Particle Physics on the Eve of LHC: Proceedings of the Thirteenth Lomonosov Conference on Elementary Particle Physics

PARTICLE PHYSICS on the Eve of LHC

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Faculty of Physics of Moscow State University

INTERREGIONAL CENTRE FORADVANCEDSTUDffiS

Proceedings of the Thirteenth Lomonosov Conference on Elementary Particle Physics

PARTICLE PHYSICS on the Eve of LHC Moscow, Russia

23 - 29 August 2007

Editor

Alexander I. Studenikin Department of Theoretical Physics Moscow State University, Russia

'lit

World Scientific

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PARTICLE PHYSICS ON THE EVE OF LHC Proceedings of the 13th Lomonosov Conference on Elementary Particle Physics 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-13 978-981-283-758-5 ISBN-IO 981-283-758-2

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v

Moscow State University Faculty of Physics Centre for Advanced Studies

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Russian Foundation for Basic Research Russian Agency for Science and Innovation Russian Academy of Sciences Russian Agency for Atomic Energy Dmitry Zimin "Dynasty" Foundation Institutions Faculty of Physics of Moscow State :::iK()be,lts,!ln institute of Nuclear Physics, Moscow State Centre for Advanced Studies Joint Institute for Nuclear Institute for Nuclear R""'"",,,,",rr-h Theoretical and i=vY,ari,rnQlnt",1 Budker Institute of Nuclear

vi

International Advisory Committee

E.Akhmedov (ICTP, Trieste & Kurchatov Inst.,Moscow), S.Selayev (Kurchatov Inst.,Moscow), VSerezinsky (LNGS, Gran Sasso), S.Silenky (JINR, Dubna), J.Sleimaier (Princeton), MDanilov (ITEP, Moscow), GDiambrini-Palazzi (Univ. of Rome), ADolgov (INFN, Ferrara & ITEP, Moscow), VKadyshevsky (JINR, Dubna), S.Kapitza (EAPS, Moscow) A.Logunov (IHEP, Protvino), V.Matveev (INR, Moscow), P.Nowosad (Univ. of Sao Paulo), L.Okun (ITEP, Moscow), M.Panasyuk (SINP MSU), VRubakov (INR, Moscow), D.Shirkov (JINR, Dubna), J.Silk (Univ. of Oxford), ASissakian (JINR,Dubna), ASkrinsky (INP, Novosibirsk), ASlavnov (MSU & Steklov Math.lnst, Moscow) ASmirnov (ICTP, Trieste & INR, Moscow), P.Spiliantini (INFN, Florence), Organizing Committee

V.Sagrov (Tomsk State Univ.), VSelokurov (MSU), VSraginsky (MSU), AEgorov (ICAS, Moscow), D.Galtsov (MSU), AGrigoriev (MSU & ICAS, Moscow), P.Kashkarov (MSU), AKataev (INR, Moscow), O.Khrustalev (MSU), VMikhailin (MSU & ICAS, Moscow) AMourao (1ST/CENTRA, Lisbon), N.Narozhny (MEPHI, Moscow), A.Nikishov (Lebedev Physical Inst., Moscow), N.Nikiforova (MSU), VRitus (Lebedev Physical Inst., Moscow), Yu.Popov (MSU) , VSavrin (MSU), D.Shirkov (JINR, Dubna), Yu.Simonov (ITEP, Moscow), AStudenikin (MSU & ICAS, Moscow), V.Trukhin (MSU)

vii

Moscow State University Interregional Centre for Advanced Studies

SEVENTH INTERNATIONAL MEETING ON PROBLEMS OF INTELLIGENTSIA "RIGHTS and RESPONSIBILITY of the NTELLIGENTSIA" Moscow, August 29, 2007

Presidium of the Meeting VASadovnichy (MSU) - Chairman VV.Belokurov (MSU) J.Bleimaier (Princeton) G.Diambrini-Palazzi (Universiry of Rome) VG.Kadyshevsky (JINR) S.P.Kapitza (Russian Academy of Sciensies) N.S.Khrustaleva (Ministry of Education and Science, Russia) A.I.Studenikin (MSU & ICAS) - Vice Chairman V.I.Trukhin (MSU)

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FOREWORD

The 13 th Lomonosov Conference on Elementary Particle Physics was held at the Moscow State University (Moscow, Russia) on August 23-29,2007 under the Patronage of the Rector of the Moscow State University Victor Sadovnichy. The conference was organized by the Faculty of Physics and SkobeJtsyn Institute of Nuclear Physics of the Moscow State University in cooperation with the Interregional Centre for Advanced Studies and supported by the Joint Institute for Nuclear Research (Dubna), the Institute for Nuclear Research (Moscow), the Budker Institute of Nuclear Physics (Novosibirsk) and the Institute of Theoretical and Experimental Physics (Moscow). The Russian Foundation for Basic Research, the Russian Agency for Science and Innovation, the Russian Academy of Sciences, the Dmitry Zimin "Dynasty" Foundation and the Russian Agency for Atomic Energy sponsored the conference. It was more than twenty years ago when the first of the series of conferences (from 1993 called the "Lomonosov Conferences"), was held at the Department of Theoretical Physics of the Moscow State University (June 1983, Moscow). The second conference was held in Kishinev, Republic of Moldavia, USSR (May 1985). After the four years break this series was resumed on a new conceptual basis for the conference programme focus. During the preparation of the third conference (that was held in Maykop, Russia, 1989) a desire to broaden the programme to include more general issues in particle physics became apparent. During the conference of the year 1992 held in Yaroslavl it was proposed by myself and approved by numerous participants that these irregularly held meetings should be transformed into regular events under the title "Lomonosov Conferences on Elementary Particle Physics". Since then at subsequent meetings of this series a wide variety of interesting things both in theory and experiment of particle physics, field theory, astrophysics, gravitation and cosmology were included into the programmes. It was also decided to enlarge the number of institutions that would take part in preparation of future conferences. Mikhail Lomonosov (1711-1765), a brilliant Russian encyclopaedias of the era of the Russian Empress Catherine the 2nd, was world renowned for his distinguished contributions in the fields of science and art. He also helped establish the high school educational system in Russia. The Moscow State University was founded in 1755 based on his plan and initiative, and the University now bears the name of Lomonosov. The 6th Lomonosov Conference on Elementary Particle Physics (1993) and all of the subsequent conferences of this series were held at the Moscow State University on each of the odd years. Publication of the volume "Particle Physics, Gauge Fields and Astrophysics" containing articles written on the basis of presentations at the 5th and 6th Lomonosov Conferences was supported by the Accademia Nazionale dei Lincei (Rome, 1994). Proceedings of the 7th and 8th Lomonosov Conference (entitled "Problems of Fundamental Physics" and "Elementary Particle Physics") were published by the Interregional Centre for ix

x Advanced Studies (Moscow, 1997 and 1999). Proceedings of the 9th , 10th , 11th and 12th Lomonosov Conferences (entitled "Particle Physics at the Start of the New Millennium", "Frontiers of Particle Physics", "Particle Phlsics in Laboratory, Space and Universe" and "Particle Physics at the Year of 250 Anniversary of Moscow University") were published by World Scientific Publishing Co. (Singapore) in 2001,2003,2005 and 2006, correspondently. The physics programme of the 13 th Lomonosov Conference on Elementary Particle Physics (August, 2007) included review and original talks on wide range of items such as neutrino and astroparticle physics, electroweak theory, fundamental symmetries, tests of standard model and beyond, heavy quark physics, nonperturbative QCD, quantum gravity effects, physics at the future accelerators. Totally there were more than 350 participants with 113 talks including 32 plenary (30 min) talks, 48 session (25-20 min) talks and 33 brief (15 min) reports. One of the goals of the conference was to bring together scientists, both theoreticians and experimentalists, working in different fields, so that no parallel sessions were organized at the conference. The Round table discussion on "Dark Matter and Dark Energy: a Clue to Foundations of Nature" was held during the last day of the 13 th Lomonosov Conference. Following the tradition that has started in 1995, each of the Lomonosov Conferences on particle physics has been accompanied by a conference on problems of intellectuals. The 7th International Meeting on Problems of Intelligentsia held during the 13 th Lomonosov Conference (August 29, 2007) was dedicated to discussions on the issue "Rights and Responsibility of the Intelligentsia". The success of the 13 th Lomonosov Conference was due in a large part to contributions of the International Advisory Committee and Organizing Committee. On behalf of these Committees I would like to warmly thank the session chairpersons, the speakers and all of the participants of the 13 th Lomonosov Conference and the 7th International Meeting on Problems of Intelligentsia. We are grateful to the Rector of the Moscow State University, Victor Sadovnichy, the Vice Rector df the Moscow State University, Vladimir Belokurov, the Dean of the Faculty of Physics, Vladimir Trukhin, the Director of the Skobeltsyn Institute of Nuclear Physics, Mikhail Panasyuk, the Directors of the Joint Institute for Nuclear Research, Alexey Sissakian, the Director of the Institute for Nuclear Research, Victor Matveev, the Director of the Budker Institute of Nuclear Physics, Alexander Skrinsky, and the Vice Dean of the Faculty of Physics of the Moscow State University, Anatoly Kozar for the support in organizing these two conferences. Special thanks are due to Alexander Suvorinov (the Russian Agency for Science and Innovations), Gennady Kozlov (JINR) and Oleg Patarakin (the Russian Agency for Atomic Energy) for their very valuable help. I would like to thank Giorgio Chiarelli, Dmitri Denisov, Francesca Di Lodovico, Hassan Jawahery, Andrey Kataev, Cristina Lazzaroni, William C. Louis, Frank Merrit, Thomas MUller, Tatsuya Nakada, Daniel Pitzl, Jacob Schneps, Claude

xi

Vallee and Horst Wahl for their help in planning of the scientific programme of the meeting and inviting speakers for the topical sessions of the conference. Furthermore, I am very pleased to mention Alexander Grigoriev, the Scientific Secretary of the conference, Andrey Egorov, Mila Polyakova, Dmitry Zhuridov, Dasha Novikova, Maxim Perfilov and Katya Salobaeva for their very efficient work in preparing and running the meeting. These Proceedings were prepared for publication at the Interregional Centre for Advanced Studies with support by the Russian Foundation for Basic Research, the Russian Agency for Science and Innovations and the Russian Agency for Atomic Energy. Alexander Studenikin

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CONTENTS Thirteenth Lomonosov Conference on Elementary Particle Physics Sponsors and Committees Seventh International Meeting on Problems of Intelligentsia Presidium Th~~

v vii ~

Fundamentals of Particle Physics The Quantum Number of Color, Colored Quarks and Dynamic Models of Hadrons Composed of Quasifree Quarks V. Matveev, A. Tavkhelidze

3

Discovery of the Color Degree of Freedom in Particle Physics: A Personal Perspective O. W. Greenberg

11

The Evolution of the Concepts of Energy, Momentum, and Mass from Newton and Lomonosov to Einstein and Feynman L. Okun

20

Physics at Accelerators and Studies in SM and Beyond Search for New Physics at LHC (CMS) N. K rasnikov

39

Measuring the Higgs Boson(s) at ATLAS C. Kourkoumelis

46

Beyond the Standard Model Physics Reach of the ATLAS Experiment G. Unel

55

The Status of the International Linear Collider B. Foster

65

Review of Results of the Electron-Proton Collider HERA V. Chekelian

67

Recent Results from the Tevatron on CKM Matr~ Elements from Bs Oscillations and Single Top Production, and Studies of CP Violation in Bs Decays J. P. Fernandez

77

Direct Observation of the Strange b Barion L. Vertogradov xiii

st 85

xiv

Search for New Physics in Rare B Decays at LHCb V. Egorychev

91

CKM Angle Measurements at LHCb S. Barsuk

95

Collider Searches for Extra Spatial Dimensions and Black Holes G. Landsberg

99

Neutrino Physics Results of the MiniBooNE Neutrino Oscillation Experiment Z. Djurcic

109

MINOS Results and Prospects J.P.Ochoa-Ricoux

113

The New Result of the Neutrino Magnetic Moment Measurement in the GEMMA Experiment A. G. Beda, V. B. Brudanin, E. V. Demidova, V. G. Egorov, M. G. Gavrilov, M. V. Shirchenko, A. S. Starostin, Ts. Vylov

119

The Baikal Neutrino Experiment: Status, Selected Physics Results, and Perspectives V. Aynutdinov, A. Avrorin, V. Balkanov, 1. Belolaptikov, N. Budnev, I. Danilchenko, G. Domogatsky, A. Doroshenko, A. Dyachok, Zh.-A. Dzhilkibaev, S. Fialkovsky, O. Gaponenko, K. Golubkov, O. Gress, T. Gress, O. Grishin, A. Klabukov, A. Klimov, A. Kochanov, K. Konischev, A. Koshechkin, V. Kulepov, L. Kuzmichev, E. Middell, S. Mikheyev, M. Milenin, R. Mirgazov, E. Osipova, G. Pan'kov, L. Pan'kov, A. Panfilov, D. Petukhov, E. Pliskovsky, P. Pokhil, V. Poleschuk, E. Popova, V. Prosin, M. Rozanov, V. Rubtzov, A. Sheifier, A. Shirokov, B. Shoibonov, Ch. Spiering, B. Tarashansky, R. Wischnewski, I. Yashin, V. Zhukov

121

Neutrino Telescopes in the Deep Sea V. Flaminio

131

Double Beta Decay: Present Status A. S. Barabash

141

Beta-Beams C. Volpe

146

T2K Experiment K. Sakashita

154

xv Non-Standard Neutrino Physics Probed by Tokai-to-Kamioka-Korea Two-Detector Complex

N. Cipriano Ribeiro, T. Kajita, P. Ko, H. Minakata, S. Nakayama, H. Nunokawa

160

Sterile Neutrinos: From Cosmology to the LHC

F. Vannucci

166

From Cuoricino to Cuore Towards the Inverted Hierarchy Region

C. Nones

169

The MARE Experiment: Calorimetric Approach to the Direct Measurement of the Neutrino Mass

E. Andreotti

174

Electron Angular Correlation in Neutrinoless Double Beta Decay and New Physics

A. Ali, A. Borisov, D. Zhuridov

179

Neutrino Energy Quantization in Rotating Medium

A. Grigoriev, A. Studenikin

183

Neutrino Propagation in Dense Magnetized Matter

E. V. Arbuzova, A. E. Lobanov, E. M. Murchikova

188

Plasma Induced Neutrino Spin Flip via the Neutrino Magnetic Moment

A. K uznetsov, N. Mikheev

193

Astroparticle Physics and Cosmology International Russian-Italian Mission "RIM-PAMELA"

A. M. Galper, P. Picozza, o. Adriani, M. Ambriola, G. C. Barbarino, A. Basili, G. A. Bazilevskaja, R. Bellotti, M. Boezio, E. A. Bogomolov, L. Bonechi, M. Bongi, L. Bongiorno, V. Bonvicini, A. Bruno, F. Cafagna, D. Campana, P. Carlson, M. Casolino, G. Castellini, M. P. De Pascale, G. De Rosa, V. Di Felice, D. Fedele, P. Hofverberg, L. A. Grishantseva, S. V. Koldashov, S. Y. Krutkov, A. N. Kvashnin, J. Lundquist, O. Maksumov, V. Malvezzi, L. Marcelli, W. Menn, V. V. Mikhailov, M. Minori, E. Mocchiutti, A. Morselli, S. Orsi, G. Osteria, P. Papini, M. Pearce, M. Ricci, S. B. Ricciarini, M. F. Runtso, S. Russo, M. Simon, R. Sparvoli, P. Spillantini, Y. 1. Stozhkov, E. Taddei, A. Vacchi, E. Vannuccini, G. Vasilyev, S. A. Voronov, Y. T. Yurkin, G. Zampa, N. Zampa, V. G. Zverev

199

xvi

Dark Matter Searches with AMS-02 Experiment A. Malinin

207

Investigating the Dark Halo R. Bernabei, P. Belli, F. Montecchia, F. Nozzoli, F. Cappella, A. Incicchitti, D. Prosperi, R. Cerulli, C. J. Dai, H. L. He, H. H. Kuang, J. M. Ma, X. D. Sheng, Z. P. Ye

214

Search for Rare Processes at Gran Sasso P. Belli, R. Bernabei, R. S. Boiko, F. Cappella, R. Cerulli, C. J. Dai, F. A. Danevich, A. d'Angelo, S. d'Angelo, B. V. Crinyov, A. Incicchitti, V. V. Kobychev, B. N. Kropivyansky, M. Laubenstein, P. C. Nagornyi, S. S. Nagorny, S. Nisi, F. Nozzoli, D. V. Poda, D. Prosperi, A. V. Tolmachev, V. I. Tretyak, l. M. Vyshnevskyi, R. P. Yavetskiy, S. S. Yurchenko

225

Anisotropy of Dark Matter Annihilation and Remnants of Dark Matter Clumps in the Galaxy V. Berezinsky, V. Dokuchaev, Yu. Eroshenko

229

Current Observational Constraints on Inflationary Models E. Mikheeva

237

Phase Transitions in Dense Quark Matter in a Constant Curvature Gravitational Field D. Ebert, V. Ch. Zhukovsky, A. V. Tyukov

241

Construction of Exact Solutions in Two-Fields Models S. Yu. Vernov

245

Quantum Systems Bound by Gravity M. L. Fil'chenkov, S. V. Kopylov, Y. P. Laptev

249

CP Violation and Rare Decays Some Puzzles of Rare B-Decays A. B. Kaidalov

255

Measurements of CP Violation in b Decays and CKM Parameters J. Chauveau

263

Evidence for DO_Do Mixing at BaBar M. V. Purohit

271

Search for Direct CP Violation in Charged Kaon Decays from NA48/2 Experiment S. Balev

276

xvii

Scattering Lengths from Measurements of Ke4 and K± Decays at NA48/2 D. Madigozhin

7m

->

1T± 1T 01T O

280

Rare Kaon and Hyperon Decays in NA48 Experiment N. M ala kana va

285

THE K+ -> 1T+ vD Experiment at CERN Yu. Potrebenikov

289

Recent KLOE Results B. Di Micco

293

Decay Constants and Masses of Heavy-Light Mesons in Field Correlator Method A. M. Badalian

299

Bilinear R-Parity Violation in Rare Meson Decays A. Ali, A. V. Borisov, M. V. Sidorova

303

Final State Interaction in K E. Shabalin

307

->

21T Decay

Hadron Physics

Collective Effects in Central Heavy-Ion Collisions G. 1. Lykasov, A. N. Sissakian, A. S. Sarin, V. D. Toneev

313

Stringy Phenomena in Yang-Mills Plasma V. 1. Zakharov

318

Lattice Results on Gluon and Ghost Propagators in Landau Gauge I. L. Bogolubsky, V. G. Bornyakov, G. Burgio, E.-M. Ilgenfritz, M. Miiller-Preussker, V. K. Mitrjushkin

326

~

and 2: Excited States in Field Correlator Method I. Narodetskii, A. Veselov

330

Theory of Quark-Gluon Plasma and Phase Transition E. V. Komarov, Yu. A. Simonov

334

Chiral Symmetry Breaking and the Lorentz Nature of Confinement A. V. Nefediev

339

Structure Function Moments of Proton and Neutron M. Osipenko

343

Higgs Decay to bb: Different Approaches to Resummation of QCD Effects A. L. Kataev, V. T. Kim

347

xviii

A Novel Integral Representation for the Adler Function and Its Behavior at Low Energies A. V. Nesterenko

351

QCD Test of z-Scaling for nO-Meson Production in pp Collisions at High Energies M. Tokarev, T. Dedovich

355

Quark Mixing in the Standard Model and the Space Rotations G. Dattoli, K. Zhukovsky

360

Analytic Approach to Constructing Effective Theory of Strong Interactions and Its Application to Pion-Nucleon Scattering A. N. Safronov

364

New Developments in Quantum Field Theory On the Origin of Families and their Mass Matrices with the Approach Unifying Spin and Charges, Prediction for New Families N. S. Mankoc Borstnik

371

Z2 Electric Strings and Center Vortices in SU(2) Lattice Gauge Theory M. 1. Polikarpov, P. V. Buividovich

378

Upper Bound on the Lightest Neutralino Mass in the Minimal Non-Minimal Supersymmetric Standard Model S. Hesselbach, G. Moortgat-Pick, D. J. Miller, R. Nevzorov, M. Trusov

386

Application of Higher Derivative Regularization to Calculation of Quantum Corrections in N=l Supersymmetric Theories K. Stepanyantz

390

Nonperturbative Quantum Relativistic Effects in the Confinement Mechanism for Particles in a Deep Potential Well K. A. Sveshnikov, M. V. Ulybyshev

394

Khalfin's Theorem and Neutral Mesons Subsystem K. Urbanowski

398

Effective Lagrangians and Field Theory on a Lattice O. V. Pavlovsky

403

String-Like Electrostatic Interaction from QED with Infinite Magnetic Field A. E. Shabad, V. V. Usov

408

xix

QFT Systems with 2D Spatial Defects 1. V. Fialkovsky, V. N. Markov, Yu. M. Pismak

412

Bound State Problems and Radiative Effects in Extended Electrodynamics with Lorentz Violation 1. E. Prolov, O. G. Kharlanov, V. Ch. Zhukovsky

416

Particles with Low Binding Energy in a Strong Stationary Magnetic Field E. V. Arbuzova, G. A. Kravtsova, V. N. Rodionov

420

Triangle Anomaly and Radiatively Induced Lorentz and CPT Violation in Electrodynamics A. E. Lobanov, A. P. Venediktov

424

The Comparative Analysis of the Angular Distribution of Synchrotron Radiation for a Spinless Particle in Classic and Quantum Theories V. G. Bagrov, A. N. Burimova, A. A. Gusev

427

Problem of the Spin Light Identification V. A. Bordovitsyn, V. V. Telushkin

432

Simulation the Nuclear Interaction T. F. K amalov

439

Unstable Leptons and ({L- e - T}-Universality O. Kosmachev

443

Generalized Dirac Equation Describing the Quark Structure of Nucleons A. Rabinowitch

447

Unique Geometrization of Material and Electromagnetic Wave Fields O. Olkhov

451

Problems of Intelligentsia

The Conscience of the Intelligentsia J. K. Bleimaier

457

Conference Programme

463

List of Participants

469

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Fundamentals of Particle Physics

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THE QUANTUM NUMBER OF COLOR, COLORED QUARKS AND DYNAMIC MODELS OF HADRONS COMPOSED OF QUASIFREE QUARKS

v. Matveev a , A. Tavkhelidze b Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Abstract. Are exposed the main stages of the early development of the hypothesis of the quantum number of color and of colored quarks

1

Introduction

At present, the dominant point of view is that all physical phenomena and processes, both terrestrial and cosmological, are governed by three fundamental forces: gravitational, electroweak and chromodynamic. The color charge serves as the source of chromo dynamic forces. In this talk we shall expose the main stages of the early development of the hypothesis of the quantum number of color and of colored quarks, put forward under the ideological influence of and in collaboration with N .Bogolubov at the JINR Laboratory of Theoretical Physics. In these works, the concept of color, colored quarks, was introduced for the first time, and a dynamical description of hadrons was given within the framework of the model of quasifree colored quarks. Introduction of the quantum number of color permitted to treat colored quarks as real physical objects, constituents of matter. Further, from the color 5U(3) symmetry, the Yang-Mills principle of local invariance and quantization of chromo dynamic fields gave rise to quantum chromo dynamics - the modern theory of strong interactions. 2

The quantum number of color and colored quarks

In 1964, when the hypothesis of quarks was put forward by Gell-Mann [lJ and Zweig [2], quarks were only considered to be mathematical objects, in terms of which it was possible, in a most simple and elegant way, to describe the properties, already revealed by that time, of the approximate unitary 5U(3) symmetry of strong interactions. At the beginning, these particles, exhibiting fractional charges and not observable in a free state, were not attributed the necessary physical interpretation. First of all, making up hadrons of quarks, possessing spin ~, led to a contradiction with the Pauli principle and the Fermi-Dirac statistics for systems composed of particles of semiinteger spin. The problem of the quark statistics was not, however, the sole obstacle in the path of theory. No answer existed to the following question: why were ae-mail: [email protected] be-mail: [email protected]

3

4

only systems consisting of three quarks and quark-antiquark pairs realized in Nature, and why were there no indications of the existence of other multiquark states? Especially important was the issue of the possible existence of quarks in a free state (the problem of quark confinement). In 1965, analysis of these problems led N.Bogolubov, B.Struminsky and A.Tavkhelidze [3], as well as LNambu and M.Hana [4], and LMiyamoto [5] to the cardinal idea of quarks exhibiting a new, hitherto unknown, quantum number subsequently termed color. From the very beginning creation of the relativistically invariant dynamical quark model of hadrons was based, first of all, on the assumption of quarks representing real physical objects determining the structure of hadrons. To make it possible for quarks to be considered fundamental physical particles, the hypothesis was proposed by three authors (Bogolubov, Struminsky, Tavkhelidze - 1965, January) that the quarks, should possess an additional quantum number, and that quarks of each kind may exist in three (unitary) equivalent states q

==

(ql,q2,q3)

differing in values of the new quantum number, subsequently termed color. Since at the time, when the new quantum number was introduced, only three kinds of quarks were known - (u, d, s), the quark model with an additional quantum number was termed the three-triplet model. Since the new quantum number is termed color, colored quarks may be in three equivalent states, such as, for example, red, blue and green. With introduction of the new quantum number, color, the question naturally arised of the possible appearance of hadrons possessing color, which, however, have not been observed. From the assumption that colored quarks are physical objects, while the hadron world is degenerate in the new quantum number, or it is colorless, it followed that solutions of the dynamic equations for baryons and mesons in the s-state should be neutral in the color quantum numbers [4,6]. From the requirement that baryons be colorless, the wave function of the observed baryon family in the ground state, described by the totally symmetric 56-component tensor

= 1/V6 ca,B")'

(0./3"1 = 1,2,3),

where

A = (a, a),

B = (b, (3),

C = (c, "I),

a, b, c are unitary quantum numbers, a, /3, "I are color quantum numbers, is totally antisymmetric tensor.

ca,B")'

5

Hence it is evident, that the Pauli principle holds valid for colored quarks, and that they satisfy the Fermi-Dirac statistics, so they can be considered real fundamental constituents of matter. From the requirement that mesons, composed of a colored quark and an antiquark, be neutral, or colorless, the meson wave function is chosen in the form

W~(Xl,X2) =

1/v'3 O~~(Xl,X2)'

where a and b are unitary indices, o~ is Kronecker symbol. The choice of baryon and meson wave functions, proposed above, leads to the conclusion that the known mesons and baryons are composed of colored quarks and antiquarks as follows: q"(l) q,,(2) - mesons; c",6'Yq,, (1) q,6(2) q'Y(3) - baryons. Subsequently, the requirement that the world of hadrons are neutral led to the discovery of the SU(3) color symmetry group. It is to be noted, here, that in his talk delivered to the conference" Symmetry principles at high energies", held in Coral Gables (1965), Nambu was the first, on the basis of SU(3) symmetry with respect to the new quantum number (color), to deal with eight vector fields, carriers of the interaction between quarks, which were the prototype of the quantum-chromodynamic gluon fields. 3

Dynamic quark models of hadrons composed of quasifree colored quarks

The introduction of colored quarks, representing physical fundamental particles, paved the way for the dynamic description of hadrons. The main obstacle, here, was the absence of quarks in a free state. Although it was evident that the issue of confinement could be ultimately settled only by experiments, a series of attempts was undertaken to provide a logically non-contradictory explanation for the "eternal confinement" of quarks inside hadrons. Thus, for example, P.Bogolubov [7] (1968) proposed the "quarkbag" model known as the Dubna bag. Later (1974), the idea of a quark bag underwent development at MIT, and the resulting model is known as the MIT bag [8]. The dynamic relativistic quasi-free quark model, the development of which was initiated in Dubna (1965 - January, April), was based on the assumption of quarks being extremely heavy objects bound in hadrons by enormous scalar forces, that on the one hand provide for the large quark mass defect in hadrons and on the other hand impedes their leaving the hadron [4,6]. In this model baryons ABC and mesons ~ are represented by a superposition of all admissible states over the quantum numbers (A, B, C), satisfying the

6

requirements of SU(6) symmetry, of quark statistics and of hadron neutrality in the color quantum number. 3.1

Hadron form factors

The dynamic composite quasi free quark model has made possible the systematic description of both the statically observed characteristics of hadrons (fl, gA/ gv etc.) and their form factors. Weak and electromagnetic interactions was introdused in a minimal manner, - electromagnetic interaction - weak interaction where AJl is the electromagnetic potential, T/f represent charged lepton weak currents, G is the Fermi weak interaction constant. For the ratio gA/gV (of the axial and vector weak interaction constants) and for the magnetic moment of the proton we obtain gA/gv ~ -5/3(1- 28),

flp

~

3/(2Mp)(1 - 8),

8 =< pi L z 1 p > where Lz is the orbital momentum of a quark bound in the nucleon with the projection of its total angular momentum equal to 8 characterizes the magnitude of relativistic corrections and amounts to 8 '" 1/6, the resulting correction for the ratio gAl gv is of the order of 30%. This example shows how significant could be the relativistic corrections. The dynamic quark model has permitted to explain the lepton decays of pseudoscalar 7r- and K -mesons and, also, the electromagnetic decays of the vector mesons into electron-positron pairs as annihilation of quark-antiquark pairs bound in the mesons. Analysis of the data on the widths of these decays resulted in a conclusion on the dependence of the scales of distances (effective sizes) on the quantum numbers of a bound system. For example

!;

1Wk(O) 12 / 1W7r (O) In the case of the decay

7r 0 ---+

12~ mk/m7r'

2/"

determined by the triangular anomaly of the axial current (J.Bell, RJackiw 1969; S.Adler 1970), the annihilation model points to the width of this decay being proportional to the number of different quark colors [9].

7

3.2

The model of quasifree quarks and the laws of scaling at high energies

Experiments, in which inclusive reactions were studied at high energies and momentum transfers, and the scaling regularities revealed, have given an impetus to further development of the theory of hadronquark structure. (Inclusive reactions were first introduced and studied theoretically by Logunov, Nguen Van Hieu and Mestvirishvili (1967); reported by Logunov [10] at the Rochester conference (1967)). Here, of essential significance was the investigation of deep inelastic processes in the inclusive scattering of electrons of nucleons, performed at the Stanford center, which in 1968 resulted in observation of the scaling properties - Bjorken scaling - indicating the existence of a "rigid" pointlike nucleonic structure. In 1969, on the basis of the quasi-free quark model, the assumption was put forward by Matveev, Muradyan and Tavkhelidze that the scaling properties of electron-nucleon interaction processes, revealed in experiments, are common for all deep-inelastic lepton-hadron processes and that they can be derived in a model-independent manner on the basis of the automodelling principle, or the principle of self-similarity [11]. The essence of the self-similarity principle consists in the assumption that in the asymptotic limit of high energies and large momentum transfers form factors and other measurable quantities of deep-inelastic processes are independent of any dimensional parameters (such as particle masses, the strong interaction radius etc.), which may set the scale of measurement of lengths or momenta. Thus, the form factors of deep-inelastic processes turn out to be homogeneous functions of relativistically invariant kinematic variables, the degree of homogeneity of which is determined by analysis of the dimensionality (it is a key property of conformal invariant theories). Application of the self-similarity principle for establishing the asymptotic behaviour of the form factors WI (q2 , v) and W2(q2 , v) of deep-inelastic scattering of electrons on protons in the Bjorken region

{'

1q 12--. 00,

V

= 2pq --. 00,

C;

= _q2/ V = const

8

results in the Bjorken asymptotic formulae derived in 1968 on the basis of certain assumptions

Application of the self-similarity principle resulted in the scaling law, being found for the first time, that describes the mass spectrum of muon pairs, produced in inclusive proton collisions [11]

~;: r

(Matveev, Muradyan, Tavkhelidze - 1969) where M is the effective mass of the muon pair, and E is the initial energy of the colliding particles.(Later this process was called Drell -Yan process). Experimental studies ofthis process, initiated in 1970 by the group ofL.Lederman at Brookhaven, confirmed this scaling law, and it was precisely in these processes that a new class of hadrons, the J /'l/J particles, was subsequently observed.

3.3

Quark counting formulae

In the case of binary hadronreactions a + b -> C + d at high energies sand momentum transfers t, application of the self-similarity principle yields for the differential cross section the following formula of the quark counting [12]

(Matveev, Muradyan, Tavkhelidze 1973) where n is the total number of quarks belonging to the particles participating in the reactions. In the case, when particle b, for instance, is a lepton, then nb = 1, and one obtains the asymptotic formula for the baryon form factor. The function f (t / s) depends only on the relation between large kinematic variables and is itself a dimensional quantity. Thus, the asymptotic power law points to factorization of the effects of large and small distances.

9

4

Summary

We described the scaling properties of elementary particle interaction processes, observed experimentally at high energies and large momentum transfers, on the basis of the self-similarity principle. At the same time the question arises concerning the extent to which scaling invariant behaviouris consistent with the main requirements of local quantum field theory. In the case of deep-inelastic electron scattering on nucleons, these problems were investigated by Bogolubov, Vladimirov and Tavkhelidze [13] for form factors, which in the Bjorkenregion have the asymptote

For the weight functions of these form factors in the Jost-Lehman-Dyson representation sufficient conditions were found that guarantee Bjorken scaling. In the case of free field the weight functions automatically satisfy these restrictions, which is precisely what provides for Bjorken scaling in the quasi-free quark model. Note, that the subsequent discovery (1973) by Gross, Wilczek and Politzer[14] of the phenomenon of asymptotic freedom in QCD. of an invariant charge, introduced by Bogolubov and Shirkov[15] in the renormgroup theory, is an essentially important step for substantiation of the picture of quasi-free quarks in hadrons. In a number of important works of the last years (t'Hooft, Maldacena, Polchinski, Polyakov, Witten) there an impressive non-perturbative derivation of the asymptotic power laws (the quark counting for form factors and the exclusive scattering cross sections of hadrons) was suggested in the framework of the conformal versions of QCD which are dual to the string theory. In this talk we have not touch upon the possibility of parafermionic character of quarks, suggested by Greenberg in 1964, as we assume that this problem shall be mentioned here by Prof. O.W.Greenberg himself. It can be noted that many of the priority works, presented in this talk, were published only as preprints or, sometimes, in the proceedings of international conferences, in accordance with the opportunities that existed at that time. The references to the original papers mentioned in the talk are given in the review "THE QUANTUM NUMBER COLOR, COLORED QUARKS AND QCD (Dedicated to the 40th anniversary of the discovery of color), by V.Matveev, A.Tavkhelidze, Particles and Nucleus v. 37, P.3, pp.576 596 (2006). References

[1] Gell-Mann M. Phys.Lett. 1964 v.8 p.214

10

[2J Zweig G. CERN Preprint TH-401 1964 [3J Bogolubov N., Struminsky B., Tavkhelidze A. JINR Preprint D-1968 Dubna 1965 Tavkhelidze A. High Energy Physics and Elementary Particles, Vienna, 1965 p.7534. [4J Han M.Y., Nambu Y. Phys.Rev.B 1965 v.139 p.10055. [5J Miyamoto Y. Progr. Theor.Phys. Suppl. Extu. 1965 N 1876. [6J Bogolubov N., Matveev V., Nguen Van Hieu, Stoyanov D., Struminsky B., Tavkhelidze A., Shelest V. JINR Preprint P-2141, Dubna 1965; Tavkhelidze A. High Energy Physics and Elementary Particles, Vienna, 1965 p.7637. [7J Bogolubov P. Ann.Inst.Henri Poincare 1968 v.VIII p.2 [8J Chodos A., Jaffe RL., Jonson K., Thorn G.B., Weisskopf. Phys.Rev. D 12 (1975) 2060 [9J Matveev V., Struminsky B., Tavkhelidze A. JINR Preprint P-2524, Dubna 1965; Struminsky B., Tavkhelidze A. Proc. of the Intern. Conf. on High Energy Physics and Elementary Particles, Kiev 1967, p.625 638 [lOJ Logunov A., Mestvirishvili M., Nguen Van Hieu. Proc. of the Intern. Conf. on Particles and Fields, Rochester 1967; Phys.Lett.B 1967 v.25 p.611 [l1J Matveev V., Muradian R, Tavkhelidze A. JINR Preprint P2-4578, P24553, P2-4824, Dubna 1979; SLAC-TRANS-0098, JINR, P2-4543, Dubna 1969,27p. [12J Matveev V., Muradian R, Tavkhelidze A. Lett. Nuovo Chim. 1973 v.7 p.712 [13J Bogolubov N., Vladimirov V., Tavkhelidze A. Theor.Mat.Fiz. 1972 v. 12 N 3 p.305 [14J Gross D.G., Wilczek F. Phys.Rev.Lett. V. 30 p.1343; Politzer H. Ibid p.134616j [15J Bogolubov N., Shirkov D. NuovoChim. 1956 v. 3 p.845

DISCOVERY OF THE COLOR DEGREE OF FREEDOM IN PARTICLE PHYSICS: A PERSONAL PERSPECTIVE

o.w.

Greenberg'

Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742-4111 Abstract. I review the main features of the color charge degree of freedom in particle physics, sketch the paradox in the early quark model that led to color, give a personal perspective on the discovery of color and describe the introduction of the gauge theory of color.

1

Introduction

Our present conception of the nature of elementary particles includes fractionally charged quarks that carry a hidden 3-valued charge degree of freedom, "color ," as fundamental constituents of strongly interacting particles (hadrons). The main features of color are (1) it is a hidden 3-valued charge degree of freedom carried by quarks, (2) it can be incorporated into an SU(3)color gauge theory, and (3) the hidden color gauge group commutes with electromagnetism. This third feature requires that the electric charges of quarks are independent of color, which in turn requires the quarks to have fractional electric charges. Quarks with fractional electric charges were introduced by Murray GellMann [ [1]] and, independently, by George Zweig [ [2]] in 1964. Also in 1964 I introduced color, using parafermi statistics of order 3 [ [3]]. This 3 is the same 3 as the 3 of SU(3)color. My work was stimulated by the SU(6) theory of Feza Giirsey and Luigi A. Radicati [ [4]] in the same year. Giirsey and Radicati placed the baryons in the symmetric 3-particle representation of SU(6). This produced a paradox: The spin 1/2 quarks must be fermions, according to the spin-statistics theorem, and can only occur in antisymmetric representations. I resolved this paradox in 1964 [ [3]] by suggesting that quarks obey parafermi statistics [ [5]] of order 3, which allows up to 3 particles to be in a symmetric state. As mentioned above, the 3 of the parafermi statistics is the same 3 as the 3 of color SU(3). Because particles with fractional electric charge had not been observed, several of the early authors chose models with integer quark charges. Such models are unacceptable both theoretically and experimentally. In models with integer quark charges electromagnetism does not commute with color so that color symmetry is broken. Such a model violates the exact conservation of color which is a crucial part of QeD. Integer charges also conflict with experimental evidence coming from the ratio O"(e+e- -+ hadrons)/O"(e+e- -+ f.L+f.L-) as well as from the analysis of jets in high energy hadron collisions. aemail address, [email protected].

11

12

I emphasize that there are two independent discoveries connected with the strong interactions: (1) color as a charge-analogous to electric charge in electromagnetism, and (2) color as a gauge symmetry-analogous to the U(l) symmetry of electromagnetism. The gauge symmetry of a theory is intimately connected with the quantities that are observable in the theory. In the context of parastatistics if only currents such as (1) [¢(x), "(JL1/!(x)) are observable, then the gauge symmetry is SU (3). With additional observables such as (2) the symmetry is SO(3). For the parafermi theory of quarks we choose only baryon number zero currents, so that only the currents of Eq.(l) are allowed. Currents such as Eq.(2) have non-zero baryon number and are not allowed. Thus the parafermi theory must be associated with the symmetry SU(3). We can make this explicit, following Oscar Klein [ [6)]' by transforming the Green components of the parafields to sets of normal fields. The choice of currents with zero baryon number leads to explicit SU(3) symmetry for the normal quark fields. To summarize, the choices of observables and of gauge symmetry are directly related. The parastatistics of H.S. Green cannot be gauged because the commutation rules for the Green components with equal values of the Green index are not the same as the commutation rules for Green components with unequal values of the Green index. Kenneth Macrae and I [ [7]] showed how to modify Green's parastatistics so that it can be gauged by reformulating parastatistics with Grassmann numbers. Further, we showed that using Grassmann numbers that obey an SU(N) (SO(N)) algebra leads to an SU(N) (SO(N)) gauge theory. In summary, the full understanding of color emerged from the work of Greenberg in 1964, and the work of Nambu in 1965 and of Han and Nambu in 1965. As is often the case in the development of new theories, neither got everything in final form at the beginning. 2

Influences leading to the discovery of the hidden 3-valued color charge degree of freedom

Here I describe the disparate influences that led me to introduce the color charge degree of freedom. In the 1950's and early 1960's I was struck by the success of very simple ideas in bringing order to the newly discovered "strange" particles. In the same period, under the influence of my thesis advisor, Arthur Wightman, I was learning sophisticated mathematical techniques based on quantum field theory. My PhD thesis on the asymptotic condition in quantum field theory gave a formalization of the Lehmann, Symanzik, Zimmermann

13

(LSZ) scattering theory. I used the theory of operator-valued distributions and gave proofs of properties of LSZ scattering theory that were mathematically rigorous according to the standards of that period. I became interested in the theory of identical particles as a graduate student in the 1950's. I wondered why only bosons and fermions occur in Nature, as well as what other possibilities might exist. Although I did not see his paper at the time, H.S. Green had introduced a generalization of each type of statistics in 1953. He generalized bose (fermi) statistics to parabose (parafermi) statistics of integer order p. Green replaced the usual bilinear commutation or anticommutation relations for bose and fermi statistics by trilinear relations. He found solutions of his trilinear relations using an ansatz. Expand a field q in p "Green" components that (for the parafermi case) anticommute when the Green indices are the same but commute when the Green indices are different,

= J(x _ y), xO = yO, t(y)]_ = O,x0::j:. yO.

[q(O:) (x), q(O:) t (y)]+

(3)

[q(O:)(x),q(!3)

(4)

(For the parabose case interchange commutator and anticommutator.) For parafermi (parabose) statistics of order p at most p identical particles can be in a symmetric (antisymmetric) state. As mentioned above, Klein gave a recipe for converting fields that antic ommute (commute) into fields that commute (anticommute). Schematically, his transformation is Q(O:)(x) = K(o:)q(O:) (x), K(o:)IO)

= 10).

(5) (6)

The Klein transformation converts the anomalous anticommutation (commutation) relations to the normal ones. Albert M.L. Messiah and I worked together on generalizations of the usual bose and fermi quantum statistics in 1962-1964. We showed that any representation of the symmetric group for identical particles is compatible with quantum mechanics in the context of first-quantized quantum theory [ [8]]. We also worked out the branching rules for changes in the number of identical particles. We formulated parastatistics without using Green's ansatz (for the case with the usual Fock vacuum) in the context of second-quantized quantum field theory [ [9]]. In addition, we derived the selection rules for interactions that change the number of identical particles. This work prepared me to address the paradox in the quark model of baryons that arose in 1964. The year 1964 was the crucial year for the discovery of both quarks and color. Quarks were suggested independently by Gell-Mann and by Zweig. Gell-Mann's quarks resembled what we now call current quarks. Zweig's quarks, which he called aces, deuces and treys, resembled what we now call constituent quarks. In early 1964 when I first heard about rumors of the idea of quarks I wondered

14

why only the combinations qqq and ijq occurred in nature. In the original models there was no reason for this. The paradox concerning the quarks in baryons arose in the SU(6) theory of Giirsey and Radicati. They generalized an idea of Wigner from 1937. Wigner had combined the SU(2)J ofisospin with the SU(2)s of spin to make an SU(4) and used this SU( 4) to classify nuclear states and derive relations for their energy levels. With a larger symmetry group he found more relations among the energy levels. Giirsey and Radicati combined the SU(3)j of the three quark flavors in the original quark model with SU(2)s to get an SU(6) that they used to classify particle states. The SU(6) theory considers a quark as a

3,...., (u, d, s) in SU(3)r

(7)

1/2,...., (t, .!.) in SU(2)s.

(8)

and the spin Giirsey and Radicati combined these as a (9)

We can also decompose the quark under SU(6) -+ SU(3)j x SU(2)s,

(10)

6 -+ (u, d, s) x (t, .!.).

(11)

For the qij mesons this works well; we have 6

(59

6*

= 1 + 35,

35 -+ (8,0)

+ (1 + 8, 1).

(12) (13)

Here the 8 and the 1 before the commas are the SU(3)j multiplicities and the 0 and the 1 after the commas are the spins of the particles. The octet of psuedoscalar mesons,

° - ° °

+ K- K- -) , (K +, K"o ,7r,7r,7r,T/"

(14)

was known, as were the singlet plus octet (or nonet) of vector mesons,

(K*+, K*o, ¢o,p+ pO, p- ,wo, j{*o, j{*-).

(15)

Both the octet and the nonet fit well in the SU(6) scheme. The analogous calculation for the qqq baryons requires decomposing the product of three 6's into irreducibles of SU(6), 6

(59

6

(59

6 = 56

+ 70 + 70 + 20.

(16)

15

This 56 is the representation that fits the data on the lowlying baryons, 56 --+ (8,1/2)

+ (10,3/2),

(17)

since there is an octet of spin-l/2 baryons, 0 AO ,L..J, "'+ ",0 "'- ~O ~-) , (P+ ,n, L..J ,LJ' ,~ ,-=

(18)

and a decuplet of spin-3/2 baryons, ( ~ ++ , ~ + , ~ 0 , ~ -

'y,H 1'

Y,*o ;::;0*0 , ......., ;::;0*- n-) . 1 , Y,*1 , .......

(19)

Giirsey and Radicati found a mass formula for these baryons that generalizes the Gell-Mann-Okubo mass formula for each SU(3) multiplet and also gives a new relation between masses in the octet and the decuplet. The 56 seemed like a compelling choice for the baryons in the quark model. However, this leads to a paradox: The permutation properties of the 56, 70 and 20 are respectively symmetric, mixed and antisymmetric. Since the quarks should have spin 1/2, the spin-statistics theorem [ [lOll requires that they should be fermions and occur in the antisymmetric 20 representation. The experimental data which places the baryons in the symmetric 56 representation conflicts with the spin-statistics theorem. When I came to Princeton in the fall of 1964 there was a lot of excitement about the Giirsey-Radicati SU(6) theory. Benjamin W. Lee gave me a preprint of a paper [ [11]] on the ratio of the magnetic moments of the proton and neutron that he had written with Mirza A. Baqi Beg and Abraham Pais. They had calculated this magnetic moment ratio using the group theory of SU(6). I translated their result into the concrete quark model, assuming the quarks obey bose statistics in the visible degrees of freedom. Both the result, that the ratio is -3/2, and the simplicity of the calculation were striking. Here is my version of that calculation: Represent the proton and neutron with spin up as (20) Ipf) = ~u+(u+dl- uld+)IO),

In~) = ~d+(u+dl- uld+)IO).

(21)

1

The (u+dl - u d+) combination in parentheses serves as a "core" that carries zero spin and isospin, so that the third quark to the left of the parentheses carries the spin and isospin of the proton or neutron. The magnetic moment is then the matrix element /-LB = (Btl/-L3IBt), where /-L3 = 2/-L0L: qQqSq, Qq = (2/3, -1/3, -1/3), the 2 is the g-factor of the quark, /-Lo is the Bohr magnet on of the quark and Qq are the quark charges in units of the proton charge. With this setup the magnetic moments can be calculated on one line, /-Lp

1

2

1

1

1

1

= 2/-L0"3{2["3. 1 + (-"3)' (-2)] + [(-"3)' (2)]} = /-Lo·

(22)

16

The analogous calculation for the neutron gives (23) The ratio is /-Ip/ /-In = -3/2, which agrees with experiment to 3%. This leads to an estimate for the effective mass of the quark in the nucleon, mN /2.79 ~ 340M eV / c2 , which is consistent with present extimates of the constituent masses of the up and down quarks. Previous calculations of the magnetic moments using pion clouds had failed. Nobody had realized that the ratio was so simple. In retrospect the calculation worked better than we would now expect, since it did not take account of quark-antiquark pairs and gluons. Nonetheless, for me the success of this simple calculation was a very convincing additional argument that quarks have concrete reality. The paradox about the placement of the baryons in the 56 representation of SU(6) was based on the spin-statistics theorem which states: Particles that have integer spin must obey Bose statistics, and particles that have odd-halfinteger spin must obey Fermi statistics. I knew there is a generalization of the spin-statistics theorem that was not part of general knowledge in 1964: Particles that have integer spin must obey parabose statistics, and particles that have odd-half-integer spin must obey parafermi statistics [[12]]. Each family is labeled by an integer Pi P = 1 is the ordinary Bose or Fermi statistics. I immediately realized that parafermi statistics of order 3 would allow up to 3 quarks in the same space-spin-flavor state without violating the Pauli principle, which would resolve the statistics paradox. To test this idea I suggested a model in which quarks carry order-3 parafermi statistics in [ [3]]. This was the introduction of the hidden charge degree of freedom now called color. With this resolution of the statistics paradox I was exhilarated. I felt that the new charge degree of freedom implicit in the parafermi model would have lasting value. I became convinced that the quark model and color were important for the theory of elementary particles. Not everybody shared my enthusiasm. It is difficult now to grasp the level of rejection of these ideas in 1964 and even for the next several years. Quarks were received with skepticism in 1964. Color as a hidden charge carried by quarks was received with disbelief. The reactions of two distinguished physicists illustrate this skepticism and disbelief. I gave a copy of my paper to J. Robert Oppenheimer and asked his opinion of my work when I saw him at a conference about a week later. He said, "Greenberg, it's beautiful!," which sent me into an excited state. His next comment, however, "But I don't believe a word of it." brought me down to earth. In retrospect I have two comments about these remarks of Oppenheimer. I was not discouraged, because I was convinced that my solution

17

to the statistics paradox would have lasting value. Nonetheless, I was too intimidated by Oppenheimer to ask why he did not believe my paper. Steven Weinberg, who contributed as much as anybody to the standard model of elementary particles, wrote in a talk on The Making of the Standard Model [ [14]] "At that time [referring to 1967] I did not have any faith in the existence of quarks." The skepticism about quarks and color can be understood: Quarks were new. Nobody had ever observed a particle with fractional electric charge. GellMann himself was ambiguous about their reality. In his paper he wrote " .. .It is fun to speculate .. .if they were physical particles of finite mass (instead of purely mathematical entities as they would be in the limit of infinite mass ... A search ... would help to reassure us of the non-existence ofreal quarks [[1]]." To add a hidden charge degree of freedom to the unobserved fractionally charged quarks seemed to stretch credibility to the breaking point at that time. In addition, parastatistics, with which the new degree of freedom was introduced, was unfamiliar. Resolving the statistics paradox was not a sufficient test of color. I needed new predictions. I turned to baryon spectroscopy to construct a model of the baryons in which the hidden parafermi (color) degree of freedom takes care of the required antisymmetry of the Pauli principle. Then I could treat the quarks as bosons in the visible space, spin and flavor degrees of freedom, with the parastatistics taking care of the necessary antisymmetry. I made a table of the excited baryons in the model using sand p state quarks in the 56, L = 0+ and 70, L = 1- supermultiplets. I followed up this work with Marvin Resnikoff in 1967 [ [13]]. This work has been continued by Richard H. Dalitz and collaborators, by Nathan Isgur and Gabriel Karl and by Dan-Olof Riska and collaborators, among others. The original fits to the baryons made in 1967 are surprisingly close to the current fits of 2008. The only evidence for color from 1964 to 1969 was the baryon spectroscopy that I proposed in 1964. It was only in 1968 that the first rapporteur at an international conference accepted the parastatistics model for baryons as the correct model. By then the data on baryon spectroscopy clearly favored the new degree of freedom. In 1969, Steven Adler, John Bell and Roman Jackiw explained the 7r -t 'Y'Y decay rate using the axial anomaly with colored quarks. This gave the first additional evidence for quarks. 3

Introduction of the gauge theory of color

Explicit color SU(3) was introduced in 1965 by Yoichiro Nambu [ [15]] and by Moo-Young Han and Nambu [ [16]]. The papers of Nambu and of Han and Nambu used 3 dissimilar triplets in order to have integer charges for the quarks.

18

This is not correct, both experimentally theoretically for reasons given above. However this paper paper includes the statement "Introduce now eight gauge vector fields which behave as (1,8), namely as an octet in SU(3)'''' [ [16)). This was the introduction of the gauge theory of color. The 1 in the (1,8) refers to what we now call flavor and is the statement that the gauge vector fields, which we now call gluons, are singlets in flavor. The 8 was what Han and Nambu called SU(3)" (which we now call SU(3)color) and states that the interaction between the quarks is mediated by an octet of gluons. Other solutions to the statistics paradox, all of which failed, were (i) an antisymmetric ground state, favored by Dalitz, (ii) the idea that quarks are not real, so that their statistics is irrelevant, and (iii) other atomic models. Adding qq pairs leads to unseen "exploding SU(3) states." The original version of the quark model did not consider "saturation," why only the combinations qqq and qq occur in nature. In 1966 Daniel Zwanziger and I surveyed the existing models and constructed new models to see which models account for saturation [ [17)). The only models that worked were the parafermi model, order 3, and the equivalent 3 triplet or color SU(3) models. The states that are bosons or fermions in the parafermi model, order 3, are in 1-to-1 correspondence with the states that are color singlets in the SU(3) model. Thus the parastatistics and explicit color models are equivalent as a classification symmetry. Some properties beyond classification agree in both models. The 7r -+ 'Y'Y decay rate and the ratio u(e+e- -+ hadrons)/u(e+e- -+ J.t+ J.t-) are the same in both cases, because it does not matter whether the quark lines in intermediate states represent Green component quarks or color quarks. Properties that require gauge theory include (i) confinement, discussed by Weinberg, by Gross and Wilczek and by Harald Fritzsch, Gell-Mann, and Heinrich Leutwyler in 1973, which explains why isolated quarks are not observed, (ii) asymptotic freedom, found by David Politzer and by Gross and Wilczek in 1973, which reconciles the quasi-free quarks of the parton model with the confined quarks of low-energy hadrons, (iii) running of coupling constants and precision tests of QeD, (iv) jets in high-energy collisions, among other things. 4

Summary

The discovery of color resolved a paradox: quarks as spin-l/2 particles should obey fermi statistics according to the spin-statistics theorem and should occur in antisymmetric states; however they occur in the symmetric 56 of the GlirseyRadicati SU(6) theory. I resolved this paradox in 1964 by introducing a new 3-valued hidden charge degree of freedom, color, via the parafermi model of quarks in which color appears as a classification symmetry and a global quantum number. I used this model to predict correctly the spectroscopy of excited

19

states of baryons. The other facet of the strong interaction, gauged SU(3)color, was introduced as a local gauge theory by Nambu and by Han and Nambu in 1965. References

[1] M. Gell-Mann, Phys. Lett. 8214 (1964). [2] G. Zweig, CERN Reports 8182/TH.401 and 8419/TH.412 (1964). The latter is reprinted in Developments in the Quark Theory of Hadrons, (Hadronic Press, Nonamtum, Mass., 1980), ed. D.B. Lichtenberg and S.P. Rosen. [3] O.W. Greenberg, Phys. Rev. Lett. 13 598 (1964). [4] F. Giirsey and L.A. Radicati, Phys. Rev. Lett. 13 173 (1964). [5] H.S. Green, Phys. Rev. 90 270 (1953). [6] O. Klein, J. Phys. Radium 9 1 (1938). [7] O.W. Greenberg and K.I. Macrae, Nucl. Phys. B 219 358 (1983). [8] A.M.L. Messiah and O.W. Greenberg Phys. Rev. 136 B248 (1964). [9] O.W. Greenberg and A.M.L. Messiah, Phys. Rev. 138 B1165 (1965). [10] W. Pauli, Ann. Inst. Henri Poincare 6 137 (1936). [11] M.A.B. Beg, B.W. Lee and A. Pais, Phys. Rev. Lett. 13 514, erratum 650 (1964). The magnetic moment ratio was found independently by B. Sakita, Phys. Rev. Lett. 13 643 (1964). [12] G.F. Dell'Antonio, O.W. Greenberg and E.C.G. Sudarshan, in Group Theoretical Concepts and Methods in Elementary Particle Physics, (Gordon and Breach, New York, 1964), ed. F. Giirsey, p. 403. [13] O.W. Greenberg and M. Resnikoff, Phys. Rev. 163 1844 (1967). [14] S. Weinberg, http://arxiv.org/abs/hep-ph/0401010vl. [15] Y. Nambu, in Preludes in Theoretical Physics, (North Holland, Amsterdam, 1966), ed. A. de Shalit, H. Feshbach and L. Van Hove, p. 133; [16] M.Y. Han and Y. Nambu, Phys. Rev. 139 BlO06 (1965). [17] O.W. Greenberg and D. Zwanziger, Phys. Rev. 150 1177 (1966).

THE EVOLUTION OF THE CONCEPTS OF ENERGY, MOMENTUM, AND MASS FROM NEWTON AND LOMONOSOV TO EINSTEIN AND FEYNMAN L.B.Okun a ITEP, 117218 Moscow, Russia Abstract.The talk stresses the importance of the concept of rest energy Eo and explains how to use it in various situations.

1

Introduction

This conference is the first in a series of conferences celebrating 300 years since the birth of Mikhail Lomonosov (1711-1765). The law of conservation of mass established in chemistry by Lomonosov and Lavoisier and seriously modified in relativistic physics two centures later is central for understanding and teaching physics today. Therefore it is appropriate to consider the evolution of the laws of conservation of mass, energy, and momentum during this period. The main message of the talk is the equivalence of the rest energy of a body and its mass: Eo = mc 2 . This equivalence is a corollary of relativity principle. The total energy of a body and its mass are not equivalent: E -I mc2 . The contents of the talk is as follows: 1. Introduction 2. XVII - XIX centuries 3.1. Calileo, Newton: relativity 3.2. Lomonosov, Lavoisier: conservation of mass 3.3. Conservation of energy 3. The first part of the XXth century 4.1. Rest energy Eo 4.2. Energy and inertia 4.3. Energy and gravity 4.4. "Relativistic mass" 4.5. Famous

VB

VB

mass

true

4.6. Einstein supports Eo = mc2 ae-mail: [email protected]

20

21

4. The second part of the XXth century 4.1. Landau and Lifshitz 4.2. Feynman diagrams 4.3. Feynman Lectures 5. Conclusions 6. Acknowledgements 7. Discussion: FAQ on mass

r:

= E2 Q1 7.2. Unnatural definition of mass E = mc2 : Q2,Q3

7.1. Natural definition of mass m 2

7.3. Equivalence of mass and rest energy: Q4-Q8 7.4. Interconversion between Eo and E k : Q9-Q12c 7.5. Binding energy in nuclei: Q13,Q14 7.6. Mass differences of hadrons: Q15-Q20 7.7. Some basic questions: Q21-Q25

2 2.1

XVII - XIX centuries Galileo, Newton: relativiy

The concept of relativity was beautifully described by Galileo Galilei in his famous book "Dialogo" (1632) as experiments in a cabin of a ship. The principle of relativity had been first formulated by Isaac Newton in his even more famous book "Principia" (1687), though not as a principle, but as corollary v. The term mass was introduced into physics by Newton in "Principia". According to Newton, the mass is proportional to density and volume. The momentum is proportional to mass and velocity. As for the term energy, Newton did not use it. He and Gottfried Leibniz called the kinetic energy vis viva - the living force.

2.2

Lomonosov, Lavoisier

In 1756 Lomonosov experimentally proved his earlier conjecture (formulated in his letter to Leonard Euler in 1748) that mass is conserved.

22

Lomonosov's handwriting in Latin: ignition of tin (jupiter) and lead (saturnus) in sealed retorts.

The 1756 report on Lomonosov's experiments which disproved the results of Robert on ignition of metals. (Written in Russian by a clerk.) "... made experiments in firmly sealed glass vessels in order to investigate whether the weight of metals increases from pure heat. It was found by these that the opinion of the famous Robert Boyle is false, for without in the external air the weight of the ignited metal remains in the same measure ... "

23

In 1773 Antoine Lavoisier independently proved the law of conservation of mass in a series of more refined experiments.

2.3

Conservation of energy

The term energy was introduced into physics in 1807 by Thomas Young. By the middle of the XIXth century a number of scientists and engineers, especially J.R.von Mayer and J.P. Joule, established the law of conservation of energy which included heat among the other forms of energy. 3

3.1

The first part of XX century

Rest energy Eo

The special theory of relativity was created by Hendrik Lorentz, Henri Poincare, Albert Einstein, and Herman Minkowski. The concept of rest energy was introduced into physics by Einstein. In 1905 Einstein proved in the framework of special relativity that the change of the rest energy of a body is equivalent to the change of its mass. In 1922 and especially clearly in 1935 he formulated the equivalence of mass m and rest energy Eo - the equation Eo = mc2 •

3.2

Energy and inertia

In relativity the energy E and momentum p of a body form the energymomentum vector Pi (i = 0,1,2,3 = 0, a). In the units in which c = 1: Po = E,Pa = p. The mass is a Lorentz scalar defined by the square of Pi: m 2 = p2 = E2 - 'j!2 . To keep track of powers of c let us define Po = E, Pa = cpo Then p2 = E2 - c2'j!2 = m 2 c4 . In Newtonian physics mass is the measure of inertia according to equations: p= mv , F = dp/dt , F = ma, where a = dv/dt. In relativity the energy is the measure of inertia: p = EiJj c2 • If the force is defined by equation F = dp/dt, then

F = m')'a + m')'3v (va) = mta + mzv( va

Jl -

where,), = 1/ v2 /c 2 • In the first years of the XXth century Hendrik Lorentz who tried to define inertial mass in terms of force and acceleration ended up with the concepts of longitudinal and transverse masses : mz = m,),3, mt = m')' which later were forgotten.

24

3.3

Energy and gravity

In Newtonian physics the source of gravity is mass. In relativity the source of gravity is the energy-momentum tensor PiPk/ E which serves as the "gravitational charge" . With the help of propagator of the gravitational field proportional to gil gkm + gimgkl _ gikglm, where gik is the metric tensor, the energy-momentum tensor can be reduced in a static gravitational field (when l, m = 0 ) to (2E2 m 2 c4 )/E. For a massive non-relativistic apple this expression is equal to mc2, while for a photon it is equal to 2E. Note the factor of 2. The energy of a photon is attracted stronger than the energy of an apple.

3.4

"Relativistic mass" vs mass

The prerelativistic commandments: 1. mass must be the measure of inertia,

2. mass must be additive. They led to the introduction of the so-called "relativistic mass" m = E / c2 which for a massive particle increases with the velocity of the particle. The idea that mass of an electron increases with its velocity had been put forward by J.J Thomson, O. Heaviside, and G. Sirl in the last decade of the XIXth century, (not so long) before relativity theory was formulated. The idea that light with energy E has mass m = E / c2 was formulated by Poincare in 1900 and was discussed by Einstein in the first decade of the XXth century. The relativistic mass increasing with velocity was proclaimed "the mass" by G. Lewis and R. Tolman at the end of that decade. A decade later it was enthroned in books on relativity by Max Born and Wolfgang Pauli.

3.5

Famous vs true

Thus the equation E = mc2 appeared and was ascribed to Einstein. This "adopted child" is widely considered as "the famous Einstein's equation" instead of the true Einstein's equation Eo = mc2 . Einstein seemed to be indifferent to this misuse.

3.6

Einstein supports Eo = mc2

In 1922 in his book "The Meaning of Relativity" Einstein formulated the equation Eo = mc2 • In December 1934 Einstein delivered his Gibbs Lecture "Elementary derivation of the equivalence of mass and energy" at a joint meeting of the American Mathematical Society and the American Physical Society.

25 In that lecture he repeatedly stressed that mass m (with the usual time unit, mc2 ) is equal to rest energy Eo. This however did not prevent Einstein's coauthor - Leopold Infeld b from stating in 1955 that the main experimental confirmation of the special relativity is the dependence of mass on velocity.

4

The second half of XX century

4.1

Landau and Lifshitz

The first monograph in which special and general relativity were presented without using the notion of mass increasing with velocity was the first (1941) edition of "Field Theory" by Lev Landau and Evgeniy Lifshitz. They wrote (in the first and the second editions in §9,§1O - in the later editions they became §8,§9) the expressions for action S, momentum p, energy E and rest energy. Unfortunately for the latter they chose the same symbol E and did not introduce Eo. The latest edition still keeps this tradition.

4.2

Feynman diagrams

A major step forward in creating the present understanding of nature were diagrams introduced by Richard Feynman. The external lines of a diagram correspond to incoming and outgoing, free, real particles. For them p2 = m 2 in units of c = 1; they are on mass shell. The internal lines correspond to virtual particles. For them p2 =I- m 2 ; they are off mass shell. Energy and momentum are conserved at each vertex of a diagram. The exchange of a virtual massless particle creates long-range force between real particles. Thus exchange of a photon creates Coulomb force (potential). The exchange of a virtual massive particle creates Yukawa potential - shortrange force with radius r = h/mc. When using Feynman diagrams, the four-dimensional momenta p and invariant masses m immensely facilitate theoretical analysis of various processes involving elementary particles. Feynman diagrams unified matter (real particles - both massive and massless) with forces (virtual particles). The role of Quantum Mechanics is crucial to this unification. A nice feature of Feynman diagrams is the interpretation of antiparticles as particles moving backword in time. b "A. Einstein,

L.lnfeld. The Evolution of Physics. 1938."

26

4.3

Feynman Lectures

The most famous textbook in physics is "The Feynman Lectures on Physics" . Several million copies of Lectures introduced millions of students to physics. In his Lectures Feynman masterfully and enthusiastically painted the broad canvas of physics from the modern point of view. Unfortunately in this masterpiece he completely ignored the Feynman diagrams and largely ignored the covariant formulation of the relativity theory. Lectures are based on the archaic notion of "relativistic mass" that increases with velocity and the relation E = mc2 . Thus millions of students were (and are!) taught that the increase of mass with velocity is an experimental fact. They sincerely believe that it is a fact, not a factoid based on a rather arbitrary definition m = E / 2. 5

Conclusions

The giant figure of Newton marked the birth of modern Science. The achievements of Science since the times of Newton are fantastic. The modern views on matter differ drasticlly from those of Newton. Still, even in the XXIst century many physics textbooks continue to use (incorrectly) the equations of Newton many orders of magnitude beyond the limits of their applicability, at huge ratios of kinetic energy Ek to rest energy Eo (l05 for electrons and 10 4 for protons at CERN), while Newton's equations are valid only for Ek/ Eo « 1. If some professors prefer to persist in this practice, they should at least inform their students about the fundamental concept of invariant mass and the true Einstein's equation:

6

Acknowledgements

I am very grateful for their help to A.A. Alehina, B.M. Bolotovsky, K.G. Boreskov, M. Gottlieb, E.G. Gulyaeva, M.V. Danilov, E.A. Ilyina, O.V. Kancheli, V.l. Kisin, V.l. Kogan, M.V. Mandrik, T.S. Nosova, B.L. Okun, E.V. Sandrakova, M.B. Voloshin. 7

7.1

Discussion: FAQ about mass

Natural definition of mass

Ql: Which definition of mass is natural in the framework of the Relativity Theory?

27 AI: The definition according to which mass is a Lorentz invariant property of an object - the 'length' of the 4-dimensional energy-momentum vector p = (E, cp). Namely m 2 = p2/c4 or in other notations m 2 = E2/C4 /c 2. This definition corresponds perfectly to the fundamental symmetry of special relativity and uses the minimal number of notions and symbols.

r

7.2

Unnatural definition of mass

Q2: Can one nevertheless introduce another definition of mass, namely, that which corresponds to the "famous Einstein's equation E = mc2 ,,? (here E is the total energy of a free body) A2: Yes. One can do this. But this cheese is not free. People who do this refer to the ordinary mass as the "rest mass" (they denote it ma). They have two different symbols for energy: E and E / c2 = m. This is confusing. This ignores the 4-dimensional symmetry of relativity theory: E is a component of a 4-vector, while E/c2 is "the cat that walkes by itself" the time component of a 4-vector the space components of which are never mentioned. Of course in any consistent theory one can introduce an arbitrary number of redundant variables by multiplying any observable by some power of a fundamental constant, like c. With proper bookkeeping that would not produce algebraic mistakes. However, instead of creating clarity, this creates confusion. It is like the well known Jewish joke on inserting the letter 'r' in the word 'haim': - What for is the letter r in the word 'haim'? - But there is no r in 'haim' - And if to insert it? - But what for to insert it? - That is what I am asking: what for? Q3: Doesn't the mass, increasing with the velocity of the body, explain why the velocity of a massive body cannot reach the velocity of light? A3: No. It does not explain: the increase is not fast enough. This follows from the expression for longitudinal mass ml = ma/(I - v 2/c 2)3/2 derived by Lorentz from F = dfl/dt.

7.3

Equivalence of mass and rest energy

Q4: Is mass equivalent to energy?

28

A4: Yes and no. Loosely speaking, mass and energy are equivalent. But the mass m of an object is not equivalent to its total energy E, it is equivalent to its rest energy Eo. Q5: What is rest energy Eo? A5: The rest energy Eo is the greatest discovery of the XXth century. Einstein discovered that any massive body at rest has a huge hidden energy Eo = mc 2 (the subscript 0 indicates here that the velocity of the body v is equal to zero). Q5a: How did Einstein discover Eo = mc2 ? A5a: In his second 1905 paper on relativity Einstein considered a body at rest with rest energy Eo, which emits two light waves in opposite directions with the same energy L/2. For an observer that moves with velocity v with respect to the body the total energy of two waves is L / Jl - v 2 / c2 • By assuming conservation of energy and by considering the case of v « c Einstein derived that /:::;.m = L/c2 • In this short note two revolutionary ideas were formulated: 1. that a massive body at rest contains rest energy Eo,

2. that a system of two massles light waves with energy L has mass L/V2. (Einstein denoted the speed of light by V.) In his publications of 1922 and 1935 Einstein cast the relation in the form Eo = mc2 . Q5b: Is it possible to prove Einstein's relation by considering emission of one wave of light instead of two? A5b: Yes, it is possible. But the proof is slightly more involved. In this case the rest energy of the body partly transforms into kinetic energy of light Land kinetic energy of the recoil body with mass m: Ek = L2 /2mc 2. Q6: Is the relation Eo = mc 2 compatible with the definition of mass given above: m 2 = E2/ c4 /c 2 ?

r

A6: Yes. It is absolutely compatible: at

v = 0 you have p = 0, while E = Eo.

Q6a: You defined Eo as the energy of a particle in its rest frame. On the other hand, photon's speed is always c. Why do you think that the concept of rest energy can be used in the case of a massless photon?

29 A6a: The experimental upper limit on the mass of the photon is extremely small (less than 1O- 16 eV/c2 ). Therefore it does not play any role in most cases, and we can safely and conveniently speak about massless photons. However even a tiny eventual mass, say 10- 20 eV / c2 , allows in principle to consider the rest frame of the photon and thus define its rest energy Eo. For all practical purposes this tiny rest energy is vanishingly small: Eo = mc2 = o. Q6b: How is it possible to put such a tiny upper limit on the mass of the photon which is negligible at any photon energy? A6b: The mass m of a virtual photon would cut off the magnetic interaction at distances larger than r = h/mc. By observing astrophysical magnetic fields at large distances one can get the upper limit on m. Q6c: Why not abandon the term "mass" in favor of "rest energy"? Why to have two terms instead of one, if we do know that mass is equivalent to rest energy? A6c: "One" is not always better than "two". The word "mass" refers to a lot of phenomena which have nothing to do with the rest energy "sleeping" in massive bodies. Such a terminological reform would be a disaster not only for Newtonian mechanics for which c is alien, but for Science in general. Q6d: Isn't it better to have both relations: Eo = mc 2 and E = mc 2 instead of one of them? Isn't "two" always better than "one"? Recall the famous wave-particle duality. A6d: Two relations (explanations) are better than one if both are correct and if each of them has its own realm of applicability. The relation E = mc2 has no separate domain of applicability. Moreover it has no domain of applicability at all. It is a consequence of introduction, along with E, of a redundant variable of "relativistic mass" E / c2 which usurped the throne of mass. Thus in this case "one" is much better than "two". Q6e: Why do you dislike the relativistic mass so strongly? A6e: I stumbled on it 20 years ago and realized how difficult it is to reeducate students and teachers brought up on the concept of mass increasing with velocity and the famous formula E = mc 2 • It selfpropagates like a virus or a weed and prevents people from understanding the essence of relativity theory. A century ago Max Planck said that the carriers of wrong views simply die out while new generations accept the truth. But it turned out that new generations come already infected. An important role in the mechanism of

30

infection was played by the authors of textbooks and popular science writers, the editors of popular magazines, like "Scientific American". It is the rare case when most of the experts know the truth, but lightly preach the non-truth. Q7: Does the mass of a box filled with gas increase with the increase of the temperature of the gas? A7: Yes, according to relativity theory, it increases. Q8: Doesn't it mean that the masses of molecules of gas increase with temperature, i.e. with their velocities? Or in other words, that energy and mass are equivalent? A8: No, it does not mean that. Such an inference would presume the additivity of masses. But according to relativity theory, the total mass of the gas is not equal to the sum of the masses of its molecules: m =I=- L mi. In fact the correct interpretation of the mass of the gas supports the relation Eo = mc 2 , not the relation E = mc2 . This can be seen from the following reasoning. The total energy E of the relativistic gas is equal to the the sum of total energies Ei of the individual particles of gas: E = LEi. Each Ei increases with temperature. Hence the total energy of gas increases. The total momentum P of the gas vanishes because the distribution of particle's momenta Pi is isotropic: P = L Pi = O. Hence the total energy of gas is equal to its rest energy. By applying the definition of invariant mass: m 2 = E2 / c4 - tp / c2 and taking into account that in this case E = Eo one gets Eo = mc 2 . This is valid both for the gas of massive particles and for massless photons.

7.4

Interconversion between rest energy and kinetic energy

Q9: Does mass convert into energy? Does energy convert into mass? A9: No. The "mutual conversion of mass and energy" is a very loose and therefore a misleading term. The point is that energy is strictly conserved in all processes. It can neither appear, nor disappear. It can only transform from one form into another. The rest energy (mass) converts into other forms of energy (e.g. kinetic energy). QlO: Does energy convert into mass in the processes of production of particles in accelerators? AlO: No. Various forms of energy transform into each other, but the total energy is conserved.

31

The kinetic energy transforms into rest energy (into masses of the produced particles) in accelerators. The colliders convert Ek into mass much more effectively than the fixed target accelerators. Qll: Did the laws of conservation of mass and energy merge into one law of conservation of mass-energy similar to the law of conservation of energymomentum 4-vector? All: Yes and No. The laws of conservation of energy and momentum of an isolated system (unified in the law of conservation of 4-momentum ) correspond to the uniformity of time and space correspondingly. There is no extra spacetime symmetry responsible for the conservation of mass. The total mass of a closed (isolated) system (the rest energy of the system) is conserved due to conservation of its energy and momentum. Q12: Doesn't the total mass change in the annihilation of positronium into two photons? Electron and positron are massive, while photons are massless. A12: No. The total mass does not change: the rest energy of the system of two massless photons is equal, in this process, to the rest energy of positronium. Q12a: What is the meaning of the term "rest energy of the system of two photons", if each of them has no rest energy and in a second after they were born the two photons are 600 000 km apart? A12a: "Rest energy of the system of two photons" means here the sum of their kinetic (or total) energies in a reference frame in which the sum of their momenta is equal to zero. In this frame they fly in opposite directions with equal energies. Q12b: Why do you refer to this rest energy as mass of the system of two photons? A12b: Because I am applying the equation Eo = mc2 . The mass of an elementary particle has a deep physical meaning because it is an important quantum number characteristic of all elementary particles of a given sort (say, electrons or protons). The mass of a nuclear or atomic level is also a quantum number. The mass of a macroscopic body is not as sharply defined because of overlap of huge number of quantum levels. As for the mass of a system of free particles, it is simply their total energy (divided by c2 ) in a frame in which their total momentum is equal to zero. The

32

value of this mass is limited only by conservation of energy and momentum, like in the case of two photons in the decay of positronium. As a rule we are unable to measure the inertia or gravity of such a system, but the self-consistency of the relativity theory guarantees that it must behave as mass. Q12c: Do I understand correctly that with this definition of mass the conservation of mass is not identical to the conservation of matter in the sense in which it was meant by Lomonosov and Lavoisier? A12c: Yes. You do understand correctly. Matter now includes all particles,even very light neutrinos and massless photons. The number of particles in an isolated piece of matter is not conserved. Roughly speaking, the mass of a body is a sum of masses of constituent particles plus their kinetic energies minus the energy of their attraction to each other ( of course, the energies are divided by c2 ). 7.5

Binding energy in nuclei

Q13: Is the mass of a nucleus equal to the sum of the masses of the constituent nucleons? A13: No. The mass of a nucleus is equal to the sum of the masses of the constituent nucleons minus the binding energy divided by c2 . Thus the nucleus is lighter than the sum of the masses of its nucleons. Q14: Can the liberation of kinetic energy in the Sun, in nuclear reactors, atomic and hydrogen bombs be explained without referring to the equation Eo = mc2 ? A14: Yes. In the same way that it is explained for chemical reactions, namely, by the existence and difference of binding energies. Rutherford considered the dependence of mass on velocity as an important fact, but neither he nor his coworkers mentioned E = mc2 or Eo = mc 2 in their works as a source of energy released in radioactive processes though they rejected the idea of perpetum mobile. 7.6

Mass differences of hadrons

Q15: Is the mass of a proton equal to the sum of the masses of two u quarks and one d quark which constitute the proton?

33

A15: No. The mass of the proton is not equal to the sum of the masses of three quarks. However, the situation here is more subtle than in the case of nucleons in a nucleus. Q16: What is the main difference between quarks and nucleons? A16: Nucleons can exist as free particles. (Hydrogen is the most abundant element in the universe, while free neutrons are produced in nuclear reactors.) Quarks exist only inside hadrons. Free quarks do not exist. Mass is defined by equation m 2 = E 2/c 4 - p2/c2 only for free particles. Therefore, strictly speaking, we cannot apply this equation to quarks. However one can use the property of asymptotic freedom of QCD - Quantum Chromodynamics. Q17: What is asymptotic freedom? A17: According to the asymptotic freedom, the higher the momentum transfer is in interaction of quarks, the weaker their interaction is. Thus, due to the uncertainty relation, at very short distances quarks look like almost free particles. In units where c = 1 the mass of u quark is 4 MeV at such distances,while that of d quark is 7 MeV. The sum of masses of three quarks inside a proton is 15 MeV, while the mass of the proton is 938 MeV. Q18: What constitutes the difference between 938 MeV and 15 MeV? A18: This difference - the main part of the proton mass, as well as of the masses of other hadrons - is caused mainly by the energy of the gluon field the vacuum condensate of gluons. Q19: Can we speak about the values of this condensate as of binding energies? A19: No, we cannot. The contribution of binding energy to the mass is negative, while the contribution of condensate is positive. By supplying enough energy from outside one can liberate a nucleon from a nucleus, but one cannot liberate a quark in that way from the confinement inside a hadron. Q20: Can we understand the source of the kinetic energy in beta decay of the neutron without invoking Eo = mc2? A20: No, we cannot. Because we cannot express the mass difference between a neutron and a proton in terms of binding energies as we did for nuclei. This is even more so for lepton masses.

34

7.7 Some basic questions Q21: Why does the velocity of light c enter the relation between the mass and the rest energy? A21: Because c is not only the velocity of light but also the maximal speed of propagation of any signal in Nature. As such, it enters all fundamental interactions in Nature as well as Lorentz transformations. Q22: Why do you claim that gravity is reducible to the interaction of energies, not masses? A22: Because a massless photon is attracted by the gravitational field of the Sun. (The deflection of light was first observed in 1919 and brought Einstein world fame.) As for the massive particle, its mass is equal to its rest energy. Thus in both cases we deal with energy. There is also another argument in favor of energy as a source of gravity. I refer here to the fact established by Galileo almost four centuries ago and cofirmed in the XXth century with accuracy 10- 12 . Namely, that all bodies have the same gravitational acceleration. It does not depend on their composition, on the proportions between different terms in their rest energy. That means that only the total rest energy of a slow body determines both its gravitational attraction and its inertia. Q23: How was this fact explained in the framework of prerelativistic physics and how is it explained by relativity theory? A23: In the prerelativistic physics it was formulated as a mysterious equality of inertial mass mi and gravitational mass mg. In relativity theory it became trivial, because both inertia and gravity of a body are proportional to its total energy. Q24: What are the main directons in the research on the concept of mass in the next decade? A24: The main experimental direction is the search for higgs at LHC at CERN. According to the Standard Model, this particle is responsible for the masses of leptons and quarks as well as of Wand Z bosons. Of great interest is also the experimental elucidation of the pattern of neutrino masses and mixings. The main cosmological direction is the study of dark matter and dark energy.

35

Q25: What was the formulation of the Corollary v in "The Principia"? A25: Here is the citation from "The Principia": Sir Isaac Newton. The Principia. Axioms, or Laws of Motion. COROLLARY V. The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion. For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line.

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Physics at Accelerators and Studies in SM and Beyond

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SEARCH FOR NEW PHYSICS AT LHC(CMS) N.V.Krasnikov a

Institute for Nuclear Research RAS, 117312 Moscow, Russia Abstract. We review the search for new physics to be performed at the Large Hadron Collider for Compact Muon Solenoid detector.

1

Introduction

The scientific programme at the LHC (Large Hadron Collider) which will be the biggest particle accelerator complex ever built in the world consists in many goals. Among them there are two supergoals: a. Higgs boson discovery, b. super symmetry discovery. LHC [1] will accelerate mainly two proton beams with the total energy VB = 14 TeV. At low luminosity stage (first two-three years of the operation) the luminosity is planned to be Ll ow = 1033 em- 2 8- 1 with total luminosity L t = 10 jb- 1 per year. At high luminosity stage the luminosity is planned to be L high = 1034 cm- 2 8- 1 with total luminosity L t = 100 jb- 1 per year. Also the LHC will accelerate heavy ions, for example, Pb-Pb ions at 1150 TeV in the centre of mass and luminosity up to 10 27 em -2 8- 1 . Bunches of protons will intersect at four points where detectors are placed. There are planned to be two big detectors at the LHC: the CMS (Compact Muon Solenoid) [2] and ATLAS (A Toroidal LHC Apparatus) [3]. Two other detectors are ALICE detector [4], to be used for the study of heavy ions, and LHC-B [5], the detector for the study of B-physics. The LHC will start to work in 2008 year. In this paper we briefly review the search for new physics to be performed at the LHC for CMS detector. To be precise we review the search for Higgs boson, the search for supersymmetry and the search for new physics beyond the SM and the MSSM. Presented results are based on the results of CMS full simulations [6]. 2

Search for standard Higgs boson at the LHC

Typical processes that can be exploited to produce Higgs bosons at the LHC are: gluon fusion: gg -+ H, WW, ZZ fusion: W+W-, ZZ -+ H, Higgs-strahlung off W, Z : qqW, Z -+ W, Z + H, Higgs bremsstrahlung off top: qq,gg -+ tt + H. a e-mail:

[email protected]

39

40

Gluon fusion plays a dominant role throughout the entire Higgs mass range of the Standard Model. The WW/ Z Z fusion process becomes increasingly important with Higgs boson mass rising. The last two reactions are important only for light Higgs masses. One of the most important reactions for the search for Higgs boson at LHC is (1) pp -+ (H -+ ')'')') + '" , which is the most promising one for the search for Higgs boson in the most interesting region 100 GeV :s; mH :s; 130 GeV. The main conclusion [6] is that CMS is able to discover Higgs boson with a mass [6]95 GeV < mH < 145 GeV at low luminosity Llow,t = 30 fb- 1 . The channel H -+ Z Z* -+ 4Z is the most promising one to observe Higgs boson in the mass range 130 Ge V -180 Ge V. Below 2Mz the event rate is small and the background reduction more difficult, as one of the Zs is off mass shell. In this mass region the width of the Higgs boson is small rH < 1 GeV, and the observed width is entirely determined by the instrumental mass resolution. For 180 GeV :s; mH :s; 800 GeV, this signature is considered to be the most reliable one for the Higgs boson discovery at LHC, since the expected signal rates are large and the background is small. The main background to the H -+ Z Z -+ 4Z± process is the irreducible Z Z production from qij -+ Z Z and gg -+ Z Z. The tt and Zbb backgrounds are small and reducible by a Z-mass cut. The use of this signature allows to detect the Higgs boson at 2: 5a level up to mH ~ 400 GeV at 10 fb- 1 [6]. The signature pp -+ H -+ WW* -+ Z+vI' - j) [6] provides the Higgs boson discovery for the Higgs boson mass region between 150 GeV and 180 GeV. Especially important is that the signature H -+ WW* -+ Z+ vI' - v allows to discover Higgs boson in the mass region around 170 Ge V where the branching ratio for H -+ 4Z is small and the use of four lepton signature for the Higgs boson discovery does not help at least for low luminosity. This signature does not require extraordinary detector performance and only requires a relatively low integrated luminosity of about 5 fb- 1 . The weak boson fusion channels qq -+ qqH lead to energetic jets in the forward and backward directions, and the absence of colour exchange in the hard process that allows to obtain a large reduction of backgrounds from tt, QCD jets, W- and Z-production and compensate the smallness of the Higgs weak boson fusion cross section compared to inclusive gg -+ H. The signature pp -+ (H -+ ')'')') + 2 forward jets was studied at full simulation level. The main conclusion is that the significance S = 5 is reached at the luminosity""" 100 fb- 1 for mH = 115 - 130 GeV [6]. Additional advantage of this signature is that the ratio of signal to background S/ B ,...., 1 in comparison with SIB ,...., 1/15 for inclusive pp -+ (H -+ ')'')') + ... reaction.

41

The H -t W( *) W -t in weak boson fusion mechanism with forward jet tagging provides the Higgs boson discovery for mH 180) GeV. H -t rr -t l + rjet in weak boson fusion mechanism with forward the boson discovery for m H GeV at The eMS standard boson discovery potential is in For the most boson mass 114.4 GeV :::; mH :::; 200 GeV the H -t "f"f and H -t -t a . [6]. Direct measurement of the sion in mass determination better than 8M boson width is possible only for mH ;::: 200 GeV where the natural width exceeds the mass resolution rv 1 GeV. Precision at the level is from H -t ZZ* -t [6J.

1

Figure 1:

discovery reach of the 8M

boson for integral Itlminosity

42

eMS

- - H-+yy cuts H-+yyopt

300 Figure 2: The del!endence of 50" discovery integral luminosity on the Higgs boson mass for eMS.

3 Supersymmetry search (SUSY) is a new type of symmetry that relates bosons and fermions [7]. Locally supersymmetric theories necessary incorporate [7]. SUSY is also an essential ingredient of superstring theories [7J. The interest in is due to the observation that measurements of the gauge constants at LEP1 are in favour of the Grand Unification in a "'11I~""''''rrnl1()",tri with superpartners of ordinary particles which are lighter than 0(1) TeV Besides supersymmetric electroweak models offer the solution of the gauge hierarchy problem [7]. In realUfe supersymmetry has to be broken and to solve the gauge hierarchy problem the masses of superparticles have to than 0(1) TeV. Supergravity provides natural of the be breaking [7], namely, an account of the ~l1T)prr:rr",,,i in hidden sector leads to soft supersymmetry breaking in observable sector. The simplest supersymmetric generalization of the SM is the Minimal SuStandard Model (MSSM) [7]. It is supersymmetric model based

43

on standard BUc (3) ® BUL(2) ® U(l) gauge group with electroweak symmespontaneously broken via vacuum expectation values of two different doublets. The MSSM consists of taking the 8M and adding the corresponding supersymmetric partners. It should be stressed that the MSSM contains two Y = ±1 Higgs doublets, which is the minimal structure for the sector of an anomaly-free supersymmetric extension of the SM.

Figure 3: eMS Discovery potential in mSUGRA model for tan fJ signature + jets + no leptons

35 and J-t =

+ for

the

all soft 8USY breaking terms are arbitrary that complicates the and spoils the prediction power of the theory. In mSUGRA model [7] the universality of different soft parameters at GUT scale is postulated. all spin 0 particle masses (squarks, sleptons, shiggses) are postulated to be equal to the universal value mo at GUT scale. All spin 1/2 particle masses (gaugino) are postulated to be equal to ml/2 at GUT scale and all the cubic and quadratic terms proportional to A and B are postulated to repeat the structure of the Yukawa superpotential. The main signature to be used for SUSY discovery at the LHC [7], [8} is (k ~ 0) leptons + (n ~ 2) jets + . This signature arises as a result of cascade decays of squarks and gluino into observable particles and LSP. The

44

main conclusion [6] is that for the mSUGRA model the LHC will be able to discover SUSY with squark or gluino masses up to 2.5 TeV for 100 The CMS mSGRA discovery potential is presented in

100G

10

600 7GO

jet+MET

1600

Figure 4: The eMS discovery potential at f1,

4

+ and

the signature

Search for new

1800

luminosiy 10 Ib- 1 lor tan,B + jets + n 2: 0 leptons

2000

35 and

beyond the SM and the MSSM

There are a lot of models different from the SM and the ",,..,.,,,,en·,, potential for search for physics beyond be found in Refs. [6], [8].

J.VJ.,-,jJJ.".l..

Conclusion

There are no doubts that at energy is the search cornerstone of the Standard boson and to check its

present the main goal of the eXI)erlmEm for the boson - the last non discovered Model. The LHC will be able to discover the basic properties. The expelmrlen boson

45

discovery will be the triumph of the idea of the renormalizability (in some sense it will be the "experimental proof" of the renormalizabilty of the electroweak interactions). The LHC will be able also to discover the low energy broken supersymmetry with the squark and gluino masses up to 2.5 TeV. Also there is nonzero probability to find something new beyond the SM or the MSSM (extra dimensions, Z' -bosons, W'-bosons, compositeness, ... ). At any rate after the LHC we will know the mechanism of the electroweak symmetry breaking(the Higgs boson or something more exotic?) and the basic elements of the matter structure at Te V scale. I thank colleagues from INR for useful discussions. This work was supported by RFFI grant N 07-02-00256. References [1 J The Large Hadron Collider, CERN / AC /95-05. [2J CMS, Technical Proposal, CERN/LHCC/94-38 LHCCPl, 15 december 1994. [3J ATLAS, Technical Proposal, CERN/LHCC/94-43 LHCCP2, 15 december 1994 [4J ALICE Technical Proposal, CERN/LHCC/95-71. [5J LHC-B, Technical Proposal, CERN/LHCC/95-XX. [6J CMS Physics Technical Design Report, Volume 2; CERN/LHCC 2006OIL [7J Reviews and original references can be found in: R.Barbieri, Riv.Nuovo Cim. 11, 1 (1988); A.B.Lahanus and D.V.Nanopoulos, Phys.Rep. 145, 1 (1987); H.E.Haber and G.L.Lane, Phys.Rep. 117,75 (1985); H.P.Nilles, Phys.Rep. 110,1(1984). [8J See, for example: N.V.Krasnikov and V.A.Matveev, UFN, v.174, 697 (2004).

MEASURING THE HIGGS BOSON(S) AT ATLAS C.Kourkoumelis a

University of Athens, Physics Dept, Panepistimioupoli, Ilissia 15171, Greece Abstmct. The ATLAS detector discovery potential and sensitivity for the SM and the MSSM Higgs bosons is reviewed. Emphasis is given to the decays into muons and to the expected discovery potentials from the first years of running.

INTRODUCTION The ATLAS detector [1] will operate at the Hadron Collider at CERN where protons will collide at energies reaching Vs 14 TeV. The detector has been designed to detect leptons, photons and jets with momenta from a few GeV to several TeV, within a pseudorapidity range of 1171 < 2.7 for leptOIls and and 1171 < 5 for jets. The detector is huge weight 7,000 tons); its very complicated installation is 1). In parallel, all the magnets of the LHC machine been installed and the first beams are expected in 2008.

Figure 1: View of the ATLAS Spectrometer in the LHC Point-l under installation,. before the insertion of the End-Cap Toroidal Magnet.

The search for the Boson(s) is a major goal of the LHC program. Extensive of the physics potential and sensitivity of the detector have been done for years, using simulated data [2]. Some of those studies are with the expectations from the first data. The current limits on the mass come from the following sources: direct search from the LEP2 [3] (mH > 114.4 GeV at 95% indirect from the precision data and the LEP working group 144 GeV if one includes the above direct limit but not including the new measurement in the conferences of last summer) and the latest Tevatron limits in [51. The theoretical upper limit for the is derived from unitarity arguments . vVF,VV.
.. e-mail: [email protected]

46

47

THE FIRST PHYSICS DATA run of LHe at the design centre-of-mass energy will be at very which is expected to soon increase to 1032 cm- 2 sec- 1 . months the emphasis will be in the study of minimum bias and the measurement of the top mass. These classes of events will be studied both as signal and as background to the new nl'>'"",,,,o channels. Table 1 shows the number of different kinds of events eXl)ected accumulated the end of 2008 with integrated luminosity of about 2 shows the number of registered events after all cuts as a function luminosity. Of course the real emphasis will be in these events to understand and calibrate the detector. A full detailed discussion of all the topics is given elsewhere [2] and cannot be covered in this decay discovery channels will be short contribution. Here only a few mentioned. nn'
11ll,111j,!U~jHV

Figure 2: The number of W and Z and events expected in the ATLAS detector for the first runs of the LHC as a function of integrated luminosity.

Table I: For the physics channels listed in the first column, the expected number of events per second and after 0.1 jb- 1 of integrated luminosity.

Process

Events per second

Recorded events per 0.1

48

3

THE STANDARD MODEL HIGGS SEARCHES

In [6] the main diagrams responsible for the Higgs production at the LHC are described. The gluon-gluon fusion via the top quark loop is dominant for low masses; the Vector Boson Fusion (VBF) is only one order of magnitude less but has a characteristic signature as will be described later [7]. Most cross-sections have been now calculated at the NNLO level, except the VBF which is at the NLO. The corresponding uncertainties are: 10-20% for the gluon-gluon, 5% for the VBF, less than 5% for the associated production with W, Z bosons and 10-20% for association production with a top quark pair. In the same reference, the branching ratios of the main decay modes of the Higgs as a function of the Higgs mass are presented. The uncertainties on the branching ratios are of the order of few % given by NLO calculations. In the following discussion of specific decay modes, the mass region for the Higgs search can be divided in three different regions: 3.1

Low mass region (115 GeV<

mH

<130 GeV)

This low mass region is the most difficult one. The natural width of the Higgs is of the order of few MeV, so one can look for a narrow resonance since the width is purely determined by the experimental resolution. The dominant decay channel is the H -> bb , but has a signal/background ratio of 10- 5 , which yields the observation impossible. The H -> 'Y'Y decay channel is rare, has a much smaller cross-section but has a better signal/background ratio. It is the "benchmark" channel for low masses. The signal/background ratio is about 10- 2 and needs a very high ( 103 @80% efficiency) experimental rejection against jets, which is provided by the ATLAS electromagnetic calorimeters. The main background in this channel is the irreducible di-photon background. Recently its cross-section has been computed at the NLO level and the discovery potential significance has increased by 1.5 (see Figure 3).The expected significance is 6.1 for integrated luminosity of 30 fb- 1 of data, for a Higgs mass of 120 GeV. Furthermore, the use of discriminating variables based on the event kinematics has been investigated and found to be promising [8] for the improvement of the significance. Besides, the direction of the photon pair can be taken from the primary vertex and its z-position can be more accurately computed if one includes the charged track information for the direction reconstruction. An accuracy of 40 f.Lm can be achieved at the low 1033 cm- 2 sec- 1 luminosity corresponding to a mass resolution of 1.4 GeV in the interval of 120-130 GeV. On the other hand, the VBF channels can be used to be combined as a discovery channel. It has the feature of having two quark-quark high PT jets in the forward-backward region. Therefore by tagging on these jets and vetoing on any jets in the central region one can achieve a big reduction of the background such as the tt . In this low mass region the most efficient search is done using

49

Figure 3: The H ...... TY discovery significance as function of mH.

as well as the associated can be added to increase >:>IF;,lUl.ll,.,CbUI..,C for

signal and backgrounds for the 8M Higgs though + ttVV for mH =120 GeV and integrated luminosity 30

50

3.2 Intermediate mass region (130 GeV< mH < 2mz) The most promising observation channel is the H -t Z Z* 4l. The main ba'C:K~[rOUna is the irreducible continuum which is known to NLO. The and tt backgrounds can be strongly reduced with isolation cuts in the calorimeter and by vertex constraints. For the region around 160 GeV the H has a very large, close to 100%, branching ratio. no mass reconstruction is possible because of the neutrinos and one has to on number counting and very precise knowledge of the The main backgrounds are the tt which can be with a mass and the WW continuum. Recently the gluon-gluon-t WW contribution has been and the single top production at the NLO leveL The VBF -t WW can also be added to obtain extra In VBF channels qqH -t qqT7 can be combined with the above two as contributes to about 3 a well. Each individual channel for a 130 GeV 1 ""ISHUIlCa,HC'''. All together give a 6 a significance at 10 fb- at 130 GeV.

Figure 5: The signal and the backgrounds for the H-+WW decay as a function of the transverse mass.

3.3

mass region (mH > 2mz)

The best discovery channel is the "golden H -t -t 41 chanwhere the background is very small due to the constraint of the two reconstructed masses to be both compatible with the Z-mass. For the very masses (mH >700GeV) the decays H -t WW -t ZZ lllllJ, H -t ZZ -t lljj, due to their will enhance the discovery sensitivity and compensate for the lower T"','V11""''' cross-sections. 6 shows the overall sensitivity covering the full mass range. lnt',PU,."j,,,,rl luminosity corresponding to one year of low data takline) almost the full range is covered, provided that the detector

51

is optimal and well understood, Figure 6 summarizes the discovery potential for ATLAS as was calculated already in 2003 based on LO cross-sections. The update of this plot is expected soon.

Figure 6: The signal significance for a SM Higgs boson in ATLAS as a function of the Higgs mass for two different integrated luminosities.

4

SUPERSYMMETRIC HIGGS SEARCHES

In most common supersymmetric scenarios, the Minimal tiupel:Symrnet;nc tension of the SM (MSSM), five bosons are odd), . The discovery limits are commonly in terms of uc;"uuU5 a where contours in mA and tan/3 are drawn. The LEP2 results have excluded a of the plane (the low tan/3 region). In addition to the SM channels described above, the most DH)mlSlIlIl: rf'"'NY""""" V,uo,.,U",.", in the tan/3 region, are the J-LJ-L channels and the charged decays. The - t 'T'T a rate compared to the J-LJ-L one, but it is more reconstructed. On the other hand, the J-LJ-L one produces a very clean O';<,H"'vUL a clear over the background in the di-muon mass distribution. 7 shows the parametric space which can be covered for one year of data ax at low luminosity in the scenario. full of the parameter space about ten times more The most difficult region is the low tan/3, < 10 and low mA, 120< mA <220 GeV. In the intermediate tan/3 one the h, is detectable.

Mr

52 d u e ' >'c *v cur yes

Figure 7: The MSSM Higgs bosons five a cr curves for the ATLAS experiment for an integrated luminosity equal to 10 fb~1.

5

MEASUREMENT OF THE MASS, WIDTH AND COUPLINGS OF THE HIGGS BOSONS

The mass of the SM Higgs boson can be measured with a high precision in the decay channels -- +• 77 and H —> 41 by a "complete" reconstruction of the mass peak. The precision is estimated to be excellent, of the order of 0.1% for an integrated luminosity of 300 fb-1 as shown in Figure 8, for masses below 400 GeV (assuming the scale uncertainty for leptons and photons to be 0.1%). For larger Higgs boson masses the precision deteriorates and becomes about 1% at 600 GeV. If the above SM channels are suppressed then the mass has to be determined from the other "indirect" channels with a a of 10 worse precision. For mH less than about 200 GeV, the natural width of the Higgs boson is dominated by the experimental resolution and hence cannot be measured directly. Above 270 GeV the expected precision of a measurement of the SM Higgs boson width, having an integrated luminosity of 300 ft}"1, is expected to be better than 10%. Since the total width of the boson below 200 GeV is not known it is not possible to extract the partial widths and coupling constants from the obserYed event rates in a CUJ:n~lC~ClY model-independent way. Under a few reasonable assumptions, the ratios of partial widths axe measurable with a few 10% accuracy [9]. Besides, the measurement of the couplings can be used for exclusion of several MSSM scenarios [10]. If the Higgs boson has a spin=l then the if —* 77 and the giuon-gluon fusion to Higgs are excluded [11]. The spin and OP'quantum

53 o

E

:; Jii 10-1

H. WHo ttH (H

-+ m

-+ bb) -+ zi" -+ 4t

A WHo itH (H

."1._1. [J H ~All

"".MoII

(_I......n 10 0.02 '"

10

10

10 10

Figure 8: Expected precision of the measurement of the Higgs boson mass for an integrated luminosity of 300 jb- 1 .

numbers of the Higgs boson can be studied in the decay channel H by studying the angular distributions of the decay leptons [12].

6

--t

ZZ

--t

41

CONCLUSIONS

The ATLAS detector has a broad physics potential. During the initial physics run for integrated luminosities as low as 1fb- 1 the Higgs boson -provided that it does not have very low mass- can be discovered. In addition, the supersymmetric particles and/or new Z' resonances -if they exist- will most likely be detected, provided that the detector is calibrated and understood and the backgrounds are well understood and measured, mainly via control samples. And of course the detector is open to surprises.

ACKNOWLEDGMENTS I would like to thank the organizers for their excellent hospitality. The project is co-funded by the European Social Fund and National Resources (EPEAEK II) PYTHAGORAS II. [1] ATLAS collaboration, "Technical Proposal", CERN/LHCC/94-43, 1994. [2] ATLAS collaboration, "Detector and Physics Performance, Technical Design Report", CERN/LHCC/99-14, 1999. [3] R. Barate, et al., "Search for the standard model Higgs boson at LEP", Phys. Lett. B 565, 2003, 61-75 [4] LEP Electroweak Working Group, http://lepewwg. web.cern.ch/LEPEWWG /

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[5] "Lepton Photon Conference", Korea, Aug. 2007, http://chep.knu.ac.kr/LP07 / [6] A. Djouadi et aI, Phys. Commun.108 (1998) 56 and hep-ph/0503172 [7] S. Asai et aI, J. Phys. G 28, 2002, 2453-2474 and Eur. Phys. J. C32, 2004,19. [8] M. Bettinelli et aI, "Search for a SM Higgs decaying to two photons with the ATLAS detector",ATL-PHY-PUB-2007-013 [9] C. Ruwiedel et al., Eur. Phys. J. C 51, 2007, 385-414 [10] M. Duhrssen, Phys. Rev. D 70, 2004, 113009 [11] C. N. Yang, Phys. Rev. 77, 1950, 242-245 [12] C. P. Buszello et al., Eur. Phys. J. C 32, 2004, 209-219

BEYOND THE STANDARD MODEL PHYSICS REACH OF THE ATLAS EXPERIMENT G.Unel a Physics Division, CERN, Geneva 23, Switzerland; and Department of Physics, University of California at Irvine, Irvine, USA. Abstract. A brief summary of the current unsolved probems of the Standard Model of particle physics is presented. The installation and commissioning status of the ATLAS experiment on the LHC accelerator is given. A phenomenological approach is followed to show the physics reach of the ATLAS experiment. Theories beyond the Standard Model are classified according to the fundamental structure they modify: Matter fields, gauge group structure, symmetry breaking mechanism and space-time.

1

The Standard Model and the ATLAS experiment

The Standard Model (SM) of particle physics is based on four pillars. The first two are the fermion fields which define the matter and the gauge group structure which describes the fermion interactions. The interactions are carried out by mediator particles, the gauge bosons. The chiral model of electroweak and strong interactions is described by massless fields. The third pillar is the mechanism with which these fields acquire mass, the Higgs mechanism. Finally, the interactions happen and the calculations are performed using a 3+ 1 dimensional space-time structure. Apart from the so far missing Higgs boson, the SM has survived all the tests that the current precision measurements could offer. The compatibility between theory and experiment is yet unshaken. The SM can even be trivially amended to accommodate the recently established existence of the neutrino masses. However, there are a number of reasons hinting that the SM can not be the final theory of fundamental particles and their interactions. The fine tuning problem in the Higgs mass, the non-unification of electro-weak and strong forces, the arbitrariness of the fermion masses and mixings, the origin of the number of generations, and the source of baryo-genesis are such hints. The ATLAS experiment [1] is being setup to investigate possible solutions to the above mentioned problems of the SM . It will take data at the LHC accelerator using 7+7 Te V proton beams. The accelerator is expected to accumulate about 1 fb- 1 integrated luminosity by the end of 2009 and then to gear up to a low luminosity regime at which 10 fb- 1 per year will be collected. The experimental setup is currently being commissioned with technical and cosmic runs [2]. ae-mail: [email protected]

55

56

2

Beyond the Standard Model

The theories proposed to solve the problems of the SM change its fundamental structure. These changes will be searched for with the ATLAS detector to validate or refute the theories Beyond the SM (BSM). This note classifies the BSM theories according to the modifications they bring to the SM. It will subsequently give the reach of ATLAS for some BSM theories as a function of the energy scale, the model parameters at which the new physics emerge and the integrated luminosity collected by the experiment. 2.1

Searches for new Matter

A possible change to the SM is the modification of its matter content. This can be achieved either by invoking a sub-structure to the elementary particles, or by adding new elementary fermions to the known ones. The following resumes these two options. New Constituents It is possible to formulate a model in which known elementary particles are composed of more fundamental constituents, for example composite or preonic models. Such models predict excited fermions both charged and neutral [3]. The reach of ATLAS for excited electrons and quarks have been studied previously [1]. A more recent work (see [4] and the references therein) concentrates on the excited neutrinos, v* which can be produced by Drell-Yan processes and decay to a boson and a lepton. The Monte Carlo study scanned the mass of v* in the range (0.5 , 2.5) TeV and investigated the possibility to fully reconstruct the v* invariant mass from its decay products (v + Z or e + W). The bosons were reconstructed in all decay modes except the tau decays. The SM backgrounds originate from the decays of pp -+ W Z and pp -+ WWW. Fig. 1 shows the experimental reach of this signature for 300 fb- 1 as a function of the v* mass and energy scale at which the compositeness appears (A). The higher (lower) curve is for 3 (5) sigma statistical significance. New Quarks Additional quarks are predicted by various models. The three most popular cases are down type iso-singlets, up type iso-singlets and iso-doublets. The fourth family iso-doublet quarks [5], suggested by the flavour democracy arguments, have been studied almost 10 years ago [1]. The down type iso-singlets (D) are proposed by E6 group inspired models [6]. The pair production, mostly independent of the extended CKM matrix elements, and decay of the D quarks were studied recently [7] using the ATLAS fast simulation software. The mass

57

o

025

05

0.75

I

1.25

15

1.75

2

2.25

Moss, TeV

Figure 1: ATLAS reach for v* using 300 fb- 1 integrated luminosity

of the new quark, which is assumed to mix with the first (or second) 8M family, was scanned. The considered decay (DD ---t ZZdd ---t 4£2j) where £ = e, Jl has a low event yield but a clean signature allowing the full reconstruction of the new quark's invariant mass. The left plot of Fig. 2 shows an example reconstructed D quark invariant mass peak compared to the background events originating from all 8M ZZjj processes. The right plot of Fig. 2 shows the 3 sigma and 5 sigma curves as a function of the new quark mass and integrated luminosity.

. .. - Signal

@

..••. 30" signal 5cr discovery

800Ge V

-

SM background

-

SM + Signal

@

800Ge V

10

400

600

JOOO

M"J~ (GoV)

1100

1200

MD (GeV)

Figure 2: Search for down type iso-singlet quarks. Left: The invariant mass reconstruction for mv = 800 GeV, Right: The discovery reach as a function of the quark mass; signal and discovery curves are shown.

The up type iso-singlet quarks, T, are predicted by the Little Higgs models [8]. The jet associated single T quark production from W exchange were considered

58

assuming maximal mixing to the third 8M family [9]. The same work studied all three possible decay modes: T -> Wb, Ht, Zt . As seen from Fig. 3, the Zt channel has small background but also a small event yield as opposed to the Wb channel which has an observability potential up to 2.5 TeV in 300 fb-1. The same integrated luminosity allows a 5 sigma discovery up to 1.4 Te V reach using the Z t channel. 4 ---

'" 0 0

~

>

'"

t!l

3.5-

ATLAS

3

0

~

t1 ~

UJ

2.5 2" 1.5

0.5 2000 Invariant Mass (GeV)

2000 Invariant Mass (GeV)

Figure 3: Invariant mass reconstruction for an up type iso-singlet quark of mass 1 TeV using 300 fb- 1 integrated luminosity, Wb on the left and Zt on the right side.

New Leptons

New leptons, L, appear in various models [5,6,10]. The work in [11] concentrates on the lepton pairs produced from quark annihilation and from gluon fusion to a quark triangular loop. In both cases, the s channel contains the Z boson and a possible Z' as propagator. The first one can also propagate with a "y. The considered decay mode is L -> Zp, / Ze. The search was performed as a function of the new lepton and new heavy neutral gauge boson (Z') mass. The experimental reach is given in Fig. 4 for a Z' mass of 700 GeV. The lower (upper) curve is the reach for 10 (100) fb- 1 of integrated luminosity. A Z' of 2 Te V would increase the 5 cr reach from 800 Ge V to 1 Te V for 100 fb -1. Leptoquarks

Leptoquarks (LQ) are predicted by GUT and composite models. The study in [12] considered their pair production from gluon fusion and quark annihilation. The same work also covers the single production. The decay modes consist of electrons (type-I) or neutrinos (type-2) and a light jet. For both scalar and vector LQs, the mass scan was performed for different coupling coefficients.

59

'I8

7

200

eoo

6OC:

4(}O

lDC{1

1200

1400

M,

Figure 4: L reach as a function of its mass, low (high) luminosity corresponds to 10 (100) fb- 1 of integrated luminosity.

Fig. 5 summarizes the reach for 300 fb- 1 , showing that about 1.2 (1.5) TeV LQ can be discovered for scalar (vector) leptoquark models. til

105'~

~

_________

~

ATLAS

fLdt=3x10'W'

10 VlQType 1 _SLQTypel VLQ Type 2 SLQ Type 2

10

10

400

600

800

1000

1200

1400

1600

1800

Leptoquark mass, GeV

Figure 5: LQ reach as a function of its mass for 300 fb- 1 of integrated luminosity.

2.2

Searches for new gauge group structure

Embedding the 8M gauge group into a larger one brings additional gauge bosons, both neutral (Z/) and charged (WI). Additionally they appear in models with extra-dimensions (ED) as the Kaluza-Klein (KK) [13] excitations of their 8M counterparts.

60

Neutral gauge Hosons

A full GEANT MC simulation study was performed to investigate the Zl discovery potential of ATLAS using a generic parameterization called CDDT [14]. The CDDT parameterization classifies Zl searches into four distinct cases, depending on its coupling to the known fermions. In this study, a 1.5 and 4 TeV Zl produced by quark anti-quark annihilation was allowed to decay into e+epairs. The left side of Fig. 6 shows the ATLAS reach for 100 fb- 1 of integrated luminosity as a function of the ratio of the new gauge boson and its gauge coupling strength (Mz/g z ) and fermion coupling modification parameter (x) . A recent study investigated the discovery reach of the KK excitations of the Z boson, zn [15]. This model uses different parameterizations (A,B,C) to reproduce the known fermion masses and mixings. The right side of Fig. 6 shows the reconstruction of the zn invariant mass from e+e- pairs using a full GEANT based simulation together with the SM Drell-Yan background. The zn discovery reach for 100 fb- 1 integrated luminosity is up to 6 TeV, depending on the parameterization.

Figure 6: Left: generic Zl search with CDDT parameterization; Right: Zl invariant mass reconstruction for different fermion parameterizations. In both cases results with 100 fb- 1 are shown.

Charged gauge Hosons

Additional charged gauge bosons, W', appear in GUT, Little Higgs and ED models [6,8,18]. The quark anti-quark annihilation produces the W' that can be studied via its hadronic [16] or leptonic [17] decays. The important parameters are the W - W' mixing angle (cot B) and the mass of the W'. Fig. 7 shows the discovery reach for the WH search (Little Higgs model) , for 300 fb- 1 integrated luminosity in the cotB - mWH plane for the WH - t tb and W H - t ev modes.

Figure 7: WI discovery reach plane. The shaded area is from hadronic decay channel, dashed line is from electron decay.

Searches

new Electro-weak symmetry

mechanisms

variants of

Scalars

o

proposed for fermion and boson mass Cf""~"T·"i'·I"Yl The studies in the context F."'Lv"", and scalars can predicted by both Little

,"U(U

additional vector bosons.

'~l~"'r"" between the matter and force carrier solve the and fine tuning problems a candidate for the Dark Matter (DM) searches of

62

14()O

Figure 8: tl,±± search reach for single production on the left and for pair production on the right.

are expected to cascade decay down to the particle (LSP), the n jets + m leptons + channels are inv'es1;igl'1ted. The large number of free parameters can be reduced to 5 in case of [23] which proposes its LSP, the lightest neutralino, as the DM candidate. the reduced parameter set should also be consistent 9, left). A recent work has investigated the reach of with WMAP data pair production in the focus point scenario and ATLAS for and [24J. The result of the study is shown in background subtraction, for 10 fb- 1 of integrated significance.

""'''''T''.;"nrnn,,,t-,'l,,

Figure 9: Left: parameter space in mSUGRA. The medium gray region is consistent with WMAP data. Right: reconstructed 9 visible invariant mass for 1 fb- 1 of data.

63

2.4

Searches for new Dimensions

If the relative weakness of the gravitational force is attributed to the existence of extra dimensions (ED), the graviton becomes the object to search for. The graviton couples to all particles and can escape undetected. The most promising channels are gluon-gluon, quark-gluon fusion and quark anti-quark annihilation yielding one jet + missing E T . The experimental reach depends on the number of EDs and also the fundamental gravity scale. A study has shown that, for 100 fb- 1 of integrated luminosity, the reach would be about 9, 7 and 6 TeV, for 2, 3 and 4 additional dimensions [25]. Large (Te V- 1 ) EDs appearing in ADD models [26] predict KK excitations of gluons, g*, which would decay into heavy quark anti-quark pairs. A study evaluated the reach of ATLAS for the decay into bb and tf pairs [27]. Depending on the mass of the g*, it is possible to discover a g* with mass up to 3.3 TeV with an integrated luminosity of 300 fb- 1 • 3

Results and Conclusions

Although this note summarized only a selection of discovery possibilities, it has shown that ATLAS has a very rich discovery potential for physics beyond the SM. The differentiation between models and the possible boost to SM process cross sections from the particles proposed by the BSM physics were also not discussed. The preparation of the experimental apparatus for data taking is well underway, the new analyses with full simulation are also ongoing. These studies will immediately be applicable to first data from LHC. Acknowledgments

The author would like to thank A. Studenikin for his hospitality in Moscow and F. Ledroit and A. Parker for useful discussions. G.U.'s work is supported in part by U.S. Department of Energy Grant DE FG0291ER40679. References

[1] ATLAS Detector and Physics Performance Technical Design Report. CERN/LHCC/99-14/15. [2] LRiu, ATL-SLIDE-2007-05, proceedings of 15th IEEE Real Time Conference (2007). [3] E.J.Eichten, KD.Lane and M.E.Peskin Phys. Rev. Lett. 50, 811 (1983); L.Abbot, E.Farhi Phys.Lett., B 101,69 (1981). [4] A.Belyaev, C.Leroy, RMehdiyev, Eur.Phys.J. C 41, 1 (2005). [5] B.Holdom, JHEP 0608, 076 (2006); B.Holdom, JHEP 0703,063 (2007).

64

[6] F.Gursey, P.Ramond and P.Sikivie, Phys.Lett. B 60, 177 (1976); F.Gursey and M.Serdaroglu, Lett. Nuovo Cimento 21, 28 (1978). [7] RMehdiyev et al., Euro.Phys.J. C. 49, 613 (2007). [8] M.Schmaltz, Nucl.Phys. B 117, 40 (2003). [9] G.Azuelos et al., Euro.Phys.J. C. 39, Suppl.2, 13 (2005). [10] S.Dimopoulos Nucl.Phys. B 168, 69 (1981); E.Farhi, L.Susskind, Phys.Rev. D 20, 3404 (1979); J.Ellis et al., Nucl.Phys. B 182, 529 (1981). [11] C.Alexa, S.Dita, ATL-PHYS-2003-014 (2003). [12] A.Belyaevet al., J.H.E.P. 09, 005 (2005). [13] E.Witten, it Nucl.Phys. B186, 412 (1981). [14] F.Ledroit, B.'frocme, ATL-PHYS-PUB-2006-024, proceedings of TeV 4 LHC workshop, (2006). [15] F.Ledroit, G.Moreau, J.Morel, J.H.E.P. 09,071 (2007). [16] Gonzlez de la Hoz, S; March, L; Ros, E; ATL-PHYS-PUB-2006-003. [17] G.Azuelos et al., Eur.Phys.J. C 39,13 (2005). [18] L.Randall, RSundrum, Phys. Rev. Lett. 83, 3370 (1999). [19] LF.Ginzburg, M.Krawczyk, Phys.Rev. D72, 115013 (2005). [20] G.Azuelos, K.Benslama, J.Ferland, J.Phys. G 32, 73 (2006). [21] Y.Hosotani, Phys.Lett. B 126, 309 (1983) ; B.McInnes, J. Math. Phys. 31, 2094 (1990). [22] H.P.Nilles, Phys. Rev. 110, 1 (1984) and references therein. [23] A.H.Chamseddine, RArnowitt and P.Nath, Phys. Rev. Lett. 49 (1982) 970; H.P.Nilles, Phys.Rept. 110 (1984). [24] U.De Sanctis, T.Lari, S.Montesano, C.'froncon, arXiv:0704.2515, SNATLAS-2007-062, Eur.Phys.J. C52, 743 (2007). [25] L.Vacavant, LHinchliffe, J.Phys. G 27, 1839 (2001). [26] N.Arkani-Hamed, S.Dimopoulos, G.Dvali, it Phys.Lett. B 429, 96 (1998). [27] L.March, E.Ros, B.Salvachua, ATL-PHYS-PUB-2006-002 (2006).

THE STATUS OF THE INTERNATIONAL LINEAR COLLIDER B. Foster" Department of Physics, University of Oxford Denys Wilkinson Building, Keble Road, Oxford, OXl 3RH, UK Abstract. The status and prospects for the International Linear Collider are summarised.

The International Linear Collider (ILC) is accepted by the international community of particle physicists as the next major project in the field. The Global Design Effort (GDE), directed by Professor Barry Barish, has been charged by the International Committee for Future Accelerators with the task of preparing all necessary design and documentation to present a fully cos ted and robust design to the funding authorities by 2010, at which point the status of the Large Hadron Collider and other relevant information will be available to allow an informed decision on construction. The current status of the project is that the Baseline Design of January 2006 has been used as the basis for a greatly developed and refined reference design, summarised in the Reference Design Report [1], which in additional to descriptions of the major parts of the project, also gives a costing in ILC units ( = US$ 1 on Jan. 1st 2007) together with an estimate of the labour necessary to realised the project. The RDR estimates are 6.62 Billion ILCUs plus 14,100 person-years of effort. The Reference Design is based around a central campus containing a single interaction hall capable of hosting two detectors in a "push-pull" configuration, a damping ring complex for both electrons and positrons, the electron source and other central services. The damped electrons from the source are transported to the end of the superconducting modules and accelerated to an energy of 250 Ge V; part of the way along their path they are diverted through a wiggler which produces hard photons which are converted to electron-positron pairs. The positrons are selected and transported to the damping ring, damped and then transported to the other end of the machine to be accelerated in their turn and then collided with electrons at the interaction point. The RDR assumes that an accelerating gradient of 31.5 MV1m can be attained by each super conducting accelerating module. Many technical developments and specifications remain to be completed. The goal of 31.5 MV1m, although reached in some cavities, has not yet been achieved with the reproducibility and the yield necessary for the industrial production for the ILC. There are many remaining technical questions in areas such as the damping rings and a whole process of value engineering, optimisation and cost containment and reduction is required. This will be carried out ae-mail: [email protected]

65

66

in a third phase of the project, under the supervision of the GDE, known as the Engineering Design Report. This will be supervised by a Project management Team currently being set up and will begin in the autumn of 2007. At the same time as the technical and engineering developments for the ILC are progressing, it is important to develop political institutions and to explain the importance of the physics of the ILC both to politicians, other scientists and to the general public. The mechanism by which the site for the ILC will be bid for and chosen needs to be investigated and defined. The GDE welcomes the strong interest recently evinced by JINR Dubna in proposing Dubna as a possible site for the ILC and the interest expressed by the Russian Federation in exploring this possibility. It is also necessary to propose models and reach agreement on how a fully international project such as the ILC can be managed and governed to ensure accountability and transparency for stake-holders. The accomplishment of the technical aims of the EDR phase resulting in a proposal to construct the ILC in 2010 could lead, if prompt approval were granted, to ground breaking in 2012 and operation by 2019, allowing a substantial period of operation overlapping that of the LHC. The GDE is committed to maintaining such a timeline, defined as it is by the available effort and likely technical progress, while in parallel assisting the resolution of political questions and preparing an atmosphere conducive to approval of the ILC project. Reference

[1] G. Aarons et al., International Linear Collider Reference Design Report, available from http://www.linearcollider.org/cms/?pid=1000025, August, 2007.

REVIEW OF RESULTS OF THE ELECTRON-PROTON COLLIDER HERA

v. Chekelian (ShekeJyan) a Max Planck Institute for Physics, Foehringer Ring 6, 80805 Munich, Germany Abstract. A review of results of the electron-proton collider HERA is presented with emphasis on the structure of the proton and its interpretation in terms of

QeD.

1

Introduction

In summer 2007, after 15 years of successful operation, the first and only electron-proton collider HERA has finished data taking. The HERA collider project started in 1985 and produced the first ep collisions in 1992. It was designed to collide electrons with an energy of 27.5 GeV with protons with an energy of 920 GeV (820 GeV until 1997). This corresponds to a center of mass energy of 320 GeV. The maximum negative four-momentum-transfer squared from the lepton to the proton, Q2, accessible with this machine is as high as 100000 GeV2 . Two ep interaction regions were instrumented with the multi-purpose detectors of the HI and ZEUS collider experiments. In 2002, placing strong super-conducting focusing magnets close to the interaction points inside the HI and ZEUS detectors, the specific luminosity provided by the collider was significantly increased. At the same time spin rotators were installed in the HI and ZEUS detector areas, and since then a longitudinal polarisation of the lepton beams of 30-40% was routinely achieved. Over the 15 years of data taking, each collider experiment collected an integrated luminosity of :::::: 0.5jb- 1 , about equally shared between positively and negatively polarised electron and positron beams. The HERA physics program covers a broad spectrum of topics such as searches of new physics, hadron structure, diffractive processes, heavy flavour and jet production, vector meson production and many others. In this paper I will concentrate on results related to neutral current (NC) and charged current (CC) deep-inelastic scattering (DIS) and the QCD aspects of these measurements at HERA. 2

Deep-inelastic NC and CC ep scattering

The deep-inelastic NC scattering cross section can be written as (1) ae-mail: [email protected]

67

68 HERA II H1 Quark Radius Limit HERA 1+11 (417 pb")

"*

H1 e+pNC03..(l4(prel.)

t.

H1 e'p NC 2005 (pre •. )

g ...

o zeus e+p NC 2004 o

1.4

. ___ .~ ___• __ .. ___ .i

"0 -C

..... 8M e+p NC (CTEQ6M)

_

~

~ 1.2

ZEUS e'p NC 04-05 (prel.

~

0.8

~ ~ ~ ~ ~~;~~::t~oa~n~ncertainty _

0_6

'*

H1 e+p CC 03-04 (pre'.)

....

H1 e'p CC 2005 (prel.)

•

ZEUS e+p CC 2004

•

ZEUS e'p CC O4-OS (pre •. )

Rq =O.74'lO,'B m (95%CL)

f-~~~~~-~~~~~-~;-------I

1_4

1_' -I

.• _-- 8M e+p CC (CTEQ6M) -

..,~_

8M e-p NC (CTEC6M)

8M e'p CC (CTEQ6M)

L

0_'

y < 0.9 Pe=O

10-7b...L~J...i~~10L3-~~---'-'-LLLl1o'~~----'--.d

Q'(GeV')

Q'(GeV')

Figure 1: The Q2 dependence of the NC and CC cross sections du/dQ2 for e±p scattering (left). The NC cross section normalised to the Standard Model expectation (right).

where a*,-c is the cross section in a reduced form, a is the fine structure constant, x is the Bjorken scaling variable, and y characterises the inelasticity of the interaction. The helicity dependence is contained in Y± = 1 ± (1 _ y2). The generalised proton structure functions, F2 ,3, occurring in eq.(l) may be written as linear combinations of the hadronic structure functions F 2 , Fi and F2~3' containing information on the QeD parton dynamics and the el~c­ troweak couplings of the quarks to the neutral vector bosons. The function H is associated with pure photon exchange term, F~f correspond to photon-Z interference, and F.f,3 correspond to pure Z exchange terms. The longitudinal structure function FL may be decomposed in a similar way. The generalised proton structure functions depend on the charge of the lepton beam, on the lepton beam polarisation, defined as P = (NR - N L) / (NR + N L), where N R (Nd is the number ofright (left) handed leptons in the beam, and on the electroweak parameters Mz and sin 2 () (or Mw):

f,

+ k( -Ve =t= Pae)Fi z + k2(V; + a; ± 2Pvea e )Fl, k( -a e =t= Pve)xF;z + k2(2veae ± P(v; + a;»xFl.

F2± = F2

(2)

XF3± =

(3)

Here, k( Q2)

=

4 sin2

~ cos2 (1 Q2~~2 determines the relative amount of z

Z to I

exchange, Ve = -1/2 + 2 sin () and a e = -1/2 are the vector and axial-vector couplings of the electron to the Z boson, and () is the electroweak mixing angle. At leading order in QeD the hadronic structure functions are related to linear combinations of sums and differences of the quark and anti-quark momentum 2

69 HERA Charged Current

*

* Hl e+p94-00

Hl e'p

a ZEUSe-p98-99

[J

ZEUSe+p99-00

-

SM e-p (CTE06D)

-

SM e+p (CTEQ6D)

Figure 2: The NC (left) and CC (right) double differential cross sections d 2uldxdQ2 in a reduced form for e+p and e-p DIS scattering.

distributions xq(x, Q2) and xij(x, Q2) of the proton: z (F2 , Fi , F2Z) = x ~)e~, 2eqvq, v; (xF:;z,xF3Z)

+ a~)(q + ij),

= 2x ~)eqaq,vqaq)(q -

(4)

ij),

(5)

where Vq and a q are the vector and axial-vector couplings of the light quarks to the Z boson, and e q is the charge of the quark of flavour q. The deep-inelastic CC cross section can be expressed as d2a6c 27l'x dxdQ2 G}

(jac

[MrvMrv+Q2]2 =__acdx,Q ± 2 _ )-

1

2'

(±

±

Y+W2 =F LxW3 -

Y2W L±) ' (6)

where is the cross section in a reduced form, G F is the Fermi constant, and Mw is the mass of the W boson. W 2, XW3 and W L are the CC structure functions defined in a similar manner as for NC. In the quark parton model, where WL == 0, the structure functions W 2 and XW3 may be expressed as the sum and difference of the quark and anti-quark momentum distributions: + + W 2 = x(U + D), XW3 = x(D - U), W 2 = x(U + D), XW3 = x(U - D). The terms xU, xD, xU and xV are defined as the sums of up-type, of down-type and of their anti-quark-type distributions. The Standard Model (SM) predicts that in the absence of right-handed charged currents the e+p (e-p) CC cross section is directly proportional to the fraction of right-handed positrons (left-handed electrons) in the beam and can be expressed as (7)

70 1

N

~M

FL extraction from H1 data (for fixed W=276 GeV)

u.. X

2,

H1 +ZEUS Combined (pre)) 0.8

Q2=1500 GeV -

1.2

L.L'

1

•

Hl preliminary • Hl e·

H1 2000 PDF 0.8

0.6

[J

Hl e-

j

-

NLOa,fit(Hl) NLO fit (ZEUS)

-

NlO MRST 2001

-

NLO (Alekhin)

:

......

NNLO (Alekhi") 1

I

0.6

0.4 0.4 0.2

t'" 10

2

10

L~ ...

1

~ I

0.2

x

-I_" .""'" ._

.. ~;' ~t--;

•

.,

...............................................................................• 10

i.

Q'/GeV'

Figure 3: The combined HI and ZEUS measurements of the structure function xF;;z (left). Summary of FL measurements by HI at a fixed photon-proton center of mass energy W = 276 GeV, W ~ .;sy (right).

The HI and ZEUS measurements of the single differential NC and CC e±p cross sections dO' / dQ2 are summarised in Figure 1 (left). At low Q2 < 100 Ge V2 the cross section of the CC process mediated by the W boson, is smaller by 3 orders of magnitude compared to the NC process, due to the different propagator terms. At high Q2 ~ the cross section measurements are approaching each other demonstrating the unification of the weak and electromagnetic forces. From a comparison of the NC measurements at highest Q2 with the SM expectation (see Figure 1, right), a limit on the quark radius of 0.74.10- 18 m is obtained [1], proving a point-like behaviour of the quarks down to about 1/1000 of the proton radius. The double differential NC and CC cross sections measurements [2] are shown in Figure 2. HERA allows to enlarge the coverage of the NC measurements by more than two orders of magnitude both in Q2 and x. The CC data provide information about individual quark flavours as can be seen in Figure 2 (right), especially at large Q2.

Mi, Mar

3

Structure functions F 2 , XF3 and FL

The NC cross section is dominated by the F2 contribution, and the reduced cross section, shown in Figure 2 (left), is essentially the proton structure function F 2 . In the figure one can see Bjorken scaling behaviour at x ~ 0.13, positive scaling violation at higher x due to gluon radiation from the valence quarks and negative scaling violation for x < 0.13 due to sea quarks originated from gluons. At fixed Q2 one observes a steep rise of F2 towards low x. The region at low x is due to quarks which have undergone hard or multiple soft gluon radiation and which carry a low fraction of the proton momentum at the time of interaction. The rise of the proton structure function at low x is

71

~'';'~Oi3 10 • HI Data • ZEUS (pre!.) 39 pb- 1 .... MRST04 10

MRSTNNLO CTEQ6HQ HVQDIS + CTEQ5F4

x=O.032

t

i=O

1O~1~0~~~1O'2~~~IO"~

10'

10

~

Q2(GeV2)

Q2/Gey2

Figure 4: The measured F!j" (left) and F~b (right), shown as a function of Q2 for various x values.

one of the most surprising observations at HERA. It can be understood as an unexpected rapid increase of the gluon density towards low x. The structure function XF3 is obtained from the NC cross section difference between e-p and e+p data, i.e. XF3 = (y+/2Y_) [a-(x,Q2) -a+(x,Q2)]. The dominant contribution to XF3 arises from "(Z interference, which allows xF;z to be extracted according to xF;z ::= -xF3/kae neglecting the pure Z exchange contribution, which is suppressed by the small vector coupling Ve. This structure function is non-singlet and has little dependence on Q2. The measured xF;z at different Q2 values can thus be averaged taking into account the small Q2 dependence. The averaged xF;z determined for a Q2 value of 1500 Gey2 is shown in Figure 3 (left) [3]. In leading order QCD the interference structure function xF;z leads to the following sum rule:

r z~ 1 r 10 xF; -;- = 310 1

1

(2u v + dv)dx

5

= 3'

(8)

Higher order corrections to this are expected to be of order as /7r. In the range of acceptance, the integral of F;z is measured to be Jo~~625 F;z dx = 1.21 ± 0.09(stat) ± 0.08(syst), which is consistent with the results of the HI and ZEUS QCD fits [4] of 1.12 ± 0.02 and 1.06 ± 0.02, respectively, for the same x interval at Q2 = 1500 Ge y2 . Non-zero values of the longitudinal structure function FL appear in perturbative QCD due to gluon radiation. According to eq. 1, the FL contribution to the inclusive cross section is significant only at high y. A direct way to measure

72 Charged Current e~p Scattering e"p-JovX • H1 2005 (prel.) 0H198-99 '" ZEUS 04-05 (prel.) f:.ZEUS 98-99

e+p---JoVX • H199-04 ... ZEUS 06-07 (prel.) f:;ZEUS 99-00

60

..

r 0.8

f

0.6 0.4 0.2

CTEQ6D .... MRST2004

-0.2 -0.4

-0.6

-O.B

DoCo','-'-'--'-c~~-'---!--~~o!oD05~~' P,

0'

• A> • A

Hl 2000 PDF ZEUS-JETS PDF

10'

10

4

Figure 5: The dependence of the e+p and e-p CC cross-section on the lepton beam polarisation P (left). Measurements of the polarisation asymmetries A± in NC interactions (right).

FL is to explore the y dependence of the cross section at given x and Q2 by

changing the center of mass energy of the interaction. Such analysis at HERA is in progress now using dedicated data collected with lower proton beam energies of 460 and 575 GeV. Data at the nominal proton energy of 920 GeV have been used by the HI collaboration to determine FL which is responsible for the observed decrease of the Ne cross section at high y. A summary of these FL measurements by HI [5] is shown in Figure 3 (right). They are compared with QeD calculations and different phenomenological models, showing that already at the present level of precision the measurements can discriminate between different predictions.

4

Charm and bottom structure functions F~c,

F!/'

Heavy quark production is an important process contributing to DIS. It is expected to be well described by perturbative QeD at next-to-Ieading order (NLO), especially at values of Q2 greater than the square of the heavy quark masses. The charm and bottom contributions to the proton structure function Fie, F~b are shown in Figure 4 [6]. They are measured using exclusive D or D* meson production and using a technique based on the lifetime of the heavy quark hadrons. In the latter case all events containing tracks with vertex detector information are used. The charm contribution on average amounts to 20 - 25% of F2 . The bottom structure function F~b is measured at HERA for the first time. It is about 1/10 of the charm contribution and amounts to ~ 2.5% of F2 at Q2 = 650 GeV 2 • The data are well described by QeD calculations. The accurate measurement of these structure functions is important to test the reliability of the theoretical framework used for the QeD analysis of

73

inclusive data and of predictions for the forthcoming LHC data, because their contribution is expected to be much increased at scales relevant for the LHC. 5

5.1

Polarisation effects in NC and CC

Polarisation dependence of the CC cross section

Measurements of CC deep-inelastic scattering with polarised leptons on protons allows the HERA experiments to extend tests of the V-A structure of charged current interactions from low-Q2, performed in the late seventies by the CHARM collaboration, into the high-Q 2 regime. ± The total CC cross sections a~d' as a function of the polarisation, measured in the range Q2 > 400 GeV 2 and y < 0.9, are shown in Figure 5 (left) [7]. The measurements agree with the SM predictions and exhibit the expected linear dependence as a function of the polarisation. Linear fits provide a good description of the data, and their extrapolation to the point P = 1 (P = -1) yields a fully right (left) handed CC cross section for e-p (e+p) interactions which is consistent with the vanishing SM prediction. The corresponding upper limits on the total CC cross sections exclude the existence of charged currents involving right handed fermions mediated by a boson of mass below 180 208 Ge V at 95% confidence level, assuming SM couplings and a massless right handed Ve.

5.2

Polarisation asymmetry in NC

The charge dependent longitudinal polarisation asymmetries of the neutral current cross sections, defined as (9)

measure to a very good approximation the structure function ratio, proportional to combinations aeVq, and thus provide a direct measure of parity violation. In the Standard Model A+ is expected to be positive and about equal to -A-. At large x the asymmetries measure the diu ratio of the valence quark distributions according to A± ~ ±k(1 + dv /u v )/(4 + dv/u v ) . The combined HI and ZEUS data are shown in Figure 5 (right) [3]. The asymmetries are well described by the Standard Model predictions as obtained from the HI and ZEUS QCD fits [4]. The measured asymmetries A± are observed to be of opposite sign and the difference 6A = A+ - A- can be seen to be significantly larger than zero, thus demonstrating parity violation at very small distances, down to about 10- 18 m.

74

6: Parton distribution functions determined at HERA (left). Results on the weak neutral of the 'It quark to the Z boson as determined at HERA in comparison with similar results by the CDF experiment and the combined LEP experiments

Partonic structure of the

nJ'ntnn

The measurements of the full set of NC and CC douhle differential cross sections at HERA allow comprehensive QCD analyses to determine the and distributions inside the proton and the constant cross inclusive HERA measurements of the NC and CC HI and ZEUS performed NLO QCD fits [4), which lead to a decmnDositioln of the parton densities. In the fit their data are and gluon distributions obtained in the HI and fits are shown in 6 (left). The results agree within the error also agree with the parton densities from fits which bands. include not HERA but also fixed target DIS data as well as data from other processes sensitive to parton distributions, such as inclusive DI'()dllct:ion and the W-lepton asymmetry in collisions. The inclusive and CC cross sections are not distribution functions but also to the electroweak the NC cross section at depends on the weak vector (v q ) and axial-vector IlDI'HHno; of up- and down-type quarks to the Z boson via the structure functions. The longitudinal polarisation of the lepton beam additional to the couplings. This has been in a combined fit the PDFs and the electroweak parameters [8]. The fitted the u are shown in Figure 6 (right) in with similar suIts obtained the CDF experiment and the combined LEP The HERA determination has a better precision than that from the

75

tho UDeert.

HERA

~'~,~,-

-'--, • ZEUS (inclusive-jet NC DIS) .,. ZEUS (inclusive-jet yp) ... ZEUS (norm. dijet NC DIS) III HI (norm. inclusive-jet NC DIS) . HI (event shapes NC DIS)

-~

- ,

-~

Judu<;jv{' jet
-,

expo uncert •

ZEUS lPhy" Ldt B 649 (l007) 12) Indusin>-jet (TO~S 'lcctions in NC DIS HI (DESY 07·073) HERA combined 2007 inclush'e-jet :'Ii"C DIS

0.15

(this

anal~'sis)

HERA averagt 2004 !

hep-('~j05060J3)

O,I~-_

10

100

~

=Q or Ei~t (GeV)

World average 2006 (So

Bethke, hep-exl060(.035)

Figure 7: Illustration of running of the strong coupling o. measured by Hi and ZEUS (left). Summary of o.(Mi) measurements at HERA (right).

in particular for the u quark. They also resolve any sign ambiguity and the ambiguities between Vu,d and au,d of the determination based on observables measured in e+e- reactions at the Z resonance.

7

Strong coupling as

Semi-inclusive processes such as jet production provide further insight into the structure of the proton. Thus, the inclusion of the jet data into the QCD analysis helps to reduce the uncertainty of the gluon distribution at medium x around 0.1. Furthermore, the jet production cross section depends on the least well-measured fundamental constant as(M~), which can therefore be extracted from the jet data. The running of the strong coupling as at HERA as a function of a hard scale defined as either Q or the mean jet transverse energy, E T , is shown in Figure 7 (left) together with the QCD predictions. The combined HI and ZEUS measurement of as(M~) [9] using the recent inclusive jet cross sections, as(M~) = 0.1198 ± 0.0019(exp)

± 0.0026(theory) ,

(10)

is shown in Figure 7 (right) together with the individual measurements, the HERA average 2004 and the world average 2006. The experimental error of this measurement is small, about 2%. A limitation of the precision of as (M~) arises from the associated theoretical error calculated by varying the renormalisation and factorisation scales by a factor of two.

76

8

Conclusions

Over the 15 years of operation of HERA a wealth of different and partly unexpected aspects of the proton structure and the fundamental forces have been uncovered, including the surprising rise of F2 towards low x as well as a significant fraction of diffractive events in DIS even at very high Q2. The structure function data and polarisation dependence of the total CC cross section are already now part of textbook knowledge. Many measurements by HI and ZEUS presented in this paper are already partially combined in order to improve the statistical precision. The final results from HERA, using all collected data from both collider experiments, are in preparation at present with the aim to provide the ultimate and lasting legacy of HERA. References [1] HI Collab., "High Q2 NC analysis using the complete HERA data", contribution to LP07, Daegu, Korea, August 2007 [Hlprelim-07-141]; ZEUS Collab., contr. to LP07, Daegu, Korea, 2007 [ZEUS-prel-07-028]. [2] HI and ZEUS Collab., "Combination of HI and ZEUS Deep Inelastic e±p Scattering Cross Sections", contribution to LP07, Daegu, Korea, August 2007 [Hlprelim-07-007, ZEUS-prel-07-026]. [3] HI and ZEUS Collab., "Electroweak Neutral Currents at HERA" , contribution to ICHEP'06, Moscow, Russia, July 2006 [Hlprelim-06-142, ZEUS-prel-06-022]. [4] HI Collab., C. Adloff et al., Eur. Phys. J. C30, 1 (2003); ZEUS Collab., S. Chekanov et al., Eur. Phys. J. C42, 1 (2005). [5] HI Collab., C. Adloff et aI., Eur Phys J C21 (2001) 33; C30 (2003) 1. [6] HI Collab., "Measurement of FrJ and F~b using the HI vertex detector at HERA", contribution to LP07, Daegu, Korea, 2007, [Hlprelim-07-171]; ZEUS Collab., "Measurement of Fi c at HERA II", contribution to HEP2007, Manchester, UK, July 2007. [7] HI Collab., " High Q2 Charged Current in polarised ep collisions", contribution to DIS2006, Tsukuba, Japan, April 2006 [Hlprelim-06-041]; ZEUS Collab., contr. to ICHEP'06, Moscow, Russia, July 2006 [ZEUSprel-06-002]. [8] HI Collab., "Combined Electroweak and QCD Fit of inclusive NC and CC Data with Polarised Lepton Beams at HERA", contribution to DIS2007, Munich, Germany, April 2007 [ Hlprelim-07-041]; ZEUS Collab., "QCD and Electroweak analysis of the ZEUS NC and CC inclusive and jet cross sections", contribution to DIS2006, Tsukuba, Japan, April 2006 [ZEUS-prel-06-003]. [9] HI and ZEUS Collab., "Precision measurements of O:s at HERA", talk of C. Glasman at HEP2007, Manchester, UK, July 2007.

RECENT RESULTS FROM THE TEVATRON ON CKM MATRIX ELEMENTS FROM B. OSCILLATIONS AND SINGLE TOP PRODUCTION, AND STUDIES OF CP VIOLATION IN B. DECAYS J.P.Fernandez a

E2PID3, Dpto. Inv. Basica, CIEMAT, Av. Complutense 22, 28040 Madrid, Spain Abstract. Recent results from the Tevatron on several elements of the CKM matrix are presented through a set of completely different technical approaches.

1

Introduction

In this review we deal with completely different analysis strategies but with the same aim: more precise knowledge of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1] at the Tevatron: Ivtbl through single top production Ivt.1 through B. mixing and arg(vt. vtb * IVc• Vcb *) through CP violation. CDF [2] and DO [3] are two multipurpose central detectors placed at the two interactions points of the Tevatron, a pP collider at a center of mass energy of y'S = 1.96TeV. This energy allows to produce and study the top quark and the properties of B mesons and baryons that leads us to determine fundamental parameters like the CKM matrix elements. 2

Single top production

Single top quark events can be used to study the Wtb coupling [4], and to measure the magnitude of the element Ivtbl without assuming only three generations of quarks [5]. The CKM quark mixing matrix must be unitary, which for three families implies Ivtbl ~ 1 [6]. A smaller measured value would indicate the presence of a fourth quark family to make up the difference. Events are choosen with two, three, or four jets, reconstructed using a cone algorithm [7] to cluster energy deposits in the calorimeter. Events are required to have itT and one isolated electron or isolated muon. Misreconstructed events are rejected by requiring that the direction of the itT is not aligned or antialigned in azimuth with the lepton or a jet. To enhance the signal content of the selection, one or two of the jets are required to be identified as originating from long-lived b hadrons by a neural network jet-flavor separator. The variables used to identify such jets rely on the presence and characteristics of a secondary vertex and tracks with high impact parameters inside the jet. A 50% average tag rate is obtanied in data. Since single top quark events is expected to constitute only a small fraction of the selected event samples, and since the background uncertainty is larger ae-mail: [email protected]

77

78 than the expected signal, a counting experiment will not have sufficient sensitivity to verify their presence. Both CDF and DO proceed instead to calculate multivariate discriminants that separate the signal from background and thus enhance the probability to observe single top quarks. Three multivariate discriminants were used: decision trees [8], a machine-learning technique that applies cuts iteratively to classify events, matrix-elements [9], a technique that calculates the probability for each event to be signal or background based on the leading-order matrix element description of each process for two-jet and three-jet events and Bayesian neural networks [10] to separate tb+tqb signal from background. Furthermore, a Bayesian approach [11] to measure the single top quark production cross section is applied. A binned likelihood is formed as a product over all bins and channels (lepton flavor, jet multiplicity, and tag multiplicity) of the decision tree discriminant, separately for the tb+tqb, tqb, and tb analyses. A Poisson distribution for the observed counts and flat nonnegative prior probabilities for the signal cross sections is assumed. The final posterior probability density is computed as a function of the production cross section. For each analysis, the cross section is measured using the position of the posterior density peak and a 68% asymmetric interval about the peak is taken as the uncertainty on the measurement. The measured posterior density distribution for tb+tqb is used to make the following measurements: DO, r7(pfj --- tb + X, tqb + X) = 4.9 ± 1.4 pb, r7(pp --- tqb + X) = 4.2~t: pb, and r7(pp --- tb + X) = 1.0 ± 0.9 pb, CDF r7(pp --- tb + X, tqb + X) = 3.0~~:~ pb, r7(pfj --- tqb + X) = 1.9~6:~ pb, and r7(pp --- tb + X) = 1.1~6:g pb. These results are consistent with the 8M expectations. The DO decision tree measurement of the tb+tqb cross section [13] is used to derive a first direct measurement of the strength of the V - A coupling Ivtbffl in the Wtb vertex, where ff is an arbitrary left-handed form factor [12]. Ivtbifl = 1.3 ± 0.2 is the result obtained. This measurement assumes Ivtdl 2 + Ivtsl 2 « Ivtbl 2 and a pure V -A and CP-conserving Wtb interaction. Assuming in addition that if = 1 and using a flat prior for Ivtbl 2 from 0 to 1, DO obtains 0.68 < Ivtbl :::; 1 at 95% C.L. CDF, using the matrix-elements method concludes: Ivtbffl = 1.02 ± 0.18experiment ± 0.07theory and using a flat prior for Ivtbl 2 from 0 to 1 Ivtbl > 0.55 at 95% C.L. These measurements make no assumptions about the number of quark families or CKM matrix unitarity. 3

E~ Mixing

Neutral E mesons (bq, with q = d, s for f3 o , f3~) oscillate from particle to antiparticle due to flavor-changing weak interactions with a frequency proportional to the mass difference flmq between the two mass eigenstates EOq, H

79 and Bqo , L'

Bqo )L, H are the linear combinations of the BOq flavor eigenstates: IBq,L,H >= plBg > ±IBg > and in turn are the eigenvectors of the 2x2 effective Hamiltonian (H) which conventionally describes the simultaneous time evolution of the Bg - Bg system. The subscripts Land H stay for "light" and "heavy". t:.ms = mBO3,H - mBoB,L is reported in inverse picoseconds (setting

= c = 1).

This frequency can be used to extract the magnitude of Vis. At present, large amounts of B~ - B~ mesons are produced only at Tevatron, providing CDF and DO experiments with unique data samples to exploit the Bs system, as well as other heavy and excited b-hadrons. The key components for the CDF & DO measurements are listed below. Precision determination of the decay point is provided by double-sided silicon-strip detectors and a singlesided layer of silicon mounted directly on the beampipe at an average radius of 1.5 cm. A drift chamber is used for both precision tracking and dE / dx particle identification. CDF incorporates Time-of-flight [14] (TOF) counters surrounding the drift chamber that are used to identify low-momentum charged kaons. In CDF, a three-level trigger system selects, in real time, pp collisions containing charm and bottom hadrons by exploiting the kinematics of production and decay, and the long lifetimes of D and B mesons. A crucial component of the trigger system for this measurement is the Silicon Vertex Trigger [15], which makes it possible to collect a large sample of B8 mesons in the fully or partially reconstructed hadronic decay modes. DO has excellent muon system and a tracking coverage in the forward region up to 1771=2.5 and triggers on leptons from semileptonic B~ - B~ decays. The B~ - B~ candidates are reconstructed from the above trigger samples in several classes of final states: partially-reconstructed semileptonic modes (CDF & DO), B~ -+ D;e- ( D; -+ ¢>(K+ K-)7r+, D; -+ K*(K+7r-)K+, D; -+ 7r-7r+7r+) fully-reconstructed hadronic modes (CDF), B~ -+ D;7r- , D;7r-7r+7rand partially-reconstructed hadronic modes (CDF), B~ -+ D;+7r-, D;+ -+ D;,/7r° in which a photon or 7r 0 from the D;+ is missing and B~ -+ D;p-, p- -+ 7r-7r 0 in which a 7r 0 is missing. The mass difference t:.m8 between the B8 mass eigenstates is measured directly in a time-dependent analysis. The measurement consists in detecting an oscillatory pattern in the proper time distribution of the B8 mesons, whose frequency is proportional to t:.m8' CDF & DO use an unbinned maximum likelihood fit to search for B8 oscillations. The likelihood combines mass, decay time, decay-time resolution, and flavor tagging information for each candidate, and includes terms for signal and each type of background. The probability distribution for a B., produced at to == 0, to decay as a B~(B8) at a later time t is given by P(t) ~ (1 ± VSvAcos(t:.m8t))e-t/rBs x G(t;(Tt)· v(t), the likelihood model for the observed proper decay time of the signal component. The first term in parenthesis describes the flavor oscillations, while the rest of the expression characterizes the proper time distribution in the absence of

1i

80 flavor tagging information. The description of the observed proper decay time includes the smearing effect cause by the detector vertex resolution, through a Gaussian resolution function. The trigger requirements and analysis selection criteria impose additional sculpting of the t distribution, which is described by a function, v(t), derived from Monte Carlo simulation of each decay. The decision of whether or not a B. candidate has mixed (shown by the signs '-'and '+' in the expression above) is made by the flavor tagging algorithms. The flavor of the B. at production is determined using both opposite-side (from the decay products of the b-hadron originated fron the other b quark in the event) and same-side (from the properties of the particles produced in the association with the reconstructed B.) flavor tagging techniques. The effectiveness E1)2 of these techniques is quantified with an efficiency E, the fraction of signal candidates with a flavor tag, and a dilution 1) == 1 - 2w, where w is the probability that the tag is incorrect. The combined tagging power of the CDF opposite-side taggers is E1)2 = 1.8%, while the same-side tagger has E1)2 = 3.7%(4.8)% in the hadronic (semileptonic) sample. DO quotes E1)2 = 2.5% for the opposite-taggers and 4.5% for a combination of oppositeside and same-side taggers. More details can be found in [16], [17] and [18]. The average statistical significance of an oscillation signal is usually approximated by the formula S = )2/SE1)2 exp(IJt!:::.m.)2)(8 + B)/8, which summarizes the crutial elements of the !:::.m. measurement: an abundant B. signal (8) with a good signal to background (B) ratio, the B. proper time measured with high resolution (IJt), a high-efficiency and high-purity identification of B. flavor a production and decay. Following the method described in [19], CDF and DO fit for the oscillation amplitude A while fixing !:::.m. to a probe value. The oscillation amplitude is expected to be consistent with A = 1 when the probe value is the true oscillation frequency, and consistent with A = 0 when the probe value is far from the true oscillation frequency. Figure 1 shows the fitted value of the amplitude as a function of the oscillation frequency. Semlleptonic+Hadronic

$

-g

-8

~1

~

1r-----------~-------r--~

.~

~

«

00 Runll Preliminary

f

2 1.5

data±1cr

Ldt =2.4 fb'1

1r---------~~---.-4 0.5

tIIltmIfIIIIIlll

-0.5

·1 -1.5

·"ti~~~1D~""15~';'2D~2&5~3~D Mn,[ps'[

t-m,(ps")

Figure 1: Amplitude scan (CDF & DO)

At !:::.m.

=

17.75 pS-l, the observed CDF amplitude A

=

1.21 ± 0.20 (stat.)

81

is consistent with unity, indicating that the data are compatible with B~ - B~ oscillations with that frequency, while the amplitude is inconsistent with zero: A/aA = 6.05, where aA is the statistical uncertainty on A (the ratio has negligible systematic uncertainties). DO obtains .6.m. = 18.6 ± 0.8 pS-l. 4

CP violation in B. decays

The interpretation of the CP violation mechanism is one of the most controversial aspects of the SM. Many extensions of the SM predict that there are new sources of CP violation, beyond the single phase in the CKM matrix. Considerations related to the observed baryon assymmetry of the Universe imply that such new sources should exist. The non-leptonic decays of B mesons are effective probes of CKM matrix and sensitive to these potential new physics effects: in the SM, the light (L) and heavy (H) eigestates of the B. system are expected to mix in such a way that the mass and decay width differences between them, .6.m s and .6.r. = r L - r H = (r even - r odd)COS¢s are sizeable. ¢s is the phase arg( -M12 /r12), which accounts for CP violation in the mixing (being M12/r12 the off diagonal elements of H, described in 3). The mixing phase ¢~M is within the SM predicted to be small [20], and thus to a good approximation the two mass eigenstates are expected to be CP eigenstates. New phenomena may introduce a non-vanishing mixing phase ¢~P, leading to a reduction of the observed .6.r s compared to the SM prediction: .6.r. = .6.r~M x Icos(¢~M + ¢~P)I [20]. CP violation in B~ -> K-7r+. The analysis was done with 1 fb- 1 and was based on the SVT trigger [15] that allowed CDF to collect a large sample of Bf.) ---> h+h- decay modes (where h == K or 7r). A Maximum Likelihood fit which combines kinematic and particle identification information is performed to statistically determine the contribution of each mode and the relative contributions to the CP asymmetries. One of the new rare modes observed was B~ ---> K-7r+ with 8.2a. The direct CP assymmetry Acp(B~ -> K-7r+) = 0.39 ± 0.15 ± 0.08. This value favors a large CP violation in B~ meson decays though it is also compatible with zero. A robust test of the SM or a probe of new physics suggested in ref. [21] is performed comparing the direct CP assymmetries in B~ -> K-7r+ and BO -> K+7r- decays. Using HFAG input [22] CDF measures r(BO -> K-7r+) - r(B O-> K+7r-)/r(B~ -> K-7r+) - r(B~ -> K+7r-) = 0.84 ± 0.42 ± 0.15, in agreement with the 8M expectation of unity. CP violation in mixing in B~ -> J/iJI¢. An untagged sample of B~ -> J /iJI ¢ candidates represents a powerful tool to measure .6.r s, since a timedependent angular analysis of the decay products allows to disentangle the heavy (B.,H) and light (B.,L) Bs mass eigenstates. The decay B~ -> J/iJI¢

rise to both CP even and CP odd final states: since it is a lJ".'UUlU"'CctlitU vector-vector the final state can either have momentum = 0, or It is possible to separate the two CP components through the and measure the lifetime difference measure lifetimes in J jW¢i, CP states by basis the polar and azimuthal where the x-axis is defined by the the ¢i -+ and the 4) frame with respect to the np,,.,,,-,,"p simultaneous unbinned maximum-likelihood fit of the reconstructed mass, the lifetime and time distributions in order to extract CP even, CP odd DO is in two ways. First the ¢is has been fixed to zero, which assumes no New contribution in A non-zero width difference of .6.r s 0.12 ±0.08(stat.) ±0.03(syst.) has been obtained. a second fit to the DO both the width difference .6.r and = 0.17 ± AVAAVVViU.!".

where letter means that parameter is allowed to float in the fit). The obtained p - value is close to 0, 0.1): 22 %. shows the confidence from this CDF and the confidence mClepen1jerlt constraints on and ¢is [24J.

Figure 2: CDF (left) and DO (right) .6.r s, t/>s confidence region.

83

5

Results

To summarize, Both CDF and DO have performed a search for single top quark production using 1.5/0.9 fb- 1 of data collected at the Tevatron collider. Both find an excess of events over the background prediction in the high discriminant output region and interpret it as evidence for single top quark production. The excess has a significance of 3 standard deviations. The first measurement of the single top quark cross section yields: O'(pp --; tb+X, tqb+X) = 4.9±1.4 pb (DO), 3.0~U pb (CDF). The cross section measurement is used to make the first direct measurement of the CKM matrix element Ivtbl without assuming CKM matrix unitarity, and find 0.68 < Ivtbl ::; 1 at 95% C.L for DO, Ivtbl > 0.55 at 95% C.L for CDF. We present CDF and DO results using 1.0/2.4 fb- 1 of data on the mixing frequency measurement in the B~ system. CDF gets t:.ms = 17.77±0.10±0.07 pS-l and DO gets t:.ms = 18.6 ± 0.8 pS-l. CDF uses its result to derive the ratio

Ivtd/vtsl

= ~

~md llms

mBg mBo

[25]. As inputs CDF uses mBo/mBos = 0.98390 [26]

with negligible uncertainty, t:.md = 0.507 ± 0.005 pS-l [25] and ~ = 1.21 ~g:gj~ [27]. CDF finds Ivtd/vtsl = 0.2060 ± 0.0007 (exp) ~g:gg~6 (theor). Finnaly, a new series of measurements by the CD F and DO Collaborations has started to give unprecedented insights into the nature of CP violation nature in the B~-B~ system. The CDF direct CP assymmetry Acp(B~ --; K-7r+) = 0.39±0.15±0.08. With the HFAG input [22] CDF measures r(BD --; K-7r+)f(BD --; K+7r-)/f(B~ --; K-7r+) - r(B~ --; K+7r-) = 0.84 ± 0.42 ± 0.15, in agreement with the SM expectation of unity. A non-zero decay width difference of t:.fs = 0.12 ± 0.08(stat.) ± 0.03(syst.) has been obtained by DO. In a second fit to the DO data, both the decay width difference t:.f and ¢. where floating parameters, which results in: t:.f s = 0.17 ± 0.09(stat.) ± 0.03(syst.), ¢. = -0.79 ± 0.56(stat.) ± O.Ol(syst.) CDF measures a t:.f. = 0.076~g:gg~(stat.)±0.006(syst.). Regarding ¢., the obtained p - value is close to SM (t:.f. = 0, ¢. = 0.1): 22 %.

Acknowledgments I thank the organizers of the XIII Lomonosov Conference for a very enjoyable conference. I also thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundations, the European Community's Human Potential Programme and the Comision Interministerial de Ciencia y Tecnologia, Spain.

84

References [1] N. Cabibbo, Phys. Rev. Lett.l0, 531 (1963); M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49, 652 (1973). [2] D. Acosta et al. Phys. Rev. D, 71, 032001 (2005). [3] V.M. Abazov et al.,Nucl. Instrum. Methods A 565, 463 (2006). [4] A.P. Heinson, et al., Phys. Rev. D 56, 3114 (1997). [5] G.V. Jikia and S.R. Slabospitsky, Phys. Lett. B 295, 136 (1992). [6] W.-M. Yao et al., J. Phys. G: Nucl. Part. Phys. 33, 1 (2006), p. 142. [7] G.C. Blazey et al., in "Run II Jet Physics" (Proceedings ofthe Workshop on QCD and Weak Boson Physics in Run II) ed. by U. Baur, R.K. Ellis, and D. Zeppenfeld, Fermilab-Pub-00/297, 47, 1999. [8] L. Breiman et al., Classification and Regression Trees (Wadsworth, Stamford, 1984); D. Bowser-Chao et al., Phys. Rev. D 47, 1900 (1993). [9] V.M. Abazov et al., Nature 429,638 (2004); Phys. Lett. B 617,1 (2005). [10] R.M. Neal, Bayesian Learning of Neural Networks (Springer-Verlag, New York, 1996); P.C. Bhat and H.B. Prosper, "Statistical Problems in Particle Physics, Astrophysics and Cosmology", ed. by 1. Lyons and M.K. Unel (Imperial College press, London, 2006), p. 151. [11] 1. Bertram et al., Fermilab-TM-2104 (2000), and references therein; E.T. Jaynes and L. Bretthorst, Probability Theory: the Logic of Science (Cambridge University Press, Cambridge, 2003). [12] G.L. Kane, et al., Phys. Rev. D 45, 124 (1992). [13] V.M. Abazovet al., Phys. Rev. Lett. 98, 181802 (2007). [14] C. Grozis et al., Int. J. Mod. Phys. A 16S1C, 1119 (2001). S. Cabrera et al., Nucl. Instrum. Methods Phys. Res. A 494, 416 (2002). [15] W. Ashmanskas et al.,Nucl. Instrum. Methods Phys. Res. A 518 (2004). [16] A. Abulencia et al., Phys. Rev. Lett. 97, 062003 (2006). [17] A. Abulencia et al., Phys. Rev. Lett. 97, 242003 (2006). [18] V. M. Abazov et al., Phys. Rev. Lett. 97, 021802 (2006). [19] H. G. Moser and A. Roussarie, Nucl. Instrum. Methods Phys. Res., Sect. A 384, 491 (1997). [20] 1. Dunietz, R.Fleischer and U.Nierste, Preprint hep-ex/0012219. [21] H. J. Lipkin, Phys. Lett.B 621, 126 (2005). [22] E. Barberio et ai, Preprint hep-ex/0603003. [23] V.M.Abazovet al.,Phys.Rev.Lett. 95, 171801 (2005). [24] V.M.Abazovet al.,arXiv:hep-ex/0702030vl. [25] W.-M. Yao et al. J. Phys. G 33, 1 (2006). [26] D. Acosta et al., Phys. Rev. Lett. 96, 202001 (2006). [27] M. Okamoto, Proc. Sci. LAT2005 (2005) 013 [hep-lat/0510113j.

DIRECT OBSERVATION OF THE STRANGE b BARION 2t L. Vertogradov a

Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980 Dubna, Russia Abstract. The first direct observation of the strange b baryon Bb (Bt) reconstructed from its decay mode Bb -t J/'!/JB-, with J/7/J -t J-t+J-t-, and B- -t A7r- -t (P7r-)7r- in pfi collisions at Vs = 1.96TeV is presented. Using 1.3jb- 1 of data collected by the DO detector 15.2 ± 4.4(stat.):!:6:~(syst.) Bb candidates at a mass of 5.774 ± O.Ol1(stat.) ± O.015(syst.)GeV were observed. The significance of the signal is 5.5a.

The quark model of hadrons predicts the existence of a number of baryons containing b quarks. Despite significant progress in studying b hadrons over the last decade, only the Ab(udb) b baryon has been directly observed. The 2{; (dsb) (charge conjugate states are assumed throughout this report) is a strange b baryon made of valence quarks from all three known generations of fermions and is expected to decay through the weak interaction. Experiments at the CERN LEP e+e- collider have reported indirect evidence of the 2{; baryon based on an excess of same-sign 2-[- events in jets [2]. We observe the decay 2{; -+ JN2-, with IN -+ f,.L+f,.L-, 2- -+ A7r-, and A -+ P7r-. The analysis is based on a data sample of 1.3jb- 1 integrated luminosity collected in pp collisions at yfs=1.96 TeV with the DO detector at the Fermilab Tevatron collider during 2002 - 2006. The DO detector is described in detail elsewhere [3]. The components most relevant to this analysis are the central tracking system and the muon spectrometer. The central tracking system consists of a silicon microstrip tracker (SMT) and a central fiber tracker (CFT) that are surrounded by a 2 T superconducting solenoid. The SMT covers the pseudorapidity region 1771 < 3 (77 = -In [tan (8/2)] and 8 is the polar angle) while the CFT for 1771 < 2. The muon spectrometer is located outside the calorimeter and covers the pseudorapidity region 1771 < 2. It comprises a layer of drift tubes and scintillator trigger counters in front of 1.8 T iron toroids followed by two similar layers behind the toroids. The topology of 2{; -+ J N2- -+ J NA7r- decay is shown on Fig.1. The reconstruction of the J / 't/J and A and their selection are validated with simulated MonteCarlo(MC) 2{; events. The PYTHIA MC program [4] is used to generate 2{; signal events while the EvtGen program [5] is used to simulate 2{; decays. The 2{; mass and lifetime are set to be 5.840 GeV and 1.33 ps respectively, their default values in these programs. The generated events are subjected to the same reconstruction and selection programs as the data after passing through the DO detector simulation based on the GEANT package [6]. MC events are a e-mail:

[email protected]

85

86 reweighted to match the J 1'Ij; transverse momentum (PT) distribution observed in the data [7]. J 1'Ij; - t /-l+ /-l- decays are reconstructed from two oppositely charged muons that have a common vertex with X2 probability greater than 1%. Muons are identified by matching tracks reconstructed in the central tracking system with either track segments in the muon spectrometer or calorimeter energies consistent with the muon trajectory. They are required to have PT > 1.5 Ge V and at least one of them must be reconstructed in each of the three muon drift tube layers. Events containing a J 1'Ij; candidate are re-reconstructed with a version of the track reconstruction algorithm that increases the effciency for tracks with low PT and high impact parameters. For further analysis, J 1'Ij; - t /-l+ /-lcandidates are required to have mass 2.80 < MJjJj < 3.35 GeV and PT > 5 GeV. A - t pn- candidates are formed from two oppositely charged tracks that are consistent with originating from a common vertex with a X2 probability greater than 1% and that have a mass between 1.105 and 1.125 GeV. The two tracks are required to have a total of no more than two hits in the tracking detector before the reconstructed pn- vertex. Furthermore, the impact parameter significance (the impact parameter with respect to the event vertex divided by its uncertainty) must exceed three for both tracks and exceed four for at least one of them. The A candidates are then combined with negatively charged tracks (assumed to be pions) to form 3- - t An- decay candidates. The pion must have an impact parameter significance greater than three. The A and the pion are required to have a common vertex with a X2 probability of 1% or better. For both A and 3- candidates, the distance between the event vertex and its decay vertex is required to exceed four times its uncertainty. Moreover, the uncertainty of the distance between the production vertex and its decay vertex (decaylength) in the transverse plane (the plane perpendicular to the beam direction) must be less than 0.5 cm. The two pions from 3- - t An- - t (pn-)n- decays (right-sign) have the same charge. Consequently, the combination An+ (wrong-sign) events form an ideal control sample for background studies. Figure 2(a) compares mass distributions of the right-sign An- and the wrong-sign An+ combinations. The 3- mass peak is evident in the distribution of the right-sign events. A Anpair is considered to be a 3- candidate if its mass is within the range 1.305 < M A7T - < 1.340 GeV. 3b" - t J 1'lj;3- decay candidates are formed from J 1'Ij; and 3- pairs that are consistent with originating from a common vertex with a X2 probability greater than 8% or better and have an opening angle in the transverse plane less than n 12 rad. The uncertainty of the proper decay length of the J 1'lj;3vertex must be less than 0.05 cm in the transverse plane. A total of 2308

87

events remains after this preselection. The wrong-sign events are subjected to the same preselection as the right-sign events. A total of 1124 wrong-sign events is selected as the control sample. Several distinctive features of the 2/; ----tJ/1/J2- ----tJ/1/JA7r- ----t(p,+ p,-) (P7r- )7rdecay are utilized to further suppress backgrounds. The wrong-sign background events from the data and MC signal 2/; are used for studying additional event selection criteria. As shown in Fig. 2(b), the proton PT of background events peaks towards lower values. Therefore, protons are required to have transverse momenta greater than 0.7 GeV. Similarly, minimum PT requirements of 0.3 and 0.2 GeV are imposed on pions from A and 2- decays, respectively. These requirements remove 91.6% of the wrong-sign background events while keeping 60.3% of the MC 2/; signal events. Contamination from decays such as B- ----t Jj1jJK*- ----t J/1/JK°7r- and BO ----t J/1/JK*-7r+ ----t Jj1jJ(Kg7r-)7r+ are suppressed by requiring the 2- candidates to have decay lengths greater than 0.5 cm and cOS(Bcol) > 0.99, as 2- baryons are expected to have significant decay lengths. Here Bcol is the angle between the 2- direction and the direction from the 2- production vertex to its decay vertex in the transverse plane. These two requirements on the 2- reduce the background by an additional 56.4 %, while removing only 1.7% of the MC signal events. Finally, 2- baryons are expected to have a sizable lifetime. To reduce prompt backgrounds, the transverse proper decay, the transverse proper decay length significance of the 2/; candidates is required to be greater than two. This final criterium retains 82.0% of the MC signal events but only 43.9% of the remaining background events. In the data, 51 events with the 2/; candidate mass between 5.2 and 7.0 GeV pass all selection criteria. The mass range is chosen to be wide enough to encompass masses of all known b hadrons as well as the predicted mass of the 2/; baryon. The distribution of M(2/;) is shown in Fig.3(a). A mass peak near 5.8 GeV is apparent. A number of cross checks are performed to ensure the observed peak is not due to artifacts of the analysis: (1) Eighteen wrong-sign background events in the mass window 5.2 - 7.0 GeV survive the 2/; selection. The J/1/JA7r+ mass distribution, shown in Fig. 3(b), is consistent with a flat background. (2) The event selection is applied to the data events in the sidebands of the reconstructed 2- mass peak. The A7r- mass is required to be in the range 1.28 - 1.36 GeV excluding the 2- mass window 1.305 - 1.340 GeV while other criteria are kept the same. Similarly, the selection is applied to the J /1/J sideband events with 2.5 < MJ1,J1, < 2.7 GeV. The high-mass side band is not considered due to potential contamination from \[If events. The (p,+ p, -) (P7r- )7r- mass distributions of these sideband events are shown in Fig.3(c-d). No evidence of a mass peak is present for either distribution. (3) The possibility of a fake signal due to the residual b hadron background is investigated by applying the final 2/;

88 selection to Me B- ----> J/'ljJK*- ----> J/'ljJK~7r-, BO ----> J/'ljJK~, and Ab ----> J/'ljJA samples with equivalent luminosities significantly greater than that of the data analyzed. No indication of a mass peak is observed in the reconstructed J /'ljJSmass distributions. (4) The mass distributions of J /'ljJ, S-, and A are investigated by relaxing the mass requirements on these particles one at a time for events both in the Sb signal region and the sidebands. The numbers of these particles determined by fitting their respective mass distribution are fully consistent with the quoted numbers of signal events plus background contributions. (5)The robustness of the observed mass peak is tested by varying selection criteria within reasonable ranges. All studies confirm the existence of the peak at the same mass.

Interpreting the peak as Sb production, candidate masses are fitted with the hypothesis of a signal plus background model using an unbinned likelihood method. The signal and background shapes are assumed to be Gaussian and flat, respectively. The fit results in a Sb mass of 5.774 ± 0.011 GeV with a width of 0.037 ± 0.008 GeV and a yield of 15.2 ± 4.4 events. Unless specified, all uncertainties are statistical. Following the same procedure, a fit to the Me Sb events yields a mass of 5.839 ± 0.003 GeV, in good agreement with the 5.840 GeV input mass. The fitted width of the Me mass distribution is 0.035±O.002 GeV, consistent with the 0.037 GeV obtained from the data. Since the intrinsic decay width of the Sb baryon in the Me is negligible, the width of the mass distribution is thus dominated by the detector resolution. To assess the significance of the signal, the likelihood, Ls+b, of the signal plus background fit above is first determined. The fit is then repeated using only the background contribution, and a new likelihood Lb is found. The logarithmic likelhood ratio J2ln(L s+b/ Lb) indicates a statistical significance of 5.5u, corresponding to a probability of 3.3 x 10- 8 from background fluctuation for observing a signal that is equal to or more significant than what is seen in the data. Including systematic effects from the mass range, signal and background models, and the track momentum scale results in a minimum signicance of 5.3u and a Sb yield of 15.2 ± 4.4(stat.)!6:~(syst.).

Potential systematic biases on the measured Sb mass are studied for the event selection, signal and background models, and the track momentum scale (see more at [1]). So, the resulting measured Sb mass is: 5.774 ± O.Ol1(stat.) ± O.015(syst.) GeV.

A lot of thanks to my DO b-Physics group colleagues, the staffs at Fermilab and collaborating institutions.

89

Figure 1: Decay topology of the :=:;; --> J /1/1:=:- where J /1/1 -> J.t+ J.t- and :=:- --> A7r- --> (P7r-)7r~. The:=:- and A baryons have decay lengths of the order of cm; the :=:;; has an estImated decay length of the order of mm (IP is the primary Interaction Point).

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Figure 2: (a) The effective mass distribution of the A7r pair before the :=:;; reconstruction. Filled circles are from the right-sign A7r- combinations showing a :=:- mass peak while the histogram is from the wrong-sign A7r+ combinations. (b) Distributions of the proton transverse momentum of the wrong-sign background events (dotted histogram) and Monte Carlo signal :=:;; events (solid histogram) after preselection. The signal distribution is scaled to the same number of background events.

90

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Figure 3: (a) The invariant mass distribution of the :=:;; candidates after all selections. The dotted curve is an unbinned likelihood fit to the model of a constant background plus a Gaussian signal. (b - d) The (f.1,+ f.1,- ) (p-rr- )-rr- invariant mass distributions of the wrong-sign background, J/'I/J sideband, and:=:- sideband events.

References

[1] V.M. Abazov et ai. (DO Collaboration), Phys.Rev.Lett. 99, 1052001(2007). [2] J. Abdallah et al. (DELPHI Collaboration), Eur. Phys.J. C44, 299(2005); D.Buskulic et al. (ALEPH Collaboration), Phys.Lett.B 384, 449(1996). [3] V.M. Abazov et al. (DO Collaboration), Nucl. lnstrum. Methods A565, 463(2006). [4] T. Sjostrand et al., Comput. Phys. Commun. 135, 238 (2001). [5] D.J. Lange, Nucl. lnstrum. Methods A462, 152(2001). [6] R. Brun and F. Carminati, CERN Program Library Writeup W5013, 1993 (unpublished). [7] V.M. Abazov et al.(DO Collaboration), Phys.Rev.Lett. 98, 121801(2007). [8] W.-M. Yao et al., Journal of Physics G33, 1(2006).

SEARCH FOR NEW PHYSICS IN RARE B DECAYS AT LHCb V. Egorychev a on behalf of the LHCb collaboration Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia Abstract.We discuss the potential of the LHCb experiment to study rare B decays and their impact on various scenarios for New Physics. Some possible experimental strategies are presented.

1

Introduction

Rare decays in the beauty sector encompass a wide range of processes offering exceedingly valuable tool in the search for New Physics (NP) as well as in precision measurements of the Standard Model (SM) parameters, e.g. the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements. The focus of this paper is made on the processes with the final states containing photons or leptons in addition to daughter hadron(s). The examples considered include the electromagnetic or electroweak penguin decays B --+ K*" Bs --+ ¢,' B --+ K*f+Cand the dilepton decay Bs --+ p,+ p,-. Most of these rare decays correspond to diagrams with internal loops or boxes leading to effective flavor-changing neutral current (FCNC) transitions. Presence of new virtual particles (e.g. the supersymmetric ones) with masses of the order of 100 GeVjc 2 may manifest itself in altering the decay rate, C P asymmetry and other observable quantities. Exciting new perspectives for the B physics emerge owing to the large statistics to be collected by the LHCb experiment, which will enable to enter a new realm of high precision studies of rare B decays.

2

LHCb experiment

The LHCb experiment features a forward magnetic spectrometer with a polar angle coverage of 15-300 mrad and a pseudo-rapidity range of 1.8 < 'f} < 4.9 [1). In order to maximize the probability of a single interaction per bunch crossing, it was decided to limit the luminosity in the LHCb interaction region to '" 2 X 1032 cm- 2 8- 1 . This has the additional advantage of limiting the radiation damage due to the high particle flux at small angles. The bb cross-section at the nominal LHCb luminosity is large enough to produce'" 10 12 bb pairs per year (10 7 s). The detector consists of a silicon vertex locator, followed by a first Ring Imaging Cherenkov Counter (RICH), a silicon trigger tracker, a 4 Tm spectrometer dipole magnet, tracking chambers, a second RICH detector, a calorimeter system and a muon identifier. One of the main features is a versatile trigger with a 2 kHz output rate dominated by pp --+ bbX events. The reconstruction of rare a e-mail:

[email protected]

91

Figure 1: The integrated luminosity required to achieve a 3
Figure 2: The invariant mass distribution for selected B| ^> K*y candidates (the blue filled histogram represents combinatorial background).

B decays at LHCb is a challenge due to their small rates and large backgrounds from various sources. The .B-mesons are separated from .the large background produced directly at the interaction point in a detached vertex analysis by exploiting the relatively long lifetime of B-meson and a large average transverse momentum (pr) of the B-meson decay products. Therefore, the signature for a B event is based upon selection of particles with high px coming from a displaced vertex. The most critical backgornnd is the combinatorial background from pp -> bbX events, containing secondary verteces and characterized by high charged and neutral multiplicities. 3 The search for Bs —> fi+fj,^ The decay Ba —> /A+ fi~ is highly suppressed in the SM, since it can only be produced through a box diagram or a Z penguin. The current SM prediction is Br{Ba —> (J,+IJ,~~) = (3.4±0.5) x 10^9 [2]. In some new physics scenarios, the branching fraction can be enhanced by a high power of tan/? (e.g. Br oc tan6 /?), where tan/? is the ratio of the Higgs vacuum expectation values. For large values of tan/?, the branching fraction could be enhanced by two orders of magnitude, which is currently within the reach of the CDF and DO experiments. The large background (unlike sign muons originating from B decays or B decays into hadrons which are misidentified as muons) expected in the search for this decay is kept under control thanks to an excellent tracking performance of LHCb (namely the invariant mass resolution for dimuons ~ 18 MeV/e2), a good particle identification and good vertex resolution. The LHCb sensitivity

93

as a function of integrated luminosity is shown in Fig 1. LHCb has the potential to claim a three standard deviation observation at the level of the 8M prediction with", 2jb- 1 whereas a five standard deviation observation would require about 10 jb- 1 [3]. 4

The search for NP in b ---; sry and b ---; s£+ £-

Phenomenologically the b ---; sry and b ---; s£+ £- decays are closely linked. 8M calculations for these rare decays are performed using an effective Hamiltonian that is written in terms of several short-distance operators [4]. The process b ---; sry is dominated by the photon penguin operator, with Wilson coefficient C7 , while b ---; s£+£- has contributions also from semileptonic vector and axialvector operators with Wilson coefficients C9 and C lO respectively. To further pin down the values of these coefficients, it is necessary to exploit interference effects between the contributions from different operators. This is possible in the exclusive decay B --7 K*£+£- decays by measuring the forward-backward asymmetry AFB(q2), the longitudinal polarisation fraction of the K*o FL and the second of the two polarisation amplitude asymmetries A~). 4·1

Electroweak penguin decay B~

--7

K* fJ.,+ fJ.,-

The decay B~ --7 K* fJ.,+ fJ.,- is loop-suppressed in the 8M, Br(B~ ---; K* fJ.,+ fJ.,-) = (1.22~g:~~ x 10- 6 ) [5]. NP contributions could drastically change the shape of the AFB(q2) curve. For example, the sign of AFB(q2) can be flipped, the zero-crossing point may be shifted, or AFB(q2) may not even cross zero [6J. The procedure is to measure the AF B asymmetry of the angular distribution of daughter fJ.,+ relative to the B direction in the fJ.,+ fJ.,- rest frame as a function of the fJ.,+ fJ.,- invariant mass. The expected number of events in one year of data taking (2 jb- 1 ) by LHCb is 7200 ± 2100 (the error is due to the branching ratio), with a background-to-signal ratio B / S < 0.5 [7]. LHCb expects to extract the C9 /C7 Wilson coefficients ratio from the value of the fJ.,+ fJ.,- invariant mass for which the AFB is equal to zero to a precision of 13% after 5 years of running (10 jb- 1 ). Taking into accout the expected background level, the resolution with 2jb- 1 of integrated luminosity is 0.016 in FL and 0.42 in A~) [8].

4.2

Radiative decays b --7 sry

The polarization of the photons emitted in the b --7 sry transition provides an important test of the 8M, which predicts most left-handed photons. In the LHCb experiment these radiative decays can be reconstructed in the modes Bd --7 K*ry, Bs --7 ¢ry or Ab --7 Ary. The reconstruction procedures for Bd,s --7 K*(¢h decays are similar. To suppress the background from Bd,s --7 K*(¢)7r° in which the 7r 0 is misidentified as a single photon, a cut on the angle between

94 Table 1: Annual yields and background-to-signal ratios for radiative Ab decays (upper limits calculated at 90 % C ... L)

channel Ab -; ky Ab -; A(1520)-"y Ab -; A(1670)-"y Ab -; A(1690)-"y

yield/2 jb- 1 750 4.2 x 10 3 2.5 x 103 4.2 x 103

B/S 42 10 18 18

< < < <

the B and the K+ in the K*(¢) rest frame is applied. The yield for 2 jb- 1 for B~ -; K*, is expected to be 68 k reconstructed events with background-tosignal ratio 0.71 ± 0.11. For B~ -; ¢, decays the annual yield is estimated to be 11.5 k with B / S < 0.95 at 95 % C.L [9J. The invariant mass distribution for selected B~ -; K*, candidates after 13 minutes of data taking is presented in Fig. 2. The expected signal yield for 2 jb- 1 integrated luminosity together with the estimate of B / S ratios for radiative Ab decays [10] are given in Table 1. 5

Conclusions

The LHCb experiment has an excellent potential for the study of rare B decays sensitive to New Physics in many Standard Model extensions. In the present work the capabilities to study the b -; decay, the asymmetry AFB in the transition b -; s£+£- and the very rare decay Bs -; M+M- have been shown.

s,

Acknowledgments

I'm very grateful to Prof. Clara Matteuzzi, Prof. Andrei Golutvin and Dr. Ivan Belyaev for many fruitful and useful discussions and comments. References

[1] [2J [3] [4J [5J [6] [7] [8] [9] [10]

LHCb Technical Design Report, CERN-LHCC 2003-030. A.J. Buras, Phys.Lett. B 566, 115 (2003). D. Martinez, J.A. Hernando, F. Teubert, CERN-LHCb-2007-033. Y. Grossman, D. Pirjol J. High Energy Phys. 0006029 (2000). W.-M. Yao et al., J. Phys. G 33, 1 (2006). A. Ali et al., Phys. Rev. D 66, 034002 (2002). J. Dickens, V. Gibson, C. Lazzeroni, M. Patel, CERN-LHCb-2007-038. U. Egede, CERN-LHCb-2007-057. L. Shchutska, A. Golutvin, 1. Belyaev, CERN-LHCb-2007-030. F. Legger, CERN-LHCb-2006-012.

CKM ANGLE MEASUREMENTS AT LHCb Sergey Barsuk a Laboratoire de l'Accelerateur Lineaire Universite Paris-Sud 11, Batiment 200, 91898 Orsay, Prance on behalf of the LHCb collaboration Abstract. Expected reach of the LHCb experiment on the CKM angle measurements is discussed on the examples of the Bd,s mixing phases and the angle "(.

1

Introduction

The unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix is visualized in terms of six unitarity triangles (UT) of equal area (Jarlskog invariant), quantifying the C P violation. Two of them, bd and tu, have comparable sides, and are the most relevant for b physics. Owing to the results from B-factories and Tevatron, the precision of UT parameters have significantly improved, the UT apex is precisely constrained. The angles are known to the precision of b.a ;::::: 6°, b.j3 ;::::: 1° and b.'Y ;::::: 20°, where 13 and 'Yare still dominated by experimental error. The precision of the sides is dominated by theoretical uncertainties. The Rb side is determined to theoretical uncertainty of rv 8% from Vub measurement, while the R t side (IVid/Visi) is known to a precision of rv 5%. For both Rb and R t limitations come from lattice calculations. For R t improvement is expected from radiative penguin studies. Comparing the precision of UT angle to the precision of the opposite side, we notice, that constraining the apex with the (13, Rb) is limited by the Rb precision, while constraining the apex with b, R t ) is limited by the precision on 'Y. Already present knowledge of the R t requires the angle 'Y to be measured with a precision of 5°. LHCb is an experiment dedicated to the study of CP violation and other rare phenomena in b decays [1,2]. The LHCb detector is a single-arm forward spectrometer, insuring efficient charged particle tracking and neutral particle reconstruction, particle identification and robust trigger. The experiment will run at a reduced LHC luminosity tuneable in the range 2 -;- 5 X 1032 cm- 2 8- 1 . At LHCb a 10 12 bb pairs will be annually produced, including all the b hadron species, with the biggest samples of Bu, B d, Bs and A b, Bu:Bd:Bs:Be:Ab;::::: 4:4:1:0.1:1. Complementing detailed studies of lighest B mesons at B-factories and Tevatron, the LHCb experiment is expected to contribute to the studies of Bs, Be and Ab. The LHCb physics program mainly comprises precision measurements of the Standard Model (SM) parameters and search for effects beyond successful SM description via CP asymmetries and rare decays, with the b physics as a central actor. The CKM angle measurements are discussed below, while for rare decays reach at LHCb, see e.g. [3]. ae-mail: [email protected]

95

96

2

Bd,s mixing phases

The exploration of the UT angles covers the mixing phases, ¢d = 213 for Bd system and ¢s = -2X for Bs system, and the 0: and, angles. Most awaited are the study of ¢s phase, which in the SM is expected to be small, 0.0037, and thus attractive for the new physics (NP) search, and the angle, to constrain the UT triangle in combination with the R t side and to search for possible NP contribution to loops by comparing the tree-mediated processes to those involving penguin loops. These studies rely on the C P (often time-dependent) asymmetry measurements. The first asymmetry study will be that of the ¢d phase, using the Bd ---. J I'l/JKs decay. The comparison with the value well-established by B-factories, provides a powerful systematics control for further asymmetry studies. Also important is to establish the direct CP violating term from the Bd ---. J I'l/JKs decay asymmetry. In one year b , having a 240k clean (SIB cv 1.4) signal events reconstructed, LHCb should be able to achieve the precision of 0.02 on the sin 213 value c. This can be compared to the BABAR and BELLE combined precision of cv 0.018 expected at the end of the B-factories, and to an error of 0.01 for 30 fb- 1 expected with ATLAS and CMS d [4]. The best channel for the ¢s phase measurement is B s ---. J I 'l/J ¢ with J I 'l/J ---. p,+ p,- and ¢ ---. K+ K-. Bs ---. J I'l/J ¢ is a P ---. VV decay, and the final state is thus a mixture of CP = +1 and CP = -1 components. To disentangle CP eigenstates, a partial wave analysis is required. LHCb is expected to reconstruct 131k Bs ---. JI'l/J¢ events with SIB ratio of 10. This results in a ¢s sensitivity of 0.01 for the integrated luminosity of 10 fb- 1 . This is better than ATLAS and CMS expected sensitivities of 0.04 for an integrated luminosity of 30 fb- 1 . Including pure CP eigenstates, Bs ---. ryc(h+h-h+h-)¢(K+K-), Bs ---. JI'l/J(p,+p,-)ry(r,), Bs ---. JI'l/Jry(n+n-n°(r,)), Bs ---. JNry'(n+n-ry(r,)), Bs ---. Dt (K+ K-n+ )D-; (K+ K-n-), despite the smaller statistics, provides a control over the method. A by-product of this analysis is the ~r s measurement. The expected LHCb sensitivity on ~r sir s of about 0.01 is small compared to the SM prediction of ~rs/rs cv 0.1. Following the Jj3NP estimation, LHCb will compare the X angle from treemediated diagram, Bs ---. JI'l/J¢, to that from pure penguin decay, Bs ---. ¢¢. This will yield an estimate of the NP contribution, JX NP = Xtree _ xpenguin, with sensitivity of 3° in one year of data taking. The NP. contribution in Bs mixing could be parametrized [5] via M12 = (1 + hs • e 2t O"s )M~M, where M~M is the dispersive part in the SM. Then ~ms bThroughout the paper a nominal LHCb year corresponds to 2 fb- 1 . CThroughout the paper only expected statistical error is quoted. dThe integrated luminosity of 30 fb- 1 corresponds to the ATLAS and CMS running at luminosity of about 1033 cm- 2 s- 1 , where these experiments expect to study b physics.

7

3 CKM angle 7

s of D,K from

R, -+ D,K decay with - with major refiectio

(c) particle i d ~ n t i ~ c a t i o n

o achieve a §tati§tic~l error of

N

98

becomes overconstrained, also to check the initial assumption of U-spin symmetry itself. This analysis requires reliable K/7r separation (Fig. 1b,c). In one year LHCb is expected to reconstruct 25k Bd ----t 7r7r events and 37k Bs ----t KK events with S / B of 2 and > 7 respectively, leading to 0'(')') rv 4° under the assumption of perfect U-spin symmetrye. 4

Outlook

With an accumulated luminosity of 10 fb- 1 LHCb will be able to measure the angle 'Y to a precision 0' stat rv 5° from tree-mediated processes, 0' stat rv 2° from processes where NP could enter DO mixing, and O'stat rv 2° (under U-spin symmetry assumption) from processes involving penguin loops, thus providing a powerful probe for NP. The Bs mixing phase 1>s will be measured to a precision O'stat rv 0.01 providing a constraint on NP by comparing tree-mediated with pure penguin processes. References [1] LHCb Collaboration, LHCb Reoptimized Detector Design and Performance TDR, CERN LHCC 2003-30. [2] B. Spaan, talk at this conference. [3) V. Egorychev, talk at this conference. [4] G.F. Tartarelli, Eur.Phys.J.direct C4S1 (2002) 35. [5] Z. Ligeti et al., hep-ph/0604112. [6] S. Cohen, M. Merk, E. Rodrigues, CERN-LHCb-2007-041. [7] M. Gronau, D. London, Phys.Lett. B253 (1991) 483; M. Gronau, D. Wyler, Phys.Lett. B265 (1991) 172. [8] D. Atwood, I. Dunietz, A. Soni, Phys.Rev.Lett. 78 (1997) 3257. [9] A. Giri, Yu. Grossman, A. Soffer, J. Zupan, Phys.Rev. D78 054018 (2003). [10] M. Patel, CERN-LHCb-2007-043; V. Gibson, C. Lazzeroni, J. Libby, CERN-LHCb-2007-048; K. Akiba, M. Gandelman, CERN-LHCb-2007050; J. Libby, A. Powell, J. Rademacker, G. Wilkinson, CERN-LHCb2007-098. [11] R. Fleischer, Phys.Lett. B459 (1999) 306. [12] J. Nardulli, talk at the 4th Workshop on the CKM Unitarity Triangle, December 2006.

eSensitivity of the method degrades with the U-spin symmetry breaking [12]. With no constraints on OTrTr,K K and 20% breaking of d TrTr = dK K, O'(-y) "" 10°. The method is believed to fail for larger U -spin symmetry breaking.

COLLIDER SEARCHES FOR EXTRA SPATIAL DIMENSIONS AND BLACK HOLES Greg Landsberg a Brown University, Department of Physics, 182 Hope St., Providence, RI02912, USA Abstract. Searches for extra spatial dimensions remain among the most popular new directions in our quest for physics beyond the Standard Model. High-energy collider experiments of the current decade should be able to find an ultimate answer to the question of their existence in a variety of models. We review these models and recent results from the Tevatron on searches for large, TeV-1-size, and Randall-Sundrum extra spatial dimensions. The most dramatic consequence of low-scale (~ 1 TeV) quantum gravity is copious production of mini-black holes at the LHC. We discuss selected topics in the mini-black-hole phenomenology.

1

Models with Extra Spatial Dimensions

A new, string theory inspired paradigm [1] proposed by Arkani-Hamed, Dimopoulos, and Dvali (ADD) in 1998 suggested the solution to the hierarchy problem of the standard model (SM) by introducing several (n) spatial extra dimensions (ED) with the compactification radii as large as ~ 1 mm. These large extra dimensions are introduced to solve the hierarchy problem of the SM by lowering the Planck scale to a TeV energy range. (We further refer to this fundamental Planck scale in the (4+n)-dimensional space-time as MD') In this picture, gravity permeates the entire multidimensional space, while all the other fields are constrained to the 3D-space. Consequently, the apparent Planck scale M p1 = l/JGN only reflects the strength of gravity from the point of view of a 3D-observer and therefore can be much higher than the fundamental (4+n)-dimensional Planck Scale. The size of large extra dimensions (R) is fixed by their number, n, and the fundamental Planck scale MD. By applying Gauss's law, one finds [1,2]: M~l = 87rM£;+2 Rn. If one requires MD ~ 1 TeV and a single extra dimension, its size has to be of the order of the radius of the solar system; however, already for two ED their size is only ~ 1 mm; for three ED it is ~ 1 nm, i.e., similar to the size of an atom; for larger number of ED it further decreases to subatomic sizes and reaches ~ 1 fm for seven ED. Almost simultaneously with the ADD paradigm a very different low-energy utilization of the idea of compact extra dimensions has been introduced by Dienes, Dudas, and Gherghetta [3]. In their model, additional dimension(s) of the "natural" EWSB size of R ~ 1 TeV- 1 [4] are added to the SM to allow for low-energy unification of gauge forces. In conventional SM and its popular extensions, such as super symmetry, gauge couplings run logarithmically with energy, which is a direct consequence of the renormalization group evolution (RGE) equations. Given the values of the strong, EM, and weak couplings at low energies, all three couplings are expected to "unify" (i.e., reach the same ae-mail: [email protected]

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100

strength) at the energy'" 10 13 TeV, know as the Grand Unification Theory (GUT) scale. However, if one allows gauge bosons responsible for strong, EM, and weak interactions to propagate in extra dimension(s), the RGE equations would change. Namely, once the energy is sufficient to excite Kaluza-Klein (KK) modes of gauge bosons (i.e., '" 1I R '" 1 Te V), running of the couplings is proportional to a certain power of energy, rather than its logarithm. Thus, the unification of all three couplings can be achieved at much lower energies than the GUT scale, possibly as low as 10-100 TeV [3]. While this model does not incorporate gravity and thus does not explain its weakness relative to other forces, it nevertheless removes another hierarchy of a comparable size - the hierarchy between the EWSB and GUT scales. In 1999, Randall and Sundrum offered a rigorous solution [5] to the hierarchy problem by adding a single extra dimension (with the size that can range anywhere from", 1/MpI virtually to infinity) with a non-Euclidean, warped metric. They used the Anti-deSitter (AdS) metric (i.e. that of a space with a constant negative curvature) ds 2 = exp( -2kRI
=

Numerous attempts to find large ED or constrain the ADD model have been carried out since 1998. They include measurements of gravity at short distances, studies of various astrophysical and cosmological implications of large ED, and numerous collider searches for virtual and real graviton effects. For detailed reviews of the existing constraints and the sensitivity of future experiments, see Ref. [6]. The host of experimental measurements conducted to date have largely disfavored only the case of two or less large ED; for any larger number of them, the lower limit on the fundamental Planck scale is only'" 1 TeV, hardly probing the most natural range of scales expected in the ADD model.

101

As was pointed out a decade ago [7], an exciting consequence of TeV-scale quantum gravity is the possibility of producing black holes (BH) in high-energy interactions, accessible at colliders or by ultra-high-energy cosmic rays. More recently, this phenomenon has been quantified for the case of TeV -scale particle collisions [8], resulting in a mesmerizing prediction that the LHC would produce mini black holes at an enormous rate (e.g., ,...., 1 Hz for MD = 1 TeV), thus becoming a black-hole factory. This observation led to an explosion of followup publications on the properties of mini-black holes produced in the lab and made this subject one of the most actively studied aspects of phenomenology of models with extra dimensions. Here we review only some of the basic facts in phenomenology of black holes. For more extensive reviews, including the latest developments, see Ref. [9]. 2

Collider Searches for Extra Dimensions

The most experimentally interesting feature of the above models with ED is rich low-energy phenomenology that originates from the KK spectrum of various particles propagating in ED. In what follows we compare the KK spectrum observed in the ADD, TeV- 1 , and RS models and discuss various experimental constraints on the model parameters. In the ADD model, the only particle that propagates in ED and acquires KK modes is the graviton. Given the size of ED (,...., 10- 3 _10- 15 m), energy spacing between the KK excitations of the graviton is ,...., 1 meV - 100 MeV. Consequently, the adjacent modes are hard to resolve and in most of the experiments the KK spectrum would appear continuous, from zero to a certain ultraviolet cutoff above which quantum gravity effects would modify this semiclassical picture. Since the fundamental Planck scale in the ADD model is MD ,...., 1 TeV, it's natural to expect this cutoff, Ms, to be of the same order: Ms ,...., MD. While each KK mode couples to the energy-momentum tensor with the gravitational strength G N, the sheer number of the available KK modes is sufficient to enhance gravitational attraction tremendously. In the TeV- 1 model with a single extra dimension of the size R, the zeroth KK mode of a gauge boson is the 8M particle of mass Mv. This mass can be either a (photon, gluon) or > a (W, Z). The mass of the i-th KK mode is given by Mi = JM~ + i 2/R2. For the compactification scale Me == l/R,...., 1 TeV, 1/ R » Mv and hence the non-zeroth KK modes for all gauge bosons are nearly degenerate in mass, as the mass of the i-th mode is approximately equal to iMe for any i > O. The KK tower of excited gauge bosons is expected to be truncated at the masses of the order of the new GUT scale (,...., 100 TeV), where new physics (e.g., brane dynamics) should take over. Finally, in the Randall-Sundrum model the zeroth KK mode of the graviton remains massless, while the higher modes have masses spaced as subsequent

102

zeros of the Bessel's function J 1 . If the mass of the first excited mode of the graviton is M 1, the subsequent excitations would have masses of 1. 83M1, 2.66M1' 3.48M1, 4.30M1, etc. The spacing between the adjacent KK excitations decreases at high masses. The zeroth excitation is coupled to the energymomentum tensor with the gravitational strength G N, while each of the higher excitations couples with the strength", 1/ A;;', i.e., comparable to the EW coupling strength. The excited modes are presumably truncated at masses'" Mpl, where brane excitations would modify the discussed behavior. Since KK gravitons couple to the energy-momentum tensor, they can be added to any vertex or line of any SM Feynman diagram accessible at colliders. Consequently, to probe the ADD model one could look for direct emission of KK gravitons, e.g., in the qq -+ g/'Y + GKK process, which results in a single jet or photon (monojet or monophoton) in the observable final state. The experimental signature for such events is an apparent transverse momentum nonconservation and an overall enhancement of the tail of the jet / photon transverse energy spectrum. Another type of effect observable in high-energy collisions is an enhancement of Drell-Yan or diboson spectrum at high invariant masses due to additional diagrams involving virtual KK graviton exchange. Searches for virtual graviton effects are complementary to those for direct graviton emission, since the former depend on the ultraviolet cutoff of the KK spectrum, Ms, while the latter depends directly on the fundamental Planck scale MD. While direct graviton emission cross section is well defined [2, 10], the cross section for virtual graviton exchange depends on a particular representation of the interaction Lagrangian and the definition of the ultraviolet cutoff Ms. We will consider two such representations [2,11]. In both of them, the effects of ED are parameterized via a single variable 'T}e = F /M'];, where F is a dimensionless parameter of order one reflecting the dependence of virtual G KK exchange on the number of extra dimensions. Ref. [2] simply fixes Fat 1, while an attempt to resolve the n-dependence is made in [11]. In both cases the interference with the SM diagrams is assumed to be constructive. LEP experiments have pioneered searches for large extra dimensions at colliders in both direct graviton emission in the e+ e- -+ 'Y / Z + GKK channel and via virtual graviton effects in fermion pair, as well as in diboson production. The best sensitvity is achieved in the e+e- -+ 'Y+G KK and in the e+e- -+ e+e- h'Y channels. For a review of LEP limits see, e.g., Refs. [6,12]. Both the CDF and D0 Collaborations at the Fermilab Tevatron pp collider sought for large ED since Run 1. D0 has pioneered searches for virtual graviton effects in the dielectron and dip hot on channels at hadron colliders [13], as well as searches in the challenging monojet channel [14] plagued by copious instrumental backgrounds from jet mismeasurement and cosmics. CDF has pioneered searches in the monophoton channel at hadron colliders [15] and also accomplished a monojet analysis [16].

103 Table 1: Most recent 95% CL lower limits on the fundamental Planck scale MD (in TeV).

Experiment and channel LEP Combined [12] CDF monophotons, 2.0 fb [18] D0 monophotons, 2.7 fb- 1 [19] CDF monojets, 1.1 fb I [20] CDF combined [18]

n=2

n=3

n=4

n=5

n=6

1.60 1.08 0.97 1.31 1.42

1.20 1.00 0.90 1.08 1.16

0.94 0.97 0.87 0.98 1.06

0.77 0.93 0.85 0.91 0.99

0.66 0.90 0.83 0.88 0.95

Table 2: Recent 95% CL lower limits on the ultraviolet cutoff Ms (in TeV) from the Tevatron Run 2. NLO QCD effects have been accounted for via a 8M K-factor.

D0 Signature ee + ", 1.1 fb 1 [21] Dijets, 0.7 fb- 1 [22]

GRW [2] 1.62 1.56

HLZ [11]

n=2

n=3

n=4

n=5

n=6

n=7

2.09

1.94 1.85

1.62 1.56

1.46 1.41

1.36 1.31

1.29 1.24

Recently these results have been updated with 1-3 fb- 1 of data collected in Run 2 in the dimuon, dielectron, diphoton, monophoton, and monojet channels [17-21] and also for the first time using the dijets [22]. These limits are listed in Tables 1 and 2. The tightest limits on MD come from the CDF combination of the monojet and monophoton channels and combined LEP limits. CDF limits exceed LEP limits for n 2:: 4. The tightest limits on the ultraviolet cutoff Ms come from the D0 experiment in the combined ee +" channel. As of the beginning of Run 2, there have been no dedicated searches for Tey-l extra dimensions. However, a number of constraints have been derived by phenomenological analysis of the existing data. For a review of indirect cosntraints see, e.g., Ref. [23]. The best limits come from LEP electroweak precision measurements; the combined limit on the compactification scale of the Tey-l dimensions Me approaches 6.8 TeY for a single Tey-l extra dimension. By now, the Tey-l extra dimensions have been looked for via virtual gauge boson excitation effects in the dielectron (200 pb- 1 [24]) and dijet (700 pb- 1 [22]) channels at D0. The preliminary lower 95% CL limits on the compactification scale Me of a single extra dimension set in these analyses are 1.12 and 1.4 TeY respectively. Both the CDF and D0 collaborations pioneered direct searches for the effects of Kaluza-Klein gravitons in the Randall-Sundrum model in Run 2 [25,26]. As discussed above, the simplest RS model is fixed by specifying just two parameters: the curvature of the AdS space k and the radius of compactification R. However, from the experimental point of view, it is more convenient to work with an equivalent set of parameters, which correspond to direct observables: M 1 , the mass of the first Kaluza-Klein excitation of the graviton, and a dimensionless parameter k = k / M PI, which governs the coupling of the gravitons to the SM fields and hence defines the internal width of the KK gravitons. Both

104

the CDF and D0 Collaborations used the dilepton and diphoton mass spectra and searched for a narrow resonance, which would indicate a production of the first excitation of the KK graviton. The best sensitivity comes from the 1 fb- I combined D0 ee + I I analysis [25] and 2.5 fb- I CDF J.LJ.L analysis, which exclude RS gravitons with masses below 0.9, 0.7, and 0.3 TeV for k = 0.1, 0.05, and 0.01, respectively. Ultimate sensitivity to large, Te V-I, and RS extra dimensions will be achieved very soon at the LHC. In most of the cases, just", 1 fb- I of data is sufficient to discover them or severely constrain parameters of these models. The LHC would also allow to search for the KK excitations of the gauge bosons directly. Undoubtedly, the most exciting possibility is production of mini-black holes at the LHC [8] possible in the ADD and RS models. 3

Black Holes at the LHC

Consider two partons with the center-of-mass energy v's = MBR colliding headon. If the impact parameter of the collision is less than the (higher dimensional) Schwarzschild radius, corresponding to this energy, a black hole (BH) with the mass MBR is formed. Therefore the total cross section of black hole production in particle collisions can be estimated from pure geometrical arguments and is of order 7rR~. BH production is expected to be a threshold phenomena and the onset is expected to happen for a minimum black hole mass'" MD (ADD) or An (RS). The total production cross section above this threshold at the LHC, ranges between 15 nb and 1 pb for the Planck scale between 1 TeV and 5 TeV, and varies by less than a factor of two for n between 1 (RS) and 7. Note that this cross section is comparable with, e.g., tt production cross section, which result in '" 1 Hz signal event rate at the nominal LHC luminosity. Once produced, mini black holes quickly ('" 10- 26 s) evaporate via Hawking radiation [27] with a characteristic temperature of", 100 GeV [8]. The average multiplicity of particles produced in the process of BH evaporation is given by [8] and is of the order of half-a-dozen for typical BH masses accessible at the LHC. Since gravitational coupling is flavor-blind, a BH emits all the;::::;: 120 SM particle and antiparticle degrees of freedom with roughly equal probability. Accounting for color and spin, we expect ;: : ;: 75% of particles produced in BH decays to be quarks and gluons, ;: : ;: 10% charged leptons, ;: : ;: 5% neutrinos, and ;: : ;: 5% photons or W/Z bosons, each carrying hundreds of GeV of energy. A relatively large fraction of prompt and energetic photons, electrons, and muons expected in the high-multiplicity BH decays would make it possible to select pure samples of BH events, which are also easy to trigger on [8]. The reach of a simple counting experiment extends up to Mp ;: : ;: 9 TeV (n = 27), for which one would expect to see a handful of BH events with negligible background.

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4

Conclusions

If TeV-scale gravity is realized in nature, its detection via Kaluza-Klein effects

as well as the possibility of production and detailed studies of black holes in the lab are just a few years away. That would mark an exciting transition for astroparticle physics: its true unification with cosmology - the "Grand Unification" to strive for. Acknowledgments

I would like to thank the organizers of the 13th Lomonosov Conference on Elementary Particle Physics for a kind invitation, warm hospitality, and a great conference. This work has been partially supported by the U.S. Department of Energy under Grant No. DE-FG02-91ER40688 and the National Science Foundation under the CAREER Award PHY-0239367. These proceedings were written while I was visiting Aspen Center for Physics, which I would like to thank for its hospitality and support. References

[1] N.Arkani-Hamed, S.Dimopoulos, and G.Dvali, Phys. Lett. B 429, 263 (1998); Phys. Rev. D 59,086004 (1999); LAntoniadis, N.Arkani-Hamed, S.Dimopoulos, and G.Dvali, Phys. Lett. B 436, 257 (1998); N.ArkaniHamed, S.Dimopoulos, and J.March-Russell, hep-th!9809124 (1998). [2] G.Giudice, R.Rattazzi, and J.Wells, Nucl. Phys. B544, 3 (1999), and a revised version hep-ph!9811291v2 (2000). [3] K.R.Dienes, E.Dudas, and T.Gherghetta, Phys. Lett. B 436, 55 (1998); Nucl. Phys. B 537,47 (1999); ibid. 567, 111 (2000). [4] LAntoniadis, K.Benakli, and M.Quiros, Phys. Lett. B 460, 176 (1999). [5] 1. Randall and R. Sundrum, Phys. Rev. Lett. 83,3370 (1999); ibid. 83, 4690 (1999). [6] D.Bourilkov, hep-ex!0103039 (2001); J.L.Hewett and M.Spiropulu, Ann. Rev. Nucl. Part. Sci. 52, 397 (2002); G.Landsberg, hep-ex!0412028 (2004). [7] P.C.Argyres, S.Dimopoulos, and J.March-Russell, Phys. Lett. B441, 96 (1998); T.Banks and W.Fischler, JHEP 9906, 014 (1999); R.Emparan, G.T.Horowitz, and R.C.Myers, Phys. Rev. Lett. 85, 499 (2000). [8] S.Dimopoulos and G.Landsberg, Phys. Rev. Lett. 87, 161602 (2001); S.B.Giddings and S.Thomas, Phys. Rev. D 65, 056010 (2002). [9] M. Cavaglia, Int. J. Mod. Phys. A 18, 1843 (2003); G. Landsberg, Eur. Phys. J. C 33, S927 (2004); J. Phys. G 32, R337 (2006); P. Kanti, Int. J. Mod. Phys. A 19,4899 (2004); arXiv:0802.2218 (2008).

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[10] E.A.Mirabelli, M.Perelstein, and M.E.Peskin, Phys. Rev. Lett. 82, 2236 (1999) . [11] T.Han, J.Lykken, and R.Zhang, Phys. Rev. D 59, 105006 (1999) and a revised version hep-ph/9811350v4 (2000). [12] S.Ask, hep-ex/0410004 (2004). [13] B.Abbott et al. (D0 Collaboration), Phys. Rev. Lett. 86,1156 (2001). [14] V.M.Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 90, 251802 (2003) . [15] T.Affolder et al. (CDF Collaboration), Phys. Rev. Lett. 89, 281801 (2002) . [16] T.Affolder et al. (CDF Collaboration), Phys. Rev. Lett. 92, 121802 (2004). [17] V.M.Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 95, 161602 (2005); ibid. 101011601 (2008); A.Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 97,171802 (2006). [18] CDF Collaboration, http://www-cdf . fnal. gOY /physics/ exotic/r2a /20071213. gammamet/LonelyPhotons/photonmet. html (2007). [19] D0 Collaboration, D0 Note 5729-CONF, http://www-dO . fnal. gOY / Run2Physics/WWW/resul ts/prelim/NP /N63/N63. pdf (2008). [20] CDF Collaboration, http://www-cdf . fnal. gOY /physics/ exotic/r2a /20070322. mono j et/public/ykk. html (2007). [21] D0 Collaboration, to be submitted to Phys. Rev. Lett.; O.StelzerChilton, talk at the 34th Int. Conf. on High Energy Physics, ICHEP 08, http://www.hep.upenn.edu/ichep08/talks/misc/schedule (2008). [22] D0 Collaboration, D0 Note 5729-CONF, http://www-dO .fnal. gov/ Run2Physics/WWW/results/prelim/QCD/Ql1/Q11. pdf (2008). [23] K.Cheung and G.Landsberg, Phys. Rev. D 65, 076003 (2002). [24] D0 Collaboration, D0 Note 4349-CONF, http://www-dO.fnal.gov/ Run2Physics/WWW/results/prelim/NP /N02/N02. pdf (2004). [25] V.M.Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 95, 091801 (2005); ibid. 100,091802 (2008); [26] A.Abulencia et al. (CDF Collaboration), Phys. Rev. Lett. 95. 252001 (2005); A.Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 99, 171801 (2007); ibid. 99, 171802 (2007); arXiv:0801.1129 (2008); CDF Note 9160-PUB http://www-cdf . fnal. gOY /physics/ exotic/ r2a/20080306.dielectron_duke/pub25/cdfnote9160_pub.pdf (2008); CDF Collaboration, http://www-cdf . fnal. gOY /physics/ exotic/r2a/ 20080710. dimuon...resonance/ (2008). [27] S.W.Hawking, Commun. Math. Phys. 43, 199 (1975).

Neutrino Physics

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RESULTS OF THE MINIBOONE NEUTRINO OSCILLATION EXPERIMENT Zelimir Djurcic a for the MiniBooNE Collaboration b Columbia University, Department of Physics, New York, NY 10027, USA Abstract.This article reports the results and status of a v/, -+ Ve oscillation search in MiniBooNE (Booster Neutrino Experiment) experiment. In the oscillation analysis we found no significant excess of events above background for neutrino energies above 475 MeV. In this energy region the data are consistent with no oscillation within a two neutrino appearance-only oscillation hypothesis. An excess of events (186±27(stat.) ±33(syst.)) is observed below the 475 MeV oscillation search cut.

1

Introduction

The MiniBooNE experiment was motivated by the result of the LSND [1 J experiment which presented evidence for neutrino DI-' --+ De oscillation with /j.m 2 at '" 1eV 2 value. If the LSND neutrino oscillation evidence confirmed, it would, together with solar, reactor, atmospheric, and accelerator oscillation data, imply Physics Beyond the Standard Model, such as the existence of a light sterile neutrino [2J. Oscillation behavior is characterized by the "LIE" ratio, where L is the distance from the neutrino source to the detector, and E is the energy of the neutrino beam. The MinibooNE configuration is tuned to have the same LIE ratio as LSND and therefore to probe similar region of oscillation parameter space, but with different systematic uncertainties. The distance the neutrinos traveled was '" 500m ('" 30m), with a neutrino energy of Ev '" 700MeV(20 < Ev < 53 MeV) in the case of MiniBooNE (LSND). While LSND observed an excess of De events in a DI-' beam, MiniBooNE performed a VI-' --+ Ve search. The initial oscillation analysis was performed within a two neutrino appearance-only vI-' --+ Ve oscillation hypothesis.

2

The MiniBooNE Experiment

MiniBooNE is a fixed target experiment currently taking data at the Fermi National Accelerator Laboratory. The neutrino beam is produced from 8.89 GeV Ic protons impinging on a 71 cm long and 1 cm diameter beryllium target inside a toroidal magnetic horn. MiniBooNE collected 5.579 x 10 20 protons on target in a data sample used for the neutrino oscillation analysis. The neutrino beam is composed of vI-' from K+ 17r+ --+ J.L+ + VI-' decays. The beam propagates through a 450 m of a dirt absorber before entering the detector. There is a small contamination from intrinsic Ve in the beam, with the flux ratio velv/, '" 0.6%. The processes that contribute to the intrinsic Ve contamination in the beam are J.L+ --+ e+veDI-" K+ --+ 7r°e+ve , and K2 --+ 7r±e±ve. The MiniBooNE detector is a e-mail:

zdj [email protected] bThis work is supported by NSF and DOE.

109

110

described in [3]. Neutrino interactions in the data sample are charged-current quasi-elastic (CCQE) scattering (39%), neutral-current (NC) elastic scattering (16%), charged-current (CC) single pion production (29%), and NC single pion production (12%). Neutrino induced events are identified by requiring that the event be observed during the beam spill, by requiring the veto to have fewer than six hits to ensure there is no cosmic muon contamination, and by requiring the tank to have greater than 200 hits to suppress decay electrons from cosmic muons. In the data analysis the times and charges of the PMT hits in the detector are used to reconstruct the interaction point, event time, energies, and the particle tracks resulting from v interactions. Particle identification is based on characteristics of the Cerenkov and scintillation light associated with different outgoing particles. Muons have a sharp outer Cerenkov ring filled by a muon travel distance; NC nO events result in two Cerenkov rings; electrons (that might be from an oscillation signal) have one fuzzy ring due to multiple scattering and the electromagnetic shower process.

3

Oscillation Analysis

MiniBooNE performed a blind analysis in order to conduct an unbiased oscillation search. Ve selection cuts and systematic uncertainties were optimized and evaluated prior to unblinding of Ve data set. An oscillation signal would correspond to an excess of candidate Ve events over expectation. Therefore, understanding of expected events was a key to the oscillation search. The uncertainties in the expectation are a dominant factor in determining the sensitivity of the experiment. Systematic errors are associated with v flux, v cross sections, and the detector model. The flux prediction includes the uncertainties corresponding to the production of n, K, and KL particles in the target. These uncertainties are quantified by a fit to external data sets from previous experiments on meson production. The cross section uncertainties are evaluated by continuously varying underlying cross section model parameters in the Monte Carlo constrained by MiniBooNE data. Uncertainties on the parameters modeling the optical properties of the oil in the MiniBooNE detector are constrained by a fit to the calibration sample of muon-decay electrons. The main analysis used a maximum likelihood method to separate CCQE ve-like interactions from the background. The events are fit under the single electron and muon ring hypotheses, and a ratio is formed with the resulting likelihoods. Events with likelihood ratios favoring the muon hypothesis are rejected. The event is then fit with two ring fits, with the mass both unconstrained and fixed to the nominal nO mass. The nO-like events are rejected based on the electron/no likelihood ratio and nO mass. The likelihood ratios and nO mass separation cuts are optimized and applied as functions of visible energy. The estimated number of background events with reconstructed v energy Ev between 475 and 1250

MiniBooNE data (stat error)

+ expected background (syst error) vp background v, background

LSND9Q% CL LSND 99% C.L

number of candidate I/e events as a function of reconstructed 1/ the data with statistical error; the histogram is the ba<3kgrOlmd systematic error. The vertical dashed line indicates the two-neutrino oscillation The background contributions form CL limit (thick solid line) for events MiniBooNE analyzed within a two-neutrino oscillation model. The limits from both the and experiments are shown as welL The MiniBooNE and Bugey l-sided upper limits on 8in z 20 corresponding to 1.64; the KARMEN curve is "uni··· fled approach" contour. The shaded areas show 90% and 99% CL allowed regions from the LSND experiment. re>rH'p<,P'"

range oscillation shows candidate

fit

= 1.64. The result is shown in is excluded at 90% OL. An excess of events is shown Table 1 lists the lJeMU,-"",,

112 Table 1: Preliminary results for the observed data and predicted background event numbers in reconstructed v energy, E v , bins. The total background is broken down into the intrinsic Ve and vI'- induced components. The vI'- induced background is further broken down into its separate components.

Ev[MeV] Total Background I/e intrinsic 1/1'- induced NC 7ru NC t:. --> N, Dirt Other Data Data - Background Data

Bac1t[lrouna

2 2 -)8 _ Data +8 Background

[er]

200-300 284±25 26 258 115 20 99 24 375±19 9l±31 2.50

300-475 274±21 67 207 76 51 50 30 369±19 95±28 2.81

475-1250 358±35 229 129 62 20 17 30 380±19 22±40 0.50

bins, separated into various components. The low Ev region is dominated by NC 1To, radiative t:. decay, and dirt events, in contrast to the region above 475 MeV, which is dominated by intrinsic I/e events. All these events are measured and constrained by the MiniBooNE data with uncertainties quoted in Table l. The excess is attributed to either electrons or gammas, as the MinibooNE detector cannot distinguish an outgoing electron from a single gamma from a 1/ interaction. The excess events originate either from a background process not included in the prediction, or a new physics phenomenon. Both possibilities are being actively pursued in current analyses. Currently the MiniBooNE collaboration is analyzing off-axis neutrinos observed in the MiniBooNE detector originating from the NuMI/Minos beam. These events will provide an important complementary analysis of the 1/ spectrum since the energy and distance is similar to the Fermilab Booster 1/ beam. The NuMI events are dominated by intrinsics I/e in low Ev region and therefore subject to different systematics. In summary, while there is presently an unexplained excess of I/e-like events at low energy, the excellent agreement in the oscillation region permits MiniBooNE to rule out the LSND result interpreted as two neutrino 1//1 --) I/e oscillation described by the standard L/E dependence. References

[1] [2] [3] [4]

A.Aguilar et al.,Phys.Rev.D64 (2001) 112007. M.Sorel, J.M.Conrad, and M.Shaevitz,Phys.Rev.D70 (2004) 073004. Z. Djurcic, Nucl.Phys.Proc.Supp1.168 (2007) 309,arXiv:hep-ex/0701017. A.Aguilar et al.,Phys.Rev.Lett.98 (2007) 231801,arXiv:0704.1500.

MINOS RESULTS AND PROSPECTS J.P.Ochoa-Ricomf for the MINOS Collaboration Lauritsen Lab, California Institute of Technology, Pasadena, CA 91125 USA Abstract. We report on the updated measurement of muon neutrino disappearance observed in the MINOS detectors. These preliminary results are determined from an exposure of 2.5 x 10 20 protons on the NuMI target and incorporate several improvements to our analysis. From a maximum likelihood fit to the reconstructed vI' energy spectra we obtain the neutrino squared-mass difference l~m~21 = (2.38:+:g:ig) x 1O- 3 eV 2 and mixing angle sin2(2023) = 1.00-0.08 with errors quoted at the 68% confidence level. We also report on the outlook for future analyses such as the searches for electron neutrino appearance and sterile neutrinos, as well as muon anti-neutrino oscillations and transitions.

1

Introduction

Neutrinos are the most enigmatic particles in the Standard Model, as the basic questions concerning their masses and mixing remain unanswered. By studying the flavor composition of a beam of muon neutrinos as it travels through the earth the MINOS experiment is able to probe the nature of these elusive particles. The neutrino beam is produced at the NuMI [lJ facility, where 120 GeV protons extracted from the Fermilab Main Injector impinge on a graphite target. The particles produced in the collisions are then focused by two parabolic horns into a 675 m long, 2 m diameter, evacuated steel pipe. The decay of these particles produces a neutrino beam with a predicted charged current (CC) neutrino event yield of 92.9%vJ.t, 5.8%vJ.t, 1.2%ve and O.l%v e . By changing the separation between the target and the horns it is possible to modify the neutrino energy spectrum. Most of the MINOS data is taken in the "low energy" (LE) beam configuration, which optimizes neutrino production in the 1-3 GeV region where the largest oscillation effects are expected. The NuMI beam is sampled by the two MINOS detectors. The 0.98 kton Near Detector (ND) is located about 1 km downstream of the NuMI target at Fermilab and is used to study the beam composition and energy spectra. The 5.4 kton Far Detector (FD) is located in the Soudan Underground Laboratory at a distance of 735 km from the target and is used to look for oscillation effects. The detectors are toroidally magnetized iron-scintillator sampling calorimeters and are functionally identical. They are described in more detail elsewhere [2J. The primary goal of the experiment is to perform precision measurements of the muon neutrino disappearance phenomenon associated with the dominant vJ.t ---> Vr oscillation mode. Results on vJ.t disappearance based on an exposure of 1.27 x 10 20 protons on the NuMI target corresponding to the data collected during the first period of NuMI operations between May 2005 and February ae-mail: [email protected]

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2006 were reported in [3]. Here we present updated results obtained from a re-analysis of this data set along with an additional data set collected between June 2006 and March 2007, corresponding to a total exposure of 2.5 x 10 20 protons on target. The outlook for future measurements, including vi-' disappearance, is also discussed here. MINOS is sensitive to sub-dominant vi-' ---+ Ve oscillations as well as oscillations to sterile neutrinos. In addition, muon anti-neutrinos and neutrinos can be separated on an individual basis due to the magnetic field that pervades the two detectors. This allows for the possibility of searching for exotic vi-' ---+ l7i-' transitions as well as for directly measuring l7i-' disappearance. 2

2.1

vi-'

disappearance results

Analysis

This updated analysis follows the same strategy as in [3]. Charged Current (CC) vi-' events leave a nearly unmistakeable signature consisting of a negatively charged muon track with some hadronic activity at the vertex. Events with these characteristics and coincident with beam spills are selected in both detectors using a multivariate algorithm that relies on kinematic and topology variables to distinguish the CC vi-' signal events from the background of neutral current (NC) events. The unoscillated CC vi-' energy spectrum is then obtained from the ND data and its comparison with the oscillated FD spectrum allows for the extraction of the oscillation parameters. No anti-neutrinos are included in the analysis and the FD data were intentionally obscured until all selection, fitting and systematic error estimation procedures were finalized. This result incorporates several improvements with respect to the previously published analysis. The neutrino interaction simulation package features more accurate models of hadronization, intranuclear rescattering and deep inelastic scattering. A new track reconstruction algorithm is used which successfully finds and fits 4% more tracks from signal CC vi-' events. A 3% increase in acceptance is also achieved by an expansion of the FD fiducial volume along the beam direction. Finally, the multivariate algorithm used to separate the signal from the NC background now combines an increased number of observables and takes advantage of their correlations with event length. The predicted unoscillated FD neutrino energy spectrum is obtained by extrapolating the observed ND spectrum using a beam transfer matrix [4] that encapsulates the kinematic and geometry effects responsible for the small (up to ±30%) differences in shape between the two spectra. This extrapolation method is largely insensitive to mis-modeling of the neutrino flux and neutrino interaction cross-sections. In addition, the hadron production model and other elements of the simulation are tuned to ND data taken in different beam configurations in order to correct for higher order effects. The beam matrix

115

Data Sample vI" (all E) vI" « 10 GeV) vI" « 5 GeV)

Observed 563 310 198

Expected (no osc.) 738 ±30 496 ± 20 350 ± 14

Observed/Expected 0.74(4.40-) 0.62(6.20-) 0.57(6.50-)

Table 1: Expected and observed CC vI' events in the FD for an exposure of 2.5 x 10 20 protons on target. The Monte Carlo expectation for the NC background in this sample is 5.6 ± 2.8 events.

FD prediction has been cross-checked by other extrapolation methods and the spread between different predictions found to be much smaller than the expected statistical error.

2.2

Oscillation results

The number of observed CC vI" events compared to the null oscillations expectation is shown in Table 1. In the low energy region of the spectrum the observed deficit of events is substantial. The oscillation parameters are extracted from a maximum likelihood fit to the observed FD spectrum. The fit is done under the framework of vI" ----> Vr oscillations and the unphysical region of sin 2(2B 23 ) > 1 is excluded. The impact of the systematic effects on the measurement is assessed by performing oscillation fits to simulated data sets with the corresponding systematics applied. The most significant sources of systematic error are found to be the uncertainty in the near to far normalization (4%), the absolute hadronic shower energy scale (10%) and the neutral current normalization (50%). These systematic uncertainties are incorporated in the oscillation fit as nuisance parameters. From the best fit values of the oscillation fit we obtain the neutrino squared-mass difference l~m~21 = (2.38:t.:g:ig) x 1O~3eV2 and mixing angle sin 2(2B 23 ) = 1.00~O.08 with errors quoted at the 68% confidence level. The best oscillation fit corresponds to X 2 = 41.2 for 34 degrees of freedom and is shown alongside the data in the left plot of Figure 1. The best fit point and the 68% and 90% confidence intervals in oscillation parameter space can be seen on the right plot of Figure 1. 3

Prospects

As the beam data continues to be collected we anticipate a significant increase to our vI" disappearance sensitivity, as shown on the left plot of Figure 2. Beyond these results, there is the possibility that MINOS could make the first measurement of a non-zero B13 if this mixing angle lies in the vicinity of the current experimental limit set by CHOOZ [8]. Even though MINOS does

116 Oscillation Results lor 2.50E20 POTs

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100

80 60 40 20 10

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Reconstructed Neutrino Energy (GeV)

Figure 1: (left) Reconstructed CC vJ.L energy spectrum in the FD for the null prediction (black), the best fit (red) and the data (points). (right) The new MINOS best fit point (star) along with the 68% and 90% contours. Overlaid are the 90% contours from the Super-Kamiokande zenith angle [5] and LIE analyses [6], as well as that from the K2K experiment [7].

not have the optimal granularity for separating between electromagnetic and hadronic showers several techniques have been devised that successfully select CC Ve events. The background consists primarily of NC and CC vI" events and is predicted using the ND data. With the existing data set our sensitivity to 813 is comparable or better to the current best limit as obtained by CHOOZ, and a first result is expected in 2008. With the full data set a discovery will be made or the current best limit reduced by about a factor of 2. By selecting for short and diffuse showers MINOS has the capability of identifying NC events with high efficiency (rv 90%) and purity (rv 60%). Given that NC events are unaffected by standard three-flavor neutrino oscillations any depletion of NC events at the FD detector would be an indication of oscillations to sterile neutrinos. The left plot of Figure 2 shows the MINOS sensitivity to the fraction of sterile mixing is defined as the fraction of disappearing vI-' 's that oscillate to sterile neutrinos. A result for this analysis is expected very soon. Its almost unprecedented ability for distinguishing between positive and negative neutrino induced muons makes MINOS an ideal ground for studying the physics of muon anti-neutrinos. For instance the FD data will be searched for exotic vI" ---+ TJI" transitions. Such transitions are predicted by some models beyond the Standard Model [9] and, it has been speculated, could explain the muon neutrino deficit observed in atmospheric neutrino experiments [10]. An anti-neutrino oscillation analysis is also in the works. Such a measurement would constitute a direct test of CPT conservation in the neutrino sector and could have a strong impact on CPT violating models introduced to, for example, expain the LSND signal [11]. In order to maximize the sensitivity to CC TJ I" disappearance we are currently studying the possibility of running with the horn current reversed for a small period of time. In such a configuration

117 MINOS Sensitivity as a function of Integrated POT

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negatively charged particles from the target would be focused, thus yielding a beam consisting primarily of anti-neutrinos. 4

Summary

We have presented the updated results on v/-L disappearance for an exposure of 2.5 x 10 20 protons on target. The best fit oscillations occur for l.6m~21 = (2.38:':8:m x 1O- 3eV 2 and sin2(2B23) = 1.00_ 0.08 with errors quoted at the 68% confidence limit. These results constitute the world's most precise measurement of l.6m~21 and improvement is expected as the data continues to be collected. Other measurements involving Ve appearance, neutral current events and antineutrinos are also progressing. Acknowledgments

This work is supported by the U.S. Department of Energy, the U.K. Science and Technology Facilities Council, the U.S. National Science Foundation, the State and University of Minnesota, the Office of Special Accounts for Research Grants of the University of Athens, Greece, and FAPESP (Fundagao de Amparo a Pesquisa do Estado de Sao Paulo) and CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico) in Brazil. We thank the Fermilab staff, the crew of the Soudan Underground Physics laboratory and the Minnesota Department of Natural Resources for their vital contributions to the project. References

[1] S.Kopp, Proc. of IEEE Part. Accel. Conj., arXiv:physics/0508001.

118

[2] to be submitted to Nucl. Instrum. Meth. [3] D.Michael et al. (the MINOS Collaboration), Phys.Rev.Lett. 97 191801 (2006). [4] M.Szleper and A.Para, hep-ex/0110001 (2001). [5] Y.Ashie et al. (the Super-Kamiokande Collaboration), Phys.Rev. D 71, 112005 (2005). [6] Y.Ashie et al. (the Super-Kamiokande Collaboration), Phys. Rev. Lett. 93, 101801 (2004) [7] M.R.Ahn et al. (the K2K Collaboration), Phys.Rev. D 74, 072003 (2006) [8] M.Apollonio et al. (the CROOZ Collaboration), Phys.Lett. B 466, 415 (1999) [9] P.Langacker and J.Wang, Phys.Rev. D 58, 093004 (1998) [10] E.N.Alexeyev and L.V.Volkova, hep-ex0504282 (2005) [11] V.Barger et al., Phys.Lett. B 576, 303 (2003)

THE NEW RESULT OF THE NEUTRINO MAGNETIC MOMENT MEASUREMENT IN THE GEMMA EXPERIMENT A.G.Beda t , V.B.Brudanina+, E.V.Demidova t, V.G.Egorov+, M.G.Gavrilov t , M.V.Shirchenko+, A.S.Starostid't, Ts.Vylov+ t Institute of Theoretical and Experimental Physics, Moscow, Russia + Joint Institute for Nuclear Research, Dubna, Russia

The new result of the neutrino magnetic moment measurement obtained by the collaboration of the Institute of Theoretical and Experimental Physics (ITEP, Moscow) and the Joint Institute for Nuclear Research (JINR, Dubna) is presented. The measurements are carried out with the GEMMA (Germanium Experiment on measurement of Magnetic Moment of Antineutrino) spectrometer. The basis of the spectrometer is a high-purity germanium detector of 1.47 kg placed at the distance of 13.95 m from the centre of the 3 GW water nuclear reactor of the Kalininskaya Nuclear Power Plant (KNPP). The detector is surrounded with the N aJ(Tl) active shielding. The antineutrino flux at the GEMMA site is fv = 2.73 X 10 13 lie cm- 2 s- 1 . The data were taken during the operation of reactor (ON period) and the reactor shutdown (OFF period) from 15.08.05 to 20.09.06 (the first run [1)) and 20.09.06 to 10.04.08 (the second run). There are totaly 9426 hours of measurements in ON period and 2965 hours in OFF period of reactor. The data were processed in the antineutrino energy region from 2.5 keV to 60 keV, the above gange was divided into two intervals: (2.5 - 9.0)keV and (11 - 60)keV. In the interval (11 - 60)keV the background was on the level of '" 1.8 events / ke V / kg / day. When processing the recoil electron spectra caused by the electromagnetic and weak interactions, we took into account the effect of the electron binding in the germanium atoms. For extraction of the electromagnetic contribution to the v - e cross section the difference method of measurements was used that implies the comparison of the electron energy spectra obtained during ON and OFF periods and two different methods of data analysis were used. In the first case, the spectrum obtained during the OFF period was approximated by a function and extracted from the spectrum obtained during the ON period. This is just the data analysis method used before by the TEXONO [2] and BOREXINO [3] collaborations. The second used in our analysis method is more conservative. It is based on the direct comparison of the spectra obtained during ON and OFF periods [1]. In both cases effects of the electromagnetic contribution to the cross section were not detected and new limits on the neutrino magnetic moment were derived. At 90% CL within the two mentioned above data analysis methods we get /.Lv < 3.1 X 10- 11 /.LB and /.Lv < 4.9 X 10- 11 /.LB respectively. ae-mail: [email protected] be-mail: [email protected]

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At present, the data taking is in progress. Simultaneously, we are preparing the experiment GEMMA II. Within the framework of this project we plan to use the antineutrino flux of '" 5.4 x 10 13 fie em -2 8- 1 , to increase the mass of germanium detector by a factor of four and to decrease the background level. These measures will provide the possibility of achieving the limit at the level of 1.5 x 10- 11 ME within the experiment GEMMA II. References

[1] A.G.Beda et al., Phys.Atom.Nucl.70 (2007) 1873, arXiv: 0705.4576. [2] H.T.Wong et al., arXiv: hep-phj0605006. [3] C.Arpesella et al., arXiv: 0805.3843v2.

THE BAIKAL NEUTRINO EXPERIMENT: STATUS, SELECTED PHYSICS RESULTS, AND PERSPECTIVES V.Aynutdinoya, A.AYrorin 3, V.Balkanoy3, 1.Belolaptikoyd, N.Budneyb, a 1.Danilchenko , G.Domogatsky3, A.Doroshenkoa , A.Dyachok b, Zh.-A.Dzhilkibaeya, S.FialkoYskyf, O.Gaponenko a , K.Golubkoyd, O.Gress b , T.Gress b , O.Grishin b , A.Klabukoya, A.Klimoyh, A.Kochanoyb, K.Konischeyd, A.Koshechkin3, V.Kulepoyf, L.Kuzmicheyc, E.Middell e , S.Mikheyeya, M.Milenin f , R.Mirgazov b , c E.OsipoYa , G.Pan'koyb, L.Pan'kov b , A.PanfiloYa, D.PetukhoYa, E.PliskoYskyd, 3 P.Pokhi1 , V.Poleschuk a , E.PopoYac , V.Prosinc , M.Rozanoyg, V.RubtzOyb, 3 A.Sheifier , A.ShirokoyC, B.Shoibonov d *, Ch.Spieringe , B. Tarashanskyb, R. Wischnewski e , 1. Yashin c, V. Zhukoy3

Institute for Nuclear Research RAS, Moscow, Russiaa Irkutsk State University, Irkutsk, Russia b Skobeltsyn Institute of Nuclear Physics MSU, Moscow, Russiac Joint Institute for Nuclear Research, Dubna, Russiad DESY, Zeuthen, Germanye Nizhni Novgorod State Technical University, Nizhni Novgorod, Russiaf St. Petersburg State Marine University, St. Petersburg, Russiag Kurchatov Institute, Moscow, Russiah Abstmct. We review the status of the Baikal Neutrino Telescope, which is operating in Lake Baikal since 1998 and has been upgraded to the 10 Mton detector NT200+ in 2005. We present selected physics results on searches for upward going neutrinos, relativistic magnetic monopoles and for very high energy neutrinos. We describe the strategy of creating a detector on the Gigaton (km3) scale at Lake Baikal. First steps of activities towards a km3 Baikal neutrino telescope are discussed.

1

Introduction

The Baikal Neutrino Telescope NT200 takes data since April 1998. On April 9th, 2005, the 10-Mton scale detector NT200+ was put into operation in Lake Baikal. Description of site properties, detector configuration and performance have been described elsewhere [1-5J. In this paper we review the current status of the Baikal Neutrino Experiment and the activities toward the km3-scale detector [6], as well as results obtained from the analysis of data taken with the Baikal neutrino telescope NT200 between April 1998 and February 2003 [7J.

2

2.1

Selected results obtained with NT200

Atmospheric neutrinos

The signature of charged current muon neutrino events is a muon crossing the detector from below. Muon track reconstruction algorithms and background

* e-mail:

[email protected]

121

122

rejection have been described elsewhere [8]. Compared to [8] the analysis of the 4-year sample (1038 days live time) was optimized for higher signal passing rate, and accepting a slightly higher contamination of 15-20% fake events [9]. A total of 372 upward going neutrino candidates were selected. From MonteCarlo simulation a total of 385 atmospheric neutrino and background events are expected. The skyplot of these events is shown in Fig. 1.

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2.2

Figure 2: Angular distributions of selected neutrino candidates as well as expected distributions in a case with and without oscillations (solid and dashed curves respectively).

Search for Neutrinos from WIMP Annihilation

The search for WIMPs with the Baikal neutrino telescope is based on a possible signal of nearly vertically upward going muons, exceeding the flux of atmospheric neutrinos. The method of event selection relies on the application of a series of cuts which are tailored to the response of the telescope to nearly vertically upward moving muons [10]. The applied cuts select muons with -1< cos(e) <-0.75 and result in a detection area of about 1800 m 2 for vertically upward going muons. The energy threshold for this analysis is Ethr ,...., 10 GeV i.e. significantly lower then for the analysis described in section 2.1 (Ethr ,...., 15 - 20 Ge V). Therefore the effect of oscillations is stronger visible. We expect a muon event suppression of (25-30)% due to neutrino oscillations assuming 8m 2 =2.5.10- 3 eV 2 with full mixing, em ~ 7r/4. From 1038 days of effective data taking between April 1998 and February 2003, 48 events with -1< cos(e) <-0.75 have been selected as clear neutrino events, compared to 56.6 events expected from atmospheric neutrinos in case of oscillations and 73.1 events without oscillations. The angular distribution of these events as well as the MC - predicted distributions are shown in Fig. 2.

123

For the MC simulations we used the Bartol96 atmospheric neutrino flux [11] without (dashed curve) and with (solid curve) oscillations. Within statistical uncertainties the experimental angular distribution is consistent with the prediction including neutrino oscillations. '",

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3

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Figure 4: Limits on the excess muon flux from the center of the Earth as a function of neutralino mass.

Regarding the 48 detected events as being induced by atmospheric neutrinos, one can derive an upper limit on the additional flux of muons from the center of the Earth due to annihilation of neutralinos - the favored candidate for cold dark matter. The 90% C.L. muon flux limits for six cones around the opposite zenith obtained with NT200 (Ethr >10 GeV) in 1998-2002 are shown in Fig. 3. It was shown [12-14] that the size of a cone which contains 90% of signal strongly depends on neutralino mass. 90% C.L. flux limits are calculated as a function of neutralino mass using cones which collect 90% of the expected signal and are corrected for the 90% collection efficiency due to cone size. Also a correction is applied for each neutralino mass to translate from 10 GeV to 1 GeV threshold (thus modifying the results as presented earlier for 10 GeV threshold [15]). These limits are shown in Fig. 4. Also shown in Fig. 3, and Fig. 4 are limits obtained by Baksan [12], MACRO [13], Super-Kamiokande [14] and AMANDA (from the hard neutralino annihilation channels) [16]. 2.3

A search for fast magnetic monopoles

Fast magnetic monopoles with Dirac charge g = 68.5e are interesting objects to search for with deep underwater neutrino telescopes. The intensity of monopole Cherenkov radiation is ~ 8300 times higher than that of muons. Optical modules of the Baikal experiment can detect such an object from a distance up to hundred meters. The processing chain for fast monopoles starts with the selec-

124

tion of events with a high multiplicity of hit channels: Nhit > 30. In order to reduce the background from downward atmospheric muons we restrict ourself to monopoles coming from the lower hemisphere. For an upward going particle the times of hit channels increase with rising z-coordinates from bottom to top of the detector. To suppress downward moving particles, a cut on the value of the time-z-correlation, Ctz , is applied: "Nhit(t· - l)(z· - z)

Ctz =

L..,,-l' , Nhit(jt(jz

>0

(1)

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MACRO

10. 10 -"'1~-'-!0."='8-'c-0~.6,....-~0•..,.4-'c-0~.2~-'-'-'='=-'-'~~~~

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Figure 5: C tz distributions for experimental events (triangles), simulated atmospheric muon events (solid), and simulated upward moving relativistic magnetic monopoles (dotted); mUltiplicity cut Nhit > 30.

17

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0.5

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B = vIc Figure 6: Upper limits on the flux of fast monopoles obtained in this analysis (Baikal) and in other experiments.

In Fig. 5 we compare the Ctz-distribution for experimental data (triangles) and simulated atmospheric muon events (solid curve) with simulated upward moving monopole events (dotted curve). Within 1038 days of live time using in this analysis, we have selected 20943 events satisfying cut 0 (Nhit > 30 and C tz > 0). For further background suppression (see [17] for details of the analysis) we use additional cuts, which essentially reject muon events and at the same time only slightly reduce the effective area for relativistic monopoles* : (1) Nhit > 35 and Ctz > 0.4 -;- 0.6 (2) X2 determined from reconstruction has to be smaller than 3 (3) Reconstructed zenith angle () > 100° (4) Reconstructed track distance from NT200 center R rec > 10 -;- 25 m . • Different values of cuts correspond to different NT200 operation configurations.

125

No events from the experimental sample pass cuts (1)-(4). The acceptances f3 =1, 0.9 and 0.8 have been calculated for all NT200 operation configurations (various sets of operating channels). For the time periods included, AefJ varies between 3· lOB and 6· lO Bcm 2sr (for f3 = 1). From the non-observation of candidate events in NT200 and the earlier stage telescopes NT36 and NT96 [18], a combined upper limit on the flux of fast monopoles with 90% C.L. is obtained. Upper limit on a flux of magnetic monopoles with f3 1 is 4.6· 1O-17cm-2s-1scl. In Fig. 6 we compare our upper limit for an isotropic flux of fast monopoles obtained with the Baikal neutrino telescope to the limits from the underground experiments Ohya [19] and MACRO [20] and to the limit reported for the underice detector AMANDA B10 [21] and preliminary limit for AMANDA II [22]. AeJJ for monopoles with

2.4

A search for extraterrestrial high-energy neutrinos

The BAIKAL survey for high energy neutrinos searches for bright cascades produced at the neutrino interaction vertex in a large volume around the neutrino telescope [3]. We select events with high multiplicity of hit channels Nhib corresponding to bright cascades. To separate high-energy neutrino events from background events a cut to select events with upward moving light signals has been developed. We define for each event tmin = min(ti - tj), where ti, tj are the arrival times at channels i, j on each string, and the minimum over all strings is calculated. Positive and negative values of tmin correspond to upward and downward propagation of light, respectively. Within the 1038 days of the detector live time between April 1998 and February 2003, 3.45 x lOB events with Nhit ~ 4 have been recorded. For this analysis we used 22597 events with hit channel multiplicity Nhit >15 and tmin >-10 ns. We conclude that data are consistent with simulated background for both tmin and Nhit distributions. No statistically significant excess above the background from atmospheric muons has been observed. To maximize the sensitivity to a neutrino signal we introduce a cut in the (tmin, Nhit) phase space. Since no events have been observed which pass the final cuts upper limits on the diffuse flux of extraterrestrial neutrinos are calculated. For a 90% confidence level an upper limit on the number of signal events of n90% =2.5 is obtained assuming an uncertainty in signal detection of 24% and a background of zero events. A model of astrophysical neutrino sources, for which the total number of expected events, N m , is large than ngO%, is ruled out at 90% CL. Table 1 represents event rates and model rejection factors (MRF) ngo%/Nm for models of astrophysical neutrino sources obtained from our search, as well as model rejection factors obtained recently by the AMANDA collaboration [23-25].

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Ig(E/GeV)

Figure 7: Left panel: all-flavor neutrino flux predictions in different models of neutrino sources compared to experimental upper limits to E-2 fluxes obtained by this analysis and other experiments (see text). Also shown is the sensitivity expected for 3 live years of the new telescope NT200+ [5,40]. Right panel: Baikal experimental limits compared to two model predictions. Dotted curves: predictions from model SS [26]' SeSi [32J and SS05 [27J. Full curves: obtained experimental upper limits to spectra of the same shape. Model SS is excluded (MRF=0.25), model SeSi is not (MRF=2.12).

For an E- 2 behaviour of the neutrino spectrum and a flavor ratio Ve : v M : V T = 1 : 1 : 1, the 90% C.L. upper limit on the neutrino flux of all flavors obtained with the Baikal neutrino telescope NT200 (1038 days) is: E 2 iJ? < 8.1 x 1O-7cm-2s-1sr-1GeV. (2) For the resonant process with the resonant neutrino energy Eo = 6.3 X 10 6 GeV the model-independent limit on ve is: iJ?ve < 3.3 x 1O-20cm-2s-1sr-1GeV-l. (3) Fig. 7 (left panel) shows our upper limit on the all flavor E- 2 diffuse flux (2) as well as the model independent limit on the resonant De flux (diamond) (3). Also shown are the limits obtained by AMANDA [23-25] and MACRO [33], theoretical bounds obtained by Berezinsky (model independent (B) [34] and for an E-2 shape of the neutrino spectrum (B(E-2)) [35], by Waxman and Bahcall (WB) [36], by Mannheim et al.(MPR) [31], predictions for neutrino fluxes from topological defects (TD) [32], prediction on diffuse flux from AGNs according to Nellen et al. (NMB) [37], as well as the atmospheric conventional neutrino fluxes [38] from horizontal and vertical directions ( (v) upper and lower curves, respectively) and atmospheric prompt neutrino fluxes (vpr ) obtained by Volkova et al. [39]. The right panel of Fig. 7 shows our upper limits (solid curves) on diffuse fluxes from AGNs shaped according to the model of Stecker and Salamon (SS, SS05) [26,27] and of Semikoz and Sigl (SeSi) [32], according to Table 1.

127 Table 1: Expected number of events N m and model rejection factors for model of astrophysical neutrino sources

Model 10 '0 X E'~ SS Quasar [26] SS05 Quasar [27] SP u [28] SP I [28] P PI [29] M PP+PI [30] MPR [31] SeSi [32]

LIe

BAIKAL + Llr ngo%/Nm 3.08 0.81 10.00 0.25 1.00 2.5 40.18 0.062 6.75 0.37 2.19 1.14 0.86 2.86 0.63 4.0 1.18 2.12

+

LlJ.L

AMANDA [23-25] ngo%/Nm 0.22 0.21 1.6 0.054 0.28 1.99 1.19 2.0 -

3 Towards a km3 detector in Lake Baikal The construction of NT200+ is a first step towards a km3-scale Baikal neutrino telescope. Such a detector could be made of building blocks similar to NT200+, but with NT200 replaced by a single string, still allowing separation of highenergy neutrino induced cascades from background. It will contain a total of 1300-1700 OMs, arranged at 90-100 strings with 12-16 OMs each, and a length of 300-350 m. Interstring distance will be R::100 m. The effective volume for detection of cascades with energy above 100 TeV is 0.5-0.8 km3 . The existing NT200+ allows to verify all key elements and design principles of the km3 (Gigaton-Volume) Baikal telescope. Next milestone of the ongoing km3-telescope research and development work (R&D) will be spring 2008: installation of a "new technology" prototype string as a part of NT200+. This string will consist of 12 optical modules and a FADC based measuring system. Three issues, discussed in the remainder of this paper, have been investigated in 2007, and will permit installation of this prototype string: (1) increase of underwater (uw) data transmission bandwidth, (2) in-situ study of FADC PMpulses, (3) preliminary selection of optimal PM. More details can be found in [6].

3.1 Modernization of data acquisition system The basic goal of the NT200+ DAQ modernization is a substantial increase of uw-data rate - to allow for transmission of significant FADC data rate, and also for a more complex trigger concept (e.g. lower thresholds and topological trigger). In a first step, in 2005 a high speed data/control tcp/ip connection between the shore station and the central uw-PCs (data center) had been established (full multiplexing over a single pair of wires, with a hot spare) [4,5,40]' based on DSL-modems (FlexDSL). In 2007, the communication on the remaining segment uw-PC - string controller was upgraded using the same approach.

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The basic elements are new string-controllers (handling TDC/ ADC-readout) with an ethernet-interface, connected by a DSL-modem to the central uw-DSL unit (3 DSL modems, max. 2 Mbps each), connected by ethernet to the uwPCs. The significant increase in uw-data rate (string to uw-PC) provided the possibility to operate the new prototype FADC system. 3.2 Prototype on a FADe based system A prototype FADC readout system was installed during the Baikal expedition 2007. It should yield input for the design of the 2008 km3 prototype string (FADC), such as: optimal sampling time window, dynamic range, achievable pulse parameter precisions, algorithms for online data handling, estimation of true bandwidth needs. These data will also be useful to decide about the basic DAQ/Triggering approach for the km3-detector: at this stage, both a complex FADC based, as well as a classical TDCI ADC approach seem feasible. The FADC prototype is located at the top of the 2nd outer string. It includes two optical modules with up-looking PM R8055, a slow control module and a FADC sphere. The FADC sphere consists of two 250 MHz FADCs, with USB connection to an embedded PCI04 computer emETX-i701, and a counter board MPCI48. The standard string trigger (2-fold channel coincidence) is used as FADC trigger. Data are transfered via local ethernet and the DSL-link of the 2nd string. Data analysis from FADC prototype is in progress.

3. 3 PM selection for the km3 prototype string Selection of the optimal PM type for the km3 telescope is a key question of detector design. Assuming similar values for time resolution and linearity range, the basic criteria of PM selection is its effective sensitivity to Cherenkov light, determined as the fraction of registered photons per photon flux unit. It depends on photocathode area, quantum efficiency, and photoelectron collection efficiency. We compared effective sensitivities of Hamamatsu R8055 (13" photocathode diameter) and XP1807 (12") with QUASAR-370 (14.6") [41], which was successfully operated in NT200 over more than 15 years. In laboratory we used blue LEDs (470 nm), located at 150 em distance from the PM. Underwater measurements are done for 2 R8055 and 2 XP1807, installed permanently as two NT200-channels, which are illuminated by the external laser calibration source [40], located 160 - 180 m away. Preliminary results of these effective PM sensitivity measurements show relatively small deviations. Smaller size (R8055, XP1807) tends to be compensated by larger photocathode sensitivities. In addition, we emphasize the advantage of a spherical shape (as QUASAR-370); we are investigating the angular integrated sensitivity looses due to various deviations from that optimum. 4 Conclusion The Baikal neutrino telescope NT200 is taking data since April 1998. The upper limit obtained for a diffuse (ve + vJ1 + vT ) flux with E- 2 shape is

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8.1 x 1O- 7 cm- 2 s- 1 sr- 1 GeV. The limits on fast magnetic monopoles and on additional muon flux induced by WIMPs annihilation at the center of the Earth belong to the most stringent limits existing to date. The limit on a 17e flux at the resonant energy 6.3x10 6 GeV is presently the most stringent. To extend the search for diffuse extraterrestrial neutrinos with higher sensitivity, NT200 was significantly upgraded to NT200+, a detector with about 5 Mton enclosed volume, which takes data since April 2005 [5,40]. The threeyear sensitivity of NT200+ to the all-flavor neutrino flux is approximately 2 x 1O- 7 cm- 2 s- 1 sr- 1 GeV for E >10 2 TeV (shown in Fig. 7). For a km3-scale detector in Lake Baikal, R&D-activities are in progress. The NT200+ detector is, beyond its better physics sensitivity, used as an ideal testbed for critical new components. Modernization of the NT200+ DAQ allowed to install a prototype FADC PM readout. Six large area hemispherical PMs have been integrated into NT200+ (2 Photonis XP1807/12" and 4 Hamamatsu R8055/13"), to facilitate an optimal PM choice. A prototype new technology string will be installed in spring 2008 and a km3-detector Technical Design Report is planned for fall 2008. E2if? =

Acknowledgments This work was supported by the Russian Ministry of Education and Science, the German Ministry of Education and Research and the Russian Fund of Basic Research (grants 05-02-17476, 05-02-16593, 07-02-10013 and 07-02-00791), and by the Grant of President of Russia NSh-4580.2006.2. and by NATO-Grant NIG-9811707(2005). References [1] 1. Belolaptikovet al. Astropart. Phys. 7, 263 (1997). [2] V. Aynutdinov et al. Nucl. Phys. (Proc. Suppl.) B143, 335 (2005). [3] V. Aynutdinov et al. Astropart. Phys. 25, 140 (2006). [4] V. Aynutdinov et al. (Proc. of V Int. Conf. on Non-Accelerator New Physics) June 7-10 (2005) Dubna Russia. [5] V. Aynutdinov et al.(NIM) A567, 433 (2006). [6] V. Aynutdinov et al. (Proc. 30th ICRC) (icrc1084), Merida, 2007; arXiv.org: astro-ph/0710.3063. [7] V. Aynutdinov et al. (Proc. 30th ICRC) (icrc1088), Merida, 2007; arXiv.org: astro-ph/0710.3064. [8] V. Balkanov et al. Astropart. Phys. 12, 75 (1999). [9] V. Aynutdinovet al. Int. J. Mod. Phys. B20, 6932 (2005). [10] V. Balkanov et al. Nucl.Phys. (Proc.Suppl.) B91, 438 (2001). [11] V. Agrawal, T. Gaisser, P. Lipari & T. Stanev Phys. Rev. D 53, 1314 (1996). [12] M. Boliev et al. Nucl. Phys. (Proc. Suppl.) 48, 83 (1996); O. Suvorova arXiv.org: hep-ph/9911415 (1999). [13] M. Ambrosio et al. Phys. Rev. D 60, 082002 (1999).

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[14] S. Desai et al. Phys. Rev. D 70, 083523 (2004); erratum ibid D, 70, 109901 (2004). [15] K. Antipin et al. (Proc. of the First Workshop on Exotic Physics with Neutrino Telescopes) Uppsala, Sweden, Sept. 20-22, 34 (2006). [16] J. Ahrens et al. arXiv.org: astro-ph/0509330 (2005). [17] K. Antipin et al. (Proc. of the First Workshop on Exotic Physics with Neutrino Telescopes), Uppsala, Sweden, Sept. 20-22,80 (2006). [18] r. Belolaptikov et. al. [Baikal collaboration] (26th ICRC) , Salt Lake City, V.2, 340 (1999). [19] S. Orito et. al. Phys. Rev. Lett. 66, 1951 (1991). [20] M. Ambrosio et. al. [MACRO collaboration] arXiv.org: hepex/02007020 (2002). [21] P. Niessen, C. Spiering [AMANDA collaboration] (27th ICRC), Hamburg, V.4, 1496 (2001). [22] H. Wissing et al. [Ice Cube Collaboration], (Proc. 30th ICRC), Merida, 2007. [23] M. Ackermann et al. Astropart. Phys. 22, 127 (2005); Astropart. Phys. 22, 339 (2005). [24] M. Ackermann et al. Astropart. Phys. 22, 339 (2005). [25] M. Ackermann et al.Phys. Rev. D 76, 042008 (2007). [26] F. Stecker and M. Salamon Space Sci. Rev. 75, 341 (1996). [27] F. Stecker Phys. Rev. D 72, 107301 (2005). [28] A. Szabo and R. Protheroe (Proc. High Energy Neutrino Astrophysics), ed. V.J. Stenger et al., Honolulu, Hawaii (1992). [29] R. Protheroe arXiv.org:astro-ph /9612213. [30] K. Mannheim Astropart. Phys. 3, 295 (1995). [31] K. Mannheim, R. Protheroe and J. Rachen Phys. Rev. D 63, 023003 (2001). [32] D. Semikoz and G. Sigl, arXiv.org:hep-ph/0309328. [33] M. Ambrosio et al. Nucl. Phys. (Proc. Suppl.), B110, 519 (2002). [34] V. Berezinskyet al. "Astrophysics of Cosmic Rays", (Elsevier Science, North-Holland) 1990. [35] V. Berezinsky arXiv.org: astro-ph/0505220 (2005). [36] E. Waxman and J. Bahcall Phys. Rev. D 59, 023002 (1999). [37] L. Nellen, K. Mannheim and P. Biermann Phys. Rev. D, 47, 5270 (1993). [38] L. Volkova Yad. Fiz. 31, 1510 (1980). [39] L. Volkova and G. Zatsepin Phys. Lett. B462, 211 (1999). [40] V. Aynutdinov et al. (Proc. 29th Int. Cosmic Ray Conf.) August 3-10 Pune India (2005); arXiv.org: astro-ph /0507715. [41] R. Bagduev et al., (NIM) A420 (1999) 138.

NEUTRINO TELESCOPES IN THE DEEP SEA Vincenzo Flaminio a on behalf of the ANTARES Collaboration Physics Department and INFN, University of Pisa, Largo Bruno Pontecorvo 3, 56117, Pisa, Italy

Abstmct. The present is a review of current experiments performed in the deep sea in a search for 1/8 of cosmic origin. After a short recollection of the historical background, we discuss experiments that are now under construction or in the data-taking phase.

1

Introduction

Our understanding of the highly energetic processes that take place in violent stellar processes, such as Supernovae explosions, Gamma-Ray Bursts, AGNs etc. has considerably improved over the last decades, thanks to the big technological progess in the field of X and l'-ray astronomy. Apart from the intrinsic limitations that to further advances in this field are placed by the absorption of X and l'-rays in the intergalactic medium, the information that electromagnetic radiation conveys is incomplete, in that such radiation is generated mainly by high-energy electrons and photons in the dense environments of stellar objects, while there is every reason to believe that, in most of these, hadronic processes play an important role. Information on such processes can only come from VB originating from the decay of shortlived hadrons produced in high-energy nuclear interactions [1-3]. So far, the only v 8 of extraterrestrial origin detected are the Solar VB [4] and a handful of VB produced in the Supernova 1987A [5]. Many groups have actively been pursuing the task of constructing large apparatus aimed at the detection of high-energy v 8 of cosmic origin. Because of the tiny cross section, and the consequent need of very large detector masses, these detectors have adopted the Cerenkov technique using as medium either large sea or lake volumes, or the Antarctic ice. The first suggestion to use sea water as a target-detector medium for high energy cosmic v 8 is due to M.A. Markov [6]. The detection principle is sketched in figure 1. Muons produced by up-going v 8 interacting in the Earth's crust underneath the instrumented volume are detected through the Cerenkov photons they emit in water. A large photomultiplier (PMT) array records position and time of arrival of the Cerenkov photons, thus allowing a precise reconstruction of the muon direction. The range (~ 1 km for a 200 GeV muon) and Cerenkov yield (about 3 x 104 photons/meter in the frequency sensitivity range of PMTs) of high energy muons in sea-water are both very large. In addition, the water transparency in this frequency range is excellent (Aab8 ~ 50 +- 60m is the typical ae-mail: [email protected]

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Figure 1: Schematic view of an underwater neutrino detector. The v charged-current interaction occurs in the Earth's crust underneath the detector. Cereakov photons are emitted by the. nation, while crossing the instrumented region. Each detector registers the position and arrival time of the photons, thus allowing a reconstruction of the muon direction.

value In the deep_sea);_The angle 9 between the neutrino and muon directions is: 9 < 1.5°/y/Wv (TeV): hence at high energy the ji and v directions coincide. Besides the obvious requirement of a large detector volume, an additional one derives from the Heed of screening the PMTs from undesired backgrounds, such as skylight and Cerenkov light from atmospheric y?. This requires the detector to be installed at large depths*. However, even working at large depths the latter background source may complicate the data analysis. To further reduce the effect of this background, experiments are therefore optimised for the detection of upgoing onions, generated by neutrinos that have crossed the Earth underneath. The advantages that this choice provide are achieved at a price: the Earth is not transparent to very high energy neutrinos. Indeed, for energies of the order of 103 TeV the neutrino interaction length becomes comparable to the Earth diameter. A further, unavoidable background comes from neutrinos originated in the decay of shortlived particles produced by cosmic rays in the upper atmosphere. These "atmospheric neutrinos" have relatively low energies and their contribution can. be reduced by cuts on energy. In this talk I will summarise the experiments of this kind that have been or

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are being carried out using sea water as the medium. It is interesting to begin this review by briefly recalling the DUMAND experiment, that led the early pioneering studies and of whose experience all subsequent experiments took advantage.

2

Pioneering developments: DUM AND

The DUMAND project aimed at the installation of an underwater Cerenkov detector at a depth of about 4500 km, near the Hawaii islands. The experiment, funded in 1990, was terminated only six years later. For a comprehensive account of the historical developments and construction steps of DUM AND we refer to a paper published in 1992 by Arthur Roberts [7]. A large number of deployments were performed, some of which provided results on the atmospheric muon vertical intensity vs depth down to 5 km and on the muon angular distribution at 4 km. The DUMAND collaboration also set what at the time were the best limits on the flux of high-energy VB from AGNs. The shore station was located at the Natural Energy Laboratory of Hawaii (NELH) at Keahole Point. A cable to connect the detector to the shore station, comprising 12 single-mode optical fibers in a stainless-steel tube and surrounded by a copper sheath capable of transmitting 5 kW of electrical power, was designed and manufactured in the early 90's. In December 1993, the DUMAND collaboration successfully deployed the first major components of DUMAND, including the junction box, environmental monitoring equipment, and the shore cable, with one complete string equipped with 16" PMTs, attached to the junction box (JB). The data system could cope, with a negligible dead time, with the background rate from radioactivity in the water (primarily from natural 40 K and bioluminescence). The counting rate for a single PMT was of the order of 60 kHz, primarily due to trace 40 K in the huge volume of seawater viewed by each tube. Noise due to bioluminescence was episodic and expected to be unimportant after the array had been stationary on the ocean bottom for some time, since the light-emitting microscopic creatures are stimulated by motion. 40 K and bioluminescence contribute mainly 1 photoelectron hits distributed randomly in time over the entire array. Bioluminescence caused spikes in the singles rate which reached 100 kHz for periods on the order of seconds, but with a very low frequency of occurrence. The deployed string was used to record backgrounds and muon events. Unfortunately, an undetected flaw in one of the electrical penetrators (connectors) used for the electronics pressure vessels produced a small water leak. Seawater eventually shorted out the string controller electronics, disabling further observations after about 10 hours of operation. Recovery of the string was accomplished between 28-30 January 1994, about 44 days after it had been deployed. The developments took a long time. In retrospective it seems that this was

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mainly caused by the lack, at least in the initial development stages, of the necessary technological means, like a reliable fiber-optical technology, pressureresistant electro-optical connectors and Remote-Operated-Vehicles (ROVs) capable to operate at depths in excess of 4000 m, for underwater connections. These became gradually available in the process of detector design, and were eventually adopted in the final setup. In mid 1996, DOE determined that further support for DUMAND-II should be terminated. The same Russian groups that contributed to DUMAND in the early stages, started an analogous enterprise under lake Baikal. This relatively small detector has operated for many years and the construction of a larger detector is now underway. Because of time constraints we shall not discuss details of this, that is not properly an "undersea" experiment. 3

Experiments in the Mediterranean sea

The construction of different prototype-detectors has been pursued over the last decades at three different locations in the Mediterranean. Chronologically, the first of these has been initiated by the NESTOR collaboration in the Ionian Sea, off the coast of Pylos, Southwestern Greece, at a depth of approximately 4000 m. The second one, NEMO is a prototype meant to be the basic building block of a km 3 detector. The chosen location is off the southern coast of Sicily, at a depth of >:;;j 3500 m. The third one, ANTARES is a medium-sized experiment (the effective area is of the same order of magnitude as the one of AMANDA [8]) now being assembled in the Mediterranean, off the southern French coast, at a depth of about 2500 m. 4

The NESTOR experiment

This has involved a large international collaboration and results obtained during the initial tests have already been published [9,10]. The chosen site is located at a distance of about 30 km Southwest of the small harbour of Methoni, at a depth of 4000 -;- 5000 m where the water quality is excellent. The detector architecture is based on what the authors call" stars". A typical star, an example of which is shown in figure 2, consists of six 16 m long arms attached to a central casing. Two optical modules (15" photomultipliers, enclosed in spherical glass housings) are attached at the end of each of the arms, one facing upwards and the other facing downwards. The electronics for each star is housed in a one-meter diameter titanium sphere within the central casing. A full NESTOR tower would consist of 12 such floors stacked vertically with a spacing od 30 m between floors. As in other undersea detectors, data and

135

power transmission is provided by electro-optical cables linking the detector to the shore station. The architecture is conceived in such a way as to avoid

Figure 2: [Left] hotmrr:~nh of a full-size NESTOR star. The test described here used a smaller detector, [right] Zenith angular distribution of atmospheric unions measured in the NESTOR prototype star in March 2003. The black dots are Me MC predictions. The inset shows the same distribution on a linear scale.

underwater operations: all connections are performed in air using dry-mating connectors before deployment. Repair operations need the recovery of the entire tower and connection cables, a formidable task for a very large detector. So far only a single smaller (5 m long arms) star has been deployed and tested for a short period in March 2003, at a depth of 3800 HI. From the total data sample collected with a four-fold coincidence trigger, 45,800 events have been selected. The resulting zenith angular distribution is shown in figure 2, where it is compared with the result of a MonteCarlo (MC) simulation of atmospheric muons, based on the Okada parameterisation [11]. The agreement is good both in shape and absolute flux. 5

The NEMO experiment

This experiment is being carried out by a large Italian collaboration. The geometry adopted is that of a series of "towers". The structure of a single tower is sketched in figure 3. It consists of 16 arms ("floors") each 18 m in length and holding a pair of 10" PMTs at its ends. The various loors are held by tensioning ropes bound to a buoy at the top. The geometry is such as to facilitate transportation and deployment, since the different loors of a given tower can by folded one on top of another, achieving a compact structure that can afterward easily be unfolded for deployment. The propsed detector geometry consists of an array of 9 x 9 such towers, interspaced by 140 m, providing an effective area of over lfern2 for energies above 10 TeV. The site

136

chosen Is located South-East of Sicily (see the inset in igure 3) at less than 80 km from shore, at a depth of 3500 m. The sea properties of this site have been studied in detail over a period of several years, and they turn out to be Ideal [12], both in terms of water properties and of biolumlnescence, A small prototype ("minitower") has been successfully deployed and tested for a few months at a somewhat shallower (« 2000 m) depth, about 20 kin away from the Catania harbour, at the end of 2006 [13]. A junction box (JB) was deployed first and connected to a pre-existing 25 km long electro-optical cable linked to a shore station. A minitower, consisting of only four "arms", each 15 m long and holding two 10" Hamamatsu PMTs at each end, was then deployed and connected to the JB. The vertical distance between arms was 40 m. Several trigger schemes were tested at the same time and a large number of events, mostly due to atmospheric muons, were recorded. Figure 3 shows a typical reconstructed muon. At the saaie time a full-scale tower is being built by the collaboration. A 100 km long electro-optical cable has beea deployed, linking the shore station, located inside the harbour area of " Portopalo di Capo Passero" with the chosen site. The building to be used as the shore laboratory for a fern,3 size detector has been acquired and is currently being equipped. Deployment of a full-sized tower is foreseen for the end of 2008.

Figure 3: [Left] The, inset shows the Capopassero site where the NEMO fern3 detector should be Installed, The two drawings illustrate the structure of the NEMO tower. [.Right] One of the first atmospheric muon tracks reconstructed In the NEMO minitower.

137

6

The ANTARES experiment

Antares is a multidiseipiiiiaxy experiment, whose main aim, of detecting neutrinos of cosmic origin, is accompanied by parallel research interests in the fields of marine biology and geophysics [15,16]. It is being carried out by a large collaboration, including Research Institutions from France, Germany, Italy, The Netherlands, Spain and Romania. Being this the experiment that has made the most impressive progress over the last couple of years, I will go here a little bit more in detail. The detector, schematically shown in figure 4, consists of 12 lines, each holding 75 10" Hamamatsu PMTs arranged in triplets (storeys) and looking downward, at an angle of 45° to the vertical. The PMTs are housed in pressure-resistant glass spheres. The separation between storeys in each line is 14.5 m, starting 100 m from the sea-floor; the distance between pairs of strings in the horizontal plane is 60-70 m. Each PMT triplet is held in place by a

Figure 4: Sketch of the Antares detector.The inset shows an indiYictaai storey, with a titanium frame holding the three glass spheres, each housing a PMT

titanium frame, as shown in the inset of figure 4, attached to a vertical electrooptical cable used for data and clock signals as well as for power transmission. Digital data transmission uses optical fibres. At the center of the frame a titanium cylinder encloses the readout/control electronics, together with compasses/tiltmeters used for geometrical positioning. Some of the storeys also house LED beacons, each containing 38 LEDs, which provide very fast pulses used for timing calibrations. For readout purposes, each group (sector) of five storeys in any given line is treated separately.

138

A laser is located at the bottom of one of the lines, while a second one is located at the bottom of the instrumentation line (see below). These provide additional means for timing calibrations. Hydrophones, attached one per sector, are used, in conjunction with sonic transmitters located at fixed locations on the sea-floor and with the compasses-tilt meters installed in the LCMs, for precise position determinations. An additional line (Instrumentation Line), equipped with instruments used to monitor other important parameters, such as temperature, pressure, salinity, light attenuation length and sound speed, is an essential component of the detector. Each line is connected, via an electronics module located at its bottom, to a JB in turn connected via a 42 km long electro-optical cable (installed in November 2001) to the shore station. All data are here collected by a computer farm, where a fast processing of events satisfying predetermined trigger requirements is performed. Precise timing is provided by a 20 MHz high accuracy on-shore clock synchronised with the GPS, distributed via the electro-optical cable and the JB to each electronics module. The expected performance of the detector has been studied in detail using MC simulations. The effective area for neutrinos reaches a maximum of :::::: 30m 2 • For v 8 at small nadir angles there is a drastic decrease at very high energies, due to absorption by the Earth. The neutrino angular resolution is dominated by electronics at high energies, where it reaches a value of :::::: 0.2 -70.3 0 • At lower energies it is dominated by the kinematics of muon production by neutrinos. Following many tests carried out over several years, the installation of the detector in its final configuration started in December 2005 and has continued in 2006 and 2007. At present five of the lines are installed and data taking is going on smoothly C. The excellent performance of the detector, both in terms of its time and space resolution has been demonstrated using data obtained with the first lines installed [14]. A very large number of triggers has been collected using the present five-line detector. These are mainly due to atmospheric muons, together with a smaller number of v 8 • Figure 5 shows the ¢ and () (zenith) distributions for atmospheric muons, compared with the MC predictions. The anisotropy in the ¢ distribution reflects the non-uniform distribution of lines in the horizontal plane. The small discrepancy present between data and MC in the () distribution is due to a still inaccurate knowledge of the angular acceptance of the optical modules d. Figure 6 shows the (z-t) plot for a reconstructed muon moving upwards (due to a neutrino interaction). The top histogram shows the measured muon angular distribution, after the application of cuts designed to reduce the contribution of atmospheric muons. The neuC At the time of writing, five additional lines and an instrumentation line have been installed and connected dThe photomultipliers look downwards at an angle of 45° in such a way as to optimise the acceptance for muons moving upwards. Their acceptance for downgoing muons is therefore more limited

139 .,-o.OO7 c - - - - - - - - - - - - - - - - - , .,- O.02C--------------~ i~~~~~ I-·_m~~

i

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0.005

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Antares Data "'"'' Monte Carlo

;; 0.016 ~ 0.014

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Antares Data .. ,"" Monte Carlo

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0.00 ·150

-100

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distributions for atmospheric muons obtained in the five-line detector.

trino sample is associated to muon events having cos () > 0; the corresponding rate is a few per day. 7

The future: conclusions

Following the pioneering DUMAND attempts and in parallel with analogous detectors now operational under the Antarctic ice, a number of undersea mediumto-large-scale experiments are under construction in the northern hemisphere. These are: NESTOR, NEMO, ANTARES. The latter, with five strings (375 PMTs) already installed e, is at present the largest running undersea experiment in the northern hemisphere. Recently the three collaborations have merged their efforts in an attempt to design and build a km 3 detector in the Mediterranean. A design study has been approved and financed by the EU [17] and work is in progress. References

M.D. Kistler and J.F. Beacom, Phys.Rev. D 74, 063007 (2006). F. W. Stecker, Phys.Rev. D 72, 107301 (2005). V. Cavasinni, D. Grasso and L. Maccione, Astrop. Phys. 26, 41(2006). For a comprehensive Review of the SSM and of the early solar neutrino experiments, see: J.N. Bahcall, "Neutrino Astrophysics", (Cambridge University Press) 1989. [5] K. Hirata et al., Phys.Rev. Lett. 58, 1490 (1987). R. M.Bionta et al., Phys.Rev. Lett. 58, 1494 (1987).

[1] [2] [3] [4]

eTen strings and 750 PMTs at the time of writing

the plot photons on the PMT and hlsl;ogl:am shows the measured Ue!llglled to further suppress atrnospn'3r1C

~Jt"j."lJI~{}V

DOUBLE BETA DECAY: PRESENT STATUS A.S. Barabash a

Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia Abstmct.The present status of double beta decay experiments is reviewed. The results of the most sensitive experiments, NEMO-3 and CDORICINO, are discussed. Proposals for future double beta decay experiments are considered.

1

Introduction

Interest in neutrino less double-beta decay has seen a significant renewal in recent years after evidence for neutrino oscillations was obtained from the results of atmospheric, solar, reactor and accelerator neutrino experiments. These results are impressive proof that neutrinos have a nonzero mass. The detection and study of Ov{3{3 decay may clarify the following problems of neutrino physics: (i) neutrino nature: whether the neutrino is a Dirac or a Majorana particle, (ii) absolute neutrino mass scale (a measurement or a limit on md, (iii) the type of neutrino mass hierarchy (normal, inverted, or quasidegenerate), (iv) CP violation in the lepton sector (measurement of the Majorana CP-violating phases). 2 2.1

Results of experimental investigations Two neutrino double beta decay

This decay was first recorded in 1950 in a geochemical experiment with 130Te [lJ; in 1967, it was also found for 82Se [2J. Only in 1987 2{3(2v) decay of 82Se was observed for the first time in a direct experiment [3]. Within the next few years, experiments employing counters were able to detect 2{3(2v) decay in many nuclei. In looMo [4-6], and 150Nd [7J 2{3(2v) decay to the 0+ excited state of the daughter nucleus was recorded. Also, the 2{3(2v) decay of 238U was detected in a radiochemical experiment [8], and in a geochemical experiment for the first time the ECEC process was detected in 130Ba [9J. Table 1 displays the present-day averaged and recommended values of T 1 / 2 (2v) from [10J (for looMo-100Ru(Ot) transition value is from [ll]). 2.2

Neutrinoless double beta decay

In contrast to two-neutrino decay, neutrinoless double-beta decay has not yet been observed b. ae-mail: [email protected] bThe possible exception is the result with 76Ge, published by a fraction of the HeidelbergMoscow Collaboration, Tl/2 ~ 1.2 . 1025 Y [12J and Tl/2 ~ 2.2· 1025 y [13J (see Table 2).

141

142 Table 1: Average and recommended Tl/2(2v) values [10). For looMo-lOORu(ot) transition value is from [11).

Isotope 48Ca 76Ge

82Se 96Zr 100Mo 100Mo-100Ru(Oi) 116Cd 128Te 130Te 150Nd 150Nd- 15 0Sm(Oi) 238U 130Ba; ECEC(2v)

1:6 .

4.3:: 10 19 (1.5 ± 0.1) . 10 21 (0.92 ± 0.07) . 1020 (2.0 ± 0.3) . 10 19 (7.1 ± 0.4) . 10 18 6.2~g:~ . 1020 (3.0 ± 0.2) . 10 19 (2.5 ± 0.3) . 1024 (0.9 ± 0.1) . 1021 (7.8 ± 0.7) . 10 18 1.4~g:~ . 1020 (2.0 ± 0.6) . 10 21 (2.2 ± 0.5) . 10 21

The present-day constraints on the existence of 2,B(Ov) decay are presented in Table 2 for the nuclei that are the most promising candidates. In calculating constraints on (mv), the nuclear matrix elements from QRPA calculations [1618] were used (3-d column). It is advisable to employ the calculations from these studies, because the calculations are the most thorough and take into account the most recent theoretical achievements. In column four, limits on (mv), which were obtained using the NMEs from a recent Shell Model (SM) calculations [19,20]. 3

Best present experiments

In this section the two large-scale running experiments NEMO-3 and CUORICINO are discussed. 3.1

NEMO-3 experiment !26,29j

Since June of 2002, the NEMO-3 tracking detector has operated at the Frejus Underground Laboratory (France) located at a depth of 4800 m.w.e. The detector has a cylindrical structure and consists of 20 identical sectors. A thin (about 30-60 mg/cm 2) source containing beta-decaying nuclei and having a The Moscow portion of the Collaboration does not agree with this conclusion [14) and there are others who are critical of this result [15). Thus at the present time this "positive" result is not accepted by the "2/3 decay community" and it has to be checked by new experiments.

143 Table 2: Best present results on 2/3(01/) decay (limits at 90% C.L.).

Isotope 76Ge

T 1/ 2 , Y

> 1.9.10 25

1.2. 1025 (7) 2.2 . 10 25 (7) > 1.6.10 25 > 3.10 24 > 5.8.10 23 > 4.5.10 23 > 2.1 . 1023 > 1.7.1023

~ ~

130Te lOoMo 136Xe 82Se 116Cd

(mvl' eV

(mvl' eV

[16-18] < 0.22 - 0.40 ~ 0.28 - 0.51(7) ~ 0.21 - 0.37(7) < 0.24 - 0.44 < 0.30 - 0.57 < 0.81 - 1.28 < 1.49 - 2.66 < 1.47 - 2.17 < 1.45 - 2.73

[19] < 0.76 ~ 0.96(7) ~ 0.71(7) < 0.83 < 0.75

< 2.2 < 3.4 < 2.1

Experiment HM [21] Part of HM [12] Part of HM [13] IGEX [22] CUORICINO [23] NEMO- 3 [25] DAM A [27] NEMO-3 [25] SOLOTVINO [28]

total area of 20 m 2 and a weight of up to 10 kg was placed in the detector. The energy of the electrons is measured by plastic scintillators (1940 individual counters), while the tracks are reconstructed on the basis of information obtained in the planes of Geiger cells (6180 cells) surrounding the source on both sides. In addition, a magnetic field of strength of about 25 G parallel to the detector axis is created by a solenoid surrounding the detector. At the present time, the investigations are being performed for seven isotopes; these are looMo (6.9 kg), 82Se (0.93 kg), 1l6Cd (0.4 kg), 150Nd (37 g), 96Zr (9.4 g), 130Te (0.45 kg), and 48Ca (7 g). The corresponding limits on Tl/2(0//) and (mv} for looMo and 82Se are presented in Table 2. T 1 / 2 (2//) for all seven isotopes have been measured (see [36]). The NEMO-3 experiment is on going and new improved results will be obtained in the near future. In particular, the sensitivity of the experiment to 2;3(0//) decay of lOoMo will be on the level of rv 2 . 10 24 y. This in turn means the sensitivity to (mv} will be on the level of rv 0.4 - 0.7 eV. 3.2

CUORICINO [24]

This program is the first stage of the larger CUORE experiment (see section 4). The experiment is running at the Gran Sasso Underground Laboratory. The detector consists of 62 individual low-temperature natTe02 crystals, their total weight being 40.7 kg. The energy resolution is 7.5-9.6 keY at an energy of 2.6 MeV. The experiment has been running since March of 2003. The corresponding limits on Tl/2(0//) and (mvl for 130Te are presented in Table 2. The sensitivity of the experiment to 2;3(0//) decay of 130Te will be on the

144

level of '" 5 . 1024 for 3 y of measurement. This in turn means the sensitivity to (my) is on the level of'" 0.2 - 0.6 eV. 4

Planned experiments

In this section, main parameters of five promising experiments which can be realized within the next five to ten years are presented. The estimation of the sensitivity in all experiments to the (my) is made using NMEs from [16-19]. Table 3: Five most developed and promising projects.

5

Experiment

Isotope

CUORE [30] GERDA [31]

130Te 76Ge

MAJORANA [32,33] EXO [34]

76Ge 130Xe

SuperNEMO [35,36]

82S e 150Nd

Mass of isotope, kg 200 40 1000 60 1000 200 1000 100-200

Sensitivity T 1L2, Y 2.1. 1026 2.10 26 6.10 27 2.10 26 6.10 27 6.4.10 25 2.10 27 (1 - 2) . 1026

Sensitivity (my), meV 35-90 70-230 10-40 70-230 10-40 120-220 20-40 45-110

Status accepted accepted R&D R&D R&D accepted R&D R&D

Conclusion

In conclusion, two-neutrino double-beta decay has so far been recorded for ten nuclei (48 Ca, 76Ge, 82Se , 96Z r , lOoMo, 116Cd, 128Te, 130Te, 150Nd, 238U). In addition, the 2f3(2v) decay of lOoMo and 150Nd to 0+ excited state of the daughter nucleus has been observed and the ECEC(2v) process in 130Ba was recorded. Neutrinoless double-beta decay has not yet been confirmed. There is a conservative limit on the effective value of the Majorana neutrino mass at the level of 0.75 eV. Within the next few years, the sensitivity to the neutrino mass in the CUORICINO and NEMO-3 experiments will be improved to become about 0.2 to 0.6 eV with measurements of 130Te and lOoMo. The next-generation experiments, where the mass of the isotopes being studied will be as grand as 100 to 1000 kg, will have started within three to five years. In all probability, they will make it possible to reach the sensitivity to the neutrino mass at a level of 0.1 to 0.01 eV.

145

References

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M.G. Inghram, J.H. Reynolds, Phys. Rev. 78,822 (1950). T. Kirsten, W. Gentner, O.A. Schaeffer, Z. Phys. 202, 273 (1967). S.R. Elliott, A.A. Hahn, M.K. Moe, Phys. Rev. Lett. 59, 2020 (1987). A.S. Barabash et al., Phys. Lett. B 345, 408 (1995). A.S. Barabash et al., Phys. At. Nucl. 62, 2039 (1999). L. De Braeckeleer et al., Phys. Rev. Lett. 86,3510 (2001). A.S. Barabash et al., JETP Lett. 79, 10 (2004). A.L. Turkevich, T.E. Economou and G.A. Cowan, Phys. Rev. Lett. 67, 3211 (1991). [9] A.P. Meshik et al., Phys. Rev. C 64, 035205 (2001). [10] A.S. Barabash et al., Czech.J. Phys. 56,437 (2006). [11] A.S. Barabash, AlP Conf. Proc. 942: 8 (2007). [12] H.V. Klapdor-Kleingrothaus et al., Phys. Lett. B 586, 198 (2004). [13] H.V. Klapdor-Kleingrothaus and LV. Krivosheina. Mod. Phys. Lett. A 21, 1547 (2006). [14] A.M. Bakalyarov et al., Phys. Part. Nucl. Lett. 2, 77 (2005); hepex/0309016. [15] A. Strumia and F. Vissani, Nucl. Phys. B 726, 294 (2005). [16] V. Rodin et al., Nucl. Phys. A 793, 213 (2007). [17] M. Kortelainen and J. Suhonen, Phys. Rev. C 75, 051303(R) (2007). [18] M. Kortelainen and J. Suhonen, Phys. Rev. C 76,024315 (2007). [19] E. Caurier et al., nucl-th/0709.2137. [20] E. Caurier et al., nucl-th/0709.0277. [21] H.V. Klapdor-Kleingrothaus et al., Eur. Phys. J. A 12, 147 (2001). [22] C.E. Aalseth et al., Phys. Rev. C 65, 09007 (2002). [23] A. Giuliani (CUORICINO Collaboration), report at TAUP'07 (Sendai, 11-13 September, 2007). [24] C. Arnaboldi et al., Phys. Rev. Lett. 95, 142501 (2005). [25] A.S. Barabash (NEMO Collaboration), hep-ex/0610025. [26] R. Arnold et al., Phys. Rev. Lett. 95, 182302 (2005). [27] R. Bernabei et al., Phys. Lett. B 546, 23 (2002). [28] F.A. Danevich et al., Phys. Rev. C 67, 035501 (2003). [29] R. Arnold et al., Nucl. Instr. Meth. A 536, 79 (2005). [30] C. Arnaboldi et al., Nucl. Instr. Meth. A 518, 775 (2004). [31] 1. Abt et al., hep-ex/0404039. [32] Majorana White Paper, nucl-ex/0311013. [33] C.E. Aalseth et al., Nucl. Phys. B (Pmc. Suppl.) 138, 217 (2005). [34] M. Danilovet al., Phys. Lett. B 480, 12 (2000). [35] A.S. Barabash, Czech. J. Phys. 52,575 (2002). [36] S. Soldner-Rembold (NEMO Collaboration), hep-ex/0710.4156.

BETA-BEAMS C. Volpe a

Institut de Physique Nucleaire Orsay, F-91406 Orsay cedex, FRANCE Abstract. Beta-beams is a new concept for the production of intense and pure neutrino beams. It is at the basis of a proposed neutrino facility, whose main goal is to explore the possible existence of CP violation in the lepton sector. Here we briefly review the original scenario and the low energy beta-beam. This option would offer a unique opportunity to perform neutrino interaction studies of interest for particle physics, astrophysics and nuclear physics. Other proposed scenarios for the search of CP violation are mentioned.

1

Introduction

The observations made by the Super-Kamiokande [1], the K2K [2], the SNO [3] and the KAMLAND [4] experiments have brought a breakthrough in the field of neutrino physics. The longstanding puzzles of the solar neutrino deficit [5] and of the atmospheric anomaly have been clarified: the expected fluxes are reduced due to the neutrino oscillation phenomenon, i.e. the change in flavour that neutrinos undergo while traveling [6]. The overall picture is now also confirmed by the recent mini-BOONE result [7]. Neutrino oscillations imply that neutrinos are massive particles and represent the first direct experimental evidence for physics beyond the Standard Model. Understanding the mechanism for generating the neutrino masses and their small values is clearly a fundamental question, that needs to be understood. On the other hand, the presently known (as well as unknown) neutrino properties have important implications for other domains of physics as well, among which astrophysics, e.g. for our comprehension of processes like the nucleosynthesis of heavy elements, and cosmology. An impressive progress has been achieved in our knowledge of neutrino properties. Most of the parameters of the Maki-Nakagawa-Sakata-Pontecorvo (MNSP) unitary matrix [8], relating the neutrino flavor to the mass basis, are nowadays determined, except the third neutrino mixing angle, usually called 813 . However, this matrix might be complex, meaning there might be (one or more) phases. A non-zero Dirac phase introduces a difference between neutrino and anti-neutrino oscillations and implies the breaking of the CP symmetry in the lepton sector. Knowing its value might require the availability of very intense neutrino beams in next-generation accelerator neutrino experiments, namely super-beams, neutrino factories or beta-beams. Besides representing a crucial discovery, the observation of a non-zero phase might help unraveling the asymmetry between matter and anti-matter in the Universe and have an impact in astrophysics, e.g. for core-collapse supernova physics [9]. ae-mail: [email protected]

146

147

Zucchelli has first proposed the idea of producing electron (anti)neutrino beams using the beta-decay of boosted radioactive ions: the "beta-beam" [lOJ. It has three main advantages: well-known fluxes, purity (in flavour) and collimation. This simple idea exploits major developments in the field of nuclear physics, where radioactive ion beam facilities under study such as the european EURISOL project are expected to reach ion intensities of 10 11 - 13 per second. A feasibility study of the original scenario is ongoing (2005-2009) within the EURISOL Design Study (DS) financed by the European Community. At present, various beta-beam scenarios can be found in the literature, depending on the ion acceleration. They are usually classified following the value of the Lorentz I boost factor, as low energy h = 6-15) [11-21,21-24]' original h ~ 60 - 100) [10,25-30], medium h of several hundreds) and high-energy h of the order of thousands) [31-35]. (For a review of all scenarios see [36].) An extensive investigation of the corresponding physics potential is being performed and new ideas keep being proposed. For example, a radioactive ion beam production method is discussed in [37] and will be investigate within the new "EuroNU" DS. Thanks to this method two new ions 8B and 8Li are being considered as candidate emitters, while the previous literature is mainly focussed on 6He and 18Ne. The corresponding sensitivity is currently under study (see e.g. [38]). 2

The original scenario

In the original scenario [10]' the ions are produced, collected, accelerated up to several tens GeV /nucleon - after injection in the Proton Synchrotron and Super Proton Synchrotron accelerators at CERN - and stored in a storage ring of 7.5 km (2.5 km) total length (straight sections). The neutrino beam produced by the decaying ions point to a large water Cerenkov detector [39] (about 20 times Super-Kamiokande), located at the (upgraded) FrEljus Underground Laboratory, in order to study CP violation, through a comparison of lie -+ 11/-1 and De -+ D/-I oscillations. This facility is based on reasonable extrapolation of existing technologies and exploits already existing accelerator infrastructure to reduce cost. Other technologies are being considered for the detector as well [40J. A first feasibility study is performed in [41]. The choice of the candidate emitter has to meet several criteria, including a high intensity achievable at production and a not too short/long half-life. The ion acceleration window is determined by a compromise between having the I factor as high as possible, to profit of larger cross sections and better focusing of the beam on one hand, and keeping it as low as possible to minimize the single pion background and better match the CP odd terms on the other hand. The signal corresponds to the muons produced by 11/-1 charged-current events in water, mainly via quasi-elastic interactions at these energies. Such events are

148 Table 1: Number of events expected after 10 years, for a beta-beam produced at CERN and sent to a 440 kton water Cerenkov detector located at an (upgraded) Frejus Underground Laboratory, at 130 km distance. The results correspond to ve (left) and Ve (right). The different 'Y values are chosen to make the ions circulate together in the ring [26] 18 Ne

6He

(r CC events (no oscillation) Oscillated (sin228l3 = 0.12, 15 Oscillated (15 = 90° ,8 13 = 3°) Beam background Detector backgrounds

= 0)

= 60) 19710 612 44

o 1

(r

= 100) 144784 5130 529 0 397

selected by requiring a single-ring event, with the same identification algorithms used by the Super-Kamiokande experiment, and by the detection of the electron from the muon decay. At such energies the energy resolution is very poor due to the Fermi motion and other nuclear effects. For these reasons, a CP violation search with 'Y = 60 - 100 is based on a counting experiment only. The beta-beam has no intrinsic backgrounds, contrary to conventional sources. However, inefficiencies in particle identification, such as single-pion production in neutral-current Ve (ve) interactions, electrons (positrons) misidentified as muons, as well as external sources, like atmospheric neutrino interactions, can produce backgrounds. The background coming from single pion production has a threshold at about 450 MeV, therefore giving no contribution for 'Y < 55. Standard algorithms for particle identification in water Cerenkov detectors are quite efficient in suppressing the fake signal coming from electrons (positrons) misidentified as muons. Concerning the atmospheric neutrino interactions, estimated to be of about 50/kton/yr, this important background is reduced to 1 event/440 kton/yr by requiring a time bunch length for the ions of 10 ns. The expected events from [26] are shown in Table 1, as an example. The discovery potential is analyzed in [10,25-30]. A detailed study of 'Y = 100 option is made for example in [29] based on the GLoBES software [42], including correlations and degeneracies and using atmospheric data in the analysis [33]. The fluxes are shown in Figure 1. Figure 2 shows the CP discovery reach as an example of the sensitivity that can be reached running the ions around 'Y = 100.

3

Low energy beta-beams

A low energy beta-beam facility producing neutrino beams in the 100 MeV energy range has been first proposed in [11]. Figure 3 shows the corresponding fluxes. The broad physics potential of such a facility, currently being analyzed, covers:

149 t..

x l 0 7 e - - - -_ _ _ _ _ _ _~

~ 8000

-

o

"'~ 7000

.€ ;>

2n 30 discovery of CP violation:

!:J.i (liep = 0, n):= 9

SPL v~

',<-SPLv!' -Beta ve (Nel~

6000

·····Beta

5000

Ve (He!)

4000 3000

2000 1000

0.2

0.4

0.6

0.8

F;, (GeY) 10-2

Figure 1: Comparison of neutrino fluxes from a super-beam (SPL) and a beta-beam. The ions circulate at the same "( = 100, independently, in the storage ring. Note that the average neutrino energies are related to the ion boost through E" "" 2"(Q{3 [27].

2

true sin 28 u

Figure 2: CP discovery reach for the b = 100) beta-beam (,6B), a superbeam (SPL), and T2HK as a function of {/13. The width of the bands corresponds to values with 2% to 5% systematical errors. [29].

• neutrino-nucleus interaction and nuclear structure studies [11,13,14,20]; • electroweak tests of the Standard Model, such as a new method to test the Conserved-Vector-Current hypothesis [15] (Figure 4), a measurement of the Weinberg angle at small momentum transfer [16] (Figure 5) or of the neutrino magnetic moment [12]); • core-collapse supernova physics [11,17]. Here I briefly mention some of the results concerning the physics potential of a low energy beta-beam. Neutrino-nucleus interaction is a topic of current great interest since the corresponding cross sections are necessary for the interpretation of neutrino detector response, for the understanding of the nucleosynthesis of heavy elements and in the search for physics beyond the Standard Model. For example, a precise knowledge of the neutrino-lead cross section [43] can be exploited to extract information on the third neutrino miwing angle that still remains unknown [44]. In spite of the numerous applications, the experimental information is scarce since only a few experiments have been performed. The best studied case, i.e. interactions on carbon still suffer from important discrepancies between experiment and theory (see e.g. [45]). The theoretical calculations of the cross sections for neutrino energies in the several tens Me V are subject to nuclear structure uncertainties due to the different choices of

150

---·_·y=6

-;-_4,OX10'

.,=10 y=14

S

-DAR

->'" Q)

e:;;

2,Qx10·

s

20

40

60

80

100

Ey (MeV)

Figure 3: Anti-neutrino fluxes from the decay of 6He ions boosted at "! = 6 (dot-dashed line),,,! = 10 (dotted line) and,,! = 14 (dashed line). The full line presents the Michel spectrum for neutrinos from muon decay-at-rest.

I'"

03.4

3.5

4.0

Figure 4: eve Test : ~X2 obtained from the angular distribution of electron anti-neutrinos on proton scattering in a water Cerenkov detector in the cases when the statistical error only (solid), with 2% (dashed), 5% (dash-dotted) and 10% (dotted) systematic errors. The 1<7 (~X2 = 1) relative uncertainty in the weak magnetism contribution J.Lp - J.Ln is 4.7%, 5.6%, 9.0% and more than 20%, respectively. The results correspond to "! = 12 [15].

the approaches and effective interactions used. (For a review see for example [46-48].) In [13] it is first shown that a devoted smaller storage ring might indeed be necessary to perform neutrino-nucleus measurement at a low energy betabeams facility, since the experiments require a close detector and only the ions close to the ends of the straight sections contribute. Alternatively, the low energy neutrino fluxes might be obtained by putting one/two detectors at offaxis of the storage ring planned for the CP violation search [19]. A comparison of the physics potential of low energy beta-beams and conventional sources is made in [14]. In [20] it is shown that interesting information on the spin-dipole as well as higher multi poles might be obtained by the vaying the 'Y of the ions. Instead of varying the ions one might take different parts of the fluxes at a detector [22]. Gathering more experimental constraints on the corresponding transition amplitudes is important since the same nuclear matrix elements are

151 27,----------,

3~O---CS-~IO-~I~S---f20 Systematic error at each '( {Ofu}

Figure 5: Weinberg angle: One signa uncertainty as a function of the systematic error at each'Y for 'Y = 12 (dotted), 'Y = 7, 12 (broken) and'Y = 7,8,9,10,11,12 (dashed-dotted line). The results are obtained considering electron anti-neutrino scattering on electrons in a water Cerenkov detector [16].

involved in the neutrinoless double-beta decay due to exchange of a massive Majorana neutrino [49]. These measurements can furnish a new constraints to the half-lives calculations that are still plagued by important discrepancies. Finally, a combination of the neutrino beams at different boost can be used to reconstruct the signal from a supernova explosion in an observatory [17]. The proposed method has the advantage that it is free from the cross section uncertainties. 4

The other scenarios for CP violation searches

Various scenarios for the study of CP violation have been proposed where the energy of the ions is much higher, the 'Y ranging from 150 [28] to several hundreds to thousands [31-35,38]. The value of 150 GeV per nucleon comes from the maximum acceleration that can be attained in the SPS. The baseline scenario in this case is the same as the original one. On the contrary, the medium and high energy options require major changes in the accelerator infrastructure, such as a refurbished SPS (or even the LHC) at CERN, as well as bigger storage rings. To match the same oscillation frequencies, such scenarios need further locations for the far detector, such as the Canfranc or the Gran Sasso Underground Laboratories. The physics potential covers the third neutrino mixing angle, the CP violating phase, as well as the neutrino mass hierarchy. Some reduction of the degeneracy problem is also expected. A specific feasibility study is still to be done in order to determine e.g. the ion intensities (that drastically influence the sensitivities) and the storage ring characteristics. Monochromatic neutrino beams produced by boosted ions decaying through electron capture are proposed in [30]. The baseline envisaged is the same as for the original beta-beam. A comparison of Ve -> vI-' oscillations (only Ve are available) at different neutrino energies is necessary. Such a configuration

152

requires the acceleration and storage of not fully stripped ions. The achievable ion rates need to be determined. 5

Conclusions

A beta-beam facility has a rich and broad physics potential. The future and ongoing feasibility studies as well as the current physics investigation will furnish the necessary elements to assess the final CP violation discovery reach. On the other hand, the availability of low energy beta-beams would open new research axis of interest for particle, nuclear and core-collapse supernova physics. This option might require either a devoted storage ring or detector(s) at off-axis. References

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T2K EXPERIMENT K. Sakashita a for the T2K collaboration Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK) , Tsukuba, Ibaraki, 305-0801 Japan Abstract. T2K (Tokai-to-Kamioka) experiment is a next generation long baseline neutrino oscillation experiment in Japan. The main physics goals of T2K experiment are to discover a finite 11I3 by observing /.Ie appearance (/.II-' --> /.Ie oscillation), and to precisely measure oscillation parameters in /.II-' disappearance. We report the status of, and prospects for T2K experiment.

1

Introduction

Neutrino oscillation is a unique tool to study neutrino mass and mixing as well as a probe of new physics beyond the Standard Model. Experimental results of the neutrino oscillation [1-6] indicate that neutrinos have mass and structure of the mixing matrix is completely different than the quark mixing matrix. These results are the first observation indicating physics beyond the Standard Model. Next step of neutrino oscillation experiment is to discover the last unknown mixing angle (0 13 ) and the leptonic CP violating phase. In particular, determination of the size of 0 13 is important because the size of 0 13 could determine the direction of future neutrino experiments; for instance, a sufficiently large 0 13 enable us to measure the leptonic CP violation phase in future experiments using a conventional neutrino beam.

T2K experiment

2

T2K experiment is the accelerator-base long baseline neutrino oscillation experiment in Japan. Muon neutrinos (lIJ.L'S) are produced by striking protons from 50GeV proton synchrotron in J-PARC on a 90 cm long graphite target. Neutrino beam is directed to the Super-Kamiokande detector located at 295 km from J-PARC. The main physics goals of T2K experiment are discovery of a finite 0 13 by observing lie appearance (lIJ.L ----> lJe oscillation), and precise measurement of oscillation parameters, ~m~3 and sin 2 (20 23 ), in 1IJ.L disappearance. There are mainly three features in T2K experiment. We will describe each feature in following subsections.

2.1

Intense narrow-band neutrino beam

We employ "off-axis" method [7] in order to produce an intense narrow-band neutrino beam. This method will enhance the experiment sensitivity by allowae-mail: [email protected]

154

155

ing tuning of the beam energy to match with neutrino oscillation maximum. The beam energy is determined by selecting the angle between the direction of parent 11"+ and neutrinos. We set the angle to 2.5 0 in order to tune the beam energy to be 0.6-0.7 GeV which corresponds to the current value of ~m~3 = (2.2 rv 2.6) X 10- 3 eV2 [1-3] with the baseline of 295 km. The number of neutrino interactions in SK detector is expected to be rv 1600 Charged Current interactions per year. It is two times larger than the case of "on-axis" beam. The off-axis beam also reduce the high energy tail and it enable us to reduce the background events for the energy reconstruction as described below. 2.2

Super-Kamiokande detector

The far detector is Super-Kamiokande (SK) detector. This is an advantage for T2K because of its good particle identification and energy resolution. SK detector is a cylindrically shaped water Cherenkov detector surrounded by 11146 20-inch phototubes. It composes water of 50 kton in total (22.5kton fiducial). We are able to identify particle using the features of the Cherenkov ring; shower-type ring for electron while sharp-edged ring for muon. The particle separation is better than 99 %. The uncertainty of the energy scale is rv2%. The more detailed description of SK detector can be found in [8]. 2.3

Energy Reconstruction

We are able to reconstruct the energy of neutrino by measuring momentum of charged lepton and assuming the Charged Current Quasi-elastic (CCQE) interaction; Vz + n ----t l- + p (where l = f-l or e). The energy of neutrino is rec

E"

=

mr

mnEz /2 - (m~ - m~)/2 ' mn - Ez + pz cos 0z

where m n , mz, and mp are mass of neutrino, charged lepton and proton, respectively, and Ez and pz is energy and momentum of charged lepton, respectively, and Oz is angle of charged lepton with respect to the incident neutrino. 3

3.1

Prospect 0 13 measurement,' vJ.l

----t Ve

appearance

We measure Ve appearance signal by detecting single electron from Ve CCQE interaction in SK detector. On the other hand, there are two kinds of background sources. One of them is the intrinsic Ve in the beam. The Ve component in the beam is 0.4 % at the peak of beam energy. Another background source is single 11"0 production in Neutral Current interactions (NC1I"° background), in

156

which nO is mis-identified as electron because two photons from nO decay hit near on the detector or one of photons undetected by the detector. In addition to the selection criteria for the single electron, we apply specific e/no separation cuts in order to further reduce the NCno background events. The total expected number of background is 22 (13 for intrinsic V e , 9 for NCnO) events for the reconstructed neutrino energy between 0.35 Ge V and 0.85 GeV while the expected number of signal is 103 events for sin 2(20 13) = 0.1, with 5 x 10 21 protons on the target (p.o. t.) measurement b. Figure 1 shows the expected sensitivity of 013 as a function of CP violating phase J. Even though the dominant term of vI' ~ Ve oscillation probability at 295 km and the neutrino energy of '" 0.6 GeV is 013 term, it also depends on the other unknown parameters, J, sin 023 and the sign of .6.m§3' We expect that our sensitivity of 013 is sin 2(20 13 ) < 0.008 at 90 % C.L. for J = 0, .6.mI3 = 2.5 x 10- 3 eV 2 (normal hierarchy) and sin2(2023) = 1 with 5 x 10 21 p.o.t. measurement. It is more than a order of magnitude improvement from the current world limit of sin2(20 13 ) [10]. In contrast, reactor experiments [9] measure Ve disappearance which purely provides sin 2(20 13 ) because its probability does not depend on other unknown parameters except for 013 . Comparison between measurements of the long baseline experiment such as T2K and the reactor experiments could provide some constraints on J, sin 023 and the sign of .6.m§3 [11]. 3.2

.6.m§3 and sin 2(20 23) " vI' disappearance

We measure vI' disappearance as both a suppression in the total number of vI' events observed at SK and a distortion of the energy spectrum compared to the expected energy spectrum from the production point. The survival probability of vI' depends on oscillation parameters, (sin2(2023), .6.m§3)' We expect that our sensitivities are J(.6.m§3) < 1 x 10- 4 eV 2 and J(sin2(2023)) '" 1% with 5 x 10 21 p.o.t. measurement. It is important to reduce systematic uncertainties less than statistical uncertainty. We studied several sources of the systematic uncertainty, and then found that shape of the expected energy spectrum should be known within 20% accuracy. In order to achieve this requirement, we have measured a production distribution of parent particles of vI' as described in next section. 4

Hadron production measurement

The energy spectrum at SK in the absence of oscillation is used to expect the number of background event in Ve appearance, and to extract oscillation bcorresponds to 5 years measurement with the designed beam intensity (O.75MW)

157

.,.

!! 150

.,.

~

~

..'"

!! 150

..'"

~ 100

~ 100

J:

a.

J:

U

U

.,

.,



a.

"- 50

"- 50

0

0

-50

-50

-100

-100

-150

-150 1 2 sin 28 13

2

1

sin 28 13

Figure 1: (Left plot) Sensitivity of non-zero 013 as a function of CP violating phase 8 for three LlmI3 values with 5 x 1021 p.o.t. measurement. Black, red and blue lines are sensitivities with LlmI3 = 2.5 x 10- 3 , 1.9 X 10- 3 and 3.0 x 10- 3 ey2, respectively. (Right plot) Sensitivity of non-zero 013 as a function of CP violating phase 8 for no-matter effect (black), normal hierarchy (red) and inverted hierarchy (blue) with the same exposure time.

parameters in vI' disappearance. The reconstructed energy spectrum in the absence of oscillation, ¢8K (Ev), is expressed as (simplified case):

(1) where E" is energy of neutrino, cp8K (E,,) is the energy spectrum of the neutrino beam at SK, (J8K (E,,) is neutrino cross-section and e8K (E,,) is detection efficiency of SK detector. Among these quantities cp8K (E,,) and (J8K (E,,) has a large uncertainty and hence we need to reduce those uncertainties. For (J8K, we carried out a measurement of neutrino cross-section. The prospects and a current status are found in the reference [12]. For cp8K (E,,), we have measured 1l'+ and K+ (hadron) production distribution because we use the production distribution in a Monte Carlo simulation (beam-MC) which is used to expect cp8K (E,,). There are also no measurements of hadron production for interactions between 30",50 Ge V protorf and graphite target. We first studied requirements on accuracy of hadron production measurements in order to achieve the requirements from T2K goals. We found that we need to measure 1l'+ momentum distribution each 20 mrad of production angle between 0 mrad and 250 mrad with accuracy less than 10 %. This measurement reduces the uncertainty of the shape of cp8K. Moreover, in order to CJ-PARC will provide 30 GeY protons at the beginning.

158 Table 1: Impact of the hadron production measurements. Uncertainties only from the hadron production with the CEARN-NA61 measurement listed in the second column is smaller than the T2K goals listed in the third column.

5(Nbkg) for Ve appearance J(sin2(2023)) J(Am~3) [xlO- 4 eV2]

Error only from the hadron production with NA61 measurement <4% 0.5 % 0.15

T2K goal 10% 1% <1

expect high energy fraction of cI>s K (Ev), we measure K+ production because most of the high energy vlJ.'s are from decays of K+. We need to measure a ratio of the number of K+ to the number of n+ with accuracy less than 10 %. This measurement reduces the uncertainty of the number of NCno events because its cross-section is large for the high energy Vw We then carried out CERN NA61 experiment [13]. The first data taking was performed in September 2007. We expect that the collected data enables us to reduce systematic uncertainty of the number of background and the oscillation parameters less than the T2K goals as summarized in Tab. 1. Analysis of the data is now in progress. 5

Construction Status

Construction of accelerator complex in J-PARC will be completed in 2008. Commissioning of some parts of accelerator has been already started. It was successful in accelerating a proton beam up to the design energy of 3 Ge V in Rapid-Cycling-Synchrotron, which provides proton beam to 50 GeV PS. Commissioning of 50 GeV PS will be started in May 2008. Neutrino beam-line has been constructed since 2004. Most of civil constructions was finished and installation of equipments has been started. Manufacture and long-term operation test of equipments are in progress and it is on schedule. Commissioning of the neutrino beam-line will be started in April 2009. Neutrino detectors at J-PARC, which measure energy spectrum, flux and flavor contents of neutrino beam, are under construction and will be installed in 2009. 6

Summary

T2K experiment aim for a discovery of 013 and precise measurement of Am~3 2 and sin (2022 ) by using a narrow-band intense vlJ. beam and Super-Kamiokande detector. T2K has a physics potential that possible to measure sin 2(20 13 ) less

159

than 0.008 with 90 % C.L. in the case of 8 = 0, ~mi3 = 2.5 x 10- 3 eV 2 (normal hierarchy) and sin2(2B23) = 1, and to measure 8(~m~3) < 1 x 10- 4 eV 2 and 8(sin2(2B23)) rv 1% with 5 x 10 21 p.o.t. measurement. Moreover, we have carried out specific hadron production and neutrino cross-section measurements in order to reduce systematic uncertainty. Construction of the accelerator complex and the neutrino beam-line are on schedule. Commissioning of the neutrino beam-line will be started in April 2009. References [1] Y. Ashie et al. [Super-Kamiokande Collaboration], Phys. Rev. D 71, 112005 (2005). [2] D. G. Michael et al. [MINOS Collaboration]' Phys. Rev. Lett. 97, 191801 (2006). [3] M. H. Ahn et al. [K2K Collaboration], Phys. Rev. D 74, 072003 (2006). [4] J. Hosaka et al. [Super-Kamkiokande Collaboration]' Phys. Rev. D 73, 112001 (2006). [5] B. Aharmim et al. [SNO Collaboration], Phys. Rev. D 72, 052010 (2005). [6] T. Araki et al. [KamLAND Collaboration], Phys. Rev. Lett. 94, 081801 (2005). [7] D. Beavis, A. Carroll, 1. Chiang, et al., Proposal of BNL AGS E-889 (1995). [8] Y. Fukuda et al., Nucl. Instrum. Meth. A 501, 418 (2003). [9] J. Cao, arXiv:0712.0897 [hep-ex] and references therein. [10] M. Apollonio et al. [CHOOZ Collaboration], Eur. Phys. J. C 27, 331 (2003). [11] H. Minakata, H. Sugiyama, O. Yasuda, K. Inoue and F. Suekane, Phys. Rev. D 68, 033017 (2003). [12] Y. Nakajima [SciBooNE Collaboration], arXiv:0712.4271 [hep-ex]. [13] N. Antoniou et al., "Study of hadron production in hadron nucleus and nucleus nucleus collisions at the CERN SPS" .

NON-STANDARD NEUTRINO PHYSICS PROBED BY TOKAI-TO-KAMIOKA-KOREA TWO-DETECTOR COMPLEX Nei Cipriano Ribeiro 1 a, Takaaki Kajita2 b, Pyungwon K 0 3 c, Hisakazu Minakata4 d, Shoei Nakayama2 e, Hiroshi Nunokawa 1 f 1 Departamento de Fisica, Ponti/icia Universidade Cat6lica do Rio de Janeiro, C. P. 38071, 22452-970, Rio de Janeiro, Brazil 2 Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan 3 School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea 4 Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan Abstract. The discovery potentials of non-standard physics (NSP) which might be possessed by neutrinos are examined by taking a concrete setting of Tokai-toKamioka-Korea (T2KK) two detector complex which receives neutrino superbeam from J-PARC. We restrict ourselves into vlJ. and vlJ. disappearance measurement. We describe here only the non-standard interactions (NSI) of neutrinos with matter and the quantum decoherence. It is shown in some favorable cases T2KK can significantly improve the current bounds on NSP. For NSI, for example, i;IJ.7" < 0.03, which is a factor 5 severer than the current one.

1

Introduction

The primary objective of the future neutrino oscillation experiments is of course to determine the remaining lepton mixing parameters, most notably CP violating phase and the neutrino mass hierarchy. Nonetheless, it is highly desirable that such facilities possesses additional physics capabilities such as exploring possible non-standard interactions (NSI) of neutrinos with matter. It will give us a great chance of discovering or constraining the extremely interesting new physics beyond neutrino mass incorporated Standard Model. Such additional capabilities are highly desirable because such projects would inevitably be rather costly, and it would become the necessity if a smoking gun evidence of new physics beyond the Standard Model is discovered in ",TeV range. Some of the present authors have proposed Tokai-to-Kamioka-Korea (T2KK) identical two detector complex which receives neutrino superbeam from JP ARC as a concrete setting for measuring CP violation and determining the mass hierarchy [1,2]. In this manuscript we report, based on [3], discovery reach to the possible non-standard interactions of neutrinos and the quantum decoherence by the T2KK setting. See [3] for the sensitivities to the Lorentz invariance violation as well as the cases which are not treated in this report. ae-mail: be-mail: ce-mail: de-mail: ee-mail: fe-mail:

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

160

161

2 2.1

Non-Standard Interactions (NSI) of Neutrinos NSI; General feature

It has been suggested [4,5] that neutrinos might have non-standard interactions (NSI) which reflect physics outside Standard Model of electroweak interactions. The possibility of exploring physics beyond the neutrino mass incorporated Standard Model is so charming that the sensitivity reach of NSI would be one of the most important targets in the ongoing as well as future neutrino experiments. The latter include neutrino superbeam experiments, reactor (h3 experiments [6], beta beam, and neutrino factory [7]. See these references for numerous other references on hunting NSL In this sense it is natural to investigate sensitivity reach of NSI by T2KK. As a first step we examine the sensitivity to NSI by using vI-' and iJl-' disappearance modes of T2KK. We of course make a comparison between discovery potentials of T2KK and Kamioka only (T2K II [8]) as well as Korea only settings. Our primary concern, however, is not to propose to use NSI sensitivity as a criterion of which setting is the best, but rather to understand how the sensitivity to NSI in T2KK is determined. (The real decision between various settings would require many other considerations.) As is now popular, the effects of NSI are parametrized in a model independent way by ca(3 parameters (0., f3 = e, j.L, T) in the matter sensitive term in the effective Hamiltonian in the flavor basis, H~~ = a(oaeo(3e + Ca(3) , where a == V2G F N e with G F being the Fermi constant and Ne electron number density in the earth. The existing constraints on ca(3 are worked out in [9]. When we restrict ourselves into the disappearance channel we can safely truncate the system into the 2 x 2 subsystem [3] as

(1) where U is the flavor mixing matrix and a == V2G F N e. Because of the form of the 2-2 element of the NSI term in the Hamiltonian, we set C 1-'1-' = 0 and simply discuss the constraint on cn' and CWT' 2.2

Sensitivity reach to NSI

We describe analysis results by skipping the details of the procedure by referring the readers [3] for it. The input values cW" and CTT are taken to be vanishing. The important point in correctly estimating the sensitivities is to marginalize over the lepton mixing parameters, in particular, 6.m~2 and (}23· To understand competition and synergy between the detectors in Kamioka and in Korea, and in particular, between the neutrino and the anti-neutrino

162 Kamioka

Korea

Kamioka + Korea

<


<

+


Figure 1: The synergy between different detectors and v and j) running. The red, the yellow and the blue lines are for the regions allowed at 10', 20', and 30' CL (2 DOF), respectively. No systematic error is taken into account and the number of energy bins considered were 36 from 0.2 to 2.0 GeV. The top, the middle, and the bottom panels are for 4 years neutrino running, 4 years of anti-neutrino running, and both neutrino-anti-neutrino combined, respectively. The left, the middle and the right panels are the constraints obtained by Kamioka detector, by Korea detector (each 0.54 Mton fiducial mass), and by both detectors combined, respectively. Note that the last one is not identical with the T2KK setting defined which is defined with each fiducial mass of 0.27 Mton, and whose sensitivities are presented in Fig. 2.

channels we present Fig. 1. We see from the figure that the Kamioka detector is more sensitive to NSI than the Korean detector, probably because of the higher event rate by a factor of ~ 10. The synergy between the neutrino and the anti-neutrino channels is striking; Neither neutrino only nor anti-neutrino only measurement has sensitivity comparable to that of 1/ and f) combined. We present in Fig. 2 the sensitivity to NSI by T2KK and its dependence on 823 . The approximate 2 a CL (2 DOF) sensitivities of the Kamioka-Korea setup 2 for sin 2 8 = 0.45 (sin 8 = 0.5) are: ICJLTI < 0.03 (0.03) and ICTTI < 0.3 (1.2). Here, we neglected a barely allowed region near IETTI = 2.3, which is already excluded by the current data. Notice that T2KK has potential of (almost) eliminating the island regions. The disparity between the sensitivities to CJLT and CTT can be understood by using the analytic formula as discussed in [3]. The figure also contain the comparison between discovery reach of NSI by the Kamioka-orily setting, the Korea-only setting, and T2KK.

163

Figure 2: The allowed regions in el'r -err space for 4 years neutrino and 4 years anti-neutrino running. The upper, the middle, and the bottom three panels are for the Kamioka-only setting, the Korea-only setting, and the Kamioka-Korea setting, respectively. The left and the right panels are for cases with sin 2 (J == sin 2 (J23 = 0.45 and 0.5, respectively. The red, the yellow, and the blue lines indicate the allowed regions at 10', 20', and 30' CL, respectively. The input value of Ll.m~2 is taken as 2.5 x 10- 3 eV 2 . The figure is taken from [3].

3

Quantum Decoherence

Though there is really no plausible candidate mechanism for quantum decoherence, people talk about it mainly because it can be one of the alternative models for "neutrino deficit", namely, a rival of the neutrino oscillation. It is well known that quantum decoherence modifies the neutrino oscillation probabilities. The two-level system in vacuum in the presence of quantum decoherence can be solved to give the vp. survival probability [lO,11J:

P(Vp.

-t

Vp.)

= 1 - '12 sin2 2(} [ 1 -

(b.m L)] 2

e-'Y(E)L

cos

2~2

,

(2)

164 v 4yr +

V4yr 4MW beams

v 4yr + V4yr 4MW beams 2

true sin 29 = 0.960

2.6

~

2.55

1, x NE

2.5

<1

w

E

2.45

2.4 Kamioka + Korea ....... Kamioka Korea

":-O--'--~-;:-~'"-O;;-.:;-1-~O:;C.1;-;C5-~~O.2 y (X10- 23 GeV) Kamioka + Korea ....... Kamioka Korea

Figure 3: The region of allowed values of'Y as a function of sin 2 2(} == sin 2 (}23 (left panel) and ~m2 == ~m;2 (right panel). The case of energy independence of 'Y. The red solid lines are for Kamioka-Korea setting with each 0.27 Mton detector, while the dashed black (dotted blue) lines are for Kamioka (Korea) only setting with 0.54 Mton detector. The thick and the thin lines are for 99% and 90% CL (1 DOF), respectively. 4 years of neutrino plus 4 years of anti-neutrino running are assumed. The normal mass hierarchy is assumed. The other input values of the parameters: ~m~l = +2.5 X 10- 3 ey2, sin 2 (}23=0.5. ~m~l = 8 X 10- 5 ey2 and sin2 (}12=0.31.

with 1'(E) > 0, the parameter which controls the strength of de coherence effect. The most stringent constraints on decoherence obtained to date are by atmospheric neutrino observation b = 1'o(E/GeV)2 < 0.9 x 10- 27 GeV, energyindependent l' < 2.3 X 10- 23 GeV), [10], and solar and KamLAND experiments b = 1'o(E/GeV)-l < 0.8 x 10- 26 GeV) [121. (A particular underlying mechanism for decoherence, if any, may have some characteristic energy dependence.) Yet, such study is worth pursuing in various experiments and in varying energy regions because of different systematic errors, and for unknown energy dependence of 1'. That was our motivation for investigating the sensitivity to quantum decoherence achievable in T2KK. In Fig. 3 presented is the allowed region of the decoherence parameter l' as a function of true values of sin 2 28 23 (left panel) and 6..m 2 (right panel). This is the case of energy independent 1'. In this case T2KK can improve the current bound on decoherence by a factor of 3. It is also obvious that the sensitivity to decoherence reachable by the T2KK setting far exceeds those of Kamioka-only setting, though the sensitivity by Korea-only setting is not so bad. For cases with alternative energy dependences of l' and for other additional non-standard physics, see [31. Most notably, more than 3 orders of magnitude improvement is expected in Lorentz-CPT violating parameter.

165

4

Conclusion

In searching for additional physics potential of the Kamioka-Korea two-detector setting which receives an intense neutrino beam from J-PARC, we have investigated its sensitivities to non-standard physics of neutrinos. It was shown that T2KK can significantly improve the current bounds on quantum decoherence and NSI in some favorable cases. Acknowledgments

H.M. would like to thank the organizers of 13th Lomonosov conference for their kind invitation. He was supported in part by KAKENHI, Grant-in-Aid for Scientific Research, No 19340062, Japan Society for the Promotion of Science. References

[1] M. Ishitsuka, T. Kajita, H. Minakata and H. Nunokawa, Phys. Rev. D 72, 033003 (2005) [arXiv:hep-ph/0504026]. [2] T. Kajita, H. Minakata, S. Nakayama and H. Nunokawa, Phys. Rev. D 75, 013006 (2007) [arXiv:hep-ph/0609286]. [3] N. Cipriano Ribeiro, T. Kajita, P. Ko, H. Minakata, S. Nakayama and H. Nunokawa, arXiv:0712.4314 [hep-ph]. [4] L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). [5] Y. Grossman, Phys. Lett. B 359, 141 (1995) [arXiv:hep-ph/9507344]. [6] J. Kopp, M. Lindner, T. Ota and J. Sato, arXiv:0708.0152 [hep-ph]. [7] N. Cipriano Ribeiro, H. Minakata, H. Nunokawa, S. Uchinami and R. Zukanovich Funchal, JHEP 12,002 (2007) arXiv:0709.1980 [hep-ph]. [8] Y. Itow et al., arXiv:hep-ex/OI06019. For an updated version, see: http://neutrino.kek.jp/jhfnu/loi/loi.v2.030528.pdf [9] N. Fornengo, M. Maltoni, R. T. Bayo and J. W. F. Valle, Phys. Rev. D 65, 013010 (2002) [arXiv:hep-ph/0108043]. S. Davidson, C. PenaGaray, N. Rius and A. Santamaria, JHEP 0303, 011 (2003) [arXiv:hepph/0302093]. J. Abdallah et al. [DELPHI Collaboration]' Eur. Phys. J. C 38, 395 (2005) [arXiv:hep-ex/0406019]. [10] E. Lisi, A. Marrone and D. Montanino, Phys. Rev. Lett. 85, 1166 (2000) [arXiv:hep-ph/0002053]. G. L. Fogli, E. Lisi, A. Marrone and D. Montanino, Phys. Rev. D 67, 093006 (2003) [arXiv:hep-ph/0303064]; [11] F. Benatti and R. Floreanini, JHEP 0002, 032 (2000) [arXiv:hepph/0002221]; Phys. Rev. D 64, 085015 (2001) [arXiv:hep-ph/0105303]. [12] G. L. Fogli, E. Lisi, A. Marrone, D. Montanino and A. Palazzo, Phys. Rev. D 76, 033006 (2007) [arXiv:0704.2568 [hep-ph]].

STERILE NEUTRINOS: FROM COSMOLOGY TO THE LHC F.Vannucci Abstract. Sterile neutrinos arise in several extensions of the Standard Model to accommodate massive neutrinos. In particular the recently proposed vMSM predicts 3 right-handed neutrinos which could explain the dark matter puzzle and the baryon asymmetry of the Universe. A summary of present limits and possible improvements are summarized.

1

The MSM model (1)

Singlet (sterile) neutrino states arise in models which try to implement massive (light) neutrinos in extensions of the Standard Model. In particular the recently developed II MSM model considers 3 singlet states N 1 , N2 and N3 associated with the 3 active neutrinos. Nl having a mass of 10 ke V has a lifetime very long compared to the age of the Universe. Because of its small mass, it is essentially stable on cosmological times. It could account for the missing mass in the form of warm dark matter. Its search can only be thought in astrophysical measurements. Limits exist from searches of monochromatic gamma-rays coming from distant galaxies. The other two states, N2 and N 3, if they are almost degenerate could solve the problem of the disappearance of antimatter in the Universe. Their masses is expected to be in the range 100 MeV-few Ge V. N2 and N3 can be searched for in the laboratory through their mixing with light neutrinos. Limits exist.

2

Production of sterile neutrinos

If heavy neutrinos exist, they mix with active neutrinos through a unitary transformation. Any neutrino beam will contain a fraction of heavy neutrinos at the level U'j:.{l where U denotes the mixing matrix element between the heavy state Nand l being either e or {l or T. At accelerators, neutrinos are emitted in pion and kaon decays. Kinematically the mass range allowed for a heavy N depends on the emission process. In 7r{l decays, sterile neutrinos can reach a mass of 30 MeV. In 7re, the range is increased to 130 MeV. Kaons allow larger potential masses of up to 450 MeV. The flux of N's accompanies the flux of known neutrinos at the level of U'j:.ll. Corrections to this straightforward result come from helicity conservation which applies differently. For massless neutrinos, it suppresses 7re decays relative to 7r{l decays. This is not true anymore for 7r -+ eN.

3

Decays of sterile neutrinos

N's will decay through weak interactions. The decay modes depend on the N mass. The first mode to open is N -+ eell, as soon as the mass is greater than 166

167

1 MeV. With increasing masses, new modes open and one can obtain ef.Lv, 1re, f.Lf.L V , 1rf.L ... The lifetime is given by the formula applying to weak decays, apart from a suppression factor coming again from the mixing U'lvl. Phase space factors are also changed.

4

Previous results

The search consists in looking for a decay signature, typically 2 tracks reconstructing a vertex in an empty volume arising in a neutrino beam. This has been attempted by the low energy experiment PS191 (2) with 5· 10 18 protons of 19 GeV on target. The figure shows the present situation with limits from above coming from lab experiments, and from below from astrophysics experiments. Between these two kinds of limits there remains a small window of existence. Soon-to-run experiments could improve these results and close the window. The MiniBoone result (3) can be interpreted as coming from a heavy neutrino of mass around 200 MeV, with mixing in the presently allowed region. This is a good reason to try to improve on the old results.

5

Future

The present result can easily be improved with the use of more modern beams. For example, the NuMI beam at Fermilab offers 100 times more neutrinos than the old PS191 measurement. The proton beam is 120 GeV. Thus the beam has a fair component of neutrinos from charm decays. This has the advantage to open the search of N's up to a mass of 1.3 GeV. A magnetized calorimeter having a good energy and direction resolution is being built: this is the Minerva detector. Thus it is enough to complement it with a vacuum volume put in front of it to test the MiniBoone excess and close a good part of the remaining window. Other possibilities exist at the LHC. First the LHCb will accumulate a huge number of B decays and this allows the search of N's up to 5 Ge V masses. Furthermore the Atlas and CMS detectors will accumulate a large number of W decays which can also be used to search for the existence of N's up to 50 GeV masses.

6

Conclusion

The vMSM model is a natural and attractive extension of the Standard Model. It predicts new states which can be searched in soon to be experiments. The fascinating possibility of the existence of sterile neutrinos in a reachable domain should be tested in the next few years.

References

FROM CUORICINO TO CUORE TOWARDS THE INVERTED HIERARCHY REGION C. Nones a on behalf of the CUORE collaboration b INFN, Sez. Milano-Bicocca, P.zza della Scienza 3, 20126 Milan, Italy Abstmct. Cuoricino is a Double Beta Decay experiment operating deep underground, at the Laboratori Nazionali del Gran Sasso (Italy) at a depth of about 3500 m.w.e. The search for the Ov/3/3 of 130Te is carried out with the bolometric technique and an upper limit of 3x1024 y (@ 90%C.L.) is set for this process. Cuoricino represents not only the largest DBD Experiment presently operating but also a prototype for a next generation experiment, CUORE (Cryogenic Underground Observatory for Rare Events). CUORE will be a tightly packed array of about 1000 Te02 bolometers placed in a special dilution refrigerator. The expected performance and sensitivity, based on Monte Carlo simulations and extrapolations of the present Cuoricino results, indicate that CUORE will be able to test the 0.020.05 eV region for the effective neutrino mass, having a great discovery potential in the inverted hierarchy region.

1

Introduction

The search for the neutrino mass is one of the most relevant and exciting field in particle physics and cosmology being the neutrino mass scale one of the key quantities in the new theories beyond the Standard Model. A physical tool able to help in answering questions about neutrinos is the nuclear process called neutrino less double beta decay (011/3/3) [1]. It is a rare nuclear transition that, if observed, will confirm that the neutrino mass is different from zero and imply that neutrinos are Majorana particles. To detect this process, the bolometric technique is very powerful. It requires very low temperatures to read appreciable thermal signals induced by the investigated decays. Cuoricino, described in the following sections, is a 011/3/3 experiment looking for the decay of 130Te and using the bolometric technique.

2

The Cuoricino detector

Cuoricino is an array of 62 bolometers of Te02 with an active mass of 40.7 kg. There are 44 bolometers with 5 x 5 x 5 cm 3 crystals and 18 bolometers with 3x3x6 cm 3 crystals. Of the small crystals, 2 are enriched in 128Te and 2 in 130Te. The array is cooled down to ,,-,8 mK in a dilution refrigerator, shielded from environmental radioactivity and energetic neutrons. It is running in the Laboratori Nazionali del Gran Sasso (LNGS), Italy, a location that guarantees a high degree of suppression of cosmic ray flux thanks to the ,,-,3500 m.w.e. depth. ae-mail: [email protected] bhttp://crio.mib.infn.it/wigmi/pages/cuore/collaboration.php

169

170 To fulfill the background requirements typical of rare events physics, a particular care was dedicated to the selection and treatment of the materials used for the construction of the Cuoricino array. Crystals were grown with low contamination materials in China and shipped to Italy where they have been polished with specially selected low radioctivity powders. The mechanical structure of the array was made exclusively in OFHC copper and PTFE, both these materials have an extremely low radioactive content. All the copper and PTFE parts of the mounting structure were separetely treated with acids to remove surface contamination. Finally, the array was assembled in an underground clean room in a nitrogen atmosphere to avoid radon contamination, closed in a copper structure and hung in vacuum inside the Inner Vacuum Chamber of the refrigerator; it is surrounded by a rv 1 cm thick roman lead cylindrical shield closed with bottom and top lead discs of thicknesses of 7.5 cm and 10 cm respectively. The refrigerator itself is shielded with low activity lead with 20 cm thickness and with borated PET with 10 cm thickness. Nitrogen is fluxed between the external lead shield and the cryostat to avoid any Rn contribution to the detector background; in addition the cryostat is placed inside a Faraday cage to reduce electromagnetic interferences. In order to prevent vibrations induced by the overall facility from reaching the detectors, the tower is mechanically decoupled from the cryostat through a stainless steel spring. Thermal pulses are recorded by means of Neutron Transmutation Doped (NTD) Ge thermistors operated in the Variable Range Hopping (VRH) conduction regime (with a volume of about rv 9 mm 3). Electrical connections are made by two gold wires, ball bonded to metalized surfaces on the thermistor, which are then attached to each bolometer by 9 spots of two-component epoxy. The front-end electronics is at room temperature for most of the channels; some channels are read-out by a Si JFET in emitter-follower configuration operating in a 150 K environment placed at the 4 K plate of the dilutione fridge. 3

Cuoricino results

Cuoricino is the most massive bolo metric experiment taking data in the world. The present limit on the half life for the 01/(3(3 decay of 130Te is

Tf/2 > 3.10 24 Y (90%C.L.)

(1)

with an exposure of 11.83 kg·y of 13oTe. This translates into a limit on the effective neutrino mass of

mv < 0.20 - 0.98 eV

(2)

where the range for the limit is spanned by the most and the less favourauble nuclear matrix elements. In the context of a number of nuclear models, these

171

Energy

Figure 1: The Cuoricino background spectrum in the region of interest for DBD of 130Te.

values fall within the range corresponding to the claim of evidence of decay by H.V. Klapdor-Kleingrothaus and his co-workers [2]. Cuoricino is still collecting data and the sensitivity to half life should be twice the present limit when it will be stopped in coincidence with the start-up of CUORE, its natural continuation. In Fig. 1 the background energy spectrum in the region of interest for DBD summed over all the channels is shown. No peak appears at the Ovf3f3 Q value and the upper limit is obtained with a Maximum Likelihood procedure. The limit is evaluated using anticoincidence sum spectra and considering separately the sum spectra of big (5x5x5 cm 3), small (3x3x6 cm 3) and 130Te enriched crystals and distinguishing two runs, one before and the other one after a long stop due to safety upgrade works in the underground laboratory; in order to account for the spread in energy resolutions of the detectors, the response function used in the fit is the sum of N gaussian (N being the number of detectors summed in the considered spectrum) centered at the same energy but with different sigma. 4

From Cuoricino to CUORE

CUORE is the natural extension ofthe Cuoricino experiment [4]. It will be also located in the Laboratori Nazionali del Gran Sasso. It is a proposed tightly packed array of 988 Te02 bolometers, each being a cube of 5 em on a side

172 with a mass of 760 g. The array will consist of 19 vertical towers, arranged in a cylindrical structure. CUORE is a funded project expected to take data during 201l. Each tower will consist of 13 layers of 4 crystals. The detector is able in principle to investigate several ultralow background searches (such as cold dark matter, solar axions, and rare nuclear decays), but by far the most prominent goal of CUORE is the search for neutrinoless double beta decay of l30Te. The temperature sensors will be NTD Ge thermistors, specifically prepared in order to present similar thermal performance. Proper resistive heaters of 300 k!l, realized with a heavily doped meander implanted on a 3x3xO.5 mm 3 silicon chip, are attached to each absorber in order to calibrate and stabilize the gain of the bolometer over long running periods. CUORE crystals will be grouped in elementary modules of four elements held between two copper frames joined by copper columns. PTFE pieces are inserted between the copper and Te02, as a heat impedance and to clamp the crystals. The bolometers will require an operational working point of "-'7-10 mK that will imply an extremely powerful dilution refrigerator. The background goal of CUORE is to achieve a counting rate in the range from 0.001 to 0.01 counts/keY /kg/y at the 0/J(3(3 transition energy of 130Te (2530 keY). A low counting rate near threshold (that will be of the order of "-'5 keY) is also foreseen and will allow to have results in the WIMP and solar axions research fields. Radioactive contaminations of individual construction materials, as well as the laboratory environment, were measured and the impact on detector performance determined by Monte Carlo computations. Although background levels of the order of 0.001 counts/(keV kg y) seem to be viable thanks to an accurate selection of construction material and to the optimization of the surface treatment, a more conservative level of 0.01 counts/(keV kg y) can be also considered for CUORE sensitivity. It is important to underline that a full Monte Carlo simulation of the CUORE background shows that all the background sources are controlled at the level of 0.001 counts/(keV kg y), with the exception of surface contamination. The presently achieved level of cleaning and shielding of copper surface in the R&D tests for CUORE allows to predict a background level from 2 to 4 times higher than the conservative scenario of 0.01 counts/(keV kg y). Therefore some work is still to be done to reach even the conservative goal of CUORE, and this work regards solely the issue of surface contamination. The radical solution of this problem would have a dramatic impact on the physical reach of CUORE. In fact, a background level of 0.01 counts/(keV kg y) would allow just to touch the inverted hierarchy region of the neutrino mass pattern, while a factor 10 lower would allow to explore deeply this region, with an obvious increase of the CUORE discovery potential. This explains why the struggle against surface background is presently the central issue in the CUORE preparation work.

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5

Conclusions

Cuoricino is the most sensitive running experiment on Oll(3(3; while being a self consistent experiment, it is also a fundamental test for the next generation CUORE experiment for what concerns detector performances and background value in the Oll(3(3 region. The good results obtained with Cuoricino have shown the feasibility of the tower-like structure and have demonstrated that detector performances are excellent even with a considerable 790 g crystal mass. The obtained results allowed us also to make an evaluation of the background for CUORE with the contamination levels measured for the materials actually at our disposal. An intense R&D with respect to material selection and surface background reduction is under progress. Recent tests and measurements have given promising results. For a background value in the 01.1(3(3 region of the order of 0.001 c/keV /kg/y CUORE could reach a sensitivity on the half life of 6.5.10 26 yin 5 years corresponding to an effective neutrino mass in the range 15-50 meV [5,6], just in the region favoured by current oscillation experiments for the inverted hierarchy. Acknowledgments

The Cuoricino Collaboration owes many thanks to the Director and Staff of the Laboratori Nazionali del Gran Sasso over the years of the development, construction and operation of Cuoricino. The experiment was supported by the Istituto Nazionale di Fisica Nucleare (INFN), the European Commission under Contract No. HPRN-CT-2002-00322 and Contract No. RII3-CT-2004506222 (ILIAS Integrating Activity), by the U.S. Department of Energy under Contract No. DE-AC03-76-SF00098, and DOE W-7405-Eng-48, and by the National Science Foundation Grants Nos. PHY-0139294 and PHY-0500337. References

[1] S. R. Elliott and P. Vogel, Ann. Rev. Nucl. Part. Sci. 52, 115 (2002). [2] H. V. Klapdor-Kleingrothaus, I. V. Krivosheina, A. Dietz, O. Chkvorets, Phys. Lett. B 586, 198 (2004). [3] C. Arnaboldi et al., Phys. Rev. Lett. 95, 142501 (2005). [4] R. Ardito et al., hep-ex 0501010. [5] V. Rodin et al., Nucl. Phys. A 766, 107 (2006). [6] V. Rodin et al., nucl-th 07064304·

THE MARE EXPERIMENT: CALORIMETRIC APPROACH TO THE DIRECT MEASUREMENT OF THE NEUTRINO MASS E.Andreotti a on behalf of the MARE collaboration b Department of Physics and Mathematics, University of Insubria C and INFN-Milano Bicocca, Italy Abstract. MARE is an experiment dedicated to the search for neutrino mass through the study of the beta spectrum of a 187Re source embedded in a low temperature calorimeter. The main aim of this experiment is to reach 0.2 eV sensitivity on neutrino mass, thus providing an alternative to KATRIN. The project will be realized thanks mainly to the convergence of three different experiences, matured within the MANU (Genova), MIBETA (Milano) and Wisconsin X-ray (NASA) groups.

1

Science motivations

1.1

Introduction

In recent years oscillation experiments provided us with the evidence of the existence of a finite neutrino mass. Nevertheless this kind of experiments can inform us only on mass square differences, but can not provide informations on absolute masses themselves. Among the available methods to investigate this issue, kinematical direct measurements based on the study of the (3 spectrum in the region close to the end-point energy, represent the only completely modelfree approach. Unfortunately it is very hard to get sensitivities much lower than ",,1 eV: for this reason (3 decay experiments are in the game only if neutrino masses are quasi degenerate. The best upper limits to date for the De mass come from electrostatic (3 spectrometers measuring 3H decay end-point (Mainz [1J and Troitzk [2]), with limits down to 2.2 eV. The next-generation experiment based on spectrometers, KATRIN, has the potentiality to reach a 0.2 eV sensitivity [3J. It will however approach the ultimate limit for this kind of devices. Furthermore these experiments suffer from difficult systematic problems, so that confidence in the results can be obtained only through confirmation by indipendent measurements with completely different systematics. The calorimetric approach to the direct measurement of neutrino mass can represent a complementary technique, able to investigate the same range of mass values as spectrometers, with no a priori limitations for the expansion of the experiment in order to increase statistics. ae-mail: [email protected] bSee web site: http://mare.dfm.uninsubria.it/frontend/exec.php cVia Valleggio 11, 22100 Como

174

175

1.2

Two complementary experimental appmaches

The experimental requirements for a direct kinematic measurement of neutrino mass may be summarized in the following: high statistics at the end-point, high energy resolution, small and well known systematic effects. At present, the two available methods involved in this search, based respectively on spectrometers and calorimeters, match these requirements in different ways. The peculiarity of spectrometers lies in the fact that the source does not correspond to the detector. In this sense 3H is the ideal candidate for this kind of experiments, thanks to its low end-point energy (18.6 keY) and its half lifetime (12.3 y), which allows to realize high specific activity sources. In addition this technique allows to select only the useful fraction of electrons. Therefore a very high statistics can be accumulated in the relevant part of the j3 spectrum, without disturbances coming from lower energy electrons. There is however a main disadvantage due to the fact that the source is external at the detector: this makes it necessary to deconvolve the response function of the instrument from the data, producing inevitable systematic uncertainties. On the other hand, since in calorimeters the source corresponds with the detector, all the energy available in the j3 decay is measured, except for that shared by the neutrino. In other terms the neutrino energy is measured in the form of a missing energy. This means that when an atom is left in an excited state during the decay, the de-excitation energy is detected together with the electron energy. This is one of the main advantages over spectrometers. However, there is an important inconvenience, connected to the fact that the whole j3 spectrum is acquired. In fact this forces to keep a low counting rate in order not to generate distortions of the spectral shape due to pulse pile-up. For this reason results with the calorimetric technique can be optimized by developing arrays of many small detectors (microcalorimeters) and improving single detector performances, by reducing rise time and increasing energy resolution. Given the previous considerations it comes out that 187Re is the ideal candidate nucleus, thanks to its transition energy of 2.47 keY, the lowest known, which guarantees high statistics at the end-point. In addition its isotopic abundance (62.8%) and half lifetime (43.2 Gy) match most of the requirements for bolometric detection, resulting in a specific activity of'" 1 Hz/mg, ideally suited to the typical size and time response of calorimeters [4]. 2

2.1

MARE: Microcalorimeter Arrays for a Rhenium Experiment

Precursor experiments

Two italian collaborations have been working on Rhenium based microcalorimeters since the ninties of the previous century: the Genova group, within the project MANU [5] and the Milano-Como group, within the project MIBETA [6].

176

These two experiences allowed to set an upper limit on neutrino mass of 15 eV (90% c.l.). The MANU group realized a single device based on metallic Rhenium (absorber mass 1.6 mg) coupled to a Ge-NTD thermistor. The experiment was run for 0.5 years with D.EFWHM = 96 eV and Tr ",200 J-tS resulting in an upper limit on neutrino mass of 19 eV (90% c.l.). Furthermore the MANU group dedicated some efforts in order to develop a specific know-how on TES (Transition Edge Sensors) for a future improvement of detector performances. The MIBETA group developed a 10 detecors array which allowed to reach an upper limit on neutrino mass of 15 eV (90% c.l.) with a total live time of 0.6 years. The single elements of the array were based on AgRe04 absorber (0.25 mg) coupled to Si-implanted thermistors, with an average D.EFWHM = 28.5 eV and Tr = 490 J-tS. The MIBETA collaboration has developed a specific know-how on Si-implanted thermistor technologies, which guarantee high reproducibility and possibility of micro machining, useful characteristics for the future expansion of the experiment.

2.2

MARE: phase I

The results obtained by the MANU and MIBETA experiments prove the potential of the calorimetric technique applied to the direct search for neutrino mass. Though the achieved sensitivity is about one order of magnitude worse than the present limit set by spectrometric experiments, Montecarlo simulations show that it is possible to reach an upper limit of about 2 eV, thus scrutinizing the Mainz and Troitzk experiments, through present technology detectors, single channel optimization and scaling up to hundreds devices. This is one of the main goals of the first phase of the MARE project, born by the convergence of the MANU and MIBETA experiences [7]. MARE-I aims also to improve the understanding about all possible systematic uncertainties and to gather further experience on Rhenium based microcalorimeters. In addition it is crucial to sustain a parallel R&D activity for the second phase of the experiment, aiming to reach a 0.2 eV sensitivity. This purpose can be attained only by the employment of new technology detectors accompanied by a brute-force expansion of the MARE-I experiment. During the first phase of the project, two experiments, directly discending from the two precursors, will be run separately. MANU2 will develop an array of about 300 detection channels, each one composed of a metallic Rhenium single crystal coupled to a TES. MIBETA2 will take advantage of the knowledge accumulated during the previous experience as well as of the collaboration with the Wisconsin X-ray (NASA) group, in order to develop a ",300 detector array based on AgRe04 single crystals (0.45 mg) coupled to Si-implanted thermistors. Tests performed with prototype detectors in Milano shows that the present detector performances (D.EFWHM = 40 eV and Tr ",400 J-ts) are an

177

acceptable baseline design for MIBETA2. Nevertheless further improvements are still possible. The whole array structure of 300 channels will be deployed through a gradual approach and a total statistics of the order of 1010 events will be accumulated during 4 years of running. Two different approaches can be used to evaluate the sensitivity: taking into account the present detector performances, Montecarlo simulations show that a mass limit of about 4 eV can be attained (conservative approach), while considering the possibility that further improvements will lead to ~EFWHM = 15 eV and Tr ......,50 I-£S, a sensitivity of about 2.5 eV can be reached.

2.3

MARE: phase II

I::C.-_~ ~-::;::2~:~:::~ 5' ~ ~

~

....... :1E=5eV,'fR =1IJ S,A=10cts

""~ <-----

~::'-m~~I

m

E:;" o. 1

150000 channels In 5 y

10000 detect... deployed per year

234

.

5

7

8

.

9

10

measuring year

Figure 1: Temporal evolution of the sensitivity to neutrino mass in MARE-II experiment, considering four different experimental configurations. The triplet of numbers that labels each curve indicates respectively flEFWHM [eV], Tr [JlSJ and single channel activity [s-lJ.

The kick-off of MARE-II will be subordinated to the final results of the previous phase. Montecarlo simulations show that in order to reach a 0.2 eV sensitivity substantial improvements over MARE-I are required: a total statistics of about 10 14 events must be accumulated and an energy resolution of......,5 eV and rise time of......,l0 I-£S are needed. New microdetector thecnologies, allowing to match these requirements and suitable to the developement of a 10000 channels array, are already under study and a further R&D activity will be performed in parallel with the MARE-I experiment.

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MARE-II data taking should start not later than 2011, in order to be competitive. The experiment will consist of a series of modular 10000 pixel arrays, which can be relatively easily installed in any available refrigerator. Figure 1 shows the temporal evolution of the sensitivity to neutrino mass, taking into account 4 different experimental configurations and the deployment of one single array per year up to 5 arrays. The final set-up will consist of a spatially distributed array of 50000 elements, 5 years after the data taking start-up, allowing to fulfil the the main purpose of the MARE project. References

[1] C.Kraus, et al., Eur.Phys.J. C 40, 447 (2005). [2] V.M.Lobashov, Nucl.Phys. A 719, 153 (2003). [3] The KATRIN collaboration, KATRIN: A next generation tritium beta decay experiment with sub-e V sensitivity for the electron neutrino mass, (hep-ex/0l09033),200l. [4] S.Vitale, et al., INFN internal report, (INFN/BE-85/2), July 10, 1985. [5] F.Gatti, et al., Nucl.Phys. B 143, 541 (2005). [6] M.Sisti, et al., Nucl.lnstr.Meth A 520, 125 (2004). [7] The MARE collaboration, MARE: MicTOcalorimeter Arrays for a Rhenium Experiment, (http://mare.dfm.uninsubria.it/frontend/exec.php), 2006.

ELECTRON ANGULAR CORRELATION IN NEUTRINOLESS DOUBLE BETA DECAY AND NEW PHYSICS A.Alia Deutsches Elektronen-Synchrotron, DESY, 22607 Hamburg, Germany A.V. Borisov b , D.V. Zhuridov c Faculty of Physics, Moscow State University, 119991 Moscow, Russia Abstract.The angular correlation of the electrons in the neutrinoless double beta decay (Ov2,B) is calculated taking into account the nucleon recoil, the Sand p_ waves for the electrons and the electron mass using a general Lorentz invariant effective Lagrangian. We show that the angular coefficient is essentially independent of the nuclear matrix element models. We work out the angular coefficient in several scenarios for new physics, in particular, in the left-right symmetric models.

1

Introduction

It is now established that the observed neutrinos have tiny masses and they mix with each other [1]. Theoretically, it is largely anticipated that the neutrinos are Majorana particles. Experimental evidence for Ov2(3 decay would deliver a conclusive confirmation of the Majorana nature of neutrinos, establishing the existence of physics beyond the standard model (SM) [2]. An extended version of the SM could contain tiny nonrenormalizable terms that violate lepton number (LN) and allow the Ov2(3 decay. Probable mechanisms of LN violation may include exchanges by: Majorana neutrinos VMS [3,4] (the preferred mechanism after the observation of neutrino oscillations [1]), SUSY particles [5,6], scalar bilinears (SBs) [7], e.g. doubly charged dileptons (the component of the SU(2)L triplet Higgs etc.), leptoquarks (LQs) [8], right-handed WR bosons [4,9] etc. From these particles light vs are much lighter than the electron and others are much heavier than the proton that gives two possible classes of mechanisms for the Ov2(3 decay: long range (with the light vs in the intermediate state) and short range mechanism. Our aim was to examine the possibility to discriminate among the various possible mechanisms contributing to the Ov2(3-decays and the various sources of LN violation using the information on the angular correlation of the final electrons. We published a preliminary study along these lines in Ref. [10] and a more detailed study in Ref. [11]. Here, we summarize the main results of Ref. [11].

2 Angular correlation for the long range mechanism of Ov2(3 decay For the decay mediated by light VMS, the most general effective Lagrangian is the Lorentz invariant combination of the leptonic ja and the hadronic J a currents of definite tensor structure and chirality [12,13] £. =

GFVud [(

v'2

Uei

.).Li J+ ",,' {3 ·i J+ H ] + EVV-A) -A,i ]V -A V -A,).L + L..t Eai] {3 a + .C. ,

(1)

a,{3

where the hadronic and leptonic currents are defined as: J:; = :UOad and = eO{3vi; the leptonic currents contain neutrino mass eigenstates, and the

j~

ae-mail: [email protected] be-mail: [email protected] ce-mail: [email protected]

179

180

index i runs over the light eigenstates. Here and thereafter, a summation over the repeated indices is assumed; (}:, /3=V=t=A, 8=t=P, TL,R (OT" = 2a l-'v Pp , al-'v = ~ [')'I-',')'v], Pp = (1 =t= ')'5)/2 is the projector, p = L, R); the prime indicates the summation over all the Lorentz invariant contributions, except for (}: = /3 = V - A, Uei is the PMNS mixing matrix. The coefficients E~i encode new physics, parametrizing deviations of the Lagrangian from the standard V - A current-current form and mixing of the non-SM neutrinos. The nonzero E~ for the particular SM extensions are collected in Table 1. Table 1.

with LQs ES p, EV +A We have calculated the leading order in the Fermi constant and the leading contribution of the parameters E~ using the approximation of the relativistic electrons and nonrelativistic nucleons. We take into account the 8 1/ 2 and the P1 / 2 waves for the outgoing electrons and include the finite de Broglie wave length correction for the 5 1 / 2 wave. Taking into account the nucleon recoil terms including the terms due to the pseudoscalar form factor we obtain the differential width in cose for the O+(A, Z) --40+(A, Z + 2)e-e- transitions: df In2 2 dcose = 21MGTI A(I- Kcose), (2)

e

where is the angle between the electron momenta in the rest frame of the parent nucleus, MGT is the Gamow-Teller nuclear matrix element and the angular correlation coefficient is

K=B/A,

-1
(3)

Expressions for A and B are given in Ref. [11 J. The analytic expressions associated with the coefficients E~~1 confirm the results of Ref. [4], while the expressions associated with E~~1, E~~~, E~~:~ transcend the earlier work. 3 Analysis of the electron angular correlation Consider the case of zero effects of all the interactions beyond the SM extended by the VMS (Le., all E~ = 0), which we call the "nonstandard" effects. The values of K = Ko == K(E~ = 0) for various decaying nuclei are given in Table 2. Table 2.

I. K I. 76Ge I 82Se I IOuMo I 13UTe I 136Xe I 0.81 . 0.88. 0.88 . 0.85 . 0.84 . We will concentrate on the case of 76 Ge nucleus in the following. The nonstandard terms with E~~1, E~~, E~;~ do not change the angular correlation (the value of K) and the terms with E~~~ give small corrections to Ko. The terms with any other parameters E~ do change this correlation.

181

Using Table 1 and taking into account the fact that IJL~I are suppressed in comparison with IE~I by the factor mdme (the chiral suppression), we find the coefficient K and the set {E} of nonzero E~S that change the 1 - 0.81 cos () form of the correlation for the 8M plus VMS, see Table 3 (the lower two entries). Table 3

{E}

8M extension

-

VM

K 0.81

E'l R

-1 < K < 1 -1 < K < 1 vM+RC EV=+=A They correspond to the following extensions of the 8M: VMS plus RPV 8USY [6], VMS plus right-handed currents (RC) (connected with right-handed W bosons [4] or LQs [8]). Hence, K can signal the presence of this new physics. Let us now consider some particular cases for the parameter space. We will analyze only the terms with E~i1 as the corresponding nuclear matrix elements have been worked out in the literature. We use various types of QRPA [14,15]. In the case of l(m)1 = 0 the current lower bound T1/2 > 1.6 X 1025 yr for the 6 7 Ge nucleus [16] yields the upper bounds on the parameters IJL~~11, IE~~11 that give bounds on the parameters of the particular models [11]. The fact that the dependence of K on the nuclear matrix elements is much weaker than the uncertainty in T1/2 from this source was illustrated in Ref. [11]. In the case of l(m)1 =1= 0, COS'lj;i = 0, where i depends on a, {3, for E~t1 =1=

vM+RPV 8U8Y

TR V+A

°

IJLI2 =

(7.9 + 10K) x 10 12 /T1 / 2 ,

with T1/2 in years, and for E~~1

IE~t112 = (5.1 - 6.3K) x 10 =1=

12 /T1/2'

(4)

°

The correlations among IE~~11, T1/2' K were used in the analysis of left-right symmetric models [17]. In the model SU(2)L x SU(2)R x U(I) we have

(6) for the mass of the right-handed W boson and its mixing angle ( with the left-handed one. The correlation among mWR ((), K, and T1/2 is shown in Fig. 1 left (right) for conservative value E = 10- 6 for the mixing parameter E = Wei Vei I. It is clear that the closer is K to 1 for the fixed value of T1/2 the stronger is the lower bound on mWR (the upper bound on (). We have also shown that the sensitivity of the angular correlation to the W R mass increases with decreasing values of the effective Majorana neutrino mass l(m)1 [11]. Acknowledgments We thank Alexander Barabash and Fedor Simkovic for helpful discussions. One of us (DVZ) would like to thank DE8Y for the hospitality in Hamburg where a good part of this work was done.

182

Figure 1: Correlation between the right-handed W-boson mass mWn (left) or the mixing (right), the angular coefficient, and the half-life Tl/2 for the Ov2f3 decay of 76Ge.

<:

References [1] Particle Data Group: W.-M. Yao et a1., J. Phys. G33, 1 (2006). [2] S.R. Elliot, P. Vogel, Ann. Rev. Nucl. Part. Sci. 52, 115 (2002); P. Vogel, arXiv:hep-phj0611243. [3] Ya.B. Zel'dovich, M.Yu. Khlopov, JETP Lett. 34, 141 (1981); Sov. Phys. Usp. 24, 755 (1981); M.G. Shchepkin, Sov. Phys. Usp. 27, 555 (1984). [41 M. Doi, T. Kotani, E. Takasugi, Prog. Theor. Phys. Suppl. 83, 1 (1985). [5 R.N. Mohapatra, Phys. Rev. D34, 3457 (1986); J.D. Vergados, Phys. Lett. B184, 55 (1987); M. Hirsch, H.V. Klapdor-Kleingrothaus, S.G. Kovalenko, Phys. Rev. Lett. 75, 17 (1995); Phys. Lett. B352, 1 (1995); Phys. Lett. B403, 291 (1997); Nucl. Phys. Proc. Suppl. B52, 257 (1997); Phys. Rev. D57, 1947 (1998); K.S. Babu, R.N. Mohapatra, Phys. Rev. Lett. 75, 2276 (1995); A. Faessler, S.G. Kovalenko, F. Simkovic, J. Schwieger, Phys. Rev. Lett. 78, 183 (1997). [6] M. Hirsch, H.V. Klapdor-Kleingrothaus, S.G. Kovalenko, Phys. Lett. B372, 181 (1996); B381, 488 (Erratum) (1996); H. Pas, M. Hirsch, H.V. KlapdorKleingrothaus, Phys.Lett. B459, 450 (1999). 78J H.V. Klapdor-Kleingrothaus, U. Sarkar, Phys. Lett. B554, 45 (2003). [ M. Hirsch, H.V. Klapdor-Kleingrothaus, S.G. Kovalenko, Phys. Rev. D54, 4207 (1996). [9] M. Hirsch, H.V. Klapdor-Kleingrothaus, O. Panella, Phys. Lett. B374, 7 (1996). 10j A. Ali, A.V. Borisov, D.V. Zhuridov, arXiv:hep-phj0606072. 11 A. Ali, A.V. Borisov, D.V. Zhuridov, Phys. Rev. D 76,093009 (2007). [12 H. Pas, M. Hirsch, H.V. Klapdor-Kleingrothaus, S.G. Kovalenko, Phys. Lett. B453, 194 (1999). [13] G. Gamov, E. Teller, Phys. Rev. 49,895 (1936); S.F. Novaes, in "Particle and Fields", Proc. 10th J.A. Swieca Summer School, Sao Paulo, Brazil, 31 Jan 12 Feb 1999 (World Scientific, Singapore, 2000) [arXiv:hep-phjOO0l283]. [14 G. Pantis, F. Simkovic, J.D. Vergados, A. Faessier, Phys. Rev. C53, 695 (1996). [15 M. Korteiainen, J. Suhonen, Phys. Rev. C75, 051303 (2007). [16 C.E. Aalseth et a1., Phys. Rev. D70, 078302 (2004). [17 J.e. Pati, A. Salam, Phys. Rev. DlO, 275 (1974); R.N. Mohapatra, J.e. Pati, Phys. Rev. DU, 566, 2558 (1975); G. Senjanovic, R.N. Mohapatra, Phys. Rev. D12, 1502 (1975); R.N. Mohapatra, G. Senjanovic, Phys. Rev. D23, 165 (1981).

NEUTRINO ENERGY QUANTIZATION IN ROTATING MEDIUM Alexander Grigoriev a Skobeltsyn Institute of Nuclear Physics, Alexander Studenikin b Department of Theoretical Physics, Moscow State University, 119992 Moscow, Russia Abstract. The exact solution of the modified Dirac equation in rotating medium is found in polar coordinates in the limit of vanishing neutrino mass. The solution for the active left-handed particle exhibit properties similar to those peculiar for the charged particle moving in the presence of a constant magnetic field. Accordingly, the particle in the rotating matter has circle orbits with energy levels, analogous to the Landau levels in magnetic field. The relevant physical realization of the problem is motion of neutrino inside the rotating neutron star. The feature of such motion to form binding states leads to the prediction of the new mechanism for neutrino trapping. The solution found can be used for detailed description of relativistic and nearly massless neutrino dynamics in neutron stars. Results obtained further develop the "method of exact solutions" in application to particle interactions in presence of dense matter.

Recently a method for calculation of various processes of elementary particles interaction proceeding in matter has been developed in a series of our papers (see [1] and references therein). The framework of the method is similar to the Furry representation in quantum electrodynamics and implies the use of exact solutions of the wave equations for particles wave functions [2,3] to account for interaction with matter. The "exact solutions method" has been already applied for description of neutrino propagation in different media and electromagnetic fields and also for evaluation of the quantum theory of the spin light of neutrino [2] and spin light of electron [1,3] in matter, the two recently proposed new mechanisms of electromagnetic radiation produced by a neutrino or an electron moving in matter. The exact solutions of correspondent modified Dirac equations in cases of a neutrino and an electron moving in matter at rest were obtained in [2] and [3] respectively. Then we we have continued these studies and investigated neutrino behavior in rotating medium with prospects for applications to neutron stars. We have found [3] (see also the second paper of [1]) the exact solution for the neutrino wave function in the case of rotating medium using the Cartesian coordinates. In this short note below we show how the same problem can be solved using the polar coordinates. The obtained below result contributes to the further development of the "exact solutions method" in application to studies of particle interactions in presence of matter. Note that the employment of polar coordinates in consideration of a neutrino motion in the rotating ae-mail: [email protected] be-mail: [email protected]

183

184

media fits the symmetry of the problem, therefore the correspondent quantum numbers that determines the neutrino quantum state have quite a natural sense. Let us consider unpolarized matter consisting of neutrons and rotating around the third axes with the angular velocity w. For definiteness, we consider below the case of an electron antineutrino motion in such a medium. The problem for different other neutrino species can be solved in a similar way. Our starting point is the modified Dirac equation [2] accounting for the neutrino interaction with matter,

(1) where

r

=

~(1 + 4sin2 Ow)j'" =

GFjI-L.

(2)

The matter current is given by jI-L = (n, nv)

(3)

where n is the invariant matter density and v being the macroscopic matter speed. In our particular case the matter current can be written as jI-L = n(l, -wy, wx, 0), where y and x are the spatial coordinates in the plane orthogonal to the rotation axis. Introducing the notation "y = GFnw and using polar coordinates we rewrite the equation (1) in components:

mWl,2 {

~ t'P ± ')'rlW4,3 - (E + GF ± P3)W3,4

=0 (E =F P3)Wl,2 + ie'fi'P[tr =F ~ t'P1w2,1 - mW3,4 = 0

+ ie'fi'P[tr

=F

(4)

where P3 is the neutrino momentum operator. The chiral representation of Dirac matrices is used. The form of the equations implies the following operators to be integrals of motion: the energy E = iot , the third momentum component P3 = -io3 , and the third component of the total angular momentum (which is the sum of the orbital and spin momenta, J 3 = L3 + 8 3 = -io'P + ~3). Therefore we write the solution

(5) where the function 'if; can be written in the following form

(6)

185

Substituting relations (5) and (6) into the system (4) one finds the following set of equations for the functions 'l/Ji:

(7)

In the case of relativistic neutrinos (i.e., in the case of vanishing neutrino mass) the fist and the second pairs of the equations decouple from each other and describe, respectively, active left-handed (L) and sterile right-handed (R) neutrino states. Now let us turn to the first pair of equations in (7). Expressing one wave function component ('l/J3 or 'l/J4) through another and introducing the substitution p = "(r2 we obtain separate equations for the each of the components,

fJ2 ~_ (l-1)2 _~_f!. (E+GFn)2-p~).I'_ 4p 2 4 + 4"( '1'3 - 0, ( p 8p2 + 8p 82 ( P8p2

~

+ 8p

_

~

_ l-1 _ f!. 4p 2 4

+

(E

+ GFn)2 - p~) 4"(

(8) .1,

_

0

'1'4 -

•

These equations are almost identical to those corresponding to the case of charged particle motion in the constant homogeneous magnetic field when the problem is considered in the frame of polar coordinates [5]. Their solutions, that posses physical meaning, are expressed via the Laguerre functions IN, s with N, s = N -l and l being the principal, radial and total orbital momentum quantum numbers. The energy-momentum relation therewith is expressed by the formula

(9) Let us consider the second equation of the system (7). Presenting the functions 'l/J3,4 in the form

(10) and taking into account the relation

iV'r [:r + ~ + r]

IN-I, s(p) =

i.JYP [2 :p + 1+ ~] IN,

s(p) =

iV 2N"(IN-I,

s(p)

(11)

186

we obtain relation between coefficients 03 and 0 4 , that leads to

(12) The remaining non-defined coefficient is determined by the normalization condition for the wave function. Finally for the relativistic active left-handed antineutrino wave function we get

(13)

} where L in the left-hand side is the normalization length. The solution for the relativistic sterile right-handed neutrino state can be presented in the plane-wave form

WR

e-~X

= -L-=-3/~2-Vr=(P=3=-===E::::);=2=+=p::;;;:r=+=p:::;;:~

(~) PI - iP2

(14)

P3 - E

with the vacuum energy-momentum relation E = p. As it follows from (9), we conclude that the transversal motion of an active neutrinos and antineutrinos is quantized in moving matter [4] very much like an electron energy is quantized in a constant magnetic field that corresponds to the relativistic form of the Landau energy levels (see, for instance, [5]). Consider again antineutrino. The transversal motion momentum is given by

P-L

= V2pN.

(15)

The quantum number N determines also the radius of the antineutrino quasidassical orbit in matter (it is supposed that N ~ 1 and P3 = 0),

R

=.J

2N

GFnw

(16)

From this it follows, for instance, that low energy (but still relativistic) antineutrinos can have bound orbits inside a rotating star. This can lead to the mechanism of low-energy neutrinos trapping inside rotating neutron stars.

187

References

[1] A.Studenikin, J.Phys.A: Math.Gen. 39 (2006) 6769; A.Studenikin, J.Phys.A: Math.Theor. 41 (2008) 164047. [2] A.Studenikin, A.Ternov, Phys.Lett.B 608 (2005) 107; A.Grigoriev, A.Studenikin, A.Ternov, Phys.Lett.B 622 (2005) 199. [3] A.Grigoriev, S.Shinkevich, A.Studenikin, A.Ternov, I. Trofimov , Grav. Cosmo 14 (2008) 248; Russ. Phys. J. 50 (2007) 596; hep-phj0611128. [4] A.Grigoriev, A.Savochkin, A.Studenikin, Russ.Phys.J.50 (2007) 845. [5] A.A. Sokolov and I.M. Ternov, Synchrotron radiation (Oxford: Pergamon Press, 1968).

NEUTRINO PROPAGATION IN DENSE MAGNETIZED MATTER E.V. Arbuzova a International University "Dubna", 141980 Dubna, Russia A.E. Lobanov b, E.M. Murchikova C Faculty of Physics, Moscow State University, 119991 Moscow, Russia Abstract. We obtained a complete system of solutions of the Dirac-Pauli equation for a massive neutrino interacting with dense matter and strong electromagnetic field. We demonstrated that these solutions can describe precession of the neutrino spin.

Neutrino oscillation phenomenon is well established now. Many interesting results were obtained and many interesting problems were discussed in literature. From the theory of this phenomenon it follows that oscillations are possible only when neutrino possesses non-vanishing mass. Consequently, it has to possess a non-zero magnetic moment [1]. In turn, the existence of the magnetic moment is associated with so-called neutrino spin oscillations. The analysis of neutrino spin oscillations was mainly based on solving the Cauchy problem for the Schrodinger type equation with an effective Hamiltonian [2-6]. In contrast the purpose of our work is to find a complete system of solutions of the wave equation which accounts for the influence of an electromagnetic field and a dense matter on neutrino dynamics. The existence of such system gives a possibility to describe the states of neutrino with rotating spin as pure states and to evaluate the probabilities of various processes with neutrino in the framework of the Furry picture. Let us discuss an equation for neutrino (its mass eigenstate). When the interaction of a massive neutrino possessing an anomalous magnetic moment /-La with background fermions is considered to be coherent, its propagation in matter and electromagnetic field is described by the Dirac-Pauli equation with the effective four-potential jJ.L. In order to avoid correlations between flavour and spin oscillations (the Mikheyev-Smirnov-Wolfenstein effect [7,8]) we shall suppose that the effective four-potential is the same for all neutrino flavours. In what follows, we restrict our consideration to the case of a homogeneous medium. Then the explicit form of the generalized Dirac-Pauli equation is:

(i8 - ~j(l +

'l) -

~/-LapiLll O'J.LV -

m) W

= O.

(1)

.Ai

Here FJ.LV is an electromagnetic field tensor, the functionjJ.L = Lf Pj ljj + Pj l is a linear combination of the currents jj and the polarizations of background fermions; summation is carried out over the background fermions. The expressions for the coefficients pjl,2 l depend on the model chosen for neutrino 1

.Ai

ae-mail: [email protected] be-mail: [email protected] ce-mail: [email protected]

188

2

189

interactions. The quantities with hats denote scalar products of the Dirac matrices with four-vectors: a == "(l-ta w We use the units Ii = c = l. Note, when propagation of neutrino through a dense matter and an electromagnetic field is studied, we should keep in mind that the matter and the field are situated in the same area. Therefore strengthes of electric and magnetic fields and average velocities and polarizations of matter should satisfy the selfconsistent system of equations including the Maxwell equations, the Lorentz equation and the classical spin evolution Bargmann-Michel-Telegdi (BMT) equation. As already mentioned, we restrict our consideration to constant velocity and polarization of matter and constant homogeneous electromagnetic field in equation (1). However, even such a choice imposes limitation on Fl-tll and fl-t: FWfll =0. (2) Let us look for a solution of equation (1). Since the functions Fl-tll, fl-t are constant, the canonical momentum operator i81-t commutes with the Hamiltonian of this equation. However, the commonly adopted choice of eigenvalues of this operator as quantum numbers is not satisfactory in our case. Particle kinetic momentum components related to its group four-velocity ul-t by the relations ql-t = mul-t, q2 = m 2 are more suitable to play this role. In the present paper we discuss the solutions described by the quantum numbers which can be interpreted as kinetic momentum components. The explicit form of a kinetic momentum operator for a particle with spin is not known beforehand. Hence, in order to find the appropriate solutions, we have to use the correspondence principle. If effects of neutrino weak interactions are taken into account, the Lorentz invariant generalization of the BMT equation for a spin vector SI-t of a neutrino moving with the four-velocity ul-t has the form [9] Sl-t =

2{ (Fl-tll + Gl-tll) SII -

ul-tu ll (F II )..

+ Gil)..) S)..},

(3)

where GW = ~el-tIlP).. fpu)... Here and further a dot denotes the differentiation with respect to the proper time T; /LoFl-t1l =} Fl-tll. Suppose the Lorentz equation is solved, and the dependence of particle coordinates on the proper time is found. Then the BMT equation transforms to an ordinary differential equation, whose resolvent determines a one-parametric subgroup of the Lorentz group. The quasi-classical spin wave function [10] is represented by a spin-tensor, whose evolution is determined by the same one-parametric subgroup. In our case this spin-tensor represents the Dirac bispinor. We choose a solution of equation (1) in the form [10,11]

'l1(x)

= e-iF(X)U(T(X), TO = O)'l1 o(x).

(4)

In this formula U(T, TO) is an operator of evolution for a quasi-classical spin wave function, e-iF(x) is a phase factor, and

190

(5) is a solution of the Dirac equation for a free particle. Here four-vector sg determines the direction of particle polarization, (0 = ±1 is a sign of spin projection on this direction and 'l/Jo is a constant bispinor normalized by the condition ~o(x)\lIo(x) = mlqo. For the operator U(T, TO) we have a relation

U( T, TO) = {

4~ 1'5(}q -

qj) +

~21'5 HI1V qv I'll q} U(T, TO),

(6)

where HI1V = - !e l1vp >, Fp>. is a dual electromagnetic field tensor. It is obvious that the solution of equation (6) can be represented as a matrix exponent. It is not difficult to verify that the solution of equation (1) can take the form (4) only if the second invariant 12 = ~Fl1v H l1v of the tensor Fl1v is equal to zero, and the vector 111 is its eigenvector corresponding to the zero eigenvalue. So we come to the condition above (2), which was obtained only on the physical attends. Substitution of the expression (4) into equation (1) taking into account (6) leads to the relation

Nil = al1T = _

m(Jcp) q 2((cpq)2 - m 2cp2) 11

+ r~+cpl1 2(cpq)

3

m (Jcp) , (7) 2(cpq)((cpq)2 _ m 2cp2)

where cpl1 = 111 12 + HI1V qv 1m. Since 111, Nil = const we obtain the proper time = (N x) and the phase factor which determines energy shift of neutrino in matter F(x) = (Jx)/2. Hence the expression for the wave function takes the form T

\lI(x) =

~

2:: e-

i

(P,x)(l_

(1' 5 Stp )(1- (01'5S0)(q + m)'l/Jo,

(8)

(=±1

where

ql1(cpq)/m - cpl1m , ...j(cpq)2 _ cp2m2 12 - (NI1...j'-(cp-q-)-2---cp-2m-2/m.

11 _

Stp -

Pt = ql1 + r

(9) (10)

It is obvious that the system of solutions (8) is a complete system of solutions of equation (1), which is characterized by the kinetic momentum of the particle ql1 and the quantum number (0 = ±1, which can be interpreted as the neutrino spin projection on the direction sg at the moment T = (N x) = O. In the general case this system is not stationary. The received solutions are stationary only when = p • In this case the wave functions are the eigenfunctions of the spin projection operator on the direction p with the eigenvalues ( = ±1 and of the canonical momentum operator ia l1 with the eigenvalues The orthonormalized system of the stationary solutions of equation (1) can be written down in the following way:

sg st

st

Pt.

(11)

191

Here J is the transition Jacobian between the variables qll and J =

Pt=

2(1 + (fIlHJ1-l'ql'/2m -2 211 (ltp) (1 + (2J(tpq)2 - m 2tp2 ) J(tpq)2 - m tp2 ).

(12)

The structure of solution (11) directly leads us to the conclusion that when neutrinos move through a dense matter and an electromagnetic field which satisfy condition (2), they can behave as free particles, i.e. move with the constant group velocity opo Vgr

= --' oP, = ~ qO

(13)

and conserve the polarization. However in interactions with other particles the channels of reactions which are closed for a free neutrino can be opened (see, for example, [11,12]), as a result of difference of the dispersion law for the free neutrino p2 = m 2 from the one for neutrino in matter and electromagnetic field:

(14) where

FJ1- = A LJ.

Pt - fll/2, •

= SIgn

(1 +

~J1- = r/2+HJ1- l' Fl' /m,

211)

rfllHlll'ql'/2m J(cpq)2 - m 2cp2

'>

(15)

.

i

Here II = Fill' Fill' is the first invariant of the tensor Fill'. Let us discuss the physical meaning of the results obtained. For this purpose we shall consider vector and axial currents constructed with help of solution

(8):

Vil ~~

= ~(XhlllJ.f(X) = qll/l,

S/1 = -Sfp(SoStp)

All

= ~(Xh5,IlIJ.f(X) =

(0 ~SIl, q

+ [st +Sfp(SoStp)] cos 2B- ~elll'PAql'SOPStPA sin 2B, m

B = (Nx)J(cpq)2 - cp 2m 2/m.

(16)

(17)

~-----

Thus, solution (8) which is a linear combination of solutions (11) describes a spin-coherent state of neutrino, propagating with the velocity v = q/ qO . In these states neutrino spin rotation takes place. Therefore, neutrino state with rotating spin is a pure state. Existence of such solutions is the direct consequence of the neutrino state description in terms of kinetic momentum. It should be stressed that as the result of calculations we obtained the complete system of neutrino wave functions, which show spin rotation properties. Introduce a flight length L of a particle and an oscillations length Lose, using the relation B = 7r L / Lose. Since the scalar product (N x) = T can be interpreted as the proper time of a particle, then the oscillation length is defined as

192

In this formula we use gaussian units and restore the neutrino magnetic moment /10·

Hence if as a result of a certain process a neutrino arises with polarization (0, (the spin vector ( can be expressed in terms of the four-vector SJ-I components as ( = S - qSO /(qO + m)), after travelling the distance L the probability for the neutrino to have polarization -(0 is equal to 2

W s! = [(0 x (tp] sin 2 (7fL/Los c). (19) Consequently, if the condition (o(tp) = 0 is fulfilled, this probability can become unity, i.e. a resonance takes place. In this way we obtained the exact solutions of the Dirac-Pauli equation for neutrino in dense matter and electromagnetic field. It was demonstrated that if the neutrino production occurs in the presence of an external field and a dense matter, then its spin orientation is characterized by the vector Sip' Due to the time-energy uncertainty relation the considered states of neutrino can be generated only when the linear size of the area occupied by the electromagnetic field and the matter is comparable with the process formation length. This length is of the order of the oscillations length. Acknowledgments The authors are grateful to A.V. Borisov, O.F. Dorofeev and V.Ch. Zhukovsky for helpful discussions. This work was supported in part by the grant of President of Russian Federation for leading scientific schools (Grant SS 5332.2006.2) . References

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[5] [6] [7] [8] [9] [10] [11] [12]

B.W. Lee, R.E. Shrock, Phys. Rev. D 16, 1444 (1977). K. Fujikawa, R.E. Shrock Phys. Rev. Lett. 45,963 (1980). J. Schechter, J.W.F. Valle, Phys. Rev. D 24, 1883 (1981). M.B. Voloshin, M.I. Vysotsky, L.B. Okun, Zh. Eksp. Teor. Fiz. 91, 754 (1986); C.-S. Lim, W.J. Marciano, Phys. Rev. D 37, 1368 (1988); E.Kh. Akhmedov, Phys. Lett. B 213, 64 (1988). A.V. Borisov, A.I. Ternov, V.Ch. Zhukovsky, Izv. Vyssh. Uchebn. Zaved. Fiz. 31, No 3, 64 (1988); M. Dvornikov arXiv:0708.2328 [hep-ph] (2007). A.Yu. Smirnov, Phys. Lett. B 260, 161 (1991); E.Kh. Akhmedov, S.T. Petcov, A.Yu. Smirnov, Phys. Rev. D 48, 2167 (1993). L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). S.P. Mikheyev, A.Yu. Smirnov, Yad. Fiz. 42, 1441 (1985). A.E. Lobanov, A.I. Studenikin, Phys. Lett. B 515, 94 (2001). A.E. Lobanov, O.S. Pavlova, Vestn. MGU. Fiz. Astron. 40, No 4, 3 (1999); A.E. Lobanov, J. Phys. A: Math. Gen. 39, 7517 (2006). A.E. Lobanov, Phys. Lett. B 619, 136 (2005). V.Ch. Zhukovsky, A.E. Lobanov, E.M. Murchikova, Phys. Rev. D 73, 065016 (2006).

PLASMA INDUCED NEUTRINO SPIN FLIP VIA THE NEUTRINO MAGNETIC MOMENT A.Kuznetsov a, N.Mikheev b Yaroslavl State P. G. Demidov University, Sovietskaya 14, 150000 Yaroslavl, Russia Abstract. The neutrino spin flip radiative conversion processes ilL -> IIR + 'Y. and + 'Y. -> IIR in medium are considered. It is shown in part that an analysis of the so-called spin light of neutrino without a complete taking account of both the neutrino and the photon dispersion in medium is physically inconsistent. ilL

1

Introduction

The most important event in neutrino physics of the last decades was the solving of the Solar neutrino problem. The Sun appeared in this case as a natural laboratory for investigations of neutrino properties. There exists a number of natural laboratories, the supernova explosions, where gigantic neutrino fluxes define in fact the process energetics. It means that microscopic neutrino characteristics, such as the neutrino magnetic moment, etc., would have a critical impact on macroscopic properties of these astrophysical events. One of the processes caused by the photon interaction with the neutrino magnetic moment, which could play an important role in astrophysics, is the radiative neutrino spin flip transition VL --t VRf' The process can be kinematically allowed in medium due to its influence on the photon dispersion, w = Iklln (here n =1= 1 is the refractive index), when the medium provides the condition n > 1. In this case the effective photon mass squared is negative, m; == q2 < O. This corresponds to the well-known effect of the neutrino Cherenkov radiation [1]. There exists also such a well-known subtle effect as the additional energy W acquired by a left-handed neutrino in plasma. This additional energy was considered in the series of papers by Studenikin et al. [2] as a new kinematical possibility to allow the radiative neutrino transition VL --t VRf' The effect was called the "spin light of neutrino". For some reason, the photon dispersion in medium providing in part the photon effective mass, was ignored in these papers. However, it is evident that a kinematical analysis based on the additional neutrino energy in plasma (caused by the weak forces) when the plasma influence on the photon dispersion (caused by electromagnetic forces) is ignored, cannot be considered as a physical approach. In this paper, we perform a consistent analysis of the radiative neutrino spin flip transition in medium, when its influence both on the photon and neutrino dispersion is taken into account. a e-mail:

[email protected] be-mail: [email protected]

193

194

2

Cherenkov process

VL ----) VR'Y

and its crossing

VL'Y ----) VR

Let us start from the Cherenkov process of the photon creation by neutrino, VL ----) VR'Y, which should be appended by the crossed process of the photon absorption VL'Y ----) VR. At this stage we neglect the additional left-handed neutrino energy W, which will be inserted below. For the VL ----) VR conversion width one obtains by the standard way:

r tot

VL-tl/R

where CrA) is the photon polarization vector, j"" is the Fourier transform of the neutrino magnetic moment current, pOI. = (E,p), p'OI. = (E',p') and qOl. = (w, k) are the four-momenta of the initial and final neutrinos and photon, respectively, A = t, C mean transversal and longitudinal photon polarizations, f'Y(w) = (e w / T _1)-1 is the Bose-Einstein photon distribution function, and Z~A) = (1 - 8II(A)/8w 2)-1 is the photon wave-function renormalization. The functions II(A) , defining the photon dispersion law:

(2) CrA) = II(A) C(A)OI.' The width r~~t--+VR can be rewritten to another form. Let us introduce the energy transferred from neutrino: E - E' = qo, which is expressed via the photon energy w(k) as qo = ±w(k). Then <5-functions in Eq. (1) can be rewritten:

are the eigenvalues of the photon polarization tensor:

<5

(qO =f w(k)) = <5 ( 2 2w(k) qo

_

IIOI.i3

2(k)) ()(± ) w qo .

(3)

Transforming the <5-function to have the dispersion law in the argument:

(4) one can see that the renormalization factor Z~A) is cancelled in the conversion width (1). Integration in Eq. (1) with the <5-function (4) can be easily performed when the function II(A)(q) is real. However, it has, in general, an imaginary part. It means, that the photon is unstable in plasma.

195

3

Generalization to the case of unstable photon

In the case, when the eigenfunction n(>,) (q) has an imaginary part, one should use instead of the 8-function its natural generalization of the Breit-Wigner type, with e.g. the retarded functions n(.X) (q) : J:

u

(2

q -

n

(A)

(») q

1 =}

7r

-Imn(>,) sign(qo) fA 2

(q2 - Ren(A))

+ (Imn(A))

2 '

(5)

where fA = +1 for oX = t and fA = -1 for oX = f. After some transformations, taking into account the additional energy W acquired by a left-handed neutrino in plasma, and changing the integration variables from the final neutrino 3-momentum to the photon energy and momentum, d 3 p' --> dqo dk (k == Ikl), one obtains: 2E+W-qo

E+W

J

2qOW) - ( 1- ~ where q2

{!(i) (qO,

dqo

J

dk [ T 1 +( f-y qo)]

(2E - qo) 2 q4

k) } ,

(6)

= q5 - k 2, and the photon spectral density functions are introduced: (7)

Our formula (6) having the most general form, can be used for neutrinophoton processes in any optically active medium. We only need to identify the photon spectral density functions {!(A).

4

Does the window for the "spin light of neutrino" exist?

To show manifestly that the case considered in the papers by Studenikin et al. [2], with taking the additional left-handed neutrino energy W in plasma and ignoring the photon dispersion, was really unphysical, let us consider the region of integration for the width r~~t--->VR in Eq. (6). In Fig. 1, the photon vacuum dispersion line qo = k is inside the allowed kinematical region (left plot), but the plasma influenced photon dispersion curve appears to be outside, if the neutrino energy is not large enough (right plot).

196

"-

"-

"-

"-

"-

"-

"-

"-

"-

"-

k

"-

o= k/

q

"-

"-

"-

"- "-

// ,

"-

,/

E

"-

"-

"-

"0 W

qo

k

"-

"-

,

"-

"-

"-

"-

"-

"-

"-

"-

"-

, /.

"-

"-

,/ ,/

':)

,// // '/

"- // 0

W

CJ)p

qo

Figure 1: The region of integration for the width r~7-+"R with the fixed initial neutrino energy E is inside the slanted rectangle shown by dashed line. The vacuum photon dispersion (if the medium influence is ignored) is shown by bold line in the left plot. The photon dispersion curve in plasma is shown by bold line in the right plot.

For the fixed plasma parameters, the threshold neutrino energy Emin exists for coming of the dispersion curve into the allowed kinematical region. Even for the interior of a neutron star this threshold neutrino energy is rather large: Emin '::::' w~/(2 W) '::::' 10 TeV, where Wp is the plasmon frequency. One could hope that the "spin light of neutrino" may be possible at ultra-high neutrino energies. However, in this case the local limit of the weak interaction is incomplete, and the non-local weak contribution into additional neutrino energy W must be taken into account. This contribution always has a negative sign, and its absolute value grows with the neutrino energy. One could only hope that the window arises in the neutrino energies for the process to be kinematically opened, Emin < E < Emax. For example, in the solar interior there is no window for the process with electron neutrinos at all. A more detailed analysis of this subject was performed in our papers [3,4J. Acknowledgements

A. K. expresses his deep gratitude to the organizers of the 13th Lomonosov Conference on Elementary Particle Physics for warm hospitality. The work was supported in part by the Russian Foundation for Basic Research under the Grant No. 07-02-00285-a. References

[1] [2J [3] [4J

W.Grimus, H.Neufeld, Phys. Lett. B 315, 129 (1993). A.Studenikin, e-print hep-ph/0611100, and the papers cited therein. A.V.Kuznetsov, N.V.Mikheev, Mod. Phys. Lett. A 21,1769 (2006). A.V.Kuznetsov, N.V.Mikheev, Int. J. Mod. Phys. A 22, 3211 (2007).

Astroparticle Physics and Cosmology

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INTERNATIONAL RUSSIAN-ITALIAN MISSION "RIM-PAMELA" 1 A.M.Galper , P.Picozza2 and O.Adriani3 , M.Ambriola4 , G.C.Barbarino 5 ,A.Basili6 , G.A. Bazilevskaj a lO , R.Beliotti 2, M.Boezio 5 , E.A.Bogomolov 9 , L.Bonechi 3 , M.Bongi 3 , L.Bongiorno 7 , V.Bonvicini 5 , A.Bruno\ F.Cafagna4 , D.Campana6 , 11 P.Carlson , M.Casolino 2, G.Castellini8 , M.P.De Pascale 2, G.De Rosa6 ,V.Di Felice2, 7 D.Fedele , P.Hofverberg 11 , L.A.Grishantseva 1 , S.V.Koldashovl, S.Y.Krutkov9 , A.N.Kvashnin 10 , J.Lundquist 5 , O.Maksumov lO , V.Malvezzi2, L.Marcelli 2 , I2 W.Menn , V.V.Mikhailovl, M.Minori 2 , E.Mocchiutti 5 , A.Morselli 2 , S.Orsi11, G.Osteria6 , P.Papini 3 , M.Pearce 11 , M.Ricci 7 , S.B.Ricciarini 3 , M.F.Runtso l , S.Russo6 , M.Simon I2 , R.Sparvoli 2 , P.Spillantini3 , Y.I.Stozhkov l , E.Taddei 3 , A.Vacchi 5 , E.Vannuccini 3 , G.Vasilyev 9 , S.A.Voronov 1 , Y.T.Yurkin l , G.Zampa5 , N.Zampa5 , V.G.Zverev l . 1 MEPhI, INFN f Roma2, 3 Florence, 4 Bari, 5 Trieste, 6 Napoli, 7 Prascati), 8 IFAC, 9 Ioffe Institute, 10 Lebedev Institute, 11 KTH, 12 University of Siegen.

Abstract. The successful launch of spacecraft "RESURS DK" 1 with precision magnetic spectrometer "PAMELA" onboard was executed at Baikonur cosmodrome 15 June 2006. The primary phase of realization of International Russian-Italian Project "RIM-PAMELA" with German and Swedish scientists' participation has begun since the launch of instrument "PAMELA" that has mainly been directed to investigate the fluxes of galactic cosmic rays. This report contains the main scientific Project's tasks and the conditions of science program's implementation after one year since exploration has commenced.

1

Introduction (scientific tasks)

a) Definitely one of the main problems of cosmology, physics of elementary particles and physics of cosmic rays, whose solutions are being worked on by many scientific groups today and who's every effort is being hampered by, is the nature of dark (latent) mass. Almost 25% of all the energy density of Universe is concentrated in the form of dark (latent) matter! The result of numerous theoretical researches comes to conclusion that dark matter consists of particles. However none of the known particles including neutrinos can not possibly be part of dark matter. They are absolutely new hypothetical particles. They can be either much lighter than electrons and they are called axions or much more heavy than nucleon and are called WIMPs (WIMP - weakly interactive massive particle). Practically all theoretical models assume that particles composing of dark matter were created early in the development of Universe. At present they appear like cooled (low energy) relic particles. A model of supersymmetry (SUSY) has had special development, there is a supersymmetric particle named neutralino and denoted by X possessing properties of WIMPs with mass 100 + 1000 bigger than a proton's mass [1]. In the case of neutralino X == X. Another model (Kaluza, Klein (KK)) is built on a multidimensional space-time and also supposes the existence of massive particles Bkk (kk- excitations) with mass :::: 5· 10 2 a proton's mass [2,3]. In the case of Bkk we have also Bkk == Bkk. 199

200

Interaction of both XX and BkkBkk can cause annihilation of WIMPs with creation of usual particles finally:

x + X -> bb, tt, TT, ZO ZO, Zo,

->,

+ ... , e± + ... ,pp + ... , dd + ...

This process is the basis for the search for X and Bkk at Project "RIMPAMELA". It is known that primary cosmic rays basically protons and nuclei are interacting with interstellar gas while going through cosmic space. This would result in a fragmentation of nuclei and creation of elementary particles including antiparticles p, e+ [4]. If to this source of antiprotons and positrons is added a source from process of annihilation of X and Bkk particles with creation of p and e+ then some peculiarities concerning process of WIMP annihilation will appear in spectra of antiparticles that are registering in near Earth cosmic. A peculiarity also could be seen in energy dependency ofp/p and e+ /e- ratios. Because masses of WIMPs are considerably bigger than mass of antiproton, some particular features should be seen at energies more than a few GeV. Precision measurements of antiprotons' and positrons' fluxes for searching of special peculiarities in spectra of antiparticles are main task of project "RIMPAMELA" - the clarification of nature of dark matter. b) The next task is searching for antihelium in cosmic rays. The probability of creation of antihelium nuclei in interaction of primary cosmic rays with interstellar gas is extremely small. Act of registration of such particles could allow us to understand better the nature of baryonic asymmetry of the Universe. It means existing antimatter's region in our Galaxy. The definition of the upper limit of c) Precise measurements of the energy spectra of known particles - protons, electrons, positrons and isotopes of light nuclei, are provided within the project RIM-PAMELA are essential for studying of the mechanisms and physical conditions of the generation in sources and acceleration during propagation of cosmic rays. Also important to study interaction of galactic cosmic rays in heliosphere and magnetosphere of the Earth. So these measurements are very interesting part of the scientific program of project "RIM-PAMELA". d) We would like to mention that we plan to study physics of acceleration processes during solar flares. e) Finally we plan to study flux of particles in nearest Earth cosmic space (albedo, captured particles). That is another important task of project "RIMPAMELA", because it is an applied aspect of the experiment.

201

2

Magnetic spectrometer PAMELA

The physical scheme of magnetic spectrometer " PAMELA" is presented in fig. 1. This set of detectors allow to measure following characteristics of particles such as charge, mass, momentum, energy, time and direction of arrival. Mode of operation of PAMELA is: when a particle fly inside the aperture of the device, it will be detected by the scintillation Time Of Flight system (TOF); the TOF system produces a signal, and gives the command to start registration of information for other detection systems of spectrometer - the coordinate system in magnet, calorimeter, shower detector, neutron detector and system of anticoincidence. This information is recording into the memory of spectrometer and then transferring to the recording device of spacecraft every three hours. The spectrometer has its own control system. This system can be reprogrammed by using commands from the Earth.

Fig. 1. The physical schematic of Magnetic spectrometer PAMELA: 1,3,7time of flight system; 2,4 - anticoincidence system; 5 - silicon strip tracker (6 double plates); 6 - magnet (5 sections); 8 - silicon strip imaging calorimeter; 9 - bottom scintillator 84; 10 - neutron detector; 11 - pressurized container.

The main physic-technical characteristics of spectrometer " PAMELA" are taken from the Monte-Carlo simulation and on-ground measurements in monochromatic beams of electrons and protons from the CERN accelerators. This data is shown in table 1.

3

Measurements. Analysis of scientific Information,

The satellite "Resurs DK" 1 was launched on the elliptic orbit with the next parameters: 380–800 km and inclination 70.4°. Magnetic spectrometer "PAMELA" was placed in hermetic container, the axis of the device is directed to the local zenith (see Fig 2). Scientific information is downloaded when the satellite is lying over the ground receiving station located in NTsOMZ (Otradnoe, Moscow). This process is performed 3-4 times per day.

Fig. 2. The spacecraft "RESURS DK" 1. The particular information from spectrometer PAMELA is extracted at the ground receiving station from the full data stream of information. Expressanalysis of data quality is performing at ground station PAMELA in NTsOMZ, If necessary it's possible to send special command to satellite for adjustments of PAMELA'S work. These commands are sending to satellite from Flight

203

Control Center (Korolev city). Raw data is transferring by the fast Internet link to MEPhI and then using the international scientific system GRID to the scientific information center of INFN in Bologna. After that, all scientific groups process the PAMELA data and perform scientific analysis (Fig.3.).

+ I

I

r-~--..,

Qualitive Separation of analysis of the PAMELA's I--____~ data being information received

-Ground - station - -of PAMELA - 1 T@Ils6!cQ. In_ rrna!iQ.n : .---'---.., Express-analysis of the whole

Preparation of Initial dati for spaaecmt's Work Program WIthIn PAMElA

I" I

The Aight Control Center

I: Express-processing I :_ __________, '--_---J

On-line

1,·,·_...,

' - -_ _ _...J

Expr.ss-diagnosHcs of detectors

ME PhI GRID

(CNAF)

*-----'

~

,---

Italy, Gennany, Sweden

I

Total processing of events

__.*___

...J

NEPhI, Ioffe institute, lebedev institute

Fig. 3. The physical scheme of receiving and processing of PAMELA data.

4

Preliminary results

Since the real beginning of experiment in July 2006 till August 2007 the spectrometer PAMELA has downlinked approximately 5 TB of raw data for analysis. Around 108 particles were chosen for the first phase of analysis. Mainly they are protons with energy range 108 -;- 2 . lOll eV, and also helium nuclei and nuclei of some other light elements, electrons, positrons and antiprotons. In contrast with previous measurements made, this magnetic spectrometer can register the fluxes of particles in the radiation belts of Earth. That allowed it to obtain the unique information about high-energy particles captured by magnetic field. The figure 4 shows 2 reconstructed events of antiproton and positron with high energy. In the diagrams (Fig. 5 and 6) are shown the distribution of particles' fluxes depending on rigidity (momentum) and dEjdx (losses of energy). This picture demonstrates the real capabilities of this scientific apparatus in flight. We consider that we will obtain new experimental information, which will be needed to help in the solving of the tasks declared in the introduction of this report.

204

Ge V antiproton (dark color of neutron detector means that there are registered neutrons), right - 92 Ge V nn.
5. Distribution

d

would like to point out several important

results.

has detected about 1000 antiprotons. It has shown that ratio of !>nt·.n'-A?"'')'"'' in the energy range 1-;-10 Ge V to the antlDroi~orls are created in thenteractions of nr.TYH.r" lU~'U-'''A''/H'. interstellar gas the next

205

7. The ratio

depending on kinetic energy

8. The flux depending on kinetic energy. It has detected several tens of thousands of leptons. That will soon make to estimate the ratio in the energy range 0.1+50 which is like the ratio with the problem of the nature of dark matter. has detected several millions of helium nuclei and has shown that ratio of antihelium to helium is rv • It is that this value will be in experiment . But the value conditions for models the there is no difference between energy Drcltoxls and helium nuclei within the range of measured

206

energies (fig. 8). But further investigation of these spectra could show the difference. Now we carefully study the ratio H e~j H e~ and BjC for different energies. These ratios are very important for understanding the process of interaction of high energy cosmic rays in galaxy with interstellar gas. In particular, it's very important for understanding of fluxes of secondary antiprotons and positrons. The fluxes He and protons have been measured in solar cosmic rays during the solar flare 13.12.2006. In particular, it has detected fluxes of He-4 nucleus with energy up to several GeV[6]. Finally by using large amounts of statistical material it is shown that there is a considerable predominance of electron fluxes over the amount of positron fluxes that appear in the energy range 50-;.-300 Me V in the radiation belts of the Earth. It is already known that positrons are dominant at the boundary of the radiation belt. 5

Conclusion

The ground station NTsOMZ is receiving approximately 15GB of scientific data information daily. This Russian-Italian Mission (RIM) will continue according to plan until the middle of 2009. However the main and new scientific results will be achieved earlier. Project RIM-PAMELA is supported by Russian Academy of Science and Russian Cosmic Agency, the Italian National Institute of Nuclear Physics, Italian Space Agency, German Space Agency, Swedish National Space Board and Swedish Research Council. 6

Acknowledgement

We would like to thank the Ministry of Education and Science of Russian Federation for financial support within grant RNP 2.2.2.2.8248. References

[1] [2] [3] [4] [5] [6]

L. Bergstrom et al., Phys.Rev. D 59 (1999) 43506. D. Hooper et al., Phys.Rev. D 71 (2005) 083503. T. Bringmann , JeAP 08 (2005) 006; astro-phj0506219 (2005). M. Simon et al., Astrophys. J. 499 (1998) 250. P. Picozza et al., Astroparticle Physics 27 (2007) 296. M. Casolino et al., in "Solar cosmic ray observations with PAMELA experiment" (Proc. 30th International Cosmic Ray Conference) ,Merida, Mexico (2007).

DARK MATTER SEARCHES WITH AMS-02 EXPERIMENT A.Malinin a, For AMS Collaboration [PST, University of Maryland, MD-20742, College Park, USA Abstmct. The Alpha Magnetic Spectrometer (AMS), to be installed on the International Space Station, will provide data on cosmic radiations in a large range of rigidity from 0.5 GV up to 2 TV. The main physics goals in the astroparticle domain are the anti-matter and the dark matter searches. Observations and cosmology indicate that the Universe may include a large amount of unknown Dark Matter. It should be composed of non baryonic Weakly Interacting Massive Particles (WIMP). A good WIMP candidate being the lightest SUSY particle in R-parity conserving models. AMS offers a unique opportunity to study simultaneously SUSY dark matter in three decay channels from the neutralino annihilation: e+, anti-proton and gamma. The supersymmetric theory frame is considered together with alternative scenarios (extra dimensions). The expected flux sensitivities in 3 year exposure for the e+ /e- ratio, anti-proton and gamma yields as a function of energy are presented and compared to other direct and indirect searches.

Introduction The first evidence for the existence of Dark Matter comes from the observation of rotation velocities across the spiral galaxies, derived from the variation in the red-shift. The rotation velocities rise rapidly from the galactic center, then remain almost constant to the outermost regions of a galaxy. The observations are consistent with the gravitation motion only if the matter in the Universe is mostly non luminous" dark matter". The recent WMAP results [1] confirm that about 83% of the matter in the Universe exists in the form of cold Dark Matter (DM). The mystery of the Dark Matter remains unsolved. Many candidates such as massive neutrino, Universal Extra Dimensions Kaluza-Klein states and Super Symmetry theory (SUSY) heavy neutralinos were proposed. If Dark Matter, or a fraction of it, is non-baryonic and consists of almost noninteracting particles like neutralinos, it can be detected in cosmic rays through its annihilation into positrons or anti-protons, resulting in deviations (in case of anti-protons) or structures (in case of positrons) to be seen in the otherwise predictable cosmic ray spectra [2]. Considering the hypothesis of a possible clumpy DM, the expected fluxes of such primary positrons, 'Y-s or anti-protons may be enhanced [3] since the annihilation rate is proportional to the square DM density contrary to the direct DM searches which will suffer from a decreased probability for the Earth to be contained into an eventual DM clump. ae-mail: [email protected]

207

208 1

AMS-02 Instrument

The Alpha Magnetic Spectrometer (AMS) is a particle physics experiment in space. Its initial space mission on board of the Space Shuttle Discovery (STS91) in June, 1998 confirmed the basic concept of the experiment [4]. During this short flight AMS measured of the GeV cosmic-ray fluxes over most of the Earth's surface [5-8], and provided the impetus to upgrade the instrument for the ISS 3 year mission (hereafter called AMS-02). These upgrades include among others a stronger, BL2 = 0.9T super conducting magnet to achieve the maximal detectable rigidity of 1 TV (the rigidity resolution better than 2% up to 20 GV) in the Silicon 'Thacker, as well as the addition of a 'Thansition Radiation Detector (TRD), a Ring Imaging Cherenkov (RICH) and an Electromagnetic Calorimeter (ECAL). The upgraded instrument will provide data on cosmic radiation in a large range of energy from a fraction of Ge V to 3 Te V with very high accuracy and free from the atmospheric corrections needed for balloon-born measurements. Its main physics goals in the astroparticle domain are the Antimatter and the Dark Matter searches as well as the cosmic ray composition and propagation study.

2

AMS-02 Sensitivity for DM search

The Monte Carlo study, based on the AMS-02 mathematical model, was performed to estimate the instrument sensitivity for the indirect DM search channels [9-13]. More than 109 events containing p+-, He, e+- and'Y at different energies have been fully simulated [14-17] passing through the detector and then reconstructed. The results of the study in the anti-proton, e+ and'Y cannels are presented in the figures 1-9. The background rejection factors up to 10- 6 necessary to extract the tiny anti-proton and e+ signals were achieved by combining the redundant information from the TRD, RICH and ECAL detectors. The selection criteria were tuned and the resulting efficiencies were used to simulate the measured spectra. Comparison with the existing data demonstrates that the AMS-02 will have an adequate sensitivity to address the enhancement in the positron fraction measurement reported by HEAT [9,10]' simultaneously constraining the DM signal parameter space by combining the anti-proton, e+ and 'Y channels [18]. The Galactic Center 'Y signal measurement by AMS-02 would provide 95% CL exclusion limits for several mSUGRA models in 3 years.

209 O.06

r------------_-.

~O.20

'5 § O.B

i

()

()

0::1 0.10

~ ·g.0.()5

Average Acc(1<E<11 GeV)=O.147 m2 Sr

~

10

Momentum (GeV/e)

10'

Momentum (GeV/e)

Figure 1: The acceptance for the anti-proton signal including the selection efficiency.

6

,

+ 10

Momentum (GeV/e)

I

I

10

Momentum (GeVie)

Figure 2: The background rejection factors for the anti-proton signal.

Hr'

I

Kinetie The simulated AMS-02 three year combined measurement of cosmic and the residua! background The of the AMS-02 eXj)ected spectrum measurement existing data Lines show dlltlelrellt secondary anti-proton flux

Mo.,ur•• Spec1rum (No Energy Un*lldlng 0'>00)

pr,a<1l.ctJ.on for the AMS-02 positron energy reconstruction three year combined measurement of the cosmic spectrum

DM Sigll1l1 frmu (E.Baltz, Phys.Rcv. 065

5: HEAT positron fraction data with an example of estimated 1 year AMS-02 measurement. Solid lines show one of the most favorable SUSY neutralino scenario and the "t".nrl,.rrl LBM prediction. SUSY signal enhancement 100 DM) is necessary to fit data.

flux from the Galactic Center as a function of for the and cuspy NFW halo the standard considered models are the scheme, AMSB scenario and The various selections were done by varying nh z cuts.

lm;egr(l,'ceu 'Y

212 ·---slandard .. gnoI S1l<)""¥l\II\

BE~S ~~

-~ 10-'

----Standanl .. gna!

10.1

('APRlCE?8

CM'IUef,QS

t

Bf-%'18

BlSSOO

-~

AMS\J!l"'>~O~l

'7'1

_'1

"1)

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flf.SSOO 10 '

~

~Hr'

~!O'

.rJ"

t

"

~IU--l

_ "_ 10'

1 ~!

10

"

-'

K""-,~-KI.jn,,glmJ

~w'.o~~

lO '

,,' 10

Kinetic energy l GeV ]

Kinetic energy [GeV]

Figure 7: Combined example. The anti-proton flux as a function of kinetic energy, assuming 150 GeV SUSY neutralino mass (left plot) and 50 GeV Kaluza-Klein boson mass (right plot) for 3 years of AMS-02 data taking.

----Standatdsignnl

10-1

10

KalILl8·KJeinsignal

10

Energy [GeV ]

AMSOl projection (3 y.)

----StBndatdsignal

AMS02pcojeclion(3y.)

Susysigl1l!J

Energy [GeV 1

Figure 8: Combined example. The positron flux as a function of energy, assuming 150 GeV SUSY neutralino mass (left plot) and 50 GeV Kaluza-Klein boson mass (right plot) for 3 years of AMS-02 data taking.

----~~"

j(r!

A~QZ"'.<e1

AM'>lI2-.n>Jt<"""(J~).,...,

"",lMZp.-""""",(3~),,",,Z

AMW1""'J...,1

1

-:"'-'.~:"""",,:"'-:.:...

---_:.o..~~:':~~_1111111

10

!lit

leY

Energy [Ge V ]

Figure 9: Combined example. The'Y flux as a function of energy, assuming 150 GeV SUSY neutralino mass (left plot) and 50 GeV Kaluza-Klein boson mass (right plot) for 3 years of AMS-02 data taking. IMBH associated DM clumps at different distance: 20 kpc (case 1) and 2 kpc (case 2) are shown.

213

Conclusions

During the 3 year mission in space, AMS-02 will perform precise, high statistics cosmic ray measurements in the 1 Ge V to few Te V energy range. It will allow to combine all indirect Dark Matter search channels, constraining the existing models and will have a high discovery potential of the Dark Matter signal. References

[1] D.N.Spergel et al., ApJS, 148, 175 (2003). [2] LV.Moskalenko and A.W.Strong, Adv.Space Res. 27, 717 (2001). [3] E.A.Baltz et al. "Cosmic-ray positron excess and neutralino dark matter",Phys.Rev. D 65 63511 (2002). [4] G.M.Viertel, M.Capell, "The Alpha Magnetic Spectrometer", Nucl. Inst. Meth. A 419, 295 (1998). [5] J.Alcaraz et al., "Search for Antihelium in Cosmic Rays", Phys. Lett. B 461, 387 (2000). [6] J.Alcaraz et al., "Protons in Near Earth Orbit", Phys.Lett. B 472, 215 (2000). [7] J.Alcaraz et al., "Leptons in Near Earth Orbit", Phys.Lett. B 484, 10 (2000). [8] J.Alcaraz et al., "Cosmic Protons", Phys.Lett. B 490, 27 (2000). [9] S.W.Barwick et al. (HEAT Collaboration), "Measurements of the Cosmic-Ray Positron Fraction from 1 to 50 Ge V", Astro-phys.J. 482, L191 (1997). [10] S.Coutu et al. (HEAT-pbar Collaboration), "Positron Measurements with the HEAT-pbar Instrument", in (Proceedings of 27th ICRC), 200l. [11] R.L.Golden et al., "Observation of cosmic ray positrons in the region from 5 to 50 GeV", A€1A 188, 145 (1987). [12] L.Bergstrom, J.Edsjo, P.Ullio, "Cosmic Antiprotons as a Probe for Supersymmetric Dark Matter", Astro-phys.J. 526, 215 (1999). [13] D.Hooper, G.Kribs, "Kaluza-Klein Dark Matter and the Positron Excess", Phys.Rev. D 70 115004 (2004) [14] R.Brun et al., "GEANT 3", CERN/DD/EE/84-1, 1987. [15] S.Giani et al., "GEANT 4 An Object-Oriented Toolkit for Simulation in HEP", CERN/LHCC/98-44, 1998. [16] V.Choutko, G.Lamanna, A.Malinin, (Trento II International Workshop on Matter Antimatter and Dark Matter Proceedings), Int. J. Mod. Phys. A 17, N12-13, 1817 (2002). [17] A.J acholkowska, et al., "An indirect dark matter search with diffuse gamma rays from the Galactic Centre with the Alpha Magnetic Spectrometer", arXiv:astro-ph/0508349, 23 May 2006. [18J S.Rosier-Lees in (Proceedings of 30th ICRC), ID0618, Merida, 2007.

INVESTIGATING THE DARK HALO R. Bernabeil,a P. Belli l , F. Montecchia l , F. Nozzoli l , F. Cappella2, A. Incicchitti2, D. Prosperi 2, R. Cerulli3,C. J. Dai4, H. L. He 5 , H. H. Kuang 5 , J. M. Ma 5 , X. D. Sheng5 , Z. P. Ye5 1

Dip. Fisica, Univ. Roma Tor Vergata and INFN Tor Vergata, 1-00133 Rome, Italy 2 Dip. Fisica, Univ. Roma La Sapienza and INFN Roma, 1-00185 Rome, Italy 3 INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy 4 IHEP, Chinese Academy, P.O. Box 918/3, Beijing 100039, China Abstract. The low background DAMA/NaI experiment (:::: 100 kg highly radiopure NaI(Tl)) at the Gran Sasso National Laboratory of the I.N.F.N. investigated the presence of Dark Matter particle in the galactic halo by exploiting the model independent annual modulation signature. This experiment has collected data over seven annual cycles for a total exposure of more than 105 kg x day and has pointed out at 6.3 0' C.L. a modulation effect satisfying all the many peculiarities of the signature. Several corollary model dependent quests for the candidate particle have also been carried out. At present the second generation DAMA/LIBRA setup (:::: 250 kg highly radiopure NaI(Tl)) is in operation. A R&D towards possible 1 ton NaI(Tl) set-up is also in progress.

1

The DAMA/NaI experiment

The low background DAMA/NaI experiment [1-10] was located in the underground laboratory of Gran Sasso and it has been part of the DAM A project. In the framework of DAMA activities let us also mention: i) DAMA/LXe [11]; ii) DAMA/R&D [12]; iii) the new second generation radiopure NaI(TI) DAMA/LIBRA set-up [13]; iv) DAMA/Ge for sample measurements [14]. DAMA/NaI was proposed [15]' designed and realized to investigate the presence of Dark Matter (DM) particles in the galactic halo by exploiting the model independent annual modulation signature [16]. In fact, as a consequence of the Earth annual revolution around the Sun, a larger flux of DM particles cross the laboratory around roughly June 2nd (when the Earth velocity is summed to the Sun velocity with the respect to the Galaxy) and by a smaller one around roughly December 2nd (when the two velocities are subtracted). This annual modulation signature offers many peculiarities that have to be simultaneously satisfied: (i) the rate must contain a component modulated according to a cosine function; (ii) with one year period; (iii) a phase roughly around 2 nd June; (iv) the modulation must only be found in a well-defined low energy range, where DM particles can induce signals; (v) it must apply just to those events in which only one detector in a multi-detectors set-up actually "fires" (single-hit events), since the probability that DM particles would have multiple interactions is negligible; (vi) the modulation amplitude in the region of maximal sensitivity has to be ::; 7% for usually adopted halo distributions, but it can be significantly larger in case of some possible scenarios [6,17J. To mimic ae-mail: [email protected]

214

215

such a signature spurious effects or side reactions should be able not only to account for the observed modulation amplitude but also to contemporaneously satisfy all the requirements. DM particle investigation has been done by DAMA/NaI also with other approaches and several other rare processes have also been studied [5].

2

The DAMAjNaI model-independent result

The DAMA/NaI experiment took data over seven annual cycles collecting an exposure of 107731 kg x day. A clear annual modulation has been observed in the measured rate of the single-hit events in the lowest energy region, satisfying the many peculiarities of a DM particle induced effect. In Fig. 1 the time behaviour of the residual rate of the single-hit events in the (2-6) ke V energy interval is reported. Fitting the data with a cosine-like function the presence 2-6 keY

~

0.1,....1

~

,....n+--

v

;0+

,....VI+-- vn ~

U 0.05 ;C

fr

'-'

O~~~--~+P~+k~~~~~~~~+d~~~~~

'" ";

:S -0.05 '"CI.I

~ -0.1 L..L..c~~~~L.L.L~~~::':"c"~~~~~~~~~

Figure 1: Experimental residual rate for single-hit events in the cumulative (2-6) keVenergy interval as a function of the time over 7 annual cycles (total exposure 107731 kg x day); end of data taking July 2002. The experimental points present the errors as vertical bars and the associated time bin width as horizontal bars. The superimposed curve represents the cosinusoidal function behaviour expected for a DM particle signal with a period equal to 1 year and phase exactly at 2 nd June. See ref. [1,4].

of annual modulation is favoured at 6.3 (J" C.L. with an amplitude equal to (0.0200 ± 0.0032) cpd/kg/keV, a phase to = (140 ± 22) days and a period T = (1.00 ± 0.01) year. The period and phase agree with those expected in the case of an effect induced by DM particles in the galactic halo (T = 1 year and to roughlyat,::, 152.5 th day of the year). Modulation is not observed above 6 keY (we remind that the set-up has taken data up to the MeV region despite the optimization was done for the keY energy range). A suitable statistical analysis has shown that the modulation amplitudes are statistically well distributed in all the crystals, in all the data taking periods and considered energy bins [1,4]. A careful quantitative investigation of all the known possible sources of systematic and side reactions has been regularly carried out and published at time

216

of each data release. No systematic effect or side reaction able to account for the observed modulation amplitude and to satisfy all the requirements of the signature has been found [1,4]. A further investigation has been performed in the last (DAMA/NaI-6 and 7) annual cycles when a dedicated renewed electronics has been installed; this has allowed to record the pulse profiles of the multiple-hits events (i.e. events in which more than one detector fire in coincidence). The multiple-hits events class - on the contrary of the single-hit one - does not include events induced by DM particles since the probability that a DM particle interacts in more than one detector is negligible. The fitted modulation amplitudes are: A = (0.0195 ± 0.0031) cpd/kg/keV and A = -(3.9 ± 7.9) . 10- 4 cpd/kg/keV for single-hit and multiple-hits residual rates, respectively [4]. Thus, evidence of annual modulation with proper features is present in the single-hit residuals (events class to which the DM particle-induced signals belong), while it is absent in the multiple-hits residual rate (event class to which only background events belong). Since the same identical hardware and the same identical software procedures have been used to analyse the two classes of events, the obtained result offers an additional strong support for the presence of DM particles in the galactic halo, further excluding any side effect either from hardware or from software procedures or from background. In conclusion, the presence of DM particles in the galactic halo is supported in a model independent way by DAMA/NaI at 6.3 a C.L .. Except the presently running DAMA/LIBRA, no other experiment whose result can be directly compared with this one is available so far in the field of Dark Matter investigation. 3

Corollary model-dependent quests

On the basis of the obtained 6.3 a C.L. model-independent result, corollary investigations can also be pursued on the nature of the DM particle candidate [1,4,6-10]. This latter investigation is instead model-dependent and considering the large uncertainties which exist on the astrophysical, nuclear and particle physics assumptions and on the parameters needed in the calculations - has no general meaning (as it is also the case of exclusion plots and of the DM particle parameters evaluated in indirect detection experiments). For simplicity, the results of the corollary analyses are presented in terms of allowed volumes/regions obtained as superposition of the configurations at given C.L. for the considered model frameworks [1,4,6-10]. The results briefly summarized here and the several other ones available in literature are not exhaustive of the many scenarios possible at present level of knowledge. Some corollary quests for the class of WIMP candidates

For the case of WIMP class candidates, it has been considered so far low (of

Figure Some examples. Left: Region allowed in the plane in the considered scenarios for pure SI coupling. Right: a slice of the 3-dlim.eruliOlGal allowed volume in the (t;CJSD, mw) plane for (1 2.435 (0 can vary from 0 to pure SD The filled region has been for no "'d.'U.lJLJ\J ref. areas enclosed by lines are obtained by HlLlULlUt:.lll!!, """,Lj,,,,,,, stream. Different lines refer to the considered possibilities for the velocity and velocity dispersion see ref. for more results on various t;UU'fl"U!s~

mass (up to many hundreds of candidates matter via: i) mixed SI&SD ii) dominant dominant SD coupling; iv) S1 inelastic ,-,,'JH:TPrlTl detailed discussion on the the for these candidates in some model frameworks has also been extended in ref. effect the presence of a non thermalized DM halo due to the Dwarf Galaxy, non-thermalized such as the Canis in the halo models with will be addressed in near future

the role of the electrom.agJ(letlc

eti'CdlJllJl!', electron of e.m. radiation electrons) arises from the rearrangement of atomic shells a atomic nucleus. This radiation is contained in detector of suitable size and because of its e.m. nature, this part of the is lost in all those approaches based OIl discrimination of the e.m. component of the measured rate. Although quite the e.m. UC;JJ'vL'C;U nature of this contribution - with respect to the behaviour of the

'1----.. --"

energy distribution of the nuclear recoils can have direct searches when in terms of low mass 'VlMP loaJl!U,"ua,","", that in order to the of the scenarios as in ref. 4] have been considered without any inclusion contribution. Some of the results in effect in the MMP quest are

allowed in the considered model rr... mpwnrlc~ SD coupling for the value () 2.435 (note for some mass and mixed Sr&SD accounting for the effect. Note that contributions of other uncertainties on parameters Sal~DJE:G contribution or more favourable form factors, would further extend increeses the sets of the best fit values. See ref. [8] for details.

GeV mass DM nAlrnr',,,,,, discussed in literature.

for the

'-'UtlUU"'uu.~

axes and lattice of the would materials, This is ~L1I,~rJlI1g in the lattice into a cn,an:t1el to electrons rather than to the nuclei the ratio between the detected and the kinetic energy of the in

to· 1

10 4

to· 1

IO"

10· J

10 4

1t}.1

10 4

(pb)

Exaulpl'0E of region/slices allowed in the considered model frameworks coupling (upper-left); ii) pure SD for the value e 2.435 (note o to iii) mixed coupling for some mass and region is obtained in abseuce of channeling effect, while OIYlal!lea when accounting for it, The dark line marks the overal external to note that the inclusion of other contributions of other uncertainties nn,'nn"',prq and the possible SagDEG and the effect raV'OlInUlle form factors, scaling laws, increases the sets of the best fit values, See ref, for more and for

to note that the results about

factor obtained

220 NaI(Tl) crystals (and in general also for other crystal detector) using neutron source and published in literature can contain channeled events, but considering the low-statistics of these measurements, the small effect looked for and the energy resolution of the detectors they cannot easily be identified. Moreover, the channeling effect becomes less important at increasing energy and its contribution results more suppressed. Therefore, there is no hope to single out the channeling effect in the already-collected neutron data [9]. The inclusion of this existing effect gives an appreciable impact in corollary analyses in terms of WIMP (or WIMP-like) candidates since, as mentioned, the quenching factor is a key quantity to derive the energy of the recoiling nucleus after an elastic scattering [9]. In particular lower cross sections are explorable in given models for WIMP and WIMP-like candidates by crystal scintillators, such as NaI(Tl). Similar situation holds for purely ionization detectors, as Ge (HD-Moscow - like experiments), while a loss of sensitivity is expected when pulse shape discrimination is used in crystal scintillators (such e.g. in KIMS) since the channeled events - having q ~ 1- are probably lost. Moreover, no enhancement can be present either in liquid noble gas experiments (DAMA/LXe, WARP, XENON, ... ) or in bolometer experiments; on the contrary some loss of sensitivity can be expected when applying discrimination procedures, based on qion « 1, since some events (those with qion ~ 1) are lost. Fig. 4 reports some examples of regions/volumes allowed by the DAMA/NaI data when the channeling effect has been taken into account. It has been considered cases of a DM particle with pure SI, pure SD and SI&SD coupling in some given model frameworks. Also for this analysis, in order to point out just the impact of the channeling effect, the same scenarios as in ref. [1] have been considered without any inclusion of the possible SagDEG contribution and the existing Migdal effect already discussed. The results further show the role of the existing uncertainties and of correct/complete description and inclusion of all the involved processes. Results for electron interacting dark matter Recently, it has also been investigated the case of a Dark Matter candidate interacting just on the electrons of the target detector [10]. In fact, some extensions of the Standard Model provide DM candidates which can have a dominant coupling with the lepton sector of the ordinary matter. These particles have been widely considered in literature and can offer possible sources for the 511 keY positron annihilation gamma's observed from the galactic bulge. These DM candidates can be directly detected only through their interaction with electrons in detectors of a suitable experiment, while they are lost by experiments based on the rejection of the e.m. component of the counting rate. The detection of electron interacting DM is based on the detectability of the energy released in the particle-electrons elastic scattering. Generally, these pro-

221

cesses are not taken into account in the DM field since the electron is assumed at rest and for a with velocity c:::: 300 km/s, the released energy is of the order of few well below the detectable energy in the detectors. also an electron bound in an atom at rest can have not momentum with these electrons can and the interaction of DM rise to detectable after the the final state ~v."'~,~.. electron and of an ionized or an excited atom pro-ele,ctl~oIllS: the produced and electrons of low energy are mostly contained in a detector of a suitable size.

10

·3

1500

xo (GeV) Figure case of electron interacting dark matter. The DAMA/NaI region allowed the plane particle mass versus its cross section on the electrons in the considered mode! frameworks The regions enclose configurations corresponding to likelihood function values distant more than 40' from the null hypothesis of modulation). We note that, although the mass region in the plot is up to 2 with larger masses are also allowed. For details see IU

terms of an electron is """'''Y''U,r the allowed in the mass versus its cross section on the electrons in some scenarios is shown. We note the mass the up candidates with This results hold for every kind of DM candidate iv>+·~~.,~~i~ with cross section a weak on electron momentum also some information on a mediator of the in literature - the U boson to the obtained allowed

~~.~~.~o~,rI

interval in agreement with the available in literature. Note that the U boson one for there are domains in

where LSP-electron interaction can dominate for

I-'

_'''''"'V one.

bosonic candidates

all the main processes involved in the detection of both in the and in the scalar interactions and

223 Furthermore, in the case of a scalar boson (here named h), only a coupling to hadronic matter can be considered in order to account for the DAM A observations [6] and the result is an allowed multi-dimensional volume in the space defined by mh and by all the ghijq, coupling constants to quarks. 4

Conclusion

The DAMA/NaI experiment pointed out at 6.3 (J" C.L. the presence of DM particles in the galactic halo by investigating the model independent annual modulation signature over seven annual cycles. No systematic effect or side reaction able to account for the observed effect has been found. Several corollary quests for the investigation on the nature of the DM candidates have also been pursued and others are available in literature considering various DM scenarios. In particular we mention the studies on the possible implications of the channeling effect in NaI(Tl) crystals [9] and the investigation on a DM candidate interacting only with electrons [10]. An investigation on the sterile neutrino as DM candidate is also under consideration. We remind that no experiment is available so far - with the exception of DAMA/LIBRA - whose results can be directly compared in a model independent way with that ofDAMA/NaI and that claims for contradictions sometimes reported in the field have intrinsically no scientific meaning [1,4,7]. We also remind that no results obtained with different target material can intrinsically be directly compared even for the same kind of coupling, although apparently all the presentations generally refer to cross section on the nucleon. As regards the indirect searches, a comparison would always require the calculation and the consideration of all the possible DM particle configurations in the given particle model, since it does not exist a biunivocal correspondence between the observables in the two kinds of experiments. However, the present positive hints provided by indirect searches are not in conflict with the DAMA/NaI result. At present the second generation DAMA/LIBRA (a '" 250 kg highly radiopure NaI(Tl) set-up) is in operation [13]. The first data release will, most probably, occur at end 2008. Finally, a third generation R&D for a possible 1 ton NaI(Tl) detector, DAM A proposed in 1996, is also in progress. References [1] R. Bernabei el al., La Rivista del Nuovo Cimento 26 n.l (2003) 1-73 and references therein. [2] R. Bernabei et al., Il Nuovo Cim. 112 A (1999) 545. [3] R. Bernabei et al., Eur. Phys. J. C 18 (2000) 283. [4] R. Bernabei et al., Int. J. Mod. Phys. D 13 (2004) 2127. [5] R. Bernabei et al., Phys. Lett. B 389 (1996) 757; Phys. Lett. B 408 (1997) 439; Il Nuovo Cim. A 112 (1999) 1541; P. Belli et al., Phys. Rev.

224 C 60 (1999) 065501; Phys. Lett. B 460 (1999) 236; R. Bernabei et al., Phys. Rev. Lett. 83 (1999) 4918; F. Cappella et al., Eur. Phys. J. direct C 14 (2002) 1; R. Bernabei et al., Phys. Lett. B 515 (2001) 6; Eur. Phys. J. A 23 (2005) 7; Eur. Phys. J. A 24 (2005) 5l. [6] R. Bernabei et al., Int. J. Mod. Phys. A 21 (2006) 1445. [7] R. Bernabei et al., Eur. Phys. J. C 47 (2006) 263. [8] R. Bernabei et al., Int. J. Mod. Phys. A 22 (2007) 3155. [9] R. Bernabei et al., Eur. Phys. J. C DOl 10. 1140/epjc/sl0052-007-0479-0. [10] R. Bernabei et al., arXiv:0712.0562v2 [astro-ph], to appear on Phys. Rev. D [11] P. Belli et al., Astropart. Phys. 5 (1996) 217; Il Nuovo Cim. 19 (1996) 537; Phys. Lett. B 387 (1996) 222; Phys. Lett. B 389 (1996) 783 (err.); Phys. Lett. B 465 (1999) 315; Phys. Rev. D 61 (2000) 117301; R. Bernabei et al., Phys. Lett. B 436 (1998) 379; New Journal of Physics 2 (2000) 15.1; Phys. Lett. B 493 (2000) 12; Nucl. Instrum. fj Meth. A 482 (2002) 728; Eur. Phys. J. direct C 11 (2001) 1; Phys. Lett. B 527 (2002) 182; Phys. Lett. B 546 (2002) 23; Eur. Phys. J. A 27 sOl (2006) 35. [12] R. Bernabei et al., Astropart. Phys. 7 (1997) 73; Il Nuovo Cim. A 110 (1997) 189; P. Belli et al., Astropart. Phys. 10 (1999) 115; Nucl. Phys. B 563 (1999) 97; R. Bernabei et al., Nucl. Phys. A 705 (2002) 29; P. Belli et al., Nucl. Instrum. fj Meth. A 498 (2003) 352; R. Cerulli et al., Nucl. Instrum. fj Meth. A 525 (2004) 535; R. Bernabei et al., Nucl. Instrum. fj Meth. A 555 (2005) 270; Ukr. J. Phys. 51 (2006) 1037; P. Belli et al., Nucl. Phys. A 789 (2007) 15; Phys. Rev. C 76 (2007) 064603; Phys. Lett. B 658 (2008) 193. [13] R. Bernabei et al., in Frontier Objects in Astrophysics and Particle Physics Vol. 90 (2004) 581. [14] P. Belli et al., Nucl. Instrum. fj Meth. A 572 (2007) 734; P. Belli et aI, in "Current problems in Nuclear Physics and Atomic energy", ed. INR-Kiev (2006) 479. [15] P. Belli, R. Bernabei, C. Bacci, A. Incicchitti, R. Marcovaldi, D. Prosperi, DAMA proposal to INFN Scientific Committee II, April 24th 1990. [16] K.A. Drukier et al., Phys. Rev. D 33 (1986) 3495; K. Freese et al., Phys. Rev. D 37 (1988) 3388. [17] D. Smith and N. Weiner, Phys. Rev. D 64 (2001) 043502 and hepph/0402065. [18] A. B. Migdal, J. Phys. USSR 4 (1941) 449; G. Baur, F. Rosel and D. Trautmann, J. Phys. B: At. Mol. Phys. 16 (1983) L419.

SEARCH FOR RARE PROCESSES AT GRAN SASSO P. Belli 1 , R. Bernabei 1 , R. S. Boiko 2 , F. Cappella.3 , R. Cerulli 4 , C. J. Daj5, F. A. Da.nevich6 , A. d'Angelo 3 , S. d'Angelo\ B. V. Grinyov 7 , A. Incicchitti3 , V. V. Kobychev 6 , B. N. Kropivyansky6, M. Laubenstein4 , P. G. Nagornyi 6 , S. S. Nagorny6, S. Nisi 4 , F. Nozzoli 1 , D. V. Poda6 , D. ProspelP, A. V. Tolmachev8 , V. 1. Tretyak6 , 1. M. Vyshnevskyi 6 , R. P. Yavetskiy8, S. S. Yurchenko 6 1

2

Dip. Fisica.. Univ. Roma Tor Veryata and INFN Tor Veryata, 1-00188 Rome. Italy Chemical department, Kyiv National Taras Shevchenko UniveTsity. Kiev. Ukraine 3 Dip. Fisica. Univ. Roma La Sapienza and INFN Roma.. 1-0018.5 Rome. Italy 4INFN. La.boratori Na.-:ionali del Gran Sa..~so. 67010 Asseryi (AQ). Italy 5 IHEP. Chinese Academy, P. O. Box 918/8, Beijing 100089, Chi.na 6 Institute for NucleaT Research. MSP 03680 Kiev, Ukraine 7 Instihde for Scintillation Materials. 61001 KhaTkov. Ukrame 8 Institute fOT Single Crystals. 61001 KhaTkov. Ukraine Abstract. New results achieved by the DAMA collaboration on the search for rare processes in the underground Gran Sasso National Laboratories (LNGS) of the I~FN are presented. In particular. the following searches have been summarized: i) search for the Q activity of 151 Eu; ii) measurement of 2{i21J decay of looMo to the first excited level of looRu; iii) search for double beta processes in 64Z n .

ot

1

Search for a decay of natural Europiulll [1,2] Both natural Europium isotopes, 15 1Eu (natural abundance 6 = 47.81(6)(}~)) and 153Eu (6 = 52.19(6)%) have a positive energy release respectively to a decay and, thus, they are potentially a radioactive. Corresponding Qo. values are: Qo. = 1.964(1) MeV for 15 1Eu and Qo. = 0.273(4) MeV for 153Eu. The first experimental limits (90% C.L.) on the half lives of the rare alpha decays of if.. 1Eu into the first excited level of 147Pm (T1/ 2 > 2.4 X 10 16 y) and of 1.53 Eu into 149Pm (T1/ 2 > 1.1 X 10 16 y) was achieved in a preliminary measurement using a Li6Eu(B03h crystal (mass::: 2.72 g) at LNGS [1]. In a second measurement, a low background CaF 2 (Eu) crystal scintillator 3"(2) x 1" (mass of 370 g), doped by Europium, was used to search for the a activity of 151 Eu [2]. The concentration of Eu in the crystal was determined with the help of the ICP Mass Spectrometry analysis: (0.4 ± 0.1)%. The detector was installed in the DAMA/R&D set-up operative at LNGS at a depth of 3600 m.w.e. The energy scale and resolution of the CaF 2 (Eu) detector for, quanta were measured with standard, sources and the response of the detector to a particles was studied with a collimated 241 Am a source - by using different sets of absorbers - from 1 MeV up to 5.25 MeV (see Fig. 1a). To discriminate events from a decays inside the crystal from the ,(/3) background, the optimal filter method was applied and the energy dependence of the shape indicators (SI) was measured. The low energy part of the background spectrum measured with the CaF 2 (Eu) crystal scintillator during 7426 h is shown in Fig.lb. There is a peculiarity in the spectrum at the energy near 250 keV 225

226 .£

~ 0.25

"ii

0.2

0.15

500

o

10 Energy of (.( particles (MeV)

a)

b)

-1

1~~__~__~____~__~

200

2jO

300

350

400

450

;00

Energy (keVl

Figure 1: Left Energy dependence of the 0:/13 ratio for the CaF2(Eu) scintillation detector. Measurements with 241 Am source using different sets of absorbers (mylar or few mm of air in some cases) are shown by circles; points shown by triangles have been obtained from the identified 0: activity in the background data. Right: Low energy part of the background spectrum measured during 7426 h in the low background set-up with the CaF2(Eu) scintillator (crosses). The peculiarity on the left of the 1478m peak can be attributed to the 0: decay of 151 Eu with the half-life Tl/2 = 5 X 10 18 y. The 0: nature of the two peaks is further supported by the pulse shape discrimination analysis; see discussion on the bottom lines given in ref. [2].

in agreement with the expected energy of the 1.51 Eu alpha decay - which gives an indication on the existence of this process. Therefore, the half-life of 151 Eu relatively to the 0: decay to the ground state of 147 Pm has been evaluated to be: Tf;2(g.S. -+ g.s.) = 5!j1 X 10 18 y, or, in a more conservative approach: Tfo,(g.s. -+ g.s.) 2: 1.7 X 10 18 Y at 68(;70 C.L. [2]. In addition, for the decay of 15 1Eu to the first excited (Ei/2+,Eexc =fll keY) level of 147 Pm a limit has also been obtained: T~/"2(g.8. -> .5/2+) 2: 6 X lO 17 y at 68%C.L. Theoretical half-lifes for 151 Eu 0: decay calculated in different model frameworks are in the range of (0.3-3.6) x 10 18 y; in particular, the measured value of half-life of 151 Eu is well in agreement with the calculations of [3]. 2

Measurement of 2/32v decay of 100Mo to the

at level of 100Ru [4]

The meAsuReMent of twO-NeutrIno 2/3 decAy of 100Mo to the first excited level of lOoRu (ARMONIA) consists of a Mo sample of:::: 1 kg enriched in 100Mo at 99.5% in form of metallic powder installed in the four low-background HP Ge detectors (about 225 cm3 each, all mounted in one cryostat) facility located at LNGS. The aim of this high sensitivity experiment is to measure the 2/32v decay of 100Mo to the first excited level of lOoRu [E(Ot) = 1130 ..5 keV] either to confirm positive results reported in [5] (with T1/2 in the range around 6 - 9 X 1020 y) or to confirm previous higher limit value of ref. [6] (T1/ 2 > 1.2 X 10 21 Y at 90% C.L.). Preliminary data have been collected deep underground at LNGS during 1927 h (see Fig. 2). Two 'Y of 590.8 keY and 539.6 keY respectively are expected in the level de-excitation of the lOoRu. The measured energy distribution in the range of 500-600 keY is reported in Fig. 2 and compared

Or

ot

at

Figure 2: Spectrum of 100 Mo sample (mass of 1009 g) measured with 4 HP Ge detectors setup at LNGS during 192? h in the range of 600-600 keV. Shaded area is background spectrum (without 100 Mo sample) normalised to the same time of measurements. Peaks at 583 keV and 511 keV are related with 20®Tl decay and the positron annihilation process, respectively.

with the background spectrum measured without the 100Mo sample (shaded area). Note that peaks at 583 keV and 511 keV are related with 2O8T1 decay and the positron annihilation process, respectively. A modest peak structure seems to be present - but at very low C.L. - around 540 keV, where one 7 searched for is expected. If this would be ascribed to the decay searched for, one gets: T1/2 = 3 x 1020 y. However, no significant statistical evidence for the peak at the energy of 591 keV is found at present and a limit on the half-life can be derived: T1/2 > 6 x 1020 y at 90% C.L.. These measurements have shown that AR.M0NIA is entering in the sensitivity region of interest. New data taking with further purified sample and larger exposure is in progress, 3 64

Search for 2/3 processes in

64

Zn with a ZNWO4 scintillator [7].

Zn is one of the few exceptions among 20+ nuclei having big natural isotopic abundance (48.268%); the mass difference between 64Zn and 64Ni nuclei is 1095.7(0.7) keV and, therefore, double electron capture (2s), and electron capture with emission of positron (s[3+) are energetically allowed. A low background ZnWO4 crystal scintillator (mass of 117 g) has been installed deep underground in the low background DAMA/R&D set-up at the LNGS for the investigation of double beta processes in 64Zn with higher sensitivity. The energy scale and resolution of the Z11WO4 detector for 7 quanta were measured with 22Na, 133Ba, 137Os, 228Th and 241Am sources. The energy spectrum measured during 1902 h is presented in Fig. 3. Comparing the simulated response functions for different 2/3 processes in 64 Zn with the experimental energy distribution accumulated with the ZnWO.4 crystal we did not find the peculiarities expected in the spectra. Therefore, lower half-life limits can be set for the 2/3 processes in decay 64Zn — 64Ni at 90% C.L. (see table 1) improving the previous results; moreover, the positive

22B

400

lOO

6()O

ROO

Energy (keV)

Instr. £.1 Meth. A A 789

44

to appear on

Rad-i.at. I8ot. 46 (1995) 455. KEK Pmc. 6 (2003) 205. arXiv: 0707.2756vl [nucl-exJ. H. Kiel et Phys. A 723 (2003) 499. Danevich et al., Nucl. Instr. & Meth. A 544

ANISOTROPY OF DARK MATTER ANNIHILATION AND REMNANTS OF DARK MATTER CLUMPS IN THE GALAXY V.Berezinskya INFN, Laboratori Nazionali del Gran Sasso, 1-67010 Assergi (AQ), Italy V .Dokuchaev b, Yu.Eroshenko C Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia Abstract.A mass function of small-scale dark matter clumps is calculated by taking into account the tidal destruction of clumps at early stages of structure formation. The surviving clumps can be disrupted further in the Galaxy by tidal interactions with stars. A corresponding annihilation of dark matter particles in clumps produces the anisotropic gamma-ray signal on the level '" 9% with respect to the Galaxy plane. It is demonstrated that a substantial fraction of clump remnants may survive the tidal destruction during the lifetime of the Galaxy if a clump core radius is rather small. The dense part of clump core produces the dominating input into the annihilation signal from a single clump. For this reason the survived dense remnants of tidally destructed dark matter clumps may provide the major contribution to annihilation signal from the Galactic halo.

1

Dark matter clumps

About 30% of mass of the Universe is in a form of cold dark matter (DM), but the nature of DM particles is still unknown. The cold DM component is gravitationally unstable and forms the gravitationally bounded clumpy structures from the scale of superclusters of galaxies and down to very small clumps of DM. The cutoff of the mass spectrum, Mmin is determined by the collisional and collisionless damping processes and typical values Mmin rv (10- 8 - 1O-6)M8 for neutralino DM. Due to uncertainties in the SUSY parameters, a numerical value of Mmin is not exactly predicted. Theoretical study of DM clumps are important for understanding the properties of DM particles, because annihilation of DM particles in small dense clumps may result in observable signals. The cosmological formation and evolution in early Universe and properties of small-scale DM clumps have been studied in many works [1-9]. Only very small fraction of these clumps survives the early stage of tidal destruction during the hierarchial clustering. Nevertheless these survived clumps will provide the major contribution to the annihilation signal in the Galaxy [9]. One of the unresolved problem of DM clumps is a value of the central density or core radius. Numerical simulations give a nearly power density profile of DM clumps. Both the Navarro-Frenk-White and Moore profiles give formally a divergent density in the clump center. A theoretical modelling of the clump ae-mail: [email protected] be-mail: [email protected] ce-mail: [email protected]

229

230

formation [10] predicts a power-law profile of the internal density of clumps

Pint(r)

=

(r )-f3 '

3 - (J _ -3- P R

(1)

where 15 and R are the mean internal density and a radius of clump, respectively, (J ~ 1.8 - 2 and Pint(r) = 0 at r > R. A near isothermal power-law profile (1) with (J ~ 2 has been recently obtained in numerical simulations of small-scale

clump formation [11]. 2

Cosmological distribution function of clumps

The first gravitationally bound objects in the Universe are the DM clumps of the minimum mass Mmin. In the case of the Harrison-Zeldovich spectrum of primordial fluctuations with CMB normalization the first small-scale DM clumps are formed at redshift z rv 60 (for 20' fluctuations) with a mean density 7 x 10- 22 g cm- 3 , virial radius 6 x 10- 3 pc and internal velocity dispersion 80 cm s-l respectively. The clumps of larger scales are formed later. The internal energy increase after a single tidal shock experienced by the clump captured into larger host clump is t::.E ~ (47r/3)'YlGphMR2, where Ph is the density of a host and the numerical factor ,1 rv 1. Let us denote by ,2 the number of tidal shocks per dynamical time tdyn' The corresponding rate of internal energy growth for a clump is E = '2t::.E/tdyn. A clump is destroyed if its internal energy increase exceeds a total energy lEI ~ G M2 /2R. For a typical time T = T(p, Ph) of the tidal destruction of a small-scale clump with density P we obtain:

T -l( p, Ph ) -_

E - 4'1,2 Gl/2 Ph3/2 P-1 . lEI rv

(2)

It turns out that a resulting mass function of small-scale clumps depends rather weakly on the value of product ,1,2. During its lifetime a small-scale clump can stay in many host clumps of larger mass. After tidal disruption of the first lightest host, a small-scale clump becomes a constituent part of a larger one, etc. The process of hierarchical transition of a small-scale clump from one host to another occurs almost continuously in time up to the final host formation, where the tidal interaction becomes inefficient. The probability of clump survival, determined as a fraction of the clumps with mass M surviving the tidal destruction in hierarchical clustering, is given by the exponential function e- J with (3)

Here t::.th is a difference of formation times th for two successive hosts, and summation goes over all clumps of intermediate mass-scales, which successively

231

host the considered small-scale clump of a mass M. Changing the summation by integration in (3) we obtain

(4) where 'Y ::: 14(r1/2/3), and t, h, tt, p, Pl are respectively the formation times and internal densities of the considered clump and of its first and last hosts. One may see from Eq. (4) that the first host provides a major contribution to the tidal destruction of the considered small-scale clump, especially if the first host density Pl is close to p, and consequently e- J « 1. Finally using the Press-Schechter formalism we obtained in [12] the following distribution function of clumps (mass fraction):

c dM d rv lJdlJ -V 2 /2f ( )dlogO"eq(M) dM ." M lJ - J27r e 1 'Y dM '

(5)

where O"eq(M) is a r.m.s. fluctuation on a mass-scale M at the time of matterradiation equality, lJ is the peak high and one may use h (r) ::: 0.2 - 0.3. Integrating Eq. (5) over lJ, we obtain eint

dM dM M ::: 0.02(n + 3) M .

(6)

An effective power-law index n in Eq. (6) is determined as n = -3(1 + 28logO"eq(M)/8log M) and depends very weakly on M. The simple M- 1 shape of the mass function (6) is in a very good agreement with the corresponding numerical simulations [11], but our normalization factor is a two times smaller.

3

Tidal destruction of clumps

Crossing the Galactic disk, a DM clump can be tidally destructed by the collective gravitational field of stars in the disk. This phenomenon is similar to the destruction of globular clusters. The kinetic energy gain of a DM particle with respect to the center of clump after one crossing of the Galactic disk is [13]

(7) where m is a constituent DM particle mass, l1z is a vertical distance (orthogonal to the disk plane) of a DM particle with respect to the center of clump, vz,c is a vertical velocity of clump with respect to the disk plane at the moment of disk crossing, gm(r) is gravitational acceleration near the disk plane and we use an

232 exponential model for a surface density of disk. As a representative example we consider the isothermal internal density profile of DM clumps. We used the adiabatic correction in Weinberg's form A(a) = (1 + a 2 )-3/2 [14], where the adiabatic parameter a = WTd with W is an orbital frequency of DM particle in the clump, T is an effective duration of gravitational tidal shock. In [12J we describe a gradual mass loss of small-scale DM clumps assuming that only the outer layers of clumps are involved and influenced by the tidal stripping. Additionally we assume that inner layers of a clump are not affected by tidal forces. In this approximation we calculate a continues diminishing of the clump mass and radius during the successive galactic disk crossings and encounters with the stars. As a criterium of a clump destruction we accept now the approaching of the radius of tidally stripped clump down to the core radius. By using the hypothesis of a tidal stripping of outer layers of a DM clump we see that a tidal energy gain Jc causes the stripping of particles with energies in the range -Jc < c < O. A corresponding variation of density at radius r is

Jp(r)

5 2

= 2 / 7r

J° Vc -

'Ij;(r) fcl(c) dc,

(8)

-/it:

where fcl(c) is the distribution of particles inside a clump over energy (in dimensionless variables). A resulting total mass loss by DM clump during one crossing of the Galactic disk is

(9) Then we calculate the tidal mass loss by clumps using a distribution of their orbits in the standard Navarro-Frank-White Galactic halo. Choosing a time interval t::..T much longer than a clump orbital period Tt, but much shorter than the age of the Galaxy to, i. e., Tt « t::..T « to, we define an averaged rate of mass loss by a selected clump under influence of tidal shocks in successive disk crossings:

1

~ (d:)d ~ t::..T L (J:)d'

(10)

The simplification in calculation of (10) follows from the fact that a velocity of orbit precession is constant. For this reason the points of successive odd crossings are separated by the same angles and the same is also true for successive even crossings. By the similar formalizm we calculate the diminishing of a clump mass due to a tidal heating by stars in the halo and bulge [9J. Combining together the rates of mass losses due to the tidal stripping of a clump by the disk and stars we obtain the evolution equation for a clump mass:

dM

dt =

(dM) dt

(dM)

d

+ dt

s'

(11)

233

3 kpc. / /' /' /'

/'

/8.5 kpc

/ 10 8

~______~________L -____~____________~______~

Figure 1: Numerically calculated modified mass function of clump remnants for galactocentric distances 3 and 8.5 kpc. The solid curve shows initial mass function.

We solve this equation numerically starting from the time of Galaxy formation at to - to up to the present moment to and calculate the probability P of the survival of clump remnant during the lifetime of the Galaxy. 4

Modified distribution function of clumps

We calculated (see details in [12]) the modified mass function for the smallscale clumps in Galaxy taking into account clump mass loss instead of clump destruction considered in [9]. One can see in the Fig. 1 that clump remnants exist below the Mmin. Deep in the bulge (very near to the Galactic center) the clump remnants are more numerous beca'use of intensive clump destructions in the dense stellar environment in comparison with the rarefied one in the halo. In [15] it was found that almost all small-scale clumps in Galaxy are destructed by tidal interactions with stars and transformed into "ministreams" of DM. The properties of these ministreams may be important for the direct detection of DM particles because DM particles in streams arrive anisotropic ally from several discrete directions. We demonstrate that the cores of clumps (or clump remnants) survive in general during the tidal destruction by stars in the Galaxy. Although their outer shells are stripped and produce the ministreams of DM, the central cores are protected by the adiabatic invariant and survived as the sources of annihilation signals.

234

5

Annihilation of dark matter in clumps

A local annihilation rate is proportional to the square of DM particle number density. A number density of DM particles in clump is much large than a corresponding number density of the diffuse (not clumped) component of DM. For this reason an annihilation signal from even a small fraction of DM clumps can dominate over an annihilation signal from the diffuse component of DM in the halo. The gamma-ray flux from annihilation of diffuse distribution of DM in the halo is proportional to rmax(()

IH =

J p~(~)

dx,

(12)

o where PH - is the density profile of halo, integration over r goes along the line of sight, ~((, r) = (r2 + r~ - 2rr8 cos ()1/2 is the distance to the Galactic center, rmax(() = (R~ - r~ sin 2 ()1/2 + r8 cos ( is a distance to the external halo border, ( is an angle between the line of observation and the direction to the Galactic center, RH is a radius of the Galactic halo, r8 = 8.5 kpc is the distance between the Sun and Galactic center. The corresponding signal from annihilations of DM in clumps is proportional to the quantity [9] rmax (()

lei

=

s

J J ~int(M) dx

o

d:: pPH(~)P(~, p)

(13)

Mmin

where p(M) is the mean density of clump, S ~ 14.5 [9]. The observed amplification of the annihilation signal is defined as T/(() = (lcl - IH)/IH is shown in the Fig. 2 for the relative core's radius Rei R = 0.1. It tends to unity at ( -> 0 because of the divergent for in of the halo profile. This amplification of an annihilation signal is often called a 'boost factor'. A boost factor of the order of 10 is required for interpretation of the observed EGRET gamma-ray excess as a possible signature of DM neutralino annihilation [16]. The usual assumption in calculations of DM annihilation is a spherical symmetry of the Galactic halo. In this case an anisotropy of annihilation gammaradiation is only due to off-center position of the Sun in the Galaxy. Nevertheless, in [15] the anisotropy with respect to the Galactic disk was discussed. A tidal heating and final destruction of clumps by the gravitational field of the Galactic disk depends on the inclination angle of a clump orbit to the disk. In the Fig. 3 the annihilation signal [9] is shown for the Galactic disk plane and for the orthogonal vertical plane (passing through the Galactic center) as function of angle ( between the observation direction and the direction to the Galactic center. For comparison in the Fig. 3 is also shown the signal from

235

250 :r: 200

H

'-

:r: 150 +

H

rl

U

H

100 50 0 25

0

75

50

100

125

150

175

Sf degree Figure 2: The amplification of the annihilation signal (Ie! - IH)/IH as function of the angle between the line of observation and the direction to the Galactic center, where fluxes are given by (12) and (13).

9J)

H

25

50

75

S,

100

125

150

175

degree

Figure 3: The annihilation signal in the Galactic disk plane and in vertical plane.

236

the spherically symmetric Galactic halo without the DM clumps. The later signal is the same in the in the Galactic disk plane and in vertical plane and therefore can be principally extracted from the observations. The difference of the signals in two orthogonal planes at the same ( can be considered as an anisotropy measure. Defined as 8 = (12 - h)/h, it has a maximum value 8 ~ 0.09 at ( ~ 39°. This anisotropy provides a possibility to discriminate dark matter annihilation from the diffuse gamma-ray backgrounds of other origin. Acknowledgments

This work was supported in part by the Russian Foundation for Basic Research grants 06-02-16029 and 06-02-16342, the Russian Ministry of Science grants LSS 4407.2006.2 and LSS 5573.2006.2. References

[1] [2] [3] [4] [5] [6J [7] [8] [9J [10] [11] [12J [13] [14J [15] [16]

C.Schmid, D.J.Schwarz and P.Widerin, Phys. Rev. D 59,043517 (1999). D.J.Schwarz and S.Hofmann, Nucl. Phys. Proc. Suppl. 87, 93 (2000). H.S.Zhao , J.Taylor, J.Silk and D.Hooper, arXiv:astro-ph/0502049v4. A.M.Green, S.Hofmann and D.J.Schwarz, JCAP, 0508, 003 (2005). J.Diemand, M.Kuhlen and P.Madau, Astrophys. J. 649, 1 (2006). A.M.Green and S.P.Goodwin, Mon. Not. Roy. Astron. Soc. 375, 1111 (2007). E. Bertschinger, Phys. Rev. D 74, 063509 (2006). G.W. Angus and H.S.Zhao Mon. Not. Roy. Astron. Soc. 375, 1146 (2007). V.Berezinsky, V.Dokuchaev and Yu.Eroshenko, Phys.Rev. D 68, 103003 (2003); Phys.Rev. D 73, 063504 (2006); JCAP 07, 011 (2007). A.V.Gurevich and K.P.Zybin, Sov. Phys. - JETP 67, 1 (1988); Sov. Phys. - JETP 67, 1957 (1988); Sov. Phys. - Usp. 165, 723 (1995). J.Diemand, B.Moore and J.Stadel, Nature 433, 389 (2005). V.Berezinsky, V.Dokuchaev and Yu.Eroshenko, arXiv:0712.3499vl. J.P.Ostriker, L.Spitzer Jr. and R.A.Chevalier, Astrophys. J. Lett. 176, 51 (1972). O.Y.Gnedin and J.P.Ostriker, Astrophys.J. 513, 626 (1999). H.S.Zhao, J.Taylor, J.Silk and D.Hooper, arXiv:astro-ph/0508215v4. W. de Boer, C.Sander, V.Zhukov, A.V.Gladyshev and D.I.Kazakov, Astron. Astrophys. 444, 51 (2005).

CURRENT OBSERVATIONAL CONSTRAINTS ON INFLATIONARY MODELS E.Mikheeva a Astra Space Centre, P.N.Lebedev Physics Institute, 117997 Moscow, Russia Abstract. We review current observational constraints on inflationary models and show that A-inflation is observationally favored.

1

Introduction

Proposed more than two decades ago inflation successfully and elegantly solved a majority of existed cosmological problems (see e.g. [1]). Now we are on the way to selecting the inflationary model that was realized in the beginning of the Universe. On this way there is a lot of troubles. The first of them is a great gap between long lasting inflation (more than 60 e-folds) and rather short interval of observable cosmological scales (from 10 Mpc to the present horizon, about 6 e-folds). So, only a short interval of inflaton values can be directly tested and a narrow part of the potential can be reconstructed. Nowadays there is a single inflationary paradigm, but a lot of inflationary models. A long list of these models includes more than two hundred items. In this paper we concentrate on available observational constraints for inflationary models with a single scalar field, which allow us to simplify the problem. 2

Model and observable quantities

If the potential of inflaton is fixed then spectra of cosmological perturbations generated during inflation can be derived. In the case of scalar field the seeds for vector mode of perturbations are absent and we deal only with scalar and tensor modes of cosmological perturbations. The former is related to density perturbations, the latter is gravitational waves. Both spectra are functions of wave number k. Testing inflation in a straighforward way means constraining predicted spectra by different sets of observational data. Observationally, it is more convenient to operate with simplified presentations of the power spectra as follows:

(1) where S(k) is the spectrum of scalar perturbations, T(k) is the spectrum of tensor perturbations, As and AT are amplitudes of the spectra, ns and nT are the spectra slopes of scalar and tensor modes, respectively. A further presentation of spectra in parameter space is acceptable for scalar mode: &ns a=&lnk' ae-mail: [email protected]

237

(2)

238 where a is a running index. The value of a is a measure of deviation of the scalar spectral index from the power law. Another observationally convenient quantity is a relative amplitude of gravitational waves, TIS. Initially it was introduced as a ratio of contributions of gravitational waves and density perturbations into large scale anisotropy of cosmic microwave background (a quadrupole or 10 angular degree anisotropy). Here TIS is a ratio of the tensor to scalar power spectrum for k c:::: 1O- 3 Mpc- 1 . So, from model functions S(k) and T(k) we pass to observable quantities ns, TIS, and a. 3

Classification of inflationary models

Inflationary models with a single scalar field can be classifying as follows ( [2]): 1) inflation on small field, with example potential V(ip) where Vo, ipo and p are positive constants;

= Va [1 - (iplipo)PJ,

2) inflation on large field, with power-law potentials as an example V(ip) '"

ipP 3) A-inflation with example potential V(ip) are positive constants.

= Vo + AipP 14, where Vo and A

This simple classification is rather convenient to design inflationary model predictions on the plot TIS - ns. 4

Current observational constraints for inflationary models

Observational cosmology deals with a long list of cosmological parameters (cosmological constant value D A , amount of cold dark matter Dcdm), Hubble constant Ho and many others). Reconstruction of parameters related to the inflationary stage of the expansion may be considered as a reduced task, but it does not simplify the problem. So, uncertainties of constraints for all these cosmological parameters will be imprinted on the values of ns, TIS and a, and their error bars, as well. Nevertheless, in the first decade of the third millennium we have a powerful tool to determine inflationary cosmological parameters. It is a joint statistical analysis of different kinds of observational data, including data on Cosmic Microwave Background anisotropy (see [5]). Of cause, CMB data alone have some degeneracies, which can be removed by taking into account available data on galaxy distributions. This approach allows one to find probability distribution contours for three classes of inflation on the plain TIS - ns. It appears that current observational data rather strongly constrain ns, which is forbidden to strongly deviate from the Harrison-Zel'dovich value (ns =

observational constraints on intlatilonlary large field inflations and between large field inflation (on the corresponds to constraints on A-inflation from to a confidence level by the line line with a confidence level The h"f'kO'r""mri case of nonzero running index.

way to detect B-mode of the CMB constraints on <.;V,O>UJlVl\JI4"''''''''

rarneters is not the

240

of problems in determining cosmological parameters caused by inflation. Another source is a priori hypotheses used during reconstruction of the parameters. Assuming a = 0 which is reasonable for some kinds of inflationary models 10- 3 ), decreases a possible value (for example, in massive field inflation lal of T / S twice. Rejection of this assumption dramatically leads from the set (ns = 0.98 ± 0.03, T / S < 0.17, see Table 7 in [5]) to the set (ns = 1.16 ± 0.1, T / S < 0.3, ibid). Statistically the difference is negligible, theoretically it is huge. In particular, it means that a real error of determination of ns is about 0.1, therefore "blue" spectra of density perturbations do not contradict available observational data. r-.J

5

Conclusions

Our conclusions are the following: 1. Massive field inflation satisfies available observational data. Chaotic inflation with potential slope p ~ 4 (see) is rejected by observations. 2. If further observations indicate ns > 1 then some variant of A-inflation was realized in the early Universe. If observations fix ns < 1 then there was inflation on massive scalar field, or small field inflation, or something else depending on the values of ns and T / S. 3. Inflationary model will be determined in the nearest future. Acknowledgment

This work was supported by Russian Foundation for Basic Research (grant number 07-02-00886). References

[1] A.Linde, "Particle Physics and inflationary cosmology", (Harwood Academic Publishers, Chur, Switzerland), 1990. [2] W.H.Kinney, A.Melchiorri, A.Riotto, Phys. Rev. D 63, 023505 (2001). [3] A.Linde, Phys. Rev. D 49, 748 (1994). [4] V.N.Lukash, E.V.Mikheeva, Int. J. Mod. Phys. A 15, 3783 (2000). [5] D.N.Spergel, ApJ Supp. 170, 377 (2007). [6] E.V.Mikheeva, V.N.Lukash, Astron. Rep. 48, 2 (2004); in Russian Astron. Zh. 81, 4 (2004).

PHASE TRANSITIONS IN DENSE QUARK MATTER IN A CONSTANT CURVATURE GRAVITATIONAL FIELD

2

D. Ebert la , V. Ch. Zhukovskyl>, and A. V. Tyukov 2 1 Institut fur Physik, Humboldt- Universitiit zu Berlin, 12489 Berlin, Germany Faculty of Physics, Department of Theoretical Physics, Moscow State University, 119899, Moscow, Russia Abstract We consider the phase transitions in dense matter with quark and diquark condensates in the static Einstein universe at finite temperature and chemical potential. The nonperturbative expression for the thermodynamic potential was obtained. The phase portraits of the system were constructed.

1

Introduction

It was proposed more than twenty years ago [2,3] that at high baryon densities a colored diquark condensate < qq > might appear. In analogy with ordinary superconductivity, this effect was called color superconductivity (CSC). The possibility for the existence of the CSC phase in the region of moderate densities was recently proved (see, e.g., the papers [4-6]). Since quark Cooper pairing occurs in the color anti-triplet channel, a nonzero value of < qq > means that, apart from the electromagnetic U(I) symmetry, the color SUc (3) symmetry should be spontaneously broken inside the CSC phase as well. The similar effect of chiral symmetry breaking in curved spacetime has been recently studied [7,8], which may be useful for the investigation of compact stars, where the gravitational field is strong and its effect cannot be neglected.

2

The extended NJL model in curved space-time

We will use the extended Nambu - Jona-Lasinio (NJL) model [1] with upand down-quarks and color group SU(3)c. This model may be considered as the low energy limit of QCD. The linearized Lagrangian with collective boson fields (J', if and .6. can be written in the form: i'

'-

t,

t,

-[.1-'"<7v I-' + J.L' 0] q - q-( (J' + .5 T7r __ ) q - 2G 3

q

- -

3

G2

A.b uA b -

u

5]

A.b [.tq t aC:E, b q

u

-

A

u

((J' 2+ 7r_2) 1

5

b [.tqC:E, - b a-t] q .

(1 )

Here, J.L is the quark chemical potential. The quark field q == qia is a doublet of flavors and triplet of colors with indices i = 1,2; Q = 1,2,3. [email protected] [email protected]

241

242 The fields a and if are color singlets, and /::,. b is a color ant i-triplet and flavor singlet. Therefore, if < a > of. 0, the chiral symmetry is broken dynamically, and if < /::,. b > of. 0, the color symmetry is broken. The effective action for boson fields Seff can be expressed through the integral over quark fields, according to

In the mean field approximation, the fields a, if, /::,.b, /::,.*b can be replaced by their groundstate averages: < a >, < if >, < /::,. b > and < /::,. *b >, respectively. Let us choose the following ground state of our model: < /::,.1 > = < /::,. 2 > = < if > = 0, and denote < a >, < /::,.3 > of. 0, by letters a, /::,.. Evidently, this choice breaks the color symmetry down to the residual group SUc (2). Let us find the effective potential of the model with the global minimum point that will determine the quantities a and /::,.. By definition Seff = - Veff dD xFg, where

J

-

Veff

3

Sq = -~'

V

=

J

d D xF9.

(3)

Static Einstein universe

We will use the static D-dimensional Einstein universe as the simple example of curved spacetime. The line element is

(4) where a is the radius of the universe, related to the scalar curvature by the relation R = (D - l)(D - 2)a- 2 . The effective potential at finite temperature or thermodynamic potential may be obtained in the following form (for more details see [9]) ,,2 O(a, /::,.) = 3 ( 2G, N

00

2: dl {El + Tin (1 + e-t3CEI-lLl) + Tin (1 + e-t3CEI+lLl)}

-::,; (Nc - 2)

-::,; E N

1Ll.12) + e;-

00

+2TIn

dl

1=0 {

J(EI - J1.)2

+ 41/::,.1 2 + J(EI + J1.)2 + 41/::,.1 2 +

(1 + e- y'CE -lLl 2+41Ll.1 2) + 2T In (1 + e- t3 y'CE t3

1

where V is the volume of the universe V(a) El

=

V1( D-1)2 - 1+ - +a a 2 ' 2

2

1 +1L)2+41Ll.1

2 ) },

= 27l'D/2a D- 1 /r( ~), 1= 0, 1,2 ... ,

(5)

(6)

243

and d _

2[(D+l)/2 j qD

+l -

l!r(D _ 1)

I -

1) ,

(7)

where [xl is the integer part of x. 4

Phase transitions

In what follows, we shall fix the constant G 2 , similarly to what has been done in the flat case [5,10], by using the relation G 2 = ~Gl. For numerical estimates, let us choose the constant G 1 = 10 such that the chiral and/or color symmetries were completely broken. Moreover, let us now limit ourselves to the investigation of the case D = 4 only. In Fig. 1, the J1, - R-phase portrait of the system at zero temperature is depicted.

6 4

2 ~

o

2 ______ L-_____________ R 10

20

30

40

Figure 1: The phase portrait at T=O. Dashed (solid) lines denote first (second) order phase transitions. The bold point denotes a tricritical point.

For points in the symmetric phase 1, the global minimum of the thermodynamic potential is at (j = 0, t. = 0 (chiral and color symmetries are unbroken). In the region of phase 2, only chiral symmetry is broken and (j =1= 0, t. = O. The points in phase 3 correspond to the formation of the diquark condensate (color superconductivity) and the minimum takes place at (j = 0, t. =1= O. Moreover, the oscillation effect clearly visible in the phase curves of Fig. 1 should be mentioned. This may be explained by the discreteness of the fermion energy levels (6) in the compact space. This effect may be compared to the similar effect in the magnetic field H, where fermion levels are also discrete (the Landau levels). In Fig. 2, J1, - R- and T - J1,- phase portraits are depicted. It is clear from Fig.2 that with growing temperature both the chiral and color symmetries are restored. The similarity of plots in R - J1, and J1, - T axes leads one to the conclusion that the parameters of curvature R and temperature T play similar roles in restoring the symmetries of the system.

244

3.5

Jl

5

Jl

3

2.5 2

3

3

1.5 2 2

0.5 L---~~----'----R

4

6

8

10

12

14

16

'--------------L--T 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Figure 2: The phase portraits at T=O.35 (left picture) and at R=3 (right picture).

We have considered the phase transitions in the static Einstein universe at finite temperature and chemical potentia!. In spite of the model character of the problem, we hope that the results of this paper may stimulate further investigations that are closer to realistic cosmological or astrophysical situations. Acknow ledgments

One of the authors (A.V.T.) is grateful to Prof. M. Mueller-Preussker for his attention and support of this work. This work was also supported by DAAD. References

[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961); ibid. 124, 246 (1961); V. G. Yaks and A. I. Larkin, ZhETF 40, 282 (1961). [2] B.C. Barrois, Nuc!. Phys. B 129, 390 (1977). [3] D. Bailin and A. Love, Phys. Rept. 107, 325 (1984). [4] M. Alford, K. Rajagopal and F. Wilczek, Nuc!. Phys. B 537, 443 (1999); K. Langfeld and M. Rho, Nuc!. Phys. A 660, 475 (1999). [5] J. Berges and K. Rajagopal, Nuc!. Phys. B 538, 215 (1999). [6] T.M. Schwarz, S.P. Klevansky and G. Papp, Phys. Rev. C 60, 055205 (1999). [7] T. Inagaki, S.D. Odintsov and T. Muta, Prog. Theor. Phys. Supp!. 127, 93 (1997), hep-th!9711084 (see also further references in this review paper). [8] X. Huang, X. Hao, and P. Zhuang, hep-ph!0602186. [9] D.Ebert, A.V.Tyukov, and V.Ch. Zhukovsky, Phys. Rev. D76, 064029 (2007). [10] D. Ebert, V.V. Khudyakov, V.Ch. Zhukovsky, and K.G. Klimenko, Phys. Rev. D 65, 054024 (2002); D. Ebert, K.G. Klimenko, H. Toki, and V.Ch. Zhukovsky, Prog. Theor. Phys. 106,835 (2001).

CONSTRUCTION OF EXACT SOLUTIONS IN TWO-FIELDS MODELS Sergey Yu. Vernov a Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119991 Moscow, Russia Abstract.Dark energy model in the Friedmann Universe with a phantom scalar field, an usual scalar field and the polynomial potential has been considered. We demonstrate that the superpotential method is very effective to seek new solutions and to construct a two-parameter set of exact solutions to the Friedmann equations. We show that the standard formulation of superpotential method can be generalized.

1

Introduction

One of the most important recent results of the observational cosmology is the conclusion that the Universe expansion is speeding up rather than slowing down. The combined analysis of the type Ia supernovae, galaxy clusters measurements and WMAP (Wilkinson Microwave Anisotropy Probe) data gives strong evidence for the accelerated cosmic expansion [1-3]. To specify a component of a cosmic fluid one usually uses a phenomenological relation between the pressure p and the energy density {} corresponding to each component of fluid p = W{}, where W is called as the state parameter. The experimental data suggests that the present day Universe is dominated by a smoothly distributed slowly varying cosmic fluid with negative pressure, the so-called dark energy. Contemporary experiments [1-3] give strong support that the Universe is approximately spatially flat and the dark energy state parameter w DE is currently close to -1: w DE = -1 ± 0.2. The state parameter wDE == -1 corresponds to the cosmological constant. As it has been shown in [4] for a large region in parameter space an evolving state parameter wDE is favoured over w DE == -1. The standard way to obtain an evolving state parameter WDE is to include scalar fields into a cosmological model. Two-fields cosmological models, describing the crossing of the cosmological constant barrier wDE == -1, are known as quintom models and include one phantom scalar field and one usual scalar field. Nowadays the string field theory (8FT) has found cosmological applications related to the acceleration of the Universe. In phenomenological phantom models, describing the case WDE < -1, all standard energy conditions are violated and there are problems with stability (see [5] and references therein). Possible way to evade the instability problem for models with WDE < -1 is to yield a phantom model as an effective one, which arises from more fundamental theory with a normal sign of a kinetic term. In this paper we consider a 8FT inspired gravitational models with two scalar fields and a polynomial potentials. We propose new formulation of superpotential method, which is more suitable to construct models with a two-parameter set of exact solutions. ae-mail: [email protected]

245

246

2

String Field Theory Inspired Two-Fields Model

We consider a model of Einstein gravity interacting with a phantom scalar field ¢ and a standard scalar field ~ in the spatially flat Friedmann Universe. We assume that a phantom scalar field represents the open string tachyon, whereas the usual scalar field corresponds to the closed string tachyon [6-8]. Since the origin of the scalar fields is connected with the string field theory the action contains the typical string mass Ms and a dimensionless open string coupling constant go:

s=

JdxA (::1;R + :~ (~gI'V(al'¢av¢ 4

-

al'~av~) - V(¢,~))),

(1)

where Mp is the Planck mass. The Friedmann metric gl'v is a spatially flat:

(2) where a(t) is a scale factor. The coordinates (t, Xi) and fields ¢ and dimensionless. If the scalar fields ¢ and ~ depend only on time, then

H2 =

_1_ ( _

3m~

if =

¢ + 3H 4> =

~~

~ 12 2'1-'

~ c2

+ 2'" +

v) ,

2~~ (4)2 - e) ,

== V,;,

~ + 3He =

~

are

(3)

(4)

-

~~ == - V(

(5)

For short hereafter we use the dimensionless parameter mp: m~ = g;M~jM:. Dot denotes the time derivative. The Hubble parameter H == a(t)ja(t). Note that only three of four differential equations (3)-(5) are independent.

3

The Method of Superpotential

The gravitational models with one or a few scalar fields play an important role in cosmology and models with extra dimensions. One of the main problems in the investigation of such models is to construct exact solutions for the equations of motion. System (3)-(5) with a polynomial potential V(¢,~) is not integrable. The superpotential method has been proposed for construction of a potential, which corresponds to the particular solutions known in the explicit form [9]. The main idea ofthis method is to consider H(t) as a function (superpotential) of scalar fields: H(t) = W(¢(t),~(t)). (6) If we find such superpotential W (¢,~) that the relations

(7)

247

(8) are satisfied, then the corresponding cp, ~ and H are a solution of (3)-(5). The superpotential method separates system (3)-(5) into two parts: system (7), which is as a rule integrable for the given polynomial W (cp,~) and equation (8), which is not integrable if V(cp,~) is a polynomial, but has exact special solutions. The use of superpotential method does not include the solving of eq. (8). The potential V(cp,~) is constructed by means of the given W(cp,~). Relations (7) and (8) are sufficient, but maybe not necessary conditions to satisfy (3)-(5). To generalize them we assume that functions cp(t) and ~(t) are given by the following system of equations:

¢=

F(cp,~),

(9)

where F(cp,~) and G(cp,~) are some differentiable functions. We consider these functions as given ones and transform system (3)-(5) into equations in W(cp, ~):

W2 = _1_ ( _

3m~

W' F ¢

~ F2 2

+ W'G = E

~G2 +2 +

v) ,

_1_ (F2 _ G2 ) 2m2p '

(11)

F/pF + F~G + 3WF = V,;, G;PF + GeG + 3WG = -

(12)

vt

We differentiate equation (10) in

W ( W/p -

(10)

(13)

cp and use (12) to exclude VJ,:

2~~ F) = 6~~ (G¢ + FE) .

(14)

Similar manipulations give also

W(W~ + 2~~ G) = - 6~~ (G For any G(cp,~) and F(cp,~) such that G¢

2m;W';' = F,

=

¢

+ FE) .

-FE we find

2m;W~

= -G.

(15)

W(cp,~) using relations

(16)

These relations are equivalent to (7). In this case the equality G¢ = -FE is g~~ = g~~. So, we have shown that the relations (7) are necessary conditions if and only if the equality G¢ = -FE is satisfied. In other case one should solve nonlinear system (14)-(15) to find the corresponding superpotential W(cp,~). Note that in our formulation we do not use the explicit form of exact solutions to find potential. Note also that for one and the same one-parameter set of exact

248

solutions we can find different form of functions F and G and, therefore, the different form of potential V. We can conclude that the proposed formulation of the superpotential method is effective to seek potential V (¢, ~), which satisfies some conditions and corresponds to a two-parameter set of exact solutions. For example, if F and G are linear functions (A, Band C are constants): (17) then we obtain the fourth degree polynomial potential

V =

A2

2 - B2 ¢2+B(A+C)¢~+ B2 - C e + -3 -2 ( A¢2+2B¢~-Ce ) 2 . (18)

2

2

16mp

and two-parameter set of exact solutions for the Friedmann equations. For example, at C = 2B + A we obtain (Cl and C 2 are arbitrary constants) and

~=

(C + C;; + l

C2t)

e(A+B)t.

(19)

More nontrivial example with the sixth degree polynomial potential and a twoparameter set of kink-like (¢) and lump~like (~) solutions is presented in [8]. Acknowledgments

Author is grateful to LYa. Aref'eva for useful discussions. This research is supported in part by RFBR grant 05-01-00758 and by grant NSh-8122.2006.2. References

[1] A. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, 1009-1038 (1998); astro-ph/9805201. [2] D.N. Spergel et al. [WMAP Collaboration]' Astrophys. J. Suppl. SeT. 170 377-408 (2007); astro-ph/0603449. [3] Tegmark et al. [SDSS Collaboration], Phys. Rev. D69 103501 (2004); astro-ph/0310723. [4] U. Alam, V. Sahni, T.D. Saina, A.A. Starobinsky, Mon. Not. Roy. Astron. Soc. 354, 275 (2004); astro-ph/0311364 [5] LYa. Aref'eva, LV. Volovich, 2006, hep-th/0612098. [6] LYa. Aref'eva, AlP Conf. Proc. 826301-311 (2006); astro-ph/0410443. [7] LYa. Aref'eva, A.S. Koshelev, S.Yu. Vernov, Phys. Rev. D72, 064017 (2005); astro-ph/0507067. [8] S.Yu. Vernov, 2006, astro-ph/0612487. [9] O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. D62, 046008 (2000); hep-th/9909134.

QUANTUM SYSTEMS BOUND BY GRAVITY Michael L. Fil'chenkov a , Sergey V. Kopylov b , Yuri P. Laptev C Institute of Gravitation and Cosmology, Peoples' Friendship University of Russia, Moscow Department of Physics, Moscow State Open University Department of Physics, Bauman Moscow State Technical University Abstract. Quantum systems contain charged particles around mini-holes called graviatoms. Electromagnetic and gravitational radiations for the graviatoms are calculated. Graviatoms with neutrino can form quantum macro-systems.

1

Introduction

As known, there exist bound quantum systems due to electromagnetic and strong interactions, e.g. atoms, molecules and atomic nuclei. If one component of the gravitationally bound system is assumed to be massive and the other is an elementary particle, then a quantum system can be formed, e.g. mini-holes in the early Universe [1-3]. Such systems are called graviatoms [4]. Another example of the quantum systems bound by gravity is macro-bodies capturing neutrinos having de Broglie's wave length of macroscopic value.

2

Theoretical solution to the graviatom problem

Schrodinger's equation for the graviatom [2]

~~

[r2 (dR pl )] r2 dr dr

2 2 _l(l r2+ 1) R P1+ 2m (E _ mc rQr g + mc r g ) 1i 2 4r2 2r

R 1 = 0 (1) P

describes a radial motion of a particle with the mass m in the mini-hole potential, where rg = 2GM/c 2 and M are the mini-hole gravitational radius and mass respectively. The energy spectrum is of hydrogen-like form

E 3

G2 M 2 m 3 21i 2 n 2

= ----,:--

(2)

Graviatom existence conditions

A graviatom can exist if the following conditions are fulfilled [4]: 1) the geometrical condition L > r 9 + R, where L is the characteristic size of the graviatom, R is that of a charged particle; 2) the stability condition: (a) Tgr < TH, where Tgr is the graviatom lifetime, TH is the mini-hole lifetime, (b) Tgr < T p , where Tp is the particle lifetime (for unstable particles); 3) the indestructibility condition (due to tidal forces and Hawking's effect) Ed < Eb, where Ed is the destructive energy, Eb is the binding energy.

249

250

L-,---~--_--

0.5

1.0

____

~

__

2.5

2.0

1.5

~

3.0

_ _ _ _ _ m·l0· 3.5

21

,g

4.0

Figure 1: The dependence of mini-hole masses on the charged particle masses satisfying the graviatom existence conditions. The light curves indicate the range of values related to the geometrical condition (the upper curve) and to Hawking's effect ionization one (the lower curve). The heavy curve is related to the particle stability condition (Tp = 1O- 22 s).

The charged particles able to be constituents of the graviatom are: the electron, muon, tau lepton, wino, pion and kaon. The conditions of existence the graviatoms reduce to the relation between the masses of the mini-hole and particle, with their product being approximately constant equal to the Planck mass squared. 4

Graviatom radiation

The intensity of the electric dipole radiation of a particle with mass m and charge e in the gravitational field of a mini-hole reads [4] 2

d

I fi

=

2fie w7f lif me3

'

(3)

where wif = (Ei - E f )/fi is the frequency of the transition i --> I and lif is the oscillator strength [5]. The electric quadrupole radiation intensity for the transition 3d --> Is is q

113

=

6fie 2 wg1 3 me

hd--+1s'

(4)

The gravitational radiation intensity for the graviatom performing the transition 3d --> Is reads 9 _ 6fiGMwg 1 113 3 hd--+1s' (5) e The mini-hole creates particles near its horizon due to Hawking's effect, its power [7].

251 Table 1: Parameters for graviatoms with the electron and wino [6].

electron 0.511

mc"'!', MeV

8

00

M, g L, cm

3.5.10 17 6.10 11 0.08 2. lOw

T,

hW12,

Id(2p

MeV 18), erg.

--t

8 ·1

wino 8 . lOt> 5. 10 -10 2.2. lOll 4.10 17 1.2. lOt> 4 .1O:.!",!.

The mini-holes being constituents of the graviatoms are formed due to Jeans' gravitational instability at the times about ~ = 10- 27 -;.-10- 21 8 from the initial c singularity. The mini-hole masses for the graviatoms involving electrons, muons and pions exceed the value of 4.38 . 1014g, which means that it is possible for such graviatoms to have existed up to now [7]. The quantity G~m = 0.608 -;.- 0.707 is a gravitational equivalent of the fine structure constant. The gravitational radiation intensities two orders exceed the electromagnetic ones. The graviatom dipole radiation energies and intensities have proved to be comparable with those for Hawking's effect of the mini-holes being constituents of the graviatoms. 5

Systems with neutrinos

De Broglie's wavelength for the neutrino with mass mv is

1) Graviatom The existence conditions: a~ = li.dB > 3rg, Tgr < energy: mvc2 rv 1 eV. Characteristic frequency:

TH.

Electron neutrino rest

Gravitational radiation: Igr =

Q9M9 m ll

c5 h lO

v

Mini-hole masses: 10 18 g < M < 10 23 g. For example, if M < 10 23 g, then hw~ < 0.2 eV,Igr size is about 10 1 -;.- 106 cm.

(8)

< 0.2 erg·8- 1 . System

252 2) Macroscopic system (comet nuclei, meteorites, small asteroids) Macro-bodies capture neutrinos onto both Bohr's hydrogen-like levels (outside the body) and Thomson's oscillatory ones (inside the body). Macro-body masses: 10 14 g < M < 10 19 g. Bohr's radius is about:

1 -;- 10 5 cm. The oscillation frequency w intensity

=

J!7fpG, the gravitational radiation (9)

where p is the macro-body density. Let consider the average density of a macro-body p equal to 4 g·cm- l . Then, we obtain the following parameters: fiw = 9.10- 19 eV, 1mb = 10- 104 erg· 8- 1 . It is of interest to note that the rotation curves of galaxies give an aI-most constant velocity v on their periphery, which for v 2 "" G M / R leads to the dependence of dark matter mass Mdm "" R, similar to the dependence of the mass of neutrinos on Bohr's radius L, since L = a~n2, and the total mass of all neutrinos on the nth level is equal to Mn = 2n 2 m v . Hence, we obtain Mn "" L. 6

Conclusion

The graviatom can contain only leptons and mesons. The observable stellar magnitude for graviatom electromagnetic radiation exceeds 23m. Stable graviatoms with baryon constituents are impossible. The internal structure of the baryons, consisting of quarks and gluons, should be taken into account. There occurs a so-called quantum accretion of baryons onto a mini-hole. The whole problem is solvable within the frame-work of quantum chromo dynamics and quantum electrohydrodynamics. Neutrinos can form quantum macro-systems. The description of gravitationally bound macro-systems with neutrinos may be helpful for solving the dark matter problem in the Universe. References

[1] [2] [3] [4]

A.B. Gaina, PhD Thesis, Moscow State University, Moscow, 1980. M.L. Fil'chenkov, Astron. Nachr. 311, 223 (1990). M.L. Fil'chenkov, 1zvestiya Vuzov, Fizika No.7, 75 (1998). Yu.P. Laptev, M.L. Fil'chenkov, Electromagnetic and Gravitational Radiation of Graviatoms/ / Astronomical and Astrophysical Transactions. 2006. v. 25, No.1, p. 33 - 42 [5] H.A. Bethe and E.E. Salpeter, "Quantum Mechanics of One- and TwoElectron Atoms", Springer-Verlag, Berlin, 1957. [6] M. Sher, hep-th/9504257. [7] V.P. Frolov, in "Einstein Col." 1975-1976, Nauka, Moscow, 1978, p. 82-151.

CP Violation and Rare Decays

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SOME PUZZLES OF RARE B-DECAYS A.B. Kaidalov a

Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia Abstract. It is emphasized that a study of rare B-decays provides an important information not only on CKM-matrix, but also on QCD dynamics. It is shown that some puzzles in B-decays can be explained by final state interaction (FSI). The model for FSI, based on Regge phenomenology of high-energy hadronic interactions is proposed. This models explains the pattern of phases in matrix elements of B ---> 'Tr'Tr and B ---> pp decays. These phases play an important role for CPviolation in B-decays. It is emphasized that the large distance FSI can explain the structure of polarizations of vector mesons in B-decays and very large branching ratio of of the B-decay to SeAc.

1

Introduction

This short review of some unusual properties of matrix elements in hadronic B-decays is based on papers with M.l. Vysotsky [1,2J. Detailed information on B-decays, obtained in experiments at B-factories [3J, provide a testing ground for theoretical models. Investigation of rare B-decays and CP violation in these decays provides not only an information on CKM matrix, but also on QCD dynamics both at small and large distances. One of the most interesting and still not solved problems in B-decays is the role of FSI. In this paper I shall demonstrate, that FSI play an important role ill hadronic B-decays and allow one to explain some puzzles observed in rare B-decays. In particular it will be shown that phases due to strong interactions are substantial in some hadronic B-decays. These phases are important for understanding the pattern of CP-violation in rare B-decays. The model for calculation of FSI will be formulated and compared to the data on B ----> 1r1r and B ----> pp decays. The model is based on regge-picture for high-energy binary amplitudes and allows also to explain a pattern of helicity non-conservation in some B-decays to vector mesons. Large distance interactions provide a simple explanation of anomalously large branching ratio of the B-decay to 3cAc. 2

B

----> 1r1r /

B

---->

pp puzzle

The probabilities of three B ----> 1r1r and three B ----> pp decays are measured now with a good accuracy. There is a large difference between ratios of the charged averaged Bd decay probabilities to the charged and neutral mesons.

ae-mail: [email protected]

255

256 It was demonstrated in refs. [1,2J that this difference is related to the difference of phases due to strong interactions for matrix elements of B ---; 1T1T and B ---; pp-decays. The matrix elements of these decays can be expressed in terms of amplitudes with isospin (I) zero and two with phases 6o and 62. To take into account differences in CKM phases for tree and penguin contributions we separate the amplitude with 1=0 into the corresponding parts A o and P. The contributions of P can be determined, using SU(3)-symmetry from decays Bu ---; KO* p+ and Bu ---; K01T+ [4] and turn out to be rather small compared to tree contributions. Note, however, that P determines magnitudes of direct CPV in hadronic decays. If we neglect the penguin contribution, then the difference of phases is expressed in terms of the branching ratios as follows

cos(oo -

on = -J3

B+_ - 2B oo

4 Jf!B+oJB+-

+ ~ I!.L B+o T+,

+ Boo - ~f!B+o

(2)

Using experimental information on branching ratios of B ---; 1T1T-decays [3J we obtain 160 - 5~1 = 48°. Penguin contributions to Bik do not interfere with tree ones because CKM angle ex = 1T - (3 - 'Y is almost equal to 1T /2. With account of P-term we get:

(3) This value agrees with the result of analysis of ref. [5J:

6o - 62 = 40° ± 7° ,

(4)

Thus the difference of phases of matrix elements with 1=0 and 1=2 is not small in sharp contrast with factorization approximation often used for estimates of heavy mesons decays. For B ---> pp-decays we obtain in analogous way: (5)

This phase difference is smaller than for pions and is consistent with zero. The fact that phases due to FSI are in general not small for heavy quark decays is confirmed by other D and B-decays. The data on D ---> 1T+1T-, D ---> 1T01TO and D± ---> 1T±1TO branching ratios lead to [6J:

(6) The last example is B ---; D1T decays. D1T pair produced in B-decays can have I = 1/2 or 3/2. From the measurement of the probabilities of B- ---> D01T-, BO ---> D-1T+ and BO ---; D01To decays in paper [7J the FS1 phase difference of these two amplitudes was determined: (7)

257

Thus experimental data indicate that the phases due to FSI are not small for heavy mesons decays.

3

Calculation of the FSI phases of B plitudes

--+

7r7r and B

--+

pp decay am-

Let me remind that for K --+ 7r7r decays there are no inelastic channels, MigdalWatson (MW) theorem is applicable and strong interaction phases of S-matrix elements of K --+ (27r) I decays are equal the phases of the corresponding 7r7r --+ 7r7r scattering amplitudes at E = m K. For B-mesons there are many opened inelastic channels and MW theorem is not directly applicable. Serious arguments that strong phases should disappear in the MQ --+ 00 limit were given by B.J. Bjorken [8J. He emphasized the fact that characteristic configurations of the produced in the decay light quarks have small size"" l/MQ and FSI interaction cross sections should decrease as 1/ M~. Similar arguments were applied in the analysis of heavy quark decays in the QeD perturbation theory [9J. These arguments can be applied to the total hadronic decay rates. For individual decay channels (like B --+ 7r7r), which are suppressed in the limit MQ --+ 00 the situation is more delicate. However even in these situations the arguments of Bjorken that due to large formation times the final particles are formed and can interact only at large distances from the point of the decay seem relevant. On the other hand a formal analysis of different classes of Feynman diagrams, including soft rescatterings [10,11]. show that the diagrams with pomeron exchange in the FSI-amplitudes do not decrease as MQ increases. Same conclusions follow from applications of generalizations of MW-theorem [12,13J. In the process of analysis of FSI in heavy mesons decays it is important to understand a structure of the intermediate multi particle states. It was shown in ref. [2] that the bulk of multiparticle states produced in heavy mesons decays has a small probability to transform into two-meson final state and only quasi two-particle intermediate states XY with masses M.k(Y) < MBA QCD « M~ can effectively transform into the final two-meson state. In refs. [1, 2J in calculation of FSI effects for B --+ 7r7r and B --+ pp decays we considered only two particle intermediate states with positive G-parity to which B-mesons have relatively large decay probabilities. Alongside with 7r7r and pp there is only one such state: 7ral. I shall use Feynman diagrams approach to calculate FSI phases from the triangle diagram with the low mass intermediate states X and Y . Integrating over loop momenta d 4 k one can transform the integral over ko and k z into the integral over the invariant masses of intermediate particles X and Y

J

dkodk z =

2~~

J

dsxdsy

(8)

258 and deform integration contours in such a way that only low mass intermediate states contributions are taken into account while the contribution of heavy states being small is neglected. In this way we get:

I M7r7r

=

M(O)I(" XY U7r XU7r Y

+ 2'TJ=O) XY->7r'lf

,

(9)

where Mflf are the decay matrix elements without FSI interactions and Tiyo....7r7r is the J = 0 partial wave amplitude of the process XY -+ 'lor (TJ = (SJ - 1)/(2i)) which originates from the integral over d2 kJ... For real T (9) coincides with the application of the unitarity condition for the calculation of the imaginary part of M while for the imaginary T the corrections to the real part of M are generated. This approach is analogous to the FSI calculations performed in paper [14]. However in [14] 2 -+ 2 scattering amplitudes were considered to be due to elementary particle exchanges in the t-channel. For vector particles exchanges s-channel partial wave amplitudes behave as sJ-1 cv sO and thus do not decrease with energy (decaying meson mass). However it is well known that the correct behavior is given by Regge theory: s"'i(0)-1. For p-exchange Cfp(O) r::::J 1/2 and the amplitude decrease with energy as 1/.fS. This effect is very spectacular for B -+ DD -+ 'WIT chain with D*(D2) exchange in t-channel: CfD*(O) r::::J -1 and reggeized D* meson exchange is damped as S-2 r::::J 10- 3 in comparison with elementary D* exchange (see for example [13]). For 71'-exchange, which gives a dominant contribution to pp -+ 71'71' transition (see below), in the small t region the pion is close to mass shell and its reggeization is not important. Note that the pomeron contribution does not decrease for MQ -+ 00, however it does not contribute to the difference of phases IJg -J~ I which we are interested in. So this phase difference is determined by the secondary exchanges p, 71' and it decreases at least as 1/MQ for large MQ in accord with Bjorken arguments. For phases J and J~ I separately the pomeron contribution does not cancel in general. If Bjorken arguments are valid for these quantities it can happen only under exact cancellation of different diffractively produced intermediate states and it does not happen in the model of refs. [1,2]. Let us calculate the imaginary parts of B -+ 71'71' decay amplitudes. In the amplitude pp -+ 71'71' of pp intermediate state in eq.(9) the exchange by pion trajectory in the t-channel dominates. It is determined by the well known constant gP->7r7r' This contribution is the dominant one for B -+ 71'71' decays due to a large probability of B -+ pp-transition. On the contrary 71'71' intermediate state plays a minor role in B -+ pp-decays. In description of 71'71' elastic scattering amplitudes in eq.(9) contributions of P, f and p regge-poles was taken from ref. [15]. Finally 71'a1 intermediate state should be accounted for. Large branching ratio of Bd -+ 71'±ai-decay ( Br(Bd -+ 71'±ai) = (40 ± 4) * 10- 6 ) is partially compensated by small P71'al coupling constant (it is 1/3 of p71'71' one). As a result the contributions of 71' a 1

o

259 intermediate state (which transforms into 7r7r by p-trajectory exchange in tchannel) to PSI phases equal approximately that part of 7r7r intermediate state contributions which is due to p-trajectory exchange. Assuming that the sign of the 7ral intermediate state contribution into phases is the same as that of elastic channel and taking into account that the loop corrections to B -. 7r7r decay amplitudes leads to the diminishing of the (real) tree amplitudes by ~ 30% we obtain: (10) The accuracy of this prediction is about 15°. For pp final state analogous difference is about three times smaller, og - o~ ~ 15°. Thus the proposed model for FSI allows us ti explain the B -. 7r7r / B -. pp puzzle. 4

Direct CPV in B -. 7r7r-decays and phases of the penguin contribution

The direct CP-violation parameter C+_ in B -. 7r7r-decays is proportional to the modulus of the penguin amplitude and is sensitive to the strong phases of Ao, A2 and penguin amplitudes. So far we have discussed phases of the amplitudes Ao, A 2. The penguin diagram contains a c-quark loop and has a nonzero phase even in the QCD perturbation theory. It was estimated in ref. [lJ and is about 10°. Note that in PQCD it has a positive sign. Let us estimate the phase of the penguin amplitude op considering charmed mesons intermediate states: B -. DD, 15* D, DD*, 15* D* -. 7r7r. In Regge model all these amplitudes are described at high energies by exchanges of D*(D 2)-trajectories. An intercept of these exchange-degenerate trajectories can be obtained using the method of [16J or from masses of D* (2007) - land D2 (2460) - 2+ resonances, assuming linearity of these Regge-trajectories. Both methods give aD- (0) = -0.8 -;- -1 and the slope a~" ~ 0.5GeV- 2 . The amplitude of D+ D- -. 7r+7r- reaction in the Regge model proposed in papers [17J can be written in the following form: 2

T DD-->1r7r (s, t) = -

g; e-

i1r

<>(t)r(l - aD" (t)) (S/ Sed)<>D" (t)

,

(11)

where r(x) is the gamma function. The t-dependence of Regge-residues is chosen in accord with the dual models and is tested for light (u,d,s) quarks. According to [17J Sed ~ 2.2GeV 2 . Note that the sign of the amplitude is fixed by the unitarity in the t-channel (close to the D*-resonance). The constant is determined by the width of the D* -. D7r decay: g6/(167r) = 6.6. Using (9) and the branching ratio Br(B -. DD) ~ 2 . 10- 4 we obtain the imaginary part of P and comparing it with the contribution of Pin B -. 7r+7r- decay probability we get Op ~ -3.5°.

95

260

The sign of op is negative - opposite to the positive sign which was obtained in perturbation theory. Since D V-decay channel constitutes only ~ 10% of all two-body charm-anticharm decays of Bd-meson, taking these channels into account we easily get (12) which may be very important for interpretation of the experimental data on direct CP asymmetry. It was shown in ref. [2], that assuming that phases satisfy to the conditions: 00 - 02 = 37°,02 > 0 and 0p > 0, it is possible to obtain the following inequality

C+_ > -0.18.

(13)

The experimental results obtained by Belle [18] and BABAR [19] are contradictory (14) c2~le = -0.55(0.09) , C2~BAR = -0.21(0.09), Belle number being far below (13). For non-perturbative phase of the penguin contribution (12) the value of theoretical prediction for C+- can be made substantially smaller and closer to the Belle result. 5

Polarizations of vector mesons in B --' VV -decays

Short distance contributions to vector meson production in B-decays lead to a dominance of the longitudinal polarization of vector meson. This is a general property valid in the large MQ- limit due to helicity conservation for vector currents and corrections should be '" M~ / M~. It is satisfied experimentally in B --' p+p_ decays, where the contribution of longitudinal polarization of p mesons is h = f L/f = 0.968 ± 0.023. On the other hand there are several B-decays to vector mesons, where longitudinal polarizations give only about 50% of decay rates. For example: for B+ --' K*op+ h = 0.48 ± 0.08, BO --' K*opo h = 0.57 ± 0.12, B+ --' ¢K*+ h = 0.50 ± 0.07, BO --' ¢K*o h = 0.491 ± 0.032 [3]. This is a real puzzle if only short distance dynamics for these decays is invoked. First let us note that in all decays, where f L ~ 50% penguin diagrams give dominant contribution. In this case a large contribution to the matrix elements of decays comes from DDs(D* V., DV;, .. ) intermediate states, which have large branching ratios. The amplitude of the binary reaction DVs --+ VV at high energies is dominated by the exchange of D* -regge trajectory and according to general rules for spin-structure of regge vertices (see for example [20]) final vector mesons are produced purely transversely polarized. Thus we expect a large fraction of transverse polarization of vector mesons in these decays. A value of h is sensitive to intercept of D*-trajectory [21]. If the

261

penguin contribution in the decays indicated above is dominant in the SU(3) limit we have:

and h in all these decays should be the same. These predictions agree with experimental data [3].

6

Puzzle of charm-anticharm baryons production

Large probability of B-decay to Ae3e has been observed recently: Br(B+ -; At3~ rv 10- 3 ) [3]. It is surprisingly large compared to the branching of Bdecay to Atf5 = (2.19 ± 0.8)10- 5 • From PQCD point of view both processes ~re described by similar diagrams with a substitution of ud (for p) by cs (for Be) and phase space arguments even favor p-production. On the other hand from the soft rescatterings point of view large DDs(.D* D s , D D; , .. ) intermediate states, considered in the previous section, can play an important role in B+ -; At3~-decays. For Atp final states corresponding two-meson intermediate states have smaller branchings and, what is even more important, have different kinematics. For DDs, .. intermediate states the momentum of these heavy states is not large (p ~ 1.8GeV) in B rest frame and all light quarks (u, d, d, s are slow in this frame. Final At3~ are also rather slow in the B-rest frame and thus all quarks have large projections to the wave functions of the final baryons. On the contrary for 7r D, pD, .. intermediate states in Atp-decays momenta of ii, d- quarks in light mesons are large and projections to the wave functions of final baryons have extra smallness. The resulting suppression can be estimated in regge-model of ref. [17] with nucleon trajectory exchange in the t-channel and is rv 10- 2 in accord with experimental observation,

7

Conclusions

FSI play an important role in two-body hadronic decays of heavy mesons. Theoretical estimates with account of the lowest intermediate states give a satisfactory agreement with experiment and provide an explanation of some puzzles observed in B decays.

Acknowledgments This work was supported in part by the grants: RFBR 06-02-17012, RFBR 06-02-72041-MNTI, INTAS 05-103-7515, Science Schools 843.2006.2 and by Russian Agency of Atomic Energy.

262

References

[1] [2] [3] [4] [5] [6] [7] [8] [9]

A.B. Kaidalov, M.I. Vysotsky, Yad. Fiz. 70,744 (2007). A.B. Kaidalov, M.l. Vysotsky, Phys. Lett. B652, 203 (2007). HFAG, http://www.slac.stanford.edu/xorg/hfag. M. Gronau, J.L. Rosner, Phys. Lett. B595, 339 (2004). C.-W. Chiang, Y.-F. Zhou, JHEP 0612,027 (2006). CLEO Collaboration, M. Selen et al., Phys. Rev. Lett. 71, 1973 (1993). CLEO Collaboration, S. Ahmed et al., Phys. Rev. D66, 031101 (2002). J.D. Bjorken, Nucl. Phys. (Proc. Suppl.) Bll, 325 (1989). M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, Nucl. Phys. B606, 245 (2001). [10] A. Kaidalov, Proceedings of 24 Rencontre de Moriond "New results in hadronic interactions", 391 (1989). [11] J.P. Donoghue, E. Golowich, A.A. Petrov and J.M. Soares, Phys. Rev. Lett. 77, 2178 (1996). [12] M. Suzuki, L. Wolfenstein, Phys. Rev. D60, 74019 (1999). [13] A. Deandrea et al., Int. J. Mod. Phys. A21, 4425 (2006). [14] H-Y. Cheng, C-K Chua and A. Soni, Phys. Rev. D71, 014030 (2005). [15] KG. Boreskov, A.A. Grigoryan, A.B. Kaidalov, I.I. Levintov, Yad. Fiz. 27,813 (1978). [16] A.B. Kaidalov, Zeit. fur Phys. C12, 63 (1982). [17] KG. Boreskov, A.B. Kaidalov, Sov.J.Nucl.Phys. 37, 109 (1983). [18] H.Ishino, Belle, talk at ICHEP06, Moscow (2006). [19] B.Aubert et ai, BABAR Collaboration, hep-ex/0703016 (2007). [20] A.B. Kaidalov, B.M. Karnakov, Yad. Fiz. 3,1119 (1966). [21] M. Ladissa, V. Laporta, G. Nardulli and P. Santorelli, Phys. Rev. D70, 114025 (2004).

MEASUREMENTS OF CP VIOLATION IN b DECAYS AND CKM PARAMETERS Jacques Chauveau a LPNHE,IN2P3/CNRS, Univ. Paris-6, case courrier 200, -4 place Jussieu, F-75252 Paris Cedex as, Prance Abstract. After a brief review of CP violation phenomenology in the standard model I depict recent measurements of the CKM angles. The emphasis is on the latest determinations of the angle (3 = 1 made using amplitude analyses of threebody final states. The results of the CKM fits to date are used to summarize the talk and put the subject into perspective.

1

CP Violation in the Standard Model

The B factories have established that the Standard Model (SM) accommodates CP violation via the Cabibbo Kobayashi Maskawa (CKM) formalism [1] where the couplings of the W boson to quarks include the elements of the fundamental unitary CKM matrix V. For three generations, V depends on four real parameters, one of which is an irreducible imaginary part that induces opposite sign weak phases for CP-conjugate transitions of quarks and antiquarks. The Wolfenstein parameterization [2], 1 - >-2/2 ->-

A>-3(1 - p - iT})

is the first term of an expansion in the small parameter A = sin Be (Be is the Cabibbo angle). By changing p and T} to 75 and 'fj defined in reference [3], an approximation good to 0(A6) is obtained which is widely used. The unitarity relation between the first and third columns of V is conveniently drawn on Figure 1 as a triangle, the Unitarity Triangle (UT), in the complex plane.

Figure 1: The CKM Unitarity Triangle UT. At 0(>-6), the apex coordinates are (p, 7i). Because of CP violation, the triangle is not squashed to the real axis. Its angles are constrained by CP violation experiments.

1\ .---------

A I

V.nv~

iVcdV.t

The apex of the UT has coordinates (75, Yj). Present measurements vindicate the SM with precisions of 0.5% on A, 2% on A, 20% on 75 and 7% on 'fj. These numbers are upper limits to possible New Physics corrections to the flavor sector of ae-mail: [email protected]

263

264 the SM. Such effects are actively searched for in an experimental programme seeking to overconstrain the position of the UT apex from all relevant measurements. In this talk I concentrate on the CP observables and the associated UT angles measurements. On August 21, 2007 the BABAR and Belle experiments had integrated luminosities of 469 and 710 fb -1. A B meson decay B ----> f ex(aJ hibits CP violation when (at least) two paths connect the initial and fib---IC nal states. One distinguishes three kinds of CP violating effects. Direct CP violation decay processes are such that the particle decay rate r(B ----> 1) differs from the antipar(bJ W ticle rate r(B ----> 1). Both neutral and charged B mesons can show b direct CP violation. CP violation in mixing stems from the misalignment of the neutral B CP eigenstates and propagation eigenstates (BH,L ex pBo ± qED, with masses mH,L). This effect which is domiFigure 2: Tree (a) and Penguin (b) b-quark nant for the neutral kaons is small decay diagrams. for the B mesons. Most important is the third category: CP violation in the interference between mixing and decay. The simplest case is a process where a neutral B decays to a CP eigenstate fop. The two paths, B O ----> fop and B O ----> EO ----> fop interfere with differing patterns for initial B O and EO resulting into a time dependent CP asymmetry. For golden modes, these asymmetries do not depend on strong phases and give clean experimental access to UT angles. Neglecting electroweak Penguin diagrams, a b-quark non leptonic decay amplitude (Table 1) is the sum of two terms which can be Tree-like or Penguin-like (Figure 2) and whose relevance is determined by the power of ..\ of their CKM factors. A decay channel where one term is dominant is a golden mode. At the B factories the B mesons are produced exclusively in BE pairs from the Y(4S) decays. The pairs of neutral B mesons are produced in an entangled quantum state. We select neutral pairs where one B mesod' decays to f and the other to a flavor specific mode. We measure 6.t the time difference between the two decays. The time dependent CP asymmetry Af is an oscillatory function of 6.t with a frequency given by the mass difference I::!.m = mH - mL: rBO--->f(l::!.t) - rB°--->f(6.t)

Af(l::!.t) bf

== r-

B0--->(

(6.) t

+ r B0--->f (6.t )

=

. Sf sm(6.m6.t) - C f cos(l::!.m6.t),

is the CP eigenstate, we drop the CP subscript for brevity.

265 'Sm>'t C 1-1>'11 2 . 9:At. . h S j -- 2 1+I>'tI 2 ' were j = 1+1>'tI 2 ' and the ratIo Aj = p At whIch compares mixing and decay amplitudes (Aj, Aj = A(BO, EO --> 1) have been introduced. For a golden mode in the 8M: Cj = 0 (no direct CP violation) and Sj = -7}j sin

ccs) and Penguin-like in the other (b --> sss). New Physics can contribute to the latter via virtual new particles in the loop.

Table 1: CKM structure of non leptonic b decay amplitudes. The amplitude for a b --> qlq-2q3 transition is written in terms of T and/or P amplitudes with the CKM factors shown explicitly. The power of >- governing the first and second terms are given. A golden channel leads to a pure measurement of a CKM phase or UT angle 'P. Only effective phases are accessible from the non golden channels. quark process Aces"" Veb Ve~Tees + Vub V':sPs Asss "" Veb Ve~Ps + Vub V':sP; Aced"" Veb Vcd Teed + "lltb '-"t'dPd Auud "" Vub V,:dTuud + Vtb Vt~Pd

2

1st term

2nd

>-~

>-4 >-4 >-3 >-3

>-2 >-3 >-3

example golden golden

Jj1/J K S,L ¢K£

D+D71"+ 71"- , pO pO

'P f3 f3 f3eff aeff

Recent Measurements of the angle /3

The most recent results are tabulated in reference [3]. Here I focus on the measurements of the b --> ccs and b --> sss channels, in particular on the golden modes. Over the last few years, much speculation was entertained by the observation that most Penguin dominated b --> sss final states were measured with sin2/3ejj lower than those from Tree dominated b --> ccs (Figure 3). A simple minded average of all the Penguin measurements fell lower than the Tree measurement with almost 3 standard deviation significance (Figure 4-b)). Figure 3 shows the latest measurements of A3'p(~t) with the golden channels B --> charmonium KO by BABAR [4] and J/1jJKo by Belle [5]: S S

= 0.714 ± 0.032 ± 0.018, C = 0.049 ± 0.022 ± 0.017 (BABAR), = 0.642 ± 0.031 ± 0.017, C = 0.018 ± 0.021 ± 0.014 (Belle),

where the first uncertainties are statistical and the second ones systematic. The average over all charmonium KO measurements is sin 2/3 = 0.678 ± 0.025 or, in the first quadrant of the (p,7}) plane: /31 = (21.3 ± 1.0)0 or /32 = (67.8 ± 1.0)0, /31 being favored by several measurements each with small statistical significance [6]. New this summer are the time dependent amplitude (Dalitz) analyses on Penguin golden channels K~h+ h- [7,8], where h refers to a 7r or a K meson.

888

Penguin channels.

direct measurements of the f3 (not a function final states as well as the non-resonant three"ll'OICl.l1lC, I focus on the After

are paraman isobar model. Each term is a ",,"ull.teA COmI)lCX (isobar) coefficient whose argument incor5 a)-c) show the two-body invariant and the components from >I",.rn11~1p·'.r"'" for the

enriched fit distributions for the time dependent Dalitz analysis of . The full fit distributions are superimposed over the data points three invariant mass spectra of the Dalitz plot. The shaded areas correspond to the background components, and the signal. Vetoes create holes the D and J /'Ij; The pO and fo peaks are in the 7r+7rThe corresponding dependent CP asymmetries for d) and are shown at the bottom of the

which makes the minded average sin over with the measurements from the Tree processes the result from the above pre4 to the older dataset 4

'AJ.tll,Jel>L,lUJlC

and measured B meson related There is no evidence for direct OP violation from the measured time

268 dependent CP asymmetries and no compelling hint for New Physics. 3

Recent Progress on the other UT Angles

Here, I have chosen to highlight the recent progress made on the GronauLondon (GL) analysis [9] of the B -+ pp channels. This charmless b -+ uud decay is not a golden mode (Table 1). The Penguin pollution introduces a phase shift on the determination of the UT angle a and one measure ael I instead of a. The GL method exploits the SU(2) symmetry to combine all charge states in B -+ pp and determine a-aell up to trigonometric ambiguities. It has been appreciated for some time that since the branching fraction for BO -+ pO pO is measurable with fair accuracy, the GL triangle can be constructed more precisely than in the founding case B -+ 7r7L Furthermore with four charged pions in the final state, the decay vertex can be accurately reconstructed and the time dependent CP asymmetry measured, an impossible feat for B -+ 7r 0 7r 0 . With high statistics it has been possible this year to measure Acp(t) for the longitudinally polarized pOpo pairs [10]. Including these results into the GL fit yields the confidence level profile for a - ael I shown on Figure 6. It is nomore fiat as was the case when no CP asymmetry measurement was included. Some discrimination between the mirror solutions already observed with previous spin-averaged measurements of Acp(t) can be seen. There is hope that an accurate determination of a will be obtained with the full data samples of the B factories.

Figure 6: Exclusion confidence level scan for Q QeJ J. The red (solid) curve corresponds to the recent measurement of the time dependent asymmetry for longitudinally polarized pO pO pairs in neutral B decays [10]. The one and two-sigma exclusion levels are shown as horizontal intermittent lines.

4

-!

~

1 ••••••. without C~Q and S~

0.8

with Cro and without S~

0 .•

0.4

0.2

·10

10

20

30

40 «-
A flavor of the CKM fits as of Summer 2007

Two groups with different cOllceptions on statistics, CKMfitter [11] and UTfit [12], perform global fits of the CKM matrix using all the relevant experimental data. As of Summer 2007, they both find the same consistency region

269 shown in Figure 7 and summarized in Table 2. The most constraint on the VT apex from CP violation comes from the sin measurements. The VT apex is also strictly bounded by the neutral B mesons oscillations. In my presentation I showed that the determined by various subsets of measurements are consistent with each other. There is no constraint from the observed direct violation in Qh';n,ypy,t

2007 results for the CKM fits fitter on the left and UTfit on the Both groups find consistency with the in a region that has been highlighted on the figures and is quantitatively described in Table 2.

Table Global CKM fits from the UTfit and CKMfitter

0.2265 ± 0.0008. The comparison of CKMfitter fit confidence inand three standard tervals deviations shows that there arc non Gaussian uncertainties, particularly for angles Oi and 'Y which more derived from the fit than experimentally determined.

UTfit

(10") 0.147 ± 0.029 0.342 ± 0.016 91.2

± 5.4

0.690 0.023 66 ±6.4

the B factories have validated the effects have not shown up. final states. Final results from 2009-2012. The TeVatron machine

270 scope of my talk) has had a significant impact on flavor physics. In the near future the LHCb experiment will become a major player in this field. New e+e- colliders (Super B, Super Flavor Factory) if built, might produce two order of magnitude bigger data samples than those from BABAR and Belle. Acknow ledgments

I am very grateful to the BABAR Collaboration on behalf of whom I gave this talk. I acknowledge the financial support of LPNHE, the University Pierre et Marie Curie, Paris-6 and the CNRS/IN2P3. My warm thanks to Prof. A. Studenikin and his team for the organization of the Conference and memorable events around it as well as their help and understanding while I prepared the writeup. References

[1] M. Kobayashi and T. Maskawa, Prog. Thear. Phys. 49, 652 (1973); N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963). [2] L. Wolfenstein, Phys. Rev. Lett. 51, 1945, (1983). [3J Heavy Flavor Averaging Group (HFAG), E. Barberio et al., arXiv:0704.3575v1 [hep-ex], and http://www.slac.stanford.edu/xorg/hfag/. [4] BABAR Collab., B. Aubert et al., Phys. Rev. Lett. 99,171803 (2007). [5] Belle Collab., K.-F. Chen et al., Phys. Rev. Lett. 98, 031802 (2007). [6J BABAR Collab., B. Aubert et al., Phys. Rev. D71 032005 (2005); BABAR Collab., B. Aubert et al., arXiv:0708.1549v2 [hep-exJ. Belle Collab., R. Itoh et al., Phys. Rev. Lett. 95 091601 (2005); Belle Collab., J. Dalseno et al., Phys. Rev. D 76, 072004 (2007). [7] BABAR Collab., B. Aubert et al., Phys. Rev. Lett. 99, 161802 (2007). [8] BABAR Collab., B. Aubert et al., contributed to Lepton-Photon 2007, arXiv:070S.2097v1 [hep-exJ. [9] M. Gronau, D. London, Phys. Rev. Lett 65, 3381 (1990). [10J BABAR Collab., B. Aubert et al., contributed to Lepton-Photon 2007, arXiv:0708.1630v2 [hep-exJ [l1J J. Charles et al., Eur.Phys.J. C 41, 1 (2005), and http://ckmfitter.in2p3.fr/. [12] M. Bona et al., JHEP 0610, OSl (2006), and http://www . utfi t. org/. [13J BABAR Collab., B. Aubert et al., Phys. Rev. Lett. 93131801 (2004).

EVIDENCE FOR Do_If MIXING AT BaBar M. V. Purohit a

Dept. of Physics and Astr., Univ. of S. Carolina, Columbia, SC 29212, USA On behalf of the BaBar collaboration. Abstract. We begin with a brief discussion of DO_Do mixing and then present results from the BaBar experiment, with a focus on recent results from DO -+ K+7[- decays where DO_Do mixing was first observed, and which remains the most sensitive mode to mixing. We also present results from the ratios of lifetimes in singly Cabibbo-suppressed (SCS) modes to the lifetime in the DO -+ K-7[+ decay mode, and results on CP violation searches in DO decays. Finally, we present DO-lf mixing results from a study of DO -+ K-7[+7[° decays. We conclude with a summary of these results and some interpretation based on present theoretical predictions.

1

-dl

The phenomenology of DO-D MIXING

Consider a neutral meson system, such as DO and If, written generically as MO and At. These mesons are unstable and decay weakly. Another basis is formed by the eigenstates of the full Hamiltonian written as H = M - if /2 where M and f are the mass and decay matrices respectively. Defining f and m to be the mean width and mass, respectively, of the mass eigenstates IM1 ) and 1M2 ), and Lif and Lim to be their differences, respectively, the decay amplitude for an initial IMO) state is given by

where,

=f

- 2im, Li,

= (y + ix)f, and x

_ Lif /2

Lim f '

x=--

-

and yare given by

y=-f-'

The net effect of this time dependence is that the wrong sign (WS) rate in, e.g., DO -+ K+7r- decays, assuming CP conservation and x, y « 1 is given by

where x' and y' are the coordinates x, y rotated by an angle <5, the strong phase difference between doubly Cabibbo suppressed (DCS) and Cabibbo favored (CF) decays. These quantities receive contributions from short- and long-distance diagrams such as the well-known quark-level box diagram for the short-distance ae-mail: [email protected]

271

272 •

1600 1400

Data

o

a)

Mixing fit

~Randomn,

a. 1200

•

~ 1000

t22J Combinatorial

~

Misrecon. CO

.... ,. No mixing fit

800

~

1 t (ps)

2

3

4

Figure 1: (a) Lifetime distribution of wrong-sign events with various contributions shown. (b) Residuals overlayed with fits with and without mixing, showing clearly that the mixing fit is preferred.

case and a diagram in which K K form an intermediate state between the DO and the lfin the long-distance case. In the Standard Model, the quantities x and yare predicted to be small compared to 1. Using the quark-level box diagrams alone yields about 10- 5 for x, which translates into a mixing rate Rmix '" 10- 10 . Standard model estimates of long distance effects give x '" 10- 3 and y '" 10- 2 . New physics models also predict a wide range of values for x and y, making it difficult to interpret DO-lf mixing. [4]

2

Studies of DO-lf Mixing by the BaBar Collaboration

The principle of measuring DO-lf mixing is simply to tag the flavor of the Dmeson at birth using the charge of the slow pion in D*+ --+ D07f+ decays, and to tag the flavor at decay time using the charge of the kaon in DO --+ K-7f+ decays. When the flavors are the same, the decays are termed right sign (RS) decays, otherwise they are called WS decays. Both DCS decays and mixing produce WS decays. These can be separated by their different lifetime dependence. Using 384 fb- 1 of DO --+ K-7f+ data, we obtain 1,141, OOO± 1,200 RS events and 4030 ± 90 WS events. A good fit to the lifetime dependence does not result unless we allow for mixing, as shown in Fig. 1. Contour plots for the mixing parameters are shown in Fig. 2. The (x,2, y') = (0,0) point is clearly excluded at roughly the 5a level using only statistical errors and at roughly the 4a level when systematic errors are

273

-20

-1.0

-0.5

0.5

1.0

Figure 2: Contour plots at various confidence levels, including systematic errors for the mixing fit to DO ---t K-1[+ data. The solid line is the fit allowing for mixing, while the dotted line is the fit assuming no mixing; the no-mixing fit is clearly excluded.

included. When we plot the ratio of WS to RS events as a function of time we verfiy that the signal behaves as a mixed signal should, with a component that increases with proper decay time. Finally, we have fit the D*+and D*-samples separately and find essentially the same parameters in each case. There is no evidence for CP violation. Our overall conclusion from this study of DO - t K'f7r± decays is that X,2

= (-0.22 ± 0.30 ± 0.21) x 10- 3 ,

y' = (9.7 ± 4.4 ± 3.1) x 10- 3 ,

where the first error is statistical and the second is systematic. Next, we turn to Do-If mixing in the ratio of lifetimes of DO - t K+ K-, 7r+7rdecays to DO - t K-7r+ decays. The DO - t K-7r+ decays are CP-mixed, while the DO - t K+ K-, 7r+7r- decays are CP-even. We denote the lifetime in the K-7r+ decays as TKrr, and the lifetimes in the K+ K- and 7r+7r- decays as Thh. Lifetimes from DO decays are denoted T+, while those from If decays are called T-. Experimentally, we determine the quantities TKrr Thh/

YcP = -(- , - 1,

where If CP is conserved, the following relations hold: YcP = Y,

~Y=O

274

Decay time fits were done to determine the lifetimes and thereby YcP and ~Y. The important result is that the lifetime in the K 7r mode is indeed different from that in the CP-eigenstate modes. Using decays in which the neutral Dmeson flavor is tagged by the charge of the slow pion, we find using 384 fb- 1 of data that YcP

= [1.24±0.39(stat)±0.13(syst)],

~y =

[-0.26±0.36(stat)±0.08(syst)],

which are in good agreement with Belle's 540 fb- 1 measurement. [1] Both BaBar and Belle exclude YC P = 0 at the > 30' level. We also search for CPV in DO -+ K+ K-, 7r+7r- decays. For CP-violation to occur, there must be at least two amplitudes with different strong and weak phases. In the standard model (SM), CPV asymmetries of the order of [10- 3 _ 10- 2 ]% are expected in these modes. [2J We determine the CP asymmetries to be afS/f = [0.00 ± 0.34(stat) ± 0.13(syst)J%, a cp = [-0.24 ± 0.52(stat) ± 0.22(syst)J% Finally, we compare the DO -+ K 7r7r 0 (WS) decay Dalitz plot with the DO -+ K 7r7r 0 (RS) decay Dalitz plot. The lifetime dependence of DO -+ K 7r7r0 (WS) decays is given by

ry(SI2' S13, t) = e- rt

{IAyI2 + IAyilAyl

[y" cos by - x" sin by] (rt) + ( Xff21yff2) IAyI2(rt)2}

where the first term describes DCS decays, the last term mixed decays and the remaining term the interference between the two. We find that x" = [2.39 ± 0.61(stat) ± 0.32(syst)J%, y" = [-0.14 ± 0.60(stat) ± OAO(syst)J%, Rmix = [2.9 ± 1.6(stat+syst)J x 10- 4 ,

in agreement with the world average.

3

Conclusions

The website http://www.slac.stanford.edu/xorg/hfag/ charm/index.html displays the latest results from BaBar combined with those from other experiments. We can state here that these various results are consistent with each other. Since the announcement of BaBar's result, various theoretical publications have emerged to interpret Do-If mixing. These use our results to constrain various models, such as certain SUSY models, Higgs models and excited vector boson models. One publication for instance, claims that "light non-degenerate squarks are unlikely to be observed at the LHC." [3J A more comprehensive recent review of a large list of models has also been published [4] and should be referred to for further details.

275

Acknowledgments Assistance of the BaBar collaboration, our PEP-II colleagues, and SLAC is deeply appreciated.

References [1] [2] [3] [4]

M. Staric et al. (Belle Collab.), Phys. Rev. Lett. 98,211803 (2007). F. Bucella, Phys. Rev. D51, 3478 (1995). M. Ciuchini, et al., Phys. Lett. B655, 162 (2007). E. Golowich, et al., Phys. Rev. D76, 095009 (2007).

SEARCH FOR DIRECT CP VIOLATION IN CHARGED KAON DECAYS FROM NA48/2 EXPERIMENT S.Balev a

Joint Institute for Nuclear Research, 141980 Dubna, Russia Abstmct.A high precision measurement of the asymmetry in Dalitz plot slopes Ag = (g+ - g-)/(g+ - g-) was performed by the experiment NA48/2. The obtained results, A~ = (-1.5 ± 2.1) . 10- 4 and A~ = (1.8 ± 1.8) . 10- 4 are based on record statistics - rv 3.1 .10 9 K± --t 7r±7r+7r- decays and rv 9.1 . 10 7 K± --t 7r±7r 0 7r 0 decays, correspondingly. The precision of the measurement is one order of magnitude better than the previous experiments and is limited by the statistical error.

1

Introduction

CP violation plays an important role in elementary particles physics: on one hand, its studies allow to make precise tests to the Standard Model (SM) and to search for new physics; on the other hand, this phenomenon is in the core of the baryogenesis according to modern cosmological models. Therefore, studies of each possible manifestation of CF-violation are tasks of fundamental importance. In kaons, besides the €' / € parameter in KO --+ 1m decays, promising complementary observables are the rates of GIM-suppressed rare kaon decays proceeding through neutral currents, and the asymmetry between K+ and Kdecays to three pions.

The K± --+ 37r matrix element can be parameterized by a polynomial expansion in two Lorentz-invariant variables u and v:

1M (u, v) 12 ex 1 + gu + hu 2 + kv 2 + ... ,

(1)

Ihl,lkl « Igl are the slope parameters, u = (83 - 80)/m;, v = (81 82)/m;, where m7r is the charged pion mass, 8i = (PK - Pi)2, 80 = L 8i/3 (i = 1,2,3), PK and Pi are kaon and i-th pion 4-momenta, respectively. The index i = 3 corresponds to the odd pion. The parameter of direct C P violation is usually defined as: Ag = (g+ - g-)/(g+ + g-), where g+ is the linear where

coefficient in (1) for K+ and g- - for K-. A deviation of Ag from zero is a clear indication for direct CP violation. The experimental precision before NA48/2 for both decay modes, K± --+ 7r±7r+ 7r- and K± --+ 7r±7r0 7r 0 , is at the level of 10- 3 [1], while SM predictions for Ag are below 10- 4 [2]. However, some theoretical calculations involving processes beyond the SM [3] predict substantial enhancements of the asymmetry, which could be observed in the present experiment.

276

277

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t

o

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10cm Vacuum He tank + tank Spectrometer /....,20=O:----~----250=-'m I

Figure 1: Schematic view of the NA48/2 experiment: beam line (TAX17,18: motorized beam dump/collimators used to select the momentum of the K+ and K- beams; DFDF: focusing quadrupoles; KABESl-3: beam spectrometer stations), decay volume and detector (DCHl-4: drift chambers; Hodo: hodoscope; LKr: electromagnetic calorimeter; HAC: hadron calorimeter; MUV: muon veto).

2

Experimental Setup

The NA48/2 experiment at the CERN SPS was designed especially to search for direct C P violation in the decays of charged kaons, and collected data in 2003 and 2004. The highest possible level of charge symmetry between K+ and K- was a crucial requirement in the choice of beam, experimental apparatus, strategy of data taking and analysis in order to reach a high accuracy in the measurement of the charge asymmetry parameter A g • A novel beam line (Fig. 1) with two simultaneous charged beams of opposite charges was designed and built in the high intensity hall (ECN3) at the CERN SPS. The charged particle beams were produced by 400 GeV SPS protons on a beryllium target. K+ and K- with momentum (60 ± 3) GeV/c were selected symmetrically by an achromatic magnet system (,achromat') which separates vertically the two beams and recombines them again on the same axis. Frequent inversion of the magnetic field polarities in all the beamline elements provides a high level of intrinsic cancelation of the possible systematic effects in the beam line. The entire reconstruction of K± ---t 7[±7[+7[- decays and the determination of the kaon charge in K± ---t 7[±7[07[0 decays rely on a magnetic spectrometer. Two drift chambers are located upstream and two downstream of a dipole magnet which deflects charged particles horizontally with a transverse momentum kick of 120 MeV/c. The magnetic field was reversed frequently in order to cancel possible left-right asymmetries in the detector system. The momenae-mail: [email protected]

278

tum resolution of the magnetic spectrometer is u(p)/p = 1.0% E9 0.044%p (p in Ge V / c). The acceptance of the spectrometer is defined mainly by an evacuated beam tube passing through its centre, with a diameter of rv 16 cm, in which travel the surviving beam particles as well as the muons from 7r ----t /-LV decays. The reconstruction of K± ----t 7r±7r0 7r 0 decays is based mainly on the use of liquid krypton calorimeter (LKr), which measures the energies of the four photons from 7r 0 decays. The LKr has an energy resolution u(E)/ E = 0.032/vEE9 0.09/EE90.0042 (E in GeV) and spatial resolution for a single electromagnetic shower U x = u y = 0.42/ JE E9 0.06 cm for the transverse coordinates x and y. The use of a priori charge symmetric detector helps to keep the result in K± ----t 7r±7r0 7r 0 mode practically unbiased. A hodoscope is used for precise time measurement of the charged particles and as a component in the trigger system for both decay modes. Detailed description of the detector components can be found elsewhere [4]. 3

Asymmetry measurement method

The asymmetry measurement is based on the comparison of the u spectra for K+ and K- decays, N+(u) and N-(u), respectively. The ratio of the u spectra N+(u)/N-(u) is proportional to [1 + 6g· u/(l + gu + hu 2 )], where g and hare the actual of the Dalitz-slope parameters [5]. The possible presence of a direct CP violating difference between the linear slopes of K+ and K- , 6g = g+ - g- , can be extracted from a fit to this ratio. The measured asymmetry is then given by Ag = 6g/2g. In order to minimize the effect of beam and detector asymmetries, we use the ratio R4(U), defined as a product of four N+(u)/N-(u) ratios:

R4(U)

=

N1+ B+ (u) . N1+ B- (u) . N1- B+ (u) . N1- B- (u) NA+B+(u) N"A+B-(u) N"A-B+(u) N"A-B-(u)'

(2)

where the upper index corresponds to the kaon charge and lower index denotes the polarity in the beam-line (A) and spectrometer magnet (B). The parameter 6g is extracted from a fit to the measured quadruple ratio R4(U) using the function f(u) = R· [1 + 6g· u/(l + gu + hu 2)]4. The measured slope difference is insensitive to the normalization parameter R, which reflects the ratio of K+ and K- fluxes (rv 1.8). The quadruple ratio method complements the procedure of magnet polarity reversal. It allows cancelation at first order of systematic biases in the beam line and in the detector system. A reduction of possible systematic biases due to the presence of stray permanent magnetic fields is achieved by the radial cuts around the average beam position, which make the geometrical acceptance to particles azimuthally symmetric. The only residual sensitivity to instrumental charge asymmetries is associated with time variations of any

279

acceptance asymmetries occurring on a time scale shorter than the magnetic field alternation period, which are studied carefully by a number of monitors recorded throughout data taking.

4

Result and conclusions

In total, 3.1 . 109 K± -+ n±n+n- and 91 . 106 K± -+ n±nono decays were selected for the analysis. The result in terms of linear slope difference b.g with only the statistical error quoted is b.g C = (0.7 ± 0.7) .10- 4 for K± ~ n±n+nand b.g n = (2.2 ± 2.1) . 10- 4 for K± -+ n±nono decay mode. These results are free of systematic biases in the first approximation due to the implemented method of cancellation of various apparatus imperfections. However, the checks of possible systematic contributions have been done, and corresponding uncertainties were obtained [6]. The obtained parameters Ag are: A~

= (-1.5 ± 1.5stat . ± 1.4syst .) .10- 4 = (-1.5 ± 2.1) .10- 4 ,

A~

= (1.8 ± 1.7stat . ± 0.5 syst .) . 10- 4 = (1.8 ± 1.8) . 10- 4

(3) (4)

correspondingly for K± -+ n±n+n- and K± ~ n±nono decay modes. The results are one order of magnitude more precise than previous measurements and are consistent with the predictions of the SM.

References [1] W.T. Ford et al., Phys. Rev. Lett. 25, 1370 (1970). K.M. Smith et al., Nucl. Phys. B91, 45 (1975). G.A. Akopdzhanov et al., Eur. Phys. J. C40, 343 (2005). [2] L. Maiani and N. Paver, The second DAipNE Physics Handbook, INFN, LNF, Vol 1, 51 (1995). E.P. Shabalin, Phys. Atom. Nucl. 68,88 (2005). A.A. Belkov, A.V. Lanyov and G. Bohm, Czech. J. Phys. 55 Suppl. B, 193 (2004). G. D'Ambrosio and G. Isidori, Int. J. Mod. Phys. A13,1 (1998). 1. Scimemi, E. Gamiz and J. Prades, hep-phj0405204. G. Fiildt and E.P. Shabalin, Phys. Lett. B635, 295 (2006). [3] G. D'Ambrosio, G. Isidori and G. Martinelli, Phys. Lett. B480, 164 (2000). E.P. Shabalin, ITEP-8-98 (1998). [4] V. Fanti et al. (NA48), Nucl. Inst. Methods A574, 433 (2007). [5] W.-M. Yao et al. (PDG), J. Phys. G33, 1 (2006). [6] J.R. Batley et al., Eur. Phys. J. C52, 875 (2007).

7f7f

SCATTERING LENGTHS FROM MEASUREMENTS OF Ke4 AND K± - t 7f±7fo7fo DECAYS AT NA48/2 Dmitry Madigozhin a (for NA48/2 Collaboration) Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Abstract. The NA48/2 experiment at the CERN SPS has collected a large sample of charged kaon decays. From the 2003 and most of 2004 statistics, the 7{"7{" scattering lengths and have been extracted through the interpretation of an anomaly (cusp) in the 7{"07{"0 effective mass distribution of the decay K± --> 7{"±7{"07{"0. Using 2003 data, the form factors of K e4 decays have been obtained, allowing the independent measurement of the same 7{"7{" scattering parameters.

ag

1

a5

Introduction

In description of 7f7f scattering at low energies, the S-wave scattering lengths (multiplied by m7T±) for states with isospin I = 0 (ag) and isospin I = 2 (a6) are the main parameters of amplitude calculation. In Chiral Perturbation Theory (ChPT) the ag is related to the size of chiral condensate and is predicted to be ag = 0.220 ± 0.005 and a6 = -0.0444 ± 0.0010 [1]. The earlier Ke4 decay studies [2,3] provided a measurement of ag with up to 10% precision. The DIRAC collaboration has presented the (ag - a6) value extracted from the pionium lifetime measurement [4]. The main goal of NA48/2 experiment was the search for the CP-violating asymmetries in decays K± - t 7f±7f+7f- and K± - t 7f±7fo7fo [5]. But the large data samples and high quality of measurements allow many other studies. Here we present the preliminary results on the 7f7f scattering lengths measurement from NA48/2 data on K± - t 7f±7fo7fo and K± - t 7f+7f-e±v(Ke4) decays.

2

Beam line and Detector

The NA48/2 setup consisted of the upgraded version of NA48 detector [6] with the new beam line that provided a simultaneous and collinear K+ and K- beams. Both beams where generated in collision of primary 400 GeV proton beam with the single Berillium target. Secondary charged particles where selected by the beam line to form the narrow momentum spectra centered near 60 GeV with RMS ~ 3 GeV. The negatively and positively charged beams than where superimposed and passed through vacuum decay volume and beam pipe in the center of detectors. The most important components of the detector for the discussed measurements where a magnetic spectrometer, Liquid Krypton electromagnetic calorimeter and the muon veto counter. The magnetic spectrometer consisted of four drift chambers and a magnet, that provided a transverse moae-mail: [email protected]

280

281

x 10 2 3000 2500 2000 1500 1000 500 0

//,

./

~/.:

0.08

0.09

0.11

0.12

Figure 1: Distribution of the 7[07[0 invariant mass squared in K± __ 7[±7[07[0 decays for NA48/2 experimental data collected in 2003 and 2004

mentum kick of 120 MeV Ic. The momentum resolution of the spectrometer could be expressed as a(P) I P = (1.0 EB 0.044P[Gev I c])%. Its resolution for transversal position of hits was about 150/1,m. The electromagnetic calorimeter depth was 27 X o of liquid Krypton. Its energy resolution is given by a(E)IE = 3.2%IJEEB9%IEEB0.42%, where E is a photon energy in GeV. Moun veto counter consisted of three planes of plastic scintillators with 80 cm iron walls upstream each of the planes. 3

Cusp Effect

During the analysis of 2003 data, a sharp change of slope in the nOno invariant mass (Moo) distribution of K± ---+ n±nono decays has been observed in the point Moo = 2m+, where m+ is the n± mass [7]. A first interpretation of this effect in terms of nn scattering in final state has been made by N.Cabibbo [8], and than a second-order calculation [9] have provided us with a formula for the experimental data fit. The decay matrix element M was assumed to be

(1) where u = (S3 - so)/m~. Ml term represents the K± ---+ n±n+n- decay with the n+n- ---+ nOno rescattering as well as the relevant next-order diagrams. The isospin symmetry breaking correction was applied in the formulae connecting ag and a6 to the five nn ---+ nn rescattering amplitudes with different charges of pions (see, for example, [10]). The study of cusp effect now is based on 2003 and majority of 2004 data (Fig. 1). Improved selection is implemented to diminish the sensitivity of the result to the quality of Monte Carlo simulation, while the simulation was tuned to

282

0.015

b-

0.005

o

Figure 2: Deviation of the experimental data (collected by NA48/2 in 2003 and 2004) from the fit by 1 result «data - fit)/fit) for the M50 distribution.

the experimentally measured variations of beam geometry, detector efficiency and resolutions. In total about 59.6 x 106 K± ........ 7r±7r 0 7r 0 events where selected using the electromagnetic calorimeter data to reconstruct the photons from 7r 0 decays and the magnetic spectrometer for charged pion tracks measurement. A matrix element (1) was used to fit the experimental one-dimensional u distribution taking into account the acceptance and resolution effects calculated by means of Monte Carlo simulation. The deviation of the data spectrum from the fit result is shown on the Fig. 2. Seven bins (0.00015 GeV each, see Fig. 2) around the cusp point were excluded from the fit in order to reduce the sensitivity to electromagnetic corrections. The excess of events in the region around the cusp can be interpreted as a poinium signature, yielding a rate of pionium formation about two times higher than the theoretical prediction for pionium production [11]. But the recent calculation of electromagnetic effects for the 7r+7r- scattering [12] has explained about a half of excess as an additional contribution of electromagnetic interaction that doesn't lead to formation of a bound state. When we have added to (1) a term !f~ to describe a possible weak m+

dependence of matrix element on another Lorentz invariant, we got by separate two-dimensional fit the result k' = -0.0097 ± 0.0003 stat . ± 0.0008 syst ' Its non-zero value is taken into account in frameworks of one-dimensional fit and lead to some small change of another slopes, but the scattering lengths is not sensitive to k' of this size. The results of the fit for the matrix element slope parameters are the following: 90 = 0.649±0.003stat.±O.004syst, h' = -0.048± 0.007 stat. ± 0.005 syst ' The scattering parameters measured values are: a6 = 0.261 ± 0.006 stat . ± 0.003 syst ± O.OOlexternaz, a5 = -0.037 ± 0.013 stat . ± 0.009 syst ± 0.002external. External errors are related to the uncertainties of PDG information used for the fit, but it doesn't include the additional 5%

ag-

283 uncertainty of the theoretical formula (1) itself (currently, this is a dominant source of uncertainty).

4

Ke4 decay

The kinematics of K± -> 7r+ 7r- e± v decay is described by the five CabibboMaksymowicz variables [13]: invariant mass squared of a dip ion Sn = M;n' invariant mass squared of dilepton Sn = M;v and angles en, ee, ¢. en is the angle between 7r- and dilepton momenta in the rest frame of dipion, ee is the angle between v and dip ion momenta in the rest frame of dilepton. ¢ is the angle between the plane of dilepton and e± momenta and the plane of dipion and 7r+ momenta in the kaon rest frame. The matrix element is defined by means of axial form factors F,G and a vector form factor H. A partial wave expansion of the form factors may be restricted to sand p waves: F = Fs ei08 + FpeiOpcosen, G = Gpe iOp , H = Hpe iOp (only phase shift 5 = 5s - 5p is observable). From the 2003 data, about 670000 Ke4 decays have been selected. Reconstructed events are distributed in 10 x 5 x 5 x 5 x 12 iso-populated bins in the (M1r1r' Mev, cos(e n ), cos(ee), ¢) space. Ten independent fits (one per M1r1r bin) of five parameters (Fp, Gp, Hp, 8, and a normalization constant, that absorbs Fs) where performed in four dimensional space using the acceptance and resolution information from Monte Carlo simulation. The value of the phase difference 5 was extracted from the measured asymmetry of ¢ distribution as a function of M 1r1r . The result of 5 measurement is shown on the Fig. 3 together with results from previous experiments [2,3]. The phase shift measurements can be related to the 7r7r scattering lengths using the analytical properties and crossing symmetry of amplitudes (Roy equations [15]). One can use the Universal Band approach [16,17] to extract alone. At the center line of the Universal Band (I-parameter fit), NA48/2 = 0.256±0.006stat±0.002syst±0.018theoTl phase measurements translate as which implies a6 = -0.0312 ± O.OOl1 stat ± 0.0004 syst ± 0.013 t heor. In the case of the fit where both ag and a6 are free parameters, the result is ag = 0.233 ± 0.016 stat ± 0.007 syst, a6 = -0.0471 ± O.Ol1 stat ± 0.004 syst (the correlation is 96.7%). Finally, resent work [18] suggests that isospin symmetry breaking effects, neglected so far in the Ke4 phase shift analysis, would lead, when taken into account, to decrease of ag by ~ 0.022, and a6 - by about 0.004, leading to good compatibility between the Ke4 and cusp analyses results for pion scattering lengths.

ag

ag

284

•

,.-....0.3 ""0

t: 0.2

~

'0

0 .1 0

-0.1

t •• t ., • t • •

. • • •

•

• NA48/2 (2003 Data) • E865 [3] ... Geneva-Saclay [2]

'

~~~~~~~~~~~~~~'----'~---,---,----,--,-I

0.28

0.3

0.32

0.34

0.36

0.38

M",,( GeV/ c Figure 3: Phase shift 8 measurements from

5

Ke4

0.4

2 )

experiments

Conclusion

Two independent results of 7r7r scattering lengths measurement, obtained by NA48/2 experiment, are compatible between each other and are in agreement with current predictions of ChPT, if the isospin symmetry breaking effects are taken into account in both analyses of experimental data. References

[1] G.Colangelo, J.Gasser, H.Leutwyler, Nucl.Phys. B 603, 125 (2001). [2] L.Rosselet et al, Phys.Rev. D 15,574 (1977). [3] S.Pislak et ai, Phys.Lett. 87,221801 (2001). [4] B.Adeva et al, Phys.Lett. B 619, 50 (2005). [5] J.RBatley et al, Eur.Phys.J. C 52, 875 (2007). [6] V.Fanty et al, Nucl.lnstrum.Methods A 574, 433 (2007). [7] J .RBatley et ai, Phys.Lett. B 633, 173 (2006). [8] N.Cabibbo, Phys. Rev. Lett. 93, 121801 (2004). [9] N.Cabibbo and G.Isidori, JHEP 503, 21 (2005). [10] K.Knecht and RUrech, Nucl.Phys. B 519, 329 (1998). [11] Z.K.Silagadze, JEPT Lett 60,689 (1994). [12] S.RGevorkyan, A.V.Tarasov, O.O.Voskresenskaya. Phys.Lett. B 649, 159 (2007). [13] N.Cabibbo and A. Maksymowicz, Phys. Rev. 168, 1926 (1968). [14] G.Amoros and J.Bijnens, J.Phys G 25, 1607 (1999). [15] S.Roy, Phys. Lett. B 36, 353 (1971). [16] B.Ananthanarayan et al, Phys.Rep. 353, 207 (2001). [17] S.Descotes, N.Fuchs, L.Girlanda, J.Stern, Eur.Phys.J. C 24, 469 (2002). [18] J.Gasser, Proc. of KAON 2007 Int. Conf., Frascati, May 21-25, (2007).

RARE KAON AND HYPERON DECAYS IN N A48 EXPERIMENT N.Molokanova U

Joint Institute for Nuclear Research, 141980 Dubna, Russia Abstract. Recent results from the experiments NA48/1 and NA48/2 are reported. The first measurement of direct emission and interference terms in K± --+ 7l'±7l'0')' and the first observation of K± --+ 7l'±e+e-')' are described. Concerning NA48/1 measurements on radiative hyperon decays are presented.

1

Introduction

The series of experiments NA48 have explored many topics in the charged and neutral kaon physics. In this paper we shall discuss some of the most recent measurements produced by two stages of the experimental program: NA48/1 and NA48/2. NA48/1 (2002) has been oriented mainly to the study of rare Ks decays and has produced also results in hyperon physics. NA48/2 (2003-2004) was designed to search for direct CP-violation in K± decays, but also many other results in rare decays have been achieved.

2 The radiative K The decay channel K± ----t 7f±7f0'f' is one of the most interesting and important channels for studying the low energy structure of the QCD. Three components contribute to K± ----t 7f±7f0'f' decay amplitude: the Inner Bremsstrahlung (IB) associated with the decay K± ----t 7f±7fo in which the photon is emitted from the outgoing charged pion, Direct Emission (DE) from the vertex and the interference (INT) between these two. The K± ----t 7f±7f0'f' decays are described in terms of two kinematic variables: the kinetic energy of charged pion in kaon rest frame (T;) and invariant variable W 2 = (PK . P",!)(P7l' . P"'!)/(mKm 1r )2, where PK , P1r , P",! are the 4-momenta of the kaon, charged pion and odd gamma, respectively. About 124.103 events were selected in the range T; < 80 Me V and 0.2 < W < 0.9. In the previous measurements a lower cut T; > 55 MeV was introduced in order to suppress K± ----t 7f±7fo7fo and K± ----t 7f±7fo background. In NA48/2 measurement these backgrounds are avoided by application of a special algorithm, which detects overlapping gamma in the detector and due to the limit ±10 MeV on the deviation ofreconstructed kaon mass from its nominal value. The upper cut on T; rejects K± ----t 7f±7fo decays. The background in the selected sample is kept under 10- 4 . The probability of the photon mistagging (i.e. choice of wrong odd photon) is estimated to be less than 0.1%. ue-mail: [email protected]

285

The preliminary Yalues for the fractions of DE and INT with respect to IB are Frac(DE) = (3.35 ± 0.35 stat ± Q.25syst)% Prac(INT) = (-2.67 ± 0.81 stat ± 0.73^,*)%This is the first measurement of a non vanishing interference term in the K^ —»• 7T 7r0/y decay. 3

First observation of the decay K^ —* ir ± e + e~7

NA48/2 experiment observed for the first time the radiative decay K^ —> 7r ± e + e~7. The signal is selected between 480 MeV/e2 and 505 MeV/c2 in the invariant ir : t e + e"7 mass and requiring the invariant e + e^7 mass to be greater that 260 MeV/e2. Fig.l displays the projections of this region on the corresponding axes. The crosses represent data while the filled distribution represent different simulated background contribution. 120 candidates were selected with 7.3 ± 1.7 estimated background. The main source of BG is the K —> 7r^~ir(L'y with a lost •*¥.

Figure 1: K~^ —• it^ze^~e~~ry decay. The invariant w^e^e y (left) and e"*"e 'y (right) masses with corresponding background distributions. Black crosses represent data distribution.

By using K^ ^ w^w0 as normalization channel the branching ratio was pr€:nnnnl:try est,lmatE:C1 to be Br{K± -* W±TT°'J) = (1.19 ± 0.12stet ± 0.04,v.t) • 10^8. More details on If* —* w±e+e~'y decay analysis could be found in [1], 4

Weak radiative H° decays

Up to this day weak radiative hyperon decays as H° —> A7 and E° —* S7 are still barely understood. Several theoretical models exist, which give

287

very different predictions. An excellent experimental parameter to distinguish between models is the decay asymmetry a. It is defined by HN _ = *0(l4W), where 9 is the direction of the daughter baryon with respect to the polarization of 5° in its rest frame. For example, the decay asymmetry for H° —» A7 can be measured by looking at the angle between the incoming 5 and the outgoing proton from the subsequent A —> pir^ decay in the A rest frame. Using this method, the measurement is independent of the unknown initial H° polarization. The NA48/1 experiment has selected 48314 5° -+ A7 and 13068 H° -* £7 candidates (fig,2). The background contributions are 0.8% for H° —•> A7 and about 3% for S° —•» S7, respectively.

Figure 2: H° —•> A"/ (left) and H° —+ £7 (right) signal together with MC expectations for signals and backgrounds.

Using these data, fits to the decay asymmetries have been performed. In case of H° —* £ 7 , where we have the subsequent decay E° —+ A7, the product cos0s~-*"£>y • cosOs-tAi n a s to be used for the fit. Both fits show the, expected linear behavior on the angular parameters. After correcting for the well-known asymmetry of A —» pn~, values of a H o^A 7 = -0.684 ± 0.0203tet ± 0.061 s v s t and as^s-y = ^0.682 ± 0.031 s t a t ± 0.0653I,st are obtained. These values agree with preYious measurements by NA48 on H° ^+ A7 [2] and KTeV on H° —* £7 [3], but are much more precise. In particular the result on S° —> A7 is of high theoretical interest, as it confirms the large negative value of the decay asymmetry, which is difficult to accommodate for quark and vector meson dominance models.

288

5

First observation of 3° —- - +

la the 2002 run of NA48/1 experiment the weak radiative decay E° ~~> Ae+e~~ was detected for the first time [4]. 412 candidates were selected with 15 C:1CgroulUO events (fig. 3) The obtained brancb:ing fraction Br(E° ~* Ae + e") = (7.7 ± 0.5 stot ± 0Asyst) • HP 6 is consistent with inner bremsstrahlung-like e+e~ production mechanism.

Figure 3: The invariant mass of Ae+e-

together with the simulated background.

The decay parameter ctEAee c a n be measured from the angular distribution dN N ^^^YCl-CHAee^COS^H), (1) where cosffps is the angle between the proton from A —» JM decay relative to the H° line of flight in the A rest frame and a_ is the asymmetry parameter for the decay A —> pw^. The obtained value aSAee = -0.8 ± 0.2 is consistent with the latest published value of the decay asymmetry parameter for 5 —> A-y. References [1] [2] [3] [4]

J.R.Batley et al, CERN-PH-EP/2007-033, accepted by Phys.Lett.B. A.Lai et al, Phys.Lett. B 584, 251 (2004). A.Alavi-Haxati et al, Phys.Rev.Lett. 86 , 3239 (2001). J.R.Batley et al, Phys.Rev. B 650, 1 (2007).

THE K+ -

rr+ vi) EXPERIMENT AT CERN Yu.Potrebenikov a

Laboratory of Particle Physics, Joint Institute for Nuclear Research, 141980 Dubna Moscow region, Russia Abstmct. The P326 proposal of a new experiment NA62 aiming to perform precise measurement of the very rare kaon decay K+ -+ 7I"+vii branching ratio at CERN is described. About 80 K+ -+ 71"+ vii events with 10% of background is planned to obtain in two years of data taking. The status of the project, current status of R&D and future plans of the experiment are discussed.

1

Introduction

The K+ -) 7r+vv decay is a flavor changing neutral current process, computable with very small theoretical uncertainty of about 5% [1]. The hadronic matrix element can be parameterized in terms of the branching ratio of the well measured K+ -) 7r°ev decay [2] using isospin symmetry. The computed value is (8.0 ± 1.1) x 10- 11 , where the error is dominated by the uncertainty in the knowledge of the CKM matrix elements. Such an extreme theoretical clarity, unique in K and B physics, makes this decay (together with KL -) 7r°vv) extremely sensitive to new physics (see for example [3,4]). Only 3 K+ -) 7r+vv events have been observed by BNL-E949 experiment [5], that gives a central value of the branching ratio higher than the SM expectation. But rv 10% accuracy measurement of the branching ratio is required to provide a significative test of new physics contributions. This is the goal of the proposed NA48/3 (or NA62) experiment at CERN-SPS [6]. The aim of the experiment is to collect about 80 K+ -) 7r+vv events with the background level of 10%. 2

Proposal of the future experiment

The NA62 experiment will use kaon decays in-flight technique, based on the NA48 apparatus and the same CERN-SPS beam line which produced the kaon beam for all NA48 experiments. The R&D program for this experiment, started in 2006, is continuing in 2007. The data taking should start in 2011. The layout of the experiment is shown in fig. 1. The goal of the experiment can be reached by having 10% signal acceptance and by using a beam line able to provide the order of 10 13 kaon decays. To study K+ -) 7r+vv decay it is necessary to reconstruct one positive pion track in the downstream detector. If a beam and a pion tracking detectors provide a precise reconstruction of the decay kinematics, the missing mass allows a kinematical separation between the signal and more than 90% of the total background (fig.2); only non-gaussian tails from K+ -) 7r+7r0 and K+ -) p,+vJ.L ae-mail: [email protected]

289

290 m

\(,\'IU)

,---~~~-' --,y~m mU
VACUUM

---'----------I--~-----

---0-

1(+-75GoV

iL-_ _ _ _---,

-1

SI'IIlES.l

S.\C

-\

',I: "

,\,>

~;::Iatm

r:,

: :: :: : IStraw "

-2

•.

--LKr

" Tube ..,

Figure 1: Layout of the experiment

in the squared missing mass resolution will present in the defined signal region. But the kinematics only cannot provide background rejection factor of 10 13 . So, different veto (photon and muon) and particle identification (CEDAR and RICH) systems are included into experimental set-up to fulfill these needs. Moreover, the detector can provide redundancy both for kinematics reconstruction and particle identification allowing to estimate background directly from the data.

-O.I.'S!-'--'--'::<_Of,.,~'-"n_o.kos~,-,-;!F'-''-'=$CLL~O.'i,""'-.~O.15 ~S!!i GeV2/c

.O.~'5~""""'-Oi7.,==;r-O.~05~,-,-;pJL;;::::~::;:;"''''''l!!O.':-'-L~O"15

4

m~ls$ GeV 2/C 4

Figure 2: Squared missing mass for krum decays

2.1

The beam line

A 400 GeV Ic proton beam from the SPS, impinging a Be target, produces a secondary charged beam. 100 m long beam line selects 75 Ge V I c momentum

291

beam with 1.1 % RMS momentum bite and an average rate of about 800 MHz integrated over an area of 14 cm 2 • The beam contains 6% of K+. The average rate seen by the downstream detectors integrated on their surface is rv 11 MHz. The described beam line provides 5 x 10 12 K+ decays, assuming 60 m decay region and 100 days of run at efficiency of 60%, which is a very realistic estimation based on the decennial NA48 experience at the SPS.

2.2

The experimental set-up and RfjD current status

The experimental set-up consists of: Beam (Gigatracker) and pion (magnetic) spectrometers. The first one consists of three silicon pixel stations across the second achromat of the beam line, produced by 300 x 300 /-lm2 pixel each. The time resolution of 200 ps is provided by 0.13 microns technology of silicon detector production. The magnetic spectrometer is designed with 6 (or 4) straw chambers with 4 coordinate views each. Chambers should work in vacuum, introduce small material contribution (0.5% Xo per chamber) and have a good spatial resolution (130 microns per view). 36 /-lm mylar straw tubes with about 10 mm in diameter welded by ultrasound machine and cowered with gold inside will be used for these reasons. This spectrometer will be used as a veto as well for high energy negative pion from Ke4 decays. The R&D program has been started in 2006, a full length and reduced-size prototype has been constructed, integrated and tested in the NA62 set-up during the 2007 run at CERN. Differential Cherenkov counter CEDAR and RICH. CEDAR [7], differential Cerenkov counter existing at CERN, will be used after its upgrade for new experimental conditions for kaon tagging to keep the beam background under control. The 18 m long RICH located after magnetic spectrometer and filled with Ne at atmospheric pressure aimed for particle identification and pion momentum measurement. It will contain about 2000 PMTs in the focal plane and has to reach a time resolution of 100 ps to provide time information for downstream tracks. A full-length prototype 60 cm in diameter and 96 PMTs has been integrated in the NA62 set-up and tested during the 2007 NA62 run at CERN. Large angle (for 10-50 mrad), medium angle (for 1-10 mrad) and small angle (for <1 mrad) veto calorimeters. Photon detection down to 50 MeV with an inefficiency of 10- 4 should be provided by large angle ring-shaped calorimeter working in vacuum. Lead-scintillator fibers and lead-scintillator tiles design of this calorimeter is now under the study. The existing liquid krypton calorimeter (LKr) [8] is planned to use as a medium angle veto with an inefficiency lower than 10- 5 • The 2006 test at SPS electron beam has confirmed the 10- 5 inefficiency for photon energies above 10 GeV. A program of consolidation and update of the readout electronics of the LKr is under way. For small angle calorimeter" shashlyk" technology will be used to reach 10- 5

292 inefficiency for photon energies above 10 Ge V. A prototype has been built and tested with electrons on the NA48 beam line in 2006. 3

Performances

Simulation has been done for preliminary analysis. The total acceptance is found to be about 17%, showing that the target of 10% of signal acceptance is safely achievable even taking into account additional losses occurring in a real data taking. RICH usage constrains the accepted pion tracks in the momentum range of 15-35 Ge V / c. The high limit on this cut is an important loss of the signal acceptance, but it assures that events like K+ ---. 7r+7r 0 deposit at least 40 Ge V of electromagnetic energy, making their rejection easier. Many sources of background have been considered and simple calculations of signal and background events in the signal regions indicates that the 10% background level is nearly achievable. 4

Conclusion

A study of the extremely rare K+ ---. 7r+vv decays is an unique instrument for searching for the new physics. The NA62 experiment at CERN-SPS proposes to follow this road by collecting about 80 events of this decay. The overall experimental design requires a sophisticated technology for which an intense R&D program has been started. Actually designed experiment is able to reach a 10 12 sensitivity per event employing existing infrastructures and detectors. References

[1] A.J.Buras, M.Gorbahn, U.Haisch and U.Nierste, JHEP 0611,002 (2006) HEP-PH 0603079. [2] W.M.Yao et al. [Particle Data Group]' J. Phys. G 33 (2006) 1.JPHGB,G33,1. [3] G.D'Ambrosio, G.F.Giudice, G.Isidori and A.Strumia, Nucl. Phys. B 645, 155 (2002) HEP-PH 0207036. (4] G.Isidori, F.Mescia, P.Paradisi, C.Smith and S.Trine, JHEP 0608, 064 (2006) REP-PH 0604074. [5] V.V.Anisimovsky et al. [E949 Collaboration], Phys.Rev.Lett. 93, 031801 (2004) HEP-EX 0403036. [6] G.Anelli et al., CERN-SPSC-2005-013, SPSC-P-326. [7] G.Bovet et al., CERN Report: CERN 82-12(1982). [8] G.Unal (NA48 Collaboration] in: IX International Conference on Calorimetry, October 2000, Annecy, France, HEP-EX 0012011.

RECENT KLOE RESULTS B. Di Micco a for the KLOE collaboration b I.N.F.N. sezione di Roma Tre, 00146 Rome, Italy Abstract.In this paper recent KLOE results on kaon and TJ physics are reviewed. The Vus determination using K -+ 7r°e±v, the lepton universality test and the limits on the SUSY Higgs will be discussed. The extraction of the SU(2) breaking parameter Q from the TJ -+ 7r+7r-7r 0 decays, the measurement of the slope parameter a of the TJ -+ 37r° decay and the correlation with the TJ mass is shown.

1

Introduction

The KLOE experiment [1] have runned at the Frascati

NE from year 2000 to 2006. It collected an integrated luminosity of 2.5 fb-1 at the

,;so

2

,;s

The tagging technique

Due to its vector nature, the

293

294

3

Vus measurement

The module of the CKM matrix elements as reported by the PDG2006 [4] is shown in the following: Vc K M =

(

0.97377 ± 0.00027 0.230 ± 0.011 (7.4 ± 0.08) x 10 3

0.2257 ± 0.0021 0.957 ± 0.0017 ± 0.093 (40.6 ± 2.7) x 10- 3

3

(4.31 ± 0.30) x 103 (41.6 ± 0.6) x 10> 0.78

)

(1)

In the hypothesis of the absence of further quarks families, the matrix VCKM has to be unitary: V6KM x VCKM = 11.. If we apply this relation and extract the diagonal elements we obtain: diag(VbKM x VCKM)

= (0.9992 ± 0.0011, 0.97 ± 0.18, > 0.61 x 10- 3 )

As evident in this relation, the most accurate determination of the unitarity condition comes from the first row of the matrix, where the unitarity is constrained at per mil level. This error comes mainly from Vud (20 %) and Vus (80%). The most effective way to look for deviation from unitarity is to improve the Vus determination. Vus can be determined from the decay width r(KL ----+ e±7r°v) and r(K± ----+ e±7r°v) from the relation:

r(Ke3(-y))

=

G2 m 5

lVusI 2If.,f°7r- (OW 1~27r: SEwCkhe()..)(l + 2b.~U(2) + 2b.~r)

(2) Here K states for the Kaon mesons: KO, K+, £ the kinematically accessible leptons: e±, f..L±, the quantities f!t 7r -, b.~~, b.~U(2) and SEW are theoretical inputs that can be evaluated with different theoretical methods [5], I Ke ()..) is the integral on the phase space of the form factor f!t 7r - (t) while f.,f°7r- (0) is 7r - (t) is a function of three parameters the form factor computed at t = O. )..~, )..:e,)..5 obtained from the fit of the K£3 decay Dalitz plot. The r(K£3(r)) can be extracted from the measurements of the Br's and life times of the kaons. KLOE has measured the Br(KL£3) [6], the KL form factors [7], the KL life time [8] the Kse3 decay [9]. The K±£3 branching ratios [10] and the K+ life time [11]. Using these measurements together with the PDG T+ life time we obtain IVus I x f.,f°7r- = 0.2157 ± 0.0006, using f.,f°7r- (0) from [12] we finally obtain lVus I = 0.2237 ± 0.0013. Using the value of Vud from [4] we find:

J!t

lVudl 2 + lVusl 2 = 0.9983 ± 0.0008 showing a small deviation from unitarity at 2.10', using a recent evaluation of Vud [13] we get lVudl 2 + lVusl 2 = 0.9991 ± 0.0008 that is at 1.10' from unitarity. K f..L3 and K e3 test lepton universality in kaon decays. Defining g( £) the quantity gee) = IG~ x V~sl in the formula (2) and using the experimental

295

measurements and theoretical inputs we obtain: g({L)jg(e) = 1.000 ± 0.008, to compare with the best lepton universality test from the T - . £VV decay: g({L)jg(e) = 1.000 ± 0.004 [14] that shows how lepton universality in KAON sector can be checked at roughly the same level of the T decay.

4

Measurement of Ke2jK{L2 and SUSY Higgs constraints

Recently the measurement of the ratio Br(K± -. e±v)jBr(K± -. {L±v) (Ke2jk{L2) is having renewed interest due to its sensitivity to lepton flavour violation theories. In particular, for large lepton mixing, suggested by the neutrino oscillation phenomenology, is sensitive to SUSY charged Higgs, for mass in the range 100-200 GeY. KLOE is performing this measurement in order to reach the maximum possible sensitivity. The analysis looks for a kink (a track topology given by two tracks with different curvature in the Drift Chamber). One of the two tracks is the charged kaon coming from the LP, the other the daughter lepton of the decay. The {L, e separation is performed studying the cluster shape in the calorimeter and the EjP ratio; E is the energy measured in the calorimeter and P the momentum of the track (E j P rv 1 for electrons). The momenta of the two tracks and the Kaon mass are used to evaluate the mass of the lepton. The mass distribution is obviously peacked at zero Mey2 in ke2 events while is peaked at 11000 Mey2 in k{L2 events. Fitting this distribution in a bidimensional plot together with the RMS of the energy deposited in each calorimeter plane (cluster shape dependent) we can count the fraction N(ke2)jN(k{L2) of events in the histogram. From our preliminary analysis we find: Br(K± -. e±v) (± ±) = 2.55 ± 0.05 stat . ± 0.05 syst . Br K -. {L v Using the whole data sample and increasing Monte Carlo statistics we expect to reach a precision of 1 %. The SUSY charged Higgs particle is also constrained by the ratio r~%:~:)). The K+ -. {L+v decay is elicity suppressed, so it is sensitive to new physics with V + A terms in the weak sector. Comparing the experimental r(K+ -. {L+v) rate to the predicted SUSY value, according different hypothesis on the Higgs mass, we can exclude a region in the (mjj, tan(3) plane. The prediction needs Vus value as an input. It can be safely evaluated from K £3 decay because, being not helicity suppressed, is largely dominated by SM contributions. Using the value of K -. {LV"'( and K£3 decays, using the lattice determination for 1+(0) and fKjf7r we pbtain:

R=

r(K -. {Lv) = 1.008 ± 0.008 rSM(K -. {Lv)

296

100 00 80 KLOE07FIN.

70

00

~

"

50 05

40

30 20 10

547

11 mass

Figure 1:

mH+) exclusion plot obtained using ke2/ktt2 ratio and decay; right: Recent measurements of the 'f/ mass.

-+

ttv"{

is shown in figure 1. Note that R exclude "" 50, mH+ 150 GeV left free from the B TV this a small discrepancy with the Standard Model the CDF eollaboration [17J. f'V

KLOE has collected about 100 milions of 1] events in its full data This r; allows to measure the fundamental of this meson, like its mass, and to test effective QCD theories based on Chiral Perturbation di,ml?~rsjon relation and unitarization.

Measurement

the

1]

mass

has measured the 1] mass at level of 5 x 10- 5 with the ¢ -) 1]"/, -+ TY. This level of accuracy is needed to among recent measurements shown in 1. Dalitz a in the also very sensitive to the 1] mass value as we will show in the result was presented in [18J. Final result was TY is a three photon final state event. The total is measured run by run events eneln!ll-nlOlnentlllm conservation on the three enler~;y-m()m.entUlm conservation constraint the AY"'TO'lA'" This allows to reach a very accuracy -+

UHIJU~;t.:

297 because the measurement doesn't depend from the absolute energy scale of the calorimeter. These fitted energies are used to build up the m')'')' mass distribution fitting which we measure the mass. Sensitivity of the measurement to several effects (detector geometry, beam energy calibration, kinematic cuts used to reject backgound and detector calibration) has been carefully taken into account. We measure: m." = 547.874 ± 0.007stat . ± 0.029 syst .MeV This value agrees with the NA48 measurement shown in fig.1 and strongly disagrees with the lower GEM measurement.

5.2

T} --t

37r decay

The T} --t 37r decay is an isospin violating hadronic decay. In a Chiral expantion its amplitude is proportional to the (mu -md) mass difference, and can be used to estimate isospin breaking in QCD. If we indicate with r the T} --t 37r width, the observed decay rate is:

The isospin conserving amplitude M(s, t, u) has been evaluated using several theoretical methods. In particular [20] showed that is possible to constrain M directly from the experimentally measured Dalitz plot parameters. The experiments expands the T} --t 7r+7r-7r0 amplitude \A\2 in powers of x = .J3 mTj-~+-Eand y = 3 mtJ-~o-m1[o , where E+, E_ and Eo are the enm7l"+-m7l"O m7r+-m7ro ergies of the 7r+, 7r- and 7r 0 . The following parametrization is used: \A\2 = 1 + ay

+ by2 + ex + dx 2 + exy + f y 3 + gx 3 + ...

Fitting our data (setting to 0 the c, e, 9 C-violating parameters) we find the following result:

a = b d =

f

=

-1.000 0.124 0.057 0.14

±0.005 stat . ±0.006 stat . ±0.006 stat . ±O.Ol stat .

+0.008 -0.019 syst.

±0.010syst +0.007 -0.016 syst. ±0.02 syst .

(3)

The Dalitz plot parameters can be used to constraint the subtraction point used in the dipsersion relation. Using our preliminary results [21], and the PDG decay width T} --t 37r, [20] found Q = 22.8 ± 0.4. A reanalysis with the final parameters would be interesting to determine the u - d quark mass difference, also important [5] to determine 6.~U(2) breaking correction in the

298

K+ £3 decay for V us extraction. Refitting the Dalitz plot using an appropriate parametrization [22J we evaluate the rJ ~ 3no amplitude. This is usually parametrized as: IA1)-+37rO 12 rv 1 + 20:z with

= ~3 ,\,3_ ( 3Ei;mry )2, where L.J~_ 1 mtJ - m7to

Ei is the energy of the nO's. We get +0.012 ( ) 0: = -0.038 ± 0.03 (stat. ) -0.008 syst .. This estimate can be compared with the direct measurement from the decay chain ¢ ~ rJ,,(, rJ ~ 3no ~ 7"(. The measurement shows big sensitivity to the value of the rJ mass used in z definition. Taking the correct rJ mass as measured by KLOE we get 0: = -0.027 ± 0.004(stat.)~g:gg:(syst.) using our preliminary mass measurement [18], in agreement with the estimate from the charged mode and with the measurement [23J. z

References

[lJ [2J [3J [4J [5J

F. Ambrosino et al., Nucl. lnstrum. Meth. A534 (2004) 403 N.N. Achasov and V.V. Gubin, Phys. Rev D64 (2001) 094016 F. Ambrosino et al., arXiv:0707.4148 W.M. Yao et al., Journal of Physics G33 (2006) 1 V. Cirigliano, Precision tests of the Standrad Model with k£3 decays, Proceedings of Science, KAON, 2007 [6J F. Ambrosino et al. Phys. Lett. B632 (2006) 43 [7J F. Ambrosino et al. Phys. Lett. B636 (2006) 166 [8J F. Ambrosino et al. Phys. Lett. B626 (2005) 15 [9J F. Ambrosino et al. Phys. Lett, B636 (2006) 173 [10J F. Ambrosino et al., arXiv:0712.3841 [l1J F. Ambrosino et al., arXiv:0707.2532 [12J P.A. Boyle et al., arXiv:0710.5136 [13J LS. Towner and J.C. Hardy, arXiv:0710.3181. [14J M. Davier, A. HTocker and Z. Zhang, Rev. Mod. Phys. 78 (2006) 1043. [15J FLAVIANET working group, http://ific.uv.es/flavianet/ [16J F. Ambrosino et al., Phys. Lett. B632 (2006) 76 [17J G. Landsberg, arXiv:0705.2855 [18J B. Di Micco et al., Acta Phys. Slov. 56 (2006) 403 [19J F. Ambrosino et al., JHEP 12 (2007) 073 [20] B.V. Martemyanov and V.S. Sopov, Phys. Rev. D71 (2005) 017501 [21J B. Di Micco et al. in Proceedings of the lCHEP 2004 conference, vol. II, p. 1064, ed. H.Chen,D.Du,W.Li,C.Lu. Hackensack, World Scientific, 2005 (hep-ex/0410072) [22J F. Ambrosiono et al., arXiv:0801.2642 [23J W.B. Tippen et al., Phys. Rev. Lett. 87 (2001) 192001 [24] M. Abdel-Bary et al., Phys. Lett. B619 (2005) 281

DECAY CONSTANTS AND MASSES OF HEAVY-LIGHT MESONS IN FIELD CORRELATOR METHOD A.M.Badalian a

State Research Center, Institute of Theoretical and Experimental Physics, Moscow, 117218 Russia Abstract. Decay constants fp, fv are calculated with the use of the path-integral representation of the meson Green's function. It is shown that a 25(20)% difference between the decay constants fD,(JB.) and fD(JB) occurs due to large differences in the pole masses of the sand d(u) quarks. The values rID = fD, / fD ~ 1.25, recently observed in the CLEO experiment, and T}B = fB, / fB ~ 1.20, obtained in unquenched lattice QCD, can be reached only if the running mass ms at low scale is ms(~ 0.5 GeV)= 170 - 200 MeV.

Recently, the leptonic decay constants in the processes D(Ds) --t JWp ( [1]), and B --t Tf.tT ( [2]) have been measured, where much better accuracy have been reached than in previous experiments. It is of interest that the ratio 'TJD = fD) fD appears to be larger than in many theoretical predictions ( [3,4]). Therefore, this ratio can considered as an important test of the accuracy of different theoretical models. In our analysis we use the analytical expressions for the leptonic decay constants in the pseudoscalar (P) and vector (V) channels, derived in ( [5]) with the use of the path-integral representation for the correlator G r of the currents

jr(x): Gr(x) = (jr(x)jdO»)vac: Jr =

J

Gdx)d 3 x = 2Nc

Ln

(Yr)nl'Pn(OW e- Mnt ,

(1)

wqnwQn

where Mn and 'Pn(r) are the e.v. and e.f. of the relativistic string Hamiltonian ( [6]), while wqn(wQn) is the average kinetic energy of a quark q(Q) for a given nS state: (2) Wqn = (Jm~ + p2)nS, WQn = (Jm~ + p2)nS, and mq(mQ) is the pole mass of the lighter (heavier) quark in a heavy-light (HL) meson. The matrix element (m.e.) (Yr)n refers the to P and V channels:

(3)

ae-mail: [email protected]

299

300

On the other hand, for the integral Jr (1) one can use the conventional spectral decomposition:

(4) Then from Eqs.(l) and (4) one obtains that

(5) where IP, IV are the relativistic corrections: IP

=

mqmQ

+ wqwQ -

(p2)

2wqwQ

;

IV

=

mqmQ

+ wqwQ + t(p2) 2wqwQ

(6)

,

which are essentially smaller than unity. The masses, m.e. W, (p2), and the w.f. at the origin, were calculated in ( [5]) and can be used to define fp and fv. The decay constants f pare given in Table 1.

fp B

Table 1: The decay constants

Ref. ( [7]) Ref. ( [3]) Ref. ( [4]) lattice ( [8]) quenched lattice ( [9]) nj = 2 + 1 this paper experiment

D 206 230(25) 234 235(22)

Ds 252 248(27) 268 266(28)

201(20) 210(10) 222.6(20)a)

(in MeV).

Bs

174 196(29) 189

Be -

-

-

216(32) 218 206(10)

-

249(19)

216(38)

259(32)

440(2)

260(10) 280(23)a)

182(8)

216(8)

438(10)

160~~~

322(42) -

b)

229(70)e) a)

b) e)

The experimental values are taken from Ref. ( [1]). BaBar data ( [2]) Belle data ( [2]) It is of interest to notice that for HL mesons the ratios

~D

=

~Ds

=

I

R

2

ID(0)1

= 0.347(3)

(7)

WqWe

appear to be equal for the D and Ds mesons with an accuracy better than 1%. Also these fractions for Band Bs mesons coincide with 2% accuracy

.:s

301

(
= EB. =

IR 1B (0)1

2

= 0.146(2).

(8)

WqWb

It is important that the equalities ED = ED. and EB = EB. practically do not depend on the details of the interaction in HL mesons. Therefore, in the ratios

rID (rIB) =

f;:,\IJ;} (t;;NJi/) 2

the factors ED (EB) cancel and one obtains

+ WsWc(b)

- (p2)D.(Bs)) MD(B) (Yp)DCB) MD.CB.) .

(msmC(b)

_

'T/D(B) -

(Yp)D(B)

(9)

In Eq. (9) the second term is close to 1.05, while the first term, proportional to m s , is not small, giving about 30-60% for different ms (below we take ms from the range 180 ± 30 MeV /c2 ). With an accuracy of ;S 2% 'T/b

=

2.708 x ms(GeV)

+ 1.07(1),

if md

= mu = 0,

(10)

i.e. in the chiral limit 'T/D

= 1.14 (ms = 85 MeV),

1.25 (m s

= 180 MeV),

1.27 (ms

= 200 MeV). (11)

For the Band B s mesons 'T/~ 'T/~o

= =

+ 1.07(1) ms + 1.07(1)

1.90 x ms 1.871 x

(md = m = 0); (md

= 8 MeV),

= 180 MeV)

1.21 (ms

(12)

which practically coincide, 'T/B

= 1.11 (m s =

85 MeV),

1.19 (ms

= 200 MeV).

(13) These values of'T/B appear to be only 3-5% smaller that 'T/D. Thus for ms = 180 Me V and md = 8 Me V we have obtained 'T/D+

= 1.25(2),

'T/B

= 1.19(1),

(14)

in good agreement with experiment and lattice data, see Table 2. In Table 2 we give also the ratio of decay constants. Our estimate of ms(0.5 GeV) = 180 MeV supports our choice of this value in the relativistic string Hamiltonian, which provides a good description of the HL meson spectra and decay constants, and gives rise to relatively large values of'T/D and 'T/B as in (11). Thus we have shown that 1. The running mass m s (P,l) at a low scale, P,1 :::::: 0.5 GeV, can be extracted from the values 'T/D and'T/B if they are known with high accuracy, ;S 5%. 2. The values 'T/D and 'T/B, as in (11), can be obtained only if the running mass ms (p,I) is inside the range 170 - 200 MeV.

302 Table 2: The ratios

fD./fD, fB./fB, and fD./fB •.

fD./ fD

a)

fB./ fB

RPM ( [6]) 1.15 1.15 1.08(1) 1.10(1) BS ( [7]) lattice ( [9]) 1.24(8) 1.20( 4) unquenched this work 1.24(3) 1.19(3) experiment 1.27(14) The ratio of the central values is taken

fD./ fB. a) 1.23 1.15(1) 1.01(8) 1.20(3) for

fD. and fB ..

References

(1] (CLEO Collaboration) M. Artuso et al., Phys. Rev. Lett. 95, 251801 (2005); G. Bonvicini et al., Phys. Rev. D 70, 112004 (2004); (CLEO Collaboration) M. Artuso et al., arXiv:hep-ex/0607074; S. Stone, hepex/0610026. (2] (BELLE Collaboration) K. lkado et al., Phys. Rev. Lett. 97, 251802 (2006); (BaBar Collaboration) B. Aubert et al., arXiv:hep-ex/0607094, hep-ex/0608019; L.A. Corwin, hep-ex/0611019. [3] Z.G. Wang et al., Nucl. Phys. A 744, 156 (2004). [4] C. Cvetic, C.S. Kim, G.-L. Wang, W. Namgung, Phys. Lett. B 596, 84 (2004). [5] A.M. Badalian, B.L.G. Bakker, and Yu.A. Simonov, hep-ph/0702157, Phys. Rev. D 75, 116001 (2007) [6] A.Yu. Dubin, A.B. Kaidalov, Yu.A. Simonov, Phys. Atom. Nucl. 56, 1745 (1993) [Yad. Fiz. 56, 213 (1993)]; hep-ph/9311344; Phys. Lett. B 323, 41 (1994); Yu.A. Simonov, hep-ph/9911237; A.M. Badalian and B.L.G. Bakker, Phys. Rev. D 66, 034025 (2002); A.M. Badalian, B.L.G. Bakker, and Yu.A. Simonov, Phys. Rev. D 66, 034026 (2002); Yu.S. Kalashnikova, A.V. Nefediev, Yu.A. Simonov, Phys. Rev. D 64, 014037 (2001). (7] Yu.A.Simonov, Z. Phys. C 53,419 (1992). [8] T.W.Chiu et al., Phys. Lett. B 624, 31 (2005); A.Yutner and J.Rolf, Phys. Lett. B 560, 59 (2003); J.Rolf et al., Nucl. Proc. Suppl. 129, 322 (2004). [9] C. Aubin et al., Phys. Rev. Lett. 95, 122002 (2005); Phys. Rev. D 70, 114501 (2004); A. Gray et al., Phys. Rev. Lett. 95, 212001 (2005); M. Wingate et al., Phys. Rev. Lett. 92, 162001 (2004).

BILINEAR R-PARITY VIOLATION IN RARE MESON DECAYS A.Alia Deutsches Elektronen-Synchrotron, DESY, 22607 Hamburg, Germany A. V. Borisov b , M. V. Sidorova c Faculty of Physics, Moscow State University, 119991 Moscow, Russia A bstmct. We discuss rare meson decays K+ -+ rr- £+ e+ and D+ -+ K- £+ f.1+ (f., f.' = e, /-£) in a supersymmetric extension of the standard model with explicit breaking of R-parity by bilinear Yukawa couplings in the superpotential. Estimates of the branching ratios for these decays are given. We also compare our numerical results with analogous ones previously obtained for two other mechanisms oflepton number violation: exchange by massive Majorana neutrinos and trilinear R-parity violation.

In the standard model (SM), the lepton L and baryon B numbers are conserved to all orders of perturbation theory due to the accidental U(lh x U(l)B symmetry existing at the level of renormalizable operators. But the Land B nonconservation is a generic feature of various extensions of the SM [1]. That is why lepton-number (LN) violating processes have long been recognized as a sensitive tool to put theories beyond the SM to the test. One of the most well known process of such type is neutrinoless double beta decay (A, Z) -+ (A, Z + 2) + e- + e- that has been searched for many years (see, e.g., [2] and references therein). In Refs. [3,4] rare decays of the pseudoscalar mesons K, D, Ds, and B of the type M+ -+ M'-C+£'+ (C,£' = e,M) (1) mediated by light (mN « ml, ml' ) and heavy (mN » mM) Majorana neutrinos were investigated. The indirect upper bounds on the branching ratios for the decays (1) have been derived taking into account the limits on lepton mixing and neutrino masses obtained from the precision electroweak measurements, neutrino oscillations, cosmological data, and searches of the neutrinoless double beta decay. These bounds are greatly more stringent than the direct experimental ones [5]. In Refs. [6,7] we considered another mechanism of the ilL = 2 decays (1) based on the minimal supersymmetric extension of the SM with explicit R-parity violation (,RMSSM, for a review, see [8]). R-parity is defined as R = (_1)3(B-L)+2s, where B, L, and S are the baryon and lepton numbers and the spin, respectively. The 8M fields, including additional Higgs boson fields appearing in the extended models, have R = 1 while R = -1 for their superpartners. The most general form for the R-parity and lepton number violating part of the superpotential is given by [8]

Wfi =

ca!3

(~)..ijkLf Lf Ek + )..~jkLfQf Dk + EiLf HE) .

(2)

Here i, j, k = 1,2,3 are generation indices, Land Q are SU(2) doublets of left-handed lepton and quark superfields (a, (3 = 1,2 are isospinor indices), ae-mail: [email protected] be-mail: [email protected] ce-mail: [email protected]

303

304

E and jj are singlets of right-handed superfields of leptons and down quarks, respectively; Hu is a doublet Higgs superfield (with hypercharge Y = 1); Aijk (= -Ajik), A~jk and €i are trilinear and bilinear Yukawa couplings, respectively. We assumed in [6, 7] that the bilinear couplings are absent at tree level (€i = 0 in Eq. (2)). As well known they are generated by quantum corrections [8] but the dominant contribution of the tree-level trilinear couplings to the phenomenology is expected. In the present report, we investigate the case of tree-level bilinear couplings: €i =I- 0 with A = 0, )..' = O. For this case, trilinear couplings cannot be generated via radiative corrections. The bilinear terms in the superpotential (2) induce mixing between the 8M leptons and the M88M charginos and neutralinos X~ in the mass-eigenstate basis and lead to the following 6..L = ±1 lepton-quark operators [9,10]:

~"'n W; f,yM PLX~ + v2g (f3fvkPRdd'R

LLH = -

+f3kv k PRu cuL

+ f3kivkPR£c£Li + f3CiJ'pR£CdL) + H.c.

(3)

Here PL,R = (1 ~ ,,(5)/2, "(5 = hO"(I"(2"(3; the constants "'n (n = 1,2,3,4), f3f, 13 k , f3t (k, i = 1,2,3) and f3c depend on the elements of the mixing matrices diagonalizing the neutrino-neutralino and the charged lepton-chargino mass matrices. The Lagrangian describing the bilinear mechanism of the decays (1) is

L = LLH

+ LSM + Lg + LX.

(4)

In addition to the LN violating part (3), it includes the 8M charged-current interactions

LSM =

~ W+M(L V£(XhMPL£(X) + L q(xhMPL Vqqlq'(X)) + H.c.,

(5) q,q' where £ = e, 11, T; q = U, c, t; q' = d, s, b; Vqql is the CKM matrix, and the M88M gluino-quark-squark and neutralino-quark(lepton)-squark(slepton) interactions £

[11]

L9 =

-v2gs (Art b (qaLg(r)iji

-

qaR9(r)ij~) + H.c.,

(6)

4

LX

= v2g L(€Ln('lj;)i[JLX~;j;L + €Rn('lj;)i[JRX~;j;R) + H.c.

(7)

n=1

Here (Ar)a b are the 3 x 3 Gell-Mann matrices (r = 1, ... ,8) with color indices a, b = 1,2,3; the neutralino coupling constants are defined as

€Ln('lj;) = -T3('lj;)Nn2 + tanOW(T3('lj;) - Q('lj;))Nnl' €Rn('lj;) = Q('lj;) tan OWNnl, where. Q('lj;) and T3('lj;) are the electric charge and the third component of the weak Isospm for the quark (lepton) field 'lj;, respectively, and N nm is the 4 x 4 neutralino mixing matrix. The ~ead~ng order amplitud~ of the decay (1) is described by 9 diagrams s~own m FIg. 1. ~he hadromc parts of the decay amplitude are calculated wlth the use of a slmple model for the Bethe--8alpeter (B8) amplitudes for mesons as bound states of a quark and an antiquark [12] (see also [3,4]). In

305 (2c)

i

(1.b)

Figure 1: Feynman diagrams for the decay M+ --+ M'- + £+ + £'+ in the bilinear .jlMSSM. Bold vertices correspond to Bethe-Salpeter amplitudes for mesons. There are also crossed diagrams with interchanged lepton lines.

addition, taking into account that the meson mass mM« mW,mSUSY, where mw is the W boson mass and mSUSY ~ 100 GeV is the common mass scale of superpartners, we neglect momentum dependence in the propagators of heavy particles (see Fig. 1) and use the effective low-energy current-current interaction. In this approximation the decay amplitude does not depend on the specific form of the BS amplitude and is expressed through the known decay constants of the initial and final mesons, fM and fk. Finally, for the total width of the decay (1) we have obtained

r ££'

Here


= r(M+ --+ M'-e+ ~m+) -- (1- 1£ )g4 f 'ttf"t"m"£rrt-.bi 2 U ££' 2 1°11"3 '*"££'

Ih-

dzz 2 [1- (h+

+ L)(2z)-1]2 [1- (l+ + L)(2z)-1]

1+

x [(h+ - z)(L - z)(l+ - z)(L - Z)]1/2

(9)

is the reduced phase space integral with h± = (1 ± mM' /mM)2 and l± = [(me ± me' )/mM]2; Nc = 3 is the number of colors. For the numerical estimates of the branching ratios (BRs), Bif' = r if' /rtotal, we have used the known values for the SM couplings 9 and g8, meson decay constants, meson and lepton masses [5], and a typical set of supersymmetric parameters [13]: a) the MSSM parameters: mo = 70 GeV, J..L = 500 GeV, 2 M2 = 200 GeV, tan/3 = 4; b) the RPV parameters: IAI (2::=1 IAiI )1/2 = 2 0.1 GeV 2 with lOA 1 = A2 = A3, IEI2 2:;=1 IEil = IAI with E1 = E2 = E3;

=

=

306

M3 = (g;/g2)M2 at the electroweak scale. Using the MSSM mass formulas [11] with the gluino mass rng = M 3 , the masses of squarks and neutralinos and the elements of the neutralino mixing matrix were calculated numerically for the above set of parameters. The results of the calculations with the use of Eq. (8) are shown in the fourth column of Table 1. In the second and third columns of the table, the present direct experimental bounds on the BRs [5] and the indirect bounds for the Majorana neutrino mechanism of the rare decays [4] are shown, respectively. We see that the BRs for the bilinear RPV mechanism are much smaller than these upper bounds. For comparison, the trilinear RPV mechanism leads to the upper limits on the BRs of order 10- 23 (10- 24 ) for K (D) rare decays with the use of conservative bounds on the trilinear couplings IA~jk'\.~' j' k' I ;S 10- 3 [6,7]. Butfor more stringent bounds IX XI ;S 5 X 10- 6 [8,14], Bu' (tri,RMSSM) ;S 10- 28 . Table 1: The branching ratios B il' for the rare meson decays M+

!tare decay KT --+ 7r KT --+ 7r KT --+ 7r DT --+ K DT --+ K DT --+ K

eTe T J-LTJ-LT eTJ-LT eT e T J-LT J-LT e T J-LT

Exp. upper bound on Bel' 6.4 x 10 -lU 3.0 x 10 -lJ 5.0 x 10 -IU 4.5 x 10 -0 1.3 x 10 -0 1.3 x 10 -4

Ind. bound on BU' (vMSM) 5.9 x 10 -:5:l 1.1 x 10 -:l4 5.1 x 10 -:l4 1.5 x 10 -:51 8.9 x 10 -24 2.1 x 10 -23

-t

M 1- £+ £1+.

Hii'

(bi,RMSSM) 3.6 x 10 -4[1 1.0 x 10 -4[1 4.2 x 10 -4[1 1.6 X 10- 48 1.5 X 10- 48 3.1 X 1O-4lf

References [1] R.N. Mohapatra, "Unification and Supersymmetry: The Frontiers of QuarkLepton Physics" (Springer-Verlag, New York, 2003). 23 A. Ali, A.V. Borisov, D.V. Zhuridov, Phys. Rev. D76, 093009 (2007). A. Ali, A.V. Borisov, N.B. Zamorin, Eur. Phys. J. C21, 123 (2001). A. Ali, A.V. Borisov, M.V. Sidorova, Phys. Atom. Nucl. 69, 475 (2006). [56 Particle Data Group: W.-M. Yao et al., J. Phys. G33, 1 (2006). [ A. Ali, A.V. Borisov, M.V. Sidorova, in "Particle Physics at the Year of 250th Anmversary of Moscow University" (Proceedings of the 12th Lomonosov Conference on Elementary Particle Physics), ed. by A. Studenikin (World Scientific Singapore, 2006), p. 215. ' [7] A. Ali, A.V. Borisov, M.V. Sidorova, Moscow Univ. Phys. Bull. 62 (1) 6 (2007). ' [8] R. Barbier et al., Phys. Rep. 420, 1 (2005). [9] A. Faessler, S. Kovalenko, F. Simkovic, Phys. Rev. D58 , 055004 (1998). 10j M. Hirsch, J.W.F. Valle, Nucl. Phys. B557, 60 (1999). 11 H.E. Haber, G.L. Kane, Phys. Rep 117, 75 (1985). 12j J.G. Esteve, A. Morales, R. Nunes-Lagos, J. Phys. G9, 357 (1983). 13 M. Hirsch, M.A. Diaz, W. Porod, J.C. Romao, J.W.F. Valle, Phys. Rev. D62 113008(2000); D65, 119901 (E) (2002). [14] S.-1. Chen, X.-G. He, A. Hovhannisyan, H.-C. Tsai, JHEP 0709, 044 (2007).

~4

FINAL STATE INTERACTION IN K

-7

27r DECAY

Evgeny Shabalin a Institute of Theoretical and Experimental Physics, 11 7258 Moscow, Russia Abstract.Contrary to wide-spread opinion that the final state interaction (FSI) enhances the amplitude < 271"; 1= OIK o >, I argue that FSI does not increase the absolute value of this amplitude.

The essential progress in understanding the nature of the t11 = 1/2 rule in K -+ 2n decays was achieved in the paper [1], where the authors had found a considerable increase of contribution of the operators containing a product of the left-handed and right-handed quark currents generated by the diagrams called later the penguin ones. But for a quantitative agreement with the experimental data, a search for some additional enhancement of the < 2n; 1= otKO > amplitude produced by long-distance effects was utterly desirable. A necessity of additional enhancement of this amplitude due to long-distance strong interactions was also noted later in [2]. The attempts to take into account the long-distance effects were undertaken in [3] - [14]. In [3], the necessary increase of the amplitude < 2n; I = otKO > was associated with liN corrections calculated within the large-N approach (N being the number of colours). In [4], [5], the strengthening of the < 27r; I = OtK > amplitude arised due to a small mass of the intermediate scalar ()" meson. One more mechanism of enhancement of the < 2n; I = otKO > amplitude was ascribed to the final state interaction of the pions [6] - [14]. But as it will be shown, the unitarization of the K -+ 2n amplitude in presence of FSI leads to the opposite effect: a decrease of the < 2n; I = OtK O> amplitude. The result obtained in [1], [2] looks as

(1) where /'i, is a function of G F, F rr , Be and other numerical ingredients of a theory. The numerical values of /'i, obtained in [1] and [2] turned out to be insufficient for a reproduction of the observed magnitude of the < 27r; 1= OIKo > amplitude. To understand, could FSI occuring at long distances change the situation, I consider at first the elastic nn scattering itself. The elastic nn scattering. The general form of the amplitude of elastic nn scattering is T =< ndp~ )nl (p~)

tni (PI )nj (P2) >= A<5ij <5kl + B<5ik <5jl + C<5il <5

j k,

(2)

where k,l,i,j are the isotopical indices and A,B,C are the functions of s = (PI + P2)2, t = (PI - p~)2 ,U = (PI _ p~)2. ae-mail: [email protected]

307

308 The amplitudes with the fixed isospin I T(O)

= 3A

+ B + C,

T(1)

= 0,1,2 are

= B - C,

T(2)

= B+C.

(3)

To understand the problems arising in description of 7m scattering in the framework of field theory, let's consider the simplest chiral a model, where 2 g,nrIT

m;;' - m 2 7f

2

mIT m;;' - s

. S -

(4)

and Band C are obtained from A by replacement s -t t and s -t u, respectively. It follows from Eqs.(3) and (4), that the isosinglet amplitude Tt~~e is a sum of the resonance part

(5) and the potential part

(6) The resonance part must be unitarized summing up the chains of pion loops. To one loop Aone-Ioop = Res

Atree(l+~rr +ir;srr) Res R

=

Atree(l+~rr +iA~::y'l4m;ls) Res R 167f '

(7)

where ~rrR is the renormalized real part of the closed pion loop [15]. Though ~rrR(S) can be calculated to leading order in gU7r7r [16], in view of very big value of this constant such a calculation does not give a proper estimate of ~rrR (s). It will be explained below, how to get a reliable magnitude of ~rrR(S). The unitarized expression for ARes is

A unitar _ Res - 1 -

Atree(s) Res ~rrR(S) - ir;srr Res

A~::(s)

1-

~rrR(S)

1 1- itantlRes'

(8)

where

4m Is = Atree(s)y'l_ Res IT . 2

tan DRes

167f(1 - ~rrR(S))

(9)

The Eq.(8) may be rewritten in the form (10) Of course, the amplitude T(O) must be unitarized including the potential part B + C too. But if this potential part is considerably smaller than the resonance one, the effect of FSI can be estimated roughly from AR~~tar. To understand

309

what gives the unitarization of Aif~~ , we use the form of the S matrix of elastic scattering with the total phase shift as a sum of the phase shifts produced by separate mechanisms of scattering [17]:

(11) Then, in terms of

==

ORes

L

and

OResj

Otot

==

ORes

+ Opot,

(12)

j

A unitar == . I V

167f 1- 4m;/s

(. s: s: sm uRes cos Upot

• s: s : ) iJ + sm UPot cos uRes e

tot.

(13)

The phase shifts ORes and Opot at s = mi< can be taken from [18], where the Resonance Chiral Theory of 7f7f Scattering was elaborated. They are ORes

Then

= 46.71

unitar

.

0

(14)

,

s:

J:

A ~ = smUPotCOS URes = AR~~tar sin ORes cos Opot

-0.156.

(15)

Therefore, the amplitude A~~~tar is small and may be neglected in a rough estimate of FSI effect. FSI in K -+ 27f decay. Basing on the result (15), the effects of FSI in the K -+ 27f amplitude may be estimated taking into account only the resonance rescattering effect. Then, in one loop approximation, the amplitude (1) is

< 7f+(P+)7f-(p_); 1= OIKO(q) Ii

[

(q2 _ p2 ) + -

At<ee( 2) Reo q (211")4i

J

>~:~-loop=

( 2 2)dn q -p p

[(p-q)Lm~)[pLm~J

+i

At,ee( Res

q

1611"

2)

(q2 _ p2 ). II _ 4m2/q2 ] -

V

11"

(16) The unitarization of this amplitude done in accordance with the prescription (8) leads to the result

°(

2 2 2 2) cos ORes I <7f7f;I=OIK q =mK»IRes=li(mK- m 1l" 1-~rrR(mi
(17)

~rrR(S = mi<) can be estimated in the framework of the phenomenological approach developed in [18], according to which an effect of ~rrR(S) may be incorporated into

(18) Then, using the parameters found in [18], one obtains ~rrR(S = mi<) = -0.12.

310

The part (17) yields 0.61 of a value of the initial amplitude (1) and the part connected with the potential rescattering, being negative, can not change the conclusion that FSI diminishes the tree amplitude.

References

[1] A.1. Vainshtein, V.1. Zakharov, M.A. Shifman, Zh. Eksp. Teor. Fiz. (JETP) 72 (1977) 1275; [Sov. Phys. JETP. 45 (1977) 670 ]i M.A. Shifman, A.I. Vainshtein, V.1. Zakharov, Nucl. Phys. B 210 (1977) 316. [2] W.A. Bardeen, A.J. Buras, J.-M. Gerard, Nucl. Phys. B 293 (1987) 787. [3] W.A. Bardeen, A.J. Buras, J.-M. Gerard, Phys. Lett. B 192 (1987) 138. [4] E.Shabalin, Yad. Fiz. 48 (1988) 272; [ Sov. J. Nucl. Phys. 48 (1988)]. [5] T. Morozumi, C.S. Lim, A.I.Sanda, Phys. Rev. Lett. 65 (1990) 404; Y.- Y. Keum, U. Nierste, A.1. Sanda, Phys. Lett. B 457 (1999) 157. [6] M.P. Losher, V.E. Markushin, H.O. Zheng, Phys. Rev. D 55 (1997) 2894. [7] E.A. Paschos, hep-ph/9912230. [8] E. Pallante, A. Pich, Phys. Rev. Lett. 84 (2000) 2568. [9] E. Pallante, A. Pich, Nucl. Phys. B 592 (2001) 294. [10] A.A. Bel'kov, et al., Phys. Lett. B 220 (1989) 459. [11] N. Isgur, et al., Phys. Rev. Lett. 64 (1990) 161. [12] G.E. Brown, et al., Phys. Lett. B 238 (1990) 20. [13] M. Neubert, B. Stech, Phys. Rev. D 44 (1991) 775. [14] S. Bertolini, J.O. Eeg, M. Fabbrichesi, Rev. Mod. Phys. 72 (2000) 65. [15] S.S. Schweber, H.A. Bethe, F.de Hoffman, Mesons and Fields, Eds.: Row, Peterson and Company, Evanston,Illinois, 1955. [16] N.N. Achasov, S.A. Devyanin, G.N. Shestakov, Yad. Fiz. 32 (1980) 1098; [Sov.J.Nucl.Phys. 32 (1980)]; Preprint TPh 109, Novosibirsk 1980; Phys. Lett. B 96 (1980) 168; Uspekhi Fiz. Nauk, 142 (1984) 361. [17] L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Non-relativistic Theory), Chapter XVIII, FizMatGiz, 1963. S.lshida et al., Prog. Theor. Phys. 95 (1996) 745. [18] E.P. Shabalin, Phys. At. Nucl. 63 (2000) 594.

Hadron Physics

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COLLECTIVE EFFECTS IN CENTRAL HEAVY-ION COLLISIONS G.I.Lykasov a, A.N.Sissakian b, A.S.Sorin c, V.D.Toneev d JINR, Dubna, 141980, Moscow region, Russia Abstract.In-medium effects on transverse-mass distributions of quarks and gluons are considered assuming a possible local equilibrium for colorless quark objects (mesons and baryons) created in central A+A collisions. It is shown that the average transverse momentum squared for these partons grows and then saturates when the initial energy increases. Within the quark-gluon string model it leads to the colliding energy dependence of hadron transverse mass spectra which is similar to that observed in heavy ion collisions. Comparison with other scenarios is given.

Experimental detection of the quark-gluon plasma (QGP) phase and the mixed phase (MP) in A+A collisions is a nontrivial task because of smallness of the space-time volume of the hot and dense system and possible contributions of hadronic processes simulating signals of QGP and MP formation. Nevertheless, recent experimental study of the transverse-mass spectra of kaons from the central Au + Au and Pb + Pb collisions revealed an "anomalous" dependence on the incident energy. The effective transverse temperature (the inverse slopeparameter of the transverse mass distribution at the mid rapidity) increases fast with incident energy in the AGS domain [1], then saturates at the SPS energies [2] and increases again approaching the RHIC energy region [3]. In agreement with expectations [4-6] this saturation was assumed to be associated with the deconfinement phase transition and indication of the MP [7,8]. The anomalous effective temperature behavior was quite successfully reproduced within a hydrodynamic model with the equation of state involving the phase transition [9]. However, this result is not very convincing since to fit data, the required incident-energy dependence of the freeze-out temperature should closely repeat the shape of the corresponding effective kaon temperature, and thereby the problem of the observed anomalous inverse-slope dependence is readdressed to the problem of the freeze-out temperature. In this paper, we propose another way to introduce a collectivity effect in a nuclear system via some in-medium effect. We consider the temperature dependence of quark distribution functions inside a colorless quark-antiquark or quark-diquark system (like meson or baryon, h) created in the central A+A collisions. A contribution of this effect to transverse momentum spectra of hadron is estimated and it is shown that it results in larger values of the inverse slope parameter and, therefore, in broadening of the transverse mass spectra. Let us assume the local equilibrium in a fireball of hadrons whose distribution ae-mail: be-mail: ce-mail: de-mail:

[email protected] [email protected] [email protected] [email protected]

313

314

function can be presented in the following relativistic invariant form:

(1) where Ph is the four-momentum of the hadron, the four-velocity of the fireball in the proper system is u = (1,0,0,0), the sign "+" is for fermions and "-" is for bosons, J.Lh is the baryon chemical potential of the hadron h, T is the local temperature, and Cr is the T -dependent normalization factor. The distribution function of constituent quarks inside h which is in local thermodynamic equilibrium with the surrounding nuclear matter, I:(x, pd, can be calculated using the procedure suggested for a free hadron in Ref. [10], see details in Ref [11],

1~(x,Pt) =

11 11 J dX1

dXh

xlt(Xh,Pht) J(x

2 d Plt d2 Pht qv(X,Pt) qr(X1,Plt)

+ Xl

- Xh) J(2)(Plt

+ Pt -

(2)

Pht) ,

where qv (pz, Pt) is related to the probability to find a valence quark with longitudinal momentum pz and transverse momentum Pt in the hadron, whereas qr (P1z, Plt) is the probability that all the other hadron constituents (one or two valence quarks plus any number of quark-antiquark qq pairs and gluons) carry a total longitudinal momentum P1z and the total transverse momentum * / VS',Xh '-I * / VS', '-I were h * * are th e S',X1 -- 2P1z Plt,. X - 2Pz* / V'-I -- 2Phz Pz* ,P1z,Phz longitudinal momenta and s' is some characteristic energy squared scale. The distribution of the hadron h in a fireball It is included in eq.(2), therefore we integrate over the longitudinal and transverse momenta of h. Assuming the factorization hypothesis qv,r(x, pd = qv,r(x) gv,r(pd and the Gaussian form for gv,r(Pt), e.g., gv,r(Pt) = V"(q/7rexp( -"(qpU2) we can get the following expression for I:(x, Pt; T) normalized to 1 [11J:

(3) where (4)

and mh(xh) = vm~ + XhS' /4. Then the averaged transverse momentum squared for the quark at x ':::' in a locally equilibrated hadron is

°

< P~,t(x ':::' 0) >~,appr.

(5)

315 T=150. MeV T=100. MeV T=O.

0.24 0.22 0.2 0.18 <X: N

c.

/\

0.16

='

0. V

0.14 0.12 0.1

o

20

40

60

80

100

120

140

Figure 1: The energy dependence of the average transverse momentum squared for the uquark in a proton in nuclear matter at temperature T.

where < PZ >~ is the transverse momentum squared for the quark in a free hadron. More accurate calculational results for this quantity of the u quark in a proton < P;,t(x ::: 0) are presented in Fig.I. Let us estimate now the Pt distribution of hadron hI produced from a collision of two hadrons h inside the fireball. We shall explore the quark-gluon string model (QGSM) [16,17] or the dual parton model (DPM) [18] based on the liN expansion in the QeD [14,15]. To calculate the transverse momentum spectrum of hadron hI in the mid rapidity region, one needs to know the pt-dependence of the fragmentation function D~l. We assume the Gaussian dependence for D~l like as for ilvApt). However, the slope of this pcdependence 'Ye can differ from the slope 'Yq for constituent quark pt-distribution

>:

where 8' has been associated with the energy squared 8hh of colliding hadrons and r = 'Yelrq. Note, that JShh is not related directly to the initial energy of colliding heavy ions. We estimated the average value of transverse momentum squared for K+mesons produced in the nucleon-nucleon < P'i+,t >~1r and pion+nucleon < P'i+,t >~~ interactions in a fireball created in the central A - A collision as a function of JShh at T = 150 MeV for two cases when 'Ye » 'Yq and 'Ye = 3'Yq [19]. In Fig.2, the curves 1 and 2 correspond to < P'i+,t >~1r and

316 0.3

:>

0.28

~

0.26

1 2

3 4 5

Q)

LO

0.24

6

"

t-

0.22 0.2 0.18 0.16

+

""

Cl..

V

0.14

//~

.....................................•..........

0.12 0.1

o

20

40

60 Shh 1/2,

80 [GeV]

100

120

140

Figure 2: Average square for the transverse momentum of K+-meson produced from the interaction of two hadrons one of them is in the equilibrated fireball as a function of "fShh at T = 150 MeV.

< P~+ ,t

>~~, respectively, when IC > > Iq , whereas the curves 3 and 4 correspond to the same quantities with IC = 3 1q . The line 5 in Fig.2 corresponds to the average square for the transverse momentum of K+ produced in the free p + p collisions < p; >~r= 0.14 GeV/c 2 . As our calculations show, the temperature dependence for < P~+,t >~t is rather weak in the interval T = 100 - 150 MeV.

As is evident from Fig.2, the obtained results are sensitive to the mass value of a hadron which is locally equilibrated with the surrounding nuclear matter at ..jShh :::; 10(GeV). We found that the quark distribution in a hadron depends on the fireball temperature T. At any T the average transverse momentum squared of a quark grows and then saturates when ..jShh increases. Numerically this saturation property depends on T. It leads to a similar energy dependence for the average transverse momentum squared of hadron hI < pt,t >~t. The saturation property for < p~ 1, t >~t depends also on the temperature T and it is very sensitive to the dynamics of hadronization. As an example, we studied the energy dependence of the inverse slope of transverse mass spectrum of K-mesons produced in central heavy-ion collisions and got its energy dependence qualitatively similar observed to one experimentally. We guess that our assumption on the thermodynamical equilibrium of hadrons given by eq.(l) can be applied for heavy nuclei only and not for the early interaction stage.

317

AcknowledgIllents

The authors are grateful for very useful discussions with P.Braun-Munzinger, K.A.Bugaev, W.Cassing, A.V.Efremov, M.Gazdzicki, S.B.Gerasimov, M.I.Gorenstein, Yu.B.Ivanov, A.B.Kaidalov and O.V.Teryaev. This work was supported in part by RFBR Grant N 05-02-17695 and by the special program of the Ministry of Education and Science of the Russian Federation (grant RNP.2.1.1.5409). References

[1) L.Ahle et. at., E866 and E917 Collaboration, Phys. Let. B476, 1 (2000); ibid. B490, 53 (2000). [2) S.V.Afanasiev et at. (NA49 Collab.), Phys.Rev. C66, 054902(2002); C. Alt et al., J. Phys. G30, S119 (2004); M.Gazdzicki, et at., J. Phys. G30, S701 (2004). [3) C.Adleret at., STAR Collaboration, nucl-ex/0206008; O.Barannikova et at., Nucl. Phys. A715, 458 (2003); K.Filimonov et at., hep-ex/0306056; D.Ouerdane et at, BRAHMS Collaboration, Nucl. Phys. A715,478 (2003); J.H.Lee et at., J. Phys. G30, S85 (2004); S.S.Adler et at., PHENIX Collaboration, nucl-ex/030701O; nuclex/0307022. [4) E.V.Shuryak, Phys. Rep. 61, 71 (1980). [5) E.V.Shuryak and O.Zhirov, Phys. Lett. B89, 253 (1980); Yad. Fiz. 28, 485 (1978) [Sov. J. Nucl. Phys. 28,247 (1978). [6) L. van Hove, Phys. Lett. B118, 138 (1982). [7) M.Gorenstein, M.Gazdzicki and K.Bugaev, Phys. Lett. B567, 175 (2003). [8) B.Mohanty, et at., Phys. Rev. C68, 021901 (2003). [9) M.Gazdzicki et at., Braz. J. Phys. 34, 322 (2004). [10) J.Kuti and V.F.Weiskopf, Phys. Rev. D4, 3418 (1971). [11) G.I.Lykasov, A.N.Sissakian, A.S.Sorin, D.V.Toneev, in preparation. [12) A.Capella, V.J.Tran Than Van, Z.Phys.ClO, 249 (1981). [13) O.Benhar, S.Fantoni, G.I.Lykasov, N.V.Slavin, Phys. Rev. C55, 244 (1997). [14) G.'t Hooft, Nucl. Phys., B72, 461 (1974). [15) G.Veneziano, Phys. Lett., B52, 220 (1974). [16) A.B.Kaidalov and K.A.Ter-Martirosyan, Phys. Lett. B117, 247 (1982). [17) A.B.Kaidalov and O.I.Piskunova, Z. Phys. C30, 145 (1986). [18) A.Capella, U.Sukhatme, C.L.Tan, J. Tran Thanh Van, Phys.Rep. 236, 225 (1994). [19) G.I.Lykasov and M.N.Sergeenko, Z. Phys. C70, 455 (1996).

STRINGY PHENOMENA IN YANG-MILLS PLASMA V.1. Zakharov a

INFN, Sezione di Pisa, Largo PontecoTVo 3, 56127, Pis a, Italy ITEP, B.Cheremushkinskaya 25, Moscow, 117218, Russia Abstract. We review the grounds for and consequences from the hypothesis that at the point of the confinement-deconfinement phase transition both electric and magnetic strings are released into the Yang-Mills plasma. We comment also briefly on the averaged Polyakov line as an order parameter of the deconfinement phase transition.

1

Introduction

The goal of this talk is to substantiate a phenomenological stringy picture for the confinement-deconfinement phase transition. The stringy picture for the phase transition was advocated first long time ago [1] and the topic is, in its generality, too broad for such a talk. Thus, we will concentrate on a recent proposal [2,3] that there exists a magnetic component of the Yang-Mills plasma at temperatures close and above the critical temperature Te. While the main ideas are presented in the original papers [2], there appeared most recently results of dedicated lattice measurements [4,5] which support the picture proposed although much more remains to be done before one could really claim observation of the magnetic component of the plasma. Electric strings

2 2.1

Action vs entropy factoTs

Consider quark and anti-quark separated by distance x. To make the construct gauge invariant one has to connect the quarks by a string: (1)

The path-ordered exponent is our first image of what we would call electric string. If quarks develop in time, the string sweeps an area A. Let us consider the most primitive dynamics of a closed string. The string carries color charge and, therefore has a divergent self-energy. To regularize this divergence, introduce finite thickness of the string, TO , TO « Ixl. Then the corresponding action is of order bare Sstring

=

CIg 2(TO )Aj TO2

=

O"bare'

A

.

(2)

To evaluate the renormalized, or physical string tension O"ren one has to subtract from (2) the entropy factor (see, e.g., [6]): ae-mail: [email protected]

318

319

where Nstring is the number of various surfaces with the same area constant C2 is of pure geometrical origin. As a result b:

A,

(3) the

(4) Consider first ro = a, where a is the lattice spacing. In the limit of the large coupling, g2(a) » 1, the bare action factor prevails and the renormalized tension is positive. We have the strong-coupling confinement. This string is infinitely thin but theory is not realistic because of the strong-coupling limit. In the asymptotic-freedom case, g2(a) -+ 0 the tension (4) is negative and the string is unstable in the ultraviolet. In the ultraviolet, on the other hand, free gluons is the right approximation and strings with a negative tension is not a viable alternative. Next, we can still consider the asymptotic-freedom case but choose the thickness of the string ro rv AQ~D' Then g2(ro) can be large enough to make the renormalized tension (4) positive. Thus, we might have 'thick' strings which could be useful effective degrees of freedom in the infrared. At large temperatures g2 is limited by g2 (T), limT -+00 g2 (T) -+ 0 since the time extension of the lattice (Euclidean space-time) is liT. Thus, at an intermediate temperature the effective tension (4) vanishes and the electric strings percolate through the vacuum. 2.2

The Polyakov line

Continuing with the finite-temperature physics, another image for the electric strings is provided by the the Polyakov line which is a Wilson line winding once through the lattice in the periodic time direction:

r

llT

P == Trn

=

TrPexp}o

Ao(x,r)dr,

(5)

where the trace is taken in the fundamental representation. Imagine that we would like to use the Hamiltonian formalism and gauge Ao = O. Unlike the case of T = 0 it is not possible to fix Ao = 0 because of the periodicity in the time direction. In other words, the non-local variable (5) is gauge invariant and cannot be eliminated by gauge transformations. It is still possible to put Ao = 0 provided that the non-local degree of freedom (5) is added explicitly [1] into the partition function:

Z[n] = b Actually,

J

DAn(x,r)exp ( -

Jd3xdr((a~k)2+Ffl))

we oversimplify the estimate of the entropy greatly, see, in particular, [7].

(6)

320 where Ak(x, (3) = n- 1Adx, O)n + n-10kn(X). Note, however, that by introducing a new variable we admit extra ultraviolet divergences into the theory and make the model non-renormalizable, for further references see, e.g., [8]. 3

3.1

Magnetic strings

Topology of the magnetic strings

In Yang-Mills theories, one expects that the magnetic strings are no less fundamental than electric strings. Moreover, condensation of magnetic degrees of freedom is commonly believed to be responsible for the confinement. The scenario is realize in the Abelian case, [9]. The magnetic degrees of freedom are identified in this case through violations of the Bianchi identities:

(7) where j::,on is the monopole current. In the non-Abelian case, the gauge potential can be expressed in terms of the field strength tensor [10]:

(8) where G~l is the matrix inverse to the matrix of the field strength tensor. As far as (8) holds, the Bianchi identities are valid automatically. There might exist, however, such field configurations that the inversion (8) is not possible because the matrix G- 1 does not exist. Actually, it was noted from the very beginning [10] that the inversion (8) fails in 2d case, see also below. Alternatively, in 4d case there can exist 2d defects [11] along which the matrix G- 1 is singular. These 2d defects is our image for the magnetic strings. Magnetic strings are to be added as new degrees of freedom to the standard YM theories which assume Bianchi identities valid. 3.2

Surface operators, monopoles

The action associated with the 2d defects can be readily guessed on symmetry grounds. In fact, such surfaces were considered, for other reasons, in Refs. [12,13] (in the latter reference they were labeled as surface operators). Namely, consider a surface, with area element da 1"'-' and introduce the action: S s'ur face

= const

J J1.vG~v da

(no summation over

j-l,

v) .

(9)

The central point is that the action (9) respects non-Abelian invariance despite of the fact that it apparently carries a color index a. The reason is that one can

321

use gauge invariance to rotate one particular component of the field strength tensor to the Cartan subgroup:

(10) where for simplicity we considered the gauge group SU(2). In other words, non-Abelian fields living on a surface are in fact Abelian. The inversion (8), on the other hand, is specific for the non-Abelian case and fails in the Abelian case. Thus, the magnetic strings replace the magnetic monopoles (7) relevant to the Abelian case. It is worth emphasizing that to carry a finite magnetic flux the surfaces (9) are to be endowed with singular fields G~v. Note that gauge fixation (10) fails if G~v = o. Such defects are trajectories living on the surfaces and correspond to non-Abelian monopoles, for related discussion see [14].

3.3

Dual pictures of confinement

There is a deep relation between magnetic and electric strings. Namely, the expectation value of the Wilson line, < W > can be evaluated either in terms of electric strings open on the heavy quarks, or in terms of the linking number between electric and magnetic strings, [15,16]. It is useful to start with 3d and consider a tube of magnetic field which pierces a surface spanned on the Wilson line. Considering, for simplicity, the Abelian case one gets for the Wilson line:

exp(ifAJ.!dxJ.!)

=

exp(i
(11)

,

where is the magnetic flux. In 4d there can be a non-trivial linking number between two 2d surfaces. In our case, one of the surfaces is spanned on the Wilson line (and corresponds to an electric string) while the other is swept by the magnetic strjng. Linear heavy-quark potential implies that

< W > ~ exp (-

Aminacon/)

,

(12)

where Amin is the minimal area spanned on the Wilson line. One can say that (12) means that the expectation value of the Wilson line is suppressed in the strongest possible way. To get such a suppression the magnetic flux carried by the magnetic strings is to be random [17]. This, in turn, implies that the magnetic strings percolate through the vacuum. For this to happen, the magnetic strings are to have a vanishing tension, (13) amagn = O. Eq. (13) means that the heavy-monopole potential is not confining, as it should be.

322 It is remarkable that we derived (13) starting with consideration of the Wilson line, not directly of the 't Hooft line. 4

Extra dimensions

The logic outlined in the preceding sections has a weak point since we mix up two different pictures for the confining string, that is, thin and thick strings. Indeed, the string which can be open on quarks is to be infinitely thin since quarks are point-like while the string which has tension in physical units, see Eq (4) has thickness of order AQ~D. This inconsistency is in fact difficult to remove and the way out which is becoming common nowadays introduces a novel notion of extra dimensions, or running string tension, for review see [18]. Roughly speaking, one is assuming the string to be infinitely thin but with tension depending on its size. For areas A :S AQ~D' (Jeff

rv

I/A

(14)

For larger areas, the string tension is frozen, (Jeff

~

(Jeanf

,

if A :::: AQ~D ,

(15)

where (Jeanf determines heavy-quark linear potential at large distances. Formulae (14), (15) are somewhat loose because area is not the only characteristic of a surface. It turns out that language of extra dimensions is much more adequate. In this framework, one postulates that there exists an extra dimension, z such that 'our' world corresponds to z = 0 while strings connecting quarks extend into z =I o. The action associated with the string is the same Nambu-Goto action which we actually discussed above but now the area is calculated with account of geometry which is a nontrivial function of z. In particular, assuming that the metric is (16) where R2 is a constant and Xi ,i = 1, .. ,4, are Euclidean 4d coordinates, one reproduces Coulomb-like heavy-quark potential. This is quite obvious from dimensional considerations and the metric (16) realizes the assumption (14). Concerning realization of (15) it is much more arbitrary since one introduces by hand a new parameter, AQCD . The following model

R2 2 ds = exp(cz 2 ) 2" (Jdt 2 +dx; +r1dz2), f(z) = 1-(nzTe)4, c ~ GeV 2, z (17) gives a reasonable description of broad variety of phenomena both at zero and finite temperatures [19].

323

As for the magnetic strings, they have another geometry and correspond to branes wrapped on extra compact dimensions, which are to be added to the five dimensions already introduced, for details see [18]. Magnetic monopoles, in this language, are Kaluza-Klein modes associated with the extra compact dimensions [11].

5

Stringy phenomena near the critical temperature

After all these preliminary remarks we are in position to make predictions specific for the string-based phenomenology of Yang-Mills theories.

5.1

Polyakov line as an order parameter

As argued first in Ref. [1], in pure Yang-Mills case (without quarks) the expectation value of the Polyakov line (5) serves as an order parameter:

(P) == 0 if T < Te .

(18)

In the explicit calculations [19] with the metric (15) the averaged Polyakov line contains at small temperatures exponentially small terms < P >~ exp( -constjT) and Eq. (18) does not hold. Although non-observance of (18) could well be a consequence of the approximations made, it might be useful to understand the reasons for this discrepancy. The proof of (18) exploits the center symmetry. Namely, the Polyakov line is changed by a phase factor under transformations belonging to the group center of the gauge group while the lattice Yang-Mills action can be formulated as symmetrical under the center transformations. However, the lattice action might not know about the center symmetry as well, (for recent discussion and references see [20]). There is no center-group symmetry in the stringy approach, based on (15) but probably there is nothing wrong about this. Thus, violations of (18) seemingly cannot be ruled out on general grounds. There are further interesting issues to discuss in this connection. In particular, the stringy formulation (15) leads to qualitative predictions which are in accord with the lattice data [22], like fast growing entropy in the system of heavy quarks towards T = Te. On pure theoretical side, dependence of continuum physics on details of the lattice regularization (whether we have the center symmetry or not) is most challenging. Because of space considerations, we cannot go into detailed discussion of these issues here, however.

5.2

Magnetic component of the Yang-Mills plasma

We have already mentioned that at the point of the phase transition, Te one expects [1] vanishing tension of the electric string:

(Jeleetr(T) 2: 0,

T 2: Te

(19)

324 On the other hand, tension of the confining string can be evaluated in terms of the magnetic strings, see subsection 3.3. Thus, Eq. (19) implies that magnetic strings acquire non-zero tension at T > Te:

(J"magn(T) 2: 0, T 2: Te .

(20)

Thus, in the deconfining phase the magnetic strings correspond to physical degrees of freedom and are to be present in the Yang-Mills plasma [2]. The question is, how to detect this effect. On the lattice, magnetic strings are identifiable directly, for details see [11]. And, indeed, the magnetic strings do not percolate at T > Te, for references see [15]. More quantitative predictions can be made in terms of the monopoles, which are, as explained above, particles living on the strings. The word 'particles' is to be perceived with some caution, however, since we are discussing now the lattice, or Euclidean formulation and the difference between virtual and real particles is not so obvious as in the Minkowski space. Nevertheless, one can argue [2] that the density of real (in the Minkowskian sense) particles is proportional to the density of the so called wrapped trajectories [21] which are trajectories stretching in the time direction from one boundary to the other:

Preal(T) '" Pwrapped(T) , T

> Te.

(21)

This relation implies, in turn, that the density pwrapped is to be in physical units, '" i\~CD and cannot depend on the lattice spacing. This is in fact a very strong constraint on the data. Which indeed turns to be true [5].

5.3

Ghost-like matter

Measurements on the magnetic strings, reveal [4] astonishingly enough, that both energy density and pressure associated with the magnetic strings are negative: tmagn(T) < 0, Pmagn(T) < 0, Te < T < 2Te . (22) There is a proposal [23] how to accommodate this observation within the stringy picture. The basic idea is that in 2d and 4d the conformal anomaly has opposite signs and this is responsible for the ghost-like sign in case of the 2d defects (22). Acknowledgments

I am indebted to O. Andreev, M.N. Chernodub, A. Di Giacomo, M. D'Elia, A.S. Gorsky for enlightening discussions. References

[1] A. M. Polyakov, Phys. Lett. B72, 477 (1978); "Confinement and liberat'ion", [arXiv:hep-th/0407209].

325 [2] M.N. Chernodub and V.1. Zakharov, Phys. Rev.Lett. 98, 082002 (2007); "Magnetic strings as part of Yang-Mills plasma ", [arXiv:hepphj0702245]. [3] Ch. P. Korthals Altes, "Quasi-particle model in hot QCD", [arXiv:hepphj0406138]; Jinfeng Liao and E. Shuryak, Phys. Rev. C75, 054907 (2007), [arXiv:hep-phj0611131]. [4] M.N. Chernodub et al., "Topological defects and equation of state of gluon plasma", [arXiv:0710.2547]. [5] A. D'Alessandro and M. D'Elia, "Magnetic monopoles in the high temperature phase of Yang-Mills theories", [arXiv:0711.1266]. [6] A.M. Polyakov, "Gauge Fields and Strings", Harvard Academic Publishers, (1987) . [7J A.B. Zamolodchikov, Phys. Lett. B117, 87 (1982). [8] J. C. Myers and M.C. Ogilvie, "New phases of finite temperature gauge theory from an extended action", [arXiv:0710.0674J; Ph. de Forcrand, A.Kurkela, A. Vuorinen, "Center-Symmetric Effective Theory for HighTemperature SU(2) Yang-Mills Theory" [arXiv:0801.1566]. [9] A.M. Polyakov, Phys. Lett. B59, 82 (1975); M. E. Peskin, Annals Phys. 113, 122 (1978). [lOJ M.B. Halpern, Phys. Rev. D16, 1798 (1977); ibid D19, 517 (1979). [l1J V.1. Zakharov, Braz. J. Phys. 37, 165 (2007), [arXiv:hep-phj0612342J. [12J M.N. Chernodub, F.V. Gubarev, M.1. Polikarpov, V. I. Zakharov, Nucl.Phys. B600, 163 (2001), [arXiv:hep-thjOOl0265]. [13J S. Gukov and E. Witten, "Gauge Theory, Ramification, And The Geometric Langlands Program", [arXiv:hep-thj0612073J. [14] G. 't Hooft, Nucl. Phys. B190 , 455 (1981). [15J J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003), [arXiv:heplatj0301023]. [16J V.1. Zakharov, AlP Conf. Proc. 756, 182 (2005), [arXiv:hepphj0501011]. [17] A. Di Giacomo, H. G. Dosch, V.1. Shevchenko, Yu.A. Simonov, Phys. Rept. 372, 319 (2002), [arXiv:hep-phj0007223]. [18J O. Aharony et al., Phys. Rept. 323,1832000, [arXiv:hep-thj9905111J. [19J O. Andreev, V.1. Zakharov, Phys. Rev, D74, 025023 (2006),[arXiv:hepphj0604204]; Phys. Lett. B645, 437 (2007), [arXiv:hep-phj0607026]; JHEP, 0704:100 (2007), [arXiv:hep-phj0611304]. [20J G. Burgio, PoS(LAT2007), 292 (2007), [arXiv:0710.0476J. [21] V.G. Bornyakov, V.K. Mitrjushkin, M. Muller-Preussker , Phys. Lett. B284, 99 (1992). [22J P. Petreczky, Nucl. Phys. A 785, 10 (2007), [arXiv:hep-Iatj0609040]. [23J A. Gorsky, V. Zakharov, "Magnetic strings in Lattice QCD as Nonabelian Vortices", [arXiv:0707.1284J.

LATTICE RESULTS ON GLUON AND GHOST PROPAGATORS IN LANDAU GAUGE I.L. Bogolubsky Joint Institute for Nuclear Research, 141980 Dubna, Russia

V.G. Bornyakov a Institute for High Energy Physics, 142281 Protvino, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia

G. Burgio Universitiit Tilbingen, Institut filr Theoretische Physik, 72076 Tilbingen, Germany

E.-M. Ilgenfritz, M. Miiller-Preussker Humboldt-Universitiit zu Berlin, Institut filr Physik, 12489 Berlin, Germany

V.K. Mitrjushkin Joint Institute for Nuclear Research, 141980 Dubna, Russia and Institute of Theoretical and Experimental Physics, Moscow, Russia Abstract. We present clear evidence of strong effects of Gribov copies in Landau gauge gluon and ghost propagators computed on the lattice at small momenta by employing a new approach to Landau gauge fixing and a more effective numerical algorithm. It is further shown that the new approach substantially decreases notorious finite-volume effects.

1

Introduction

The gauge-variant Green functions, in particular for the covariant Landau gauge, are important for various reasons. Their infrared asymptotics is crucial for gluon and quark confinement according to scenarios invented by Gribov [1] and Zwanziger [2] and by Kugo and Ojima [3]. They have proposed that the Landau gauge ghost propagator is infrared diverging while the gluon propagator is infrared vanishing. The interest in these propagators was stimulated in part by the progress achieved in solving Dyson-Schwinger equations (DSE) for these propagators (for a recent review see [4]). Recently it has been argued that a unique and exact power-like infrared asymptotic behavior of all Green functions can be derived without truncating the hierarchy of DSE [5]. This solution agrees completely with the scenarios of confinement mentioned above. The lattice approach is another powerful tool to compute these propagators in an ab initio fashion but not free of lattice artefacts. So far, there is no consensus between DSE and lattice results. For the gluon propagator, the ultimate decrease towards vanishing momentum has not yet been established in lattice computations. Lattice results for the ghost propagator qualitatively agree with the predicted diverging behavior but show a substantially smaller infrared exponent [6]. The lattice approach has its own limitations. The effects of the finite volume might be strong at the lowest lattice momenta. Moreover, gauge fixing is ae-mail: [email protected]

326

327 not unique resulting in the so-called Gribov problem. Previously it has been concluded that the gluon propagator does not show effects of Gribov copies beyond statistical noise, while the ghost propagator has been found to deviate by up to 10% depending on the quality of gauge fixing [7,8]. Recently anew, extended approach to Landau gauge fixing has been proposed [9]. In this contribution we present results obtained within this new method and using a more effective numerical algorithm for lattice gauge fixing, the simulated annealing (SA) algorithm. Results for the gluon propagator have been already discussed in [10], while results for the ghost propagator are presented here for the first time. 2

Computational details

Our computations have been performed for one lattice spacing corresponding to rather strong bare coupling, at f3 == 4/ g6 = 2.20, on lattices from 84 up to 324. The corresponding lattice scale a is fixed adopting ..j(ia = 0.469 [ll] with the string tension put equal to a = (440 Me V)2. Thus, our largest lattice size 32 4 corresponds to a volume (6.7 fm)4. In order to fix the Landau gauge for each lattice gauge field {U} generated by means of a Me procedure, the gauge functional

(1) is iteratively maximized with respect to a gauge transformation g(x) which is usually taken as a periodic field. In SU(N) gluodynamics the lattice action and the path integral measure are invariant under extended gauge transformations which are periodic modulo Z(N),

g(x + Lv) = z"g(x),

z"

E Z(N)

(2)

in all four directions. Any such gauge transformation is equivalent to a combination of a periodic gauge transformation and a flip Ux " ----* z" Ux " for a 3D hyperplane with fixed X". With respect to the flip transformation all gauge copies of one given field configuration can be split into N 4 flip sectors. The traditional gauge fixing procedure considers one flip sector as a separate gauge orbit. The new approach suggested in [9] combines all N 4 sectors into one gauge orbit. Note, that this approach is not applicable in a gauge theory with fundamental matter fields because the action is not invariant under transformation (2), while in the deconfinement phase of SU(N) pure gluodynamics it should be modified: only flips in space directions are left in the gauge orbit. In practice, few Gribov copies are generated for each sector and the best one over all sectors is chosen by employing an optimized simulating annealing algorithm in combination with finalizing overrelaxation.

328 3

Results

Thus, we are looking for the gauge copy with the highest value of the gauge functional among gauge copies belonging to the enlarged gauge orbit as defined above. It is immediately clear that this procedure allows to find higher local maxima of the gauge functional (1) than the traditional ('old') gauge fixing procedures employing purely periodic gauge transformations and the standard overrelaxation algorithm. Obviously the two prescriptions to fix the Landau gauge, the traditional one and the extended one, are not equivalent. Indeed, for some modest lattice volumes and for the lowest momenta it has been shown in Ref. [9] that they give rise to different results for the gluon as well as the ghost propagators. Comparing results for different lattice sizes we found that the results seem to converge to each other in the large volume limit. It is important that results obtained with the new prescription converge towards the infinite volume limit much faster. In Fig. 1 the gluon propagator D(p2) is 9~~~~~~~~~~~~

••

......

8 7

N

>6 o --5 Q)

L = 1.7 fm 0 L = 2.5 fm L = 3.4 fm v L = 5.0 fm '" eL = 6.7 fm 0 <)

'.~

,...-I

-----. 4 No.. 3

o

'-'

•

2 1

o

0

0.2

0.4

0.6

0.8

1

p, GeV Figure 1: Comparison of the gluon propagator computed with new (empty symbols) and old (filled symbols) procedures on lattices of various sizes. Error bars which are in most cases smaller than symbol sizes are not shown for clarity. The curve corresponds to eq. (3).

shown. b One can see that the Gribov copy effects are strong up to p rv 0.6 GeV. Furthermore, results obtained with the new procedure show no finite volume effects while these effects are clearly seen for results obtained with the old procedure. We have also checked, whether our result can be seen in agreement with the expectation D(p -+ 0) = O. We made a fit for 0 < p < 0.5 GeV with the function

(3) bFor definition of D(p2) as well as of the ghost dressing function J(p2) see Refs. [9,10].

329 and found a = 0.09(1) which is in qualitative agreement with the DSE result [4]. In Fig. 2 we show our results for the ghost dressing function J(p2). Again, strong Gribov copies effects are seen. On the other hand, the finite-volume effects are not strong for both procedures. We confirm earlier SU(3) results [6] that the lattice ghost propagator is much less diverging in the infrared limit than it is predicted by DSE solutions [4]. Moreover, within the new procedure this disagreement becomes even stronger. This is one of the problems to be resolved in future studies.

3.5

o

L = 3.4 fm v L = 5.0 fm " L = 6.7 fm 0

. All

.....

o ~

N

3

0..

'--"

to-,

2.5

~

..

" . .,

".

VA ..

"t

. )[

2 1.5 0

" ., 0.2 0.4

0.6

0.8

1

.""

1.2

"

1.4

p, GeV Figure 2: Ghost dressing function for two procedures. Symbols are as in Fig. 1.

References [1] [2] [3] [4] [5] [6] [7]

V. N. Gribov, Nucl. Phys. B 139, 1 (1978). D. Zwanziger, Nucl. Phys. B 412, 657 (1994). T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66, 1 (1979). R. Alkofer, Eraz. J. Phys. 37, 144 (2007). C. S. Fischer and J. M. Pawlowski, Phys. Rev. D 75, 025012 (2007). E.-M. Ilgenfritz, et al., Eraz. J. Phys 37, 193 (2007). T. D. Bakeev, E. M. Ilgenfritz, V. K. Mitrjushkin and M. MuellerPreussker, Phys. Rev. D 69 (2004) 074507 [arXiv:hep-lat/0311041]. [8] A. Sternbeck, E.-M. Ilgenfritz, M. Miiller-Preussker, and A. Schiller, Phys. Rev. D 72, 014507 (2005). [9] 1. L. Bogolubsky, G. Burgio, M. Miiller-Preussker, and V. K. Mitrjushkin, Phys. Rev. D 74, 034503 (2006). [10] 1. L. Bogolubsky, et al., arXiv: 0707.3611 [hep-lat]]. [11] J. Fingberg, U. M. Heller, and F. Karsch, Nucl. Phys. B 392, 493 (1993).

~

AND S EXCITED STATES IN FIELD CORRELATOR METHOD

Ilya Narodetskii a, Alexander Veselov b Institute of Theoretical and Experimental Physics, 117258 Moscow, Russia

Abstract.The energies of the P-wave hyperon excited states have been computed employing the field correlator method in QCD. The hyperon spectrum appears to be expressed through two parameters relevant to QCD, the string tension (T and the strong coupling constant as, and the bare strange quark mass ms. Using these parameters a unified description of the ground and excited hyperon states is achieved. In particular, we find that both the nucleon and hyperon states have the similar cost (in 6.L) ~ 460 MeV.

1

Introduction

A comprehensive knowledge about the baryon mass spectrum from first principles is important for our understanding of quantum chromodynamics. Ground state spectroscopy on the lattice is by now a well understood problem and impressive agreement with experiments has been achieved. However, the lattice study of excited states is not so advanced, see [1] and references therein. The purpose of this contribution is to present a consistent treatment of the P-wave hyperons within the alternative method in QCD, the field correlator method (FCM) [2J. Our investigation was initially motivated as an attempt to extend an approach of ref. [3] for low-lying orbitally excited baryons.

2

The Effective Hamiltonian in FCM

In the FCM one derives the Effective Hamiltonian (EH), which comprises both confinement and relativistic effects, and contains only universal parameters: the string tension a, the strong coupling constant as, and the bare (current) quark masses. In what follows we compute only the confinement energies (corrected by the perurbative one gluon exchange potential) and disregard the spin-spin and spin-orbit interactions. The application of the method for the baryons was described in detail elsewhere [5J. Here we give only a brief summary important for this particular calculation. The EH has the following form:

H=

t

i=l

-----------------------ae-mail: [email protected]

(_m_; + /1i) + Ho + V, 2/1i

2

be-mail: [email protected]

330

(1)

331

where Ho is the kinetic energy operator. V is the sum of the string potential Vy(rl, r2, r3) and a Coulomb interaction term arising from one gluon exchange. The string potential is

(2) where a is the string tension and rmin is the minimal length string corresponding to the Y-shaped configuration. The one gluon exchange interaction is VCoulomh = -

2 -3 Q s

1

(3)

"""' - ,

~r" i<j

'J

where Q s is the strong coupling constant and rij are the distances between quarks. In Eq. (1) mi are the bare quark masses, while J-li are the constant auxiliary einbein fields, initially introduced in order to get rid of the square roots appearing in the relativistic Hamiltonian [4]. The dynamics remains essentially relativistic, though being non-relativistic in form. The einbein fields are eventually treated as variational parameters. The eigenvalue problem is solved for each set of J-li, then one has to minimize (H) with respect to J-li. Such an approach allows one a very transparent interpretation of einbeins: J-li can be treated as constituent masses of quarks of current mass mi. In this way the notion of constituent masses arises. The baryon mass is given by the formula (4) where EO(mi, J-li) is an eigenvalue of the operator Ho the condition

+ V,

J-li are defined from

(5) and C is the quark self-energy correction which is created by the color magnetic moment of a quark propagating through the vacuum background field [6]. The effect of the quark self-energy is to shift the mass spectrum by a global negative amount C : (6) where the function 1](t) has been calculated in [6], l/Tg is the gluon correlation length. In this paper we use Tg = 1 GeV. clts negative sign is due to the paramagnetic nature of the particular mechanism at work in this case.

332

3

Results

We do not perform a systematic study in order to determine the best set of parameters (J, as and ms to fit the hyperon spectra. Instead, we employ some typical values of (J and as that have been used for the description of the ground state baryons: (Js = 0.15 GeV2 and as = 0.39. We use the values ofthe current light quark masses mu = md = 7 MeV, ms = 100 and 175 MeV. The three-quark problem has been solved using the hyperspherical approach. Our main results for the p and A excitations ofthe P wave qqs and ssq hyperons are displayed in the Table 1. The p excitations correspond to the excitations of the like quarks (ud for ~ or ss for 2), while the A excitations correspond to the excitations of the odd quark. We present the dynamical quark masses, zero-order eigenenergies and the hyperon masses for ms = 100 and 175 MeV. The dynamical quark masses J.1i corresponding to the excitations p and A are somewhat different. This is not surprising because these quantities can be considered as the average kinetic energies of the current quarks, which are larger for the quarks in the P-wave and smaller for the quarks in the S-wave. The eigenenergies Eo of the p and A excitations are degenerate, the masses of the corresponding states are nearly degenerate: they differ no more than 10 MeV. This difference is due to the difference of the dynamical masses J.1i which enter the mass formula (4) and the self energy contribution (6). The excitation energies t::.=M(L=l) - M(L=O) are of the order of 460 MeV both for ~ and 2 and also coincide with the excitation energies for the nucleon. The chiral physics effects, missing our approach, can shift nucleon and A states. However, the chiral effects are less important for the ~ states. Therefore we identify two approximately degenerate qqs excited states of negative parity with the P-wave ~ resonances. The PDG lists in the considered mass region the three ~ resonances, D 13 (1670), 8 11 (1750) and D 15 (1775), with JP = (3/2)-, (1/2)-, (5/2)- respectively. The latter state corresponds to the A - excitation with 8 = 3/2. The results are compatible with the known states, showing discrepancies with the experimental data of order 5% or less. We finally remark that our result for the negative parity ground state in the 2 channel M (L = 1) = 1781 Me V agrees well with the recent finding from the lattice quenched calculation

[lJ. 4

Conclusions

In this paper we have extended our previous study of the ground states of the qqq, qqs and ssq baryons to the description of their first angular excitations. We use the EH method. For each baryon we calculated the dynamical quark masses J.1i from Eq. (5), energy eigenvalues, and the baryon masses (4) with the self-energy corrections (6). Our study suggests that a good description of

333

Hyperon

ms

qqs

100 100 175 175

ssq

100 100 175 175

Excitation

J.Ll = J.L2

J.L3

Eo

M

p P A

479 438 482 440

431 509 457 532

1627 1629 1612 1616

1724 1717 1774 1758

P A P A

491 452 518 480

419 500 423 506

1621 1620 1594 1591

1745 1752 1829 1845

,\

Table 1: Masses of the p and>" hyperon excitations. Shown are the dynamical quark masses JLi. the confinement energies Eo and the hyperon masses M (all in units of MeV).

the P-wave baryons can be obtained with a spin independent energy eigenvalues corresponding to the confinement plus Coulomb potentials. Moreover this comparative study gives a better insight into the quark model results where the constituent masses encode the QCD dynamics.

Acknowledgment This work was supported by RFBR grants 05-02-17869 and 06-02-17120. References [1] T. Burch et al., Phys, Rev. D 74, 014504 (2006) [2] H. G. Dosch, Phys. Lett. 190, 177 (1987), H. G. Dosch, Yu. A. Simonov, Phys. Lett. 202, 339 (1988) [3] 1. M. Narodetskii and M. A. Trusov, Phys. Atom. Nucl. 67, 762 (2004); Yad. Fiz. 67, 783 (2004) [4] A. Yu. Dubin, A. B. Kaidalov, and Yu. A. Simonov, Phys. Lett. B323, 41 (1994) ; B343, 310 (1995) [5] O. N. Driga, 1. M. Narodetskii, A. 1. Veselov, arXiv: hep-ph/0712.1479 [6] Yu. A. Simonov, Phys. Lett. B 515, 137 (2001) ; A. DiGiacomo and Yu. A. Simonov, Phys. Lett. B 595, 368 (2001)

THEORY OF QUARK-GLUON PLASMA AND PHASE TRANSITION E.V.Komarov a, Yu.A.Simonov b ITEP, Moscow Abstract.Nonperturbative picture of strong interacting quark-gluon plasma is given based on the systematic Field Correlator Method. Equation of state, phase transition in density-temperature plane is derived and compared to lattice data as well as subsequent thermodynamical quantities of QGP.

1

Introduction

The perturbative exploring of quark-gluon plasma (QGP) has some difficulties in describing the physics of QGP and phase transitions. However, it was realized 30 years ago that nonperturbative (np) vacuum fields are strong ( [1]) and later it was predicted ( [2]) and confirmed on the lattice ( [3]) that the magnetic part of gluon condensate does not decrease at T > Te and even grows as T4 at large T. Therefore it is natural to apply the np approach, the Field Correlator Method (FCM) ( [4]) to the problem of QGP and phase transitions, which was done in a series of papers ( [5]- [9]). As a result one obtains np equation of state (EoS) of QGP and the full picture of phase transition, including an unbiased prediction for the critical temperature Te(P,) for different number of flavors nt.

2

Nonperturbative EoS of QGP

We split the gluonic field AIL into the background field BIL and the (valence gluon) quantum field aIL: AIL = BIL + aIL both satisfying the periodic boundary conditions. The partition function averaged both in perturbative and np fields is

Z(V, T) = (Z(B

+ a))B,a

(1)

Exploring free energy F(T,p,) = -Tln(Z(B))B that contains perturbative and np interactions of quarks and gluons (which also includes creation and dissociation of bound states) we follow so-called Single Line Approximation (SLA). Namely, we assume that quark-gluon system for T > Te stays gauge invariant, as it was for T < Te , and neglect all perturbative interactions in the first approximation. Nevertheless in SLA already exist a strong interaction of gluons (and quarks) with np vacuum fields. This interaction consists of colorelectric (CE) and colormagnetic (CM) parts. The CE part in deconfinement phase creates np self-energy contribution for every quark and gluon embedded ae-mail: [email protected] be-mail: [email protected]

334

335

in corresponding Polyakov line. An important point is that Polyakov line is computed from the gauge invariant qij (gg) Wilson loop, which for np Df interaction splits into individual quark (gluon) contributions. As for CM part - its consideration is beyond the SLA, because as has been recently shown in paper ( [10]) strong CM fields are responsible for creation of bound states of white combinations of quarks and gluons. To proceed with FCM we apply the nonabelian Stokes theorem and the Gaussian approximation to compute the Polyakov line in terms of np field correlators Lfund = tr Pexp (i9 B 4 (z)dz 4 ) = c

It

J

J tr c

with

exp ( -~

ISnISn daJ.L" (u)dO)'o.a (v)DJ.L"')..cr ) (2)

DJ.L"')..cr == g2 (FJ.L" (u)(u, v)F>.a(v)(V, u)) Df and DE arise from CE field strengths:

[E +DIE +U420D f] oDf ou~ +UiUk oiP

1 Nc DOi,Ok = 6ik D

(3)

As a result the Polyakov loop can be expressed in terms of "potentials" VI and VD VI(T)+2VD) 9/4 (4) Lfund = exp ( 2T ,Ladj = (Lfund) , with Vl(T) == VI(oo,T), VD == VD(r*,T) ([5])

1 21

00

VI(r, T)

=

dv(l - vT)

00

VD(r, T)

=

dv(l - vT)

1T d~ ~Df( J~2 + v2)

lT d~

(r -

(5)

~)DE( Je + v2 )

(6)

In what follows we use the Polyakov line fit ( [8,9]) Lfund

(x = ~, T) = exp (- (1.~~:5~~~2T )

(7)

The free energy F(T) of quarks and gluons in SLA can be expressed as a sum over all Matsubara winding numbers n with coefficients Ljund and L~dj for quarks and gluons respectively. For nonzero chemical potential one can keep L fund,adj independent of 1-", treating np fields as strong and unchanged by in the first approximation. The final formulas for pressure of qgp are ( [7,9])

I-"

I-"

SLA

p pq == ~4

= n1f~

[(I-"-Yl.) " T + " (I-"+Yl.)] -T

(8)

336 12

l,...,

DB

D~

2 D.4

02

0 1~

O~

2.5

100

3..5

°

=

300

400

500

600

Figure 2: Analytic (8), (10) and lattice ( [11]) curves for pressure of QGP with nf = 0,2+ 1,3 from ([9]).

Figure 1: Fit (7) of Polyakov line for nf = and nf = 2)(black curves) to the lattice data ( [11]).

where v

200

mq IT and

1

00


=

o

z4

..jz2 +

1 v2

(9)

(ev'z2+i/2 + 1)

(10) The energy density is € = T2 8~ (~) v and the speed of sound in plasma is c; = ~~. In Fig.5 c; is shown calculated with the use of (8), (10) and compared to lattice data for /1 = 0 from ( [11]). No lattice calculations has yet been done for sound speed at nonzero baryon density, though our theory allows to do that and as is shown in ( [12]) the result for /1 > 0 does not differ much from the case /1 = O. 3

Phase transition

To obtain the curve of phase transition one needs to define pressure PI in the confined phase and PI I in the deconfined phase, taking into account that vacuum energy density contributes to the free energy, and hence to the pressure:

Having formulas for pressure (which contain parameter of may write for the phase transition curve T c (/1):

Lfund(X)

(7)) we

(12)

337

16

EfT4

14

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Figure 4: The phase transition curve Tc(J.!) (in GeY) from (9) as function of quark chemical potential J.! (in GeY) for nf = 2 (upper curve) and nf = 3 (lower curve) and f1G2 = 0.0034 Gey4 from ( [8]).

Figure 3: Analytic and lattice ( [11]) curves for energy density of QGP with nf = 2 + 1 and nf = 3 from ( [9]).

0.40 0.35 0.30 0.25

nF2, Wilson: mps/mv=O.65. mpslmv=O.95 I SU(3) nF(2+1): staggered, mq•• >O nr=2: staggered, isentr. EoS

0.20 0.15

• ...

0.10 0.05 0.00 0.5

T ITO

1.0

Figure 5: Sound speed for J.!

=

1.5

2.0

2.5

3.0

0 and n f = 3 (blue dashed curve) compared to lattice data from ( [11]).

here D.cvac = Icvac - c~~~1 = (11 - ~nf )/32D.G 2 • In particular, for the expected value of D.G 2 /G 2 (stand) ;::::: 0.4 one obtains Tc = 0.27 GeV (nf = 0), 0.19 GeV (nf = 2), 0.17 GeV (nf = 3) in good agreement with lattice data.

338

4

Summary

The EoS of QGP is written, where the only np input is the Polyakov line. It should be stressed, that only the modulus of the Polyakov line enters in EoS due to gauge invariance. The phase transition curve Tc(J.l) and speed of sound c; (T) are obtained and agree well with lattice data. An important point of the work is that the only parameter used to receive the final physical quantities from the initial QeD Lagrangian is the Polyakov line taken from lattice data, and is in agreement with analytic estimate for T = Tc ( [5]). Acknowledgments

The financial support of RFFI grant 06-02-17012 is acknowledged. References

[1] M.Shifman, A.Vainshtein, V.Zakharov, Nucl. Phys. B147, 385, 448 (1979). [2] Yu.A.Simonov, JETP Lett. 54 (1991) 249. [3] M.D'Elia, A.Di Giacomo, E.Meggiolaro, Nucl. Phys. B 483, 371 (1997). [4] A.Di Giacomo, H.G.Dosch, V.I.Shevchenko, Yu.A.Siomonov, Phys. Rep. 372, 319 (2002). [5] Yu.A.Simonov, Phys. Lett. B 619, 293 (2005). [6] N.O.Agasyan, Yu.A.Simonov, Phys. Lett. B 693,82 (2006). [7] Yu.A.Simonov, Nonperturbative equation of state of QGP, arXiv:hepphj0702266v2, Ann. Phys (in press). [8] Yu.A.Simonov, M.A.Trusov, arXiv:hep-phj0703228v2, Phys. Lett.(2007) B, 650 (1), p.36-40. [9] E.V.Komarov, Yu.A.Simonov, arXiv:0707.0781v2, Ann. Phys (in press). [10] A.V.Nefediev, Yu.A.Simonov, arXiv:hep-phj0703306. [11] F.Karsch et al., hep-Iatj0312015, hep-Iatj0608003. [12] E.V.Komarov, Speed of sound in QGP (in preparation).

CHIRAL SYMMETRY BREAKING AND THE LORENTZ NATURE OF CONFINEMENT A.V.NefedieV'

Institute of Theoretical and Experimental Physics, B.Cheremushkinskaya 25, 117218 Moscow, Russia Abstract.The Lorentz nature of confinement in a heavy-light quarkonium is investigated. It is demonstrated that an effective scalar interaction is generated selfconsistently as a result of chiral symmetry breaking, and this effective scalar interaction is responsible for the QeD string formation between the quark and the antiquark.

The question of the Lorentz nature of confinement is one of the most longstanding problems in QCD. Indeed, this question is very important for understanding the phenomenon of chiral symmetry breaking, for establishing the correct form of spin-dependent potentials in heavy quarkonia, for understanding the relation between chiral symmetry breaking and the QCD string formation between quarks, and so on. The latter question is discussed in this talk at the example of a heavy-light quarkonium. The approach of the QCD string [1] appears very convenient and successful in studies of various properties of hadrons, both conventional and exotic. This approach follows naturally from the Vacuum Background Correlators Method (VCM) [2]. The key idea of the approach is description of the gluonic degrees of freedom in hadrons in terms of an extended object - the QCD string - formed between colour sources. Nonexcited string is approximated by the straightline ansatz, so that only radial scratchings and rotations are allowed, whereas excitations of the string are described by adding constituent gluons to the system, so that a hybrid meson, for example, can be represented as the quarkantiquark pair attached to the gluon by two straight-line segments of the string. Consider the simplest case of the quark-antiquark conventional meson consisting of a quark and an antiquark connected by the straight-line Nambu-Goto string with the tension (T. The Lagrangian of such a system is L

= -m1 V1- £~ - m2V1- £~ - (Tr 11 d(3V1- [ii x ((3£1 + (1 - (3)£2)]2, (1)

r=

Xl - X2

n = rlr,

and one can proceed along the lines of Ref. [1] in order to arrive at the Hamiltonian. An important ingredient which distinguishes the given approach from the naive potential approach is the proper dynamics of the string which is encoded in the velocity-dependent interaction (the last term in Eq. (1)) and which translates into an extra inertia of the system with respect to the rotations. This ae-mail: [email protected]

339

340

extra inertia leads to the decrease of the (inverse) Regge trajectories slope and brings it to the experimentally observed value of 27f0" or 7f0", for a light-light and heavy-light systems, respectively [1,3]. This effect is unimportant for our present purposes and thus we neglect it, arriving a simple Salpeter equation (which is exact for the case of vanishing angular momentum, l = 0): H'Ij; = M'Ij;,

H

=

ViP' + mi + ViP' + m§ + O"r,

(2)

which is celebrated in the literature. The question we address in this talk is whether the confining interaction in Eq. (2) is of the scalar of vectorial nature. In order to make things as simple as possible, we consider the one-particle limit of Eq. (2) setting ml -* 00 and m2 == m. In this case, the resulting effectively single-particle system can be described by a Dirac-like equation and the question posed above translates into the form of the Dirac operator in this equation, that is whether the confining interaction is added to the energy or to the mass term in this operator. We choose the following strategy [4]. We start from the Euclidean Green's function of the given heavy-light quarkonium, SqQ(X, y) x exp

=

{-~

~c

J

D'Ij;D'lpt DAJJ- 'lj;t(x)SQ(x, YIA)'Ij;(y)

J

J

4 d xF:; -

d4x'lj;t( -if) - im - A)'Ij;} ,

(3)

and fix the modified Fock-Schwinger gauge [5],

(4) which leads to the static particle decoupling from the system. Then we perform integration over the gluonic field in Eq. (3) and arrive at the Dyson-Schwingertype equation derived in the Gaussian approximation for the QeD vacuum [2], that is only the bilocal correlator of gluonic fields is retained:

(-if)x - im)S(x, y) - i

J

d4zM(x, z)S(z, y) = O(4)(x - y),

(5)

with the mass operator -iM(x, z) = KJJ-v(x, z)')'JJ-S(x, z)')'v, Following Ref. [6] we approximate the interaction kernel as K 44 (X, y) == K(x, y) ~

O"(lxl + 1Y1 - Ix - YI),

K 4i (X, y) = 0, Kik(X, y)

= 0 (6)

and rewrite Eq. (5) in the form of a Schrodinger-type equation (in Minkowskii space):

«if; + i3m)lJI(x) + i3

J

d3 zM(x, Z)1JI(i) = EIJI(x),

(7)

341

with the mass operator

M(x, Z) =

-~K(X, £)(JA(x, Z),

A(x, Z) = 2i

J~~

S(w, x, Y)(J.

(8)

The question of the Lorentz nature of confinement can be now formulated as the question of the matrix structure of the quantity A [4]. Indeed, the phenomenon of spontaneous breaking of chiral symmetry (SBCS) means that a piece proportional to the matrix (J appears in A in a selfconsistent way. A convenient technique to deal with the phenomenon of chiral symmetry breaking is given by the chiral angle approach [7]. Let us parametrise the quark selfinteraction, described by the term V(x - Y) = alx - YI in the kernel (6),

"L,(j5) =

J

d4k

-iV(p - k)

7r

1'oko -1'k - m - "L,(k)

(2 )41'0

_-

- 1'0

-~

= [Ap - m] + ({j5)[Bp - p], (9)

by means of the so-called chiral angle '{Jp. Then the selfconsistency condition of such a parametrisation, Api Bp = tan'{Jp, takes the form (explicit expressions for the Ap and Bp follow from the term by term comparison of the l.h.s. and the r.h.s. of Eq. (9)):

psp-mcp =21

J

d k - [cksp-(pk)SkCp ~ :. ] ,splcp=sin/cos'{Jp, (10) (27r)3V(p-k) 3

which is known as the mass-gap equation [7]. Nontrivial solution to this equation behaves as '{Jp(p = 0) = 7r 12 and '{Jp(p ---4 (0) = O. This allows us to make definite conclusions concerning the Lorentz nature of confinement in Eq. (7) depending on the value taken by the quark relative momentum. Indeed,

and thus p---40 P ---4

00

scalar confinement vectorial confinement.

(12)

In the meantime, Eq. (7) admits a Foldy-Wouthuysen transformation, which can be performed in a closed form [8],

The resulting equation reads:

342

where Cp/Sp = cos/sin~(~-'Pp) and Ep = Ap sin 'Pp + Bpcos'Pp is the dressed-quark dispersive law which can be reasonably well approximated by the free-quark dispersion (this approximation fails for the chiral pion - see, for example, the discussion in Ref. [9]). For small values of the interquark momentum, when 'Pp :::::; 7r /2, Eq. (14) reduces to the one-particle limit of the Salpeter Eq. (2). Notice that, according to Eq. (12), this limit exactly corresponds to the purely scalar confinement in Eq. (7). We conclude therefore that the intrinsic Lorentz nature of the QCD string is scalar. As soon as chiral symmetry is broken spontaneously, the selfconsistently generated scalar part appear in the effective interquark interaction. If this interaction dominates, the system can be described by the Salpeter Eq. (2) or by its more sophisticated version which takes the proper string dynamics into account. Acknowledgments

This work was supported by the Federal Agency for Atomic Energy of Russian Federation and by grants NSh-843.2006.2, DFG-436 RUS 113/820/0-1(R), RFFI-05-02-04012-NNIOa, and PTDC /FIS /70843 /2006-Fisica. References

[1] A.Yu. Dubin, A.B. Kaidalov, and Yu.A. Simonov, Phys.Lett. B 323, 41 (1994); Phys.Lett. B 343, 310 (1995). [2] H.G. Dosch, Phys.Lett. B 190, 177 (1987); H.G. Dosch and Yu.A. Simonov, Phys.Lett. B 205, 339 (1988); Yu.A. Simonov, Nucl.Phys. B 307, 512 (1988). [3] V.L. Morgunov, A.V. Nefediev, and Yu.A. Simonov, Phys.Lett. B 459, 653 (1999). [4] Yu.A. Simonov, Yad.Fiz. 60, 2252 (1997); Phys.Rev. D 65, 094018 (2002). [5] I.I. Balitsky, Nucl.Phys. B 254, 166 (1985). [6] A.V.Nefediev and Yu.A.Simonov, Pisma Zh.Eksp. Teor.Fiz. 82, 633 (2005); Phys.Rev. D 76, 074014 (2007). [7] A.Amer, A.Le Yaouanc, L.Oliver, O.Pene, and J.-C. Raynal, Phys. Rev.Lett. 50, 87 (1983); A.Le Yaouanc, L. Oliver , O.Pene, and J.-C. Raynal, Phys.Lett. B 134, 249 (1984); Phys.Rev. D 29, 1233 (1984); A.Le Yaouanc, L.Oliver, S.Ono, O.Pene, and J.-C. Raynal, Phys.Rev. D 31, 137 (1985); P. Bicudo and J. E. Ribeiro, Phys.Rev. D 42, 1611, 1625, 1635 (1990); P. Bicudo, Phys.Rev.Lett. 72, 1600 (1994). [8] Yu.S. Kalashnikova, A.V. Nefediev, and J.E.F.T. Ribeiro, Phys.Rev. D 72,034020 (2005). [9] A.V. Nefediev and J.E.F.T. Ribeiro, Phys.Rev. D 70,094020 (2004).

STRUCTURE FUNCTION MOMENTS OF PROTON AND NEUTRON M.Osipenko a INFN, Sezione di Genova, 16146 Genova, Italy, Moscow State University, Skobeltsyn Institute of Nuclear Physics, 119992 Moscow, Russia Abstract. QCD-inspired phenomenological analysis of experimental moments of proton and deuteron structure functions F2 have been presented. The obtained results on the diu ratio at large-x, isospin dependence of higher twists and comparison with Lattice QCD calculations were discussed. We remind shortly these results: the obtained ratio is consistent with the asymptotic limit diu __ 0 at x -- 1, the total contribution of higher twists is found to be isospin independent and the non-singlet moments are in excellent agreement with the Lattice data. We present here some details of the analysis triggered by the public discussion.

Measurements of the nucleon structure function F2 provide the information about the longitudinal momentum distribution of partons. These distributions being governed by soft strong interactions cannot be described by perturbative QeD methods. Only Lattice QeD simulations allow to evaluate these quantities. Recent measurements of proton and deuteron structure function moments over wide Q2-interval [1,2] and the evaluation of neutron moments [3] allowed to improve the knowledge of these non-perturbative distributions. Detailed descriptions of these analyses are given in papers mentioned above, whereas in the present proceeding we develop further two arguments selected by the public discussion. Experimentally extracted moments of the proton and deuteron structure functions F2 were analyzed to separate leading twist (LT) and higher twist (HT) terms. This was performed by fitting the data Q2-dependence with the following expression:

where LTn is the LT part of the n-th moment evaluated at NLL accuracy, as is the running coupling constant, /1 2 is an arbitrary scale (taken to be 10 (GeV /C)2), a~ is the matrix element of corresponding QeD operator, I~ is its anomalous dimension, f30 = 11 - ~ N F with N F being number of active flavors, 7 is the order of the twist and k is the maximum HT order considered. The number of HT terms (k) in the expansion 1 is, of course, arbitrary because we don't know at which I/Q2 power the series converges. Moreover, anomalous dimensions of perturbative coefficients in front of HT terms are known in a very few cases [5,6]. Most of x-space analyses neglect this dependence assuming I~ = ae-mail: [email protected]

343

344

ofor T

> 2. In the presented analyses the anomalous dimensions were varied as free parameters and extracted from the best fit to the data. The results show a very strong sensitivity of the fit to the values of HT anomalous dimensions at low-Q2. Indeed, it can be seen in the comparison of two twist expansions shown in Fig. 1: one using HT anomalous dimensions as free parameters and another one assuming them to be zero. The lower limit of the fitted Q2_ interval was taken to be 1.2 (GeV jC)2 for the full fit. In the fit with fixed anomalous dimensions it was increased to 3.6 (GeVjc)2 by the condition of having the same X 2 per number of degrees of freedom. It is evident that only the variation of anomalous dimensions permits to describe the data until Q2 = 1.2 (GeVjc)2 by two HT terms. This observation emphasizes that the knowledge of perturbative anomalous dimensions of HT terms is crucial to single out individual HT operator matrix elements. 0.005 , . , . - - - , - - - - - - - - - - - - ,

0.005 r ; r - - . - - - - - - - - - - - - ,

0.004

0.004

0.003

0.003

0.002

0.002

LT\'........\ ....... .

J'.'" 0.001 HT '"

... . ~

O~----.~ ...~::~.:'~···~·------~ ,.....

-0.001 -0.002

/1"

!TW-6

o~--~~~~-------~ .................

TW-6 -0.001 -0.002

Figure 1: Fit of the structure function moment Ms with Eq. 1 using higher twist anomalous dimensions "I::; as free parameters (left) and assuming "I::; = 0 (right): dashed line - the leading twist contribution, dotted lines - twist-4 and twist-6 contributions, dot-dashed line - the total higher twist contribution, solid line - the total fit.

In the extraction of neutron moments we assumed the dominance of the Impulse Approximation (IA) in the LT part of deuteron moments and treated other nuclear effects as, model dependent, corrections to this approximation. This allowed for a simple extraction of LT neutron moments from the following algebraic relation:

(2) where MJ:, M;: and M;: are LT moments of the proton, neutron and deuteron, respectively. N;: is the moment of the nuclear momentum distribution fD

345

i.e. the structure function of the deuteron composed of point-like nucleons (see Ref. [3] for details). The dominance of IA in LT moments, however, implies that processes beyond IA contribute mainly to HT terms. These processes are the scattering off correlated nucleons (Final State Interaction (FSI)) and the scattering off a nuclear constituent different than the nucleon (Meson Exchange Current (MEC)). Only the lowest n = 2 LT moment, sensitive to the low-x dynamics, carries a small contribution from FSI and MECs estimated to be < 0.5 %, whereas it is found to be negligible for higher (n > 2) LT moments. In fact, the LT part of FSI shown in Fig.2 contaminates structure functions at very low x (x < 0.1) values because the nucleon spectator has a long time ~o ::::; 1/M x (here M is the nucleon mass) to interact with nuclear environment while awaiting return of the active quark [7]. However, for higher moments (n > 2) the mean x value is close to unity and therefore the time left for the interaction ~o ::::; 1/M < < 1/ m1f. Here we assume that nuclear interactions are carried mostly by pions. y

q

q

N

D

in

N

D

Figure 2: Example of FSI mechanism in the inclusive electron-deuteron scattering. The nuclear interaction between the nucleon spectator and nuclear medium is likely carried by a colorless pion.

Hence, the bulk of FSI was expected to give a contribution to the HT term of the moment expansion, where the current quark rescatters from the nuclear spectator. In order to test this assumption phenomenologically we used calculations of FSI in the quasi-elastic peak region based on the model from Ref. [4J. To this end, we computed moments of the modeled deuteron structure function F2 including and excluding FSI. The ratios between these two calculations for n = 2 and n = 8 are shown in Fig. 3. From the figure it follows that, indeed, FSI contribution disappears at large Q2 generating an additional HT term, while LT part is unaffected by this contribution. Moreover, the size of FSI contribution in moments does not exceed few % at lowest analyzed Q2, and it is much smaller than the total nucleon HT contribution estimated to be about ~ 25% at the same Q2. We speculate, therefore, that also the HT term has not more than 20% contamination from FSI, and Eq. 2 is also applied to this term within ~ 20% accuracy. This accuracy is comparable to the precision of the extracted total HT term [1,2].

346 1.05

1.04 1.03

1.02

......,

1.01 H

III

'" rr.~

\,

1

0.99 0.98 0.97 0.96 0.95 10

1 Q2

(GeV/c)

2

Figure 3: Ratio of the deuteron structure function moments calculated using the parameterization of Ff from Ref. [4J including and excluding FSI in the quasi-elastic channel: the solid line - n = 2, the dashed line - n = 8.

Summarizing, the presented analysis of the experimental moments of proton and deuteron structure functions F2 showed that: • knowledge of perturbative anomalous dimensions of higher twist terms is crucial to single out individual higher twist operator matrix elements; • for n > 2 FSI mechanism appears in the nuclear structure function moments as an additional higher twist term. Its partial estimates indicate that the relative contribution of FSI to the total higher twist term does not exceed 20%, comparable to the precision of the total higher twist extraction. References [1] [2] [3] [4] [5] [6] [7]

M. Osipenko et al., Phys.Rev. D 67, 092001 (2003). M. Osipenko et al., Phys.Rev. C 73, 045205 (2006). M. Osipenko et al., Nucl.Phys. A 766, 142 (2006). C. Ciofi degli Atti and S. Simula, Phys. Rev. C 53, 1689 (1996). E. Shuryak and A. Vainstein, Nucl. Phys. B 201, 141 (1982). H. Kawamura et al., Mod. Phys. Lett. A 12, 135 (1997). R.L. Jaffe, in M.B. Johnson and A. Pickleseimer, editors, Relativistic Dynamics and Quark-Nuclear Physics, pp. 1-82, Wiley New York, 1985.

HIGGS DECAY TO bb: DIFFERENT APPROACHES TO RESUMMATION OF QCD EFFECTS A.L. Kataev a Institute for Nuclear Research, 117312 Moscow, Russia V.T. Kim b St. Petersburg Nuclear Physics Institute, 188300 Gatchina, Russia Abstract.The comparison between parameterisations of the perturbation results for the decay width of the Standard Model Higgs boson to lib-quarks pairs, based on application of M S-scheme running quark mass and pole b-quark mass, are presented. In the case of the latter parameterisation taking into account of order O(a~) term is rather important. It is minimising deviations of the results obtained at the O(a;) level from the results, which follow from the running quark mass approach.

Decay widths and production cross-sections of scalar bosons are nowadays among the most extensively analysed theoretical quantities. In the case if the Standard Electroweak Model Higgs boson has the mass is in the region 115 GeV:S MH :S 2Mw , where the lower bound comes from the searches of Higgs boson at LEP2 e+e- -collider, it can be detectable in the LHC experiments through the mode H -; "("( and at Tevatron through the main decay mode H -; bb (see e.g. the review [1] ). Moreover, the decay H -; bb may be seen at TOTEM CMS LHC experiment, which is aimed for searches of Higgs boson through its diffraction production (see e.g. [2]). The detailed study of this mode is also useful for planning experiments at possible future linear e+e-colliders [3]. In the mentioned region of masses theoretical expressions for r(H -; bb) is dominating over expressions for other decay modes of the SM H-boson, and therefore is dominating in the denominators of various branching ratios, including Br(H -; "("(). In view of all these topics it is useful to estimate theoretical error bars of r Hbb. To consider this question we will compare parameterisations of QCD predictions for r Hbb' expressed through the running MS-scheme mass mb(MH ) and pole quark mass mb (at the order O(a;)-effects of perturbation theory the similar studies were made in Ref. [4]). Consider now the basic formula for r Hbb in the case of N f =5 number of active flavours [5]

ae-mail: [email protected] supported by RFBR Grants N 05-01-00992, 06-02-16659 be-mail: [email protected]

347

348

b

where rb ) = ¥J-GFMHml, mb=mb(MH), a s=a s (MH)=a s /7r, [3i, 'Yi are the coefficients of the QCD [3 and mass anomalous dimension functions ~r 1 was calculated in [6]. ~r2' ~r3, ~r4 were evaluated in [7], [8], [5]. The huge negative value of ~r 4 indicates, that the structure of perturbation series in the Minkowski region differs from the sign constant growth of perturbation QCD coefficients in the Euclidean region. The possibility of the manifestation of this effect at the a!-level was demonstrated previously in [9], [10]. Consider the renormalisation group (RG) equation

(4) The RG functions are known up to 4-loop level. The solution of Eq.( 4) is

_ (M) - ( ) (as(MH))"Yolf30 AD(as(MH)) mb H = mb mb as(mb) AD(as(mb))

(5)

and the coefficients ofthe polynomial AD(a s ) are expressed through the coefficients of RG-functions (see Ref. [11]). We will use the QCD coupling constant expanded in inverse powers of In(M~/ A~~5) 2) at the NLO, NNLO and N 3LO. At the a~-level the expression for r Hbb in terms of the quark pole mass and the MS-scheme coupling constant can be obtained by three steps. First, one should use the RG equation, which translates mb(MH ) to mb(mb). Second, one can use the relation

)3)

mb(mb)2 = ml(l- 2.67a s(mb) -18.57a s(mb)2 -175.79a s(m b

(6)

where the O(a;)-term was obtained in [12] and the order a~-term was calculated by semi-analytical methods in [13] (this result was confirmed soon in [14] by complete analytical calculation.) Finally, as(mb) should be transformed to as(MH)' The coefficients of the truncated in as(MH) series for r Hbb have the following numerical forms:

r Hbb = rbb) ( 1 + ~riOS) as + ~r~OS) a; + ~r~OS) a~ ) where ~riOS)

(7)

= (3 - 2L) wih L == In(MJdml) and

~r~OS)

= ( _

4.52 - 18.139L + 0.083L 2)

(8)

were previously taken into account in [4], while we will be interested in the effect of the next term. It reads

~r~OS) =

( -

316.906 - 133.421L - 1.153L 2 + 0.05L 3 )

.

(9)

349 The inclusion of the expressions for the two-loop diagrams with massive quark loop insertions, tabulated and taken into account in the RunDec Mathematica package of [15] leads to slight modification of the and a~-corrections to Eq.(7): c

a;-

( - 5.591 - 18.139L + 0.083L 2 )

(10)

( - 322.226 - 132.351L - 1.155L 2 + 0.05L 3 )

(11)

The constant term of .6.r~OS) is also affected by the contributions to the a~­ coefficient of Eq.(6) of the diagrams with massive quark loop insertions, evaluated in [16]. However, even in the case of charm-quark loop, these extra terms are rather small. We will neglect these massive-dependent effects. Fig.1 demonstrate the of the ratio Rb(MH ) = r(Ho - t bb)/r~b) both in the case of running quark mass and pole quark mass parameterisations. The QeD parameters are fixed as: mb = 4.7 GeV and mb(mb) = 4.34 GeV [17], NLO: A<:)s = 253 MeV and NNLO and N3LO: A<:)s = 220 MeV, which correspond to the central values of the results, obtained in Ref. [11] ~ 1.2

~

H

~

t ' j

1

0 .•

-+ bbar in

RG-improved approach

•.•.•.....................•.•••.•.•.•••.•••.•.••..•.•. .f!'!'.r:!'! .•....•.••.••.•

1.2 , - - - - - -_ _ _ _ _ _ _- - - ,

;:::..

H'--" bbcu" in pole-mo •• approach mo •• I••• co ••

:8

I!

,

~

~ D. " D.'

r-----.--.-._ ~

.•••••••••...••..••-.-••.•-•.. •. •. •. . .•. •. . .•. . •. •.••.•.•.-•...•.....-...~~....................•.. :

r

______

..........................

Figure 1: The comparison of the expressions for Rb(MH). To get the impression of scale variations, MH is varied in the lowerer than experimentally interesting region also.

One can see, that within the application of the "running" quark mass approach, or as it sometimes called RG-inspired approach, the results do not depend essentially from the values of the coefficient functions of Eq.(l), calculated within MS-scheme. Almost the whole effect of decreasing of Rb(MH) ratio is described by effects of running of mb(MH)2. Another observation is that in the pole-mass parameterisation the large logs In(MJdm~) are important and at the a~-level their interference produce negative correction, which decrease CNote, that Eq.(13) in [15J contains an obvious misprint for the a~-term proportional to

nr: instead of the first quadratic term the linear log term should be typed (and used).

350

the results for Rb(MH ) ratio, making it comparable with the results of the RGinspired approach. Another interesting problem, related to resummation of the 1[2 terms either through fractional variant of Analytic Perturbation Theory approach of Ref. [18] (see Ref. [19]) or through the variant of contour-improved perturbation theory of Ref. [20] (see Ref. [21]), will be considered elsewhere. References [1] N. V. Krasnikov, V. A. Matveev, Phys. Part. Nucl. 31, 255 (2000) . [2] A. De Roeck, V. A. Khoze, A. D. Martin, R. Orava, M. G. Ryskin, Eur. Phys. J. C 25, 391 (2002). [3] A. Droll,H. E. Logan, Phys. Rev. D 015001 (2007). [4] A. L. Kataev, V. T. Kim, Mod. Phys. Lett. A 9, 1309 (1994). [5] P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn, Phys. Rev. Lett. 96, 012003 (2006). [6] E. Braaten, J. P. Leveille, Phys. Rev. D 22, 715 (1980); N. Sakai, Phys. Rev. D 22, 2220 (1980); T. Inami, T. Kubota, Nucl. Phys. B 179, 171 (1981); S. G. Gorishny, A. L. Kataev, S. A. Larin, Sov. J. Nucl. Phys. 40, 329 (1984). [7] S. G. Gorishny, A. L. Kataev, S. A. Larin, L. R. Surguladze, Mod. Phys. Lett. A 5,2703 (1990) 2703; Phys.Rev. D 43, 1633 (1991). [8] K. G. Chetyrkin, Phys. Lett. B 390, 309 (1997). [9] A. L. Kataev, V. V. Starshenko, Mod. Phys. Lett. A 10, 235 (1995). [10] K. G. Chetyrkin, B. A. Kniehl, A. Sirlin, Phys. Lett. B 402,359 (1997). [11] A. L. Kataev, G. Parente, A. V. Sidorov, Phys. Part. Nucl. 34, 20 (2003). [12] N. Gray, D. J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C 48, 673 (1990). [13] K. G. Chetyrkin, M. Steinhauser, Nucl. Phys. B 573, 617 (2000). [14] K. Melnikov, T. v. Ritbergen, Phys. Lett. B 482, 99 (2000). [15] K. G. Chetyrkin, J. H. Kuhn, M. Steinhauser, Comput. Phys. Commun. 133, 43 (2000). [16] S. Bekavac, A. Grozin, D. Seidel, M. Steinhauser, JHEP, 0710, 006 (2007). [17] A. A. Penin, M. Steinhauser, Phys. Lett. B 538, 335 (2002). [18] D. V. Shirkov and I. L. Solovtsov, Phys. Rev. Lett. 79, 1209 (1997). [19] A. P. Bakulev, S. V. Mikhailov and N. G. Stefanis, Phys. Rev. D 75, 056005 (2007). [20] A. A. Pivovarov, Sov. J. Nucl. Phys. 54, 676 (1991) [21] D. J. Broadhurst, A. L. Kataev, C. J. Maxwell, Nucl. Phys. B 592, 247 (2001).

A NOVEL INTEGRAL REPRESENTATION FOR THE ADLER FUNCTION AND ITS BEHAVIOR AT LOW ENERGIES A. V. N esterenko BLTPh, Joint Institute for Nuclear Research, Dubna, 141980, Russia Abstract. The Adler function is studied by employing recently derived integral representation. The latter embodies the nonperturbative constraints on the Adler function, in particular, it retains the effects due to the nonvanishing pion mass. In the framework of the developed approach the Adler function is calculated by making use of its perturbative approximation as the only additional input. The obtained result agrees with the experimental prediction for the Adler function in the entire energy range and possesses remarkable higher loop stability.

1

Introduction

The Adler function D( Q2) [1] plays a crucial role in various issues of elementary particle physics. Specifically, this function is essential for the theoretical description of both, some strong interaction processes and hadronic contributions to some electroweak observables. The latter represents a decisive test of the Standard Model and imposes strict restrictions on new physics beyond it. Perturbation theory remains the only reliable tool for calculating the Adler function at high energies. Namely, in the asymptotic ultraviolet region D (Q2) can be approximated by the power series in the strong running coupling O!s(Q2). However, spurious singularities of the latter invalidate this expansion at low energies. This significantly complicates theoretical description of the low-energy data and eventually forces one to resort to various nonperturbative approaches. An important source of the nonperturbative information about the hadron dynamics at low energies is provided by the dispersion relations. The latter supply one with the definite analytic properties in a kinematic variable of a physical quantity at hand. The idea of employing this information together with perturbation theory and renormalization group method forms the underlying concept of the analytic approach to QCD [2]. Some of the merits of this approach are the absence of unphysical singularities and a fairly good higherloop and scheme stability of out coming results. The analytic approach was successfully employed in studies of various aspects of strong interaction [2-5]. The primary objective of this paper is to study the infrared (IR) behavior of the Adler function by employing integral representation [6]. The latter was derived in a general framework of the analytic approach to QCD, the effects due to the pion mass being retained. It is also of a particular interest to examine the stability of the calculated D(Q2) with respect to the higher loop corrections. The layout of the paper is as follows. In Sect. 2 the dispersion relation for the Adler function and its interrelation with R-ratio of e+e- annihilation into hadrons are overviewed. In Sect. 3 a novel integral representation for D(Q2) is discussed and the calculation of the Adler function within the developed approach is presented. In Sect. 4 the obtained results are summarized. 351

352 2

The Adler function

The Adler function D (Q2) [1] naturally appears in the theoretical description of the process of electron-positron annihilation into hadrons. Specifically, the measurable ratio of two cross-sections is proportional to the discontinuity of the hadronic vacuum polarization function II(q2) across the physical cut: R (s) =

(J"

(e+e- -+ hadrons; s) 1 1·1m [II( s - zs .) - II( s + zs .)] = (e+e- -+ p,+ p,-; s) 27fi E--+O+

(1)

(J"

with s = q2 > 0 being the center-of-mass energy squared. It is worth noting here that R( s) vanishes identically for the energies below the two-pion threshold due to the kinematic restrictions, see also Ref. [7]. The mathematical implementation of the latter condition consists in the fact that II(q2) has the only cut q2 2': 4m; along the positive semiaxis of real q2 [1,7]. For practical purposes it is convenient to deal with the Adler function [1]

(2) where Q2 = _q2 2': 0 denotes a spacelike momentum. This function plays a crucial role for the congruous analysis of hadron dynamics in spacelike and timelike domains. In particular, the experimentally measurable R-ratio (1) and theoretically computable Adler function (2) can be expressed in terms of each other by making use of the relations (see Refs. [1,8]) S iE 1 de R(s) = -2. lim D( -C) -(' (3) 7fZ E--+O+ S+iE

l

where m7r :::::: 135 MeV is the mass of the 7f 0 meson. Although there are no direct measurements of the Adler function (2), it can be restored by employing the data on R-ratio (1). Specifically, in the integrand of the dispersion relation for D(Q2) (3) one usually approximates R(s) by its experimental measurements at low and intermediate energies, and by its theoretical expression at high energies. Computed in this way experimental prediction for the Adler function is presented in Fig. 1 by shaded band, see Refs. [6,9] for the details. As it has been noted above, the high-energy behavior of the Adler function (2) can be approximated by the power series in the running coupling oA Q2) in the framework of the perturbative approach. Specifically, at the £-loop level

(4) where a~l)(Q2) is the i-loop perturbative invariant charge and d 1 = 1/7f. As one may infer from Fig. 1, this approximation is reliable for the energies Q :2: 1.5 Ge V only. Besides, expansion (4) is incompatible with the dispersion relation (3) due to unphysical singularities of a 8 (Q2) in the IR domain. The latter also causes certain difficulties for processing the low-energy data.

353

3

Novel integral representation for Adler function

For practical purposes it proves to be convenient to express the Adler function (2) and R-ratio (1) in terms ofthe common spectral function. This objective can be achieved by employing relations (3), the parton model prediction Ro(s) = e(s - 4m;) [7], and the fact that the strong correction to the Adler function vanishes in the asymptotic ultraviolet limit Q2 --+ 00. Eventually one arrives at (see Refs. [6,9] for the details) (5) (6)

Here the spectral function PD (CJ) can be determined either as the discontinuity of a theoretical expression for the Adler function across the physical cut PD (CJ) = 1m D ( -CJ + iO+) /1r or as the numerical derivative of data on R-ratio PD (CJ) = -dR(CJ)/d InCJ. It is worth noting that Eq. (5) embodies the nonperturbative constraints on Adler function arising from the dispersion relation (3). Besides, Eq. (6) by construction properly accounts for the effects due to the analytic continuation of spacelike theoretical results into timelike domain. In order to compute the Adler function in the framework of the approach at hand, one first has to determine the spectral function PD(CJ). In what follows we restrict ourselves to the study of only perturbative contributions to the latter, namely p~fJrt(CJ) = ImD~?rt(-CJ+iO+)/1r, where D~?rt(Q2) is given by Eq. (4). Note that in the limit of massless pion (mrr = 0) the obtained expressions (5) and (6) become identical to those of the so-called Analytic perturbation theory (APT) [2], the perturbative spectral function Ppert(CJ) being assumed. For the illustration of the significance of the pion mass within the approach at hand, it is worth presenting Adler function (5) computed by making use of the perturbative spectral function Ppert(CJ) for both, massless and massive cases. The obtained results are presented in Fig. 1 by solid curves. In the case of the massless pion (which is identical to the APT [2]), one arrives at the result which is free of IR unphysical singularities, but fails to describe the Adler function for the energies Q ;S 1.0 GeV, see Fig. 1 A (see also paper [6] and references therein). At the same time, as one may infer from Fig. 1 B, for the case of the nonvanishing pion mass the representation (5) is capable of providing an output for the Adler function, which agrees with its experimental prediction in the entire energy range [6]. Moreover, the Adler function (5) is remarkably stable with respect to the higher loop corrections. Namely, the relative difference between the i-loop and (£+ I)-loop expressions for D(Q2) (5) is less than 4.9%, 1.5%, and 0.3% for £ = 1, £ = 2, and £ = 3, respectively, for 0 :::; Q2 < 00, see Refs. [6,9] for the details. It is worth noting also that

354 1.5

D(Q2) 1

1.0

0

0.5

Q,GeV 0.5

1.0

1.5

2.0

2.5

Q,GeV 1.0

1.5

2.0

2.5

Figure 1: The Adler function (5) (solid curves) calculated by making use of the perturbative spectral function ppe't(a) in the massless (A) and massive (B) cases. Numerical labels correspond to the loop level considered. The experimental prediction for D(Q2) is shown by the shaded band, whereas its perturbative approximation is denoted by the dashed curve.

the obtained results are supported by recent studies of meson spectrum in the framework of the Bethe-Salpeter formalism [4]. 4

Summary

The IR behavior of the Adler function is examined by employing representation (5), which retains the pion mass effects. The approach at hand possesses all the appealing features of the massless APT [2]: it supplies a self-consistent analysis of spacelike and timelike data; additional parameters are not introduced into the theory; the outcoming results possess no unphysical singularities and display enhanced higher loop stability. In addition, the developed approach provides a reliable description of the Adler function in the entire energy range. Acknowledgments

This work was supported by grants RFBR 05-01-00992 and NS-5362.2006.2. References

[1] S.L. Adler, Phys. Rev. DID, 3714 (1974). [2] D.V. Shirkov and I.L. Solovtsov, Phys. Rev. Lett. 79,1209 (1997); Theor. Math. Phys. 150, 132 (2007). [3] A.V. Nesterenko, Phys. Rev. D 62, 094028 (2000); 64, 116009 (2001). [4] M.Baldicchi, A.V.Nesterenko, G.M.Prosperi, D.V.Shirkov, and C.Simolo, Phys. Rev. Lett. (in press); arXiv:0705.0329 [hep-ph]. [5] A.C.Aguilar, A.V.Nesterenko, J.Papavassiliou, J. Phys. G31, 997 (2005). [6] A.V. Nesterenko and J. Papavassiliou, J. Phys. G 32, 1025 (2006); A.V. Nesterenko, arXiv:0710.5878 [hep-ph). [7) R.P. Feynman, "Photon-Hadron Interactions" (Benjamin, Mass.), 1972. [8) A.V.Radyushkin, Rep. JINR-2-82-159 (1982); JINR Rapid Comm. 78,96 (1996); N.V.Krasnikov and A.A.Pivovarov, Phys. Lett. B116, 168 (1982). [9] A.V. Nesterenko, in preparation.

QCD TEST OF z-SCALING FOR 1I"°-MESON PRODUCTION IN pp COLLISIONS AT HIGH ENERGIES M. Tokarev a and T. Dedovich Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia Abstract. Results of the next-to-Ieading order pQCD calculations of inclusive cross sections of nO mesons produced in pp and pp collisions over a wide range of collision energy (up to the LHC energy) and transverse momentum are compared with ISR and RHIC data. The dependence of the spectra in PT and z presentations for different parton distribution and fragmentation functions is studied. The sensitivity of obtained results to the choice of the renormalization (J.!R) , initial-state factorization (J.!F) and final-state factorization (fragmentation) (J.!H) scales is verified. It is shown that self-similar features of particle production dictated by the z-scaling give strong restriction on the asymptotic behavior of the inclusive spectra in high-PT region.

1

Introduction

Different approaches to description of particle production are used to search for regularities (Bjorken, Feynman and KNO scalings, quark-counting rules etc.) reflecting general principles in lepton, hadron and nucleus interaction at high energies. One of the most basic principles is the self-similarity of hadron production valid both in soft and hard constituent interactions. Other general principles are locality and fractality which can be applied to hard processes at small scales. These investigations have shown that the interactions of hadrons and nuclei can be described in terms of the interactions of their constituents. Fractality in hard processes is a specific feature connected with sub-structure of the constituents. This includes the self-similarity over a wide scale range. One of the scalings in high energy inclusive reactions is z-scaling observed in high-PT particle production (see [lJ and references therein). It is based on principles of locality, self-similarity and fractality of hadronic interactions. The scaling function 'I/J(z) and scaling variable z used for data presentation are expressed via experimental quantities such as the inclusive cross section Ed 3 (j/dp 3 and the multiplicity density dN/d'T/. Data z-presentation reveals properties of energy and angular independence with power law, 'I/J(z) rv z-f3, at high z. In the report we present results of analysis of new data on high-PT spectra of 11"0 mesons produced in PP collisions at the Relativistic Heavy Ion Collider (RHIC) in the framework of z-scaling. Properties of z presentation are used to predict spectra of 11"0 mesons produced in PP and PP collisions over a wide ae-mail:[email protected]

355

356

vIS and PT range. The obtained results are compared with NLO QeD calculations performed with different parton distribution and fragmentation functions (PDFs and FFs). The sensitivity of '¢(z) to renormalizaion, factorization and hadronization scales are studied as well. The results could give new constraints on PDFs and FFs, more deep understanding of phenomenological aspects of QeD and verification of flavor dependence of z-scaling [1,2]. 2

z-Scaling

The idea of z-scaling is based on the assumptions that gross feature of inclusive particle distribution of the process M1 + M2 -> m1 + X at high energies can be described in terms of the corresponding kinematic characteristics of the constituent subprocess written in the symbolic form (x1Mt) + (X2M2) -> m1 + (x1M1 + X2M2 + m2) satisfying the condition

(1) The equation is the expression of locality of hadron interaction at a constituent level. Here Xl and X2 are fractions of the incoming momenta P1 and P2 of the colliding objects with the masses M1 and M 2. They determine the minimum energy, which is necessary for production of the secondary particle with the mass m1 and the four-momentum p. The parameter m2 is introduced to satisfy the internal additive conservation laws (for baryon number, isospin, strangeness, and so on). The quantity n is introduced to connect kinematic (X1,2) and structural (8 1,2) characteristics of the interaction. It is chosen in the form

(2) where m is the mass of nucleon and 81 and 82 are factors relating to the fractal dimensions of the colliding objects. The fractions Xl and X2 are determined to maximize the value of n(X1,X2), simultaneously fulfilling the condition (1)

(3) The fractions X1,2 cover the full phase space accessible at any energy. According to the self-similarity principle the scaling function '¢ (z) is constructed as the function depending on the single dimensionless variable z expressed via dimensionless combinations of Lorentz invariants. It is written in the form 7rS -1 d3 (T (4) '¢ (z) = - (dN/dTJ )(T,n . J E dP3 Here, Ed3 (T/dp 3 is the invariant cross section, s is the center-of-mass collision energy squared, (Tin is the inelastic cross section, J is the corresponding Jacobian. The factor J is the known function of the kinematic variables, the

357

momenta and masses of the colliding and produced particles. The function 'I/J(z) is normalized as follows

1a';)O 'I/J(z)dz =

1.

(5)

The relation allows us to interpret the function 'I/J(z) as a probability density to produce a particle with the corresponding value of the variable z. According to the fractality principle the variable z is constructed as a fractal measure z = zon- 1 for the corresponding inclusive process. It reveals the property zen) -. 00 at n- 1 - . 00. The divergent part n- 1 describes the resolution at which the collision of the constituents can be singled out of this process. The n(X1,X2) represents relative number of all initial configurations containing the constituents which carry fractions Xl and X2 of the incoming momenta.

3

QeD test of z-scaling

Here we analyze the new data obtained by the STAR and PHENIX Collaborations [3,4] on high-PT spectra of nO mesons produced in PP collisions at VB = 200 GeV. The results are compared with the NLO QCD calculations in PT and z presentations. Figure lea) shows nO meson PT-spectra obtained at ISR (see [2] and references therein) and RHIC energies [3,4]. The strong dependence of cross sections on collision energy was experimentally observed. The scaling function 'I/J(z) for the same data are presented in Figure 1 (b). The shape of the scaling function for RHIC data (*,6) is found to be in good agreement with 'I/J(z) for the ISR data shown by the dashed line. The asymptotical behavior of 'I/J(z) is described by the power law, 'I/J(z) rv z-i3. The value of the slope parameter f3 is independent of kinematical variables. 10'~--------

10 • 10 1

10

~,

10

_2

"''''~'.,.

p-p

"~..

11'0

...';'....

10 -.

'M'

10'" . . 10 .... 10~

10 ., 10~ 1O~

s'IJ, GeV b.

200

" __ 90 G

"'.

--..,.~.

PHENIX

...

* 30-62 200 STAR \"'" ... ISR 10 -.. 10 _II 10 -.~~-"--~~.uL-~~.u.J 10

a)

10

I

b)

Figure 1: The PT (a) and z (b) presentations of experimental data on inclusive cross sections of rro mesons produced in pp collisions at the ISR and RHIC [3,4].

358

..

" ....-----------, ,,' , p+p",,,,O+X

"

10-'

..u 10-11

'> ~

::'"

17)1<0.35

NLO OCD

z,~,;:~

1

10-

:g;

E 10'" • 10-'

). ~g~

;;:g:~

. . . . . o.

~ IO-fa

10 -II 10'"

'"

10'"

+

R~,~···· ISR

'\

10' Of

10·0> 1t 10· O~,~,,:-,O"';,':-',"';2~0"';2~'"';30~'':-'5"":"'0 p,.. GeV/e

b)

a) '0·...-----------, '0'

:u '>

10

1 10 "

10'"

~

p+p ... .,..O+X

p+p ....nO+X

~. ~::'.

",1<0.35

"'1<0.35 NLO QeD .111·23-14000 fUI ~~~~

.111 , GoV

~:~.! \~~~i~·. ~~ ~:

10'''0

-"\ * + spp~·······-._•..

"

5

10

15 ~o 25 Pr. GaV/e

c)

JO

35

~~'\l~~

40

d)

Figure 2: The PT (a,c) and z (b,d) presentations of data on inclusive cross sections of 71"0 mesons produced in PP and PP collisions at the ISR, RHIC and SppS. Experimental data are taken from [3-5].

The invariant cross sections of nO-meson production in PP as a function of the collision energy JS and transverse momentum PT calculated in the NLO QCD are shown in Figure 2 (a ) . The calculations depicted by the dashed lines were performed with the parton distribution functions CTEQ5m and the fragmentation functions KKP. The renormalization, factorization and fragmentation scales were set to be equal each other J.lR = J.lF = J.lH = C . PT and C = 1. The spectra were calculated over a wide range of the transverse momentum PT = 1-100 GeVjc and the energy JS = 30 -14000 GeV at Berns = 90°. As seen from Figure 2( a) the dependence of the spectra on the collision energy enhances with PT. The difference of the cross sections at JS = 23 and 14000 Ge V reaches about 5 and 8 orders of magnitude at PT = 5 and 10 GeV Ic, respectively. Experimental data on cross sections obtained at the ISR and RHIC [3,4] are shown in Figure 2(a) by the symbols (+) and (0, *), respectively. The same data are plotted in the dependence on the variable z in Figure 2 (b). The solid line represents asymptotic behavior of the scaling function 'IjJ( z) for high z. The z presentation of the NLO QCD calculations is shown by the dashed curves. A reasonable agreement between experimental data and the corresponding NLO QCD calculations presented in the framework of the z-scaling is observed. Figure 2 shows PT (c) and z (d) presentations of the inclusive spectra of nO mesons produced in pP collisions over a range PT = 1 - 40 GeV Ic , JS = 23-14000 GeV and 11J1 < 0.35. The NLO QCD calculations with the CTEQ5m

359 parton densities and KKP fragmentation functions are plotted by the dashed lines. Experimental data are depicted by points (*, +). The NLO QCD calculations with other parton (MRST) and fragmentation (BKK) functions as well as with the variation (c=1/2,1,2) of the renormalization ({LR), factorization ({LF) and fragmentation ({LH) scales give similar results. We found that {3pp > (3fiP. As seen from Figures 2(b) and 2(d) the NLO QCD predictions demonstrate dramatic deviation from the asymptotic behavior of 'ljJ(z) predicted by z-scaling as the collision energy and transverse momentum increase in the region where experimental measurements are not performed yet. Thus we conclude that self-similar features of nO-meson production dictated by the z-scaling give strong restriction on the asymptotic behavior of the scaling function 'ljJ(z). The behavior is not reproduced by the NLO QCD evolution of cross sections with the phenomenological parton distribution and fragmentation functions used in the present analysis. 4

Conclusions

The QCD test of z-scaling for nO-meson production in pp and pp collisions was performed. Based on the results we conclude that z-scaling is a regularity which reflects the self-similarity, locality and fractality of the hadron interaction at a constituent level. It concerns the structure of the colliding objects, interactions of their constituents and fragmentation process. Self-similar features of particle production dictated by the z-scaling give strong restriction on the asymptotic behavior of the inclusive spectra in high-PT region. We consider that experimental verification of the NLO QCD calculations and z-scaling predictions are important for further tests of the QCD and more precise specification of the phenomenological ingredients which enter the pQCD calculations at small scales. Acknowledgments

The investigations have been partially supported by the special program of the Ministry of Science and Education of the Russian Federation, grant RNP.2.1.1.5409. References

[1] 1. Zborovsky, M.V. Tokarev, Yu.A. Panebratsev and G.P. Skoro, Phys. Rev. C 59, 2227 (1999); 1. Zborovsky and M.V. Tokarev, Phys. Part. Nucl. Letters 3, 68 (2006); Phys. Rev. D 75, 094008 (2007). [2] M.V. Tokarev, et al., J. Phys. G: Nucl. Part. Phys. 26, 1671 (2000). [3} P.G. Jones et al., Hadron Collider Physics Symposium, May 20-26,2007, La Biodola, Isola, dElba, Italy. [4] A. Adare et al., hep-ex/0704.3599, 26 April 2007. [5] M. Banner et al., Phys. Lett. B 115, 59 (1982).

QUARK MIXING IN THE STANDARD MODEL AND THE SPACE ROTATIONS G.Dattoli a Gruppo Fisica Teorica e Matematica Applicata Unita Tecnico Scientifica Tecnologie Fisiche Avanzante (FIS-MAT), ENEA - Centro Richerche Jilrascati. Via Enrico Fermi 45, 1-00044 Jilrascati, Rome, Italy K.Zhukovsky b Faculty of Physics, M. V.Lomonosov Moscow State University, Vorobyovy Gory, 119991, Moscow, Russia Abstract. The rotation matrix and the Cabibbo-Kobayashi-Maskawa (CKM) matrix are discussed. The exponential parameterisation of the Cabibbo-KobayashiMaskawa (CKM) matrix is viewed as the rotation matrix in Euler angles with Pitch-Roll-Yaw convention for the angles and as the angle-axis representation of the rotation matrix. The account for the CP-violating phase in CKM and the 0(3) rotation matrix in the angle-axis form are discussed in the context of such view on the mixing matrix. The generation of the new parameterisations of the CKM matrix by the exponential form with distinguished CP violating part is demonstrated.

In the framework of the Standard Model of Particle Physics [1], [2], [3] the quark mass eigenstates (q) are different from the weak interaction quark eigenstates (q/), which are a linear superposition of the mass eigenstates. For 3 generations of quarks the quark mixing is expressed via this matrix V, called Cabibbo-Kobayashi-Maskawa (CKM) matrix [4], which by agreement acts on the charge --e/3 physical mass states (d, 8, b) and transforms them into new interaction eigenstates (d', 8', b'). The CP violation is implemented via the complex phase in the CKM matrix elements. According to Euler's rotation theorem the rotation is given by three rotation angles (
where the parameter t5 accounts for the violation of the CP symmetry and the parameter>. is responsible for the quark flavour mixing, whereas the unitarity remains preserved [51. For the real entries for the A matrix, it turns into the angle-axis form of a three dimensional rotation M, defined by the angle of ae-mail: [email protected] be-mail: [email protected]

360

361

:fi =

rotation <.P and the direction unit vector rotation is performed: M

~ <.P ) ( n,

= e-PN ,

N=

(nx, ny, nz), around which the

(~z-ny

(2)

The Cabibbo case takes place when the unit vector is turned against the z-axis, i.e. :ficablnbo.= (0,0, -1). Comparing formula (2) for the rotation matrix in axis-angle presentation with the exponential quark mass mixing matrix (1), we find the expression for the single rotation angle <.P in terms of the parameters >., 0 and (3 of the mixing matrix:

The coordinates of the unit vector :fi = (nx, ny, n z ) are determined by the parameter >. = sin ee and the parameters 0 and j3 of the exponential parameterisation as follows: (4)

With <.P = 0, (2) yields the unit matrix for the rotation M (:fi, <.p) I-p=o = 1= diag {I, 1, I}, cancelling the rotation and, respectively, the mixing between quarks fades out. Decompose the exponent (1) of the matrix V in the sum of the Real and Imaginary parts, A = A1 + A 2 , as followS': AZFIAI

A1 =

=

>. aA' -(3 >.2 ) , ->. 0 _0>.3 (3 >.2 0

C

C°

0>.3 (1- e- iO )

(5)

0 o >. 3 (-1 + ei6 ) 0 0 0

°

),

where A2 is the matrix (1) with 8=0. Then the transformation following form:

V ~ PRotPCP (1 - ~ [A2' All),

PRot

= e A2 ,

~6)

V takes the

PCP =

e

Al

,

(0 1)

(7)

CThe other possibility to separate the real and the imaginary parts would be the following: A2

0 =).,

(

1

-1 _0).,2 cos

(8)

0 (3).

0).,2

cos (8) )

-(3).,

0

and Ai

=

io).,3 s in8

0 1

0

0 0

0 0

362

where the commutator is of the order 0(>,4). Thus, in this approximation the transformation, corresponding to the quark mixing with CP violation, is composed of the purely rotational part PRot, which is related to the rotation matrix (2) via the formulae (3), (4) and the CP violating part PCP

PRo'

~ exp (~),

>.

-a >.3

a),3

-(3 >.2 0 (3 >.2 0

)

,

PCP

'in2~) COS 2~ 01 <+ o ,

= 0

",- sin 2D.. 0 cos 2D..

(8) where D..

. J

= a >.3 SIll 2'

",±

= ie±i!.

(9)

The above CP violating matrix reminds the Cabibbo mixing matrix, acting on the quarks d and b with the weights", for the entries (1,3) and (3,1):

d' = dcosD.. + ",+bsinD..,

b' = ",-dsinD.. + b cos D...

(10)

The transformation (8) preserves the norm and the orthogonality:

(d'i d') = (b'l b') = 1,

(d'i b') =

WI d') = 0

(11)

The CP violating phase J can be separated from the other parameters in in (8) as follows:

nofooooJn (2A) cos e~n PCP

=

~ :+00.1::00 I

( ",- 2: I n(2A) sin (~c5) o n=-oo

2:

n

(2A) ,in

PCP

(~') 1' (12)

I n (2A) cos (~c5)

n=-oo

where A = a>. 3 . There exist the Hermitian-conjugated and inverse with respect to PCP matrices: pc~· PCP = P(jp. PCP = I. (13) Another parameterisation of the mass mixing matrix, involving the rotation matrix (8), can be generated

V = PCP/2 . PRot· P CP / 2 ,

P CP / 2

= exp (

~1 )

.

(14)

This matrix V has the same entries as the matrix (1) at least up to >.9 power. Matrix PCP / 2 , accounting for the complex term contribution, i.e. for the CP violating part, writes as follows: P CP / 2

=

cosD.. 0 ( ",- sinD..

0 ",+ sin D.. 1 o 0 cosD..

),

(15)

363

where D. and Ii are defined by (9). The expression (15) can also be written in terms of the Bessel functions to distinguish the CP phase {j as follows: 00

L:

I n (A) cos ('~n

n=-oo

PCP/2

=

0 (

Ii-

n=~oo I n (A) sin (~6)

Direct check of the unitarity of the matrix V, given by formulae (14) - (15), confirms that the new parameterisation V = PCP/2 . P . P C P/2 is exactly unitary. Thus, in the case of conserved CP, the mixing of the quarks in the SM can be viewed as the rotation around a fixed axis in 3D space in the angle q,. The presence of a CP-violating phase breaks this symmetry and such simple geometric picture gets lost. In the exponential mass mixing parameterisation allows the geometric interpretation of the CP violation - the rotation with the complex "y" component of rotation direction vector v in angle-axis presentation. Formulae (14) - (15) define a new unitary mixing matrix parameterisation with distinguished CP violating part. The CP violation is effectively separated in the coefficients of the expansion into series of Bessel functions in (12) and (16). Further exact treatment of the CP violation problems is cumbersome and can be performed on the base of the theories with extended symmetries and Gell-Mann SU(3) matrices. On the other hand, the mixing matrix V in the found form (14) includes the corrections up to O().9) and abundantly satisfies the present level of experimental confidence so that it is appropriate for the analysis of experimental data at present and in the near future.

Acknowledgments The authors express their gratitute to the Organizing Committee of the 13th Lomonosov Conferences on Elementary Particle Physics for the invitation to the Conference. One of the authors (K.Zh.) is grateful to A.V.Borisov and A.E.Lobanov for useful discussions of the results of the work.

References [1J S.Weinberg, Phys. Rev. Lett., 19, 1264 (1967). [2J A.Salam, "Elementary PfLrticle Theory", Ed. by N.Svartholm (Almquist ForIag AB), 1968. (3] S.L.Glashow, Nucl. Phys., 22, 579 (1961). [4] M.Kobayashi and T.Maskawa, Progr. Theor. Phys., 49, 652 (1973). [51 G.Dattoli and K.Zhukovsky, Eur. Phys. J. C, 50, 817. (2007).

ANALYTIC APPROACH TO CONSTRUCTING EFFECTIVE THEORY OF STRONG INTERACTIONS AND ITS APPLICATION TO PION-NUCLEON SCATTERING A.N. Safronov a

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia Abstmct. Analytic approach to constructing effective theory of strong interactions is developed and its application to pion-nUcleon scattering is considered. The model-independent definitions of renormalized coupling constants of contact interactions are proposed. The information on these constants from available data of the phase shift analysis of pion-nucleon scattering is extracted.

1

Introduction

The pion-nucleon dynamics is one of the most fundamental problems in nuclear and particle physics. It is now widely believed that QCD is fundamental theory of strong interactions. On this basis all hadron-hadron interactions are completely determined by the underlying quark-gluon dynamics. However, due to the formidable mathematical problems raised by the non-perturbative character of QCD at low and intermediate energies, we are still far from a quantitative understanding hadron-hadron interactions from this point of view. The path-integral method together with the idea of spontaneous chiral symmetry breaking leads to effective field theory (EFT) of strong interactions [1,2]. The EFT formulates the theory of hadron-hadron interactions in terms of meson and baryon (anti-baryon) degrees of freedom. The Lagrangian of EFT is highly nonlinear and has rather complicated form. Therefore in practice the decomposition of this Lagrangian in power series of particles momenta and pion mass factor is used. In this approximation the theory refers to as chiral perturbation theory (ChPT). ChPT suffers from some inconsistencies. In this approach different regularization and renormalization methods (for example, a dimensional regularization [1] and a regularization based on introducing cutoff factors in loop integrals [3], or so called infrared regularization (IR) scheme [4, 5] lead to different predictions for transition amplitudes [6,7]. Recently the approach to constructing effective interaction operators between strongly interacting composite particles has been proposed [8] on the basis of analytic S-matrix theory and methods for solving the inverse quantum scattering problem. We define effective potential (or potential matrix in multi-channel case) as local operator in the partial-wave quasipotential equation (LippmannSchwinger type equation), such that it generates an on-mass-shell scattering amplitude which has required discontinuities at dynamical cuts. ae-mail: [email protected]

364

365

In the present work we have constructed the spectral and potential functions in S-, P- D- and F-partial-wave channels of pion-nucleon scattering, using the data of the energy-dependent phase shift analysis (on-line computer code SAID [9]). The information on renormalized coupling constants of ChPT effective Lagrangian was extracted from this analysis. 2

Analytic approach to constructing effective theory of pion-nucleon interaction

In the developed analytical approach to constructing effective theory of hadronhadron interactions we manifestly take into account the nearest to physical region dynamical cuts of scattering amplitudes and also the contact interactions generated by effective Lagrangian of ChPT. Conceptually contact interactions take into account contributions of distant dynamical singularities (interactions at small distances). The discontinuities of the partial-wave S-matrix along dynamic cuts are determined by model-independent quantities - specifically, renormalized coupling constants and on-mass-shell amplitudes of elementary sub-processes (Cutkosky cutting rules [10]). It can be calculated in framework of the relativistic quantum field theory with the help of various dynamic approaches (for example, EFT). The contributions of loop diagrams are considered by methods of dispersion integration. Renormalized (physical) coupling constants of contact interactions are determined by subtraction constants in dispersion relations. The basic advantages of the suggested approach in comparison with schemes for constructing ChPT available in the literature consist in the following. 1) The theory from the beginning is under construction in terms of renormalized ( physical) coupling constants and on-mass-shell scattering amplitudes. 2) The equations are formulated in a manifestly Poincare-invariant form. 3) There is no ambiguity problem, connected with regularization of divergences in loop diagrams. 4) Hadron-hadron scattering amplitudes on construction have correct analytical structure of the dynamic cuts nearest to physical region. 3

Model-independent definitions of renormalized coupling constants

In the limit t -> 0, v -> 0 (where v = (8 - u)/4m, m is the nucleon mass) the invariant amplitudes of pion-nucleon scattering A(±), f3(±) (in which contributions of one nucleon exchange mechanism are subtracted [5]) have the form

(1)

(2)

366 The model-independent dimensionless renormalized coupling constants of contact interactions up to fourth order of ChPT we define by equations

(3)

(4)

(5)

A3

2 m = 8J-L2

[

2 (-)

1 - 2f", ao

+ bo(-))] ,

_ A4 -

1

2 (+)

2mf",a2

,

_ A5 -

2 2b(-) 2 ,

m f",

(6)

where f", is the pion decay constant, gA is axial coupling constant and J-L is pion mass. It can be shown that our definitions (3)-(6) of coupling constants Kl, K2, K"" f3"" AI-A5 are identical at tree diagram level to definitions given in Ref. [7]. One has to stress here that our definitions of coupling constants are model-independent, while definitions of low energy constants in different formulations of ChPT available in the literature depend on a used regularization and renormalization methods and in this sense they are model-dependent. The results of our analysis for coupling constants based on fixed-t dispersion relations (A) [11] and partial-wave dispersion relations (B) are presented in Tab. 1. Table 1. Coupling constants (3)-(6) obtained in present work using energydependent phase-shift analysis of pion-nucleon scattering [9]. Method K", Kl K2 f3", A2 A3 A5 A4 A 6.43 -0.60 2.04 -2.32 -20.56 -0.96 6.91 20.56 B 6.37 -0.67 2.01 -2.39 -7.04 -1.51 6.91 21.52 Often used in the literature coupling constants Ci (i = 1..4) of heavy baryon (HB) ChPT [12] are related in the one loop approximation to constants (3), (4) by relations

(7)

(8)

367 Constants C;, (i = 1..4) obtained in our analyses (in one loop approximation (OLA) of HB ChPT for specified above methods A and B) and also results of other calculations [13-18] are presented in Tab.2. Table 2. Coupling constants Ci ( in Gey-l, i = 1..4) obtained in our analyses and data from Ref. [13-18]. [13] [14] A B [15] [16] [17] [18] Cl -0.95 -1.03 -0.94 -1.23 -0.81 -0.76 -0.81 -0.93 3.20 4.73 4.76 3.28 8.43 3.20 2.80 3.34 C2 -6.58 -6.72 -5.40 -5.94 -4.70 -4.78 -3.20 -5.29 C3 3.96 3.45 3.47 3.49 3.47 3.40 5.40 3.63 C4 4

Conclusion

The concept of the analytical approach to constructing the effective theory of strong interactions at low and intermediate energies is developed and its application to a pion-nucleon scattering is considered. The information on spectral and potential functions in S-, P- D- and F-partial-wave channels of scattering and also on renormalized coupling constants of ChPT effective Lagrangian is extracted from the available data of the pion-nucleon phase shift analysis. References J. Gasser and H. Leutwyller, Ann. Phys. 158,142 (1984). S. Weinberg, Nucl. Phys. B 363, 3 (1991). J.F. Donoghue, et al., Phys. Rev. D 59, 036002 (1999). T. Becher and H. Leutwyler, Eur. Phys. J. C 9, 643 (1999). P.J. Ellis and H.-B. Tang, Phys. Rev. C 57, 3356 (1998). S.R. Beane, Nucl. Phys. A 632, 445 (1998). K. Torikoshi and P.J. Ellis, Phys. Rev. C 67, 015208 (2003). A.N. Safronov and A.A. Safronov, Yad. Fiz. 69, 408 (2006). RA. Arndt, et al. Computer code SAID, on line program at http://gwdac.phys.gwu.edu. [10] RE. Cutkosky, J. Math. Phys. 1, 429 (1960). [11] G. Hohler, et al., Nucl. Phys. B 39, 237 (1972). [12] E. Jenkins and A.Y. Manohar, Phys. Lett. B 255, 558 (1991). [13] M. Mojzis, Eur. Phys. J. C 2, 181 (1998). [14] N. Fettes, et al., Nucl. Phys. A 640, 199 (1998). [15] P. Biittiker and U.-G. Meissner, Nucl. Phys. A 668, 97 (2000). [16] M.C.M. Rentmeester, et al., Phys. Rev. C 67, 044001 (2003). [17) D.R Entem and R Machleidt, Phys. Rev. C 68, 041001 (2003). [18] Y. Bernard et al., Nucl. Phys. A 615, 483 (1997). [1] [2] [3] [4] [5] [6] [7] [8] [9]

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New Devolopments in Quantum Field Theory

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ON THE ORIGIN OF FAMILIES AND THEIR MASS MATRICES WITH THE APPROACH UNIFYING SPIN AND CHARGES, PREDICTION FOR NEW FAMILIES N.S. Mankoc Borstnik a , Department of Physics, Faculty for Mathematics and Physics, University of Ljubljana, J adranska 19, 1000 Ljubljana, Slovenia Abstract. The Approach unifying all the internal degrees of freedom-the spins and all the charges into only the spin-is offering a new way of understanding the properties of quarks and leptons, that is their charges and their couplings to the gauge fields as well as the appearance of families and their mass matrices. The starting Lagrange density for spinors in d (= 1 + 13), which carry nothing but two kinds of spins (no charges) and interact with only the gravitational field through vielbeins and two kinds of spin connection fields-the gauge fields of the two kinds of the Clifford algebra objects manifests in d = (1 + 3) the properties of fermions and bosons, postulated by the Standard model of the electroweak and colour interactions, with the Yukawa couplings included. In this talk the Approach is presented and as well as a possible way of spontaneous breaking of the starting symmetry, which leads to the properties of the observed fermions. Rough predictions for not yet measured families of fermions is made, with the dark matter candidates included.

1

Approach unifying spins and charges

The Standard model of the electroweak and strong interactions (extended by assuming nonzero masses of neutrinos) fits with around 25 parameters and constraints all the existing experimental data. It leaves, however, unanswered many open questions, among which are questions about the origin of U(I), SU(2) and SU(3) charges, the families and the Yukawa couplings of quarks and leptons, as well as about the corresponding Higgs mechanism. I am assuming a simple Lagrange density [1-6] for a Weyl massless spinor, which in d = (1 + 13)-dimensional space carries two kinds of spins and no charges and interacts correspondingly with only the vielbeins and the two kinds of the spin connection fields b

S

J

L

~(E?j!,apoa'l/J) + h.c. = ~(E?j!,a r

POc<

dd x L,

Pc< -

'12 sabWabc< - '12 §ab-Wabc<'

aPOc<'l/J)

+ h.c., (1)

ae-mail: [email protected] bLatin indices a, b, '" m, n, '" s, t,. denote a tangent space (a fiat index), while Greek indices a, {3, '" Jl., V, "CJ, T" denote an Einstein index (a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c,. and a, (3, ". ), from the middle of both the alphabets the observed dimensions 0,1,2,3 (m, n,,, and Jl., v, ,,), indices from the bottom of the alphabets indicate the compactified dimensions (s, t,,, and CJ, T, ,,). The signature ryab = diag{l, -1, -1,· .. , -I} is assumed.

371

372

Here fa a are vielbeins (inverted to the gauge field of the generators of translations eaa , eaafab = 6/:, eaa f f3 a = 6a (3 ), with E = det(e aa ). I am proposing two kinds of the Clifford algebra objects [2,9,10]: the Dirac "(a operators, which in the Approach unifying spins and charges take care of the ordinary spin and all the charges, and the second kind of the Clifford algebra objects :ya, which take care of the families of spinors. Accordingly there are two kinds of the spin connection fields: Waba and Waba, the gauge fields of sab and [;ab, respectively. 1

2ha"(b - "(b"(a), 2TJab = {ia, :yb}+,

S-ab

1 C-a-b

=2"("(

-b-a) ,

-"("(

{:ya,"(b}+ = 0 = {sab,[;cd}_.

(2)

One Weyl spinor representation of sab in d = (1 + 13) (a left handed one, for example) with the spin as the only internal degree of freedom, manifests in four-dimensional "physical" space (if analyzed in terms of the subgroups SO(1,3) x U(l) x SU(2) x SU(3)) as the ordinary (SO(l, 3)) spinor with all the known charges of one family of the left handed weak charged and the right handed weak chargeless quarks and leptons of the Standard model. To manifest this we make a choice of TAi = Ls t cAi st sst, with CAi st chosen in such a way that TAi fulfil the commutation r~lations of the SU(3), SU(2) and U(l) groups: {TAi,T Bj }_ = i6 AB fAijkTAk, with the structure constants f Aijk of the corresponding groupS'. After appropriate breaks of the symmetry of the (simple) starting action the observed families of quarks and leptons coupled to the known gauge fields and carrying masses, determined by a part of the starting Lagrange density, follow. It is the second kind of the Clifford algebra objects [7,9,10]' which defines families and determines together with the corresponding gauge fields the off diagonal elements of mass matrices, and together with the first kind of the Clifford algebra objects and the corresponding gauge fields the diagonal part of mass matrices (for which in the Standard model the Higgs field is needed). We make a choice of the Cartan sub algebra set with d/2 = 7 elements in d = 1 + 13 for both kinds of generators' S03 S12 S56 S78 S910 S1112 S1314 and [;03, [;12, [;56, [;78, [;910, [;1112, [;1314'. Th~n w~ ex~res~ the basis f~r one Weyl in d = 1 + 13 as products of nilpotents and projectors, which are binomials of "(a [3] (3) CThe index A identifies the charge groups: A = 1 denotes SU(2), A = 2 denotes one of the two U(l) groups-the one following from SO(l, 7)-A = 3 denotes the group SU(3) and A = 4 the group U(l) following from 80(6), and index i identifies the generators within one charge group.

373

II

Octet, r(1,7) = 1, r(6) = -1, of quarks 03

U CR UR

dR dR

12

56

78

12

56 78

12

56 78

12

56

78

03

12

56

78

[-i](+) 1(+)[-]11 [-] [+] (-) 03

UL

12 12

56 56

78 78

::2 1

1

?

-1

1

2" 1

-1

'2

9 1011 1213 14

U CL C

1

9 1011 1213 14

(+i)[-]I [-](+) II [-] [+] (-) 03

-1

9 1011 1213 14

dCL

2"

9 1011 1213 14

1

-1

9 1011 1213 14

-1

?

0

1

?

I ~

k k

~

0

-1

0

-1

?

k

-1

0

1 R

k

-1

7.

0

1 R

1 R

1

0

1 R

k

0

1 R

1 R

'2

?

-1

1

7.

1

0

?

-1

-1

(+i)[-]I (+)[-]11 [-] [+] (-)

I

'2

9 1011 1213 14

[-i](+) I [-](+) II [-] [+] (-)

I 1

1

9 1011 1213 14

[-i][-]I [-][-]11 [-] [+] (-) 03

dCL

I I

9 1011 1213 14

(+i)(+) 1[-][-]11 [-] [+] (-) 03

C

78

[-i][-]I (+)(+) II [-] [+] (-) 03

C

56

(+i)(+) I (+)(+) II [-] [+] (-) 03

C

12

?

2"

1 R

-1

T

T

-1

Table 1: The 8-plet of quarks-the members of SO(l, 7) subgroup, belonging to one Weyl left handed (r(1,13) = -1 = r(I,7) x r(6» spinor representation of SO(l, 13). It contains the left handed weak charged quarks and the right handed weak chargeless quarks of a particular colour (-1/2,1/(2V3». Here r defines the handedness in (1 + 3) space, S12 defines the ordinary spin (which can also be read directly from the basic vector), 7 13 defines the weak charge, 7 2 defines the U(l) charge from SO(1,7), 7 33 and 7 38 define the colour charge and 7 4 defines another U(l) charge, which together with the first one defines Y = 7 4 + 7 2 . The vectors are eigenvectors of all the members of the Cartan sub algebra set ({ S03, S12, S56, S78, S910, Sl112, S1314}). The reader can find the whole Weyl representation in ref. [12].

respectively, which all are eigenvectors of ab

sab

(k): ab

Sab

(k)

sab

and

tt),

sab

[k]:=

~ (~),

sab

ab k ab [k]= -- [k] .

~

sab

ab

k

ab

"2

[kJ, 2

(4)

We choose the starting vector to be an eigen vector of all the members of the 03

12

56

78

9 10 11 12 13 14

Cartan set. In particular, the vector (+i)(+)(+)(+) [-] [+] (-) has the following eigenvalues of the Cartan sub algebra set sab: (~,~,~,~, -~,~, -~), respectively. With respect to the charge groups it represents a right handed weak chargeless u-quark with spin up and with the colour (-1/2, 1/(2V3))(= (7 33 ,7 38 )). Accordingly we may write one octet of the left handed and the right handed quarks of both spins and of one colour charge as presented in Table 1. Quarks of the other two colour charges and the colour chargeless leptons distinguish from this octet only in the part which determines the colour charge

374

and 7 4 (7 4 = 1/6 for quarks and the relations

7

4

=

-1/2 for leptons). Taking into account

ab

ab

ab

ab

"'t (k)

rr [-kJ, "l (k)= -ik [-kJ,

ab la [k] ab ria (k) ab ria [k]

ab (-k), ab -iT/aa [kJ, ab i (k),

ab Ib [k]= -ikrr ab ab ;;b (k)= -k [kJ, ab ;;b [k]= -kT/ aa

ab

ab (-k), ab (k) .

ab

ab

ab

one easily sees that la transform (k) into [-kJ, ;ya transform (k) into [kJ, with unchanged value of sab (Eq.(4)). It is accordingly self evident that sab can (must, since it exists?!) be used to

generate families. Let us assume that breaking of the starting symmetry SO(1,13) (1) to SO(1,7) x SU(3) x U(l) occurs at some scale at around 10 17 GeV or higher, then at a lower scale the break of SO(l, 7) x SU(3) x U(l) to SO(l, 3) x SO(4) x U(l) x SU(3) occurs, and that at some lower scale at around 10 13 GeV one further break occurs leading to the symmetry SO(l, 3) x SU(2) x U(l) x SU(3). (The break is accompanied by the al~pearance of new fields A~ (the hyper field of the Standard model) and A~: A~3 = A~ sin tl2 + A~ cos tl2' A~ = A~ cos tl2 - At sin tl2' A~± = ~(A~ =f iA~2), a = m, s the gauge fields new operators Y = 7 4 + 7 23 , Y' = 7 23 - 7 4 tan 2 tl2' 7 2 ± = 7 21 ± i7 22 and similarly in the sab sector A;3 = A; sin 2 + At cos 2, A; = A; cos 2 - At sin 2, A;± = ~(A;l =fiA;2).) The last break occurs at the weak scale, leading to SO(l, 3) x U(l) x SU(3).(The measurable gauge fields Am, Zm, W± finally appear: A~3 = A sin til + Za cos til, A~ = Aa cos til - Za sin til, W~± = ~ (A~l =fiA~2) a = m, s, as the gauge fields of the operators Q = 7 13 + Y = S56 + 7 4 , Q' = - Y tan 2 til + 1 13 11 7 ,7 ± = 7 ± i7 12 with the coupling constants e = gY cos til, g' = gl cos til and tan til = Similarly also new fields in the sab sector appear A;3 = As sinel + Zs cosel , A; = As cos e l - Zs sin e l , TV! = ~(A~l =f iA~2), as the gauge fields of the operators Q = i 13 + Y = S56 + i4, Q' = - y tan 2 e l + i 13 , il± = ill ± ii 12 with the coupling constants e = gY cos e l , g' = gl cos e l -y and tan til = Then the starting action manifests as

e

e

e

e

a;.

Tt-.)

1:- = ~hm[Pm - g3

L

1

73iA~ - e QAm - g' Q'Zm - gM7+W~ +7-W':;:; v2

i

l yl -gy y' A m

2

LV2 7 2+ A 2m+ + 7 2- A 2m-]+

IS [ps - eQAs - g' Q' ZS - gyl y' At eQAs - g'Q' Zs - gyl Y' At -

375

with m,m' E {0,1,2,3}, s,s',t E {5,6,7,8}. The first row of Eq.(5) manifests the action of the Standard model after the break at the weak scale, the second row with very heavy fields A;;" and A~± can at low energies be neglected. The last two rows determine mass matrices. They can be written as

,0 {( +) 78

1/J t

78

Po+

+ (-)

Po- }1/J, Po±

=

(P07 =t= i P08),

while POs =

Ps -

~5abwabs

78

78

ac bd

~SabWabs. We can rewrite - 2:(a,b) ~ (±) sabwab± = - 2:(ac),(bd), k,l (±)(k)(l) A'fl((ac), (bd)), with the pair (a,b) in the sum before the equality sign running over all the indices which do not characterize the Cartan subalgebra, with

a, b = 0, ... ,8, while the two pairs (ac) and (bd) in the sum after the equality sigh denote only the Cartan subalgebra pairs (for 50(1,7) we only have the pairs (03), (12); (03), (56) ;(03), (78); (12), (56); (12), (78); (56), (78) ); k and I run over four possible values so that k = ±i, if (ac) = (03) and k = ±1 in all other cases, while I = ±l. After the break of 50(1,13) to 50(1,7) x 50(6) at some high scale, we end up with 28 / 2 - 1 = 8 of the lowest mass families. The 8 x 8 mass matrices decouple after the break at the next scale (~ 10 17 - ~ 10 13 GeV) to 50(1,3) x 50(4) x 5U(3) x U(l) into two times 4 x 4 families. The break of 50(4) x U(l) into 5U(2) x U(l) (at ~ 10 13 GeV)) determines the scale of masses of the upper four families, while the last break at the weak scale determines the masses of the lower four families. The lowest of the higher four families is stable. In [14] we discuss the possibility that this family forms the clusters which demonstrate as the dark matter. The lowest four families demonstrate [8] as the known three families and the fourth one, which might be seen at the new accelerators. We use Eq.(5) to evaluate the mass matrices on a tree level. Taking the unknown fields from (5) as free parameters and fitting them using the Monte-Carlo program to the experimental data within the known accuracy (to the known masses and mixing matrix). We assume at this stage that mass matrices are real (neglecting the C P violating terms). Although it turns out that there are only two independent off diagonal matrix elements for each type of quarks or leptons, we still have too many free parameters to make any prediction. Assuming the values for the fourth family masses we get the mass matrix in MeV for the u-quarks

(-15~ -83)

(-150,-83) (1211,1245) (-306,304)

(-306,304)

o

(9,22)

(

o (-306,304) (171600,176400) (-150, -83)

(-30~, 304) (-150, -83) 200000

)

376 and for the d-quarks (5,11) (8.2,~4.5)

(8.2,14.5) (83,115) (174,198)

o

( (174,198)

o (174, 198) (4260,4660) (8.2,14.5)

(174,198) ) (8.2,014 .5) 200000

,

corresponding to the following values for the masses of the quarks

mujGeV mdjGeV

(0.005,1.220,171.,215.) , (0.008,0.100,4.500,285. )

and the mixing matrix for the quarks -0.974 0.226 0.0055 ( 0.00215

-0.226 -0.973 -0.0419 0.000414

-0.00412 -0.0421 0.999 -0.00293

0.00218 ) -0.000207 0.00294 . 0.999

By enlarging the masses of the fourth family of quarks, the fourth family (very) slowly decouples from the first three families. 2

Concluding remarks

In this talk I presented how does the Approach unifying spins and charges, postulating two kinds of spins, answer some of the open questions of the Standard model of the electroweak and colour interactions. i. We started with one Weyl spinor in d = 1 + 13, which carries two kinds of spins (no charges) and interacts correspondingly with vielbeins and gauge fields of the two kinds of the generators: sab and sab, corresponding to ,a and "fa-the two kinds of the Clifford algebra objects. ii. A simple starting Lagrange density for a Weyl spinor in d = (1 + 13)dimensional space manifests in d = (1 + 3)-dimensional space the properties of all the quarks and the leptons of the Standard model (with the right handed neutrinos included) and the families. iii.There are sab, which determine in d = (1 + 3) the spin and all the charges. One Weyl spinor representation includes (if analyzed with respect to the Standard model groups) the left handed weak charged quarks and leptons and the right handed weak chargeless quarks and leptons, coupled to all the corresponding gauge fields. iv. There are sab, which generate (an even number of) families. It is a part of the starting action which manifests as Yukawa couplings of the Standard

377

model'ljJt,o,Spos'ljJ, s = 7,8, with POs = _~8abWabs - ~§abwabs contributing to diagonal and off diagonal elements of mass matrices. v. There are several possibilities of breaking the starting 80(1,13) symmetry to the Standard model symmetry after the electroweak break. We assume the break of 80(1,13) to 80(1,7) x 8U(3) x U(l) (80(1,7) manifests a left handed weak charged quarks and leptons and the right handed weak chargeless quarks and leptons). One of possible breaks predicts the fourth family of quarks and leptons at the energies still allowed by the experimental data [11]. In this paper the results for the break from (80(1,7) x U(l)) x 8U(3) to (80(1,3) x 80(4) x U(l)) x 8U(3) and further to 80(1,3) x (8U(2) x U(l)) x 8U(3) and then to 80(1,3) x U(l) x 8U(3) are presented: the masses of quarks and the corresponding mixing matrix for the "lightest" four families. Letting the fourth family mass growing, it very slowly de couples from the first three. vi. If the Approach is the right way beyond the Standard model, then there are chances that the fourth family will be observed at new accelerators, the stable fifth family seems to be a very promising candidate for forming the dark matter [13,14]. To decide whether or not the way of breaking symmetries presented in this paper is the right one, further studies going beyond the tree level are needed. References [1] N. S. Mankoc Borstnik,Phys. Lett. B 292 (1992) 25-29. [2] N. S. Mankoc Borstnik, J. Math. Phys. 34 (1993) 3731-3745. [3] A. Borstnik Bracic, N.S. Mankoc Borstnik, hep-ph/0512062, Phys Rev. D 74 (2006) 073013-16. [4] N.S. Mankoc Borstnik, hep-ph/071 1.4681, p.94-113, 53-113, hep-ph/0401043, hep-ph/0401055, hep-ph/0301029. [5] N. S. Mankoc Borstnik, Modern Phys. Lett. A 10 (1995) 587-595. [6] N. S. Mankoc Borstnik, Int. J. Theor. Phys. 40 (2001) 315-337. [7] N. S. Mankoc Borstnik, H. B. Nielsen, Phys. Rev. D 62 (2000) (04010-14). [8] G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoc Borstnik, hep/ph-07082846, hep-ph/071 1.4681, 53-113. [9] N.S. Mankoc Borstnik, H.B. Nielsen, J. of Math. Phys. 44 (2003) 4817-4827, hep-th/0303224. [10] N.S. Mankoc Borstnik, H.B. Nielsen, J. of Math. Phys. 43 (2002) 5782-5803, hep-th/0111257. [11] S.S. Bulanov, V.A. Novikov, L.B. Okun, A.N. Rozanov, M.I.Vysotsky, Phys. Atom.Nue!. 66 (2003) 2169; Yad.Fiz. 66 (2003) 2219, hep-ph/0301268v2. [12] A. Borstnik, N. S. Mankoc Borstnik, hep-ph/0401043, hep-ph/0401055, hepph/0301029. [13] M. Khlopov, N.S. Mankoc Borstnik, hep-ph/0711.4681,p.195-201. [14] G. Bregar, N.S. Mankoc Borstnik, hep-ph/0711.4681, p.189-194.

Z2 ELECTRIC STRINGS AND CENTER VORTICES IN SU (2) LATTICE GAUGE THEORY M. I. Polikarpov a ITEP, B. Cheremushkinskaya str. 25, 117218 Moscow, Russia P. V. Buividovich b JIPNR, Akad. Krasin str. 99, 220109 Minsk, Belarus Abstract.We study the representations of SU (2) lattice gauge theory in terms of sums over the world sheets of center vortices and Z2 electric strings, i.e. surfaces which open on the Wilson loop. It is shown that in contrast to center vortices the density of electric Z2 strings diverges in the continuum limit of the theory independently of the gauge fixing, however, their contribution to the Wilson loop yields physical string tension due to non-positivity of their statistical weight in the path integral, which is in turn related to the presence of Z2 topological monopoles in the theory.

It is often believed that Yang-Mills theory can be entirely reformulated in terms of string degrees of freedom, since the basic property of non-Abelian gauge theories - quark confinement - is explained by the emergence of confining string which stretches between test quark and antiquark. Even the simplest string models for Yang-Mills theory turn out to be very successful in reproducing the spectrum of bound states of the theory. One of the most successful recent developments is the AdS/QCD, the description of QCD bound states in terms of string theory on five-dimensional anti-de Sitter space or its modifications [1,2]. In this description the Wilson loop behaves as W [0] rv exp (-S5D [C]), where S5D [C] is the area of the minimal surface in five-dimensional space spanned on the loop C. The loop C is assumed to lie on the boundary of this five-dimensional space. However, up to now there is no exact representation of Yang-Mills theory in terms of electric strings, Le. the strings which open on Wilson loops. In fact, the only string one usually encounters in non-Abelian gauge theories is a chromo electric string of finite thickness at finite lattice spacing, which is observed in numerical simulations as a cylindric region with higher energy density between two test colour charges [3,4]. Recently a different type of strings has been discovered in lattice gauge theories, namely, Z N magnetic strings or center vortices. Although the existence of such strings in Yang-Mills theories was predicted a long time ago [5], they have been actually observed in lattice simulations only several years ago [6,7]. The simulations has shown that center vortices are infinitely thin and have a finite density in the continuum limit. Moreover, lattice results suggest that center vortices are the effective degrees of freedom in the infrared domain of Yang-Mills theory [6,7], since removing center vortices from lattice configurations destroys all its characteristic infrared properties, such as confinement ae-mail: [email protected] be-mail: [email protected]

378

379

or spontaneous breaking of chiral symmetry. Effective action of center vortices and their geometric properties were extensively studied in [8]. Detection of center vortices in lattice configurations of gauge fields is based on the separation of SU (N) link variables into SU (N) /ZN variables and the variables which take values in the center of the gauge group, Z N. Configurations of these ZN variables can be exactly mapped onto configurations of closed self-avoiding surfaces, correspondingly, summation over Z N variables can be represented as a sum over all vortex configurations [9]. For ZN lattice gauge theories there exists a duality transformation [10] which allows to represent the observables either in terms of center vortices (ZN magnetic strings) or in terms of surfaces which open on Wilson loops. Since Wilson loop represents the worldline of an electric charge, it is reasonable to call such surfaces Z N electric strings. The aim of this paper is to investigate the representation of SU (2) lattice gauge theory in terms of a sum over all surfaces of such Z2 electric strings and to compare the properties of electric strings and center vortices. It turns out that in contrast to center vortices, electric Z2 strings are gauge-independent, thus the area of such strings can not be minimized by a procedure similar to the maximal center projection, which ensures the physical scaling of the density of center vortices [6,7]. The density of Z2 electric strings diverges in the continuum limit, consequently, they can not be described in terms of continuum theory. Nevertheless, there is a mechanism which makes the contribution of the minimal surface dominant - namely, the statistical weight of electric strings in the partition function is not positively defined due to the presence of topological Z2 monopoles. In order to separate Z2 center variables in SU (2) lattice gauge theory, each lattice link variable should be represented as gl = (-1) ml 91, ml = 0, 1. By definition the variables 91 are the elements of the coset manifold SU (2) /Z2 = SO (3). It will be assumed that multiplication of all "tilded" variables is a multiplication in SO (3) group. Note, however, that the product of two SU (2) group elements can not be in general expressed as the product of (-1) ml and the product of 91 in SO (3). In order to characterize this deviation, it is useful to define the Z2-valued function m (91,92) as: (1)

where gl,2 = (_1)m 1 ,2 91,2' Such function can easily be generalized for any number of SU (2) variables. For the purposes of this paper it is convenient to define the Z2-valued plaquette function mp and the functional m [0] which characterize the difference of products of SU (2) and SO (3) link variables over lattice plaquettes and over arbitrary closed loops, respectively:

380

II 91 = (-1)

m[GJ+LmL LEG

lEG

II 91

(2)

lEG

By definition mp and m [0] depend on the SO (3) variables 91 only. The functions m Uh, 92), mp and m [0] are characterized by the homotopy classes 7l'1 (SU (2) /Z2) C:o:' Z2 [11]. In order to make this statement more precise, consider, for instance, the function m (91,112). The points 91 E SO (3) and 9192 E SO (3) can be connected by two different geodesics '1'1 and '1'2 in the following way: '1'1: 9 (s) = 9192' S '1'2: 9 (s) = 9i- ?h,

[0,1]

(3)

s E [O,IJ

(4)

s E

The geodesics '1'1 and '1'2 form a loop on the group manifold, which is characterized by some element of 7l'1 (SO (3)) C:o:' Z2 which is precisely (_l)m(91,92). Further analysis is most easily performed using the notations of external calculus on the lattice [12]. p-forms on the lattice are associated with p-simplices, e.g. scalar functions are associated with lattice sites, I-forms - with lattice links, 2-forms - with lattice plaquettes etc. Correspondingly, scalar functions will be denoted by a subscript s, I-forms - by a subscript land 2-forms - by a subscript p. External and co-external differentials are denoted as d and 0, respectively. Scalar product of two p-forms f and 9 is denoted as (1,9) and the Hodge operator is denoted as *. Hodge operator on the lattice acts between p-forms defined on the simplices of the original lattice and D - p-forms defined on the simplices of the dual lattice. It can be shown that the operators d, 0, * and the scalar product of p-forms on the lattice have the same properties as in the continuum theory [12]. In all operations on Z2-valued forms summation is understood as summation modulo 2. Note that gauge transformations in SO (3) gauge theory 91 --t h s9tf"s' affect mp and m [0]: mp --t mp + dml, m [0] --t m [0] + L ml, where

lEG ml = m (hs, 91) +m (hs91' h-;,1). A gauge-invariant conserved current of topological Z2 monopoles can be defined as follows [13]: (5)

The Wilson loop in the fundamental representation of SU (2) gauge group can be expressed in terms of the new variables 91 and ml as:

Z (,8) W [0] =

J II SU(2)

I

d91 Tr

(II 91) II exp (,8/2 Tr 9p) = lEG

P

381

J

=L

ml SO(3)

II d91 Tr

(II

91)

lEG

I

II exp (,6/2 (_1)dm l +m p Tr 9p)

(6)

p

where Z ({3) is the partition function of the theory, gp =

TI gl

and 9p =

lEp

TI 91. lEp

The sum over Z2-valued variables ml in (6) can be represented as a sum over self-avoiding surfaces in two different ways, using the original or the dual lattice. One of these representations converges in the weak coupling limit, while the other is more suitable for the strong-coupling expansion [lOJ. Configurations of center vortices (Z2 magnetic strings) can be directly constructed from ml's. Namely, the worldsheets of center vortices are the surfaces on the dual lattice which consist of plaquettes p* for which *dml = 1 [6,7J. It is straightforward to show that such surfaces are indeed closed if one notes that for Z2 - valued 2-forms co-external derivative yields the I-form which is equal to I only if the link belongs to the boundary of the surface made of plaquettes for which this 2-form is nonzero. Since o*dml = *d**dml = *ddml = 0, the surfaces constructed from dual plaquettes with nonzero *dml are always closed. Such one-to-one correspondence between co-closed 2-forms and closed surfaces allows one to rewrite the sum over ml 's as a sum over all closed non-intersecting vortex worldsheets ~m on the dual lattice:

Z ({3) W [C]

=

L

J

ml SO(3)

1]

exp

(N2

(-1) dm, +m" Tr m[G]+

( -1)

II d91Tr

(II

91)

ii,) ~ m". ;'f". "OBi"

L

(-I)l~ ml+m[G]

lEG

I

If

diiil'<

'm p '

pEEC

II exp ({3/2 (-1) mpd

mp

Cu iii) Tr 9p) =

p

L Em; 8Em =O

J

(_l)L[E m ,c]

SO(3)

II d9l Tr I

(II 91) lEG

(-I)m[G]IIexP({3/2 (_l)mpTrgp) II exp(-{3 (-1) mpTr 9p) p

(7)

'pEE m

where mp' = *dml' ~G is an arbitrary surface spanned on the Wilson loop and L [~m' C] = L *mp' is the topological winding number ofthe surface ~m and pEEc

the loop C. The factor L [~m, CJ in (7) implies that the Wilson loop changes

382 sign each time it is crossed by center vortex. Note that the representation (7) is gauge-dependent, since the action associated center vortices depends on the non-invariant functions mp. The factor (_l)m[Gj is also gauge-dependent. Although the Wilson loop remains gauge-independent in any case, one can try to fix the gauge in such a way that the contribution of center vortices to the expectation value (7) becomes dominant and the contribution of the terms (_l)m[Gj and Tr

(n 91)

to the string tension can be neglected. It turns

lEG

out that such gauge-fixing procedure is exactly the maximal center projection [6,7, 14J, which rotates all link variables as close as possible to some element of the group center. It is also interesting to note that the presence of topological monopoles changes the action of center vortices, which indicates that monopoles can induce some nontrivial dynamics on the vortex worldsheet. Localization of Abelian monopoles on center vortices and the associated two-dimensional dynamics were indeed observed in lattice simulations [7,8J. The expression (6) can be also represented as a sum over worldsheets of Z2 electric strings - closed self-avoiding surfaces I;e which open on the loop G [9J. This is achieved by expanding the statistical weight of each plaquette in powers of (_l)dml using the identity exp((-l)mx) = chx (l+(-l)m t hx). The ~roduct of all the weights is then expanded into the sum of products of ( -1) ml over different sets of lattice plaquettes. After summation over ml each such product contributes to the sum only if each ml enters the product an even number of times. In this case the corresponding set of plaquettes forms a surface I;e bounded by the loop G [9], i.e. the sum over all sets of lattice plaquettes reduces to a sum over all such I;e:

Z(f3)W[GJ =

J II d91Tr (II91) (_l)m[Gj+Ee~in

L

Ee; 8E e=G80(3)

I

mp

lEG

L:

II ch (f3/2 Tr 9p) II th (f3/2 Tr 9p) (-l)/'EV

jl'

(8)

p

where I;e min is the surface of the minimal area spanned on the loop G and the Stokes theorem was used to represent L: mp as a sum over all links dual pEEe,E e

min

to cubes which belong to the volume V bounded by I;e and I;e min. Note that both combinations of Z2 variables in (8), m [GJ + L: mp and L: jl', are Ee

min

l'EV

gauge-invariant, therefore the representation (8) is gauge-independent. Thus the Wilson loop in the fundamental representation of SU (2) gauge group (or, more generally, in all representations with half-integer spin) can be represented as a sum over all surfaces of Z2 electric strings I;e which open on it. It is not difficult to derive similar representations for the partition function of the

383

theory or another observables such as t'Hooft loop or the correlators of Wilson loops. In order to see whether the representations (8) and (7) can be used in the continuum limit of the theory, one should study the scaling of the total area, or, equivalently, of the density of center vortices or Z2 electric strings. Consider first the density of center vortices. According to (7), some lattice plaquette *p belongs to center vortex if for the dual plaquette p 11 sign Tr gl = -1, IEp

correspondingly, the probability that a given plaquette belongs to center vortex is: 1-

P = (

11 sign Tr gl

1- (

)=

IEp

2

11 sign Tr gl ) IEp

2

(9)

P is nothing but the density of vortices in lattice units, which is by definition gauge-dependent. Consider first the theory without gauge-fixing and integrate 11 sign Tr glover gauge orbits gl ---; hsglh;l. A simple IEp

calculation which involves character decomposition of sign Tr g and using the "gluing" formula for the integrals of group characters gives the result J 11 dh s 11 sign Tr (hsglh;l) L: O'.kXk (gp), where the coefficients sEp

IEp

k=1/2,3/2, ... (_I)k+1/2 k- 6 .

O'.k behave as Thus integrating (9) over gauge orbits yields some gauge-invariant function of the plaquette variable gpo It can be shown that this function takes values in the range [-1/3 + 2/7r 2 , 1/3 - 2/7r 2 ]. As 1/3 - 2/7r 2 ;:::j 0.130691 < 1, the density of center vortices in lattice units remains finite at weak coupling, and their physical density diverges as a -4. On the other hand, any perturbative calculation around the vacuum gl = 1 yields exactly zero density of center vortices, which thus appear as truly nonperturbative objects. The situation changes dramatically after the maximal center gauge is imposed - in this case the total area of center vortices is minimized and their density in lattice units goes to zero as a 2 , thus their physical density remains finite in the continuum limit [6,7J. Since statistical weights of Z2 electric strings in the sum (8) are not positive, strictly speaking they can not be interpreted as random surfaces [15J. Nevertheless, one can define the vacuum expectation value of their density by introducing the" chemical potential" ?te for electric strings and by differentiating the partition function Z ({3, ?te) over it. This amounts to multiplying the statistical weight of each surface by an additional factor exp (-?te lEe I), where IEel is the total area of Ee. The partition function of the theory can be calculated from (8) by shrinking the loop C to zero. After such modification of (8) one can reverse all the transformations which led to this representation and write the partition function of the theory at nonzero ?te in terms of original

384

link variables as:

Z ((3, Me)

=

J SU(2)

dgl

IT (ch (f3/2 Tr gp) + e-JLesh (f3/2 Tr gp))

(10)

p

Correspondingly, the average total area of I;e in lattice units is: I z(a )1 _ =",I-(exp(-(3Trgp)) (I I; e I)=-~ {) n 1-', Me JLe- O ~ 2 Me

(11)

p

The expectation value (exp (-(3Tr gp) ) tends to zero as (3 -> 00 and the continuum limit is approached, therefore in the continuum limit of the theory Z2 electric strings occupy half of all lattice plaquettes and their density diverges as a- 4 . Nevertheless, the sum (8) remains well-defined at a -> 0 and in fact sums up to exp (-alI;e min!), which can only be explained by the exact cancellation L jt* of contributions from different surfaces with opposite signs of (_I)t*EV , i.e. due to the presence of Z2 topological monopoles. Indeed, if the term with jl* is omitted, the expression (8) can be considered as the partition function of Z2 lattice gauge theory with fluctuating, but always positive coupling. It can be shown that in the weak coupling limit electric strings in Z2 gauge theory also occupy half of all lattice plaquettes. The sum over such creased surfaces with positive weights can only lead to perimeter dependence of the Wilson loop, which is indeed the case for ZN lattice gauge theories in the weak coupling m[G]+

L

mp

limit. The terms Tr ( IT ) 91 and (-1) Ee min are also not likely to lEG contribute to the full string tension, since it can be shown that the expression (8) yields physical string tension even when these terms are omitted [16]. Thus it is reasonable to conjecture that in the weak coupling limit Z2 electric strings are confining due to the presence of topological monopoles with currents jl*. It could be interesting to study numerically the properties of such topological monopoles. To conclude, it was shown that unlike Z2 center vortices, which remain physical in the continuum limit [6,7], their duals - Z2 electric strings - can not be consistently described as random surfaces in the continuum theory. Instead, electric strings condense in a creased phase with infinite Hausdorf dimension, but nevertheless due to some cancelations between surfaces with positive and negative statistical weights the minimal surface I;e min dominates in the Wilson loop. In fact the formation of some creased structures is typical for subcritical strings [15]. For instance, sub critical Nambu-Goto strings exist only as branched polymers [15]. It was conjectured in [1] that such subcritical strings can be described as strings on AdS5 background, which hints at some possible relation with AdS/QCD.

385

Acknowledgments This work was partly supported by grants RFBR 05-02-16306, 07-02-00237-a, by the EU Integrated Infrastructure Initiative Hadron Physics (I3HP) under contract RII3-CT-2004-506078, by Federal Program of the Russian Ministry of Industry, Science and Technology No 40.052.1.1.1112 and by Russian Federal Agency for Nuclear Power. References [1] Alexander M. Polyakov. Confinement and liberation, 2004. [2] A. Karch, E. Katz, D. T. Son, and M. Stephanov. Linear confinement and AdS/QCD. Physical Review D, 2006. [3] G. S. Bali, K. Schilling, and C. Schlichter. Observing long colour flux tubes in SU(2) lattice gauge theory. Physical Review D, 51:5165, 1995. [4] P. Yu. Boyko, F. V. Gubarev, and S. M. Morozov. On the fine structure of QCD confining string, 2007. [5] Gerardus t' Hooft. On the phase transition towards permanent quark confinement. Nuclear Physics B, 138:1 - 25, 1978. [6] L. Del Debbio, M. Faber, J. Giedt, J. Greensite, and S. Olejnik. Detection of center vortices in the lattice Yang-Mills vacuum. Physical Review D, 58:094501, 1998. [7] F. V. Gubarev, A. V. Kovalenko, M. I. Polikarpov, S. N. Syritsyn, and V. I. Zakharov. Fine tuned vortices in lattice SU(2) gluodynamics. Physics Letters B, 574:136 - 140, 2003. [8] P. V. Buividovich and M. I. Polikarpov. Center vortices as rigid strings. Nuclear Physics B, 786:84 - 94, 2007. [9] A.Irback. A random surface representation of Wilson loops in Z(2) gauge theory. Physics Letters B, 211:129 - 131, 1988. [10] A. Ukawa, P. Windey, and A. H. Guth. Dual variables for lattice gauge theories and the phase structure of Z(N) systems. Physical Review D, 21:1013 - 1036, 1980. [11] P. Goddard, J. Nuyts, and D. Olive. Gauge theories and magnetic charge. Nuclear Physics B, 125:1 - 28, 1977. [12] P. Becher and H. Joos. The Dirac-Kahler equation and fermions on the lattice. Z. Phys. C, 15:343, 1982. [13] E. T. Tomboulis. 't Hooft loop in SU(2) lattice gauge theories. Physical Review D, 23:2371 - 2383, 1981. [14] V. G. Bornyakov, D. A. Komarov, and M. I. Polikarpov. P-vortices and drama of Gribov copies. Physics Letters B, 497:151, 2001. [15] J. Ambj0rn. Quantization of geometry. Lectures presented at the 1994 Les Houches Summer School, 1994. [16] J. Greensite, M. Faber, and S. Olejnik. Center projection with and without gauge fixing. JHEP, 9901:008, 1999.

UPPER BOUND ON THE LIGHTEST NEUTRALINO MASS IN THE MINIMAL NON-MINIMAL SUPERSYMMETRIC STANDARD MODEL S. Hesselbach, G. Moortgat-Pick IPPP, University of Durham, Durham, DH1 3LE, U.K. D. J. Miller, R. N evzorov a b Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, U.K. M. Trusov

Theory Department, ITEP, Moscow, 117218, Russia Abstract.We consider the neutralino sector in the Minimal Non-minimal Supersymmetric Standard Model (MNSSM). We argue that there exists a theoretical upper bound on the lightest neutralino mass in the MNSSM. An approximate solution for the mass of the lightest neutralino is obtained.

Super symmetric (SUSY) models provide an elegant explanation for the dark matter energy density observed in the Universe. To prevent rapid proton decay in SUSY models the invariance of the Lagrangian under R-parity transformations is usually imposed. As a consequence the lightest supersymmetric particle (LSP) is absolutely stable and can play the role of dark matter. In most super symmetric scenarios the LSP is the lightest neutralino, which provides the correct relic abundance of dark matter if it has a mass of 0(100 GeV). In this article we explore the neutralino sector in the framework of the simplest extension of the minimal SUSY model (MSSM) - the Minimal Nonminimal Supersymmetric Standard Model (MNSSM). The superpotential of the MNSSM can be written as follows [1-3]

(1) where WMSSM(/L = 0) is the superpotential of the MSSM without /L-term. The superpotential (1) does not contain any bilinear terms avoiding the /Lproblem. At the same time quadratically divergent tadpole contributions can be suppressed in the considered m<,:>del so that ~ ::::; (TeV)2 [1,2]. At the electroweak (EW) scale the superfield 8 gets a non-zero vacuum expectation value ((8) = s/V2) and an effective /L-term (/Lett = AS/V2) is automatically generated. The neutralino sector of the MNSSM is formed by the superpartners of the neutral gauge and Higgs bosons. In the field basis (iJ, W3, iI~, iI~, S) the aO n leave of absence from the Theory Department, ITEP, Moscow, Russia. be-mail: [email protected]

386

387

neutralino mass matrix reads Mxo

= MI

0

-MzswcfJ

Mzsws fJ

0

0

M2

MzcwcfJ

-MzcwsfJ

0

-MzswcfJ

MzcwcfJ

0

-Meff

--sfJ

Mzsws fJ

-MzcwsfJ

-Meff

0

--cfJ

0

0

--sfJ

.xv

.xv --cfJ

0

V2

V2

.xv

~

(2)

V2

where MI and M2 are the U(l)y and SU(2) gaugino masses while Sw = sin ew , Cw = cosew , sfJ = sin/3, cfJ = cos/3 and Meff = .xs/V2. Here we introduce tan/3 = V2/VI and v = Jvr +v~ = 246 GeV, where VI and V2 are the vacuum expectation values of the Higgs doublets fields Hd and H u , respectively. From Eq.(2) one can easily see that the neutralino spectrum in the MNSSM may be parametrised in terms of

.x,

Me!! ,

tan/3 ,

(3)

In supergravity models with uniform gaugino masses at the Grand Unification scale the renormalisation group flow yields a relationship between MI and M2 at the EW scale, i.e. MI ~ 0.5M2 • The chargino masses in the MNSSM are also defined by the mass parameters M2 and Me!!' LEP searches for SUSY particles set a lower limit on the chargino masses of about 100 GeV restricting the allowed interval of IM21 and IMeffl above 90 -100GeV. In contrast with the MSSM the allowed range of the mass of the lightest neutralino in the MNSSM is limited. In Fig. 1 we plot the lightest neutralino mass Imx~ I in the MSSM and MNSSM as a function of M2 for different values of Me!!' From Fig. 1 it becomes clear that the absolute value of the mass of the lightest neutralino in the MSSM grows when IM21 and IMe!!1 increase while in the MNSSM the maximum value of Imx~1 reduces with increasing IM21 and IMeffl· In order to find the upper bound on Imx~ I it is convenient to consider the matrix MxoM1o whose eigenvalues are equal to the absolute values of the neutralino

masses squared. In the basis

(B, W -iI~sfJ + iI~cfJ' iI~cfJ + iI~sfJ' S) 3,

the

bottom-right 2 x 2 block of MxoM1o takes the form [4] (

IMe!!I~ + 01I Me!!

2

lI*l~elf!

),

(4)

where 0- 2 = M1 cos 2 2/3+11I12 sin 2 2/3, 1I = .xv/V2. Since the minimal eigenvalue of any hermitian matrix is less than its smallest diagonal element the lightest

388

neutralino in the MNSSM is limited from above by the bottom-right diagonal entry of matrix (4), i.e. Imx~ I ::; Ivl. At the same time since we can always choose the field basis in such a way that the 2 x 2 submatrix (4) becomes diagonal its minimal eigenvalue JL6 also restricts the allowed interval of Imx~ I, i.e.

Imx~ 12 ::; 11-6 = ~ [IJLe f f 12 + &2 + Ivl 2

(5)

.---------------~--------

12 + &2 + Iv12) 2- 4IvI2&2] .

(1l1-e f f

The value of JLa reduces with increasing IJLeff I. It reaches its maximum value, i.e. 11-6

min{ &2,

=

Ivl 2},

when l1-eff

O. Taking into account the restriction

-+

on the effective JL-term coming from LEP searches and the theoretical upper bound on the Yukawa coupling A which is caused by the requirement of the validity of perturbation theory up to the Grand Unification scale (A < 0.7) we find that Imx~ I does not exceed 80 - 85 Ge V at tree level [4,5].

"

\ 250 200

"..

70 r-----------'---,------------.-,

60

/

\ I

\

-----'~

I

\

150

100 50

-1000

-.-

Imx~1

-500

"..

-

,r 1/

./

.~

\

50

----

40 30 _ . _ . _ . _ . _ . -

-

20

10

,1

500

1000

-1000

-500

500

1000

Figure 1: Lightest neutralino mass versus M2 in the (a) MSSM and (b) MNSSM for tan.B = 3, >. = 0.7, Ml = 0.5Ml. Solid, dashed and dash-dotted lines correspond to J.LeJ J = 100 GeV, 200 GeV and 300 GeV, respectively.

Here it is worth to notice that at large values of JLef f the allowed interval of the lightest neutralino mass shrinks drastically. Indeed, for IJLef f 12 » M~ we have

1m 01 2< X,

-

2-2

IV 1 (J

(IJLe ff l2

+ &2 + IVI2)

•

(6)

Thus in the considered limit the lightest neutralino mass is significantly smaller than M z even for the appreciable values of A at tree level.

389 When the mass of the lightest neutralino is small one can also obtain an approximate solution for mx~' In general, the neutralino masses obey the characteristic equation det (Mxo - ",1) = 0, where", is an eigenvalue of the matrix (2). However, if '" - t 0 one can ignore all terms in this equation except the one which is linear with respect to '" and the ",-independent one which allows to solve the characteristic equation. This method can be used to calculate the mass of the lightest neutralino when M I , M2 and Ilef f » Mz because then the upper bound on Imx~ I goes to zero. We get in this limit (see [4,5]) \Ileff \v sin 213 Il;ff + v 2 2

Im Xl I 0

~

(7)

According to Eq.(7) the mass of the lightest neutralino is inversely proportional to Ilef f and decreases when tan 13 grows. At small values of A the lightest neutralino mass is proportional to A2 because the correct breakdown of electroweak symmetry breaking requires Ilef f to remain constant when A goes to zero. At this point the approximate solution (7) improves the theoretical restriction on the lightest neutralino mass derived above because for small values of A the upper bound (5)-(6) implies that \mxo\ ex A. Note, however, that the lightest neutralino is predominantly singlino if M I , M2 and Ileff » Mz which makes its direct observation at future colliders quite challenging.

Acknowledgment RN acknowledge support from the SHEFC grant HR03020 SUPA 36878.

References [1] C. Panagiotakopoulos, K. Tamvakis, Phys.Lett. B 469, 145 (1999). [2] C. Panagiotakopoulos, A. Pilaftsis, Phys.Rev. D 63, 055003 (2001). [3] A. Dedes, C. Hugonie, S. Moretti, K. Tamvakis, Phys.Rev. D 63, 055009 (2001). [4] S. Hesselbach, D. J. Miller, G. Moortgat-Pick, R. Nevzorov, M. Trusov, in preparation. [5] S. Hesselbach, D. J. Miller, G. Moortgat-Pick, R. Nevzorov, M. Trusov, arXiv:0710.2550 [hep-ph].

APPLICATION OF HIGHER DERIVATIVE REGULARIZATION TO CALCULATION OF QUANTUM CORRECTIONS IN N=l SUPERSYMMETRIC THEORIES K.Stepanyantz a

Faculty of Physics, Moscow State University, 119992 Moscow, Russia Abstract. We discuss the structure of quantum corrections in N = 1 supersymmetric theories, obtained with the higher covariant derivative regularization. In particular, we argue that all integrals, defining the Gell-Mann-Low function in supersymmetric theories, are integrals of total derivatives. As a consequence, there is an identity for Green functions, which does not follow from any known symmetry of the theory, in N = 1 super symmetric theories.

Investigation of quantum corrections in super symmetric theories is an interesting and sometimes nontrivial problem. For example, in N = 1 supersymmetric theories it is possible to suggest [1] the form of exact ,B-function

a2 ,B(a)

=-

[30

(1 - 'Y(a))]

OCR) 27r(1 - 02 a / 27r ) 2 -

.

(1)

To derive it by usual methods of the perturbation theory is a complicated problem. Here we will show that a derivation of this ,B-function (more exactly, the matter contribution to this function) can be made if a special new identity for the Green functions of the matter superfield takes place. We consider N = 1 super symmetric Yang-Mills theory, which is described by the action

Quantization of this model can be made by standard methods. In particular, we use the background field method, which allows preserving the background gauge invariance and considerably simplifies the calculations of quantum corrections. In order to regularize model (2) we add to its action the higher derivative term, which is invariant under the background gauge invariance, but breaks the BRST-invariance. Therefore, calculating quantum corrections it is necessary to use a special subtraction scheme, which restores the Slavnov-Taylor identities in each order [2]. In order to cancel the remaining one-loop divergences we should also insert to generating functional the Pauli-Villars determinants [3]. We will calculate a matter contribution to the Gell-Mann-Low function. If V denotes the background gauge field and

ae-mail: [email protected]

390

391

+41

J

4

d p d4 () ( ¢ + (-p, ()) ¢(p, ())¢-+ (-p, ()) ¢(p, -) (27f)4 ()) ZG(o:, J1-/p),

(3)

where 0: is a renormalized coupling constant and Z is the renormalization constant for the matter superfield, then the Gell-Mann-Low function {3(0:) and the anomalous dimension ')'(0:) are defined by

{3(d(O:,J1-/p))

= a~pd(O:'J1-/P);

')'(d(O:,J1-/p))

= - a~p In ZG(o:,J1-/p). (4)

Calculation of the matter contribution to the Gell-Mann-Low function can be made substituting solutions of Slavnov-Taylor identities to the SchwingerDyson equation for the two-point Green function of the gauge superfield [4]. The result (without subtraction diagrams) can be written as

(5)

where dots denote contributions of the gauge field and ghosts and (PV) denotes a contribution of the Pauli- Villars fields. The function G is defined by Eq. (3) and the function f is related with the three-point function

(6) where ¢o and ¢o are introduced in the generating functional by adding to the action the term

Actually the Green function (6) is very similar to the usual Green function, but one of the matter ends is not chiral. The first term in Eq. (5) is an integral of the total derivative and can be easily calculated using the identity

However, the explicit calculations [5,6] always show that the second term in Eq. (5) is also an integral of the total derivative and is always equal to O. This allows suggesting existence of the new identity

392

(9)

which is not a consequence of supersymmetric or gauge Slavnov-Taylor identities. (The derivative with respect to In A is needed in order to make all integrals well defined.) The new identity is nontrivial starting from the three-loop approximation (or the two-loop approximation for the Green function (6)). Its verification for the Abelian theory was made in three- and partially four-loop approximation [6]. A sketch of a possible proof exactly to all orders in Abelian case is made in Ref. [7]. However, it is necessary to verify if the new identity takes place in the non-Abelian case. For this purpose we consider [8] the three-loop diagram

and construct the corresponding function J, calculating the diagrams, which are obtained from it by cutting the matter line and attaching an external line of the background gauge field by all possible ways. After substituting the result to the left hand side of identity (9), we obtain

d4q d J(q2) (211")4 dInA q2 G(q2)

J8 {A x 8qJ.l

X

d [ dA (k

q2k2[2(k

2 2

=

a

11" O2

1 (02(R) - "202)

(k+q+l)J.I + q)2 (k + q + l)2 + k2n / A2n)

+ l)2 (1

(1

~ (k + l)2n / A2n ) ] } = O.

J

d4q d4k d4l (211")12

(1 + l2n / A2n)

X

X

(10)

Therefore, the new identity and the factorization of integrands to total derivatives also take place in the non-Abelian case. In order to check if the factorization of integrands to total derivatives is a general feature of supersymmetric theories, we calculate the two-loop ,6-function for the N = 1 supersymmetric Yang-Mills theory without matter. It well known that in this case

(11) Calculating two-loop diagrams, defining the Gell-Mann-Low function (so far without diagrams with insertions of counterterms), in the limit p -+ 0 we find (in the Euclidean space after the Weak rotation)

393 d - 8 2-

6

d

rr· rrQodlnA

x { ( (q

J

4 d kId (2rr)4k 2 dk2

J

4 d q (

2

(2rr)4 q (l+q

2n

2n) -1

/A)

+ k) 2 (1 + (q + k) 2n / A2n )) -1 [2 (n + 1) ( 1 + k2n / A2n ) -1

-2n( 1 + k 2n / A2n)

-2]}.

x

_

(12)

This integral can be calculated by Eq. (8). Then the two-loop Gell-Mann-Low function agrees with Eq. (1). After taking into account diagrams with counterterms insertions [9] with the higher derivative regularization we find that divergences are only in the one-loop approximations similar to the supersymmetric electrodynamics, while the Gell-Mann-Low function has corrections in all loops. This agrees with results of Ref. [10]. Therefore, factorization of integrands to total derivatives seems to be a general feature of all supersymmetric theories. However, the reason is so far unclear. Actually new identity for Green functions (9) is a consequence of this fact. Acknowledgment

This paper was supported by the Russian Foundation for Basic Research (Grant No. 05-01-00541). References

[1] V.Novikov, M.Shifman, A.Vainstein, V.Zakharov, Phys.Lett. 166B,329, (1985). [2] A.A.Slavnov, Phys.Lett. B 518, 195, (2001); Theor. Math. Phys. 130, 1, (2002); A.A.Slavnov, KV.Stepanyantz, Theor. Math. Phys. 135, 673, (2003); 139, 599, (2004). [3] L.D.Faddeev, A.A.Slavnov, "Gauge fields, introduction to quantum theory, second edition", Benjamin, Reading, 1990. [4] KV.Stepanyantz, Theor.Math.Phys. 142, 29, (2005); 150, 377, (2007). [5] A.A. Soloshenko, K V. Stepanyantz, hep-th/ 0304083. (Brief version of this paper is Theor.Math.Phys. 140, 1264, (2004).) [6] A.Pimenov, KStepanyantz, Theor. Math. Phys. 147, 687, (2006). [7] KStepanyantz, Theor.Math.Phys. 146, 321, (2006). [8] A.Pimenov, KStepanyantz, hep-th/0710.5040. [9] A.Pimenov, KStepanyantz, hep-th/0707.4006. [10] M.Shifman, A.Vainstein, Nucl.Phys. B277, 456, (1986).

NONPERTURBATIVE QUANTUM RELATIVISTIC EFFECTS IN THE CONFINEMENT MECHANISM FOR PARTICLES IN A DEEP POTENTIAL WELL K.A. Sveshnikov a , M.V. Ulybyshev b

Department of Physics, Moscow State University, 119991, Moscow, Russia Abstract. The properties of relativistic bound states of bosons and fermions confined in the deep potential well are considered within the framework of covariant hamiltonian formulation of the quasipotential approach. It is shown, that the main properties of such relativistic bound states like wavefunctions and the structure of energy spectrums turn out to be appreciably different from corresponding solutions of differential Schrodinger or Dirac equations for the static external potential of the same form.

1

Introduction

In this report the spectral problem for relativistic bound states of bosons and fermions in the one-dimensional potential well is considered within the framework of quasi potential approach [1] in the relativistic configuration representation (RCR) [2,3]. In this approach the kinetic term of the Hamiltonian contains operators of pure imaginary shift in the radial argument instead of differential ones [2,3], what allows for the study of some nonperturbative properties of relativistic bound states, that are invisible in the standard decription based on differential Schrodinger /Dirac equations in the static external potential of the same form. For a scalar particle in the RCR the quasipotential equation looks like the finite-difference analogue of the Schrodinger equation [2,4]:

~[¢(~ -

n

in) +

¢(~ + in) - 2¢(~)] + V(~)¢(~) = E¢(~)

,

(1)

where V(~) = V02 B(I~I

-

~o) .

In (1) the origin of the variable ~ is the group one, actually it is the eigenvalue of the (first) Lorentz group Kasimir operator and plays the role of the (dimensionless) particle relative coordinate in the center-of-mass system [2,3]. In our l+l-dimensional case the Lorentz group consists of only one Lorentz boost generator, whose spectrum coincides with the real axis, whence -00 :::; ~ :::; +00. In this case is also dimensionless and is nothing else, but the effective Planck constant of the system, which enters the r.h.s. of field and group algebra commutators and defines the magnitude of quantum effects with respect to the classical ones, that by definition are chosen to be 0(1). The regular dimensional c.m.s. coordinate x is connected with ~ via relation ~ = M x, where

n

a e-mail:

[email protected] be-mail: [email protected]

394

395 M is the total mass of the system. For x being the wavefunction argument, the imaginary shift in (1) should be hiM and so coincides with the (effective) Compton wavelength of the system. Here we'll use the first, more formal dimensionless treatment of ~ and h, thence all the other parameters should be considered as dimensionless too. The correct formulation of the problem (1) requires for a consistent definition of pure imaginary shifts in the argument of the wavefunction. For these purposes we'll consider in (1) only analytical in the strip 11m ~I < h functions [4]. In addition, we demand for the bound states wavefunctions to be square-integrable on the real axis. The physical sense of the analyticity condition is the convergence of the finite-difference analogue of the Schrodinger kinetic energy [4]: K[¢] = -

= -

;2 Jd~ [¢(~ -

;2 Jd~ [¢(~ +

ih) -

ih) -

¢(~)]*[¢(~ -

ih) -

¢(~)]

¢(~)]*[¢(~ + ih) - ¢(~)]

=

.

For a potential well this requirement removes also the ambiguity in solutions of finite-difference equations of such type. 2

The solution and main results

The general approach to solution of the spectral problem (1) is quite similar to the corresponding differential Schrodinger equation. It is solved firstly in spatial regions, where the potential V(~) is a constant, afterwards the obtained solutions are sewed together in order to provide analyticity of the wavefunction in the whole strip 11m ~I < h. The general form of solutions inside and outside the well is represented by the Dirichlet series with the following structure (here we show explicitly the even wavefunction inside the well): +00

¢in = 2

L

An cos(Wn~),

n=-oo

where the following parametrization of the energy is accepted: E

. 2 hw 2 4 . 2 hI), = -42 smh - = Vo - - 2 sm h

2

h

2 '

.

.

zWn = ZW+

27m

T.

(2)

The most effective method of treating the eq.(l) is transforming it to the integral form, what after some algebra combined with the analyticity condition mentioned above leads to the following infinite set of algebraic equations for

396 coefficients An [4]:

/1,8

=

27rS

/1,+

h'

S

= 0 ... 00.

(3)

The principal difficulty of algebraic systems like (3) is the existence of exponentially increasing with Inl factors in the coefficients. As a consequence, there doesn't exist any general method of solution for such systems. However, in the case under consideration it is possible to elaborate a specific nonperturbative method of "quasi-exact" solution [4], which allows to study the properties of spectrum and wavefunctions for a wide range of parameters of the problem in detail. The main condition for such" quasi-exact" solution to be valid reads

(4) what is quite consistent with the hadron structure at low energies. In particular, for ~o '::::!. n (the pion) we get oX '::::!. 10- 3 . Dropping the details of this solution given in [4], we present the final result: for the low-lying levels in a sufficiently deep well, for which the following conditions hold 2

Vo

4

»n2

7r

'

/1,

.

= h + za,

w«

a ,

na

-» 1 27r

(5)

the energy spectrum is defined from e2iwaeff

= -1,

aeff = 2~0 - 2~~0 ,

n na

~~o = - I n - . 7r

27r

(6)

The eq. (6) looks like the corresponding spectral equation for even solutions of ordinary differential Schrodinger equation for a particle in the infinitely deep well, but now the geometric width of the well 2~0 is replaced by the effective aeff, which is smaller than 2~0. As a result, the energy levels of relativistic bound states in a deep well lye higher, than their nonelativistic analogues. Although this effect can be rigorously proven for only such and parameters of the potential, which satisfy the following relation

n

1«

nVr)2 (T «

~

e2rren -

,

(7)

actually the interval, where the result (6) is valid, covers a rather wide range of values Vo. For example, for ~o '::::!. 2n the upper limit for (nVo)2 can be estimated as 10001000. In the fermionic case it is convenient to deal with the squared Dirac equation, since it provides the most straightforward way to achieve the energy spectrum.

397 Under the same conditions for low-lying levels in a deep well (5) we can show, that the spectral equation looks like the bosonic case, except additional phase factor S, which enters the r.h.s. of (6). Under our conditions it becomes approximately equal to ±1, namely

S = ±1

+ 0 [(In Va/Va)2]

.

So with increasing Va the spectral equation for fermions approximately coincides with the bosonic one for odd (even) levels, depending on the sign in S.

3

Conclusion

To conclude let us compare the bosonic and fermionic spectrums in the case of ordinary differential and finite-difference RCR-equations. In the case of the differential equations the energy levels in the fermionic case remarkably differ from the bosonic one by a constant shift due to different boundary conditions for infinitely deep potential well. Another picture we observe in the case of finite-difference RCR-equations for sufficiently, but not infinitely, deep well, which meets the conditions (7). Namely, with increasing Va the energy levels of fermions coincide with those of bosons, and an effective contraction of the well and the corresponding growth of the energy levels compared to the differential case takes place as well. The effect of coincidence of bosonic and fermionic spectrums for relativistic bound states can be qualitatively understood as follows. Boundary conditions here are replaced by condition of analyticity in the strip, therefore there is no discontinuity in the first derivative on the well boundary and the wavefunctions penetrate always into the forbidden region. This effect is quite similar both for bosons and fermions, so instead of different boundary conditions in the relativistic case for low-lying bound states of bosons and fermions in a deep well we have almost similar behavior of the wavefunctions with the same depth of penetration in the forbidden region, hence almost identical energy levels.

Acknowledgment This work has been supported in part by the RF President Grant NS-4476.2006.2.

References [1] A.A.Logunov, A.N.Tavkhelidze, Nuov. Cim. 29, 380 (1963). [2] V.G. Kadyshevsky, R. M. Mir-Kasimov, N. B. Skachkov, Part. and Nucl. 2, 635 (1972). [3] N.B.Skachkov, I.L.Solovtsov, Part. and Nucl. 9, 5 (1978). [4] K.A.Sveshnikov, P.K.Silaev, TMF 132, 408 (2002).

KHALFIN'S THEOREM AND NEUTRAL MESONS SUBSYSTEM a Krzysztof Urbanowski b University of Zielona Cora, Institute of Physics, ul. Prof. Z. Szafran a 4a, 65-516 Zielona Cora, Poland. Abstract.The consequences of Khalfin's Theorem are discussed.we find, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. Within a given model we examine numerically the Khalfin's Theorem and show in a graphic form how the Khalfin's Theorem works.

1

Introduction

One of the most interesting two state (or two particle) subsystems is the neutral mesons complex. The standard method used for the description of the properties of such complexes is the Lee-Oehme -Yang (LOY) approximation [2,3]. The source of this approximation applied by LOY to the description and analysis of the decay of neutral kaons is the well known Weisskopf-Wigner theory of the decay processes. The rigorous treatment of two particle complexes shows that there are some inconsistences in the LOY method. This problem is connected with the so-called Khalfin's Theorem [4-8].

2

Khalfin's Theorem and its implications

According to the general principles of quantum mechanics transitions of the system from a state I'lPI) E 11. at time t = to the state l'lh) E 11. at time t > 0, 11P1) ~ 11P2), are realized by the transition unitary unitary transition operator U(t) acting in 11.. The probability to find the system in the state l1Pj) at time t if it was earlier at instant t = in the initial state l1Pk) is determined by the transition amplitude Ajk(t),

°

°

(1)

where (j, k

= 1,2).

Khalfin's Theorem If [4-8]

= f 21 (t) ~f A21(t) A12(t) P=

const.

(2)

then there must be

R

= Ipi = 1.

aThis paper is a shortened version of [lJ be-mail: [email protected];[email protected]

398

(3)

399 As it was pointed out in [7] the only problem in the proof of this Theorem is to find conditions guaranteeing the continuity of hI (t) at t = O. This problem can be solved by taking into account properties of U(t). Namely quantum theory requires U(t) to have the form, U(t) = e- itH , (using units Ii = 1), where H is the total hermitian Hamiltonian of the system, (or, in the interaction picture

UI(t) = 11' e -i J~ H I (T) dT, where 11' denotes the usual time ordering operator and HI(T) is the operator H in the interaction picture). Using this observation one can easily verify that to assure the continuity of hI (t) at t = 0 it suffices that there exists such n ::::: 1 that ('l/J2I Hk l'l/JI) ('l/J2I Hn l'l/JI)

0,

(0

:s: k < n),

=I 0 and I('l/J2 IHnl'l/JI) I <

(4) 00.

(5)

Note that for the neutral meson complexes according to the experimental results the particle-antiparticle transitions I'l/JI) ~ 1'l/J2) exist, which means that there must exist n < 00 such that the relation (5) occurs. This means that in fact for the neutral meson complexes, where the transitions I'l/JI) ~ 1'l/J2) take place, only the assumption of unitarity of the exact transition operator U(t) assures the validity of the Khalfin's Theorem. Let us assume now that vectors I'l/JI), 1'l/J2) are orthonormal and that the twodimensional subspace 'HII of'H is spanned by them. If one assumes that the evolution operator UII (t) acting in this 'HII has the form UII (t) = e -itHII and that the operator HII is a non-hermitian time-independent (2 x 2) matrix acting in 'HII' then denoting Ujk ~ ('l/Jj lUll (t)I'l/Jk)' (j, k finds that in the considered case

U2I (t) UI2(t)

=

= 1,2) after some algebra one

h2I def = r = const hI2 '

-

(6)

and

Ull(t) = U22(t)

<=}

hll

= h22,

(7)

where hjk = ('l/JjIHIII'l/Jk), (j,k = 1,2). The conclusion following from Khalfin's Theorem, (2), (3) and from (6) seems to be important,

Conclusion 1 If Irl =I 1 and the time-independent effective Hamiltonian HII is the exact effective Hamiltonian for the subspace 'HII of states of neutral mesons, so that (8) where j =I k, (j,k = 1,2), r is defined by (6) and Ujk(t) = (jle-itHlllk), and Ajk(t) is given by (1), then the evolution operator U(t) for the total state space

400

'H can not be unitary. With the use of this exact transition operator A(t), (the matrix elements Ajk(t) of A(t) are given by (1) ) for the subspace 'HII the exact effective Hamiltonian HI! governing the time evolution in 'HII can be expressed as follows [9,10] HII

=

.8A(t)

HII(t) == 2-at[A(t)]

-1

.

(9)

It is easy to find from (9) the general formulae for the diagonal matrix elements, hjj(t), as well as for the off-diagonal matrix elements, hjdt) of the exact HII(t) [9]. Using these formulae and taking into account CPT- and CPsymmetry properties of the system considered and the Khalfin's Theorem one finds that the following property must hold for the exact effective Hamiltonian for neutral meson subsystem:

Conclusion 2 If [8, H] = 0 and t

rep, H] =/: 0, that is if All (t)

= A22(t) and

I~~~m I =/: 1 for

> 0, then there must be (hll(t) - h22(t)) =/: 0 for t > O.

So, within the exact theory one can say that for real systems, the property (7) can not occur if CPT symmetry holds and CP is violated. This means that the relation (7) can only be considered as an approximation.

3

Model calculations

In this Section we discuss results of numerical calculations performed within the use of the symbolic and numeric package "Mathematica" for the model considered by Khalfin in [4,5], and by Nowakowski in [8] and then used in [11]. This model is formulated using the spectral language for the description of Ks,KL and KO, K, by introducing a hermitian Hamiltonian, H, with a continuous spectrum of decay products (for details see [1]). Assuming that CPT symmetry holds but CP symmetry is violated and using the experimentally obtained values of the parameters characterizing neutral kaon complex make it possible within this model to examine numerically the Khalfin's Theorem as well as other relations and conclusions obtained using this Theorem (for details see [8, lID. The results of numerical calculations of the modulus of the ratio ~~~m for some time interval are presented below in Fig. 1. Analyzing the results of these calculations one can find that for x E (0.01,10),

Ymax(X) - Ymin(X) ':::: 3.3 where, Ymax(X)

= Ir(t)lmax

and Ymin(X)

X

10- 16 ,

= Ir(t)lmin-

(10)

401

~~d~W~--~W~I~I~'i~-

··C.2:

..

4

10

Figure 1: Numerical examination of the Khalfin's Theorem. Here y(x)

= Ir(t)1 == 1~~~m I, x = If .t,

and x E (0.01,10).

Similarly, using "Mathematica" and starting from the amplitudes Ajk(t) and using the formulae for hll (t), h22(t) and the condition All (t) = A22(t) one can compute the difference (h ll (t) - h22 (t) for the model considered. Results of such calculations for some time interval are presented below in Fig. 2. An expansion of scale in the left panel of Fig. 2 shows that continuous fluctuations, similar to those in the right panel of Fig. 2, appear.

2.01' 10.

11

1.99"0-"+~--""--;;2-""'3--:----"5

x

1. 9S .10. 13 1.97 '10. 13

-1.5'10. 16

1.96 '10"l) 1.95-10. 13

Figure 2: The real part (left) and the imaginary part (right) of (hll(t) - h22(t))

*' .

There is y(x) = 3{(h ll (t) - h22(t) and y(x) = <J(hll(t) - h22(t) in Figs 2 t, x E (0.01,5.0) and 3{ (z) and <J (z) respectively. In these Figures x = denote the real and imaginary parts of z respectively and units on the y-axis are in [MeV]. 4

Final remarks

From (10) and Fig. 1 the conclusion follows that if one is able to measure the modulus of the ratio ~~~m only up to the accuracy 10- 15 then one sees this quantity as a constant function of time. The variations in time of I ~~~m I become detectable for the experimenter only if the accuracy of his measurements is of order 10- 16 or better. Within the standard approach the following parameters are used to describe the scale of CP- and possibly CPT - violation effects [3,12]' C ~f ~(cs

+ cl) ==

402 h - h A D- 1 (h 12 -h 21 ), and 0 clef = '12 ( Cs-Cl ) =~, where D clef = h 12+ h 21 +u/-L, and ~/-L = /-LS - /-LL· According to the standard interpretation following from the LOY approximation, C describes violations of CP-symmetry and 0 is considered as a CPT-violating parameter [12]. Such an interpretation of these parameters follows from the properties of LOY theory of time evolution in the subspace of neutral kaons [2,3,12]. From Conclusion 2 and from the results of the model calculations presented in Sec. 3 it follows that the parameter 0 should not be considered as the parameter measuring the scale of possible CPT violation effects: In the more accurate approach [13] and in the exact theory one obtains 0 =1= 0 for every system with violated CP symmetry and this property occurs quite independently of whether this system is CPT invariant or not. What is more, from the Conclusion 2 one finds that if CP symmetry is violated and CPT symmetry holds then there must be cl =1= Cs contrary to the standard predictions of the LOY theory. These conclusions are in full agreement with the results obtained in [14] within the quantum field theory analysis of binary systems such as the neutral meson complexes.

References [1] K. Urbanowski, arXiv: [hep-ph]0712.0328. [21 T. D. Lee, R. Oehme, C. N. Yang, Phys. Rev., 106 (1957) 340. [3] S. Eidelman et al, Review of Particle Physics, Phys. Lett. B 592, No 1-4, (2004). [4] L. A. Khalfin, Preprints of the CPT, The University of Texas at Austin: DOE-ER-40200-211, February 1990 and DOE-ER-40200-247, February 1991; (unpublished, cited in [6]), and references one can find therein. [5] L. A. Khalfin, Foundations of Physics, 27 (1997), 1549. [6] C. B. Chiu and E. C. G. Sudarshan, Phys. Rev. D42 (1990), 3712 [7] P. K. Kabir and A. Pilaftsis, Phys. Rev., A53, (1996), 66. [8] M. Nowakowski, Int. J. Mod. Phys. A14, (1999), 589. [9] K. Urbanowski, Phys. Lett., B 540, (2002), 89; hep-ph/0201272. [10] K. Urbanowski, Acta Phys. Polan. B 37 (2006) 1727. [111 J. Jankiewicz, Acta. Phys. Polan. B 36, (2005), 1901; Acta. Phys. Polan. B 38, (2007), 2471. [12] L. Maiani, in The Second Daif>ne Physics Handbook, vol. 1, Eds. L. Maiani, G. Pancheri and N. Paver, SIS - Pubblicazioni, INFN - LNF, Frascati, 1995; pp. 3 - 26. [13] J. Jankiewicz, K. Urbanowski, Eur. Phys. J. C 49, (2007), 721. [14] B. Machet, V. A. Novikov and M. 1. Vysotsky, Int. J. Mod. Phys. A 20, (2005), 5399; hep-ph/0407268. V. A. Novikov, hep-ph/0509126.

EFFECTIVE LAGRANGIANS AND FIELD THEORY ON A LATTICE Oleg V. Pavlovskya Institute for Theoretical Problems of Microphysics, Moscow State University, 119992 Moscow, Russia Abstract.esent developments in the Random Matrix and Random Lattice Theories give a possibility to find low-energy theorems for many physical models in the BornInfeld form [1]. In our approach that based on the Random Lattice regularization of QeD we try to used the similar ideas in the low-energy baryon physics for finding of the low-energy theory for the chiral fields in the strong-coupling regime.

1

Why we need in the Random Lattice QCD

The attempts to obtain a chiral effective lagrangian from lattice QCD had been performed many times a long ago. Using of the well-known Brezin&Gross trick [4] it could be possible to perform the link's matrix integration in strong coupling regime and obtain the various first order chiral effective theories [5]. In spite of first great success this approach had not been very popular and origin of this stems from the fact that the approaches from [5] does not take the possible to obtain any corrections to the first order results. The lattice regularization breaks a rotational symmetry of the initial theory from the continues rotation group to a discrete group of rotations on fixed angles. And the lattice regularization approach gives the correct results only such tensors which are invariant by respect to this discrete group. In particular using the ordinary Hyper-Cubical (HC) lattice one can obtains the only first order effective theory and for corrections this method generates non-rotational (non-lorentz) invariant terms. For generating of the high-order effective field theories more symmetrical lattice must be considered. Fortunately this conception is known for a long time and is called the Random Lattice Approach [6]. The ideas of the Random Lattice was proposed by Voronoi and Deloune and today this method is very widely used in the modern science and technology. For the quantum field theory method was modified by Christ, Friedberg and Lee [6]. In these articles have been shown that in order to obtain the restoration of the Lorentz (rotational) invariance, it is necessary to perform an average over an ensemble of random lattices. As result one get the averaging over all possible directions and it is intuitively clear that this procedure leads to the disappearance of the artifacts connected this violation of the group of the space rotation. But how to perform such random discretization? This procedure has the tree steps: 1) Draw N sites Xi at random in the volume V; 2) Associate with each Xi a so-called Voronoi cell Ci Ci = {xld(x, Xi) ~ d(x, Xj), Vj i- i} where d(x, y) is a distance between points X and y. It is means that Voronoi cell Ci ae-mail: [email protected]

403

404

consists of all points x that are closer to the center site Xi than any other site; 3) Constrict the dual Delaune lattice by linking the center sites of all Voronoi cells which share a common face. After this if one consider the the big ensemble of such Voronoi-Deloune random lattices based on various distributions of sites Xi, it possible to prove the origin rotational symmetry will restored [6]. In our work we use this procedure for obtaining of an effective chirallagrangian from lattice QCD. This methodological point of view this is a modification of the method proposed in [7] on the case of the Random Lattice approach.

2

From Lattice QeD to chiral lagrangians: step by step

Now let me briefly remand a general steps of the algorithm of the chiral lagrangian derivation from the lattice QCD that was proposed in [7]. The starting point of our analysis is a standard lattice action with Willson fermions

Z = j[DG][Di[J][D'¢] exp{ -Spl(G) - Sq(G, i[J, '¢) - SJ} where: 1) plaquette gauge field term is

BpI =

'7 2:

pl

Gx,J.t

[1 =

~c ReGx,J.tGx-J.t,vG~+v"Pt,v]

,

(1)

exp{ig flink dx~AJ.t(X')};

2) link fermions term is

Bq

=

I: Tr(AJ.t (x) GJ.t (x) + Gt(x)AJ.t(x)) x,J.t

AJ.t(x)g = i[Jb(X + /L)P:,¢a(x), AJ.t(x)g = i[Jb(x)P;:'¢a(x + /L)

P;-

and = ~(r ± IJ.t); 3) source term is

In order to realize the strong-coupling regime on the lattice let us consider the limit of the large coupling constant 9 (g -; 00). Main result is that in such limit integral over the gauge field can be performed. Let us consider the leading order contribution in this strong-coupling expansion. The integrals over the gauge degrees of freedom can be calculated into the large N limit by using of the standard procedure [5] and result of these calculations is following

Z = jlDi[J][D,¢]exP{-NI:Tr[F(),.(x,v))l- SJ} x,v

(2)

405 where Al/ = -M(x)P;; M(x

+ /I)P;;

and

~)]- Tr[log(l- ~~)]

F(A) = Tr[(l -

2

Now it would be very interesting to point out that the function F(A) has the typical form of the Born-Infeld action with first logarithmic correction. Our next step is the integration over the fermion degrees of freedom in (2). Using the source technics it was shown [7] that integral (2) can be re-written into the form of the integral over the unitary bosons matrix Mx

z=

J

(3)

DM eXPSeff(M).

As a matter of principle, we already perform the transformation from the color lattice degrees of freedom (G and 'IjJ) to the boson lattice degrees of freedom (M). Now our task is to realize the continuum limit of expression (3). The nest step of our analysis correspond with the studying of the stationary points of the lattice action Seff. Fortunately this is very well studied task [8]. This problem is connected with well-known investigations of the critical behavior of the chiral field on the lattice and with the problem of the phase transformation on the lattice (for references see the issue [9]). In [7], it was shown that for our task the stationary point is

Mo = uoi, uo(m q = 0, r = 1) = 1/4. Now one can expressed M(x) in terms of the pseudoscalar Goldstone bosons

1 + 15

M = uoexp(i7ri Ti'Y5/!7r) = uo[U(x)-2-

1 - 15 + U + (x)-2-]

and the effective action is given in the form of the Taylor expansion around this stationary point

(4) Let us consider the expansion of the chiral field U = exp(i7riTi/!7r) on the lattice around point x (by respect to the small step of the lattice a), U (x + /I) =

U(x) + a(ol/U(x)) +

a; (o~U(x)) + ....

And for components of the Taylor expansion (4) one obtain

Tr[(Al/(X) - AO)] Tr[(Al/(X) - Ao)2] Tr[(Al/(X) - AO)3]

- 2AoTr(a) 2 A'02Tr(",2) .... -2A~Tr(a3)

-

4,2Tr() AO a

(5)

406 Expressions (5) are very essential because these are a simplest illustration of all aspects of the violation of the rotational symmetry on the lattice. For this moment we specially say nothing about the structure of our lattice. The basic idea of the RL is the averaging over the big ensemble of various lattices with random distributions of sites and it possible to show that such averaging leads to the restoration of the rotational invariance. The basis of = = Sij / lij where Sij is a vectors in the CFL method is following volume of the corresponding 3-dimensional boundary surface of the Voronoi cells and lij = li'i - fj I is the length of link. Using the summation formulas from [6] one get that after the averaging only pairs are survive

1/

et

1/0

(6) a

pairings pairs

At other hand, the result (6) could be obtained by means of the following trick [10,11]. For beginning let us consider a lattice with fixed position (for simplicity it possible to use the trivial HC lattice) in a flat space. Now let us consider small deformations of the geometry of this space (Tij -; 9ij). Using of this idea one can rewrite the problem of the random lattice averaging in the terms of the random surface [10]. This is the standard quantum gravity task and using the methods of the Matrix Theory one can show that our result (6) is just the direct consequence of well-known Wick's Theorem about the pairings [12]. The expression (6) gives us possibility to calculate all terms in expansion (5). Let us consider just the first column in the expression (5). It is easy to show that either of these is proportional to some power of the leading order contribution tr [o/-,UoJ.L U+] tr[(oJ.L UoJ.LU+)2]

+ +

tr[(o/-,UoJ.LU+)n]

+

(7)

Substituting (7) into the (4) and collecting of all terms which depend on the power of the prototype lagrangian one obtain following expression for the effective chirallagrangian .ceff

rv

tr [IOg(l-

-tr [1- y'1- l/fJ2o/-,UoJ.LU+] -

~(1- y'1- 1/,820J.L UOJ.LU+))] + ...

(8)

where ... are all another terms (in particular the Skyrme term) and ,8 is a effective coupling constant that depend on the value of our stationary point Uo·

407 3

Conclusions

The aim of this paper is to derive the chiral effective lagrangian from QCD on the lattice at the strong coupling limit. We find that this theory looks like a Born-Infeld theory for the prototype chiral lagrangian. Such form of the effective lagrangian is expected. From the methodological point of view our consideration is very similar with the low-energy theorem in string theory that lead to the Born-Infeld action [1]. Moreover, in [13], it was shown that Chiral Born-Infeld Theory (without logarithmic corrections) has very interesting "bag" -like solution for chiral fields. It was additional motivation of our work. Acknowledgment

This work is partially supported by the Russian Federation President's Grant 195.2008.2. References

[1] A. A. Tseytlin, Nucl. Phys. B 501 (1997) 4l. [2] J. Ashman et al. [European Muon Collaboration], Phys. Lett. B 206, 364 (1988). [3] K. Kikkawa, T. Kotani, M. a. Sato and M. Kenmoku, Phys. Rev. D 19 (1979) lOll. [4] E. Brezin and D. J. Gross, Phys. Lett. B 97 (1980) 120. [5] H. Kluberg-Stern, A. Morel, O. Napoly and B. Petersson, Nucl. Phys. B 190 (1981) 504. [6] N. H. Christ, R. Friedberg and T. D. Lee, Nucl. Phys. B 202, 89 (1982). [7] S. Myint and C. Rebbi, Nucl. Phys. B 421, 241 (1994) (arXiv:heplat/9401009]. [8] D. J. Gross and E. Witten, Phys. Rev. D 21, 446 (1980). [9] P. Rossi, M. Campostrini and E. Vicari, Phys. Rept. 302, 143 (1998) [arXiv:hep-lat/9609003]. (10] F. David, "Simplicial quantum gravity and random lattices," arXiv:hepth/9303127. (11] L. Bogacz, Z. Burda, J. Jurkiewicz, A. Krzywicki, C. Petersen and B. Petersson, Acta Phys. Polon. B 32, 4121 (2001) [arXiv:hep-lat/0110063]. (12] P. Di Francesco, arXiv:math-ph/9911002. (13] O. V. Pavlovsky, arXiv:hep-ph/0312349.

STRING-LIKE ELECTROSTATIC INTERACTION FROM QED WITH INFINITE MAGNETIC FIELD A.E. Shabad a P.N. Lebedev Physics Institute, Leninsky prospect 53, Moscow 119991, Russia

V.V. Usov b Center for Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel Abstract. In the limit of infinite external magnetic field B the static field of an electric charge is squeezed into a string parallel to B. Near the charge the potential grows like JX3J (In JX3J + canst) with the coordinate X3 along the string. The energy of the string breaking is finite and very close to the effective photon mass.

It is known that the attraction force between two colored charges, when calculated using the Wilson loop method on a lattice, is concentrated in a string with its width being of the order of the lattice spacing l. The string potential consists of three additive components: (i) the Coulomb term 1/ R that dominates at small distance R from the charge, (ii) a constant term, corresponding to the infinite mass renormalization (recall that l is the ultraviolet cutoff parameter in the lattice theory), turns to infinity as l/l, when l is taken close to zero, (iii) the term linearly growing with R corresponding to a constant string tension and providing the confinement according to the Wilson area criterion (see, e.g., [1]). We show that in quantum electrodynamics with external magnetic field B the Coulomb potential of a point-like static charge, when corrected by the vacuum polarization, acquires a similar string-like form in the infinite-magnetic-field limit, the electron Larmour radius LB = (vIeB)-l = (l/mv1J) ----) 0 playing, in a way, the same rOle as the spacing l does in the lattice theory. Here b stands for the magnetic field, b = B / B o, measured in the units of Bo = m 2 / e = 4.4 x 10 13 G, e and m are electron charge and mass, resp. Electric potential AD of a point charge q when calculated as a sum of chains of electron-positron loops in a magnetic field B, so strong that the Larmour radius is much smaller than the electron Compton length, LB « m- 1 (this implies B » B o), is represented as a sum Ao(x)

= A s .r . (x) + Al. r . (x).

(1)

The analytic expressions for these functions are given in [2,3]. The argument x here is the radius-vector with its origin in the charge, its components across and along the magnetic field being x = (Xl., X3). The function As.r. (x) has the exact scaling property A(x)

_________________A_s_.r_(X_)_=~, ae-mail: [email protected] be-mail: [email protected]

408

(2)

409 where the dimensionless function A contains the magnetic field through its argument x = xLi31 only. The function As.r.(x) is short-range: for distances from the charge, large in the Larmour scale, IX31 » L B, or Ix ~ I » LB, it reduces to the Yukawa law

As r (x) ..

~

_q_exp{ - (~)! Jii 41fLB Jii + ~

+ xD = .!L exp{ - (2ab/1f)! mlxl} 41f Ixl '

(3)

where a = e2/41f = 1/137. This equation reflects the Debye screening of the charge by the polarized vacuum. The effective photon mass (inverse Debye radius) in (3) M = (2a/1f)1/2 Li3 1 tends to infinity together with the magnetic field. The» photon mass M » in the dimensionless function A(x) is finite, M = (2a/1f)1/2, and corresponds to the topological photon mass in the twodimensional massless electrodynamics of Schwinger [4]. Anisotropic corrections to (3) were pointed in [5]. The second term in (1) is long-range: it slowly decreases at the distances of the order of the Compton length m- 1 and larger, following the anisotropic Coulomb law

(4) It represents the whole potential Ao(x) (1) there, since A s.r. (x) is already negligible at such distances. This potential decreases with the growth of perpendicular distance x~ from the charge faster than along the field. The equipotential surface is an ellipsoid (x~)2 + x~ = const, which contracts to a string in the limit b = 00. The family of electric lines of force, parameterized by the angle 2 ¢, is for each value of the magnetic field given as X3 = (x ~ )f3- tan ¢ (see Fig. 1). All of them gather together inside a string passing through the charge and directed along the external magnetic field. The short-range part of the potential has the following asymptotic expansion near the point X3 = X~ = 0, where the charge is located,

As.r.(x)

~ 4~

C!I -

2mCs.r .

+ o(x~) + O(X}J) .

(5)

For large magnetic fields b » 21f/a rv 10 3 we have 2mCs.r . ~ 0.9595 Li3 1J2a/1f. It is infinite for LB = 0 or b = 00, as stated at the beginning. The corresponding constant in the expansion of A(x) is finite: 28 = 0.9595 J2a/1f. The constant 0.9595 is calculated using the experimental value of a. For a = 0 it turns into unity. The growth of Cs.r. with the magnetic field following the square root law provides the narrowing of the potential and, in the end, the finiteness of the ground state energy of a hydrogen-like atom in infinite magnetic field (see [2,3]

410

(b)

1~~ 1

Figure

1:

Lines

2

345

of force of electric field coming from (a) No magnetic field, B=O, (b) B = 104 Bo

and also the discussion [6]). In the limit b potential becomes the Dirac 5-function:

=

a

6

point

charge.

00 the short-range part of the

q

(6)

As.r.(O,x)lb=oo = 2.178 27r5(x).

The behavior of the long-range part Al. r. (X3, X~ = 0) at the string X~ = 0 is shown in Fig.2. (See [3J for analytical equations). The limiting form of this function at LB = 0 is finite. It decreases in agreement with Eq. (4) at large distances, and has the following behavior near the charge, IX31 « m -1,

Al.r.(X3,0)lb=oo ~ Al.r.(O,O)lb=oo

+ ~:

(1 - ;f(a)) 2mlx31 [In(2m1X31) -

~ In 2 + l' -

1] ,

(7)

where l' = 0.577 is the Euler constant and the coefficient f depends on the fine structure constant, f (a = 1/137.036) = 4.533. It is nonanalytic in a : f(a)la-+o ~ -Ina. We see that the growth is not linear. It provides "confinement" within the Compton distances, where the approximation (7) is valid. The first constant term in (7) is defined by an integral depending on the fine structure constant. If calculated with its experimental value, a = 1/137.036, it makes Al. r. (0, O)lb=oo = 1.4152(qm/27r). The numerical coefficient here is rather close to v'2 = 1.4142. Bearing in mind that Al. r.(00,0) Ib=oo = 0 we note that Al. r.(0,0) Ib=oo is the increment ofthe long-range part of the potential along the string between the point where the charge is located and the infinitely remote point, i.e. is equal to the work needed for removing a unit test charge to infinity. It may be referred to as the energy of the string breaking. It is remarkable that this quantity is finite. If we set q = e, we find that the energy density

411 -0.8

-0.9

SN

-1

ts

~-1.1

---

" --,./. i-1.2 I

-1. 3

,;

-'1

-1. 4 I

o

0.2

0.4

0.6

0.8

IX31 [(2m)-']

Figure 2: Energy -eAl.r. (X3, 0) of the electron in the long-range part of the potential of the point charge q = eZ for b = 106 , 10 5 , 3 X 104 , 104 (dashed lines from bottom to top). The dashed thick line corresponds to the limit b = 00 and represents the string potential.

of breaking of a string associated with the electron is with surprising accuracy equal to the dimensionless photon mass: Al.r.(O,O)lb=oo 1m = 1.0007 M. The coincidence would be exact for the value of the fine-structure constant equal to 1/121. By postulating this coincidence as a a physical principle we may obtain an approach for calculating a, to which end an approximation, better than one-loop approximation referred to here, would be needed. Formation of a string in QED discussed in the present paper is not completely unexpected, since it was noted [1 J that the Abelian theory does have a topologically nontrivial two-dimensional sector (described by a nonlinear (J"model).

Acknowledgments Supported by Program No. LSS-4401.2006.2, RFBR Project No. 05-02-17217, and by the Israel Science Foundation ofIASH. One of the authors (A.E.S.) expresses his gratitude to Professor D.B.Melrose for hospitality at the University of Sydney and valuable discussions.

References [1] [2J [3J [4J [5J [6J

Kei-Ichi Kondo, Phys.Rev. D 5S, OS5013 (199S). A.E. Shabad and V.V. Usov, Phys. Rev. Lett. 98, lS0403 (2007). A.E. Shabad and V.V. Usov, arXiv: 0707.3475 [astro-ph]. J. Schwinger, Phys. Rev. 128, 2425 (1962). N. Sadooghi and A. Sodeiri Jalili, Phys.Rev. D 76, 065013 (2007). S.-Y. Wang, Phys. Rev. Lett. 99, 22S901 (2007); A.E. Shabad and V.V. Usov, Phys.Rev.Lett. 99, 22S902 (2007).

QFT SYSTEMS WITH 2D SPATIAL DEFECTS LV. Fialkovskya, V.N. MarkoV', Yu.M. Pismalf High Energy Physics Department, St-Petersburg State University, Russia Abstract.We present the general treatment of delta-potentials in the framework of quantum field theory. The path integral calculation technique is sketched by the example of scalar fields. Results of generalization of this approach to realistic fields are given: mean electromagnetic field for fermion system and Casimir energy for photo dynamics in presence of cylindrical shell are presented.

1

Introduction

In 1948 a prediction was made of a macroscopical attractive force between two neutral perfectly conducting parallel planes, the Casimir effect [1]. It appeared that imposing boundary conditions (BC) on quantum field changes its ground state, and consequently its (infinite) vacuum energy. The difference between unperturbed vacuum energy (i.e. as in empty space) and one of constrained vacuum (subject to boundary conditions) is finite and causes measurable force. Nowadays it is confirmed experimentally with 0,5% of total error [2]. Since the pioneering Casimir work, there was a lot of research done on the subject, for a review see for instance [3]. However, up to now the main approach to description of matter with sharp boundaries used in the literature was based on fixing the values of quantum fields (and/or their derivatives) on the surface with help of BC. From the physical point of view, BC cannot be fully accepted as they constrain all of the modes of the field while in nature high-frequency modes propagate freely through any material boundary. The most obvious generalization of BC in modeling sharp boundaries is to introduce into the theory interaction of quantum fields with static (classical) background. The simplest background is of delta-function profile supported on the defect, describing a thin film present in the system. Such delta-potentials in the framework of local, lorenz and gauge (if applicable) invariant, renormalizable QFT were first investigated by Simanzik in 1981 [4], our works [5,6] follow his approach. Since then, there were quite a number of Casimir calculations with delta-potentials done. However, until very lately the issue of renormalizability of such theories still invoked contradictions [7-9]. Still all of the existing papers are concerned with simplified scalar models treated via zeta-function technique or heat kernel expansion, which by definition are applicable in one-loop calculations only. Until [5,6] there were no attempts to construct a self-consistent QED model with a deltapotential interaction satisfying all QFT principles and allowing one to describe self-consistently all possible observable consequences of the presence of a defect. a e-mail:

[email protected] [email protected] c [email protected]

412

413

In this paper we will first sketch the basic ideas for calculations with deltapotentials by example of a scalar field. Then we will present our results for fermion fields interacting with an infinite plane, and for the electromagnetic field coupled to infinite cylindrical shell.

2

Scalar models

Any QFT model is completely determined by its action. Following Simanzik we construct the action of a model in two parts S = So + Sdef where So is the (standard) volume contribution as for quantum fields in empty space, and Sdef is the action of the defect with a delta-function potential. Both parts must satisfy all basic principles of QED - locality, gauge and Lorenz invariance, renormalizability. For scalar case it reads (1)

where K = -82 + m 2 - is standard kinetic operator for massive scalar fielcP) , and>' is the coupling constant of the interaction with the defect. Its surface S is defined by equation 4>(x) = 0, x = (XO,Xl,X2,X3). All physical phenomena can be easily described if the generating functional of the Green functions is known

Z[J] = N

J

Dtp exp {-S[tp]

+ Jtp}, N

= Det 1 / 2 (K)

(2)

for normalization of Z we choose the condition Z[O]I>.=o = 1 For explicit integration of functional integral we represent the contribution of the defect action with help of auxiliary fields 'l/J(x), defined in the surface of the defect xES. Then it is possible to perform integration coming to

here S - modified propagator of the model, D = K-l is free scalar propagator in empty space. Surface operator Q = 1 + 2>'(ODO) determines the Casimir energy of the system, Ecas = 1/ (2T) Tr Ln Q. Geometrical properties of the system are encoded in operator 0 which is a projector onto the surface of the defect J dyD(x, y)O(y, z) == D(x, z). This technique is generalized to other fields in our works [5,6,10,11]. dWe operate in Euclidian version of the theory, where

a2 == L::=o al

414

3

Quantum Electrodynamics

For QED fields, the principles of action construction lead unambiguously to the defect action as [5,6] SdeJ

=

J

d4xJ(<J>(x))

(~Q'!j!+aEJLVPO"aJL<J>(x)AVaPAO")

(4)

with Q - a linear combination of all 16 Dirac matrices with constant coefficients, and a - dimensionless constant. The standard QED part is obviously

(5) where A = ')'JLAJL, FlLv = avAIL - aIL A" , e and m - are unit charge and electron mass respectively. We can see that interaction of the electromagnetic (EM) field with the defect unavoidably brings parity violation into the theory. Sticking to parity even fermion models, one however should expect that all effects of the presence of the boundary will be suppressed at the distances much larger then inverse electron mass m. To prove this estimation, we consider [6,10J pure fermionic model with a= 0, Q = A+')'JLqJL and the defect on an infinite plane <J> = X3. We calculate (directly observable) mean electromagnetic field in the leading order in e. Extending the technique of the previous paragraph to spinor fields it is possible to derive the augmented free fermion propagator of the theory with defect. Then the tadpole graph gives non-trivial contribution to the current of fluctuating fermion fields, which produce non-vanishing mean EM field. The resulting electric and magnetic fields are constant at large distances from the defect plane as it should be in classical electrodynamics for uniformly charged plane with constant currents. This gives the normalization condition establishing correspondence between parameters of the model (A, qlL) and classical charge and current density distributions. At the distances of the order of picometers '" lO-lOcm and less the mean EM field possesses quantum corrections proportional to l/x§. These corrections are exponentially suppressed at larger scales with factor e- m \X3\ similar to the behavior of radiative corrections to the Coulomb law. For full details of the calculations see [10]. Thus, massive fermion fields cannot indeed contribute to the Casimir force which has macroscopical (experimentally verified) values at the scale of 10100nm. This means that to establish theoretically the Casimir effect in QED we unavoidably must consider parity-violating Chern-Simon term for EM field in (4). For the case of planar geometry it was considered in [5]. The nonuniversality of the Casimir force between two infinite planes and its sign change (depending on the value of the coupling constant a) was predicted.

415

xI

For the defect on infinite circular cylindrical shell = + x~ - R2 we considered [11] pure photodynamical model (4) with Q = 0, regularized within PauliVilars approach. Constructing vector analog of operator Q(x) from (3), we can derive Ecas as an integral over the corresponding phase space of In det Q(P). Application of the Abel-Plana summation formula and its generalizations [12] let us finally present the energy as 1 E = 47fR2 f (a)

M

+ RM3A3 + RAl

where M -+ 00 is the regularization parameter, f(a) is a particular finite function given explicitly in [11], and A l ,2 determine the counter-terms. Divergencies are removed by renormalization of corresponding parameters of classical part of the energy. In the limit a -+ 00 the finite part of the energy reproduces the r.esults for perfectly conducting defect [13]. Acknowledgments

This work is supported in part by RFRB grant 07-01-00692 (V.N. Markov and Yu.M. Pismak). References

[1] H. B. G. Casimir, Pmc. K. Ned. Akad. Wet. 51, 793 (1948). [2] G. L. Klimchitskaya, R. S. Decca, E. Fischbach, D. E. Krause, D. Lopez and V. M. Mostepanenko, Int. J. Mod. Phys. A20, 2205 (2005). [3] G. L. Klimchitskaya, V. M. Mostepanenko, Contemp.Phys. 47 (2006) 131-144, arXiv:quant-ph/0609145vl; [4] K. Symanzik, Nucl. Phys. B 190, 1 (1981). [5] V. N. Markov, Yu. M. Pis'mak, arXiv:hep-th/0505218; V. N. Markov, Yu. M. Pis'mak, J. Phys. A39 (2006) 6525-6532, arXiv:hep-th/0606058. [6] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis'mak, Int. J. Mod. Phys. A21, No. 12, pp. 2601-2616 (2006), arXiv:hep-th/0311236. [7] N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, M. Scandurra and H. Weigel, Phys. Lett. B 572, 196 (2003), arXiv:hep-th/0207205. [8] K. A. Milton, J. Phys. A37 (2004) 6391-6406, arXiv:hep-th/0401090. [9] M. Bordag, D. V. Vassilevich, Phys. Rev. D70 (2004) 045003. [10] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis'mak, J. Phys. A: Math. Gen. 39 (2006) 6357 - 6363. (11] I. V. Fialkovsky, V. N. Markov, Yu. M. Pis'mak, arXiv:0710A049. [12] A. A. Saharian, arXiv:0708.1187. I. V. Fialkovsky, arXiv:0710.5539. [13] L. L. DeRaad, Jr. and K. Milton, Ann. Phys. (N.Y.) 136, 229 (1981); K.A. Milton, A.V. Nesterenko, V.V. Nesterenko, Phys.Rev. D59 (1999) 105009, arXiv:hep-th/9711168v3.

BOUND STATE PROBLEMS AND RADIATIVE EFFECTS IN EXTENDED ELECTRODYNAMICS WITH LORENTZ VIOLATION LE.FrolovO, O.G.Kharlanov b , V.Ch.Zhukovskyc Faculty of Physics, Moscow State University, 119992 Moscow, Russia Abstract.Using the extended electrodynamics introducing the Lorentz violation of the minimal CPT-odd type, we discuss the electron bound states in a central potential and in a homogeneous magnetic field (taking the electron anomalous magnetic moment into account), including the corresponding eigenstate problems and the radiation angular distributions, in particular, for the synchrotron radiation.

1

Introd uction

In the present investigation, we will focus on the extended electrodynamics with the Lorentz violation of the minimal CPT-odd form [lJ:

,5

_ir°,l,2,3

with = and the electron charge qe = -e. b~ is the constant axial vector coupling (condensate) that encapsulates the Lorentz violation. Its timelike component bo has at present one of the weakest constraints compared to other similar couplings, e.g. lbol.$ 1O-2eV while Ibl .$ 1O-1ge V. Working within this theory, we will investigate the electron bound states both in the Coulomb potential and in a constant homogeneous magnetic field, discussing the one-particle integral.;; of motion, eigenstates, and spectrum in the external field, and the spontaneous mdiation (spectml-)angular di..;;tributions. 2

Hydrogen-like bound state

Let c = n = 1,0: = e2 /41[, ~ = {b o, O}, and consider the eigenstate problem for A~(x) = {cp(r), O}, i.e. in a central potential. Make the unitary tran..'lformation: (2) (3)

where HD is now an (approximately) P-even operator. The function 'lj;nljmj[bo=O, ·th quant urn numb 1 3 .-. 1 . Wl ers · J = 2' 2'···' mj = --J, J, = J. ± 21 and panty a e-mail: frolov_ieGmail.ru be-mail: okharlGmail.ru ce-mail: zhukovskOphys.msu.ru

416

417

Hn;

hO,

p = (_1)1, L"I the eigenfunction of both Hnlbn=O and and thus of this gives the solutions for Hn in the b5-approximation (O(b~) are omitted): e -ibn2\.

.1. 'f'nljm;

Ib,,=O , e-

-

E nlj = En1jlbn=o

2i bo A p'.1.
b5 .

(

-

1• 1)1.'f'nCjm; ;

(4)

.

± m (J + 1/2) for I = J ± 1/2.

(5)

The degeneracy over 1 typical for the Coulomb field is thus removed in the presence of bo; the splitting is:S 105 Hz for hydrogen (Z = 1) and j = 1/2, and increases linearly with j. The bo = 0 eigenfunctions in thL"I field are known, so (4) gives us the explicit approximate expressions for them in the bo i=- 0 case. The bo i=- 0 eigenstates can also be found using the 1/e-power expansion up to the second order which gives the quasirelativistie Hamiltonian [2]:

,

P,'2) en en ( 11,) ~_'J +-uH-eAo+~ u[EP]+-2divE , (6)

pl2 ( 1-

h=_b_ 2m

4m

2mc

L-

4m e

with p~2 = (P + bt u)2 - 2b~, bt = bo/e, and n,e i=- 1. In the Coulomb field AI' = Lfn-~, O}, the Hamiltonian bo-linear terms, which are P-odd, can be removed with the parity-nonconserving unitary transformation U:

, "I

h=Uh

b,=O

' t +O(bt2 )+O(l/e); 3 U

U=exp

{ib-,;-

t (

1+ zr 2r

e)

O'·r } , (7)

1£

where re = an/me. Using U in the formula for the radiation angular distribution to transform the in/out-states )i)bO and {f)bO to )i)o and {flo, we obtain: 3

dWfi dO k

°

k \ e (T)* . (II ( -er, + ie(k = 27fn "2 . r')'r - kl[k'])I) I-L i E;-Ef

k=

where k and

lie e(T)

'

-

I-'

eli (1" " = --2+ 0' ) + I-'A' I-'A = me

°\2

(8)

ebo [ -) ---::? o-r,

(9)

m~

define the photon momentum and polarization

T

=

(T,7f. In search of leading-order bo-induced terms

in (8), consider a special case of atomic tran.."Iition: 12p1/2, mj = -1/2) -'lls1/ 2, mj = 1/2) (polarized states). After summation over T, the distribution [2] is shown in fig. 1 where we chose ~ = 0.05 (,y) for vividness and z is the angular moment quantization axis. The distribution asymmetry of the order ;S 10- 8 relative to the (xy) plane appears due to the parity-nonconserving radiation processes in- Figure 1: An example of volving jL~ and the interference of the correspond- radiation angular distribution ing radiation with the El radiation. The total rates (8); dotted line demonstrates are unaffected within the linear order in boo For un- the bo = 0 case

S

418

polarized atoms, the spherically-symmetric distribution i., restored; we did not consider the methods of preparing the polarized atomic states in our analysis. 3

Electron in a homogeneous magnetic field. Synchrotron radiation

Consider now an electron with anomalous magnetic moment 1IL = ::1r 1in a Tn constant homogeneous magnetic field H = He", AI' = {a, -:dHr]}; assume b = and H « He = ~2 ~ 4.41 X 1013 Gauss (the Schwinger critical field strength). The one-particle Hamiltonian then reads (in Heaviside units):

°

fIn

= fIn(bo,Pz, m,IL) = a.P+'--/m+JLH'--/E3 -bo'Y5,

P == {Pl , P2,Pz}. (10)

In the subspace with definite pz where '!f;( r) '" eiP%z, the unitary transformation cos 1) (Pziii.) = (-sin1)

sin 1)) cos1)

(m) pz'

(11) 1) = arctan

bo

pH' (12)

formally reduces the problem to that without Lorentz violation, solved in [3], except f~r the change (m,pz,IL) ..... (m,pz,p,). Taking

3z

fL.

and

== mE3 +

i,O'5[~.P]z as the integrals of motion, with the eigenvalues n - s - 1/2 and II == (v'm2 + 2eHn, with n, S = 0, 1,2, ... ,( = ±1, and rolling back to the initial r,epresentation, we see that the electron tmnsversal spin polarization operator ll-L transforms to a mixed (partially longitudinal) spin polarization opemtoT

The solutions for'!f; and the spectrum have the form (see details in [4]): e-~1'3ei(p%-fi%)Z.I' I"(r', o/€,pz,n,s,.,

'!f;e,p.,n,s,( r; bo, m, p) EE,p.,n,(bo,m,p)

=

EE,fi.,n,do, in,

° ,

mil) , ,

p,) = fJ(ll + J1H)2

+

(14)

rz,

(15)

where E = ±1 is the sign of the energy and ll'!f;E,p%,n,s,( = ll'!f;E,p.,n,s,(. Using these solutions, we apply the quasiclassical theory of synchrotron radiation quite analogous to the bo = 0 case, so we discuss it only briefly. The electron motion becomes quasiclassical for H « He and m/ E == .\ « 1. We neglect the ratio jiB / E (quite natural under typical laboratory conditions, E '" IGeV, H rv 104 Gauss, if bo » 1O- 20 eV) and assume the initial momentum pz = O. However, 1'J is not assumed to be small. In spherical coordinates with the polar angle (J measured from the z-axis, the total radiation power is

W=Wcl

! .

27 y2 dysm(}d(}641T2.\5(I+ey)4q>, k E

(,y 1 + (,y'

0

< y < +00;

8

2

Wcl=27(eme) , 3 H 1

(, = "2 He);'

(16) (17)

419 where y is a dimensionless variable defining the mdiation frequency, and ~ estimates the quantum efJect.'i. Some of the functions q>;«(J) [4] for polarization T and spin (' = ±(, are shown in fig.2. The distribution asymmetry relative to the orbit plane (} = '1£/2 caused by the 10ngitudinal admixture to its in/out spin polarization when bo i= 0, also maintains for unpolarized electrons. 4t~(O), (.

=

-1; a:::::: -1.2 - 10- 8

cl>i(6),

<; = -1;

a

~ -5.4 _ 10- 5

4>';(8), (.

=

-I: a:::::: 0.11

Figure 2: Normalized angular distributions <1>$=(9) for k = IMeV, E = IGeV, H = 104 Gauss , bo = 1O-g eV. a is the factor of asymmetry of the distribution relative to the () = ~ plane.

Experiment confirms the 'transversality' of electron states, therefore we can conclude that 1J « 1. Taken under laboratory conditions, this implies: IboI « ILH", 1O-6 eV. Moreover, if reliable data would be obtained for the observation of the radiation of the electron anomalous magnetic moment, demonstrating no signature of 1J i= 0, that would imply 1J;S jLH/E, and thus, Ibol ;S 1O-2o eV.

4

Conclusion

The investigation of the two systems discussed above showed that the Lorentzviolating coupling bo manifests itself in the modified electron spectrum and integrals of motion (parity or polarization), in the nonperturbative interaction with its anomalous magnetic moment, and in the asymmetries of the radiation distributions, especially for polarized electrons. The results obtained gave UIl stringent constraints on 60 and seem promising in suggesting new experiments.

Acknowledgements The authors are grateful to A.V.Borisov, D.Ebert, and A.E.Lobanov for helpful discussions.

References [1] D.Colladay and V.A.KosteleckY, Phys.Rev.D 58, 116002 (1998). [2] O.G.Kharlanov and V.Ch.Zhukovsky, J.Math.Phys. 48, 092302 (2007); arXiv:0705.3306v3(hep-th). [3] I.M.Ternov, V.G.Bagrov, and V.Ch.Zhukovsky, Vestnik Mosk. Univ., Fiz. Astron. 7, No.1, 30 (1966). [4J I.E.Frolov and V.Ch.Zhukovsky, J.PhYlI.A 40, 10625 (2007); arXiv:0705.0882(hep-th).

PARTICLES WITH LOW BINDING ENERGY IN A STRONG STATIONARY MAGNETIC FIELD E.V. Arbuzova" International University "Dubna", 141980 Dubna, Russia G. A. Kravtsova b Faculty of Physics, Moscow State University, 119991 Moscow, Russia V. N. Rodionov c Physics Department, Russian State Geological Prospecting University, 118816 Moscow, Russia Abstract. The equations for the bound one-active electron states based on analytic solutions of the Schrodinger and Pauli equations for the uniform magnetic field and a single attractive 5(r)-potential are discussed. We show that binding energy equations for spin-l/2 particles can be obtained without using the language of boundary conditions in the 5-potential model. We use the obtained equations to calculate the energy level displacements analytically and to demonstrate nonlinear dependence on the field intensity.

The effect of an external electromagnetic field on nonrelativistic charged particles systems (like atoms, ions and atomic nucleuses) has being investigated systematically for a long time (see, for example, [1-8]). Although this problem has a long history, a set of questions still requires additional study. As usual the exact solutions of Schrodinger equations with Hamiltonians taking a particle bound a short-range potential in the presence of external field into account are used. Furthermore, there is a rather common opinion that the role of magnetic field in decays of quasistationary states is invariably stabilizing [1, 7,8]. This view arises because the spinor states of electrons in an external electromagnetic field are usually neglected in nonrelativistic treatments, which is often inadequate [9]. We consider charged spin-O and spin-1/2 particles bounded by a short range potential (a o-potential) and located in an external stationary magnetic field of an arbitrary intensity. Energy level displacements can be seen for a particle in a o-potential and a magnetic field. The binding-energy equation is most appropriate for investigating such states [4,6]). Our main purpose is to derive equations for the binding energy of a fermion in the field contained an attractive singular potential and a stationary external magnetic field. We apply standard quantum mechanical methods using the expansion of the unknown wave function in a series at the eigenfunctions obtained for the fermionic system in the pure magnetic field. It is very important that our approach permits developing the consistent investigation of the spin effects arising in the external magnetic field. ae-mail: [email protected] be-mail: [email protected] ce-mail: [email protected]

420

421

Let us consider a charge in a uniform magnetic field B specified as

= (-yB, 0, 0).

(1)

The particle wave function in field (1) has the form 'l/J (t r) = e-iE"t/neixp,/fl+izpz/nu (Y)

(2)

B

= (0,0, B) =

\7 x A,

np,pz 'En = nw ( n

where

A

+ ~) + ~~

n

,

(3)

is the energy spectrum of the electron; m and e are the particle mass and charge, w = leBI/me, Px and pz are the momenta of the electron in the x- and z-directions. The functions Un(Y) are expressed in terms of the Hermite polynomials Hn(z), the integer n = 0, 1,2, ... indicates the Landau level number. We now study a simple solvable model. We consider the motion of a scalar particle in the three dimensional case in a single attractive 8(r) potential, where 8(r) is the Dirac 8-function, in the presence of a uniform magnetic field. In fact, we need to solve the following Schrodinger equation: -1 [ ( 2m

. fJ -~h-

fJx

eB) +y 2e

1

h 2 -fJ2 - h 2 -fJ2 -"Ah 2 o(r) wE,(r) fJy2 fJ z 2

'

= E wE,(r).

We can take solutions of Eq.( 4) in the form

WE' (r) =

L

CE'npxpz 'l/Jnpxpz (r) ==

n,px,pz

fJ

dpxdPzCE'npxpz 'l/Jnp,pJr),

(4)

(5)

n=Q

where 'l/Jnpxpz (r) is the spatial part of the wave functions (2). Coefficients CE'np,pz can be easily calculated, and after integrating over pz we get the equation 00 lIE' pz 2 (6) 1 = N7n!2mnw ~ (n + A)1/2' A = 2" - nw + 2mnw where N is the normalized coefficient independent of the field. This equation implicitly defines the energy of a bound localized electron state in the magnetic field and can be analytically reduced to a simpler form. As a result we have 00 x rr; ~ vIE e- ( a x ) (7) v E - V Eo = 2';; X 3/ 2 sinh(ax) - 1 dx, o where a = nw/2E, E = IE'I ;: : .0, Eo is the absolute value of the binding energy of the particle in the o-potential without the action of the external field. This result is consistent with analogous equation obtained by the wellknown method using boundary conditions of wave functions in the model of o-potential [4,7, 10J. Expanding the integrand function in Eq.(7) in the weak field limit nw « 2Eo , we obtain E =E ~ h 2 w2 h4w4) . o 48 E02 + 576 E 0 4 (8)

J

(1 _

_1_

Note that the square term in (8) coincides with the analogous result of [4].

422

The equation that explicitly determines the bound state energy in the strong field limit !U.u > 2Eo has the form:

E'

=

!U.u (0.205 - 0.452/fi - 0.367

~)

(9)

.

It is very important that we can use the present approach to study the spin effects in magnetic fields by the same way. The case of spin-l/2 particle can be calculated based on exact solutions of the Pauli equation. The Hamiltonian in the field (1) has the form

H = _1_

(-ih~ + eB y)2 _ ~

2

8 2m 8y2

_

~~ + JL0"3 B ,

(10)

2m 8x c 2m 8z 2 where JL = lelh/2mc is the Bohr magneton, 0"3 is the z-component of Pauli matrixes. The last term in (10) describes the interaction of the spin magnetic moment of the electron with the magnetic field. The electron wave function in field (1) has the form 1 ( 1+s ) '!/Jnpxpzs(t, r) = 2'l/Jn pxpz (t, r) 1 _ s ' (11) where 'l/Jnpxpz (t, r) is the solution (2) of the Schrodinger equation.

= !U.u ( n + ~) + ~~ + s!U.u~ (12) is the electron energy spectrum, s = ±1 is the conserved spin quantum number. Ens

Taking the interaction of the electron spin magnetic moment with the magnetic field into account, we obtain the equation

{E = 2[ - 12Eo -

VEo

V3J~48-E-o-4-+-s(-h-3w-3-E-o---4-8h-LW-E--:l ]

o

h2 w2 _ 48Eo2

.

(13)

where s = ±1 corresponds, respectively, to spin orientations along or against the magnetic field direction. Expansion (13) can be written in the weak field limit as

E !U.u 1 h2w2 Eo = 1 - s 2Eo - 48 Eg .

(14)

It is easily seen from Eq. (14) that the energy level Eb= -IEol existing in the o-potential without a perturbation is shifted under the magnetic field action by +!U.u/2Eo (upwards) in the case s = 1 and by -!U.u/2Eo (downwards) for s = -1. But the depth of an arrangement of energetic levels with respect to the continuous spectrum boundaries is the same in these two cases. We note that it has the same depth in the case of the spin-O particles. In the strong field limit we obtain for different spin values s = 0, + 1, -1

£'

~ hw (0.205 + ~ - 0.452{fi - 0.367;: )

.

(l5)

We emphasize that the dependence of energy level shifts on the particle spin does not disappear in the strong-field limit. The continuous spectrum

423 boundaries are shifted in the cases s = 0 and s = 1. However, the perturbative displacements of the binding-energy levels (as in the weak-field limit) are at the same distances from the continuous spectrum boundaries in all cases. We have shown that in the case of weak intensity a magnetic field indeed plays a stabilizing role in the consi:dered systems because the depth of the perturbative binding-energy levels from the continuous spectrum boundaries are shifted downward under the field action independently of the particle spin. But our results show a nonlinear dependence on the field intensity in the strong field limit. In superstrong magnetic fields, the binding-energy levels can approach the continuous spectrum boundaries. The system instability increases in strong magnetic field. This conclusion therefore disproves the opinion that a magnetic field always plays a stabilizing role in systems of bound particles. Acknowledgments This work was supported by the Russian Foundation for Basic Research (No. 05-02-16535-p) and by the Program for Leading Russian Scientific Schools (Grant No NSh-5332.2006.2). References

Ya. B. Zel'dovich, Zh. Exsp. Tear. Fiz. 39, (1961) 776. L. V. Keldysh, Zh. Exsp. Tear. Fiz. 34, (1958) 1138. W. Franz, Z. Naturf. A. 13, (1958) 484 Yu. N. Demkov and G. F. Drukarev, Zh. Exsp. Tear. Fiz. 47, (1964) 918; Yu. N. Demkov and G. F. Drukarev, Zh. Exsp. TeaT. Hz. 49, (1965) 257. [5] 1. V. Keldysh, Zh. Exsp. Tear. Fiz. 47. (1964) 1945; D. E. Aspnes, Phys. Rev. 147, (1966) 554; V. 1. Ritus, Zh. Exsp. Tear.. Fiz. 51, (1966) 1544; G. F. Drukarev and B. S. Monozon, Zh. Exsp. Tear. Fiz. 61, (197l) 956; A. 1. Nikishov, Zh. Exsp. TeaT'. Fiz. 62, (1972) 562; Ya. B. Zel'dovich, N. L. Manakov and L. P. Rapoport, Usp. Fiz. Nauk 117, (1975) 569. [6] A. 1. Baz, Ya. B. Zel'dovich and A. M. Perelomov 1971, ScatteT"ing, React'ions and Decays in Nonr-elativistic Quant'urn Mechan'ics 2nd edn (Moscow: Nauka) (in Russian) [7] V. S. Popov, B. M. Karnakov and V. D. Mur, Zh. Exsp. Tear. Fiz. 113, (1998) 1579. [8] N. 1. Manakov, M. V. Frolov, A. F .. Starace and 1. 1. Fabrikant, J. Phys. B: At. Mol. Opt. Phys. 33, (2000) R141. [9] V. N. Rodionov, G. A. Kravtzova and A. M. Mandel, Zh. Exsp. Tear. Fiz. Lett. 75,435 (2002); 78, 253 (2003). [10] V. N. Rodionov, G. A. Kravtzova and A. M. Mandel, Proceedings of the Third Intemational Sakharov Conference on Physics,(Moscow 2002), Moscow, Nauchnyi Mir, 785-793, 2003. [1] [2] [3] [4]

TRIANGLE ANOMALY AND RADIATIVELY INDUCED LORENTZ AND CPT VIOLATION IN ELECTRODYNAMICS A.E. Lobanov a, A.P. Venediktov b Faculty of Physics, Moscow State University, 119991 Moscow, Russia Abstract. We discuss the triangle anomaly and its application to the extended electrodynamics with Lorentz and CPT violation. We show that the requirement of the translational invariance allows to eliminate the ambiguity in the expression for the radiatively induced Chern-Simons term.

We start with recalling some facts about the axial anomaly [1,2]. It is enough for our purposes to consider only one-loop triangle graphs with one axial and two vector vertexes. There are two such graphs. Feynman rules lead to the following expression:

Here PI and P2 are the outgoing vector momenta, and the incoming axial vector momentum is PI + P2. This integral is linearly divergent and this is usually treated as the origin of an essential ambiguity in the final result. The standard way to eliminate this ambiguity and to obtain the unique value for the anomaly is using the conservation of the vector current or the axial current due to corresponding symmetries. It is well known that we cannot get the conservation of both currents, but once we choose which symmetry should be preserved, the ambiguity is eliminated. It was stressed by Jackiw in his work titled "When radiative corrections are finite but undetermined" [3] that the "correct" answer for the triangle graph is not intrinsic to it, but depends on the context. The situation is even more uncertain in the extended electrodynamics with an axial vector term. The corresponding fermionic lagrangian is:

where btL is a constant 4-vector. An important question about this theory is whether the Chern-Simons term ~CtLctLa,e'"( Fa,eA'"( could be induced through radiative corrections [3-6]. It is natural to do the calculations in the lowest order in bw Then one meets triangle graphs with the axial vertex contracted with vector btL carrying zero momentum. The coefficient of the induced ChernSimons terms is determined by btL TXAtL ( -p,p) p 2=O' Unlike the case when the axial vector carries nonvanishing momentum, the requirement of gauge invariance does not fix a value for the graphs in this theory. The nonperturbative

I

ue-mail: [email protected] be-mail: [email protected]

424

425

resolution of this problem proposed in [4] was proved mathematically incorrect [6] so the question is still open. In our approach we use the translational invariance to compute these graphs. It allows to eliminate the ambiguity and to get the unique result. We have to fix a value of the following integral:

This integral is undetermined so we need a special procedure to obtain a unique final result with necessary properties. These properties include analyticity in the outer vector momentum p and the full translational invariance. Now we make some formal transformation of the integral shifting the integration momentum k. Due to the Lorentz invariance the shift must have the following form: k/1 -+ k/1 + XP/1' where x is a constant independent of p/1. It leads to the expression:

SXA/1(-P,P) = d4k tr (l'x(k +(x + l)p + mh 5 1'/1(k +(x + l)p + mhA(k +xp + m)) (271")4 ((k + (x + 1)p)2 - mZ'y2 ((k + Xp)2 - m 2) .

f

The analyticity of the result in p let us expand the integrand AXA/1 (p, k) into a power series in p: (p k) = AXA/1 (0, k) AXA/1'

+

aAXA/1(p, k) a '" p

I . p'" + ... p=o

Evaluating the degree of divergence of the terms we see that only the first two of them lead to divergent integrals, so we need to keep only terms at most linear in p. The integral of the first one

should be set equal to zero due to the Lorentz invariance. The second term leads to the integral:

f

d4k aAXA /1(p,k) I (271")4 ap'" p=o

.f

d 4k xk 2C: XA /1",-2(2x 42 - (271")4

+ l)k",ki3C: xA/1i3 + 2k/1ki3 C:XAi3"'-(x + 2)m 2C:XA /1'" (k2 _ m2)3

.

426

We can replace kO/k f3 with k28~/4 and kl-'k f3 with k28~/4 in the integrand, so this expression is simplified to

J

d4k 8A xA I-'(p, k) (271")4 8 p O/

I - .x p=;;

4~ ( +

4 r d k ) XAI-'O/ }(271")

2c

4

(

2 1 _ k ) (k2 _ m2)2 (k 2 _ m2)3 .

However the result should be independent of x to satisfy the requirement of the translational invariance, so we should set the integral in the above expression equal to zero. This also sets the complete expression equal to zero and removes the induced Chern-Simons term. It is interesting to note that the formal calculation of this expression gives for the initial integral SXAI-' ( -p,p) I

p2=O

= - 8 12 (x

71"

+ 2)CXAI-'O/PO/ .

It leads to results obtained previously in various papers, provided an appropriate choice of x is made. In particular, the result of [4,7] is

To summarize, we showed that the requirement of the full translational invariance eliminates ambiguity in the expression for the radiatively induced Chern-Simons term in the extended electrodynamics with Lorentz and CPT violation. Acknowledgments

The authors are grateful to A.V. Borisov and V.Ch. Zhukovsky for helpful discussions. This work was supported in part by the grant of President of Russian Federation for leading scientific schools (Grant SS - 5332.2006.2). References

[1] J.S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969); S. Adler, Phys.Rev. 177,2426 (1969). [2] B.L. Ioffe, Int. J. Mod. Phys. A 21, 6249 (2006). [3] R. Jackiw, Int. J. Mod. Phys. B 14, 2011 (2000). [4] R. Jackiw, A.KosteleckY, Phys.Rev.Lett. 82, 3572 (1999). [5] S. Coleman and S. L. Glashow, Phys.Rev. D 59, 116008 (1999). [6] G. Bonneau, Nucl.Phys. B 593, 398 (2001); G. Bonneau, Nucl.Phys. B 764, 83 (2007). [7] M. Perez-Victoria, Phys.Rev.Lett. 83,2518 (1999).

THE COMPARATIVE ANALYSIS OF THE ANGULAR DISTRIBUTION OF SYNCHROTRON RADIATION FOR A SPINLESS PARTICLE IN CLASSIC AND QUANTUM THEORIES V. G. Bagrov, A. N. Burimova, A. A. Gusev Department of Physics, Tomsk State University, 634050, Tomsk, Russia. Tomsk Institute of High Current Electronics, SE RAS, 634034 Tomsk, Russia. Abstract. We present the comparative analysis of the angular distribution properties for linear components of the synchrotron radiation, calculated in terms of classic and quantum theory for spinless particle. It is shown, that the output computed relying on quantum theory is always lower than the classical one.

Introduction The synchrotron radiation (SR) is one of the most studied physical phenomenon. It is known for its wide application in different industrial fields and medicine. Most of its properties were firstly predicted theoretically and only then gained their experimental confirmation. In particular the angular distribution features of SR are completely studied [1] in classical theory. In quantum theory there are also quite simple representations of SR characteristics in the most practically interesting ultrarelativistic approximation. These representations particularly imply the following fact: in terms of the above approximation the output of SR calculated by quantum methods (the quantum output) is lower then the corresponding output calculated by classical electrodynamics (classical output) in any direction. The same situation takes place in the other extreme case of nonrelativistic particle. Therefore the suggestion that at any value of particle's energy and in any fixed direction the quantum output is lower then the classical one (at least for spinless particles) seems to be obvious. This suggestion is generally accepted but its proof has not been still presented. With the evident of the recent papers [2, 3] it became relevant to find out the region of physical parameters that ensures the correctness of the above suggestion. This papers show that for the angular distribution of SR separate spectral components this suggestion fails and at high particle's energy values this distribution always contains a set of angles where the quantum output exceeds the respective classical - one. It is also known (see [4] and references there in) that the 7f - component of the SR for a fermion (electron) doesn't vanish in the plane of particle's orbit, while in classical theory this component does not radiate and therefore in this direction the quantum output (for the electron) always exceeds the classical one; as for a spinless particle the quantum output is lower then the classical one. In the present paper we give the comparison of the total (summed over the spectrum) angular distribution for the classical and quantum SR outputs. In 427

428 quantum case we consider the radiation of the spinless particle (boson). The comparison is made for (J" - and 7f - SR components and for nonpolarized SR (summed over polarizations). It is shown that in the latter case the classical output exceeds the quantum one.

Starting theoretical expressions Consider a particle with the rest mass m and the charge e. We assume that the particle moves in a constant and uniform magnetic field with the strain H > 0 and the velocity along the magnetic field vector is zero. The energy E of the particle is determined in terms of /3 = v / c, where c is the speed of light in vacuo and v is the module of particle's velocity

In quantum theory apart from /3 the main quantum number n = 0,1,2,3, ... (the number of the initial energy level) should be shown. Let us introduce functions Pu (n, /3, B), Prr(n, /3, B), P(n, /3, B) positive everywhere in 0 ~ /3 < 1 and defined by the following expressions:

P ( /3 B) u n"

= dWr = Fu (n,/3,B) dWgl

P(n /3 B) "

iJ>u(/3,B)'

P ( /3 B) rr n"

= dW,ru = Frr (n,/3,B) dW,il

iJ>rr(/3,B)'

(2)

= dWr +dW,ru = Fu(n,/3,B) + Frr (n,/3,B) cos 2 B dWgl+dW,il

iJ>u(/3,B)+iJ>rr(/3,B)cos 2 B

Where dWqu, dWcl are the quantum and the classical angular distributions of SR respectively, B is the angle between the vector of the magnetic field and the direction of the radiation propagation. In equations (2) the following notations are used:

4+3q

4+q

iJ>u(/3, B) = 16(1 _ q)5/2' iJ>rr(/3, B) = 16(1 _ q)1/2 ' n

Fu,rr(n,/3,B) = LFu,rr(vjn,/3,B), v=l

8v 2 x[' 2

(x)

F ( /3 B) n n-v (/3) u V j n" = (2n + 1); (1 + p)2' Frr v; n, , B = 1- p

x

= v 1 + p'

P=

./

2vq

V1- 2n + l'

q

= /3 2 sin2 B,

4 2[2

v qp

()

X (in n-v + p)2

0~ q~

,

(3)

/3 2 < l.

Where [n,n-v(X), [~,n-v(x) are the Laguerre function and its derivative [1].

429

The remarkable thing is the following: at each fixed n the functions Pa ,7r (n, (J, 8) depend on one variable q only, where q = /3 2 sin 2 8:

Pa ,7r(n,{J,8)

= pJ':)(q).

(4)

The function P(n, /3, 8) dos not possess this property. At each fixed n these functions are strongly depend on both /3 and 8 and satisfy the inequality following from (2):

min{pJn)(q), pJn)(q)} ~ P(n,{J,8) ~ max{pJn) (q), pJn)(q)}.

(5)

The study of pJ~) (q) functions properties It is easy to see that in the nonrelativistic limit the (2, 3) from formulas (2, 3) imply

/3 « 1 (i. e.

q

< (J2

«

1)

(n)( 2n [ 37q ] (n) 2n [ 35 q ] Pa q):::::: 2n + 1 1 - 4(2n + 1) , P7r (q):::::: 2n + 1 1 - 4(2n + 1) , (6) P(n, /3, Therefore at

8) : : 2n2: 1 [1 -

q 4(2n + 1)

(35 + 1 + :OS2 8) ].

/3 « 1 we have p(n) (q) < P(n a

-

(.I

, p,

8) < p(n) (q) < ~ < l. 7r 2n + 1

(7)

The functions pJ~)(q), P(n,{J,8) decrease while q increases at fixed n. While n increases at fixed q these functions increase but stay lower than unity and at n -t 00 tend to unity. Replacing the summation on v by the corresponding integration in (3) and using well-known approximations [1] of Laguerre functions by McDonald's functions K 1/ 3(X), K 2/3(x) we could easily find the following expressions in the ultrarelativistic case:

2 3

Ho

m c = --. eli

It evidently follows from (5) that the pJ~) (q) are monotonically decreasing functions on J.L (therefore at fixed q monotonically increasing functions on n

430

and at fixed n monotonically decreasing functions on q) tending to zero at J.L -+ 00 follows from (8). At J.L < < 1 we have

p(n)( ) ~ 1 _ 320J.L p(n)() ~ 1- 256J.L • u q 2l7rv'3 ' ,.. q 157rv'3

(9)

It is obvious that inequality (7) changes to the opposite one:

1> pJn)(q) ~ P(n,(3,O) ~ pJn) (q).

(10)

It is obvious from (3) that the functions Fu,,..(n, (3, 0) are finite at any values of q (including q = 1). Hence, at q -+ 1 the following asympthotics always take place: pJn)(q) ~ Au(n)(l- q)5/2, pJn)(q) ~ A,..(n)(l- q)7/2. (11) Here Au,,..(n) are some numbers depending on n. This guaranties that at 1 q < < 1 the inequalities (10) hold and pJ~ (q), P(n, (3, 0) tend to zero at q -+ 1. The obtained results prove the validity of the following inequalities 0< min{pJn) (q), pJn)(q)} :S P(n,(3,O) :S max{pJn) (q), pJn)(q)} < 1. (12) In conclusion on the figures below we present the graphs of these functions for different n. 1.

1.

0.8

0.8

P<:(q) 0.6

0.6

0.4

0.4

0.2

0.2

0.6

0.8

q

1.

0.4

0.6

0.8

1.

Thus we prove that the SR output computed relying on quantum theory is always lower than the classical one.

431

Acknowledgments This work was partially supported by RFBR grant 06-02-16719 and Russia President grant SS-871.2008.2. References

[1] V. G. Bagrov, G. S. Bisnovatyi - Kogan, V. A. Bordovitsyn et all. Synhrotron Radiation Theory and Its Development. Editor V. A. Bordovitsyn. World Scientific. 1999. [2] M. V. Dolzhin, A. T. Yarovoi. Russian Physics Journal. 2005, v. 48, 8, p. 833-838. [3] M. V. Dolzhin, A. T. Yarovoi. Russian Physics Journal. 2005, v. 48, 12, p. 1270-1278. [4] V. G. Bagrov, M. V. Dolzhin. Nuclear Instruments & Methods in Physics Research. A. 2007, v. 575, No 1-2, p. 231-233.

«Using visual classical models is a usual method with the help of which one searches ways to obviate difficulties and which is used to find new representations in the field of microphysics» v. L. Ginzburg

PROBLEM OF THE SPIN LIGHT IDENTIFICATION

v. A. Bordovitsyn', V. V. Telushkin Tomsk State University, 36 Lenin Avenue, Tomsk 634050, Russia Abstract. Here the spin light identification problem when the spin radiation caused by intrinsic magnetic moment of an electron proceeds against the background of powerful synchrotron radiation, recoil effects, and other relativistic phenomena in detail considered. 1. Introduction After the rehabilitation of relativistic classical spin theory on the basis of Bargmann-Michel-Telegdi theory [1] and successful approbation of this equation in high-precision experiments of measuring of the anomalous magnetic moment of an electron [2], a question about the possibility of constructing a classical theory of intrinsic magnetic moment of an electron, which is adequate to a more rigorous quantum theory, have arose. This resulted in discovery of a new physical phenomenon [3-8] - spin light. The theory developed in these papers become useful in a variety of new field of applications of relativistic radiation theory (see, for example, [9-11 D. However, establishment of the correspondence principle of classical and quantum theory for the intrinsic magnetic moment radiation of a relativistic electron turned out to be much more complicated task than in the problem about spin precession on the basis of classical Bargmann-Michel-Telegdi theory. Though the problem of point-like magneton radiation is known long ago [12-16], no exact accordance of classical and quantum interpretation of spin light was existed for a long time. The point is that in this case it is essential to consider a lot of physical effects competing with each other and proportional to the Planck constant. Thereupon the accurate determination of spin light as effect caused by radiation of intrinsic magnetic moment of an electron (in short, a magneton) take on great importance. • e-mail: [email protected]

432

433

It is clear, that the classical theory can't pretend to construction of a general theory of spin light. However, we will show here, that the correspondence principle in classical and quantum theory of synchrotron radiation of intrinsic magnetic moment of an electron to a first approximation by Plank constant is fulfilled absolutely exactly. Here we will show also, that this principle by removal of side effects in the radiation of true magnetic moment is fulfilled also up to a second approximation by Plank's constant.

2. Semi-classical identification of the spin light for synchrotron radiation. It turned out that in semiclassical approximation even in the simplest case of electron radiation with spin directed along magnetic field, the total synchrotron radiation power is a very complex expression in which together with charge radiation and radiation recoil effects magnetic moment radiation itself disintegrates into a number of terms fundamentally different from a physical point of view:

w =wSR (Ie +fr +1e)J +fL-)J+IL-Th+fTh+fa) )J )J )J )J. tol

Here (1)

is a well known expression for the synchrotron radiation power and the terms in brackets correspond to the contribution of this or that physical effect in total radiation power:

fe =1 - charge radiation, j: + 175 f r = - 55.J3 24 '=' 9

j: 2

'='

recoil effects on radiation,

f. p

i>'[- >( q- 24!SF3 q')]

= 1+

- interference of the charge radiation and radiation of intrinsic magnetic moment,

f )JL =

[1+'" 1+1-"'(~+ 2

9

2

9

385.J3 432'

J];:2

- magnetic moment radiation due to the Larmor precession

fTh

)J

= (1 + ,,' 7 + 1- ,{ 1 2

9

2

9

);2

'='

434 - magnetic moment radiation due to the Thomas precession,

=[_ 1+ ~~' 1 + 1- ~~' (_! _35-13 J];:2

jL-Th

2

fl

3

2

3

216 r;

':>

- interference of the Larmor and Thomas radiation,

fa

= 1+r;r;' [ 2

I'

a(,

r; 3

245-13 72

,2J_!!",2] + 1-r;r;' (49+ 175-13 J!!..e 9 2 6 r; 9

- radiation due to anomalous magnetic moment of an electron:

g-2 l1a = 11- 110 = -2-110 where Po formulae

=en / 2m oc is the Bohr magneton. Besides, everywhere in these

q=3110Hr=~nr2 =~~r 2 m oc

2 mocp

(2)

2H

is a quantum parameter well-known in the synchrotron radiation theory, and factors

=

r;,r;' ±1 correspond to the spin quantum numbers, magnetic field.

H*

is the Schwinger critical

3. Spin light in the classical theory of synchrotron radiation Here we will show that the purely classical theory can explain completely the origin of the spin light. The radiation of electrical charge, possessing also an intrinsic magnetic moment in the classical theory, is described by the Lienard- Wiechert potentials and Hertz tensor polarization potentials [17]

Aa

eva, QafJ =_~nafJ. c RP v P RP v P

=_!

Corresponding tensor of electromagnetic field is calculated by the formula

=L.~_A[,IlnV]_..!.. d Q[,IlU n c d'i c 2 d'i 2 2

HUV

n V]

U'

Using then the standard technique of classical theory of radiation with equations of an electron motion and spin precession in the homogeneous magnetic field in the linear approximation by 11 one can find the spectral-angular distribution of radiation power in the form

435

2 2 2 3 n { cos e 2 12 f.loOJ COS e '} dO. = 47r y4 f32 sin 2 eJ n + J n + 4~ eoc f3 sin e nJ n J n WSR eL

dwn

=

'

(3)

=

Here OJ eoH / mocy is the cyclotron frequency coincident at g 2 with the frequency of spin precession. This formula is a generalization of the Schott formula for spectral-angular distribution of synchrotron radiation with respect to radiation of intrinsic magnetic moment of an electron. It can be shown that this formula reproduces exactly all properties of spin light concerned with radiation at Larmor precession and described by the semi-classical theory (see also Ref. [8]). Here we will show this on the example of calculating of total radiation power in the most actual ultrarelativistic case when

f.loOJ _ f.lo _ 1 liOJ _ 1 H _ 1 ~ eoc - eop - 2 moc 2 - 2y H* - 3y2 . Towards that purpose one should sum up the expression in formula (3) over the spectrum and integrate it over the angles. As a result we find the same formulas for total synchrotron radiation power and its polarization components as in the semiclassical theory but without recoil effects, Thomas precession and anomalous magnetic moment of an electron

W = (1 +~~~ )W eL

SR '

eL ( 7 1 .;:) eL ( 1 1 ';:) Wa = 8+6~~ WSR ,W1/" = 8+6~~ WSR '

W;L =

(~ + ~ ~~ ) W

SR '

W;L

(4)

i

= ( + ~ ~~ ) WSR .

This result does not depend on sequence of foregoing operations. Thus, this radiation is non-polarized as one could expect from the origin of the spin light. The formula for spectral-angular distribution is eL

dW 27 2( 2)2{K213 2 +--2 x2 2 - - = - - 2 Y l+x K1I3 dxdy 167r 1+ X 2 f.lo y2 yx } + 6--~ ( )112 K1I3K2/3 WSR ' eOp 1+x2 Integration over the spectrum in this expression gives the angular distribution of synchrotron radiation power with additional term for the spin light

436

{3 [7 ---;;;- = 32 (1 + eL

dW

X2 y12

+

(1

2 5X + X2

Y'2

]

35

+ 16 ~~ (1 + X2 t 2 WSR ' X2}

If the expression (15) is integrated over the angles, one can find the spectral composition of this radiation eL

f

dW 9..[3 {co - - = - y K 5/3(X)d.x dy 81T y

f

2 co } +-~~ K I/3(X)d.x W SR '

3

y

The terms for the spin light in the last two formulas are equal to doubled components for linear polarization of radiation. Naturally, further integration in the last formulas over the angular parameter x or over the spectrum leads us to the formulae (4). This result can be shown by another method. According to the general theory of relativistic radiation of point like magnetic moment, the part of energy corresponding to the mixed synchrotron radiation emitted per unit proper time is determined by expression ([5],see also Ref. [18], formula (6.17»

ap 2 dpaJ = 2 eof.1o (d n w _~va dwp nJ'O'w __ 1 naPw w w p ). ( d-r 3 c4 d-r 2 p c2 d-r CT c2 P p Here nap is the dimensionless classical tensor of spin, pa is the four-dimensional momentum of radiation. Its zero component gives the power of mixed radiation

W eL

=.:.. dpo r d-r

Substitution of the corresponding solution of equation of motion, and averaging over period of charge motion and over the spin precession gives

W

eL

r JwSR'

z =(l+.!.;=n 3'='

where

n z = r~ . As to recoil effects and Thomas precession they can be completely

described by classical methods but with use of quantum laws of conservation.

4. Conclusion

Thus, we have shown that the classical and quantum theory of spin light are in agreement with each other at the first approximation by Plank's constant. A question is arising: is the correspondence principle fulfilled in higher-order approximation with respect to the Plank constant? According to the method described earlier the answer to this question is fairly evident: all depends on the possibility of neglecting of the quantum effects and other factors like the Thomas precession.

437

An extraordinary example is radiation of a neutron in a homogeneous magnetic field which arises exclusively due to the spin flip in the quantum theory. Relativistic quantum theory of neutron radiation was developed by the group of Russian scientists (I. M. Ternov, V. G. Bagrov and A. M. Hapaev) [21]. The classical theory of neutron radiation emitted at the spin precession, which was developed by V. A. Bordovitsyn with coauthors [17-20], turned out to be in full accordance with the quantum theory but differs by a constant coefficient equal to 4, which, as it turned out, is connected with specific properties of quantum transition with spin-flip ([8]). However such radiation in the classical theory does not exist in the common interpretation. Therefore the correspondence principle in this case is inapplicable. With regard to the synchrotron radiation of an electron the correspondence principle applied to radiation of the intrinsic magnetic moment works very well in the limit case p ~ 00 and on assumption that the value of anomalous magnetic moment is large enough to neglect the Thomas precession. Note that in the mixed synchrotron radiation the terms which are proportional to Plank constant and contain the anomalous magnetic moment are in full accordance with the classical theory. Apparently, this is connected with the fact that the anomalous magnetic moment does not undergo Thomas precession (see [22]) . It is easy to show that the developed here classical theory gives the same terms 2

for radiation without spin-flip and proportional to h as are derived by the semiclassical theory for spin radiation caused by Larmor precession. Thus, we have in detail considered here the spin light identification problem when the spin radiation proceeds against the background of powerful synchrotron radiation, recoil effects, and other relativistic phenomena. In its pure form the spin light contributes to the synchrotron radiation power as a small correction 2

proportional to h • At the present time the problem of spin light radiation of the relativistic magnetic moment is particularly urgent in connection with the construction of ultrahigh energy accelerators. The procedure for experimental observation of spin dependence of synchrotron radiation power was proposed in Budker Institute of Nuclear Physics (Novosibisk), and this experiment itself was described in [23-25]]. In this experiment synchrotron radiation power proportional to h was for the first time observed to be dependent on the spin orientation of a free electron moving in a macroscopic magnetic field. Now it is possible to carry out more detailed investigation of spin light. Acknowledgments

We thank Prof. Yu. L. Pivovarov. for interesting discussion on these problems and Prof. V.Ya. Epp for his help in improving of the paper. This work was supported by RF President Grant no. SS 5103.2006.2, and by RFBR grant no. 06-02-16 719.

438 References

[1] V. Bargmann, L. Michel, V. L. Telegdi, Phys. Rev. Lett. 2 (1959) 435. [2] A A Schupp, R. V. Pidd, H. R. Crane, Phys. Rev. 121 (1961) 1. [3] V. A Bordovitsyn, I. M. Ternov, V. G. Bagrov, SOY. Phys. Usp. 165 (1995) 1083 (in Russian). [4] V. A Bordovistyn, V. S. Gushchina, I. M. Ternov, Nucl. Instr. Meth. A 359 (1995) 34. [5] VA Bordovitsyn, Izv. Vuz. Fiz. 40, N22 (1997) 40 (in Russian). [6) G. N. Kulipanov, A E. Bondar, V. A Bordovitsyn et aI., Nucl. Instr. Meth. A405 (1998)191. [7] I. M. Ternov, Introduction to Spin Physics of Relativistic Particles, MSU Press (1997) 240 (in Russian). [8] Synchrotron Radiation Theory and its Development. Ed.V.ABordovitsyn, World Scientific, Singapore, 1999. See also: Radiation Theory of Relativistic Particles, Fizmatlit, Moscow, 2002 (in Russian). [9] VA Bordovitsyn, V.Ya. Epp, Nucl. Instr. Meth. A 220 (1998) 405. V. A Bordovitsyn, [10] A Lobanov, A Studenikin, Phys. Lett. B 564 (2003) 27. [11] A E. Lobanov, Phys. Lett. B 619 (2005) 136. [12] G. J. Bhabha, G.C. Corben, Proc. Roy. Soc. 178 (1941) 273. [13] A Bialas, Acta Phys. Polon, 22 (1962) 349. [14] M. Koisrud, E. Leer, Phys. Norv. 17 (1967) 181. [15] J .Cohn, H.Wiebe, J.Math. Phys. 17 (1976) 1496. [16] J. D. Jackson, Rev. Mod. Phys. 48 (1976) 417I. [17] V. A Bordovitsyn et aI., Izv. Vuz, Fiz.21, N25 (1978) 12; N210 (1980) 33. [18] V. A Bordovitsyn, G. K. Razina, N. N. Byzov, Izv. Vuz, Fiz. 23, N210 (1980) 33. [19] V. A Bordovitsyn, R. Torres, Izv. Vuz., Fiz. 29 N25 (1986) 38. [20] V. A Bordovitsyn, V.S.Guschina, Izv. Vuz., Fiz. 37, N21 (1994) 53. [21] I. M Ternov, V.G.Bagrov, A M. Khapaev, Zh. Exp. Teor. Fiz.48 919650 921 (in Russian), SOY. Phys, JETP 21 (1965) 613. [22] VA Bordovitsyn, V.V.Telushkin, Izv. Vuz., Fiz. 49, N2 (2006). [23] V.N.Korchuganov, G.N.Kulipanov, M.N.Mezentsev, et aI., Preprint INP 7783, INP, Novosibirsk (1977) . [24] AE.Bondar, E.L.Saldin, Nucl. Instr. Meth.195 (1982) 577. [25] S.ABelomestnykh, AE.Bondar, M.N.Yegorychev, et al. Nukl. Instr. Meth., 227 (1984) 173.

SIMULATION THE NUCLEAR INTERACTION Timur F. Kamalov a Physics Department, Moscow State Open University, 107966 Moscow, Russia Abstmct. Refined are the known descriptions of particle behavior with the help of Lagrange function in non-inertial reference systems depends of coordinates and their multiple derivatives. This entails existing of circumstances when at closer distances gravitational effects can prove considerably stronger than in case of this situation being calculated with the help of Lagrange function in inertial reference systems depends of coordinates and their first derivatives. For example, this may be the case if the gravitational potential is described as a power series in sir where s is a constant correspondence for the nuclei scale.

1

Simulation in real reference frame

1.1

Particles in real reference frame

Classical physics usually considers the motion of bodies in inertial reference systems. This is a simplified and approximate description of the real pattern of the motion, as it is practically impossible to get an ideal inertial reference system. Actually in any particular reference system there always exist minor influences. Let us consider the precise description of the dynamics of the motion of bodies taking into account complex non-inertial nature of reference systems. For this end, let us consider a body in a non-inertial reference system, denoting the position of the body as r and time as t. Then, expanding into Taylor series the function r = ret), we get

_ r - ro

at 2

1.

1 ..

3

1 . (n)

4

+ vt + - 2 + ,at + ,at + ... + ,a 3. 4. n.

t

n

+ ...

(1)

Let us compare this expansion with the well-known kinematical equation for inertial reference systems of Newtonian physics relating the distance to the acceleration a,

rNewton

at 2

= ro + vt + T·

(2)

Denoting the hidden variables accounting for additional terms in non-inertial reference systems with respect to inertial ones as qr, we get 1 .

3

qr = 3! at

1 ..

4

1 . (n)

+ 4! at + ... + n! a

n

t

+ ...

(3)

Then

r

=

rNewton

ae-mail: [email protected]

439

+q

(4)

440

For inertial reference systems the Lagrangian L is the function of only the coordinates and their first derivatives, L = L(t, r, r) For non-inertial reference systems, the Lagrangian depends on the coordinates and their higher deriva. .....

·(n)

tives as well as of the first one, i.e. L = L(t, r, r, r, r, ... , r ) Applying the principle of least action, we get [1]

J ......

JL.) ~

n dn 8L (5) -1) dt n (--:(;0 ) Jrdt = O. n=O 8 r Then, the Euler - Lagrange function for complex non-inertial reference systems takes on the form JS = J

·(n)

L(r, r, r, r, ... , r )dt

=

(6) Or

(7) Denoting

p = p(2)

aL p

Or'

=

aT

= a~, p(3) =

=aL

a.(4)' r

p(5)

= fLk-

, a·(20<) r

ar

=

·(20<)

p(20:)

a.I:.

ar

·(4)

p(4)

aL

p(20:+1)

aL

a·(5) r

= ~. a·(20<+1) r

we get the description of inertial forces for complex non-inertial reference systems. The value of the total force taking into account the Coriolis force may be expressed through momentums in non-inertial reference systems and their derivatives:

P

+p

(2)

+p

(4)

+ ... + P

(20:) _ dp

-

dt

d 3p(3)

d5p(5)

+ dt3 + di5 + ... +

Let's accept the denote of the generalized derivation ·(n)

V' = E:=o(-l)n( .~n»)'

·(n)

ar

V',a = E:=o(-l)n( .~n»)·

ar i3

d 20:+1 p (20:+1) dt 20:+1 (8)

441

1.2

Scalar potential in the phase space of coordinates and their multiple derivatives

The Generalized Poisson's equation for the scalar potential cp(.n) of gravitational field in this case from the sources with density distribution of the source p and factor k depending on the system of units shall take on the form 2:N_ (_l)n(_£!£_-)'(n) = xp n-O a(';l or, in our denote, Generalized Poisson's equation is ·(n)

2:~=o \1 2 cp =

xp.

Than the solution of Generalized Poisson's equation is

(9)

Its solution is _

cp -

cp

.(O)(_S_)

r-ro

S2) S3) SN+l ) + cp '(1)( (r-ro)2 + cp '(2)( (r-ro)3 + ... + cp '(N)( (r-ro)Nfl·

The generalized metrics is a function of ·(n) _ 9ij

-

as

_s- ,

r-ro

·(n+1) (

S

)

(r-ro)'

9ij

This series diverges for .(n)( kn+l) 9ij (r-ro)n+l

>

.(n-1)( kn ) 9ij (r-ro)n ,

that is, at low distances. The series converges provided , that is ·(n) ( sn+l) 9ij (r-ro)n+!

.(n-1) (

< 9ij

sn

)

(r-ro)n ,

at higher distances we shall obtain the Riemann metrics. The same refers to the scalar potential cp'(n), which diverges in case cp

.(n) (

(r ro)n+l

sn+l)

>

cp

sn+l) (r-ro)n+l

<

cp

.(n-1) (

sn

)

sn

)

(r-ro)n

and converges provided cp

.(n) (

.(n-1) (

(r-ro)n .

Than the result formula for the gravitational potential is

cp

where mass.

cp-

= GM exp sir,

potential, Goo gravitational constant, s- constant, M

(10)

f pdv

-

442 1.3

Nuclear interactions and gravitational forces

From (10) follow, that the phase space of coordinates and there multiple derivative gives the corrected Newton's formula for gravitational potential c.p of mass m is

(ll) where ag,bg,cg ,... - constants. Here s is the unknown constant which have the seance of distance. For example, if s '" 1O- 15 m for the atom's size and more we have always Newtonian low. For the long distances r > > s, we have the equation for the gravitational potential c.p = Gm~a, where a = l. For the short distances r < < s, we have the equation for the gravitational potential (11). For particles described by the generalized Lagrange function at small distances, i.e. when the series diverges, there shall be much stronger gravitational forces acting than it is usually considered in calculations employing the Lagrange function. This model of short-range gravitational interaction allows one to compare nuclear and gravitational interactions at small distances. Acknowledgments

I thank Professor A. Studenikin and Doctor A. Grigoriev for providing me the possibility of fruitful discussions. Reference

[lJ M. V. Ostrogradskii, M'emoires de l'Academie Imp'eriale des Sciences de Saint-P'etersbourg v. 6, 385 (1850).

UNSTABLE LEPTONS AND (p, - e - 7)- UNIVERSALITY O.Kosmachev a Joint Institute for Nuclear Research, 141980, Moscow Region, Dubna, Russia Abstract. Main advantage and virtue of proposed method is a possibility to describe and enumerate all possible types of free equations for stable and unstable leptons in the frame work of homogeneous Lorentz group by means of unique approach.

1

Introduction

Free states are necessary for description of interactions. As it is known they play the role of initial and final states. Free states equations are unique way to introduce in theory quantum numbers identifying any leptons. Such quantum numbers characterize an equation structure. They will be called structural quantum numbers. The proposed method succeed from those fundamental requirements as Dirac equation [1]: invariance of the equations relative to homogeneous Lorentz group taking into account four connected components; formulation of the equations on the base of irreducible representations of the groups, determining every lepton equation; conservation of four-vector of probability current and positively defined fourth component of the current; spin value of the leptons is proposed equal to 1/2. One can show [2] that a totality of enumerated physical requirements are necessary and sufficient conditions (together with some group-theoretical requirements) for formulation of lepton wave equation out of Lagrange formalism. As it was shown formerly [2] Dirac equation is related with three different irreducible representations of homogeneous Lorentz group.It follows from the fact that Dirac ,-matrix group contains two subgroups d" b, and dual property of d'Y. In this case standard (proper, orthochronous) representation is realized on d, group, T-conjugate representation is realized on b, group, P-conjugate representation is realized on I, group, Corresponding algebras (six-dimensional Lie algebras of homogeneous Lorentz group) are characterized completely by their commutative relations (CR). They are of the form for d, group [ai, ak]

= Cikl2a!,

rbi, bk )

=

-cikI 2a !,

[ai, bk)

= cikl 2b l,

where Cikl is Levi-Cevita tensor, i, k, l = 1,2,3; ai, bi are infinitesimal operators of three-rotations and boosts respectively and al '" ,3,2, a2 '" ,I'Y3 a3 == a}a2 '" a2ala2"1 = all, bl '" ,}, b2 '" ,2, b3 '" Here following definitions are used [1)

,2,},

[i( 'J.tPJ.t)

+ me]'ll =

,3·

0,

'J.t'v

+ 'v'J.t

ue-mail: [email protected]

443

= 28J.tv,

p" V =

1,2,3,4.

444

Commutative relations on the base of b')':

[ar, a2l

=

2a3,

[b~, b~l = 2a3, [aI, b~l = 0, [aI, b~l = 2b

3

[a2' b3l = 2b~, [a3, b~l = 2b~, where;b~ '" -/'1'/'4,

[a3, all = 2a2, [a2,a3] = 2al, [b3, b~l = 2a2, [b~,b3] = 2al, [a3, b3l = 0, [a2' b2l = 0, [al,b 3l = -2b~, [a2' b~l = -2b3, [a3, b~l = -2b~,

b~ '" -/'2/'4,

(1)

b3 '" -/'3/'4· Commutative relations on

the base of f')'-group:

[aI, a2l = 2a3' [b~, b~l = - 2a 3' [aI, b~l = 0, [al,b 2l = 2b3,

3l =

[a~, b

-2b~,

[a 3,b~] = 2b~,

[a~,a3l = - 2a l, [b~,b3] = 2al, [a~, b~l = 0,

[al,b 3l = -2b 2, [a~, b~l = -2b3, [a3' b~] = 2b~.

[a3' all = 2a~, [b3, b~] = -2a~, [a3' b3l = 0,

(2)

The last connected component c" was obtained with following commutative relations:

[aI, a~l = 2a3'

[b"I' b"] 2 -- 2a '3 , [ar,b~l = 0, [aI, b~] = 2b~ [a2,b~] = -2b~,

[a'3' b"l I

--

2b"2'

[a2, a3l = -2ar, [b~, b~] = -2ar, [a~, b~l [aI, b~l

[a'2' b"l I [a3' b~l

[a3' ad = 2a~, [b"3' b"] I -- 2a'2'

[a 3, b~l = 0, = 0, = -2b~,

--

-

(3)

2b"3'

= 2b~.

The last three types of CR (1),(2),(3) ap to lately [2] were not represented in physical literature. Now we have the complete and closed set of constituents for description of lepton wave equations. 2

Equations for stable leptons

The base of every lepton equation is a corresponding /,-matrix group. Each of the /,-matrix group are produced by four generators. Three of them anticommute and ensure Lorentz invariance of different kinds. The fourth generator is a necessary condition for the formation of wave equation. The distinct nonidentical equations are became by virtue of different combinations of the four subgroups d,),' b,),' c')" fT Structural content of the groups for every type of equation has the form. 1. Dirac equation -

D')'[IIl: d,)" b,),' fl'.

445

2. Equation for doublet massive neutrino -

D'Y[I): d'Y' C'Y' IT

3. Equation for quartet massless neutrino -

D'Y[I I I): d'Y' b'Y' C'Y' IT

4. Equation for massless T-singlet -

D'Y[IV]: bT

5. Equation for massless P-singlet -

D'Y[V): cT

Every equation has its own structure allowing to distinguish one equation from other. All equations have not physical substructures, therefore leptons are stable. Obtained method allows to calculate full number of the stable leptons in the framework of starting suppositions. 3

Extensions of the stable lepton groups

Is it possible to obtain additional lepton equations on the bas of previous suppositions? This problem is attained by introducing additional (fifth) generator for new group production. As it turned out there are exist three and only three such possibilities. Each of them is equivalent to introduction of additional quantum characteristics (quantum numbers). The extension of Dirac ,-matrix group (D'Y(IJ)) by means of anticommuting generator r5 such that rg = I leads to 6.1-grouP with structural invariant [4],[3J equal to In[6.1J = -1. The extension of Dirac ,-matrix group by means of anticommuting generator r~ such that r~2 = -I leads to 6.3-grouP with structural invariant equal to In[6.3) = o. The extension of neutrino doublet group (D'Y(I)) by means of anticommuting generator r~ such that r~2 = -I leads to 6.2-groUP with structural invariant equal to In[6.21 = 1. al-grOUP has the following defining relations r JLr v

+ r vr JL =

28JLv,

(/-L, v

= 1,2,3,4,5)

(4)

One can show on the bas of (4) that 6. 1 contains 3 and only 3 subgroups of 32-order. As a result we have following content

6.t{D'Y(IJ),

D'Y(IIJ),

D'Y(IV)}

(5)

Relation (4) together with structural invariant In[6. 11 = -1 identify 6. 1 in physical sense. a 3 -group is obtained under extension of Dirac group by similar defining relations rsrt + rtr s = 28st , (s,t = 1,2,3,4), (6) (s = l,2,3,4),rg =-1. rsr5 + r5rs = 0, The group content was changed in this way

6.3{D'Y(IJ),

D'Y(I),

D'Y(IIJ)},

(7)

446

This corresponds to structural invariant In[~31 A 2 -group and it defining relations.

rS t + rtr s

rsr4 + r 4r s r ur5 + r5r u

=

2b s t,

= 0, = 0,

= O.

(s,t=1,2,3), (s = 1,2,3),r~ =-l. (u = 1,2,3,4),r~ =-1.

(8)

The group content differs from two previous cases ~2{D,(I),

D, (III) ,

D,(V)},

(9)

Structural invariant is equal to In[~21 = 1. 4

Conclusion

All examined equations have its own mathematical structure. These structures are not repeated, therefore they may be used for theoretical identification of the particles in free states. The first five equations including Dirac one have not physical substructures. Objects without structure can not disintegrate spontaneously , therefore all they are stable. The last three equations (AI, A 2 , A 3 ) have internal structures allowing of physical interpretation. If we suppose that the mass of the new particles is more than sum of masses of its constituents, they become candidates for unstable leptons. It is evidentally that equations on the base of Al and A3 may be interpreted as the equations for the massive charged leptons such as M± and T±. Their structural distinctions are the base for solving of the (M - e - T)universality problem by means of interaction descriptions. It is possible to relate A 2 -group with massive unstable neutrino. References [1] P.Dirac, Proc.Roy. Soc. A vol.117, 610 (1928). [2J O.Kosmachev, Representations of the Lorentz Group and Classification of Stable Leptons (Preprint JINR, P2-2006-6) Dubna, 2006. [3J A.Gusev, O.Kosmachev, Structural Quantum Numbers and Nonstable Leptons (preprint JINR, P4-2006-188) Dubna, 2006. [4J J.S. Lomont Applications of finite groups, (Academic Press, New York, London) 51, 1959.

GENERALIZED DIRAC EQUATION DESCRIBING THE QUARK STRUCTURE OF NUCLEONS A.Rabinowitch Abstract. We consider a generalization of the Dirac equation to describe the quark structure and anomalous magnetic moments of nucleons. The suggested generalization contains two 3 by 3 matrices consisting of the quark charges and describes a wave function of a nucleon having 12 components. It is shown that the magnetic moments of nucleons determined via the generalized Dirac equation accord with their experimental values.

As is known, the Dirac equation for the relativistic electron cannot be applied without substantial modifications to describe nucleons, since it does not give their anomalous magnetic moments. That is why a generalization of the Dirac equation was proposed in which an additional term describing non-minimal interaction of nucleons with electromagnetic fields was introduced [1,2]. However, this well-accepted generalization has two serious disadvantages. Namely, it does not describe the quark structure of nucleons and the experimental values of the anomalous magnetic moments of protons and neutrons cannot be deduced from it. Because of these reasons we seek another equation for nucleons which could be free of the two disadvantages. For this purpose let us consider the following generalization of the Dirac equation to describe nucleons:

(1) where F is the column consisting of three bispinors \{1k, k = 1,2,3, a and b are 3 x 3 matrices characterizing the quark structure of nucleons, 1 is the unit 3 x 3 matrix, are the Dirac matrices, n = 0,1,2,3, An are potentials of an external electromagnetic field, ep is the proton's charge, m is the rest mass of a nucleon when

"r

447

448

a+ = a, b+ = b, a 2 = 1, ab = ba,

(2)

where the sign "+" denotes the Hermitian conjugation. As can be readily verified, the following matrices a and b satisfy relations (2) and all the four requirements: for the proton they have the form

-1/3 2/3 2/3

2/3 ) 2/3 -1/3

(3)

and for the neutron they have the form

2/3 -1/3 2/3 ) 2/3 , a= -1/3 2/3 ( -1/3 2/3 2/3

b=

-1/3 2/3 -1/3 ) 2/3 -1/3 -1/3 . ( -1/3 -1/3 2/3

(4)

Consider the 4-vector .r of current densities of a nucleon. Analogously to the Dirac equation, from equation (1) we derive

(5) where F is the Dirac conjugate and the components JTt satisfy the differential equation of charge conservation ~r=Q

~

Let us multiply equation (1) by an arbitrary 3 x 3 matrix k satisfying the relations

k+ =k, ak=ka, bk=kb.

(7)

Then analogously to (6), from equation (1) we derive the charge conservation equation

(8) The current densities .r can be represented as the following sum of the quark current densities OIJa):

(9)

for the proton for the neutron

()1

=

02

= 2/3,

01

=

O2 = -1/3,

03 = -1/3; 03 = 2/3,

(10)

449 where ()le p are the quark charges and the quark current densities are defined as

Here e is some number and the matrices q, u, v have the form

q

1(101)

=2

0 1

1 1

1 0

u=

,

011) ( 110 1 0

1

,

V=

0 10) ( 001 1 0

0

.

(12)

As can be readily verified, the matrices k = u, k = U and k = v satisfy (7) and hence equation (8). This gives us the following three invariant integrals presenting three normalization conditions for the wave function F:

J

(13)

F+ qFcfx = 1,

where the integrals are taken over the three-dimensional space. The values of these three invariant integrals follow from the condition that the quark charges should be (2/3)e p , (2/3)e p and -(1/3)ep for protons and -(1/3)ep , -(1/3)e p , (2/3)e p for neutrons. Consider now the interaction of nucleons with a constant and homogeneous magnetic field. Then neglecting the nuclear potential <1>, from equation (1) we derive

(lilW/8t-~)FI

=

e(aIPI+a2P2+a3P3)F2,

(lilW/8t+~)F2

=

e(alPI +a2P2+ a 3P3)F I ,

(14)

where t = XO/e is time, ak are the Pauli matrices, Fland F2 are the columns consisting of the first, third, fifth 2-spinors and of the second, fourth and sixth 2-spinors of the column F, respectively, and the 3 x 3 matrices Pk have the form

(15)

Ao = A3

= 0,

Al

= (1/2)yH,

A2 = -(1/2)xH,

H

= const.

(16)

Here H is the magnitude of a magnetic field aligned with the axis z and x, y, z are rectangular spatial coordinates. Consider the nonrelativistic case. Then putting

450

Fl = exp (-iamCAt/h) GI,

F2 = exp (-iamCAt/h) G2,

(17)

from (14)-(16) and (2) we obtain the following equation: 2 2 2 2 P = PI +P2 +P3'

(18)

Consider the energy E of a nucleon. For it we have the well-known formula

(19) From (17)-(19) we derive the following formula for the intrinsic magnetic moment J.l of a nucleon:

(20) From (4) and (12) we obtain the expressions (ab)p, (ab}n of ab for the proton and neutron: (ab)p= 1 = 2q + v - u, (ab)n= -(1/3)(4q + 2v - 3u).

(21)

Therefore, using (13), from (20) we find the values J.lp and J.ln of J.l for the proton and neutron:

(22) Here mI, m2 are the values of m for the proton and neutron. For them we have 'mp/mi = 1 - c5 i , i = 1,2, where c5 i = l¥i/('mpc2)I and ¥i are mean values of the nuclear potential <1>. The values of c5 i are about several percent of l. As is well known, the experimental values of J.l are 2.79J.lo for the proton and -l.91J.lo for the neutron. Therefore, the obtained expressions (22) for the magnetic moments of nucleons accord with their experimental values. References

[1] A.Sokolov, I.Ternov, V.Zhukovsky, "Quantum Mechanics", (Nauka, Moscow), 1979.

[2J J .Bjorken, S.Drell, "Relativistic Quantum Mechanics", (McGraw Hill, New York), 1964. [3] N.Bogolubov, D.Shirkov, "Introduction to the Theory of Quantized Fields", (Nauka, Moscow), 1984. [4] L.Shiff, Phys. Rev. 84 1 (1951). [5J A.Rabinowitch, Int. J. Theor. Phys. 33 2049 (1994); 36 533 (1997).

UNIQUE GEOMETRIZATION OF MATERIAL AND ELECTROMAGNETIC WAVE FIELDS O.Olkhov a , N.N. Semenov Institute of Chemical Physics, Russian academy of sciences, Kosygin Str.,4, 119991 Moscow, Russia Abstract. For the first time the unique geometrical model of quantum particles and electromagnetic waves is suggested. This model explains all "irrational" properties of quantum objects: stochastic behavior, wave-corpuscular duality, EPR-paradox and light velocity invariance.

1 Main idea The Dirac equation for particle with spin 1/2 has the form [1]

i')'1-'81-''l/J=m'l/J, where 81-' X3

=

Y,

(1)

= 8/8xl-" J1 = 1,2,3,4, 'l/J(x) is the Dirac bispinor, Xl = t, X2 = X, = Z, ')'1-' are Dirac matrices, 'h = c = 1. For definite values of

X4

4-momentum PI-" the solution to Eq. (1) has the form

'l/J = u(pl-') exp( -iPl-'xl-') = u(pl-') exp 21Ti( -xl-' A~l),

AI-'

= 21Tp~l,

(2)

where u(pl-') is a normalized bispinor. There is a correspondence between tensors and geometrical objects: these tensors define invariant properties of above objects. For example, usual vectors correspond to simplest geometrical objects-to points [2], and this is the reason why Newtonian mechanics uses vectors within its formalism. Spinors correspond to nonorientable geometrical object (see, e.g., [3]). So, we suppose that spinors are used in Eq. (1), because this equation describes some nonorientable geometrical object and "spin = 1/2" is a formal expression of the nonorientable property of the object. To define properties of the proposed geometrical object more exactly we consider more precisely the symmetry properties of the solution (2). Function (2) is an invariant with respect to coordinates transformations XiI-'

=

xI-'

+ nl-'AI-"

nl-'

= 0, ±1, ±2, ...

(3)

Transformations (3) are elements of the group of translations in the 4-space where wave function (2) is defined. Function (2) can be considered as a vector realizing this group representation. Being the Dirac bispinor solution(2) realizes also the representation of group of reflections along any three of four axes in the space-time (this group is represented by four Dirac matrixes ')'1-' [4]). Above two groups form a group of four sliding symmetries (translations plus corresponding reflections [5]). The physical space-time does not have such symmetry. So, this group may operate only in some auxiliary space. From the other hand, it is known that discrete groups operating in some space can ae-mail: [email protected]

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452

reflect a symmetry of geometrical objects that have nothing in common with this space. It will be the case when such space is a universal covering space of some closed topological manifold [6]. Above properties of solution (2) leads us to the hypothesis: the Dirac Eq. (2) describes some closed nonorientable topological space-time 4-manifold and space-time plays also the role of an universal covering space for above manifold. At the present time, only two-dimensional euclidean closed manifolds are classified in details, and their fundamental groups and universal covering planes are identified [6]. So, we have no opportunity for rigorous consideration of specific properties of suggested psudoeuclidean 4-manifold. But qualitative properties, explaining main ideas of new interpretation, can be investigated using low-dimensional analogies, and we will show within elementary topology that the above 4-manifold represents propagation of the topological defect of threedimensional euclidean space and that propagation of this defect demonstrates specific properties of quantum particle. 2 Topological defect Let us consider a simple analogy-two-dimensional manifold homeomorphic to torus. Torus is a orientable manifold, but for the time being it does not matter for our consideration. In euclidean 3-space such torus is denoted as topological production of two circles-S 1 x S1. The closed topological manifold is representable by any of its possible deformations (without pasting) that conserve manifold's continuity. But now we restrict our consideration to one simplest configuration of above torus when one of S1 is a not deformed circle in the plane XY and another is a not deformed circle in the plane X Z (we denote the latter as SD. In pseudo euclidean plane circle looks like a hyperbola [2]. Therefore, our torus looks like hyperboloid in the three-dimensional pseudo euclidean space (after replacing Z = iT and replacing the circle Sf by a corresponding hyperbola). Positions of the geometrical object described by our pseudoeucliden torus are defined by time cross-sections of the hyperboloid. These positions form an expanding circle in two-dimensional euclidean plane XY. But we need to have in mind that two-dimensional pseudoeuclidean torus describes the object existing into two-dimensional space-time with one-dimensional euclidean "physical" space. This means that an observable part of the object is represented in our example by the points of intersections of above circle with OX axis though, as a whole, the circle is "embedded" into two-dimensional, "external" space XY. This circle can be considered as a topological defect of the physical onedimensional euclidean space OX. Just an affiliation of the intersection points to the topological object differs these points geometrically from neighboring points of the one-dimensional euclidean space. It can be shown that in case of nonorientable pseudo euclidean Klein bootle its time cross-sections perform periodical movement relative to the intersection point, so just this periodical movement

453

of the space topological defect attribute the phase (and wave properties) to propagating intersection points and just this movement are described by the wave function. So, in pseudo euclidean four-dimensional physical space-time the suggested object described by the Dirac equation looks like a topological defect of physical euclidean 3-space that is embedded into 5-dimensional euclidean space, and its intersection with physical space represents an observable quantum object. Measurable corpuscular properties of the physical object (4-momentum) are defined through the wave characteristic of the topological object by relation (4)

Notice that within suggested approach the notions of the less general, macroscopic theory (4-momentums) are defined by (4) through the notions of more general microscopic theory (wave parameters of the defect periodical movement). This looks more natural than the opposite definitions (4) within traditional interpretation. Suggested interpretation gives also a simple explanation for non local correlation between noninteracting particles in EPR-experiments. Noninteracting particles divided by macroscopic distance could be the intersection points of the same topological defect that could serve as an information channel. 3 Stochastic behavior Consider the simplest example of closed topological manifolds-one-dimensional manifold homeomorphic to a circle with given perimeter length A. The closed topological manifold is representable by any of its possible deformations (without pasting) that conserve manifold's continuity, and we will see that just this property explains appearance of probabilities in quantum formalism. To use concrete simple calculations, we consider only all possible circle deformations that have a shape of ellipse with perimeter length A. The equation for the ellipse on an euclidean plane has the form X2/a 2 + y2/b 2 = 1, (5) where all possible values of the semiaxes a and b are connected with the perimeter length A by the known approximate relation

A ~ 11'[1, 5(a + b) - (ab)1/2].

(6)

This means that the range of all possible values of a is defined by the inequality amin :::; a :::; amax ~ A/I, 511', amin « amax . In the pseudoeuclidean two-dimensional "space-time," the equation for our ellipses has the form (after substitution Y = iT) X2/a 2 - T2/b 2 = 1, (7) and this equation defines the dependence on time T for a position of the point X of the manifold corresponding to definite a. At T = 0, X = ±a; that is,

454 our manifold is represented by the two point sets in one-dimensional euclidean space, and the dimensions of these point sets are defined by all possible values of a. So, at T = 0, the manifold is represented by two regions amin S; IXI = as; amax , and it can easily be shown that at T =1= 0 these regions increase and move along the X-axis in opposite directions. All other deformations of our circle will be represented by points of the same region, and every such point can be considered as a possible position of the "quantum object" described by our manifold. All manifold's deformations are realized with equal probabilities (there are no reasons for another suggestion). Therefore, all possible positions of the point-like object into the region are realized with equal probabilities. So, this example shows the possibility of the consideration of above object as a point with probability description of its positions as it is suggested within standard representation of quantum particles. Geometrical interpretation of Maxwell equations is fulfilled in the same manner, and will be presented in details in subsequent publications. 4 Conclusion It is shown that quantum objects can be described as specific distortions of the space euclidean geometry (space topological defects). This explains all "irrational" properties of quantum formalism. Suggested approach can be also considered as an attempt to create a nonlocal model with hidden variables given by ensemble of various homeomorphisms of the topological space manifold that represents the same single particle. Acceptance of this idea means refusal of existing physical paradigma where all matter is considered as consisting of more and more small elementary particles. Notions of particles and waves does not exist into microworld represented as the curved space: they appears only as a result of classical interpretation of the contact of regions of curved space with macroscopical devices. Preliminary results see in [7]. References [1] J.D. Bjorken, S.D. Drell, "Relativistic Quantum Mechanics and Relativistic Quantum Fields", (McGraw-Hill, New York) 1964. [2] P.L. Rachevski, "Riemannian geometry and tensor analysis", (Nauka, oscow) 1966. [3] V.A. Jelnorovitch, "Theory of spinors and its applications", (AugustPrint, Moscow) 200l. [4] A.I. Achiezer, S.V. Peletminski, "Fields and Fundamental Interactions", (Naukova Dumka, Kiev) 1986. [5J H.S.M. Coxeter, "Introduction to Geometry", (John Wiley and Sons, N.Y.-London) 1961. [6J B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, "Modern geometry", (Nauka, Moscow) 1986. [7J O.A. Olkhov, Journ.of Phys.:Conj.Ser. 67, 012037 (2007).

Problems of Intelligentsia

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THE CONSCIENCE OF THE INTELLIGENTSIA John Kuhn Bleimaier a 15 Witherspoon Street, Princeton, New Jersey 08542 USA

Our conscience represents our sense of moral compass. It dictates our judgment as to what is right and what is wrong. It influences, often decisively, the choices which we make. While the law codifies morality in the macrocosm, the conscience enforces morality in the individual microcosm. Society may compel the observance of its regulations of conduct by means of the police power of the state. However, it is the conscience which makes social organization possible on the basis of individual internalized compunction. There are never enough gendarmes to insure compliance with the law. It is the presence of conscientious scruple which causes the citizen to conduct himself with rectitude even when nobody is looking. Which comes first, morality or conscience? This is not a chicken or egg intellectual exercise. Morality actually does precede conscience. Upbringing, socialization and education form the conscience of the individual. Thus the moral values of the social environment form the conscience of the members of society. Man is not born with a developed conscience. If morality comes before conscience we may be tempted to enquire, whence comes morality? Either morality is revealed by God or it is dictated by the founders of society. Emergence from the Hobbsian state of nature and the formation of interactive society requires the presence of a community of moral values. While the conscience is the product of morality, it is actually more. The conscience is an extrapolation of morality. Some activity may not be plainly recognized as immoral and yet engaging in that activity my engender compunctions of conscience. Thus the conscience unconsciously draws conclusions as to rightness and wrongness applying the sense or spirit of revealed morals. The conscience also helps us to weigh moral imperatives in relation to one another. The conscience allows for the selection of the lesser of two evils. Just as the individual person possesses a conscience, society too may be considered to have a conscience. Society has morally compelled laws, but it also has a conscience which helps it prioritize between competing moral values and legal obligations. Society is not a monolith and is composed of diverse individuals and communities. However, in order for the society to function there must be a sufficient commonality of values for the societal conscience to exist. The intelligentsia may be viewed as the repository of the conscience of society. The intelligentsia constitutes that group of men and women in society who, on the basis of intellect and education, think deeply about the meaning of things. The intelligentsia may not be an elite in the political science sense of that word. Their official, temporal power may be limited. However, the intelligentsia develops the theoretical underpinnings of society's institutions. For an intelligentsia to be "of' the society, it must share the society's basic moral values. As the intelligentsia works out the theoretical basis for the society's institutions it enshrines the conscience of the society. Indeed, the situation of the intelligentsia in relation to the conscience of society renders the intelligentsia a critical component of the social order even in the absence of officially aB.A.; Master of International Affairs; Juris Doctor; member of the New York, New Jersey and US Supreme Court bars; Fellow of Mathey College, Princeton University

457

458

enshrined power. The history of the world is replete with examples where societies have lived through enormous political upheavals and periods of alarming instability only to survive intact as a result of the continuity of its conscience and consciousness as vouched safe by its intelligentsia. Germany presents a classic example. In the course of a century the German people experienced fragmentation, unification, monarchy, republicanism, fascism, communism, victory, defeat and revolution. Throughout the conscience of the German nation remained inviolate in the custody of the intelligentsia. The German example illustrates that the intelligentsia is not always, or perhaps ever, unified. Certainly the dominant voices of the intelligentsia during the empire, the Weimar period and the Third Reich were very different. However, the fundamental assumptions underlying the social order remained constant allowing the society to survive. The experience of Germany is paralleled by that of Russia. The Russian people at the dawn of the 20th Century were the dominant national group in a vast multinational monarchy. During the communist era Russia ceased to exist in the context of a Marxist, Soviet monolith where Russian national feeling was associated with counterrevolutionary, bourgeois reaction. Finally, at the close of the century a Russian state reemerged, denuded of its imperial greatness and self conscious of its national identity. Nevertheless, the future of Russia is vouched safe by the continuity of its intelligentsia, the repository of its art, literature and culture in general. And what has made possible the enduring viability of embattled intelligentsias in Germany and Russia? It is precisely the conscience of the intelligentsia which has compelled it to persevere. The sense of obligation to the community at large, the sense of moral imperative, these are the well springs of the nation which are uniquely entrusted to the intelligentsia. Viewed in this light it is obvious that the intelligentsia is not only the conscience of society, the intelligentsia's own sense of conscience represents the intelligentsia's raison d' etre. What makes the intelligentsia a distinct group within the social structure is the fact that common intellectual analysis has lead to common moral consciousness. Of course this does not result in a monolithic group, sharing opinions and attitudes. Far from it. The intelligentsia is almost always in a state of intellectual foment and riven by factious bickering. However, in a healthy society, the controversy within the intelligentsia does not relate to fundamental principles but to the means appropriate to the attainment of agreed upon goals. Thus the factions within the intelligentsia do not ordinarily divide over the long term, generalized goals for society but rather over the means to be used for their attainment. As with individuals, so with groups, the conscience imposes burdens. It compels us to take action or to refrain from taking action. Thus the conscience of the intelligentsia engenders responsibilities for the intelligentsia in society. There is a responsibility to promote virtue and to eradicate vice, as variously as these terms my come to be defined. There may be a responsibility to ones neighbors to alleviate poverty, to preserve the natural environment, to conquer disease. These are the responsibilities of the intelligentsia. They are self imposed. Does the intelligentsia have any specific rights? In a democratic and egalitarian society no social group has a right to arrogate unto itself any entitlement to privilege. People whose intellectual inclination predisposes them to analysis

459

have no inherent claim to any preferment within the social unit. The intellectual toiler is not due any superior protection or preferment in relation to the manual laborer. The economic system may reward achievements as it will, but membership in the intelligentsia does not in and of itself confer a unique right. However, society as a whole has a right to have an intelligentsia. In the pantheon of evolving fundamental human rights, the right of a nation to cultivate its own intellectual elite, to develop its culture, to enshrine its conscience, is absolute. The notion of national self determination subsumes the concept of national intellectual advancement and confers the right on every society to possess an intelligentsia. That intelligentsia is the repository of its national conscience and, indeed, of its national consciousness. When we posit that every nation, every people, have the right to possess an intelligentsia, does that imply that every society already has an intelligentsia? In my opinion, it does not. Perhaps every social group contains within itself an intellectual elite composed of educated and thinking individuals. However, we have defined the intelligentsia as that cadre which is the repository of the society's conscience. Not every society possesses a societal conscience, A society may fail to possess a societal conscience if it is so heterogeneous as not to possess a commonly held morality. It may also not have a societal conscience if the predominantly held value system is not sufficiently nuanced as to rise to the level of a societal conscience. Such societies may include intellectuals and great thinkers but an internal group worthy of the title, intelligentsia, may be absent. When we realize that the existence of an intelligentsia and the existence of a societal conscience are intimately bound together, it behooves us to seek out the origin of the intelligentsia and societal conscience. The individual conscience is universal. Man in society is a moral creature and his morality gives rise to his individual conscience. It is this presence of conscience which distinguishes man from the savage beast. Man controls his instinct and lives by a code of conduct within his society. He conforms his behavior to that which is acceptable to society by means of his individual conscience. In the primitive social unit, society has laws and morals, the individual has a conscience. The emergence of societal conscience comes with the Christian era. When Christ abolished the law in regard to man's relationship with God, He placed a burden upon all individuals to conduct themselves with rectitude not based upon rules but motivated by faith. The un-attainability of perfection coupled with the categorical imperative of its pursuit engendered a new sense of conscience. A conscience which could be societal in a community of individuals espousing Christian values. Christian spirituality is uniquely soul searching and analytical. To the extent that a society can be Christian, Christian dominated or Christian influenced, it can and must have a societal conscience. The Christian society establishes priorities not motivated by utility but based upon conscience. The intelligentsia is the critical component within the society which analyses social alternatives and weighs them in relation to conscience. A religion divorced from the law makes critical analysis imperative in governing conduct. Conscience rather than regulation controls behavior. In the society at large the group we call the intelligentsia speaks for the societal conscience. Articulating the societal conscience is the function of the intelligentsia.

460 Only a society whose religious heritage requires a societal conscience has a role for an intelligentsia to play. All this does not imply that members of the intelligentsia are all Christians, or that only avowedly Christian societies have an intelligentsia. What we are talking about is societies which are rooted in a Christian tradition. Morality is derived from religion. But morality continues to exist for a long time, even in the absence of religion. Once again Russia provides a fine example. In the 19th Century Russia was an avowedly Christian country with an officially sanctioned Christian church and a government which claimed legitimacy on the basis of divine right. It was precisely in this context that the word and the concept of the intelligentsia was born. However, following the Bolshevik revolution and during the entire existence of the atheist Soviet state, the intelligentsia continued to exist. Many members of the intelligentsia were Marxists. Whether or not they would recognize it, those Marxists, atheists and anti-Christians had their conscience formed in the context of Christian morality. Notions of social justice have their genesis in the teachings of Christ. In the post Soviet era, when Russia continues to be governed by unbelievers, the intelligentsia is still the embodiment of a conscience formed in the crucible of Christian morality. Does this mean that Christianity represents nothing more than the fertilizer which prepares the soil of a nation for the growth of societal conscience and the development of an intelligentsia? I think not. Indeed, if a society espoused anti-Christian policy for a long enough period, its societal conscience might wither away and its intelligentsia might become extinct. However, the fact that this has not happened in Russia after nearly a century tends to reveal that the societal conscience which underlies the intelligentsia tends to push the pendulum back toward Christianity, notwithstanding transitory political, ideological and economic phenomena. Thus the Russian intelligentsia not only survived the Soviet state, it precipitated the demise of that entity. If we analyze the history of Christendom we realize that the early Christians did not constitute a society at all. They were a systematically oppressed minority within the Roman empire. However even at this historic stage the position of conscience was of inordinate importance to Christians. The dictates of individual conscience led believers to accept martyrdom. The absence of an organized Christian community meant the absence of a Christian intelligentsia. However, when Christianity became the official religion of the Roman empire during the time of Constantine a societal conscience was born. The newly Christian empire had to balance the pacifist, anarchist teachings of Christ with the practical requirements of governing a vast political domain. The Church came to accept a civil reliance on the strictures of a hybrid Roman/Mosaic law, while recognizing the preeminence of faith and grace in the spiritual sphere. At this stage we see the emergence of the first Christian intelligentsia. Thinking people had to balance Christian forgiveness with social regulation. The quest for the attainment of this balance created the societal conscience. The class of people bearing the burden of assessing this balance became the intelligentsia. Western jurisprudence has enshrined the quest for balance in the attainment of justice with the recognition of a distinction between law and equity. The legal system enforces the laws of society. However, when the strict enforcement of the law offends the conscience of the tribunal, the doctrine of equity may intervene so as to prevent an unjust result. The concept of equity in western

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jurisprudence is the embodiment of societal conscience. Next in the historic evolution ofthe intelligentsia the Protestant Reformation gave birth to the idea of freedom of conscience. The Reformation terminated the Catholic Church's monopoly in relation to matters of the spirit. It affirmed the original Christian sense of individual spiritual accountability. Freedom of conscience in relation to matters spiritual expanded the role of the intelligentsia from that of doctrinal elucidation to the maintenance of dialogue on matters of conscience. In the 18th Century when Thomas Jefferson promulgated the American Declaration of Independence he established the supremacy of conscience. Jefferson's ringing words posited for the first time that when a sovereign establishes regulations which contravenes the laws of God and nature, those regulations are a nullity. This is the ultimate extrapolation of the power of societal conscience. And, once again, it is the role of the intelligentsia to deliberate what regulation is consistent with societal conscience. In the 21st Century the intelligentsia continues to exercise its role as societal conscience in that part of the world which has come under the historic influence of Christian teachings. To undertake conscientious analysis is the duty of the intelligentsia. To possess an intelligentsia is a fundamental human right of all societies.

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Programme ofthe 13th Lomonosov Conference on Elementary Particle Physics and the 6 th International Meeting on Problems of Intelligentsia 23 August, THU Registration (Auditorium 1-31, Faculty of Physics, Moscow State University) 09.30 - 09.45 Opening (Conference Hall) AStudenikin, MSU V.Belokurov, Vice Rector of Moscow State University ASlavnov, Head of Department of Theoretical Physics, Faculty of Physics ofMSU V.Savrin, Vice Director ofInstitute of Nuclear Physics ofMSU V.Matveev, Director ofInstitute of Nuclear Research

08.00 - 09.30

09.45 - 13.50 MORNING SESSION (Conference Hall) Chairman: V.Belokurov 09.45 V.Matveev, ATavkhelidze (INR) Quantum number color, colored quarks, QCD (30') 10.15 O.W.Greenberg (Univ. of Maryland) The discovery of color, aparticipant viewpoint 10.45 ASlavnov (Steklov Math. Inst & MSU) Local gauge invariant infrared regularization for Yang-Mills field (30 min) 11.15 L.Okun (ITEP) The evolution of concepts of mass, energy and momentum from Newton and Lomonosov to Einstein and Feynman (30 min) 11.45-12.20 Tea break Chairman: ASlavnov 12.20 N.Krasnikov (INR) Search for new physics at LHC (30 min) 12.50 V.Chekelian (MPI) Review of the results of the electron-proton collider HERA (30 min) 13.20 F.Vannucci (Univ. of Paris-VII) Sterile neutrinos, from cosmology to the LHC (30 min) 13.50 - 15.30

Lunch

15.30-18.15 AFTERNOON SESSION (Conference Hall) Chairman: O.W.Greenberg 15.30 B.Foster (Univ. of Oxford) The status of the International Linear Collider (30 min) 16.00 l.P.Fernandez (CIEMAT, for CDF Coil.) Recent results from the Tevatron on CKM matrix elements from Bs oscillations and single top production and studies ofCP violation in Bs decays (30 min) 16.30-17.00 Tea break Chairman: L.CeInikier 17.00 G.Gutierrez (FNAL, for DO Coil.) Measurements ofthe top quark and W boson mass and Standard Model Higgs searches at the Tevatron (30 min) 17.50 G.Borisov (JINR, for DO Coil.) Discovery of cascade b baryon (25 min) 18.20 - 23.00 SPECIAL SESSION (40 0) Reception banquet will be held on board of a ship that will stream along the river across the central part of Moscow; the conference buses to the ship will depart from the entrance to the Faculty of Physics at 18.20.

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464 24 August, FRI 9.00 -14.00 MORNING SESSION (Conference Hall) Chainnan: F.Vannucci 9.00 V.Gavrin (INR) Solar neutrino experiments (30 min) 9.30 S.Mikheyev (INR) Oscillations o/natural neutrinos (30 min) 10.00 B.K.Fujikawa (LBNL, for KamLAND Coli.) An update on KamLAND (30 min) 10.30 B.Shaybonov (JINR) Baikal neutrino experiment: status and perspectives (30 min) 11.00 - 11.20 Tea break Chainnan: S.Mikheyev 11.20 Z.Djurcic (Columbia Univ.) Results o/MiniBooNE experiment (20 min) 11.40 H.Ray (LANL) Searches/or physics beyond the standard model with MiniBooNE (20') 12.00 P.Ochoa (California Inst. of Technology) MINOS results and prospects (25 min) 12.25 K.Sakashita (KEK, for T2K Coli.) Status o/the T2K experiment (25 min) 12.50 K.Graf(Univ. of Erlangen-Nuremberg) Acoustic particle detection with the ANTARES neutrino telescope (20 min) 13.10 T.Kajino (Univ. of Tokyo) Big-Bang cosmology, nucleosynthesis and neutrino oscillation (25 min) 13.35 V.Lukash (Lebedev Phys.lnst.) Dark matter: from initial conditions to structure formation in the Universe (25 min) 14.00 - 15.00

Lunch

15.00 - 19.00 AFTERNOON SESSION (Conference Hall) Chainnan: H.Gemmeke 15.00 H.Minakata (Tokyo Metropolitan Univ.) Kamioka-Korea two detector complex/or determining neutrino parameters (25 min) 15.25 ABarabash (ITEP) Double beta decay: present status (25 min) 15.50 C.Nones (Centre de Spectrometrie Nucleaire et de Spectrometrie de Masse) From Cuoricino to CUORE towards the inverted hierarchy region (20 min) 16.10 AAIi (DESY), ABorisov (MSU), O.zhuridov (Moscow lnst. of Electronics & Math) Electron angular correlation in neutrinoless double beta decay and new physics (15 min) 16.25 E.Andreotti (Univ. of Insubria) The MARE experiment: calorimetric approach/or the direct measurement o/the neutrino mass (20 min) 16.45 E.Mikheeva (Lebedev Phys.lnst.) Observational constraints/or inflationary models 17.00 - 17.30 Tea break Chainnan: AShabad 17.30 AStarostin (ITEP) Status o/the experiments on the neutrino magnetic moment measurement (30 min) 18.00 AStudenikin (MSU) Neutrinos and electrons in dense matter: method 0/ exact solutions 0/ quantum wave equations (15 min) 18.15 E.Arbuzova (Int. Univ. "Dubna"), E.Murchikova, ALobanov (MSU) Neutrino propagation in a dense magnetized matter (15 min) 18.30 AKuznetsov (Yaroslavl State Univ.) Plasma induced neutrino spin-jlip in a supernova and new bounds on the neutrino magnetic moment (15 min) 18.45 E.Gavryuseva (lnst. of Astrophys. and Space Research, Arcetri) Where the global topology o/solar magnetic field is originatedfrom and how does it interact to the variability o/neutrinoflux? (15 min) 19.00 - 22.00 Sight-seeing bus excursion in Moscow

465 25 August, SAT 9.00 -14.00 MORNING SESSION (Conference Hall) Chairman: G.Borisov 9.00 R.Partridge (Brown Univ.) Search/or new phenomena and non-SM Higgs at the Tevatron (30 min) 9.30 A.Pranko (FNAL, for CDF CoiL) Jet and photon production at the Tevatron (20 min) 9.50 V.Bunichev (SINP MSU, for DO CoiL) Top quark properties and EW measurements at the Tevatron (25 min) 10.15 B.Di Ruzza (Univ. of Trieste & INFN Trieste for the CDF Coil.) Tevatron results on B spectroscopy, lifetimes and rare decays (25 min) 10.40 J.Chauveau (Univ. Paris-VUVII) Measurement o/CP violation in B decays and CKM parameters (30 min) 11.10 - 11.30 Tea break Chairman: B.Esposito 11.30 C.Kourkoumelis (Univ. of Athenth, for ATLAS Coil.) Measuring o/Higgs boson at ATLAS (30 min) 12.00 G.Une1 (CERN & Univ. of Califomi a, for ATLAS CoiL) Physics Beyond the Standard Model in ATLAS (30 min) 12.30 M.Siebel (CERN, for ATLAS CoiL) QCD studies in ATLAS (20 min) 12.50 B.Spaan (Univ. of Dortmund) LHCb: physics, status and perspectives (30 min) 13.20 S.Barsuk (LAL, Orsay) CKM angle measurements at LHCb (20 min) 13.40 V.Egorychev (ITEP) Search/or new physics in rare decays at LHCb (20 min) 14.00 -15.00 Lunch 15.00 - 19.25 AFTERNOON SESSION (Conference Hall) Chairman: R.Partridge 15.00 T.Cuhadar Donszelmann (Univ. of British Columbia) Rare B Decays at BaBar (25 min) 15.25 M.Purohit (Univ. of South Carolina) DO mixing at BaBar (25 min) 15.50 S.Balev (JINR) Search/or direct CP-violation in charged K decays from NA48/2 experiment (20 min) 16.10 D.Madigozhin (JINR) Pion-pion scattering lengthes/rom NA48 data on Ke4 and threepion decays 0/ charged kaons (25 min) 16.35 N.Molokanova (JINR) Rare kaons and hyperons decays in NA48 (20 min) 16.55 - 17.15 Tea break Chairman: G.Gutierrez 17.15 F.Nozzoli (Univ. of Rome-II) Search/or rare processes at Gran Sasso (20 min) 17.35 B.Di Micco (Univ.Rome-IIl) Recent Kloe results (25 min) 18.00 Y.Potrebenikov (JINR) The!C -> 7r+ VV experiment at CERN (20 min) 18.20 E.Shabalin (ITEP) Final state interaction in K->27r decay (15 min) 18.35 K.Urbanowski (Univ. of Zielona Gora) Khaljin's Theorem and neutral meson subsystem (20 min) 18.55 AAIi (DESY), ABorisov , M.Sidorova (MSU) Bilinear R-parity Violation in Rare Meson Decays (15 min) 19.10 O.Kosmachev (JINR) Nonstable leptons and (p-e-r)-universality (15 min) 26 August, SUN 9.00-19.00

Bus excursion to Sergiev Posad

466 27 August, MON MORNING SESSION (Conference Hall) 9.00 - 13.50 Chairman: ADelia Selva 9.00 D.Gorbunov (INR) Status o/UHECR (30 min) 9.30 H.Gemmeke (lnst. for Data Processing and Electronics, Research Center Karlsruhe) The Auger experiment (30 min) 10.00 H.Gemmeke (lnst. for Data Processing and Electronics, Research Center Karlsruhe) Radio detection o/ultra high energy cosmic rays (30 min) 10.30 V.Flaminio (Univ. of Pis a) Neutrino telescopes in the deep sea (30 min) 11.00 C.Volpe (lPN CNRS) Beta-beams (30 min) 11.30 - 11.55 Tea break Chairman: V.Flaminio 11.55 G.Landsberg (Brown Univ.) Search/or extra dimensions and black holes at colliders 12.25 _D.Polyakov (Center for Adv. Math. Sci. & American Univ. of Beirut) New discrete states in two-dimensional supergravity quantum systems bound by gravity (20 min) 12.45 M.Fil'chenkov, S.Kopylov, Yu.Laptev (Peoples' Friendship Univ. of Russia) Quantum systems bound by gravity (20 min) 13.05 R.Nevzorov, S.Hesselbach, D.l.Miller, G.Moortgat-Pick, M.Trusov (Univ. of Glasgow) Lightest neutralino in the MNSSM (20 min) 13.25 C.Heusch (Univ. of California, St.Cruz) High-energy e-e-, gamma-e-, gamma-gamma interactions (25 min) 13.50 - 15.00 Lunch 15.00 - 19.15 AFTERNOON SESSION (Conference Hall) Chairman: C.Heusch 15.00 A.Isaev (lINR) Algebraic approach to analytical evaluation 0/ Feynman diagrams 15.20 KStepanyantz (MSU) Application o/higher covariant derivative regularization to calculation 0/quantum corrections in N= 1 supersymmetric theories (20 min) 15.40 O.Kharianov, LFrolov, V.Zhukovsky (MSU) Bound state problems and radiative effects in extended electrodynamics with Lorentz violation (20 min) 16.00 ALobanov, AVenediktov (MSU) Triangle anomaly and radiatively-induced Lorentz and CPT violation in electrodynamics (15 min) 16.15 - 16.55 Tea break Chairman: ABorisov 16.55 S.Vernov (SlNP MSU) Construction o/exact solutions in two-fields models (20 min) 17.15 KSveshnikov, M.Ulybyshev (MSU) Nonperturbative quantum relativistic effects in the confinement mechanism/or particles in a deep potential well (15 min) 17.30 AMykhaylov, Yu.Mykhaylov (MSU) Linearized gravity in a stabilized brane world model in five-dimensional Brans-Dicke theory (15 min) 17.45I.Fialkovsky, V.Markov, Yu.Pis'mak (St. Petersburg State Univ.) Parity violating thin shells in the framework o/QED (15 min) 18.00 T.Kamalov (Moscow State Open Univ.) Simulation the nuclear interaction (15 min) 18.15 O.Olkhov (Semenov Inst. ofChem. Phys.) Unique geometrization o/material and electromagnetic wave fields (IS min) 18.30 Yu.Rybakov (Peoples' Friendship Univ. of Russia) Open and closed cosmic chiral strings in general relativity (15 min) 18.45 M.Georgieva (Offshore Tech. Development Pte Ltd, Singapore) The size 0/ a parton

467 19.00 S.Gladkov (Moscow State Regional Univ.) On nonlinear dispersion of electromagnetic spectrum (IS min) 28 August, TUE 9.00 - 14.00 MORNING SESSION (Conference Hall) Chairman: B.Spaan 9.00 A.Kaidalov (ITEP) Some puzzles in B-decays (25 min) 9.25 V.Zakharov (ITEP) Nonperturbative physics at short distances (25 min) 9.50 M.Polikarpov (ITEP) Low dimensional manyfolds in lattice QCD (25 min) 10.15 Yu.Simonov (lTEP) Dynamics ofQCD at nonzero T and density (20 min) 10.35 G.Lykasov, AN.Sissakian, AS.Sorin, V.D.Toneev (JINR) Thermal effects in heavy-ion collisions (20 min) 11.55 ABadalian (ITEP) Decay constants ofheavy-light mesons (15 min) 11.10 - 11.30 Tea break Chairman: AKaidalov 11.30 N.Mankoc (Univ. of Ljubljana) Properties offour families ofquarks and leptons within the approach unifying spins and charges (25 min) 11.55 ANesterenko (JINR) Adler function within the analytic approach to QCD (20 min) 12.15 I.Narodetskii (ITEP) P wave baryons within the Field Corre/ator Method in QCD (20') 12.35 ASidorov (JINR) Polarized parton densities and higher twist corrections in the light of the recent CLAS and COMPASS data (20 min) 12.55 ANefediev (ITEP) Chiral symmetry breaking and the Lorentz nature of confinement 13.10 M.Osipenko (SINP MSU & INFN) Experimental moments of the structure function F2 ofproton and neutron (15 min) 13.25 V.Braguta (IHEP) Double charm onium production at B-factories and charmonium distribution amplitudes (20 min) 13.45 V.Bomyakov (IHEP) Lattice results on gluon and ghost propagators in Landau gauge 14.00 -15.00

Lunch

15.00 - 19.20 AFTERNOON SESSION (Conference Hall) Chairman: G.Diambrini Palazzi 15.00 AKataev (INR), V.Kim (INP, Gatchina) Higgs-+bb decay and different QCD corrections (20 min ) 15.20 I.Bogolubsky, E.-M.Ilgenfritz, M.Muelier-Preussker, AStembeck (JINR) Gluon and Ghost propagators in SU(3) gluodynamics on large lattices (15 min) 15.35 M.Tokarev (JINR) QCD test ofz-scalingfor piO-mesonproduction (15 min) 15.50 ASafronov (SINP MSU) Analytic approach to constructing effective theory ofstrong interactions and its application to pion-nucleon scattering (IS min) 16.05 D.Ebert, ATyukov, V.Zhukovsky (MSU) Phase transitions in dense quark matter in a constant curvature gravitational field (15 min) 16.20 K.Zhukovskii (MSU) Quark mixing in the standard model and the space rotations (15 min) 16.35 O.Pavlovsky (MSU) Effective Lagrangians andfield theory on a lattice (15 min) 16.50-17.10

Tea break

Chairman: V.Zhukovsky

468 17.10 AShabad (Lebedev Physics Inst.) String-like electrostatic interaction in QED with infinite magnetic field (20 min) 17.30 V.Skvortsov (MIPT), N.Vogel (Univ. of Tech. Chemnitz) Nuclear reactions and accompanying physical phenomena in plasma oflaser-induced discharges (15 min) 17.45 E.Arbuzova (Intern. Univ. "Dubna"), G.Kravtsova (MSU), V.Rodionov (Russian State Geological Prospecting Univ.) Particles with low binding energy in the strong stationary magnetic field (15 min) 18.00 V.Bagrov (Tomsk State Univ.) New results of synchrotron radiation theory (20 min) 18.20 V.Telushkin, V.Bordovitsyn (Tomsk State Univ.) Coherent Spin Light (15 min) 18.35 V.Sharikhin (Moscow Power Engineering Inst.) Microdrops condensation ofsolar photons in strong magnetic field (15 min) 18.50 ARabinowitch (Moscow State Univ. ofInstrument Construction and Informatics) A new generalization ofDirac's equationfor nucleons (15 min) 19.05 V.Belov, E.Smimova (Moscow Inst. of Electronics and Math.) Semiclassical soliton type solution of the nonlocal Gross-Pitaevsky equation (15 min)

29 August, WED MORNING SESSION (Conference Hall) Round Table Discussion on "Dark Matter and Dark Energy: a Clue to Foundations of Nature" Chairman and convener: AStarobinsky 9.00 AGalper (MEPhI), P.Picozza (Univ. of Rome-II) Antimatter and dark matter research in space (30 min) 9.30 AStarobinsky (Landau Inst.) Dark energy: present observational status, scalar-tensor andf(R) models (30 min) 10.00 R.Bemabei (Univ. of Rome-II) Investigating the dark halo (30 min) 10.30 AMalinin (Univ. of Maryland) Dark matter searches with AMS-02 (30 min) 11.00 -11.30 Tea break 11.30 V.Dokuchaev (INR) Anisotropy of dark matter annihilation in the Galaxy (20 min) 11.50 V.Berezinsky (LNGS), Yu.Eroshenko (INR) Remnants of dark matter clumps in the Galaxy (20 min) 12.10 - 13.00 Discussion and conclusion 13.00 -14.30 Lunch

9.00-13.00

SEVENTH INTERNATIONAL MEETING ON PROBLEMS OF INTELLIGENTSIA: "Rights and Responsibility of the Intelligentsia" 14.30 -14.40 Opening (Conference Hall) Chairman: AStudenikin 14.40 V.Trukhin (Dean of the Faculty of Physics, Moscow State University) 15.00 S.Filatov (Found. for Social, Economical and Intellectual Programs) Rights and Responsibility of the Intelligentsia (30 min) 15.30 I.Bleimaier (Princeton) The Conscience of the Intelligentsia (30 min) 16.00 Discussion and conclusion Closing ofthe 13th Lomonosov Conference on Elementary Particle Physics and the 7th International Meeting on Problems of Intelligentsia SPECIAL SESSION (40 0)

List of participants of the 13th Lomonosov Conference on Elementary Particle Physics and the 7th International Meeting on Problems of Intelligentsia Andreotti Erica Arbuzova Elena Astafurov Vladimir Badalian Alia Bagrov Vladislav Balev Spasimir Barabash Alexander Barsuk Sergey Belokurov Vladimir Bernabei Rita Bleimaier John Bogolubsky Igor

Univ. of Insubria Int.Univ. ofDubna Group REI ITEP Univ.ofTomsk JINR ITEP LAL,Orsay MSU DAMA Princeton JINR

Borisov Anatoly Borisov Gennady Bornyakov Vitaly Braguta Victor Bunichev Viacheslav Chauveau Jacques Celnikier Ludwik Chekelian Vladimir Cherepashchuk Anatoly Cuhadar Donszelmann Tulay Curatolo Maria Della Selva Angelo Djurcic Zelimir Di Micco Biagio Di Ruzza Benedetto Diambrini Palazzi Giordano Dokuchaev Vladislav Egorychev Victor Eroshenko Yury Esposito Bellisario Fernandez Juan Pablo Fialkovsky Ignat Filatov Sergey

MSU Lancaster Univ. IHEP IHEP SINP Univ. Paris-VINII Observatoire de Meudon MPI SAl Univ. of British Columbia

Fil'chenkov Michael Flaminio Vincenzo Foster Brian Fujikawa Brian

INFN Frascati Univ. ofNeaples Columbia Univ. Univ .Rome-III Univ. of Trieste & INFN Trieste Univ.ofRome-I INR ITEP INR INFN Frascati CIEMAT St. Petersburg State Univ. Found. for Social, Economical and Intellectual Programs Peoples' Friendship Univ. of Russia Univ.ofPisa Univ.ofOxford LBNL

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[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected], [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] giordano.diarnbrini@ romal.infn.it [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

[email protected] [email protected] [email protected] [email protected]

470 Galper Arkady Gavrin Vladimir Gavryuseva Elena Gemmeke Hartmut

Gladkov Serguey Gorbunov Dmitry GrafKay Grats Yuri Greenberg Oscar Grigoriev Alexander Gutierrez Gaston Heusch Clemens Isaev Alexey Kadyshevsky Vladimir Kaidalov Alexei Kajino Toshitaka Kamalov Timur Kataev Andrei Kharlanov Oleg Kosmachev Oleg Kourkoumelis Christine Krasnikov Nikolay Kuznetsov Alexander Landsberg Greg Lobanov Andrei Lukash Vladimir Lykasov Gennady Madigozhin Dmitry Malinin Alexander Mankoc Borstnik Nonna Matveev Victor Mikhailin Vitaly Mikhailov Yuri Mikheyev Stanislav Mikheeva Elena Minakata Hisakazu Molokanova Natalia Murchikova Elena Narodetskii Ilya Nefediev Alexey Nesterenko Alexander Nevzorov Roman Nikishov Anatoly

MEPhl INR Inst. of Astrophys. and Space Research, Arcetri Inst. for Data Processing and Electronics, Research Center Karlsruhe Moscow State Regional Univ. INR Univ.ofErlangen-Nuremberg MSU Univ.ofMaryland MSU FNAL Univ. of California, St.Cruz JINR JINR ITEP Univ.ofTokyo Moscow State Open Univ. INR MSU JINR Univ. of Athenth INR Yaroslavl State Univ. Brown Univ. MSU LPI JINR JINR Univ.ofMaryland Univ.ofLjubljana INR MSU SINP INR LPI Tokyo Metropolitan Univ. JINR MSU ITEP ITEP JINR Univ.ofSouthampton LPI

amgalper@mephLru [email protected] [email protected] [email protected]

[email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] mikheyev@pcbail O.inr .ruhep.ru [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] nikishov@lpLru

471 Nones Claudia

Nozzoli Francesco Ochoa-Ricoux Juan Okun Lev Olkhov Oleg Osipenko Mikhail Panasyuk Mikhail Partridge Richard Pavlovsky Oleg Polikarpov Mikhail Polyakov Dimitri Potrebenikov Yury Pranko Alexander Purohit Milind Rabinowitch Alexander Ray Heather Rodionov Vasily Rybakov Yuri Safronov Arkady Sakashita Ken Savrin Vladimir Shabad Anatoly Shabalin Evgeny Sharikhin Valentin Shaibonov Bair Shirkov Dmitry Sidorov Alexander Sidorova Maria Siebel Martin Simonov Yuri Sissakian Alexey Skvortsov Vladimir Slavnov Andrey Smirnova Ekaterina Spaan Bernhard Spillantini Piero Starobinsky Alexei Starostin Alexander Stepanyantz Konstantin Studenikin Alexander Tavkhelidze Albert

Centre de Spectrometrie Nucleaire et de Spectrometrie de Masse Univ.ofRome-II California Inst. of Technology ITEP Semenov Inst. of Chern. Phys. SINP, INFN SINP& MSU Brown Univ. MSU ITEP Center for Adv. Math. Sci., American Univ. of Beirut JINR FNAL Univ. of South Carolina Moscow State Univ. ofInstrument Construction and Informatics LANL Moscow State Geological Prospecting Acad.

Peoples' Friendship Univ. of Russia SINP KEK SINP, Kurchatov Inst. Lebedev Phys. Inst. ITEP Moscow Power Engineering Inst. JINR JINR JINR MSU CERN ITEP JINR MIPT, Univ. of Tech. Chemnitz Steklov Math. Inst & MSU Moscow Inst. ofElectr. & Math. Univ.ofDortmund INFN-Florence LITP ITEP MSU MSU INR

[email protected], [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] shabad@lpLru [email protected] [email protected] bairsh@yandex. [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] smimova_ [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] a\[email protected]

472

Urbanowski Krzysztof

Tomsk State Univ. JINR Faculty of Physics, MSU MSU MSU CERN, Univ.ofCalifornia Univ. of Zielona Gora

Vannucci Francois Venediktov Artur Vemov Sergey Volpe Cristina Zakharov Valentin Zhukovsky Konstantin Zhukovsky Vladimir Zhuridov Dmitri

Univ. Paris 7 MSU SINP IPN CNRS ITEP MSU MSU MSU

Telushkin Valeriy Tokarev Mikhail Tmkhin Vladimir Tyukov Alexander Ulybyshev Maxim Unel Gokhan

[email protected] [email protected] [email protected] alex_ [email protected] [email protected] [email protected] [email protected]. zgora.pl [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

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