61-,

all

(2.15) .il

L: a(G+++a + G+_-.. + G_+-a + G_-+

.

G rrr •

=

G....

= ~

.

a( G---a + G-++a

+

G+-+a

+

~

41-,

G++-a ) ~

4 1-•

a)

(~~mIT;6'~13)l~~ ffia9r~?H.lIJ~':::R..~~ GC+l, GC-l -~.)

:tm*

G p (1234) ~1f1J¥:~IU~a9.i¥:IS]m:,:tm (2.1) ~,!i!IJ~IJm (1.3)-(1.10) i;i~;fO!(Ej(

(2.15) ~"iiH~

== G< = G<+l = G<-l, G zm = Gr ... = G.... ~ 4 :p:;, Gzzu = Grr .. = G.... ~ 6 :p:;, Gzzn .... G..... = G.... ~ 4 :p:;,

G uu

=

0,

Gzm

(2.16)

1052 29 !(1j:

882

L

G211l = i

~,

8(lpzP3;':.)([[[1. Pz], p,l". p,J)

.p,

GUll

=

L

i

{8(PIP~P3P.)([ [{PI' Pz}, p,],p.l>

(;, p',) (t, p')

+ 8( PIP3PZP.)( [ { [PI' p,], PZ}, p.]) + 8(PIP3P.Pz)({[[PI, P3], p.]. Pz})} Gml

=

L

i

~,

{8(PIP2P3 4 )([{{PI> PZ}, P3}, 4])

P,

+ + G, =

8(PIP z4p3)({[{PI, Pz}, 4], P3}) 8(P I4pzP3)( {{ [PI' 4], Pz}, P3}

L

I

>}

~,

(2.17)

8(PIP2P3P.)({{{PI, Pz}, P3}, P.}).

p.

~I!:!;J;\;-f-tfJ'E!*7=.il*i;~x-.t ~T [,] fO&X1~T {,} l¥J-tml3:~ ,X1~IZ31'lJ;ftitl~

tJJCJ fml¥J ita;ru;flHfEjlSt-IF;TJ (" - F.a tEiW", "WL.az - :tEiW", fat! /lY •.a 1t~5fO S iW i?IJ !#U\s:tE~ t:jJ T. kJ....t~J;\;m~}g'1'ffie~**~~tfE~~.

"=..aZ-

:tEiJiJ"~~ ) ,12SI J!t ~

X;jTffi~~**~~f.F:tE~jC~ ~ ffi{ ag 1C

~. .R~i:l:~M~J(

G (1 p-

' ..• n)

==

i

=

lJ" IV [J] I1 ' lJJ(1 )lJJ(2)' • ·lJJ(,,) ./=0

,-"

? ••• G <, ( 1 , -, .n )

IV[J] =

(...tft.lW- c ~ffi"a~",

Z[J]

G~(I)

G;(l2) G;(l23)

5fO

I

lJ" Z [ J 1 I lJJ(1)lJJ(2)" ·lJJ(n) 1/=0'

loZ[J]

(2.18)

W[J] ~ffiEI¥J~J?jGiZ~l3.1J)~3§troff~~:

= G,(1), == G,(l2) + iG,(1)G,(2), == G,(lZ3) + i[G p (l)G/23) + G,(2)G r(l3) + G,(3)G,(12)] - 2G,(l)G,(2)G,(3), (2.19)

~~~.ili: ~F.a~~t:jJ.R~X1~~*~~~~~ G~(I2)= -i({I,Z}) - {(1),(2)}); G~1l(I23)

== G2I\(IZ3) = -

~ 8(IPzP3)([[I,Pz],P3])

(2.Z0)

(2.z1)

P,

li~~{t,ifij

G6(123)

=-

~ {8(PIP31)([{pz, P3}, 1]) - ([(PZ)P3' 1]) P,

- «(pz(p,), 1]) + 8(pzlp,)«{[Pz, 11, p,}) - ([pz, IJ)(p,»},

1053 7

J1ll

883

G~(l23)

.L: B(PIPzP3)«{{PI' PZ}, P3}) -

=-

({PI, P2D(P3)

1',

+ 2{{ (PI), (PZ) }.

(2.22)

(P3! }),

~~; E9.o?.i?Bj~CJ"i!k:tl!!~ /'iJ G~m

=

Gnll

+ Gkll

=

+

2i[G(l)G~II(~H)

2i[G,(13)G,(:?4)

G lCl

+

+

+

G(2)G!II(I34)]

G,(l-l)G,(23)],

+ G(2)G zz1 (l3-l) + G(3)Gm (I24)] Gi13)G,(H) + G<(23)G,(l-l)]

2i[G(l)G m (13-1)

+

2i[G<(l2)G,O-l)

+

-

8[G(l)G(2)G,(34)

+

GO )GO )G.( 1-1 ) ],

+

G(l)GO)G/2-I) (2.23)

~~. tl:~,~~ffiX1~-f-iIX~ffijJ$Gssfi:i8~~ G~II.··1

{t~t~ If;j 51 Ass r - 05i ~, :If' ~ ~ ~

=

G~II ...I' ;f§§-=f LSZ

.Era i¥J ~n!A.J , iTiJPJT 1f u.:e: 1& 35. ,.~ Wi;fa

*

rsl

.fig~~lm

lI1C !£iJi,Z ~~ 11ti£

~ Jfj;f§ ii 1~¥t- P& ~~5(jjij -1f JiJf;;r: ~ •

f'F7'i1=-~i£il~!G[m~~SS~·WtJ,~(f1~-tttl;r.j~J:~ .o?.~;ttza~~W~Ji!iJC:)(~ [-I] rjJ Btlt~u-@)!~J;il.Ji!IJ). 1t~)!@~Ji!IJ, CJUilij£:!:t!!.h\ffi G p ;&~ffJ1Clf-, ~ Dyson

~,~jtlE13 G 9X;

1J

G~~a'g*~.

L~ • • ~$~*~£~~.~~~~~SS~~

0.1 ) ;?trjJ

J = (1+,

J_),

~ = ( ~:) ~~. UJ§1li'~~?tf.f-,%, X>j:m:![tlHJ\!ss§l~m:§lWJ>.It

~?t • • Gp~~~m-=f~5~~*,X1G~Gss~?tCJ:If'~~~.~WJI.~. ~If;j ~§I~.~1li'~~~. ~~,~~*~ R.(l) = G p (l2)J.(2)

mlJ:fr t\jiE ~ D'l5tllPm¥A U3 ~IS$, ~(I)

:ax;

R.

=

GrJ p ,

~!j.

= G(I2)u3!(2)

R(l) =

~ ~

G(l2)uli(2) ~

:tm* J £5'~:ijlj,;/jl~E'Z~;tttEiE,tQ5tJ:m~,

= GU 31 ,

R = GuJ.

(3.2) ~zt~

G) = (G~)' tEX17'~:ijlj ~ IlfP.J $I g:r EI WJ 1M lJi!.1fE:i& i£i!Mt:.

2. m.¢.i.i?jg~~$~*lfD p(l2) = Ap(13)Bp(32)

0.2)

1054 29 ;'iff

884

= (

(0 D.) D, D.

0

A,.B, A,B.

A.B.) + AcB4 •

(3.3)

~~.~.~.r~~~~A~B®.~,~

z, =

A~l) A~~)·

•• A~d)

(3.4) n

Z.

=

2:

A~l)··· A~~-l) A~~) A~~+l) ••• A~·tl.

(3.5)

t=1

~N.{f4!!,~jmi2!i:~:2

= Q-1ZQ ~ Zp

=

.

(2.12)~, tiJ~~

~ A(l)· •• A(~-I) .'Ct)AC~+l) ••• AC.t) .L.J I" r .'1~ Q: CI J

(3.6)

t=\

:!tt:jJ~=+-~-+.

"?Tr1r~,ra~~111 ,miAui ~W~ij:!t~.IIIDi~. ~j5!ll=tliMJ9i1b~B f,(l23)

=

if,(14)f,(25)f,(36)G,(456)

(3.7)

t( 123) = i(ta3)( 14 )(ta3)(25)(ta3)(36)G( 456),

fm

=

0, fzu

=- if,f.f.G zu ,

f m = i(f.r,f.G LZl

+ f,fcf.GZll + f,f,f.Gm)

~~.

R~£~~t:jJ~~ili~®~£~~®*~,~~~~~~~ili~~~®B~~T. ~j5!llg!liIiIi!JYl1b~11c r~4) l3~JJi!~;f>t:~11c w~·) ®*~[71

(3.8)

f(4)

=

-W(4)

R1r-~Aj;J:J:&"1&~"ili*®*l\:tm

+

3W(3)U\G(2)U\W(3).

1055

71m

883

~~I±ll!{mI2SlT 1/2.

ago 3.

*

:i2:ml!'j G jfQ G ~-~~~~J;\;

.. -1

(2.3) $, f5i-f- 2-' ~Ji!;ff:*x;j

8-iiijtf{l¥!J$IIJt*~ G p(13)Tp(32)

=

Tp(13)G p(32)

=

O'p(l2),

~ G,~ T, RX:Jgf-m~..ta9l2!~I!$. ~~Ja~

Gu3f GUtf'

= =

fU3G = 8, fulG = 8.

"i'iJ ill U3GU3 ~ f, G 5fO Utf'UI ~~, ?t~~jj~5( ragl2!~iI$. G++ + G __ = G ..._ + G_... !ll(; G n =

(3.9) R~ Gp *"f;!,j~~iI$", &P 0 (3.10a)

(E13~==rH~l~2:~ Gp *~JgJl;j"~~iRSJZf.i;Joo:agJa:~:) ,i!-~JJllgt6:i (3.9) t\;f{t~ fp: r ++ + T __ T..._ + T -of- !ll(; Tn = Q. (3.10b) 0=

[2;IJIt f

"i'iJ~~RX:~1t{ (2.11) ~a9~-r

_= (0 fo) ,

T

(3.11)

Tr T.

2:JI;j"#~~* Tp "i'iJI/.J.~RX:II;j~~~agSJZf.i:Jm.

ill

Gp (12)

= G p (21) .. ~~~I!$ p jfQ~:W-:l:fJ.JcPagm:*'ft_t,~&P$i'11!11-,% Tr ra9M G,lf1liJE

£j\~~f!':E, "i'iJ~iiEl!'j

(3.12a)

G(k) = GT ( -k) = -O"i G*( -k)ul = -uIG+Ck)ul, G(k) = GT ( -k) = -O",G*( -k)O", = -0"3G+(k)U3. l!m..t;f,jTj!~J1 .. *::!Ul;t;W .. +~m:*;t;W.

(3.12b)

lS:~ttJ9J:~$lmi:t (3.9) ~ffl~l2!~1!$

f ;fa f , ffii:if'~>1<:9initi r, a9~-f*~~. ;film (3.7), (3.8) 1:f~"i'iJ~ iiEOO, ~ .~~**iiijltl¥!J 1~!sIHt &l: , Wi] ~ ffl~ ~ ~a9 ~ .#. IDi1f.l iiij It. ~llmffiJml2!~~iE"i'iJ~~ I±l ~WT~ffi~i!~~?T~J;\;(~:t: "'ffl"Jt"[2;I-f-)

f [dcPp]exp (- ~ cPpTpcPp + JpcP p) =

f [dcP...dcP-]exp (- ~ cPO"i0"3cP + l u3¢)

= ...

00exp(-l..JGJ) . z p p p •

(3.13)

4. #~*~:tm A,(12)B p(l2)C p(l2) ~f'FJg-1'12-'%~1JD~~. ~~~*~:S::A ~ffi~~I!$.J;\;,lm."i'iJill~~OO.I±l*. ~:tm

S,(l2)

==

G~(l2)

(3.14)

1056 886

29 ~

iiJRlt~jj}:j.Bii!i~~*~W7t(~1iUIH~t), bi~:A1fI~U~tt~. .~J::., EE (3.10a) ~~ G,(12)G.(12) = 0

i'iJl.IE

+

G!.A12)

G~_(12)

=

+

G!._(12)

G~+(12),

G.[G!+3G~] G<[ G!

+

3( G.

+

)

(3.15)

G,)Z] •

~=l5"tPfl~~mltMil1,m~~~m~,:A1f-@~iS~~,j![] G(T) G+

=

+ G __ = + G__+ =

G++ G+++

+

G+ __

+

G_+_

G_

(If!.¢.ij~~),

G+_

+ G_+ (M.¢.ij~~), + G_++ + G+_+ + G++_

G ___

=

GH,

l'lP

(=-.~i?Ei~),

. ..•. ... ....

~

•.

~~*~~

(4.1)

•• a.~•• ~

G1 = 0, G n = 0, G\ll = 0, .R£~.!E~# (4.1) t\;~ (4.2) ~~~ .~mf{, M'ffiiBr£~~fflj~.~,

~j1}Lr!?-i®nPJTtfrXEj«(f.J1tj.

-,%j![] (2.9), (2.15)

(4.1) ~(f.J~-J§*, .~XEj(

(4.2)

:i!£:>CifiX [2] tP

G ~/l$$JC~lltsIAa~i2

if~,~.!E3!I('f~.Jt:

G,.

=

= G... , G, ... = G" .. = G.... ,

G•. , G, ..

= G.... , G,,,. = G.... , • • . • =l5"tP~ttK~~• • • 7~@*~. G".

( 4.3)

G.... ,

(4.4) (4.5)

~~~n.B~#ii!i~Mlli-~l.IEOO.

~jj~#~f{~RlZiZ:~~!J3-~!J3~~#[7)

Z[l-r, J-1IJT=J_=J = 1, W[l-r,J-1IJ+=J_=J

~i'iJEE~n~XEX~~lli*.

1:, J2 = -1- (h + 2

J-)

=

(4.6)

0,

(4.7)

~.#~~*J=0,~£~~7m*~~~••.

£~lIj'~~, J1 = 1.. 2

(l-r - J-)

•

:;fm~~~~~RlZiZI1.i§1f;j51 A

1¥.J~~j'~it,~~ (2.4) ;ftJ (3.2) ~.

~ w[J+, W[l-r,

1-1. ~ I,), =

J-l =

Jo + 8J,)"

:tE J"o mllt.li7f.

(8W 8l-r _ 8W 8J-) 8h 8J8zW

+ 8 J - - - 8J- 81-8J_

+ 1.. (8l-r 2!

2

~Jm~=-1jtP~jWj{ti-a~,.

z

8 W 8J+ 8l-r8J+

8W 81+ - - 81_ 8J+81-

~

Z

8 W) 8J- - - 81+ 81-81+

+"',

1057

7,M

887

:A:tj:I~~7.1-~~:l:'J1I1ittE

y:lhl3

h =

1-

=

10

ilI:-~~ 81+

5d:.

=

81-

=

81. ~~

h =

1-"

81 zff~~1~¥tl-ffi:*~: 8W

8W

8J(.,,+) = 8J(x_)

I1+-'-=1'

I

2

_-----=8:....:.lr~v_

2

8J(x+)8J(y+)8J(z+)

8J(x+)8J(y_)81(z-)

8 W 82~V 8 W + = + , 8J(x+)8J(y+) 8J(x_)8J(y_) 8J(x+)8J(y_) 81(x_)8J(y+) 1+=1_=1 _ _-=8.-:.::~H:' + 8 W + 8~W 3

8J(x_)8J(y+)8J(z-)

(4.8) ~~.

1± (4.8) ~tj:I~ J =

P1SiatJi*~ W[Jl

ftA

=

t.

n\

0, &Pf~

(4.1):A:.

L·· -L G~(l,

:tuJ*:;r:.~ J

=

0,

rralCL

W [1]

f'F;g f81!1 tIT iZ

2,· .. , n)1(I)J(2)·· ·J(n)dld2···dn

(4.9)

(4.8) ~at~-~,i:l::it

8J(Xi) ( ) - - = 8 p -"i-Y± 81(y±)

fI:l

Gj, atM.fiJ\tt,fiifJ(~~~7.1-f.f-5)

i: ( -=2

n -

1 ) [ G~ ( 1 !

+,

1, 2, .. ·71

-

G~ ( - , 1, 2 , .•• n -

1) -

1)]

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[1]

[2] [3] [4]

J. Schwinger, J. Math. Phys., Z (1961), 407. mI:l'tt:/, ;Jj:~!Ijc, f.ij2!H&*"iifl:fIJ-atE~F.ijZ~!1Cit!lt4lmtPll9am. 1Di). p.a:l'tt:/,;Jj:a!ljc.-";ftB~l1l'!;;~~l1l. 3 (1979). 314.

c~it~l1li1l:1i»~IiJ:l:. f4!!f::flJl!H:f:(I~:fl

[ 5] [6]

D. Langreth, In eli near and Nonlinear Transport in Solids», ed. by J. Devreese and V. Van. Doren, Plenum, N. Y. (1976). 11. Kenllblw, )J(3TCfJ, 47 (1964), 1515. R. Kubo, Rei's. PrOIlT. Phys •• 29 (1966), 255.

[7]

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I8]

H. Lehmann, K. Symanzik, W. Zimmerlnann, N14oIJO Cime1Jto. 6 (1957). 319.

TRANSFORMATION PROPERTIES OF THREE SETS OF CLOSED TIME PATH GREEN'S FUNCTIONS Z:aou

GUANG-ZHAO

Yu Lu

HAO BAr-LIN

(Institute of TIIBorBtical Phytri.cB, Academia Sinico)

ABST.IlACT

In this article, we derived the transformations among three sets of closed time path Green's functions and some calculation rules, from which a general definition of arbitrary multipoint retarded and advanced Green's functions follows naturally. Some algebraic identities among multipoint functions have shown to be the consequences of the property W [J+.J-l I J+~J_~J = 0 for the generating functional on the closed time path.

1059 ~ 29 ~

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ml7f ,ifii83M~l"t~~l:i1: (r-'(t»;".,y = o;;(U)",y + ".,

(4.12)

~* Qi ifii83~~jcil!iiW:tJ;fO~Iff('~WiOi!:~. ~f$CJkJ.

(r-'(t»;",.ay

= ifL7;Q;(x,

I)OC3)(X

-y)

f •• ~.~~~~~,tExtQ;~~~Iffi'R.~••.

+

83 xt ~ ~ m ~ ili~7f

Qi ~~Iffi'!.ili

(4.13)

1i1T~~f~.,~.R~~T

*~f&~&. ~at(3.10)t\;tj:I~ ret) ~P$kJ.&~$?t1i1;t, 1?;~~;€P$1:B~lCJ.&~$?t1i1

;to

}j~(3.13 )~~7ft\;~tr ~!jliJ55Z~&~~ ,T~xttt( 4.11 );fIJ( 4.13)j;\;,~ili

f = 1.

(4.14)

!l!t~( 4.13 )j;\;~-:qIi;;$:*~R~;e*,1f~mcff:tEat( 4.14 )j;\;1JJ~RjG:lz:. ~~tE-ruH'R~ ~Ji7ft\;( 4.1Z);fIJ( 4.13 ) liECJ.1fjt1i!!!.ili ,~cp-E1Fc~~lEiW1:3llJ~lEiW~~atf'Fffl.

;;$:

)(1f~M~1?;{n. :a:k~lli{~~ ,J¥~:I:~1f~mR.IG1i1

J.) _ iL~.., -0,.

aQj = -rr (IJF _ at OQi'

(4.9);fIJ (4.15)

(IJF - Ja). IJqa

(4.15)

j;\;~.~(f1mw.~IIiii~~:tJ$~;;$:1f~m. tEl~Hl~~**~.1f~tj:I,

1065

8M

967

~@:Ii.titpag J *13 J+

=

J-, JJ~~iE~~l!I!7'~:I;b,;1t$fullJ~ 'El*~&Jl9cjtt?;*it A

~ ~:l:ag i3 EB l:tffiim 1J[] ~ \liB m7'~:I;b .

I2SI tt , ~ {f] ¥.J;~ -:Ii ~ ~ll ~ j )(. ~ I~}tz EJj~.

~~m.~m~M4,£~j)('M~7JJj.titpi3M • • ~T~.~~,~~~A~ (2.4) ~~~J:t. ~f¥:~Hi:;fO<:j=lliii~;t!l~~tp I±IIJ\!.~~*~~~,:a:<:j=lli:l.:~ra:JI¥J.~

~tpl±l!J\!.~~~~#;}'m"~.

~X;j Q; &. qG ~~~i!I@{r, u{~~-~~~ifiX7!J~. :~.f;~

:W!~, llJ~~~$A ?t~gg~. f¥:~m:~lbl~-#!: A;;

.~. f¥:~it-'=3<:j=f][:I:~rJ3]~-#!:

A;G

=

iL7iQi,

m:i:13f¥:~:S:lit ~lli:i:~ra:J~iW *=?t §J1j;9 Aa;

0, &P~~~t?;(f]~IS]~~F*fIf{~

=

-'=3'iTt][]i:5E*,PJrItJ. -a!L qa

=

-iL7iQi

lit

AG/I

=

A;G

=

O.

'iT

-ifG/I.,q.,. EB=.F~

~~JS$lit~~'m"!J&~&ffJ\tt,.~~;9~. ruI*~.l!tX;j Q9,I(; q ~~F~tt*~,~IJm~F~tt w~lli.~llJltJ.*~mmfi~~. ~-~~n~~~~nM~. f'FJ9Af*"JT,~~~~~ttl¥J&~S~W ,&pmw C ISI • ~m.'El*-'tEB=-~tE~ID:: ~~a-'J~F<:j=f][IF~:I: Q, t?;f\;~~tn~~w~. iE~ -'t =-tffi~:l:fi;]nX:a!J<:j='tgIiW~ q, f\;~~#ca!J ,~~~.m.t@ ~ ~*~ [qG, q/l] [Q;, qG]

= =

igoEGhq."

( 4.16)

igoE;a;Qi,

/Gdr = igoE.dr>

(4.1i)

L7i=-ig oEiGi,

.~.ffi

E«d.,

.,~~&X~· r,jar-&5~L1:.

19i!fJ,&P~ F -

F -

,ftA ( 4.15) Et( 4.9);::\.,

j:j:;;47'~~:rwIgtL&~fJ§j

J;Q; - lG'/G, *~~ ao; -=. at

=

£.t& = at

IJF IJQ;

-u-

IJF + goE all··0·- IJqG ' I

IG#Yz _IJF _ goEGil IJF Qi IJq/i IJQ; - goEG/I., -IJF q.,. IJq/i

lfllit

C1

r=

(4.l8 )

To. lall = ).oIJa/l> #~~~:B:%~,&P~~i!U8 ]$a!JJj~m

-aQ = at

-8q = at

A, B,

ai'iH;;:ftj

2 IJF loV - IJq

IJF To IJQ

+

goQ

IJF IJq

+ goQ x

--,

x -IJF +

IJF goq X - . IJq

IJQ

(4.19)

c ~~tp7i~~F~fk~t~U/!~, m%j!Iii!fJ., jt* E, F, H, J :m~[81lit sss ~~1.141

.!'i:IllJn:jG~~~~Jj~~~1±I ,j~JI:!~¥P&'f~.

(1] J. gehwinger, J. Math. PhY8•• 2(1961),407; 11. KenllblIll, )I(3T(/J, 47(1964),1515. [ 2 J D. Dubois. In " Lectures in Th~oretieal Physics", vol. IX C, ed. by W. E. Brittin, Gordon and Breach. N. Y., (1967); V. KOTcnman . .dnn. PI/ys. (N. Y.), 39(1966), 7~; D. Lnngrcth, In "Linear a.nd Nonlinear Transport in Solids", ed. by J. Devreese a.nd V. Va.n Doren. Plenum.

1066 29 ~

968

N. Y., (19,6); A.. M. Tremblay. B. Patton, P. C. 1Inrtin, P. Malduquc. Ph1/s. Rer., A19(lD79), 17:!l.

[3]

IJ.'IJ't?:L;5:~17!:,~ttfuJlll~m,~li~,fWtl1F,.H±l!P~tll~.

[4]

JijJJ'tfl,;ij:~i1.i<, r;1j~~!It4Jl1l~~~JlIl,3(H17fl),3H.

[5]

}l;1J'tB,i1J.~,~;'):,/.',\~fullll~u~Pl!,3(l!ji!)),304.

[ 6] )7JJ'ttL;lj:~iii:j: ,!f.1;jiJ~ta ,29(1980) ,(il~. [ 7 1 F. C. Marlin. E. D. Siggia, H. A. Rosc. Ph.lIs. Ra., A8 (1973), 4:!3. [S) P. C. Hohl'l,hcrg, B. I. Halperin, Rell . .llod. Pity., .. ,49(197i). 435.

[ 9) [10) [11]

K. Kawasaki. In "Critical PhCllOlllCI\rl", Inter. Sell. of Physics" Enrico Fcrmi' '. LI. <,d. !J~' M. S. G rccn, Acad. Prcss. (19i1). DJ~Ili!J: B. I. Halperin, P. C. Hohenberg, S. K. Ma, Phys. Rev .. B10(1974), 139; S. K. Ma. G. F. Mazenko. Phys. Rev., Bll (1975). 4077. H. K. Janssen. Z. Ph 1/Sil.;, B23(1976). 377; n. Bausch, H. K. Janssen. H. Wegner, Z. Physil:. B24(1976), 113; C. Dc Dominicis, L. Pcliti, Pilys. HeL· .. B18(1978), 353.

[1~1

IJ.'IJ'tB ,il$tEti:, T'i,!':, ;j:f!] ~l~, 969.

[13]

[HJ

R. Grnhalll. H. Hakcn, Z. Ph!l8i~·. 243 (I!ln). :!S!l. L. SaS\'ari, F. Schwab!. P. Sz('pf:tlllsy. PlI!lSictr, A81(19i5). lOS.

[15]

JiJ:J'ti3 .T;;jl:"T;;f8.j;i:,!fJdl!ll~tli, 29 (1980),

~;8.

NON EQUILIBRIUM STATISTICAL FIELD THEORY AND CRITICAL DYNAMICS (I) GEi\"'ERA.LIZED LAXGEYlN EQ"C.A.TION lIAo B.\l-LIX Yr: Lu

ZHOU GUAXG-ZIIAO RT..~ ZIIAO-nIX

(lllstitll.le of Thcol"cti.c(!l PlI!lSic.•• .:1cac/emia Sillica)

ABSTRACT

Stnrtiuj! from the equations satisfied b~- thr Yertt'x functions on the closed time path, we deriYed the generalized Lange\"ill equations for the order parameters and the conserved yaria.bles. The proper form of the equations for the conserved varia.bles, including automatically the mode coupling terms, was determined from the Ward-Takahashi identities and the linear response theory. All existing d~'naInic models "Were recovered by assuming the corresp-onding sYlllllletr~· properties of the system. The whole theoretical framework is also applicable for describing the systems near steady states fur from equilibrium.

1067 ~ 29 ~

~ 8

AA

!j:bJ

1980 ~ 8 fj

m

~

4-11

Vol. 29,

ACTA PHYSICA SI:-
No.

~;

Auq., 1980

,I, 71

tJ;1

J~J]G p

(
:IF 9 }j

1i E ifi§tl

*~M~~~~@ft~.~~~~.~~E~ili2,~~~~~~~~~~.~A2.~ ~«.~rnB. ~ffi.~mR**

••• =.~~,.~~~~~~n~~~~~~.

%~

1~ ill ~.!{ lZi:JJiI. 1'4" 1ll! ti?: Il~
~M-m~.w~,Mffig~~~.~~Sili~, ~M~lli~~OO~~~~~~W ~,m~rnT~m~-~D~~~~~~~~rXM~linE. ~k$~m~n~~~~ ~~B~0*~.~~W~~$~~~~~.ili.

ti:1;;Li ~L;fl:!~~ FF~:§l:;fQ~·tlUi:.t'.:JiPJ,j,@r XWtLli~~ aQi(l)

at

l!rn.

;i (t)

= K.(. -0) + ~.-, (I) ,

(1.1 )

~jg!bA.r.IJWT7T:ffi~~JH)l.~. r~WTct~'5Jillict1f;{$iZffirr.'i~Cll. (1.1) ~tiJlCJ.~

nJG~j\1ijWrct~ gi (t) fi'ij~~ctW Qi (t) ~n:k:5Jt ,;tEit€t~:fR?1~f'F~F~'~JZr~,&PtiJ;J~1~rJi J ~ct~ Qi (t) ~1!;t*,iZ~(JI. 9!H~i¥-Jf'F~£;fljm~~J{~ T 8 ~!tZa~ 93-1k{!f C' )[dQ]B

(~-

K(Q) - g)c.(Q)

=

1.

(1.2)

EET(1.2)~~ B iill!itt~ ~~ii;r:;:~ Q ,ffij~(1.1)~,&:,~9i~I.iZi£t~iE
:i!~~mriffi:fJHIjlY:J~F~ft~¥k: gi ~ Qi S~$tiJttfi7IJ~. itjIj1lrt'{ltSl-F, -e:~T[JJ (1.3)

:i!EI! dr =

dxtll :¥:ggg!E~7Tji:;.

J[dQ]

fltfE(1.2):A;$i¥-J 8 ~:t..zillictii§~~:5t.~,~nJG

[~~]exp(Jdr [ird (~; -

K(Q) -

s) -

~[J*:tE~7T(1.4)J:\:-F1iAI!I-F e:
~ :~]) = 1.

t(x)Q(x) ]),

(1.4)

~~~~.f!jI*~

1068 29 ~

970

SjZ.l;ijma9~fiJGi:Z~(~*itcf:ra9!M=fiEi:Z~~~~,gX:72:~)

Z~[j, I] =

r[dQ]

[{~]expGdx [if) (~- K(Q)

_.l.

8K 2 8Q

Wfi.':,

Z~[O, 0]

=

1.

-

g)

iJO - ilrJ J).

(u)

-

; ~.bJl.rJ:d9T7t1\j

W[g]OCexp ( - ~ gu-tg ),

(1.6)

;!tcf:r 0'-1 ;li!;;*~~JI$ 0' a9~. tE(15)t\:cf:riOniGx-f; a9r~WTSjZi1:])§~ Z[j, /] =

r[dQ] [~~] expGdx [- ! f)uf) + if) (~f- - K(Q») 8K_ i ]O_iTA]) 2 8Q 1t.

_.l.

(Li)

~m:;li!;;tLS;;~t\:a9~~§Jcit~it~niGiZ~. Martin, Siggia ;fO Rose ~fJJit~~~~!~(;'1i

;ffr; ~rSR ~tt)1I"119::tr*ml!~~t\:[5J. Jg71ii{tf.tttt.lt¥i§~f.:Jf9!T~.!E.ft,

{fuii1§IA T

-ffi:.!:3JJR1f~:I::;r:x-f £a9"P!PJ~~" ,gt;li!;;2:m.a9 f). l:3:i:T~ita9fR%~ffi{ ,tEe~~?J ""f, 'f::ffJ;li!;;AJx-f£a9:B:. slA f) ~~~TJtI§1.J[l-m-, tEl;fj~~~~Jtcf:r1fIl"1Ii3]a9iExfIJ ~~,~TJtI§~~M • • n.tE~""fa9~~cf:r~~,tEffig~~~~a9~~ • • ~~~ I§! ~ ;t·tiH~ ~Utll ~ ~ t\:a9 i\[SR ±hit, Ii:iJ 11"1 fi ill , ~ T a9:;r: ~ x-f ~ ·~:;r:;li!;;A Jg:it!!. J:\:.A!! sIA

,*

a~ ,'f::PJO~&P.9!: 7#citi5~p,rr;a!J*Jjji.

(1.i)t\:cf:rx-f f) Z[J]

==

ageg~~?J-X;li!;;~JI)T~Ut'~,~?J-)§~~

)[dQ] expOdx [-

-

! (~-

K(Q)

-T)u-

1

...

ii1i~tJ9~*~:Z:;li!;; CPi(';)' i == 1, 2,···, e, :M~1¥~I:~;j=fKI:a9:i:-€;~T;li!;; f);(cp(,;», i==I,2,···,n+m, 1f®~*~:m:*~~~niG1¥~it. Jg1ii~~m,11!ire: 'f::{fl~;li!;;_ts~T. ""fffi.T:;r:*fjjlj~ill,;It*5(Jy..ITxcf:riiJfiill. IttfJJ~JJ;j~J5eJru;f§fIz:lli(~,@lt~~tE

1

== 101l"1X;Jjij

» = P(ep'(%), 10)8(ep'(%, ' 0) -

(ep'(%, '0) !p!ep"(%, ' 0 ~~. Qi(:r:) tJ91JJ~?t1\jlitrm:~

P(Qi(X), ' 0)

== tr{8(Q;(%) =

1

ep"(%, 0». '

(2.1)

- !M'P(x»)p}

[dp(x) ]8(Qj(%) - Qj(cp(:r:))P(ep(x), 10).

(2.2)

1069

sM

971

(2.3)

~

JQ

== 2.:::

.. P

a(rp+ -

rp_)

==

i

r d;.TJi(X) Qi( rpC:,), = to) - ep'(:c»8(rp(x, t_ = to)

)dq/(X)8(rp(X, t+

- rp'(x»

T r ~~i~U~It1Ff.:f.lr.7tJJfT.

(2.4)

Jp

P(rp'(x) , 10).

X

L

(2.5)

ml;jjfJ:m7t.

~j:t(2.3);fi~*~I~l~L 81ili~{lj3-~T 1

=) [dQ]8(Q+- Q_)8(Q(:,) -

~~~7tl:XFf.:J§tiJ~~(2.3)i\;r:j:1at Q(rp(x»

~Jt~ Q(:,).

Q(cpCx»),

(1.6)

:p:J:fIJJ:fi03:t (2.7)

(2.8)

.,

C"clf

In] W

==

)[dl] -2,. c iJOI-iJVU] .

:iXffix-t~~:fR7tf'F7~~~¥UIJ&~~.

EBrxt

(2.9)

I (x) ~l£rrii~:fR7t.tiJ~ W[I]

;fjJi!(;tEM;gL~H5J$~!3 EB~~p.x:iZi!Bl. ~*ffl WKB Jj~tt1J:~7t(2.9)~, ffE~U*W100

lli il~L ~ffi § rxt IlifHIL7'~#O >.It f.lJ Wi SjLfi;] , 1~ ~U

*

~jtat 1f ~ f'F m:l:

Seff [

:iXmfl~at~*~~~~~~Tat~%. ~**~.~£*#O

Q 1.

•. ~-$e~£*

#O~,tE.%:i:-~. ~f-filJla.jjll8 ~~5IA-1-iTrB;f;5Ji1i:. i'l~jU,E§.~mMl7t::ffi'tiJ~ Ji!(;~j;5J.7t::ffi'at*~ P(ep', to)

=

. II P,(ep:, to), i=J

"""-

.!II..

I.mtEfltt~~f'Fm:i:~-~M:&t. 1!l~5Z:~itm:;F!m;*~:atf5~-T ,1~~~~iZ~:A1f"'F~J~&t[7-9]: W[l+(X), I-(x)] I,+(z)='_(z]

W*[1+(x), I-(x)]

=

= 0,

-WeI-ex), hex)],

(2.10) (2.11)

1070 29 ~

972

~!l!ag J±(X)

fO""fooSg

!Il4:e.M-7Zffi~~,f1:}-0-

Q±(x) ?HJljtt*iEfft:xag7'~:thfO~~.i:.

~ CUO) t\;:l;j 1+,1-

1-, "iiJlc.1JfE1~-~:7Jj**:r.t. t-lHI§(2.11) &,(2.9)t\;, "iiJ*~ S:ii[ Q+(x), Q_(x)] = -Sefi[ Q_(x), Q+(x) 1. (2.12) Q+ = Q- Ilt, Seff ~§1i!Eli!~. ~ Q±(x) = Qo(X) + b.Q±(X), :r.J. (2.l:!) :;\;:tE Qo F>ftiti:fF' 7Z ffi JPllf , j(l~:tE Q a :J: ~~ PJ't· rz ffi ~!!tt rSl Sg * as )'~ 0.13) I ...

=

*

=

8Q+(x)

= S±ii(r, y) =

Sm(;r:, y)

(8S 8Q_(:r:)

Sm(Y, r)

-Stji(Y, .r:),

=

S'fii(Y' r)

;

=

(2.11-)

-S:ii(Y, r).

(:;.15) ;tt~~rtE.

~*~~AW~~WGag~~~,M

111,

L

r£'

CPi(X) -

cpf(x)

Q.(cp) -

Qf(x)

= Uij(g)IPj(X), = Vij(g)Qi(X)

btm~:5t:ffi"~~,Jj!UH W[P(x)] = rv[J(x)], IKe) =I,(x)vtCg),

Seff[Qg(:e)] =Seff[Q(X)], QfCf:) = Fij(g)Qj(cp).

:t!D*i±C2.9)JtcPlf.{

WKB ~f~~~L ~n.rt-t[~pr fIJ,

0= oW 81 ' Setf[Q]

:l!H'.t

(2.16)

Seft ~1fI;a:j~TIiij§~WZ7Z~ag-tJ}t!JW[HI,Ei3":f 1+

ScH[QO, Qo]

I

8Soff 8Q+ 0+=0-=0.

SF

eu i)

= -nQJ.

=

=

=

1-,

1!iJ.jlj

Q+

0,

I

8S.ff 8Q_ 0+=0_=0.'

+ SF =

S+

+ S_,

8' 5=Off_..,..(-:-) = jl-1(T/QiI(r\)'" Qi/(xl)))\P.r. ___ 8QiI(r\)· . ·8QiI XI

m:tm

-is±

= iT ±

=

Q-,

CUt!)

(2.19)

0·20) (2.21)

&TIii~ffI~agttml:[7.8J

if ±(k)

-is±(k)

>

0

> o.

(2.2:!)

~!!. T(·n ~£f'F7~a:;~~lfiagTIii~i~lUl. tE~-¥~~mil![7·81

T+ii(k)

= L i ;CVe-/1k.,

S-ii(k) - S+ii(k) -

... -00

-(Jk~-ii(k) •

(2.23)

1071 8

Wl

973

n-~*~~.~fflm~M~A~

••• ~,~~M~~~~~~~Wffi~,~M~ -=

S.d Q ] ,ll!, OJ ~ flH~ t:: ~ iiW ,@a~'~ FvH±J ~ !'tE ~ ~ l~ ~~'L 1i: th~~ Ii H- Z~ . ;1~ =jj" :if iiE fV.l ,1£ [d!] a~ liE ~jt!R)t rlt=llf.~ ~ rm ill: (l::( , 5\17~ ~.ru: ft fit n~ il=" Pi. 7.: ~ j5~ i~ i[.z i:J ~R , I1HmJ~13~ SJZ J$])j~ FJ

ff,.t~~jU!.ll!~f:i'L[£;~~a~ :-"ISR :i;tj~~.

$I:; tt ~1:~?t (2.9) ~,~:;Hm tt.t I2SI T 1£ i'~1U~\ Pf1 ill: ~ 7f A

==

r QI -

W

JP

=

-f

_l.. ~ AnV(~;AI + .... 2 •p

(3.1)

~m~gm~~n~~~m,~~~.~~~ti~~~)t

=

A

IvA.C'_1 =

_l.. ~ Al(]"J~f;(2)CT!AI,

-f

(3.2)

2 •

(TV ++ W+-) ,

AI

TV _+ TV __

=

+),a, = ( 01 _ !l1 \.l'

(AI t:.1_

it ~ji&j WT ~ 71 , /-tEjU mn:em ,31( I~ t;'iscfllQI

=

t;'-irwll

det(CT,I·f;C!)a;) I-f.

(3.3)

(3.-1)

t,:!\ =

(ff ++ fr +-) __ _+

~=~rnjij0Si~j.E:;$.

fr:!:I1.!DtmR[9] $~~l:IJa~~~:

detf(2)(r, y)

= detT(2)(r, y) = dc:tf,,(r, y)detf.(.t", y)

= ~l! Q(x)

-'=3 A(x)

2

[det ( 8f \8Q(X)8A(Y)

1¥J1E~!x!'(3.12)t~.

-:~I~=u =~

)]2 ,

(3.5)

mff.fZ:&H 1] rp:Y-Jtti~

(:;+ +:;J~=~

.!:3(l.l)Et(1.2):tUt$X,"EJ!.2J.~ffi,{(nU.~~f;2!4

=-y 2

8f

y,

8Q8A

~;- :~.

h[t~ffi~~j*,!€~$, j:tIT37'L;'\;

g[;~:J1E~t:tIT71j~. j;t.jlj(3.5)~cjJ~s!z-)jiE~.!:3(3.4)~$i¥-.J 1 / 2 ;r~iM,>jtl!iJ

is

If

•

[0] -

=

-inO] -

_l.. r ~K 2

J 8Q

(M)

dr

•

1072 29 ~

974

(3.8)

(3.9) Q(:c, t+

:tiO*Ji~

h = 1- = I,

=

to)

= Q(X,

t_ = to).

(3.10)

x'.t~m±iT lex) ag·R71;:pIf;{ft-ti13l!I(~,flH~:X:~ [1] ~a~~-;:;iiE

:fO(2.1i)~

(3.11 ) ~)~t~d~1-fJlliim:tJlfJj .:x.eJl:LE1.f~(&P TDGL }j~).

m~~.~~~n~_m~IfJ- •. ~ffi.M#rnn~~~,.~~ • .:x.~~.~, iliictf±b:i::fElEtQlt-trS];ct~~. ft~~~7t(2.8)~r:jJt'F~~Jt~, §IA Q(:.:)

= ~

A(x)

=

+

(Q+(x) Q+(x) -

[dQ+(x) ][dQ_(x)]

=

Q_(.-c) ,

Q_(x) ,

[dQ(x) ][dA(x)

1.

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NONEQUILIBRIUM STATISTICAL FIELD THEORY AND CRITICAL DYNAMICS (II) LAGRAl~GIAN

FIELD THEORY FOK\IULATION

ZHOU GUANG-ZHAO

HAO BAI-LIN

Yu Lu

(Institllte of Theoretical Phys'ics, ..dcademio; S'inica)

ABSTRACT

The expression of the effective action for the order parameters is derh'ed from .the continuous integral representation for the generating functional on the closed time path in the one loop approximation for the Fourier transforms. The Lagrangian formulation af the critical dynamics is recovered in the second order ap·proximation of fluctuations in the closed time path continuous integral. The various possibilities of improving the existing theory of critical dynamics are considered.

1076 ~ 30 ~

~ 2 JOj

1981 ~ 2

Vol. 30,

A

No.2

Feb., 1981

ACTA PHYSICA SINICA

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== po(B,t)

(2.13)

(2.15)

(2.16)

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1079 2

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tE

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(2.19 )

l:tllt, film 7 ~#taglltrSJ!iiPl:~;J¥l';rn R !39&i.iE·[j:. n$!'!iag

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(2.25)

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(2.27)

1080 30 ~

168

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29 (1980).

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[l~J

.[,'3:l'tZ3" ,;ij;~i~i,j(,

~~JJ1l!"f:~, Wli:~&.

TIME REVERSAL SYMMETRY AND NON-EQUILIBRIUM ST ATISTICAL STATIONARY STATES (I) 7,uou

GUANG-ZHAO

Su

ZHAO-BI~

(Institutc of theoretical [J/lysic8, .1cademia Sillica)

..lnsTRAcT

This is the> fil'st part of om' work on time reyersnl ~~'mmetry applied to non-equilibrium statistical station.lry states fl"llll 11 unified microscopic quantum statistical point of view. In this paper, a formalism for time reversal symmetry is constructed in the framework of the Closed Time Path Green '5 Functions (CTPGF), which can be applied both to equilibrium and non-equilibrium stationary states. By using the generating functional technique of the CTPi.}F. symmet.ry relations for the statistical Green's functions and vertex l'ullctions are derh-ed for systems inyariant under time reversal.

1084 ~ 30!ffi: 1981

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Vol. 30,

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ACTA PHYSICA SINICA

1980 ~

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No.3

March, 1981

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(.,-. 1)

= 0; Q, 1] = 0 Q, 1], 1f = __ ~

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(3.41) ~~

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[ 2 J Zhou Gu:mg·zhao, Su Zhao·bin, Yu Lu, Hao Bai·lin. ASITP· i9003 ; [3]

[4] [5]

~ 7i; tL 0: 1H1<, jjij; ii3 f~, T ;ii{, ~J1l!~I!i!, 29(1980), 96l. mJ'tB,iifII.fl3f.~,Tjj!:, ~1lI~m., 29(1990),969. rnIJ'tE,;lJSll1c, ~it~l!I!ilI:li, m.n::tlt, '4~1:!lJ!ii±, 1!Il~1:!lJ!i.

N. R. L. In

Van Kampen, Pliysica, 23 (195i) , 70i, Slli; L. l'hlhorn, .:I.rkit· foer Fysik, 17(1960), 361; Grah:un, H. Eaken, Z. Phys•• 243(19il), 259; 215(19il), HI. K:mdanoff, G. Bayro, Quantum Statistical :\Jecbanics Benjamin ~. Y. (196~); D. Dubois, •• Lectures in Theoretical Ph~·sics," Yol. IXC, eds. W. Brittin, et al., Gordon and Breach,

N. Y. (1961).

[6] [7] [8] [9]

H3J'tB ,;lJ1!.I11c, i9i~~l1I!~~~J!, 3(1979),3l.J,. R. Kubo, Bep. Progr. Pllys•• 29(1966), 25.5. ~l1I!~ta, 29(1980), 618. f1J~ S. K. Ma, G. :lfazenko, PllY8. Bell., Bll(19;5), 4077; C. DeDominicis, L. Peliti, PliY8. Bell., B18 (1978), 353.

m1J'tB ,i!j; !il1i.lc ,

1092 3Wj

409

TIME REVERSAL SYMMETRY AND NON-EQUILIBRIUM STATISTICAL STATIONARY STATES (II) Zuou

GCAXG-ZII_W

Su

ZII.\O-I3lX

(Institute of theoretical Physics • ..:!.cal/cmia Sini,'a)

ABSTRACT

This is the 2nd part of our discussion on time reyersal symmetry applied to the non'l'quilibrium statistical stationary states (NESS) ft'nul a micros(,'opic quantum statistical point of view, ·With the application of the main results of 1. a systematic inyestigation on the general properties of the NESS is ::riYC'n. For s,'stems inyariant under time rt'wrsal, thc e:s:ist('nee of a gC'neralizNl p()h>lltial allll thl' fluctuation-dissipation theorem in the low frcquenc~-limit are establislll'tl in the XESS. Thc Onsager's reciprocity rclations for the local thermodynamical equilibrium systems are also generalized to that for the NESS im'ariant under time reyersal symmetry. Finally, the time dependent Ginzburg-Landan equations for order parameters and consern:-d densities haxe been ('xpressed in a general form with time irrewrsible and reversible parts similar to that met in the literatures studying critical dynamics.

1093 ~ 33!l6

1984 ~ 7

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Jul., 1984

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H. Kleinert, Phys. Lett. B, 69 (1977), 9; H. S. Levit, PII}'S. Rev. C, 21 (1980), 1594. [ 3] A. Kerman, S. Levit, T. Troudet, MIT mEl!*-, [4] ]. Hubbard, PI.ys. Rev. L(!/t., 3 (1959), 77; [ 5] t-~HliI: A. Kerman, S. Levit, Phys. R~v. C. 24 ( 1981).269. [6] C. R. Hu, Phy.r. Rev. B, 21 (1980), 2775. [ 7] Zhou Guang·zhao, Su Zhao·bin, HaD Bai-lin, 1 (1982), 295; 307; 389. [ I] [ 2]

[8]

Reinhardt, Nucl. Phys. A, 298 (1978), 77. CfP-998, (1982). R. Stratonovich, Sov. PII)'S. Dok/.,2 (1958),416. (1981), 1029; H. Reinhardt, Nllc/. Phys. A, 367

Yu Lu, Camm/III in Tllear Phys. (Beijing China),

;5:./ijc.T~.~:l'GB,!ftJl!I!"FID, 33(1984),

[9] Guang·zhao Zhou, Zhao-bin Su, &i-lin Hao, Lu Yu, Phys. Rev. B, 22 (1980), 3385; fflI fun, ~.l1j( ,#tit!ll
mJ:),(;B,i'H'/JlC,!It.!J!l!~I!l, 30(lnl), 16~; 401.

THE GENERALIZED MEAN FIELD EXPANSIONS FOR MANY FERMION SYSTEMS Su

ZHAO-BING

Yu Lu

Zuou GUANO-ZHAO

(Instilllte 01 Theoretical Physics, Academia Sinica~

ABSTRACT

The set of coupled, seU-consistent ~lquations for the order pa.ramt'ter, the fermion field and the collective excitations developed previously in the framework of the closed time-path Green'8 functions (CTPGF) is applied to systems wit.h four·fermion int.eraction. The Hartree, the Hartree-Pock (HF) and t.he random phase approximations (RPA) for the order parameter a.nd its fluctuations are derived in a systematic way. The thermodynamical potential for the fermion system is calculated explicitly in these approximations. The formalism presented here ca.n be applied to nuclear as well as other lnany f('rmion systems in both equilibrium and nonequilibriwll states.

1102

~ 33

-If

~ 6

1984 ~ 6

M

Vol. 33. No.6

A

Jun .• 1984

ACTA PHYSICA SINICA

1983

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1106 6

M

809

(32) (33)

(34) BWe BK(y, r)

=- _ (cP,(r)cPt(y)

+

iG(r, y».

(35)

~;f§Ji!Z~;r~~j$j~$fJ;j", Q,(r), cP,(r) ~ cPt(r) 7.HjIj~ Q(r), cP(r) ~ cP+(x) ~m

tt3J2-j$jm:,ifij :f1}X<.t W p

A(r, y) ~ G(r, y) JilIJ7HjIj~Q~~cP~~=llfl"lm~ CTPGF.

f1;

Legendre ~1fR:, 51A.~tLT~iiJt9(ffij~ 2PI) TIli.~mtt~JiX;iZm

f,[Q<> cPt, cP,; A, G] =- W,[Ir, ]+, ]; M, K] - irQ, - ]+cP, - cPt] -

1-

Tr [/1.1 (Q,Q, Tr

~ Tr [M(Q,Q,

+ iA)]

[K(cP,cP~ +

iG)l

+

iA)]

L == L

"""

~

+

Tr lK(cP,cPt

+ iG)],

(36)

+

(37)

d'rd·yM(r, y)(Q,(y)Q,(r)

d 4xd'yK(r,

Y)(cP,(Y)cP~(r) +

iA(y, x»,

iG(y, x».

(38)

~tlm (31 )-(35):tt, ~~ r:8 (36):ttH~.!$1::i:l

(39) (40)

(41)

~,-

BA(r, y)

=

~ M(y, r),

(42)

21

~,- - iK(y, r).

(43)

BG(r, y)

.R~~~ f, (f.jAf*iZPEI~~, (39)-(43)~fi;JJiX;iffI~if;~ii: Q,(r), cP~(x), cP«r) lk =M" CTPGF A(r, y) ~ 6(x, y) tE!3~)j~m. li',t, tEmi!;T;r~?Jj~*fJ;j"~F~ :mi!;Tiji~tElJi!~,!!P ]+ - ] = 0 fJ;j" cP, = cPt = o. ~-)jiIff, (~*:ii.lti:~tE)f..tfS ~11-J iB'jiH~: r:8 if;~:it Q,(x) j!3! tE • ret?!.i ftULT tE ~ iI.. ~ ik~tL -Tit 7t1Jj ~ r:8 =ImCTPGF ~~tE,(39)~( 42), (43) j.\;gt~~{fl;lt~:I1£.!$tE~it)j~m,t1'-)jil,t1'-* ~~~. 1m*III&*~ik,#~.!;Tmz Hartree ;zIiP1,it:Ji!dm1lt)(ii.t$W~tE Bogo1iuboyde Geones )jilOl • )jilm (39)-( 43) iiJ~~~tE:I1£j. lJi!tE}JIrFtEl'ii]~H;:1m{iiJAf*fi;J ~ f" r-=f.i$~.&-f!P~1.W f, (f.j~~IIII!1Ji7f)j~.

1107 810

33 ~

~;tc'ilB!!i't~, ""FOO~~ J = J+ = c/J. - c/J;;

Tp[Q.(x) ,

A, G] == Tp[Q.(x) ,

= o.

WijijiZ~

c/J;;(x), c/J(x);

..

A,G]lo/>,..p+..,..

~ffl!lTA:fOGi~~3!lltLT~IlJ~1't-J (2PI). ~1l*tEBIt!3Wlt~WI't-J

(44)

W: §fi5ftl~:t&fF

m:i:$ Icff[c/J+, c/J, Q] -/o[c/J+, c/J] + 10[Q] + lin,[c/J+, c/J, Q] + Wnc/J+, c/J, Q],

(45)

CTPGF .!:3jj1lt:l:T~~$~#~!i!n19~1JUit~It..tIa]~IIIr.]!li~. CTPGF $J!;jIi3]j)]!~~m

lE3't:( -

00

,+ 00 )1it fftx ( +

,-oo)~lL~~ . .R~i±~ftlRtIa]~I't-J~1JIJ,:iiT~i~$

00

CJTCSJ ~~~rnjij~nJ(;iZPEi;!OOM7f '&*i~m:~IlJ1tJftlPf.lJmL.

2:m][~~

ill t~

*

(1~}l~EHt h);

Tp[Q .. A,

G1

=1[0.] -

ih Tr {ln[Li;'A] _Li;'A+ l} 2

+ in Tr{ In [S;'G] 1[Q.]

==

I.ff [c/J+, c/J,

Li-~) o

+

l}

+

TlP[Q.. A, G],

0=0.

(48)

1

BQ(x)BQ(y) l.-v+-o=o' -

G-o l ( :c, ) y - -

B~I
1

Bc/J+(x)Bc/J(y) ~-·~+=o=o Bc/J

BlId! +() () 1",~",~=o x Rc/J y OsO.

(49)

,

(50)

•

Sol ~~, 'BfrJZIa]I't-J~~~ SOI(:c, y) = Go'(r, y)lo.=o.

:;}97f4J~ TlP ' ~.:m

l.ff

(46) (4i)

Q11",=",+=0,

B~ill

:c, y -

-o- l:c,(y) S

{f~i±~, G;l.!:j

- GO'G

(51)

$~~:rn= Q(x)

.1C.tJ:(f.Jl.ili:;}9Wijij(~* Q.(x),

1't-JJ]( ..a~fIJ Q.(x)~. Tzp £1C.t leff $Jmm:W=-~ Id. A(r, y) :fO G(:c, y) ·:;}9pg~~J:iJT~3!lltLT~IlJ~~~

~mz:fO. (46)~$ Tr, In. ~ii:i~W~tEiZPEi;@J(.J:j£ffl't-J, §~xtlK~rm:a~pg~3

!3 mlt1itf!;j~~t,f- >It:fO. ~~T:fO.~T~!3~.~~M~J(.:;}9

J:(

r, y

) - - i

n( r, y )

=T = 2i --

ar~p

(52)

BG(y, :c)' BTlI'

(53~

n BA(y, :c)

m15~(39),(42)1it(43),*~45-fII!?~I{~~, 1lJ~~tH~~. Q.(x)

1it=M-

CTPGF

A

~Gm{jl.i'@'I't-J15~

~ _ 81[Q.] + 8Q.(x)

8Q.(::)

iA Tr {8G;1

8Q.(x)

G} + ~ 8Q.(::)

=

0,

(54)

1108 6

»l

811

-_Ie x, y) + lI(x, y) =

2i 8fp _ _'( ) " 8A(y, x) - A x, Y - Ao -i

or

-,,- 8G(y: x)

G-'(x, y) -

=

-

Ga'(x, y)

+

(55)

0,

l:(x, y) =- O.

(56)

:f[U~Jti&l:[ 1 ]-[ 4] tP ~tI:l ~nil!ilU, El3 Pf.IJm@]¥tllilitl¥:Jstla]~:i:, 15~(54 )-(56)Ztst

jJU~5jjj:tE~~:i: {?cCx) ~~!Mla]~ Ginzburg-Landau 15~I?I.NJtE5.'S.~ Mfiw~**2EI~15li :fQfttI-Tst:t5~~jE15fj. ~~I Qc(x) w.§~a915fjiiJ~JiX:X1~I¥:J~~

1.. S., 2

s

Of p == 1.. {OJ[ Q.(x) ] OQc(X.,) 2" oQkc.,) ] Of '" } - i" Tr [ 8 8(;-' G + Qc(X.,) OQ,.(r~)

=

Q

~tP

II =

r+=r_

(57)

0 '

±~a1rl3]*d!~""f*ffi,,§, S:l!

(58)

== 1.

1S~ ,MiiW~*~~**~.M!l.~=IlJi- CTPGF ~:;:~;~p-'l:

Ar(x, y)

=

~ S"7J pA(x,,, Yp),

(59.1)

A.(x, y)

=

~ 7J";pA(x,,, yp),

(59.2)

A.(r, y) =- ~ ;";pA(r,,, yp),

7J:l!

(59.3)

(60)

== ±l.

1lJ&~**~~JiJT~.EE.~15~ilJ:E1ipj(;

Jd4y[A;;;'(r, y) -

IIlx, y)]Ar(Y, .11') - 04(r - .11'),

(61)

Jd 4y[G;;;'(x, y) -

l:r(x, y)] Gr(y, z) = O'(:r - z).

(62)

~(61),(62)J;\;llirm:W;tt~,Zt~~¥UmiW~**~.Mt7~UE~15fi. *~~ttJiJT~~~15~~ ft~-Tst:t5JiJT~.EE.~~~15fjWWffi*. *m~~~J;\;,~~15~ilJ~~ A. = -

Ar(A;" -

IIc)A.,

= -

Gr( GO;;' -

l:JG •.

G.

(63) (64)

Jtiit[ 6] tP ~~ tI:l, 1!tlfiiJEl3*l9C~~~~~15fj1tEtI:l (fU~-T~~jE15fj.

:ti~Hi!ii~*, ~«11tE~lli7 Q.(x), Art A., A" Gr , G.;fO G. ~ 71-i?iI~Jfrfvl9~~ 7 1-15~, El3~ 71-:i:~ff1iiJld.)j<:tI:l~~:i:~ffilliZlttrT~~lt~*€ti:~~T~st:t5. ¥U!JUtE~.ll:,~ff1i2JS79:1ftFft1..3fr~. fzp ~1II00nHF~~..t~~ Planck 1it~" ~ ~.tt~*m. ~~~m~ •• 3fr~~~~~R,R~.~*~iW~.~~7. ~.~ ~~~ Tm ~Jtltl9~SjZ:lSj~3frM.

W:

~T§ffi~SjZJjst;flj:f£P'i1t19*~~~J.llL.l:.1!~5~1~~, .liR1*lI1.agRtt~ft;A 1fiiiliWTst:t5. jltlJt~[3] tPFJT1iEW3, ~®if~""fagRttis.~,iiJJmM~f.E-*€ti:~1m§~:(E fflfl-=f So ~ Ao ZtP. ~~,~t-=f~~tI9~ltit"iiJ~~~~~·

1109 812

33

!IS

1i . . afim.iliT-m~ • • ~.#~~~~~~B~~fim. ~~ • • *.~.~~, 1E.~~ fa 1t:J, ~tIHJt T -fill ~ !JOIMtli ~ sp.Jr~lit ~PS¥-ij ~~ • • ~~ *-r;fO~Ht;:1i~Z;b ~IDi fj! m ili, l.lltt~ :it:LFJr I?Ull:lJlt f;fj lji , fa M(z, y), #X1~fn{'FT Legendre ~~.

~ ~ ~ It:J liis~.

K(z, y)

lllHHlg~ EI3 T 51 A Ti'R 2

5'Hij{

!Ji!tEii.i~i~f!ll~~ mrJL~)[.[l-41a!JII*~. ~rz:1H~~m,tt~51 Aa!J:i:fa" ,"-l5-1if-ili.

)[fttI:[2-4]~5IAS'9~.:i:mtt~l:JilGiZ.i?iSJ Z;[h] ~*)[~5IAIt:J Zp a!J*~~ Z;[h] = Z .. fh, J+, J, M, KJ I/=/+=AI=I\==O.

(65)

5a~~,~~mmlt:J~.iZ.~:LOOS'9*~.

KJ I/=/+=M=I\==O,

W;[hJ

=

W p[h, J+, J, M,

r~[Q<3

=

rp[Q" A, G] / : : = a:~

£.G.[Q,] = [8r~[Q" A, G) 5Q,(x) 8Qkr)

_,

I ] /arIi: .Il..G

(66)

(67)

dr =

.~ =G.

(68)

J1t;r~, )[fttI:[ 4] ~ ~51A Ttt~r; 1t:J~~f'Fmiii:~i'!. ~rn]fjififB~~jtlas, I1~fiII~;i!

1t:J~~Z~.-*~.~~m-r~iZ.~~~~~,ffi~M~.*-r~m~~~~. ~m 5IAIt:J:r.n~RijIi!T lK-tJ/:~~, ftitr-rfo~f*i!t~~fa 1S.Jf-F:~~~lmaS .

•

fiB~mlK.~~S'9tt*M~T~-.~~~-r~~~R~m-r.~~R~~~

*7Cij;~m,

*-i=§5U .7CS*:t.J~ I31JiEf'&'~~m!lt:JiW:t.J$fr ~UIl, ttm:y -+-5SHff-r:fiEi fF m~m~~~.~~. ~*,~#~~~Tm~-* •• f*~m,*-i=M.~ • • ~~a-r S'9iW~.fM~f~ltJ!at ,~fl

[1] [ :) ] [31 (4

J

[5] [6] [71 [8] [9] [10] [11]

[12]

mlt:J.

Guang-zhao Zhou, Zhao-bin Su, Bai-lin Hao, Lu Yu. Phys. Rev. B, 22(1980), 3385. Zhou Guang-zhao. Su Zhao-hin, Hao Bui-lin, Yu Lu, Commun, in TlteoT. Pllys. (Beijing, China), 1(1982),295. Zhou Guallg-zhao, Su Zhao-bin, Hao Bai-lin, Yu Lu, Commun.. in Tlteor. PlIys_ (Beijing, China), 1(1982), 307. Zhou GUl1llg-zhao, Su Zhao· bin, Hao Bai-lin, Yu I.,u, Commun. ill Theor. Phys. (Beijing, Chino), 1(1982), 389. J. Cornwall, R. Jaekiw, E. Tomboulis, Phys. Ret'. D, 10(1974), :!4!!8. m~-B,~g/Jj:,t1ftt~l1!l1j!:i1I:lil,~1i..,iFIlfl3l*,Tiit~::£fAiI"';!!j!:lliJllittt, (1951). J. Hubbard, PlIY8. Re·c. Lett., 3(1959), 77, R. L. Stratonovieh, Dol;l. Acad. Nauk. SSSR, 115(1957), 10!!7. fIl:llD: J. Hubbard, Phys. RII'V. B, 19(1979), 2626. C.-R. Hu, Phys. Ret'. B, 21(1980), 2775 &Tn!!I:tiii.(. Su Zhao-bin, Wallg Ya-xin and Yu Lu, ~~~. Su Zhao-bin, Yu Lu and Zhou Guang·zhao, Commun. in T1160'1'. PlIyB. (Beijing, China), 2(1983), 1181, 1191. pj:.l1]c,Tti,mlj'f;iMB.1j!:nl.~jta.

1110

6M

813

ON A SET OF COUPLED EQUATIONS FOR THE ORDER PARAMETER -STATISTICAL GREEN'S FUNCTIONS Su

ZHAO-BIN

Yu Lu

ZHOU GUANG-ZHAO

(Inatit'ute of Tlleoretical Physica, .J.cademia Sinica)

ABSTRACT

A set of coupled equations for a quantum statistical system that determine self-consistently the order parameter, the energy spectrum, the dissipation and the distribution for both fermion field and collective excitation is suggested with a loop expansion formalism. They are applicable to non-equilibrium as well as equilibrium systems.

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