Advances in Nuclear Physics, Volume 27 (Advances in the Physics of Particles and Nuclei)

ADVANCES IN NUCLEAR PHYSICS VOLUME 27

CONTRIBUTORS TO THIS VOLUME Stefan Scherer

Igal Talmi

Institut für Kernphysik Johannes Gutenberg-Universität Mainz Mainz, Germany

The Weizmann Institute of Science Rehovot, Israel

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ADVANCES IN NUCLEAR PHYSICS Edited by

J. W. Negele Center for Theoretical Physics Massachusetts Institute of Technology Cambridge, Massachusetts

E. W. Vogt Department of Physics University of British Columbia Vancouver, B.C., Canada

VOLUME 27

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ARTICLES PUBLISHED IN EARLIER VOLUMES

Volume 1 The Reorientation Effect • J. de Boer and J. Eicher The Nuclear Model • M. Harvey The Hartree-Fock Theory of Deformed Light Nuclei • G. Ripka The Statistical Theory of Nuclear Reactions • E. Vogt Three-Particle Scattering—A Review of Recent Work on the Nonrelativistic Theory • I. Duck

Volume 2 The Giant Dipole Resonance • B. M. Spicer Polarization Phenomena in Nuclear Reactions • C. Glashausser and J. Thirion The Pairing-Plus-Quadrupole Model • D. R. Bes and R. A. Sorensen The Nuclear Potential • P. Signell Muonic Atoms • S. Devons and I. Duerdoth

Volume 3 The Nuclear Three-Body Problem • A . N. Mitra The Interactions of Pions with Nuclei • D. S. Koltun Complex Spectroscopy • J. B. French, E. C. Halbert, J. B. McGrory, and S. S. M. Wong Single Nucleon Transfer in Deformed Nuclei • B. Elbeck and P. O. Tjøm Isoscalar Transition Rates in Nuclei from the Reaction • A. M. Bernstein

Volume 4 The Investigation of Hole States in Nuclei by Means of Knockout and Other Reactions • Daphne F. Jackson High-Energy Scattering from Nuclei • Wieslaw Czyz Nucleosynthesis by Charged-Particle Reactions • C. A. Barnes Nucleosynthesis and Neutron-Capture Cross Sections • B. J. Allen, J. H. Gibbons, and R. L. Macklin Nuclear Structure Studies in the Z = 50 Region • Elizabeth Urey Baranger An s-d Shell-Model Study for A = 18 – 22 • E. C. Halbert, J. B. McGrory, B. H. Wildenthal, and S. P. Pandy

Volume 5 Variational Techniques in the Nuclear Three-Body Problem • L. M. Delves Nuclear Matter Calculations • Donald W. L. Sprung Clustering in Light Nuclei • Akito Arima, Hisashi Horiuchi, Kunihara Kubodera, and Noburu Takigawa

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Articles Published in Earlier Volumes

Volume 6 Nuclear Fission • A. Michaudon The Microscopic Theory of Nuclear Effective Interactions and Operators • Bruce R. Barrett and Michael W. Kirson Two-Neutron Transfer Reactions and the Pairing Model • Ricardo Broglia, Ole Hansen, and Claus Riedel

Volume 7 Nucleon-Nucleus Collisions and Intermediate Structure • Aram Mekjian Coulomb Mixing Effects in Nuclei: A Survey Based on Sum Rules • A. M. Lane and A. Z. Mekjian The Beta Strength Function • P. G. Hansen Gamma-Ray Strength Functions • G. A. Bartholemew, E. D. Earle, A. J. Ferguson, J. W. Knowles, and M. A. Lone

Volume 8 Strong Interaction in A-Hypernuclei • A . Gal Off-Shell Behavior of the Nucleon-Nucleon Interaction • M. K. Strivastava and D. W. L. Sprung Theoretical and Experimental Determination of Nuclear Charge Distributions • J. L. Friar and J. W. Negele

Volume 9 One- and Two-Nucleon Transfer Reactions with Heavy Ions • Sidney Kahana and A. J. Baltz Computational Methods for Shell-Model Calculations • R. R. Whitehead, A. Watt, B. J. Cole, and I. Morrison Radiative Pion Capture in Nuclei • Helmut W. Baer, Kenneth M. Crowe, and Peter Truöl

Volume 10 Phenomena in Fast Rotating Heavy Nuclei • R. M. Lieder and H. Ryde Valence and Doorway Mechanisms in Resonance Neutron Capture • B. J. Allen and A. R. de L. Musgrove Lifetime Measurements of Excited Nuclear Levels by Doppler-Shift Methods • T. K. Alexander and J. S. Forster

Volume 11 Clustering Phenomena and High-Energy Reactions • V. G Neudatchin, Yu. F. Smirnov, and N. F. Golovanova Pion Production in Proton-Nucleus Collisions • B. Holstad Fourteen Years of Self-Consistent Field Calculations: What Has Been Learned • J. P. Svenne Hartree-Fock-Bogoliubov Theory with Applications to Nuclei • Alan L. Goodman Hamiltonian Field Theory for Systems of Nucleons and Mesons • Mark Bolsterli

Articles Published in Earlier Volumes

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Volume 12 Hypernetted-Chain Theory of Matter at Zero Temperature • J. G. Zabolitzky Nuclear Transition Density Determinations from Inelastic Electron Scattering • Jochen Heisenberg High-Energy Proton Scattering • Stephen J. Wallace

Volume 13 Chiral Symmetry and the Bag Model: A New Starting Point for Nuclear Physics • A. W. Thomas The Interacting Boson Model • A. Arima and F. Iachella High-Energy Nuclear Collisions • S. Nagamiya and M. Gyullasy

Volume 14 Single-Particle Properties of Nuclei Through (e, Jean Mougey

) Reactions • Salvatore Frullani and

Volume 15 Analytic Insights into Intermediate-Energy Hadron-Nucleus Scattering • R. D. Amado Recent Developments in Quasi-Free Nucleon Scattering • P. Kitching, W. J. McDonald, Th. A. J. Maris, and C. A. Z. Vasconcellos Energetic Particle Emission in Nuclear Reactions • David H. Boal

Volume 16 The Relativistic Nuclear Many-Body Problem • Brian Serot and John Dirk Walecka

Volume 17 P-Matrix Methods in Hadronic Scattering • B. L. G. Bakker and P. J. Mulders Dibaryon Resonances • M. P. Locher, M. E. Saino, and A. Švarc Skrymions in Nuclear Physics • Ulf-G. Meissner and Ismail Zahed Microscopic Descriptions of Nucleus-Nucleus Collisions • Karlheinz Langanke and Harald Friedrich

Volume 18 Nuclear Magnetic Properties and Gamow-Teller Transitions • A. Arima, K. Shimizu, W. Bentz, and H. Hyuga Advances in Intermediate-Energy Physics with Polarized Deuterons • J. Arvieux and J. M. Cameron Interaction and the Quest for Baryonium • C. Amsler Radiative Muon Capture and the Weak Pseudoscalar Coupling in Nuclei • M. Gmitro and P. Truöl Introduction to the Weak and Hypoweak Interactions • T. Goldman

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Articles Published in Earlier Volumes

Volume 19 Experimental Methods for Studying Nuclear Density Distributions • C. J. Batty, E. Friedman, H. J. Gils, and H. Rebel The Meson Theory of Nuclear Forces and Nuclear Structure • R. Machleidt

Volume 20 Single-Particle Motion in Nuclei • C. Mahaux and R. Sartor Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics • B. D. Keister and W. N. Polyzou

Volume 21 Multiquark Systems in Hadronic Physics • B. L. G. Bakker and I. M. Narodetskii The Third Generation of Nuclear Physics with the Microscopic Cluster Model • Karlheinz Langanke The Fermion Dynamical Symmetry Model • Cheng-Li Wu, Da Hsuan Feng, and Mike Guidry

Volume 22 Nucleon Models • Dan Olof Riska Aspects of Electromagnetic Nuclear Physics and Electroweak Interaction • T. W. Donnelly Color Transparency and Cross-Section Fluctuations in Hadronic Collisions • Gordon Baym Many-Body Methods at Finite Temperature • D. Vautherin Nucleosynthesis in the Big Bang and in the Stars • K. Langanke and C. A. Barnes

Volume 23 Light Front Quantization • Matthias Burkardt Nucleon Knockout by Intermediate Energy Electrons • James J. Kelly

Volume 24 Nuclear Charge-Exchange Reactions at Intermediate Energy • W. P. Alford and B. M. Spicer Mesonic Contributions to the Spin and Flavor Structure of the Nucleon • J. Speth and A. W. Thomas Muon Catalyzed Fusion: Interplay between Nuclear and Atomic Physics • K. Nagamine and M. Kamimura

Volume 25 Chiral Symmetry Restoration and Dileptons in Relativistic Heavy-Ion Collisions • R. Rapp and J. Wambach Fundamental Symmetry Violation in Nuclei • H. Feshbach, M. S. Hussein, A. K. Kerman, and O. K. Vorov Nucleon-Nucleus Scattering: A Microscopic Nonrelativistic Approach • K. Amos, P. J. Dortmans, H. V. von Geramb, S. Karataglidis, and J. Raynal

Articles Published in Earlier Volumes

Volume 26 The Spin Structure of the Nucleon • B. W. Filippone and Xiangdong Ji Liquid-Gas Phase Transition in Nuclear Multifragmentation • S. Das Gupta, A. Z. Mekjian, and M. B. Tsang High Spin Properties of Atomic Nuclei • D. Ward and P. Fallon The Deuteron: Structure and Form Factors • M. Garçon and J. W. Van Orden

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PREFACE This volume contains two major articles, one providing a historical retrospective of one of the great triumphs of nuclear physics in the twentieth century and the other providing a didactic introduction to one of the quantitative tools for understanding strong interactions in the twenty-first century. The article by Igal Talmi on “Fifty Years of the Shell Model – the Quest for the Effective Interaction”, pertains to a model that has dominated nuclear physics since its infancy and that developed with astonishing results over the next five decades. Talmi is uniquely positioned to trace the history of the Shell Model. He was active in developing the ideas at the shell model’s inception, he has been central in most of the subsequent initiatives which expanded, clarified and applied the shell model and he has remained active in the field to the present time. Wisely, he has chosen to restrict his review to the dominating issue: the choice of the effective interactions among valence nucleons that determine the properties of low lying nuclear energy levels. The treatment of the subject is both bold and novel for our series. The ideas pertaining to the effective interaction for the shell model are elucidated in a historical sequence. In a massive article, which will be valued both for its completeness and its sound judgment of the various contributions to the subject, Talmi succeeds without use of a single figure comparing model results with each other or with experimental data. Instead, the compelling flow of ideas and results for the Shell Model are described in a manner that makes clear why this magnificent edifice has had such great resilience. Many of the ideas of the Shell Model are very elegant. They converge on a picture of nuclear states that is so all-encompassing that there has been very little room, so far, for the intrusion of subnucleon physics. Talmi takes us right up to the most recent decade in which very large computers have been able to verify and expand upon earlier views of the effective interaction by directly diagonalizing the matrices pertaining to the myriad of nuclear states that arise from several valence nucleons beyond closed shells. This review is then a celebration of a most successful model of atomic nuclei and of its underlying ideas.

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Preface

The second article of this volume is also massive and pertains to a subject that has become very important for the field. Chiral Perturbation Theory has been one of the few quantitative tools for coping with the theory of the strong interactions, QCD, which is both remarkably simple to express as a fundamental Lagrangian and notoriously intractable to solve. Chiral Perturbation Theory is particularly useful for the nucleon-nucleon and the nucleon-meson interactions. Stefan Scherer has worked on the theory for more than a decade and has taught a course on it at the University of Mainz. He has chosen in his present review to write a didactic article which should be especially helpful for nuclear physicists who are entering the field. The subject has had major critics who have worried about the convergence of the theory and also about its relation to effective interactions. These subjects are dealt with clearly in this review. Our series has now produced twenty-seven volumes and the technology producing it, which began thirty five years ago with an article set in the hot lead of linotype machines, has now evolved to electronic formatting and production. Having recently concentrated our effort in the successful transition to this new technology, we are now positioned to devote renewed effort to the commissioning of review articles. With the ever increasing breadth on contemporary nuclear physics and related fields, the need for insightful, pedagogical reviews has never been greater.

J.W. NEGELE E.W. VOGT

CONTENTS

Chapter 1 FIFTY YEARS OF THE SHELL MODEL – THE QUEST FOR THE EFFECTIVE INTERACTION Igal Talmi 1.

Introduction

2.

The (Re)Emergence of the Shell Model 2.1. Single Nucleon Orbits 2.2. The Supermultiplet Scheme

11 12 19

3.

Early Calculations 3.1. Energy Levels of Simple Configurations 3.2. Nuclear Magnetic Moments 3.3. Electromagnetic Moments and Transitions. Beta Decay 3.4. Seniority and the to Even-Even and Odd-Odd 3.5. Applications of the Nuclei 3.6. Simple Potential Interactions. Configuration Mixing 3.7. The Beta Decay of 3.8. Simple Potential Interactions. Ignoring the Hard Core 3.9. The Pb Region I

26 26 37 44 53

4.

2

Effective Interactions from Experimental Nuclear Energies 4.1. Simple Configurations 4.2. Protons and Neutrons in Different Orbits. 4.3. The Zr Region I xiii

58 62 68 73 85 89 91 102 109

Contents

xiv

4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10. 4.11.

The Shell Mixed Configurations in the Shell The Nickel Isotopes The Pb Region II The Zr region II The Shell The Complete and Beyond “Pseudonium Nuclei”

115 121 133 140 144 168 175 179 181 181

Some Schematic Interactions and Applications 5.1. The Pairing Interaction 5.2. The Surface Delta Interaction, the Quasi-Spin Scheme and Extensions 5.3. The SU (3) Scheme

193 202

6.

Seniority and Generalized Seniority in Semi-Magic Nuclei 6.1. The Seniority Scheme and Applications 6.2. Generalized Seniority

215 215 220

7.

Large Scale Shell Model Calculations and Beyond 7.1. The 7.2. The 7.3. Nuclei in which N = 20 Loses Its Magicity

232 232 241 249

8.

What have we learned? Where do we stand?

253

References

265

5.

Chapter 2 INTRODUCTION TO CHIRAL PERTURBATION THEORY Stefan Scherer 1.

Introduction 278 1.1. Scope and Aim of the Review 278 1.2. Introduction to Chiral Symmetry and Its Application to Mesons and Single Baryons 279

2.

QCD and Chiral Symmetry 2.1. Some Remarks on SU(3)

290 290

Contents

2.2. 2.3. 2.4.

xv

The QCD Lagrangian Accidental, Global Symmetries of Green Functions and Chiral Ward Identities

293 297 310

3.

Spontaneous Symmetry Breaking and the Goldstone Theorem 323 3.1. Degenerate Ground States 324 3.2. Spontaneous Breakdown of a Global, Continuous, Non-Abelian Symmetry 330 Goldstone’s Theorem 335 3.3. 3.4. Explicit Symmetry Breaking: A First Look 338

4.

Chiral Perturbation Theory for Mesons 339 340 4.1. Spontaneous Symmetry Breaking in QCD 4.2. Transformation Properties of the Goldstone Bosons 347 352 4.3. The Lowest-Order Effective Lagrangian 4.4. Effective Lagrangians and Weinberg’s Power Counting Scheme 360 365 4.5. Construction of the Effective Lagrangian 370 4.6. Applications at Lowest Order 380 4.7. The Chiral Lagrangian at Order 384 4.8. The Effective Wess-Zumino-Witten Action 388 4.9. Applications at Order 405 4.10. Chiral Perturbation Theory at

5.

Chiral Perturbation Theory for Baryons 5.1. Transformation Properties of the Fields 5.2. Lowest-Order Effective Baryonic Lagrangian 5.3. Applications at Tree Level 5.4. Examples of Loop Diagrams 5.5. The Heavy-Baryon Formulation 5.6. The Method of Infrared Regularization

417 418 423 427 440 449 483

6.

Summary and Concluding Remarks

496

Appendix A.1. Green Functions and Ward Identities A.2. Dimensional Regularization: Basics A.3. Loop Integrals in SU(2) × SU(2) A.4. Different Forms of

501 501 507 512 522

References

529

Index

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ADVANCES IN NUCLEAR PHYSICS VOLUME 27

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Chapter 1

FIFTY YEARS OF THE SHELL MODEL – THE QUEST FOR THE EFFECTIVE INTERACTION Igal Talmi The Weizmann Institute of Science Rehovot, Israel

2

1.

Introduction

2.

The (Re)Emergence of the Shell Model

11

3.

Early Calculations

26

4.

Effective Interactions from Experimental Nuclear Energies

89

5.

Some Schematic Interactions and Applications

181

6.

Seniority and Generalized Seniority in Semi-Magic Nuclei

215

7.

Large Scale Shell Model Calculations

232

8.

What have we learned? Where do we stand?

253

References

265

1

2

Igal Talmi

1. INTRODUCTION In 1999 we celebrated 50 years of the modern version of the shell model for nuclei. In this model it is assumed that the nuclear constituents - protons and neutrons - move independently in a central potential well. This well should be due to the average interaction between these constituents, the nucleons. To obtain the observed spacings between single nucleon orbits, the potential well should include a rather strong interaction between the spin and orbital angular momentum of each nucleon. The strong spin-orbit interaction turned out to be an essential ingredient. It was introduced by Maria G. Mayer (1949) and independently by Haxel, Jensen and Suess (1949) and led to the observed single nucleon orbits and the correct magic numbers for which major shells are completely filled. The shell model revolutionized the theory of nuclear physics. The field mostly affected was nuclear structure theory which deals with energy levels of nuclei, their wave functions and electromagnetic and beta transitions between them. Another wide field, of various reactions between nuclei and between nucleons and nuclei, was also strongly affected. The shell model was imposed on nuclear theorists by a large variety of experimental data summarized by M.G. Mayer (1948). Theoretical developments inspired experimentalists and, more frequently, novel experiments presented challenges to theorists. To review the impact of the shell model on nuclear physics in the first 50 years, a real encyclopaedia would be necessary. This review has a rather modest aim. It deals with the rather limited goal of calculation of certain energies and wave functions within the shell model. The calculation of nuclear energies is a very difficult problem. The special stability of nuclei with magic proton and neutron numbers follows directly from their energy levels in the potential well. Still, the average potential is due to two-nucleon interactions and, hence, a better approximation to the potential energy could be obtained by calculating them using shell model wave functions. This, however, is very difficult for several important reasons. First, the interaction between free protons and neutrons is not sufficiently well known. Results of scattering experiments and properties of the deuteron have been fitted by several, rather phenomenological, “realistic” interactions. Still, in the absence of an established theory for these interactions, no definite prescription is available. The internal structure of protons and neutrons in terms of quarks and gluons adds much to the complications. This problem, however, is only part of the difficulty of calculating nuclear energies “from first principles”. The solution of the many body problem is usually extremely complicated. The strong and singular interaction between free nucleons makes ordinary perturbation methods useless. Motivated by the apparent success of the shell model, nuclear many body theorists have tried to derive a renormalized (effective) in-

Fifty Years of the Shell Model

3

teraction which could be used with shell model wave functions to calculate nuclear energies. Much extensive and intensive theoretical research has been carried out in this direction. Most results were obtained for 3 and 4 nucleon systems or for nuclear matter. Properties of the latter have been extrapolated from data on heavy nuclei since this species is not available for experiment. The results are encouraging but not definitive. It is still not clear what role is played by effective 3 body interactions and those involving more nucleons. Such effective interactions are not excluded by many body theory even if they do not appear in the original interaction between nucleons. Starting from the shell model, it is possible to aim at a simpler goal. If we consider a nucleus in which there are several nucleons outside closed shells (valence nucleons), energies of states can be divided into three parts. The first is the binding energy of the closed shells, the second part is the sum of single nucleon energies, which includes their kinetic energies and their interactions with the core nucleons, and the third is the mutual interaction of the valence nucleons. Calculating binding energies of closed shells is the most difficult task. It amounts to solving the nuclear many-body problem. Single nucleon energies are probably easier to calculate since they are much smaller. Still, this involves the kinetic energy of the valence nucleons and the effective interaction between them and core nucleons (those in closed shells). The easiest part to calculate is the interaction between valence nucleons. If they occupy a single orbit, only the matrix elements of the effective interaction between nucleons in this orbit are required. The energy of this part is rather small and hence, the matrix elements need not be very accurate. If valence nucleons occupy several orbits, differences between single nucleon energies are also needed. These can be usually taken from experiment. Since early days of the shell model, theorists tried to calculate energies due to the interaction between valence nucleons in the various states allowed by the Pauli principle. Such calculations are necessary not only to obtain energies of nuclear states but also to determine wave functions of these states. Adopting a central potential well, states of closed proton and neutron shells, or states with one nucleon outside or missing from closed shells, have well determined wave functions. For these nuclei, the predictions of the shell model, at this stage, are only qualitative. Determination of the central potential well yields magic numbers at shell closures, the order and spacings of single nucleon orbits as well as radial functions. This determination, however, involves reliable solution of the nuclear many body problem. If there are several valence nucleons (outside closed shells), there are usually several possible states allowed by the Pauli principle. Eigenstates of the shell model Hamiltonian can be determined only by taking into account the mutual interaction of the valence nucleons. The calculated eigenstates should

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Igal Talmi

have energy eigenvalues and other observables like moments and transition rates which agree with experiment. Shell model theorists have tried for several decades to obtain by quantitatively significant calculations, these eigenvalues and wave functions. The calculation of nuclear energies, which also yields the shell model wave functions, is greatly facilitated by use of symmetries. The isospin symmetry, which will be described in the following, is rather strictly observed in nuclei in spite of their great complexity. With certain conditions on the nature and range of the two-body interaction, other symmetries have been found. The supermultiplet (SU(4)) symmetry was introduced by Wigner and by Hund, in the earlier version of the shell model. The system of nucleons moving independently in a potential well turned out to be fertile ground for other, more specific symmetries. The mathematical expression of symmetries is by introducing groups of transformations which act on the Hamiltonian and on wave functions. The interacting boson model, which can be viewed as an approximation to the shell model, has several symmetries corresponding to simplified physical situations. That model, in which extensive use is made of group theory, will not be described in this review. Some of these groups, however, may occur also in the shell model and they will be briefly described, without going into details. The main difficulty in shell model calculations was the choice of the effective interaction between valence nucleons to be used. Over the years different approaches to this problem were developed which will be described in the following pages. The evolution of ideas about the determination and use of effective interactions is the subject of this review. Its aim is to show how physicists have been grappling with this problem, the ways they developed to solve it and some results of these efforts. An attempt was made to review all relevant papers. It is difficult to believe that this has been achieved. Some prejudice may have been applied in the choice of papers, but most of those which are not included, are simply victims of oversight. Their authors are asked to accept a profound and sincere apology. The results of most papers included, are presented without comments or criticism. It is easy to be critical in retrospect but the papers should be judged by taking into account the time and circumstances. Even this rather limited subject is sufficiently wide so that some interesting and important developments will not be included. For example, very little will be mentioned of effective interactions calculated in many body theory from the interaction between free nucleons. Other important topics which will not be considered are listed below.

Fifty Years of the Shell Model

5

The Collective Model A very important development in nuclear structure physics was the introduction of the collective or unified model. This model is very successful in describing nuclei which exhibit collective spectra like the well known rotational bands. Nevertheless, this model will not be considered in this review for several reasons. The very large quadrupole moments and very strong electromagnetic E2 transition probabilities observed in certain nuclei, attracted the attention of Aage Bohr (1950). He thought that such effects cannot be simply described by the motion of a few nucleons and they must be due to collective motion of most nucleons in the nucleus. Bohr and Mottelson (1953) developed the theory of the collective model. The relevant degrees of freedom were taken to be the shape and orientation of the nuclear surface. They limited the deformation from the spherical shape to be given by the spherical harmonic. A differential equation which describes the dynamics of this quadrupole deformation was written. It is expressed in terms of the Euler angles specifying the direction in space and derivatives with respect to them. In the intrinsic frame of reference two more variables are required to obtain 5 variables which completely characterize the system. These are the angles and Also deformations defined by higher values, and also have been later considered. Spins of individual nucleons could be coupled to the nuclear surface. In their 1953 paper, Bohr and Mottelson considered various possibilities of collective motion, one of which was the case of strong, axially symmetric, deformation of the surface. Such a shape could rotate around an axis perpendicular to the axis of symmetry giving rise to states of the nucleus with energies proportional to J(J+1). In the same year, such rotational bands were discovered by Heydenburg and Temmer (1953) and the collective model scored a major success. In order to incorporate motion of individual nucleons into the collective model, it was extended and further developed (the “unified model”). It starts with a potential well which may be deformed. The motion of nucleons inside the well is considered to be fast while changes of the nuclear shape, by either rotations or vibrations, are considered to be adiabatic. Thus, for every value of the deformation parameter(s) of the potential, the single nucleon orbits can be determined. The equilibrium deformation is found by minimizing the sum of energies of occupied single nucleon orbits. This procedure which was introduced by Nilsson and Mottelson (1955), leads for magic numbers to a spherical equilibrium shape. Other nuclei turn out to be deformed. An anisotropic harmonic oscillator, with axial symmetry, was used by Nilsson (1955) to calculate single nucleon orbits which are named after him. Whereas the shell model was motivated by atomic structure, the unified model resembles to some extent, the theory of molecular structure. Unlike the situation in nuclei, in molecules

6

Igal Talmi

there are large energy separations between rotations, vibrations and electronic excitations. A better way to calculate equilibrium deformations is the use of HartreeFock calculations without the constraint of a spherical self-consistent field. Whenever there are valence nucleons outside closed shells, the self-consistent field does not have spherical symmetry. The eigenstates of a rotationally invariant Hamiltonian have definite values of J. Projecting such states from the Hartree-Fock wave function would certainly result in better wave functions. If the deformation is small, the projected states may correspond to shell model states and no rotational band is expected to emerge. On the other hand, if the deformation is large and there are many valence nucleons, the situation may be different. Wave functions with even slightly different orientations of the deformation axis will be close to orthogonal. In such cases, projected states with definite angular momenta may form a rotational band whose energies are close to being degenerate. In actual cases it would be too complicated to carry out exact projection of states with definite J and approximation methods have been developed. Still, the notion of states of valence nucleons determined by their effective interactions is confined to the shell model whose central potential well is spherically symmetric. In this case, the degeneracies or near degeneracies of single nucleon energies are removed by introduction of the mutual effective interaction. The latter is usually taken as a perturbation and its diagonalization yields wave functions of the various states. In the collective model, nucleons are assumed to move independently in a deformed potential well. Hence, the portion of the inter-nucleon interaction included in the common potential well is larger than in the spherical case. Even without considering further interactions, wave functions with definite angular momenta are determined by projecting from states of nucleons in the deformed potential. As pointed out above, the problem of projection is usually very complicated and is carried out by using some approximation. For these reasons, not much effort was aimed at looking for the correct effective interaction. A very interesting question which has occupied the interest of many theorists, is the relation between the collective model and the (spherical) shell model. In principle, the answer is straightforward since shell model states form a complete set of states. Wave functions of the unified model, as any wave function of A bound nucleons, can be expanded as a linear combination of them. Such expansions, however, are extremely complicated. The situation is actually somewhat simpler but still far from simple. Even in the collective model, closed shell nuclei are considered to be spherical and so are also nuclei near them. There is no simple way to describe nuclei in the transition region by using the collective model. Strongly deformed nuclei, however, have a simple

Fifty Years of the Shell Model

7

description in that model. But even in most of them, it is the orbits of valence nucleons which are considerably affected by the deformation of the potential well. The effect of deformation on nucleons in closed shells may then be approximated by renormalization of the effective charges of the valence nucleons. Thus, it should be possible, in principle, to express states of rotational bands by certain linear combinations of states of valence nucleons. These should be calculated from the effective interaction between valence nucleons. The number of the latter in relevant cases is much too big and approximate methods should be developed to handle this problem. This point is important for understanding the relation between the shell model and the collective model. For actual calculations, the collective model may be considered as an approximation method which is very efficient for certain regions of nuclei. It is indeed remarkable that very complicated states of the spherical shell model may be successfully approximated by wave functions of nucleons moving independently in a deformed potential well.

Hartree-Fock Calculations The very foundation of the shell model is, in principle, the Hartree-Fock (HF) approach. If nucleon wave functions describe independent motion in a central potential well, there is a simple prescription for the state which minimizes the total energy. This occurs when the potential well acting on a nucleon in any given orbit, is equal to its average interaction with the other nucleons in their various orbits. In other words, the minimum energy is obtained when the potential well is self-consistent. The HF ground state is the best variational wave function of all states which are antisymmetrized products of single nucleon wave functions (Slater determinants). The HF wave functions can describe precisely ground states of nuclei with closed orbits or those with one nucleon outside (or missing from) closed orbits. In all other cases, a single determinant cannot be an eigenstate of the shell model Hamiltonian since it does not have a definite angular momentum. It is a linear combination of several states with various angular momenta. If the deformed potential well is determined by variation of some parameters, a more accurate procedure is to project states with good angular momentum and then vary the parameters (“variation after projection”). This procedure, however, is very complicated in actual cases. As mentioned above, Hartree-Fock calculations may yield in certain cases self consistent fields which are deformed, as assumed in the collective model. In certain cases, where the deformation of the HF potential well is large, such states may form a rotational band. In such cases, variation after projection is expected to yield the same deformation parameters for all members of the band.

8

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In such cases, it is possible to restore the rotational symmetry of the system, remove the degeneracy and obtain a J(J+1) dependence of energy levels by “cranking”. A component of spin perpendicular to the symmetry axis of the potential well is added to the Hamiltonian multiplied by a Lagrange multiplier This combination is then minimized by using perturbation theory. Eigenvalues of the Hamiltonian turn out to depend quadratically on Thus,

and the rotational frequency, is related to the angular momentum of a state J by where I is the resulting moment of inertia of the band. Clearly, this approximation may work only for strongly deformed nuclei which exhibit rotational bands. In other cases, the Hartree-Fock theory cannot be directly applied to nuclear states. Nevertheless, there were many attempts to apply the Hartree-Fock method to nuclei. These calculations have not played an important role in the quest for the effective interaction between valence nucleons. They are based on some phenomenological interaction which yields the binding energies and nuclear radii of doubly magic nuclei. Closed shell nuclei turn out to be spherical in HF calculations. The presence of one or more valence nucleons leads to self consistent potentials which are deformed. In certain cases, described above, this deformation may have physical significance, as in nuclei exhibiting collective rotational spectra. In other cases, it just exhibits the inadequacy of the HF approach. In most shell model calculations, it is simply assumed that nucleons occupy certain orbits in a central and spherical potential well. Interactions used in Hartree-Fock calculations are usually inadequate to reproduce eigenstates due to the interaction between valence nucleons. Hartree-Fock calculations attempt to calculate global properties of nuclei. The interactions used in them have a sufficient number of parameters to fit some properties of doubly magic nuclei. The fact that these interactions are unsuited for calculating nuclear level spacings raises doubts about their physical meaning.

Relativistic Mean Field Theory Another development in nuclear structure physics, which will not be discussed, is the relativistic mean field theory. The usual shell model and the collective model are non-relativistic. The velocities of nucleons in nuclei are small compared to the speed of light and hence, it is assumed that relativistic effects may introduce only small corrections. Possible such effects were identified and discussed. An apparently different approach was suggested by Walecka (1974). He introduced a relativistic mean field theory (RMF) which

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is based on certain meson exchange interactions. In the past, exchange of pions, the lightest mesons, was considered as the most important contribution to the interaction of free nucleons at larger distances. In the RMF theory, pion exchange interaction was not considered since “this interaction essentially averages to zero in nuclear matter because of its strong spin and isospin dependence”. Instead, exchanges of sigma and omega mesons were introduced. The neutral scalar meson cannot be found in tables of mesons. It was not discovered experimentally and there are no reliable theoretical estimates of its mass. It is widely used, however, in the description of the interaction between free nucleons in terms of meson exchanges. It gives rise to an attractive, Yukawa type interaction between any pair of free nucleons. Its mass is assumed to be smaller than the mass and hence, the range of the potential due to exchange is larger than the range of the repulsive potential due to exchange of The combined interaction resembles the free nucleon interaction with its hard core and strong attraction at longer range. Still, if a non-relativistic mean field calculation is used, ignoring two-body correlations between nucleons, no stable nuclei result. If the attraction is stronger, the nucleus shrinks to a point and if the repulsion is dominant, the nucleus has zero density. The reason is that the source of the (static) fields of both mesons is the same nucleon density. This is no longer the case in the relativistic treatment of nucleons. There, the scalar mesons are coupled to the scalar density of the Dirac nucleons whereas the vector mesons are mainly coupled to the 4th component of the nucleon current vector. The finite binding energy emerges in such calculations from the slight difference between these two sources. In this way, the interactions which bind the nucleus are completely separated from interactions between valence nucleons. In particular, it is acknowledged that “some properties of nuclei (such as the spectrum of two-nucleon states near the Fermi surface) may depend on finer details like the one-pion-exchange tail” but, as quoted above, “it essentially averages to zero in nuclear matter”. To obtain sufficient binding, the average scalar potential S and the average vector potential V must be rather large. This leads to an attractive feature of the model. The spin-orbit interaction which arises naturally in the relativistic model, is determined essentially by the of S and V. The latter is very large and the resulting spin-orbit interaction is of the order of magnitude needed. On the other hand, the single nucleon wave functions are determined essentially by the of S and V which is rather small, like the non-relativistic potentials in common use. The question which is relevant here is whether this RMF model is in contradiction with the non-relalivistic shell model which has been successfully used since 1949. It turned out that this is not the case. In application to nuclei, calculated lower components of the Dirac wave functions of single nucleons turn out to be rather small compared with the upper ones, as is seen in the paper of

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Lalazissis et al., (1998). Hence, non-relativistic wave functions are good approximations of the single nucleon wave functions in the RMF theory. Thus, the RMF leads to a central potential well, which determines single nucleon energies and wave functions, which is to a very good approximation, a potential well used in the non-relativistic model. Another attractive feature of the relativistic model is a bunching of certain single nucleon energies. This bunching, experimentally observed, was shown by Ginocchio (1997) to be due to a certain symmetry of the Dirac Hamiltonian which is an exact symmetry when S+V = 0.

Coulomb Energy of the Protons A very important symmetry of the interaction between nucleons is charge independence. Apart from the Coulomb interaction between protons, the interaction between two nucleons depends entirely on their space and spin state. Any state of two protons and of two neutrons is antisymmetric in the space and spin variables. A proton and a neutron when in the same antisymmetric state have the same interaction as a proton pair or a neutron pair. A proton and a neutron may be also in space and spin states which are symmetric. In such states, the nuclear interaction may well be different from the one in antisymmetric states. This symmetry of the Hamiltonian leads to a certain classification of states according to their symmetry properties under exchanges of proton and neutron variables. Charge independence has a mathematical description in terms of isospin quantum numbers. Proton and neutron are taken to be the same particle in two different charge states (the small mass difference is ignored). In analogy to spin, they are assigned an isospin 1/2 with projections +1/2 and –1/2 respectively. Each nucleon is assigned an isospin vector t in an abstract space. The value of t·t is equal to with Isospin vectors of several nucleons may be combined to form the total isospin vector T. The space-spin antisymmetric states are assigned total isospin T = 1 with projection for a proton-neutron pair. The space-spin symmetric state of a proton neutron pair is assigned isospin T = 0 so that protons and neutrons obey a generalized Pauli principle. Their wave functions are fully antisymmetric in space, spin and isospin (charge) variables. Charge independence of the nuclear Hamiltonian implies that its eigenstates may depend on the total isospin T but not on its 3-projection That is, the eigenstates are independent of the charge of the states which is determined by the number of protons among the A nucleons. Looking at levels of light to medium nuclei with the same isospin T, their spacings are indeed fairly independent of or charge. Ground state energies (binding energies) depend on and their differences are fairly consistent

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with those calculated by using the Coulomb energy of the protons as a perturbation. Both experimental tests and theoretical estimates indicate that the isospin T is a very good quantum number. Admixture of different isospins is extremely small and has usually a negligible effect on the calculation of nuclear energies. Such slight admixtures have a measurable effect in various weak transitions. The differences in binding energies have an important effect on the stability of heavier nuclei. Although weaker than the nuclear interaction, the Coulomb energy is roughly proportional to the square of the proton number Z. For higher Z values it becomes comparable to the nuclear interaction since the latter has saturation properties which make it roughly proportional to the number of nucleons A. As a result, in heavier nuclei which are stable enough to be studied, protons and neutrons occupy different major shells in ground states and low lying levels. In such states, the isospin T does not help in classification of states, even if it is a good quantum number. All such states have the same isospin T = (N – Z)/2 and its use does not introduce any simplification. In such nuclei, protons and neutrons can be considered independently which avoids the complications due to isospin mixing expected in such nuclei. Hence, this review will not deal with Coulomb energies of nuclei even though comparison of calculated and experimental Coulomb energies is important for our understanding of certain aspects of nuclear structure.

2.

THE (RE)EMERGENCE OF THE SHELL MODEL

In 1999 we celebrated 50 years of the nuclear shell model. Actually it was proposed much earlier than 1949. Nuclear structure physics started with the discovery of the neutron and the historical paper by Heisenberg (1932), in which he suggested that protons and neutrons can be considered as two states of the same particle and that they are the building blocks of atomic nuclei. Physicists who tried to understand this system had in front of their eyes the successful model of electron shells in atoms. In the shell model of the atom, electrons move independently in a central potential well in orbits characterized by a radial quantum number (number of nodes of the radial part of the wave function in the finite-range of ) and orbital angular momentum The central potential is due to the positive charge of the nucleus and the average repulsive interaction of the electrons. In certain atoms in which electrons occupy all states, allowed by the Pauli principle, in orbits which are close in energy, a major shell is closed. Such atoms have high excitation energies and high ionization energies. This is the reason why such noble gas atoms barely participate in chemical reactions.

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2.1. Single Nucleon Orbits In spite of the clear differences between atoms and nuclei, physicists looked for similarities and managed to find them. On the basis of the few binding energies which were measured at that time, Bartlett (1932), in the same year of Heisenberg’s paper suggested that, in analogy with electron shells in the atom, in the orbit is completely filled with two protons and two neutrons “while is obtained by adding on a closed with six protons and six neutrons”. Elsasser (1933) started a serious study of proton and neutron shells in nuclei. In addition to 2 and 8 he identified the magic numbers, for which a major shell is fully occupied, 50, 82 and even 126. He also suggested some order and bunching of single nucleon which reproduce these numbers. It was recognized that the order of single nucleon orbits in nuclei must be different from the one in atoms. There, it is roughly derived from a Coulomb potential, due to the central nuclear charge and modified by the repulsion of the electrons. In nuclei, however, the possible order of levels was similar to that in a (3-dimensional) harmonic oscillator. In it, major shells are characterized by and contain degenerate orbits with or 0. Orbits will be denoted by and are replaced, for historical reasons by Harmonic oscillator levels start with the orbit which closes at proton or neutron number 2. The next is the orbit, yielding the magic number 8 when it is fully occupied. The next shell, with degenerate and orbits, closes at magic number 20. To reproduce shell closures at 50, 82 or 126, drastic shifts of single nucleon orbits were required. It is not difficult to understand why the shell model for nuclei seemed hard to understand. In the atom, there is a massive nucleus whose large electric charge supplies a strong central potential acting on the light mass electrons. Although the potential is modified by the mutual interaction of the electrons, the approximation of a central field seems to be very good. It is the basis of approximation methods like the Hartree-Fock approach. In nuclei, nothing resembles this simple picture and the mutual interaction of protons and neutrons is rather strong, as seen from the large binding energies. Adopting the shell model implies that in nuclei, protons and neutrons move independently in a central potential well. A justified question is how could such motion take place in view of strong interactions between the consistent nucleons. In a series of 3 papers published in Reviews of Modern Physics, Bethe and Bacher (1936) reviewed the status of nuclear physics at that time. They present a fair and balanced view of the shell model, its successes and short comings. They argue that although the order of single nucleon orbits proposed by Elsasser and others reproduces the magic numbers, their model lacks a “theoretical foundation”. A deeper argument which they present, concerns the effect of nucleon-nucleon

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interactions on the single nucleon picture of the shell model. It is fair to admit that this problem has remained with us until now. It is in this aspect that the shell model still “lacks theoretical foundation”. A more serious argument against the shell model for nuclei was raised by Niels Bohr. In a paper published in Nature, Bohr (1936) wrote: “It is, at any rate, clear that the nuclear models hitherto treated in detail are unsuited to account for the typical properties of nuclei for which, as we have seen, energy exchanges between the individual nuclear particles is a decisive factor. In fact, in these models it is, for the sake of simplicity, assumed that the state of motion of each particle in the nucleus can, in the first approximation, be treated as taking place in a conservative field of force, and can therefore be characterized by quantum numbers in a similar way to the motion of an electron in an ordinary atom.” In the shell model, a nucleon moves in an orbit which is essentially undisturbed by the interaction with other nucleons. The mean free path of a nucleon in the nucleus, as estimated from nuclear reactions, is much smaller than nuclear dimensions. Bohr continues with a devastating conclusion: “In the atom and in the nucleus we have indeed to do with two extreme cases of mechanical many-body problems for which a procedure of approximation resting on a combination of one-body problems, so effective in the former case, loses any validity in the latter...”. In their review article, Bethe and Bacher just argued that the interaction between nucleons should be introduced into the shell model. Niels Bohr argued that in view of the strong interaction, observed in nuclear reactions, the shell model cannot be used even as a first approximation. His criticism had a profound influence on many young physicists (e.g., Racah). Only those who were deeply involved, like Wigner and Hund, were not discouraged and continued their work using the shell model. Their work, however, was mostly limited to nuclei lighter than for which the order of single nucleon orbits could be simply understood. The SU(4) scheme or the supermultiplet theory which they developed independently, could be applied to nuclei also without adopting explicitly the shell model. Some of its predictions agreed with experiment but others could not be made with sufficient accuracy without deeper understanding of nuclear states. In 1948, Maria G. Mayer published a paper on closed shells in nuclei. She presented there the “experimental facts...to show that nuclei with 20, 50, 82 or 126 neutrons or protons are particularly stable.” She quotes the 1934 paper of Elsasser who suggested “that special numbers of neutrons or protons in the nucleus form a particularly stable configuration. Then she writes, “The complete evidence for this has never been summarized, nor is it generally recognized how convincing this evidence is”. The experimental evidence in the paper concerns isotopic abundances, number of isotones, edges of the stability

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curve, delayed neutron emission, neutron absorption cross sections and also, tentatively, asymmetric fission. She mentions the fact that shell closure at 20 is understood but does not speculate about the other numbers. It was her paper which forced the shell model on the physics community. The realization that there are magic numbers was in sharp contrast with a statistical approach to the nucleus or to the liquid drop model. In those models, there could be no distinction between nucleon numbers 50 and 52 or between 82 and 84. The solid experimental facts which Mayer presented became a challenge to physicists. Maria Mayer’s paper was noticed by Nordheim and by Feenberg and Hammack who at the end of the same year submitted their versions of the shell model. Feenberg and Hammack (1949) followed essentially the level scheme of Elsasser. In addition to shell closures they discuss spins and magnetic moments, as well as “islands of isomerism” - concentration of electromagnetic isomeric transitions just before shell closure. Nordheim (1949) has another scheme which also agrees with observed shell closures and with observed spins and magnetic moments. He makes explicit use of the results of Schmidt for nuclear magnetic moments. The model of Lande (1934), in which “ particle only, one proton or one neutron, is responsible for the total spin and magnetic properties of the whole nucleus”, was taken up by Schmidt (1937) who had by then, more experimental data. Both level schemes of Nordheim and Feenberg and Hammack, list the various orbits by their orbital angular momentum and characterize the states by the total L and total intrinsic spin S like in Russell-Saunders coupling of atomic electrons (LS-coupling). Maria Mayer was trying to understand the origin of the shell closures which she so convincingly demonstrated. She was also aware of the experimental data which Nordheim and Feenberg presented in favour of their schemes. She must have discussed the situation with some colleagues since Enrico Fermi, listening to her, asked “Is there any indication of spin-orbit coupling?”. In her paper, in which she introduced the shell model, she thanked him for this remark, “which was the origin of this paper.” In some other place she said that she thought for a moment and answered “Yes and it explains everything”. In that paper, she considered the effect of a strong spin-orbit interaction on order and spacing of single nucleon orbits. She shows how in her scheme the observed shell closures are reproduced as well as most observed nuclear spins as presented by Nordheim and Feenberg. She tentatively suggests shell closure at 28 and also mentions how, in her scheme “the prevalence of isomerism towards the end of the shell, noticed by Feenberg and Nordheim, is easily understood”. This paper, Mayer (1949), sent as a “letter to the editor” of the Physical Review, was delayed by the editor who asked Feenberg, Hammack and Nordheim (1949) to write a letter in which they compared their schemes and Mayer’s scheme. These two letters were published in the same issue. Mayer’s

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paper is “On Closed Shells in Nuclei” and in the title of the other one the name “Nuclear Shell Models” appears explicitly, perhaps for the first time. Before both letters were published, a letter submitted two months later was published in the Physical Review by Haxel, Jensen and Suess (1949) “On the “Magic Numbers” in Nuclear Structure”. This letter was preceded by a short paper by Jensen, Suess and Haxel (1949). They independently discovered that shell closures at the numbers observed experimentally are reproduced if “one assumes that a strong spin-orbit coupling...leads to a strong splitting of a term with angular momentum into two distinct terms A more detailed discussion of spins of various nuclei according to their scheme, was published in three short papers in Naturwissenschaften (1949) in which the authors are listed according to cyclic permutations of their names and in a longer paper in Z. f. Physik 128 (1950) 1295. In one of those they state explicitly that instead of LS-coupling the scheme should be used in nuclei. In this scheme, orbits are denoted by and is replaced by a letter as explained above. In the model suggested by Mayer and Jensen et al., single nucleon states with and have different energies. It was known that in nuclei some spin-orbit interaction was observed and its sign is opposite to the sign in atoms. Hence, the single nucleon level with was expected to be the lower of the two. If there is strong splitting between the two states, the energy of the orbit with the highest value in an oscillator shell is lowered down to the lower oscillator shell. The orbit is lowered into the oscillator shell, yielding shell closure at 28. Similarly, the magic number 50 is obtained by the orbit joining the orbits left in the shell. Magic numbers 82 and 126 are due to the lowering of the and respectively. The splitting should be sufficiently large so that two nucleon interactions will not seriously mix states of nucleons with those of ones. In atoms, the prevalent coupling scheme is Russell-Saunders coupling or LS-coupling. In its eigenstates, the orbital angular momenta of the particles are coupled to a definite total orbital angular momentum L. The spins are coupled to a definite total spin S which is coupled to L to yield a definite total angular momentum J. If the interactions can be expressed by space coordinates, all possible states obtained from coupling those S and L are degenerate. The degeneracy of the various J states can be removed by certain spin interactions like the spin-orbit interaction. In the 1949 version of the shell model the spinorbit interaction is the dominant one and should be diagonalized first. Thus, the individual values of the nucleons are good quantum numbers. They are combined to form the total angular momentum J of eigenstates. This scheme is the scheme which occurs also in some cases of atomic spectra.

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The two body interactions are then treated as a perturbation. The definitive paper on the new version of the shell model was published by Maria Mayer (1950I). The model is explained in detail and its predictions of spins, isomeric states and even beta decay are presented, compared with the experimental data. In that paper she states explicitly certain assumptions which were tacitly made in her papers and those of Jensen et al. They concern ground states of several nucleons in a An “even number of identical nucleons...always couple to give a spin zero”. “An odd number of identical nucleons in a state will couple to give a total spin There were only two exceptions to this rule. The J = 3/2 spin of which should be equal to that of a nucleon and the J = 5/2 of where the orbit is She explained that “three protons in a state can couple to give a spin 3/2, although as a rule the energy for the state with spin 5/2 is lower.” A similar explanation is given for the other case and the calculated magnetic moments of these exceptional states agree with the measured values better than the Schmidt moments of and protons. The title of that paper was “Nuclear Configurations in the Spin-Orbit Coupling Model I. Empirical Evidence”. In continuation of it, Mayer (1950II) published a second paper, “II. Theoretical Considerations”. The rules of coupling which she used to reproduce the observed ground state spins are stated and an attempt is made to derive them from a mutual interaction between nucleons. These rules are: 1. The lowest state of a configuration ( identical nucleons is J = 0 if is even. 2. It is

if

nucleons in the

) of

is odd.

3. Of two in the same major shell, the one with the larger has a larger pairing energy.

The last rule was needed to explain why ground states with large in odd nuclei are rather rare, she suggested that higher tend to “fill in pairs”. “The object of this paper is to investigate if there are theoretical reasons for these empirical rules”. A given ordered set of single nucleon orbits determines the magic numbers. They are the maximum numbers of protons or neutrons, allowed by the Pauli principle, which may occupy the various orbits in a major shell. Nuclei with closed major shells are particularly stable as summarized by Mayer (1948). States of one nucleon outside closed shells are also uniquely determined by the central potential well. The single nucleon may occupy any of the states in the higher shell(s). States of a nucleus with one nucleon missing from closed

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shells, one hole states, are similarly determined. The central potential well of the shell model, should somehow emerge from the solution of the nuclear many body problem. Fifty years ago, this solution was not available, nor is it now. Thus, these predictions of the shell model, as unexpected and strikingly successful as they turned out to be, are only qualitative. Quantitative results were expected for calculations of the mutual interaction (“residual interaction”) between valence nucleons. All states which several valence nucleons can form, have the same energy in the central potential well. This degeneracy may be removed by considering the mutual interaction of the valence nucleons. Eigenvalues and eigenstates are obtained by diagonalization of the sub-matrix of the Hamiltonian constructed from all states of the configuration (occupation of the various orbits). For experimental reasons, the low lying states, and particularly the ground state, are of special interest. The Mayer rules specify the spins of ground states of such configurations. The rules of coupling that Mayer suggested are very simple. In atoms, where LS-coupling prevails, Hund’s rule of coupling is that in the configuration, the lowest state has the maximum value of spin S. Such states have maximum symmetry in electron spins and hence, minimum spatial symmetry and therefore, minimum repulsion of the electrons. Among all states with maximum S, the lowest has the highest value of the total orbital angular momentum L. In nuclei, the interaction is mostly attractive and relevant coupling rules for were not known. If the shell model is adopted, spins of nuclei with closed shells are J = 0. If there is one “valence” nucleon, or one hole, in the outside closed shells, the spin of that state is The energy of a single nucleon is equal to the eigenvalue of the Hamiltonian, with a central potential and a one-body spin-orbit interaction, for the In principle, it should be equal to the expectation value of the kinetic energy of the nucleon in the and its interaction with all nucleons in closed shells. The configuration of several valence identical or non-identical, has several states allowed by the Pauli principle. They are all degenerate in the central potential, they all have the same energy equal to times the single nucleon energy of the given In such cases, the ground state and order of other levels are determined by the mutual interaction of the It is considered as a perturbation and thus, calculated by diagonalizing its matrix constructed from wave functions in the configuration. The two nucleon interaction is assumed to be rotationally invariant and hence, eigenstates of this interaction matrix have definite values of the total spin J. Apart from the Coulomb interaction between the protons, the interaction is assumed to be charge independent and hence, eigenstates are also characterized by definite values of the total isospin T.

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In pre-1949 papers, some authors tried to use two body interactions to calculate binding energies of nuclei. Mayer’s approach was much more modest. Mayer assumed that there is an attractive interaction between identical nucleons due to a potential which “for reasons of definite and easy evaluation it was assumed to have the shape of a . .” Her method of calculation is indeed “straightforward but inelegant”. She calculated energies of states of configurations up to where there is only one state with J = 0 or and thus, the sum method could be used. Mayer carried out a “straightforward but tedious calculation of the interaction” and obtained detailed results. Nevertheless, she found that these states are indeed the lowest and found empirically an elegant expression for their energies. It is the correct one also for higher values (for seniorities and see Sect. 6). These expressions are

where I is an integral of multiplied by radial parts of the wave function. “However, no general proof, for all values, of the simple result...is given”. She pointed out that these expressions reproduce the oddeven variation in binding energies. If the integral I does not strongly depend on then the pairing energy is indeed proportional to This was the first time that two body interactions were included in the shell model with strong spin-orbit interaction and Maria Mayer was well aware that using a zero range potential is only a crude approximation of a short range interaction. Yet, “nevertheless, it might be interesting to note that this assumption can explain qualitatively the empirical rules.” In fact, this paper introduced the into nuclear structure physics to such an extent that sometimes people would take it as an integral part of the shell model. In some pre-1949 papers, the opposite approximation was used. The range of the two body interaction was taken to be larger than nuclear dimensions and the potential was approximated by a constant in the region where nucleon wave functions are large. If this long range approximation is adopted and the interaction has a large component of space exchange (Majorana interaction) the results for configurations are drastically different. Racah (1950) calculated the energies in such cases and found them equal to an attractive constant multiplied by

Hence, the ground states have lowest possible values of J consistent with the Pauli principle. For odd that value of J is equal to if (or ),

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i.e., a single nucleon (or hole). For (or ) it is equal to 3/2 and for any other odd it is equal to 1/2. Racah took it as “an argument in favour of the LS coupling against the coupling model”. Like others, it took him some time to accept the fact that is the prevalent scheme for nuclei. In these calculations and in many to be performed later, first-order perturbation theory was used. Matrix elements of the two-body interaction were calculated between shell model wave functions. Disagreement with experiment was often blamed on effects due to admixtures of higher configurations. The re-emergence of the shell model raised again old arguments against the independent nucleon picture of the nucleus. The question asked was whether such a picture could hold in view of the short-range strong interaction between nucleons. At that time it was not known how singular this interaction really is. The new version of the model raised also a new problem - the origin of the strong spin-orbit interaction. Several attempts have been made to explain this effect but even now, 50 years later, the definitive answer is still missing. It is interesting to note that most objections were directed towards the description of nuclear states by The independent nucleon picture was somehow accepted in spite of its difficulties. A strong spin-orbit interaction could not be compatible with a very elegant theory developed by Wigner (1937) and independently by Hund (1937). It is based on nucleon-nucleon interactions which are due to a potential (“Wigner force”) or a potential multiplied by a space exchange operator (“Majorana force”). Eigenstates of Hamiltonians with such interactions could be classified according to irreducible representations of the SU(4) group. This theory was used in pre-1949 years and could account for some important properties of nuclei, like symmetry energy. A particularly attractive feature of this theory was the clear distinction between favoured (“allowed”) and unfavoured (“super-allowed”) beta decays. It may be interesting to have a look at the SU(4) theory and at some of its predictions. This can serve as a brief introduction to applications of group theory to nuclear spectroscopy, as well as describe some of the challenges facing the initial introduction of

2.2. The Supermultiplet Scheme The SU(4) scheme is the first extensive application of group theory to nuclear physics. It is the first of a long list of groups which have been applied over the years to the theory of nuclear structure physics. The use of group theory is based on symmetry properties of the nuclear Hamiltonian. Any such symmetry has direct and important consequences for the eigenstates. If the Hamiltonian has a certain symmetry, it is invariant under certain operations which define the symmetry. Acting twice with such an operation on the Hamiltonian it remains

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unchanged and hence, the composition of two such operations is another symmetry operation. Thus, the symmetry operations form a group and the mathematical theory of groups can be applied to the eigenstates and eigenvalues of the Hamiltonian. A symmetry operation, which leaves the Hamiltonian invariant, may be applied to an eigenstate. The result is an eigenstate with the same eigenvalue of the Hamiltonian. If P is the symmetry operation applied to a state of the system, it can be applied to the eigenvalue equation

Using the invariance of H,

i.e.,

the result is

Thus, is also an eigenstate of H with the same eigenvalue. To demonstrate the implication of symmetry, the best known example of a rotationally invariant non-relativistic Hamiltonian H may be used. The symmetry operations in this case are 3-dimensional rotations which form the rotation group O(3). Eigenstates of H are also eigenstates of the square of the angular momentum with eigenvalues J(J + 1). They may be expressed as linear combinations of 2J+1 independent states, characterized for instance, by the M. Under such rotations, eigenstates transform among themselves. These transformations are irreducible - no subsets can be formed which transform among themselves under all rotations. The matrix of the Hamiltonian constructed from such eigenstates has a special form. Its non-vanishing elements form submatrices along the main diagonal, each submatrix is proportional to the unit matrix and is characterized (usually, not uniquely) by the total angular momentum J. The matrices of rotations expressed in this basis have non-vanishing matrix elements only in sub-matrices along the diagonal and each submatrix is uniquely characterized by J. The submatrices with given J of all rotations form an irreducible representation of the rotation group O(3). The eigenstates which define a given submatrix of the Hamiltonian form a basis of the given irreducible representation. In general, eigenstates of a Hamiltonian with a given symmetry belong to certain bases of irreducible representations of the symmetry group. The labels of these irreducible representations can serve as quantum numbers which characterize eigenstates, like J in the example described above. Wigner, in his papers, considered the SU(4) theory only as a first approximation. He assumed that the mutual interaction between nucleons depends only on the space coordinates of the nucleons. Such interactions include besides an ordinary potential (Wigner force) also the space exchange Majorana interaction. The spin and isospin operators need not appear in the Hamiltonian

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which is then symmetric under any exchange of space coordinates of the nucleons. As a result, any function of space coordinates which is an eigenfunction of the Hamiltonian, is transformed under a permutation of these coordinates into an eigenfunction with the same eigenvalue. If the transformed function is different, another permutation may be applied to it etc. All such functions may be expressed as linear combinations of an independent set of functions. Even if there are several such independent functions they do not correspond to independent states of the system. They are all needed for construction of eigenstates which are fully antisymmetric in space, spin and isospin variables of the nucleons. An eigenfunction fully symmetric in space coordinates should be simply multiplied by a fully antisymmetric state of spin and isospin variables. In other cases, linear combinations should be formed of spatial eigenfunctions which transform among themselves under permutations multiplied by spin-isospin states with the dual (“opposite”) symmetry. Thus, the space symmetry which is allowed by the Pauli principle is limited by the available states of spin and isospin. These latter states transform among themselves under permutations of spin-isospin variables of the nucleons. Hence, they form a basis of an irreducible representation of the group of permutations. A set of orthogonal states can be obtained from them and any permutation can be expressed as a matrix in this basis. These sets of spin-isospin states are also bases of irreducible representations of the unitary group of transformations in the 4-dimensional space of spin and isospin states, constructed from single nucleon states with Transformations of this unitary group U(4), mix states with different spins and isospins but the Hamiltonian which does not contain spin and isospin operators, is trivially invariant under them. The eigenvalues depend only on the spatial parts of the eigenstates and, in particular, on their symmetry properties. Due to the Pauli principle, the latter are determined by the symmetry of the spin-isospin functions. This way, eigenvalues of interactions which do not have any dynamic dependence on spin or isospin, depend on quantum numbers which characterize the symmetry of spin-isospin states. In the case of identical particles, like atomic electrons, and interactions which are independent of spins, eigenvalues are determined by the spatial parts of the wave functions. The latter are characterized by their orbital momentum L, and other quantum numbers if needed, while their symmetry properties are determined by the total spin value S. Energies of states with given L and S are then degenerate and do not depend on the total angular momentum J. They form multiplets in which eigenvalues are split by spin dependent forces, like the spin-orbit interaction. In the U(4) scheme, all states with given spin-isospin symmetry form a It may contain states with different spins S and isospins

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T. Since there are nuclei in which only some of the states of a given supermultiplet may appear. The irreducible representations of the U(4) group of spin-isospin states are characterized by 4 numbers. These can serve as quantum numbers to characterize the eigenstates of the Hamiltonians considered here. One of them is simply the total number of nucleons which is shared by all states of a given nucleus. The U(4) irreducible representations are also irreducible representations of its special subgroup, the SU(4) group, with zero traces of its generators (infinitesimal elements of the group like and in the case of O(3)). The latter representations are characterized by 3 numbers which together with determine the irreducible representations of U(4). The 3 numbers include S and T in a special way. The 3 numbers are integers or half integers, usually denoted by P, and which satisfy the inequality as well as In a given supermultiplet, P is the largest value which T or S can assume in any of its states. In states for which T = P (or S = P), is the largest value which S (or T) can assume. In the SU(4) scheme, T and S play equivalent roles in spite of their different physical meanings. To see which supermultiplets are predicted to be the lowest, Wigner introduced a drastic approximation. To obtain “qualitative estimates” he assumed that the range of the interaction is “large compared with nuclear dimensions” and in calculating matrix elements, it could be replaced by a constant. The constant was taken as the value of the potential at which was reasonable for the Gaussian potential used in those years. In this approximation, all states of a given supermultiplet, those with the same spin-isospin symmetry, are degenerate. We saw above how this approximation was applied by Racah to a Majorana interaction in the case of identical nucleons. Due to the Pauli principle, the space exchange operator may be replaced by the spin exchange operator multiplied by –1. In this case, the Majorana interaction may be replaced by

If the potential is replaced by a constant, the eigenvalue is a simple function of the spin S. It is a negative constant multiplied by

An ordinary potential interaction is simply proportional to in this approximation. Thus, in this approximation, the eigenvalues of the interaction which is independent of spins, are given by the eigenvalues of the spin vector squared. The operator is the quadratic Casimir operator of the SU(2) group of the spin space (an operator constructed from the generators of the

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group, here and which commutes with all elements of the group. Its eigenvalues characterize the irreducible representations of the group). Similarly, in the SU(4) case, the Majorana interaction can be replaced by a potential multiplied by the negative of the exchange operator in spin-isospin space. The latter is the product of spin exchange and isospin exchange operators

This operator can be evaluated by using the (quadratic) Casimir operator of the SU(4) group whose eigenvalues are simple functions of the numbers P, The resulting eigenvalues of the Majorana interaction in the Wigner approximation are given by a negative (attractive) constant multiplied by

From this expression it follows that the lowest supermultiplets are those with minimum values of P, and The isospin of the ground state of a given nucleus is usually equal to |N – Z|/2 and cannot be smaller. Hence, the lowest states of such a nucleus are expected to have P = T = |N – Z|/2 and the supermultiplet characterized by (T, S, ). In even-even nuclei, T is an integer and the lowest value of S is 0. Also vanishes in this case and the supermultiplet is specified by (T,0,0). In odd-even nuclei, with half integral T, the lowest value of S is 1/2 and the value of is either 1/2 (for odd proton nuclei) or –1/2 (for odd neutron nuclei). The interaction energy in this approximation, for a Wigner interaction with coefficient and a Majorana force with coefficient is thus given by the expression

multiplied by a negative constant which is the value of the (Gaussian) potential at the origin. The expression of the interaction energy contains the term T(T + 4) which accounts for the symmetry energy term in various mass formulae. It includes a linear term in T which is sometimes called the “Wigner term”. Such a term is indeed necessary for reproducing nuclear binding energies. These results are very general and completely independent of any assumption on the spatial parts of the eigenfunctions. Other observables of the lowest states in this approximation, like nuclear magnetic moments calculated by Margenau and Wigner (1940), can also be evaluated. To describe this calculation, it is convenient to express explicitly the operators whose eigenvalues are P, and in

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the state with highest values of and If P is the eigenvalue of and is the eigenvalue of then is the eigenvalue of The magnetic moment of the total spin S may be expressed as (the proton state has and the neutron )

where is the gyromagnetic ratio of the proton spin, 2×2.79 n.m. (nuclear magnetons) and is that of the neutron spin, –2×1.93 n.m. Taking the expectation value of the component of the µ operator in the state with and inserting the values of S and Y mentioned above, we obtain the following values for the magnetic moment of the spin

The spin S should be coupled to the total orbital angular momentum L, which has a definite value in any eigenstate of SU(4) Hamiltonians, to form the total spin J. It follows that nuclear magnetic moments fall into 4 sets, odd proton nuclei with J = L + 1/2 and J = L – 1/2 and odd neutron nuclei with J = L + 1/2 and J – L – 1/2. The spin are the same as in the Schmidt-Lande model. The slopes of the Margenau-Wigner lines, however, depend on and may be different. Under the assumption that protons and neutrons contribute equally to L, Margenau and Wigner took to be equal to Z/A. In 1940 there were not many measured magnetic moments and no definite conclusions could be drawn. Later it became clear that the slopes of measured magnetic moments follow the Schmidt lines rather than those of Margenau and Wigner. It is interesting to note that in a shell model without spin-orbit interaction, magnetic moments in the SU(4) scheme could be identical to the Schmidt-Lande moments. If there are identical nucleons in a interacting by a short range attractive potential, ground states have L = 0 for even and for odd. A small spin-orbit interaction would make the states with lower for up to and the states with lower for the rest of the orbit. A very nice feature of the SU(4) scheme is the clear distinction between favoured and unfavoured allowed beta decays. The favoured decays have rather large matrix elements of the operator between initial and final states. The unfavoured decays are also allowed ones, but the matrix elements are much

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smaller. As an example, we can look at beta transitions in A = 43 nuclei. The transition from the J = 7/2 ground state of to the ground state of has logft=3.5 (the ft value is inversely proportional to the square of the matrix element). On the other hand, the transition from the J = 7/2 ground state of to the ground state of which has the same spin and parity, is much less favoured, with logft=5.0. The decay proceeds also by a Fermi transition but even when this contribution is subtracted, the Gamow-Teller transition is still much stronger than the decay. The operator whose matrix elements determine the allowed Gamow-Teller transitions is given by

This operator is independent of space coordinates and hence has vanishing matrix elements between eigenstates in different supermultiplets. Spatial functions with different symmetries are orthogonal. Allowed Fermi beta transitions, due to the operators or can take place only between states in the same supermultiplet which have the same isospin. Hence, beta decays between states in different supermultiplets may take place only if there are some admixtures of other supermultiplets. If the SU(4) scheme gives a good description, such admixtures are expected to be small. Thus, favoured transitions are those between states in the same supermultiplet and all other allowed decays are expected to be unfavoured. This result of SU(4) symmetry agrees in most cases with experiment. There is no similar distinction between favoured and unfavoured beta decays in the shell model. Some systematic deviations from extreme can yield such a distinction but not in such an elegant way. It is worth while to mention that there is one case of a highly attenuated Gamow-Teller beta decay between two states which should be in the same supermultiplet. This is the decay of the J = 0 ground state of to the J = 1 ground state of with log ft=9. (The attenuative of this decay created a major problem for nuclear physics but was also important in archaelogy as the basis of “carbon dating”.) Both states should belong to the (1,0,0) supermultiplet and within it, the lowest state is expected to have L = 0. In that state should have S = 0, T = 1 and in S = 1, T = 0. Attempts were made to overcome this glaring discrepancy by assigning the ground state L = 2( J = 1) rather than L = 0 but this would be rather artificial and in contrast to the assumptions made for SU(4) symmetry. In case of exact SU(4) symmetry and the approximation made above, the L = 0 and L = 2 states should be degenerate in both nuclei. Use of a more realistic potential would lead to equal splittings of these levels in and The lowest J = 2 level in lies at 7 MeV above the J = 0 ground state. The assumption that the order of these levels is reversed in is in clear disagreement with the assumptions of the SU(4) scheme. In any case, there are

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stronger arguments against applicability of the SU(4) scheme to nuclei. The most important assumption of the SU(4) scheme is that interactions which depend on the spin operators are absent from the Hamiltonian or at least that they are much smaller than other interactions. This refers to true spin dependent terms which cannot be eliminated, unlike the Majorana space exchange interaction considered above. In the SU(4) scheme, both S and L have definite values. The strong spin-orbit interaction, which determines the order of single nucleon orbits and magic numbers, breaks SU(4) symmetry in a major way. Apart from the case of a single nucleon or hole, neither S nor L have definite values and instead, the total spins of individual nucleons are good quantum numbers. In view of the elegance and mathematical beauty of the SU(4) scheme, as well as its great past contributions to nuclear structure theory, it is not difficult to imagine the emotional impact on its creator, Eugene Wigner, when it had to be abandoned. This was one case in which Nature did not follow the trail which he blazed. After that, Wigner never turned active attention to nuclear structure physics. He followed some attempts to attribute shell closures to tensor forces which are present in the nucleon-nucleon interaction. They failed, however, to explain even the meagre experimental data available then. Yet, even until recent years, there were several attempts to resurrect the SU(4) scheme and apply it to nuclei. Some physicists have realized its merits and tried to find some evidence for its validity in nuclear physics.

3. EARLY CALCULATIONS 3.1.

Energy Levels of Simple Configurations

The first calculations of nuclear energies due to two body interactions were motivated by the results of Mayer’s calculations. In her paper she suggested that the ground state spins which do not follow her simple rules, are due to states of the predicted configurations but with A possible reason for this “cross over” of levels could have been that the assumption of zero range of the nuclear potential was too drastic. Calculation of energy levels in the and configurations of identical nucleons were carried out by her student Dieter Kurath (1950) who used a Gaussian potential and harmonic oscillator wave functions for the nucleons. The use of a Gaussian potential was a time honored tradition, starting with Heisenberg and Weizsacker and continuing with Wigner and Feenberg. It was supposed to reproduce results of proton-proton and proton-neutron scattering and the binding energy of the deuteron. Kurath found for longer ranges of the potential, that in the configuration the ground state spin was indeed J = 3/2 instead of J = 5/2.

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This, he pointed out, is in agreement with the argument of Racah, described above. In the configuration the situation was less clear. At the rather long range of the potential where the J = 5/2 becomes the lowest state, the J = 3/2 is very close and for slightly larger range it becomes the ground state. Similar calculations of the configuration of identical nucleons were carried out by Talmi (1951a) who used a Yukawa type potential and hydrogen like wave functions. The results indicated that a cross over between the J = 5/2 and J = 3/2 levels occurs only at a rather very long range of the potential. This was attributed to the fact that even at very long range, the Yukawa potential goes over to a Coulomb potential rather than to a constant. It was pretty clear that these calculations could not explain the exceptional ground state spins of and In addition to experimental information and theoretical discussion of ground states, excited states were considered. In the seminal paper of Goldhaber and Sunyar (1951) on classification of nuclear isomers, they stated the rule that the first excited state of even-even nuclei has “usually spin 2 and even parity”. The experimental information was considered by Horie, Umezawa, Yamaguchi, and Yoshida (1951) who found evidence for several cases in which the first and second excited states of even nuclei had spins 2 and 4. They also calculated, but gave no details, levels of some configurations of identical nucleons and found that the short range attractive (they write “repulsive”) interaction of Mayer yields this order of levels. They also noticed that the spacings of these levels were not well reproduced by the The systematic study of experimentally measured excited states of even-even nuclei which revealed important regularities was carried out by Gertrude Scharff-Goldhaber (1952). The problem of the exceptional ground state spins continued to occupy the attention of young nuclear theorists. In the April 1952 meeting of the American Physical Society, Kurath (1952a) reported on his calculations using harmonic oscillator wave functions for the nucleons and a Gaussian potential. He calculated in ground states of nucleons in the and orbits. Unaware of the work cited above, he calculated energies of excited states of configurations, even, of identical nucleons and found the order 0,2,4. This agreed with the situation reported by Scharff-Goldhaber in the same session (chaired by E. Fermi) who quoted the paper of Horie et al. Kurath also calculated the ground state spin of a configuration, odd, of protons and neutrons and found it to be These results were obtained in the limit of vanishing range of the potential. He also calculated in that limit the levels of a configuration of one proton and one proton and found the ground state spin to be In the case of one neutron and one proton hole, he found the ground state spin to be These ground state spins of odd-odd nuclei agreed with the observed ones in and Kurath also looked at

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the change of order of these level as the range was increased. The problem of cross over of levels as a function of range occupies a certain part of a long paper by Talmi (1952). In his search for a simple way to calculate matrix elements of a he was looking for a transformation of two nucleon wave functions into functions of the relative and center-ofmass coordinates. Such a transformation which leads to a finite expansion can be carried out only for eigenstates of the harmonic oscillator potential. This transformation is particularly useful for calculation of matrix elements of noncentral interactions like mutual spin-orbit interaction and tensor forces. For central interactions, this method reduces the evaluation of matrix elements to simple integrals of the potential multiplied by radial parts of harmonic oscillator wave functions. This leads to a convenient comparison of various potentials. As applications, relative positions of the levels with and were calculated for configurations in and also In all cases it was found that the cross over strongly depends on the shape of the potential. Of special interest was the case since several J = 7/2 levels in several nuclei were discovered, close to expected J = 9/2 levels and sometimes even lower than the latter. Goldhaber and Sunyar assigned them to configurations. The calculations produced for certain potentials low lying J = 7/2 levels but not as ground states for any range. There were cross over of other levels like J = 3/2 for very large range of the potentials. That paper deals also with the origin of the single nucleon spin-orbit interaction. From results of scattering experiments, Case and Pais (1950) deduced the existence of a mutual spin-orbit term in the nucleon-nucleon interaction. The question was raised whether such an interaction could yield sufficient splitting between single nucleon orbits with and The results were encouraging. For example, the splitting between and states of a single nucleon outside the and closed shells was calculated to be 3.5 MeV. Still, the potential was very singular and hence the results were very sensitive to the nuclear parameters. Results of similar calculations were reported by Flowers (1952a) in a letter to The Physical Review. Details of this work were published later by Edmonds and Flowers (1952a, 1952b). In those calculations it was found that there is a competition between high spin states and the state for being the ground state. Flowers concludes that “the complete absence of such high spins among the observed ground states raises considerable doubt whether the conditions of are properly established in the light nuclei with masses <50. The work of Jahn and the present author on the other hand, shows that the correct spin values for light nuclei are obtained without ambiguity in LScoupling, assuming small spin-orbit coupling”. Some of these results were presented by Flowers (1952b) in the Amsterdam Conference on Nuclear Physics

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(September, 1952). The study of energy levels carried out by Edmonds and Flowers (1952b) was detailed and systematic. They used the seniority scheme which was originally introduced by Racah (1943) for classifying the states of atomic electrons. Flowers (1952c) generalized it to and to include isospin (Racah’s paper with this generalization was published somewhat later). For isospins smaller than Flowers introduced another quantum number besides the seniority As will be described in the following, a state of the configuration of identical nucleons with seniority and spin J has a corresponding state in the configuration. In that state, there are no pairs. The former state can be obtained from the latter by coupling to it pairs with J = 0 and antisymmetrizing. Thus, the resulting state has the same value of the spin J. Similarly, starting from a state with given value of J and a given isospin which contains no pairs with J = 0, and adding pairs with J = 0, a state of the configuration is obtained with seniority and spin J. Since the J = 0 pairs carry isospin 1, the isospin of the latter state need not be equal to To characterize states in the seniority scheme with isospin, both and are needed in addition to and T. In the special case of identical nucleons, the isospin T is equal to and the reduced isospin is equal to Edmonds and Flowers carried out detailed calculations of energy levels of all configurations of identical nucleons for the and orbits. For protons and neutrons they calculated the energy levels of configurations and for nucleons, levels of configurations and also for the levels with T = 0. They used harmonic oscillator wave functions and studied the spacings of levels as a function of the range of the Gaussian potential which they adopted. For maximum isospin, the results of Edmonds and Flowers on the order 0, 2, 4 levels for even and cross over of the and other J values, are similar to those obtained earlier (“In this way the earlier results of Mayer, Kurath and Talmi have been extended and generalized”). In the configurations of identical nucleons there are two J = 2 and two J = 4 levels with seniorities and They find that the non-diagonal matrix elements between the and states are “small enough not to affect at all the ordering of levels, nor their spacings, significantly” (actually, they vanish exactly). In general, they claim that nondiagonal matrix elements between states with different values of and are of negligible importance. For a reasonable short range of a Yukawa potential those matrix elements give rise to admixtures of the order of The results of Edmonds and Flowers for lower isospin values in the short range limit turned out to be rather similar to the The ground state spins for even turned out to be J = 0 with seniority quantum numbers and The levels were calculated to lie higher in

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the order 2,4,6. Positions of levels in odd-odd nuclei, with and changed as a function of range and no conclusive predictions could be made. For longer ranges of the potential they find some strange cross over cases. In the configuration with T = 1/2, the J = 5/2, is the ground state up to the range ( is the range of the Gaussian potential and is the oscillator constant) where it crosses with the J = 13/2 level In the the J = 19/2 level crosses the J = 7/2 level for greater than 0.8. The authors were baffled by the fact that such high spin levels were not found experimentally. “There are many cases of cross overs of the kind predicted by Kurath, even those take place theoretically at a rather longer range than is desirable (at about twice the actual range, in fact)” whereas, their “cross overs, levels of higher angular momentum” were not observed. They suggest various possible reasons “but a simpler explanation is that the approximation is not valid for light nuclei...If we assume that holds only for masses greater than about 50 the difficulty is removed...”. The authors had really great faith in the results of their calculations and little faith in In all calculations mentioned above, the radial parts of the wave functions were adjusted to nuclear dimensions. The range of the Yukawa potential was determined by the pion mass and an equivalent range was chosen for the Gaussian potential. The nucleon-nucleon interaction was supposed to reproduce the meagre data about proton-proton and neutron-proton scattering as well as the binding energy of the deuteron. This interaction should have also reproduced binding of actual nuclei and in particular, saturation of nuclear binding energies. Saturation was attributed to exchange forces in the thirties, and several such interactions were considered. In addition to ordinary potential interaction (“Wigner force”), space exchange (“Majorana force”), spin exchange (“Bartlett force”) and charge exchange (“Heisenberg force”) were considered. The general nuclear interaction was written as an attractive potential which is multiplied by the operator

Edmonds and Flowers adopted the exchange operator suggested by Rosenfeld (1948) in his book Nuclear Forces, with and The “Rosenfeld mixture” of exchange operators was very popular in those years. The potential for space symmetric states is attractive for both T = 1 and T = 0 states. If T = 1 such states have S = 0 and the coefficient multiplying the attractive potential is –.13 + .93 – .46 + .26 = .6. If T = 0 than S = 1 and the coefficient is –.13 + .93 + .46 – .26 = 1. An overall attractive interaction cannot saturate and the Rosenfeld interaction is repul-

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sive in space antisymmetric states The coefficient for T = 1, S = 1 states is –.13 – .93 + .46 + .26 = –.34 and for T = 0, S = 0 states, it is equal to –.13 – .93 0 .46 – .26 = –1.78. If we adopt the long range limit for the potential, all diagonal matrix elements are independent of L. Then the interaction in nuclei with total T = 0 and total S = 0 is determined by the statistical weights of the states described above. These weights are proportional to (2T + 1)(2S + 1) and the total interaction is proportional to This is probably one of the conditions for saturation which led to the Rosenfeld prescription. In the case of a non-vanishing matrix elements are obtained only for spatially symmetric states for which can be replaced by 1 and, due to the Pauli principle, can be replaced by The exchange operator is then reduced to A study of the complete shell was taken up by Kurath (1952b), a subject which occupied his attention for many years. This was perhaps due to the presence of David Inglis at Argonne National Laboratory, who was very much interested in light nuclei, both experimentally and theoretically. Inglis, as well as Kurath, followed the work of Feenberg and Wigner (1937). Even in the face of strong criticism of the shell model, at that time, it was still considered applicable for nuclei up to oxygen. Feenberg and Wigner as well as Feenberg and Phillips (1937), considered the complete in LS-coupling. They calculated energies of various states and compared them with experiment. When the shell model was introduced, Inglis looked at the beginning of the and concluded that neither LS-coupling nor fitted the data. He tried to interpolate between these two limits and introduced the name of “intermediate coupling”, a term which remained a trade name for many years. In Kurath’s paper, however, he just considered the description of nuclei and compared it to the LS-coupling description published earlier. He compared, in these two descriptions, spins of nuclear ground states, magnetic moments and even binding energies. The latter calculated values are adjusted to fit the experimental ones by adding a linear term, as was done in previous LS-coupling calculations. On the basis of his calculation, he concludes that there is no clear preference for either scheme. At the end, he makes an important suggestion. “It is possible that there is a transition from LS-coupling in the early part of the shell to in the latter part”. In his paper, Kurath used harmonic oscillator wave functions. The potential of the interaction is a Gaussian potential a la Feenberg and Wigner

This potential may be multiplied by 1 or by space exchange, spin exchange or space-spin exchange operators. Within the matrix elements of any

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potential interaction are given in terms of two radial integrals. In the standard approach, the two-body potential is expanded in a series of Legendre polynomials whose argument is the cosine of the angle between the two vectors and leading from the origin of the potential to the two nucleons. The coefficients in this expansion are functions of the lengths of these vectors and

Use of the addition theorem of spherical harmonics yields the expansion

Matrix elements of this expression between states with and and may be conveniently carried out by first integrating over the angles of the two nucleons. The results for each are the same for all potentials. They are multiplied by radial integrals, called Slater integrals which are specific to the potential and the radial parts of the wave functions considered. After the angular integration, the only non-vanishing terms are those for which and Within the all values are equal to 1 and hence The term vanishes when products of 3 spherical harmonics are integrated as follows from parity considerations. In general, from parity consideration it follows that matrix elements of between states with and vanish unless is even. Thus, only two Slater (radial) integrals appear in matrix elements, and Feenberg and Wigner defined two linear combinations of them

In his paper, Kurath used the same notation, but his K and L are larger by a factor 9. A more ambitious calculation was carried out, in his Ph.D thesis, by E.A. Crosbie (1953), apparently unaware of the work of Kurath. Unlike him, he used the “Rosenfeld exchange mixture” described above. The form of the potential is not specified since he assumed L/K = 6 and hence, the strength of the two body interaction was determined by the value of K. In addition to the two body interaction he considered a one body spin-orbit interaction

The parameter which determines whether wave functions are close to LScoupling or is the ratio Crosbie calculated energies of several configurations in both limits. He then tried to interpolate between these

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coupling schemes by using second order perturbation theory. He found some agreements between calculation and experiment as well as some disagreements. He also found some evidence that the spin-orbit interaction is more important at the end of the Nuclei in the offered a convenient subject for calculations. The wave functions are rather simple and yet there were several puzzles. Ground state spins of and were known to be 1 and 3 respectively. In pure they should be equal. The difference was usually attributed to the gradual change of coupling scheme with mass number, from LS- to The difference between and ground state spins was considered by Zeldes (1953) while working with Racah on his Ph.D thesis. These spins were also considered by Racah and Zeldes (1950). Zeldes diagonalized the full matrices in the space of the (both an orbits) of the spin-orbit interaction as well as the mutual central interaction (“Rosenfeld mixture”). Thus he found that by using intermediate coupling the difference in spin values could be obtained with the same strength of the spin-orbit interaction. Calculations in the shell were carried out by Schulten (1953). He used harmonic oscillator wave functions and a Gaussian potential with an exchange mixture He explains that if non-central interactions which are present in the interaction between free nucleons are neglected, those parameters may be just fitted to experimental data in the nuclei considered. He used a single nucleon spin-orbit interaction and calculated energy levels, magnetic and quadrupole moments. For nuclei in the middle of the shell, he is applied an approximate calculation. He seemed to obtain reasonable agreement with experiment of magnetic and quadrupole moments. This is surprising since his calculated energy levels are in rather poor agreement with the observed ones. He also mentions that by adding a mutual (two-body) spinorbit interaction he could obtain the very much attenuated beta decay. As will be shown below, this could not be true. Following Inglis (1953), intermediate coupling was seriously taken up by Lane (1953). Instead of the interpolation between the LS-coupling and limits, he carried out exact calculations. He developed the theoretical expressions which must be used to calculate various observables if the coupling scheme lies between the extreme limits. Lane applied his formalism to the odd parity states in the mirror pair of nuclei and He explained that since the results depend “directly upon the (unknown) detailed nature of the nuclear interactions”, he tried to compute observables “that are not expected to depend very strongly on the detailed nature of the interaction forces, but only upon the mode of coupling he concludes”. These are magnetic dipole moments and transitions and “reduced level-widths for nuclear emission”. The two nucleon central interaction which he used had the Rosenfeld exchange prescription. The

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nature of the potential need not be specified since Lane used the prescription L = 6K. In addition, he included a one body spin-orbit interaction The eigenfunctions of the shell model Hamiltonian are then determined by the single parameter This parameter is varied and the observables listed above are plotted as its functions. In discussing the results, Lane writes: “It is very satisfying to find that all four experimental magnitudes can be explained with the shell-model in intermediate coupling. It would be even more satisfying to find that all the magnitudes were consistent with the same value of the intermediate coupling parameter”. Still, given the uncertainties in the approximations made in the paper, the author concludes that the results are “consistent with one value” of of about 4.4. This paper is the first in a series entitled “Studies in Intermediate Coupling” I to IV. In the second paper, Lane and Radicatti (1954), derived detailed expressions for electric and magnetic multipole transitions. They used them to calculate transition probabilities for E1, Ml and E2 transitions in the both in LS-coupling and in They compared these predictions with available data and concluded that neither coupling scheme could successfully reproduce the experimental values. They concluded that only intermediate coupling could account for the data. They applied their formulae to the E1 transition from the lowest state and the Ml transition from the lowest state to the ground state of They made the same assumptions as made by Lane (1953) and plot transition rates as a function of Correcting an error in that paper, they found that a value 5 for fit very well both transition rates. They calculated rates of E2 transitions and were satisfied to see that some of them are much stronger than single nucleon transitions. From their results, which could not yet be compared with experiment, they conclude “that the shell-model can, in fact, predict such strong transitions, and so it is not necessary to use a surface oscillation model (Bohr and Moltelson 1953) to obtain this result”. In the third paper of the series, Lane (1955a) applied this analysis to various observables in and The calculations are carried out with the interaction used in his 1953 paper, as a function of the parameter The agreement between calculated values in intermediate coupling and the data was fair. The value of the parameter for which the agreement was best is about 3. It is considerably smaller than the value 5 which Lane used for A = 13 nuclei and indicates eigenstates rather close to those of LS-coupling. This is in agreement with other observations that the coupling scheme changes from LS-coupling to as more nucleons enter the In the fourth paper, Lane (1955b) re-examines the and mirror nulcei. In addition to the observables considered in his 1953 paper, the ft-value of “the allowed mirror transition was examined in intermediate coupling and was shown to be

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in excellent agreement with previous results”. Lane checked the robustness of his results by varying the exchange mixture of the interaction. He replaced the Rosenfeld mixture by the “Serber mixture (Serber 1947) which fitted the high energy scattering data” whose exchange part is given by The effect on the calculated observables was rather small. This was also the case if the ratio of the radial integrals L/K was varied between 6 and 5. Amos de-Shalit and Maurice Goldhaber (1953) wrote a paper on “Mixed Configurations in Nuclei” in which they discussed several effects of such mixing. Some of the discussion was rather straightforward but much of it concerned interesting examples and novel conjectures. The authors expect that near closed shells, configuration mixing will be minimal and hence, they consider beta decays of rather pure configurations. If in the configurations of the initial and final states, two or more nucleons are in different orbits, the transition matrix element vanishes. They consider the possibility that “one group of nucleons (protons or neutrons) influences the other in such a way as to make a particular configuration energetically more favourable”. This effect, is due to the proton-neutron interaction, which some authors tended to neglect, as is the case in the “odd group model”. In this paper, the authors explained that the effect which they discuss (“stabilization”) may lead to transitions which are highly forbidden or highly favourable. The authors presented several examples of such cases. In A = 85 nuclei, as noticed earlier, there is an allowed beta decay from the ground state of to the state of whose logft value is very high, 9.5. The authors explanation is that in the state of the proton “stabilizes” the neutron configuration. If it is rather pure, such a configuration cannot be reached by neutron decay into a proton, from the neutron configuration, the only one which is possible in the state. On the other hand, in the corresponding transition from the ground state of the initial neutron configuration could be a mixture of and so that the transition need not be very hindered. The experimental result is logft=6.2. There are several such examples in this region but the purity needed in these cases seemed to be too high. A calculation of these ft value was carried out more than 10 years later by Gloeckner, Lawson and Serduke (1976). It was based on a more systematic determination of the mixed configurations of the slates involved and will be mentioned later on. Other interesting cases, near the closed shells of were considered by de-Shalit and Goldhaber. The shell model configurations in the ground states of and are uniquely determined for both protons and neutrons. In the first forbidden beta decay from the ground state of to the ground state of a single proton decays into a single

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neutron and “the transition probability should obtain its highest value”. This is the explanation for the very low logft value of only 5.1. There are several other highly favoured first forbidden transitions in that region. The authors quote the transition from the ground state of to the ground state of with logft=5.5, where a neutron decays into a proton. There is another, more recent, example offered by the decay of the ground state of to a state of A neutron from the core decays into a proton and the rate is given by logft=5.2. This should be contrasted with the allowed transition of the ground state of to a excited state of for which logft>8.3! It seems that the weight of neutron pairs in the ground state is very small or the overlap of the radial functions of a proton and a neutron is very small. The effect of proton neutron interaction on states of even-even nuclei was the second subject of the paper by de-Shalit and Goldhaber. They believed that “the idea of configuration mixing seems to be especially useful, at least qualitatively, in even-even nuclei”. What they actually do first, is just to consider correctly the same configuration, of protons and neutrons, by including the proton-neutron interaction. If valence protons and neutrons are in the same orbit, this is mandatory, due to charge independence, even though it was occasionally ignored (“odd group model”). The authors considered the lowest J = 2 level and explained that it contains contributions of both proton and neutron excitations which lower its position relative to just proton (or neutron) excitations. The orthogonal state is pushed higher by the proton-neutron interaction. They tried to explain the empirical rule of Kraushaar and Goldhaber that “for a sequence of states the matrix element for the E2 cross over transition is in many cases considerably smaller than that for the E2 fraction in the transition”. They suggested that it is due to the seniority selection rule, The authors considered the effects of real configuration mixings where they explained the “regularities observed for the separations in even-even nuclei”. “As one approaches a magic number from either side, the number of different possibilities for states with a total J = 2 becomes smaller and smaller and this reduces the repulsion of the lowest state”. In semi-magic nuclei, only protons or neutrons can be easily excited which “naturally results in a maximum for the separation in that region”. The authors explained that the “largest number of possibilities of creating a state occurs in the middle between closed shells with a corresponding minimum in the separation”. They realized that this can also happen to the J = 0 ground state but explained that the “states of J = 0 which may mix and lead to a depression of the ground state are usually higher and fewer in number than those of J = 2 and their effect may thus be expected to be smaller”. The authors correctly

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pointed out that the “strong mixture of configurations is probably responsible for a “smearing out” of most of the sub-shell effects”. They even wrote that “the possibility cannot be excluded on these considerations alone that the state eventually may cross the state to become the ground state”. It may be worth while to mention that a similar argument was made by Bethe and Bacher (1936) for the extra stability of nuclei with magic numbers like 50 and 82. These did not have a natural explanation as shell closures of They tentatively suggested that ground states of nuclei with protons and neutrons in half filled shells are pushed down due to the interaction with many higher J = 0 states. In the following, where generalized seniority will be described, it will be shown that an argument based on the number of interacting states cannot be generally true. The nature of the interaction may be very important.

3.2.

Nuclear Magnetic Moments

An important way to learn the structure of nuclear wave functions is to study their magnetic moments, as well as other electromagnetic moments. In contrast with the nuclear interaction which was, and still is to some extent unknown, the theory of the electromagnetic field and its interaction with matter was well established and understood. From its beginning, an important aim of experimental nuclear physics was to measure magnetic moments of nuclei and an important aim of nuclear theory was to calculate them. In view of the early successes of the shell model, it was a challenge to understand magnetic moments within its framework. The model introduced by Lande (1934) and used by Schmidt (1937) is remarkably simple. Lande assumed that “one particle only, one proton or one neutron, is responsible for the total spin and magnetic properties of the whole nucleus”. From this assumption it follows that magnetic moments of odd-even nuclei should fall into four groups: odd-proton nuclei with and and odd-neutron nuclei with and As a function of the spin moments in these groups should lie on straight lines given by

where for odd proton nuclei, for odd neutron ones and the units are nuclear magnetons (nm). The measured magnetic moments lie indeed on bands, roughly parallel to those “Schmidt lines”. Where the ground state spin agreed with the shell model, the value of as determined from the magnetic moment agreed with the predicted value.

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Still, the deviations of magnetic moments from the Schmidt values became the subject of intensive theoretical research. The model of Lande, called later the single particle model, is very simple and hence, attractive. It can be valid, however, only in the case of a single valence nucleon outside closed shells or one nucleon missing from closed shells (one hole state). When there are several valence nucleons or holes, their single nucleon energy is the same in all states of the configuration. The degeneracy can be removed and definite eigenstates determined, as well as the order of levels, if their mutual interactions are taken into account. Thus, all valence nucleons contribute to “the spin and magnetic properties of the whole nucleus”. In special cases, however, ground states may have the Schmidt moments. If there are several identical nucleons, either protons or neutrons, in the same and the total spin is the predicted moment has the Schmidt value. This follows from the simple summation

This result holds in cases where all other nucleons are coupled to a state with J = 0. In other cases, the magnetic moment depends on the structure of the ground state. This was clear to users of the SU(4) scheme but this realization was abandoned by some physicists along with the SU(4) scheme itself. The simplicity of the single particle picture was very attractive and led physicists to look for simple mechanisms which cause the deviations from its predictions. Also the apparent regularity of observed magnetic moments may have hinted for a simple and general explanation. As mentioned above, measured magnetic moments lie in bands which are roughly parallel to the Schmidt lines. Thus, these bands could be obtained by single nucleon moments as expressed above, with the orbital of free nucleons and reduced absolute values of the spin This observation was made independently by A. de-Shalit (1951), F. Bloch (1951) and H. Miyazawa (1951). They even suggested possible mechanisms for the quenching of the free nucleon spin (Schmidt values) towards the “Dirac ” which are for protons and for neutrons. Actually, Lande used his model already in 1934 to calculate the magnetic moment of the neutron which he found to be –.6 nm. Today, such an approach could be described as using effective spin and it is used in current shell model calculations. Now, however, it is used with shell model wave functions which are determined by a two body interaction. Effective were introduced by Talmi (1951 b) who noticed that in odd-odd nuclei with N = Z, magnetic moments agree well with the Schmidt values. This was attributed to a cancellation of deviations from the proton and neutron moments. In however, the measured moment could be

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reproduced by using proton and neutron magnetic moments taken from nearby nuclei. This approach was taken up explicitly by H.M. Schwartz (1953) who used “semi-empirical factors” to calculate magnetic moments of a number of odd-odd nuclei. Some of the nuclei he considers have indeed one valence proton (or hole) and one valence neutron (or hole). In other odd-odd nuclei, where the configurations are more complicated, he still uses a simple picture with one valence proton and one valence neutron. In detailed shell model calculations, the Hamiltonian containing two-nucleon interaction is diagonalized in a given space. The results include energy eigenvalues and eigenstates which are shell model wave functions. Using the latter, magnetic moments could, and have been, calculated in specific cases. In the rest of this section, attempts at reaching a general prescription for magnetic moments will be described. There were other, more sophisticated, suggestions to give a global explanation of the observed features of nuclear magnetic moments. It is clear that a linear combination of states with and would have a magnetic moment which lies between the Schmidt lines. These states, however, if due to a single nucleon, have opposite parities and could not be combined. The major component could be a single nucleon slate, or more precisely, a state with seniority The other state would then be a many nucleon state or simply a state with higher seniority. Both states could have the same parity and could then be combined to form an eigenstate. Such states with definite values of S = 1/2 and with or are usually not states. In LS -coupling, however, such states can be simply defined. Such a scheme was used by Davidson (1951) to explain deviations of magnetic moments from the Schmidt values. In his analysis, he used the Schmidt moments for the simple state but for the other one he used the corresponding Margenau-Wigner values for the orbital Z/A. The deviations from the Schmidt moments occupied much attention of deShalit (1953a). His work was based on the shell model and considers “departures from the single particle model” and their effects on nuclear magnetic moments. He started with a single nucleon state and added to it a state in which the core is excited to a state with This is then added to to form a state with He assumed that only core nucleons identical to the single nucleon are excited, the other kind of nucleons whose number is even, remain unperturbed and coupled to spin 0. On the basis of this assumption (the “odd group model”), he deduced a simple relation between deviations of moments of odd proton and odd neutron nuclei with the same numbers of identical nucleons in the odd group. This relation is

In spite of the clear violation of charge independence, which is implied by the

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odd group model in many cases, this prediction agrees with the observation that deviations of moments of odd proton nuclei are indeed larger than the deviations for corresponding odd neutron nuclei. A. de-Shalit also found which core excited states cause deviations of the right sign. An approach which takes into account isospin symmetry was taken up by Mizushima and Umezawa (1951). They considered configurations of valence protons and neutrons and used wave functions with definite isospins. If in an odd proton nucleus there is an even number of charge independence forbids their coupling to J = 0 states only. They contribute to the spin of the nucleus as well as to its magnetic moment. The authors calculated moments of several nuclei where this situation takes place. Their calculated values are indeed in belter agreement with measured magnetic moments than the Schmidt moments. Flowers (1952d) took up this approach and calculated a larger number of magnetic moments. He adopted ground state wave functions with lowest seniority In this way he could choose a definite state in cases where Mizushima and Umezawa were unable to do so. He also derived a formula for the moments of ground states of the configuration with T = 1/2. Also Flowers finds improvements over the Schmidt moments, and his results are in better agreement with experimental moments. For some disagreements he blamed the scheme. Flowers mentioned that already in Rosenfeld’s book it is pointed out that quadrupole moments of odd neutron nuclei are as large as nearby odd proton nuclei. Using correct isospin wave functions, Flowers calculated quadrupole moments of odd neutron nuclei which he found “arise quite naturally from the coupling between neutrons and protons.” He mentioned that Rosenfeld remarks that quadrupole moments of several nuclei are about 10 times larger than the single nucleon estimates. Such moments led eventually to the collective model. Both de-Shalit and Davidson did not consider possible cross terms of the magnetic moment operator between the major component and other components admixed to it. Thus, the deviations are proportional to the squares of the admixed amplitudes. To obtain the observed deviations, rather large mixing is needed. This approach was carried out to a rather extreme limit by Volkov (1953) who considered magnetic moments as well as M1 “forbidden” transitions (those that could not take place in pure ). In order to obtain the measured magnetic moments, he tried to replace the simple shell model state by “an equally weighted admixtures of states formed from all possible odd-particle configurations within the odd-particle open shell of the Mayer-Jensen shell model”. First he assumed that “the even-type nucleons are all coupled to zero angular momentum” but later he relaxed that restriction. Following these assumptions, he considered only average magnetic moments of sets of nuclei where the odd group of nucleons is in the same major shell.

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He explains that “interference terms between the different configurations are neglected because their signs are expected to be as often negative as positive so their net contribution to the average should be small”. He claimed “a considerable improvement over the Mayer-Jensen model”. In view of his assumptions it is not surprising that he found that “the Mayer-Jensen shell model probably gives a rather poor description of the ground-state wave function of complex nuclei”. The simple shell model state is only a fraction of the ground state in his “Modified Shell Model of Odd-Even Nuclei”. In the SU(4) scheme described above, magnetic moments and the allowed beta decay operator do not have non-vanishing matrix elements between states inside and outside the model space. These operators are diagonal in the single nucleon orbital angular momentum This is no longer the case in the current version of the shell model. Of the orbit with the largest value of in a given major shell only the part is included whereas the part lies in the next higher major shell. The spin operator of a single nucleon has rather large matrix elements between and states (“spin flip”). Looking for admixtures into the shell model wave functions which may strongly affect magnetic moments and beta decay matrix elements, it may be necessary to include excitations between major shells. If such admixtures are small, they will hardly affect energies of states and will not upset the general picture of the shell model. Admixture due to spin flip matrix elements may yield interference terms with the main part of the shell model wave functions and hence, be very effective. The fact that interference terms may be important for calculations of magnetic moments was pointed out by Ross (1952). In his paper he tried to find evidence for effects of the interaction between nucleons on magnetic moments. If this interaction is due to meson exchanges, there should be some meson exchange currents. Meson exchange corrections are difficult to calculate since no reliable theory is known. This situation has not basically changed even today. Ross looked at the measured moments and tried to see whether those effects can be disentangled from effects due to some changes in the shell model wave functions. He realized that “a very substantial modification of the moment can be associated with a very small modification of the wave function”. His point was “that the expectation value of the moment may be linear in the amplitude of small admixtures to the wave function”. Further in the paper he pointed out explicitly which admixtures have this feature. He tried to find evidence also from “forbidden” M1 transitions. Those transitions cannot take place if pure wave functions are used. The experimental situation was not sufficiently clear to draw concrete conclusions, yet his approach turned out to be very fruitful. Blin-Stoyle (1953) made explicit use of this approach to calculate the ef-

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fects of such configuration mixing on nuclear magnetic moments. In his paper, he derived general expressions for such admixtures. He considered a special case where there is one valence nucleon in the and two nucleons in another orbit are coupled to J = 0. The admixed configurations are those in which one nucleon is raised to the orbit. He then looked into the simple case in which the odd nucleon is in a orbit. The admixtures he considered are due to a zero range potential a la Pryce of whose formulae and interaction he made use. Blin-Stoyle found that calculated deviations from the Schmidt moments are in the right direction and about the right size for nuclei with valence nucleon. For a nucleon his correction vanishes for any value of In that case there are no first order corrections which contribute to magnetic moments. This feature is due to the zero range of the assumed interaction and would not arise for other forms of the potential. He remarks that in actual nuclei, the deviations for valence nucleons are indeed much smaller than for ones. Blin-Stoyle and Perks (1954) extended this approach to nuclei with other valence They mentioned the work of Flowers but ignored isospin completely. They considered various excited states for which the corrections to magnetic moments are linear in their amplitudes. They calculated these amplitudes with a zero range potential and derived expressions for the deviations from the Schmidt moments. They show that the corrections have the right sign and order of magnitude. In “view of the approximation of a interaction” “and uncertainty in the form of the radial dependence of the nucleon wave functions” they did not attempt a detailed calculation. They introduce a few parameters with which they fit well the general trend of measured moments and explain that the fitted values of the parameters are reasonable. In one case, however, they carry out a detailed calculation. This is the case of the magnetic moment of in which, according to the shell model, there is one proton outside closed shells of the doubly magic nucleus. Its measured moment however, is 4.1 nm which is very much higher than the Schmidt value of 2.6 nm. They make use of the results of Pryce for this region of nuclei and derive bounds for the moment which nicely bracket the measured moment. At the same time and independently, Arima and Horie (1954) extended and generalized the work of Blin-Stoyle. In their paper, they used explicitly many nucleon wave functions and coefficients of fractional parentage as well as other concepts due to Racah. They considered all admixtures whose amplitudes contribute linearly to magnetic moments. To calculate the various admixtures in perturbation theory they used a zero range potential. Its exchange nature is determined by two constants which determine the strengths of the interaction in spin singlet, S = 0 (i.e., T = 1) and spin triplet, S = 1 (i.e., T = 0) states.

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The exchange operator which multiplies the

is defined as

The coefficients and calculated from the Rosenfeld prescription are proportional to .6 and 1 respectively. Arima and Horie adopted similar values, taking both and to be negative and Arima and Horie considered states in which one nucleon is raised from a orbit into a higher orbit. The magnetic moment operator has non vanishing matrix elements between the ground state and this excited state only if both arise from the same Some such excitations do not involve the odd group nucleons in the A nucleon may be raised from the closed orbit with into the orbit with If there were nucleons in that orbit they were originally in a J = 0 state and in the excited state they are expected to couple to their lowest state with and To obtain a non vanishing matrix element of the magnetic moment vector operator, the resulting hole state should couple with that state to a state with spin J = 1. Other excitations which contribute first order corrections to magnetic moments are obtained if in the valence orbit a nucleon is raised into it from the closed orbit. To obtain a state with total spin the hole state should be coupled to a state with J > 0, even, of the If, however, in the valence orbit there are more than one nucleon and one of the may be raised into the empty orbit. This must be coupled to a state of the remaining with J > 0, even Arima and Horie calculated magnetic moments of the amplitudes of the various excited configurations in first order perturbation theory as well as their effects on magnetic moments. The elegant and simple formulae which they obtain are derived separately for protons and neutrons. Ignoring isospin is probably a reasonable approximation, certainly for heavier nuclei where valence protons and neutrons occupy different major shells. In their analysis of measured moments this assumption is justified for most of the proton and neutron configurations which they adopt. For example, they did not consider nuclei in the nor in the beginning of the Only in a few cases they seem to ignore the necessary corrections considered by Mizushima-Umezawa and Flowers. For example, in the configuration is one proton and two neutrons in the orbit. In the ground state with J = 3/2 and T = 1/2, the two neutrons are coupled to J = 0 only in about 83% of the unperturbed state. In 17% of it they are coupled to a state with J = 2. As a result, the pure shell model moment is which is .26 nm rather than the . 13 nm according to Schmidt. With their interaction they obtained the deviations of magnetic moments in the right direction. In the first order approximation which they used,

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no corrections are possible for nuclei with one nucleon outside closed or one nucleon missing from them. Also, all corrections vanish if there is one valence nucleon as Blin-Stoyle has found. Arima and Horie calculated magnetic moments of actual nuclei. They assume certain values for the coefficients of the and the spin-orbit interaction. By adopting some simple configurations they manage to obtain rather good agreement with measured magnetic moments. For the magnetic moment they obtained the value of 3.4 nm and they remarked that it could be further raised if slight adjustments are made in the splitting due to spin-orbit interaction. Even if their very simple approximations could be challenged, the paper of Arima and Horie is an important landmark in understanding nuclear magnetic moments. They made efficient use of a mechanism in which slight deviations from pure wave functions cause deviations of magnetic moments from the Schmidt values in the right direction and of the right order of magnitude. The important point of their work is the demonstration that calculated corrections to magnetic moments, which are proportional to amplitudes of the admixed configurations, agree with the observed deviations. Their calculated amplitudes of about .1 and even .2 lead to deviations of wave functions from the limit of only a few percent. Their results should have put an end to futile attempts to attribute deviations of moments to some global fictitious unknown effects. Still, some important deviations of shell model magnetic moments from measured ones, cannot be accounted for. These are moments of nuclei with one valence nucleon, or hole, and closed and orbits of protons and neutrons. Measured moments of and are indeed very close to the Schmidt moments. This is not the case for the closed shells of N = Z = 20. Magnetic moments of and deviate from the expected Schmidt values by . 1 to .3 nm.

3.3. Electromagnetic Moments and Transitions. Beta Decay Detailed information on nuclear wave functions may be obtained from various moments and even more detailed information, from rates of transitions between various states. In electromagnetic transitions, the nucleus goes from one state to another by emission of a real photon or a virtual one which is absorbed by the atomic electrons (internal conversion). In beta transitions, a nucleus goes into a neighbour by emission of an electron and a (anti)neutrino or a positron and a neutrino. The latter case may proceed by absorption of an atomic electron by the nucleus and emission of a neutrino (electron capture). The operator whose matrix elements determine the rate of the transitions described above, is a single nucleon operator. The rates are simply proportional

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to the squares of the matrix elements of the operator, taken between initial and final states. Transition probabilities are even more sensitive to details of the wave functions than moments of single states. In many cases, small admixtures into simple wave functions may have a large influence on transition probabilities. The energy has a stationary value near an eigenstate of the Hamiltonian. Moments and transition probabilities do not share this property. To calculate them it is necessary to possess detailed knowledge of nuclear wave functions. Contributions of small admixtures of other states to energy eigenvalues are proportional only to the squares of mixing amplitudes. This, however, is not the case for moments and transition rates. In the case of magnetic moments which was considered above, small admixtures into wave functions of simple shell model configurations were considered by Blin-Stoyle and by Arima and Horie. They looked at admixtures whose contributions to magnetic moments were proportional to the amplitudes of these admixtures. Rather small amplitudes of such admixtures led to sufficiently large corrections to single nucleon (Schmidt) magnetic moments. It was natural for Blin-Stoyle and for Arima and Horie to consider such corrections also to electric quadrupole moments and certain transition probabilities. Most electromagnetic transitions observed in atoms are electric dipole ones. Higher multipole radiations are much weaker since the ratio between atomic dimensions and the wave lengths of emitted radiations is very small. In the case of nuclei, that ratio , although still small, is considerably larger, so that higher multipole radiations were expected and indeed observed by experiment. An early calculation of nuclear transition probabilities was published by Weisskopf (1951). He derived general expressions for transition probabilities in nuclei and then estimated their values by using an “exceedingly crude method”. This was done by assuming “that the radiation is caused by a transition of one proton which moves independently within the nucleus”. The expressions derived by Weisskopf are for electric multipole radiation of order where the parity carried by the emitted photon is and for magnetic radiation where it is equal to The electric radiation is essentially emitted by the electric charges of the protons whereas the magnetic one is due to the magnetic moments of the nucleons. The latter include the orbital magnetic moments of the protons and the spin magnetic moments of both protons and neutrons. The approximate calculation of Weisskopf has been extensively used by experimentalists and his estimated values became known as Weisskopf units. A comprehensive report about classification of nuclear isomers was then published by M. Goldhaber and A. Sunyar (1951)where spin differences of states were determined by measuring rates and conversion coefficients of the emitted radiation. Maurice Goldhaber used to joke that he made Weisskopf famous...

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He certainly made the Weisskopf estimates well known and appreciated. More detailed calculations and comparison with experimental data were carried out by Moszkowski (1952). The radial integrals which were only estimated by Weisskopf, were calculated numerically by Moszkowski assuming single nucleon wave functions of a spherical square well. Early experiments have shown that most electromagnetic rates are of the order of magnitude expected for transitions of a single nucleon. A notable exception were electric quadrupole transitions (E2). Also in cases where the E2 transitions were expected to be due to a single neutron, the rates were comparable to those due to a proton. Along with enhanced E2 transitions, also quadrupole moments of nuclei, specially those far removed from magic numbers, turned out to be very large and predominantly positive. On this evidence, A. Bohr (1950) suggested that such large moments and transition rates must be due to collective effects in which many nucleons participate, Bohr and Mottelson (1953) further developed these ideas and in a comprehensive paper discussed various scenarios of collective effects. One of those was the possibility of equilibrium (prolate) quadrupole deformation of the nuclear surface. When rotational levels were first observed by Temmer and Heydenburg (1954), Bohr and Mottelson applied very successfully the collective model mainly to strongly deformed nuclei. In view of the impressive success of the collective model in certain regions of nuclei, attempts to calculate quadrupole moments and transitions using shell model wave functions were limited to nuclei which were considered to be spherical. In cases where wave functions were obtained from shell model calculations, E2 moments and transitions rates were usually calculated. Still, there were attempts to look for a general mechanism for the observed enhancement of E2 moments and transition rates. This was the approach of Horie and Arima (1955) in their paper on “Configuration Mixing and Quadrupole Moments of Odd Nuclei”. To calculate quadrupole moments of odd-even nuclei, they made the same assumptions as in their paper on magnetic moments. They assumed that in the unperturbed ground state, the even group of nucleons couple to a state with J = 0 (and seniority ) and the odd group couples to a state with Here they realized, however, that such assumptions are not always consistent with charge independence. Still, they explained, quoting Umezawa (1953), that “introduction of isospin does not improve the agreement with respect to quadrupole moments”. They also pointed out that for medium heavy and heavy nuclei use of isospin “does not change the wave function”. Horie and Arima took for the two-body interaction a zero-range attractive potential which, for T = 0 states, is stronger by factor 3/2 than for T = 1 states. The authors added a word of caution about this choice: “this agrees

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with the experimental data of two-nucleon systems at low energy, it remains an assumption since the interactions in larger nuclei might differ from those in two-nucleon systems”. The authors considered excitation of a proton in a closed orbit to a higher one due to its interaction with the odd group nucleons. They calculated matrix elements of this interaction between the unperturbed ground state and states in which one proton is raised from one orbit to another one. There are non vanishing matrix elements of the quadrupole operator between such states and the initial state if the hole and particle couple to J = 2 and their orbital angular momenta satisfy or Admixtures of such states contribute to the quadrupole moment linearly in their amplitudes. Thus, it is expected that small admixtures may make large contributions to the calculated quadrupole moment. The authors explained that, unlike the case of magnetic moments, it is necessary to calculate explicitly the radial integrals. They carried it out using harmonic oscillator wave functions for the nucleons. A difficult problem is to determine the energy differences between the unperturbed ground state and the admixed states. The authors relied mostly on empirical data coupled with some theoretical considerations. In spite of these difficulties, calculated quadrupole moments by Horie and Arima are in fair agreement with many measured values with the clear exception of strongly deformed nuclei. The change from the single nucleon values for odd proton nuclei is in the right direction and order of magnitude necessary to fit the data. It is significant that some disagreements between their calculated values and measured ones disappeared with more recent and accurate experiments. The authors looked at the dependence of the corrections to quadrupole moments on the number of odd valence nucleons in the configuration. They pointed out that for some corrections this is the same as the dependence on of the quadrupole moment due to the nucleons. The same approach is presented in a very short paper by Perks (1955) on “The Quadrupole Moments of Odd A Nuclei”. He used the same interaction and assumptions as in his paper with Blin-Stoyle (1954) on magnetic moments. He quoted results obtained in his Ph.D thesis, which showed that the corrections to quadrupole moments were in the right direction. He found those corrections “to be only of the same order of magnitude as the single particle values of the quadrupole moments”. He concluded that “they cannot explain the large deviations in the region 100< A <200. They did however explain those of some odd-proton nuclei and how quadrupole moments can arise in odd-neutron nuclei”. The differences between single nucleon (Schmidt) magnetic moments and measured moments of most nuclei are not very large. They are usually of the same order of magnitude as the moments or smaller. The situation of beta decay matrix elements is different. As described above, in the SU(4) scheme al-

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lowed Gamow-Teller beta transitions can take place only between states which belong to the same supermultiplet. This introduces the distinction between favoured or super-allowed decays and others, unfavoured ones. The latter can take place only via small admixtures of states of other supermultiplets. In the version of the shell model, such distinction is completely lost. The spin operator and hence, the Gamow-Teller operator, is no longer diagonal in shell model wave functions. This is true, in particular, for single nucleon states with maximum in the major shell. That is equal to whereas its partner with lies in the next higher shell. Experimentally, the distinction between favoured and unfavoured beta decays is well established. Whereas the former have logft values of about 3.5, the latter have logft values about 5 and higher. Some logft values are very very large like 9 of the decay which is described below in detail, 7.3 of the analog decay and 8.5 for the decay of to ground state. The logft values are inversely proportional to the sum of squares of matrix elements of the transition operator between the initial and final states. In (favoured) transitions between ground states of mirror nuclei the sum of squares includes also that of the Fermi matrix element. This one has non-vanishing matrix elements only between states with the same isospin T and identical space-spin parts. Its value for allowed transitions, is which is equal to 1 for mirror nuclei. Still, the Gamow-Teller matrix elements of favoured transitions, allowed in the SU(4) scheme, are much bigger than those of other transitions. Blin-Stoyle and Caine (1957) considered beta decays in A = 43 nuclei. These are the mirror transition between the T = 1/2, J = 7/2 ground states of and and the one between the ground state and the T = 3/2, J = 7/2 ground state of The logft of the former is 5.03 (it was quoted then as 4.8) and the logft of the latter is 3.50 (quoted as 3.4). Simple shell model configurations of the configuration lead to one J = 7/2 with T = 3/2 but to 3 states with J = 7/2 and T = 1/2. The authors take for the T = 1/2 ground states of and the wave function with seniority and reduced isospin From this choice they calculate the predicted logft values of 4 and 3.6 respectively. For N = Z = 20 the shells are closed also in LS-coupling. Hence, the only states which have non-vanishing matrix elements of the beta transition operator between them and the simple shell model slate, belong to the configuration in which one nucleon is excited into the orbit. Admixtures of such states may contribute to matrix elements of the transition operator proportionally to their amplitudes. The authors calculated these admixtures by using a delta potential like the one which they used for the magnetic moments. This, rather schematic calculation leads to predicted logft values of 4.8 for the unfavoured decay and 3.4 for the mirror transition.

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In the first years of nuclear physics, rates of beta decays served as a useful tool in assigning spins and parities to measured levels. For the reasons explained above, as seen from some examples quoted, this is no longer the case. There are cases where beta decay rales depend critically on delicate details of the wave functions, which are far beyond what shell model calculations can reproduce. For this reason, beta decay rates are not considered as crucial tests of successful shell model calculations. This is particularly important in regions where both and orbits are not included in the valence space. The effect of small configuration mixing on quadrupole moments and E2 transitions was considered by Amado and Blin-Stoyle in a short note (1957) and in a detailed paper by Amado (1957). Both papers are on “Weak Collective Effects in the Nuclear Shell Model” and in both, the authors considered configuration mixing of core states excited by interaction with valence nucleons. Such admixtures are used instead of coupling of valence nucleons to collective surface vibrations of the core as considered for instance by True (1956). “From this configuration mixing point of view, then, collective effects are due to mixtures of excited states of the core caused by interaction with the odd particle”. To obtain contributions to quadrupole moments and E2 transition probabilities which are linear in the amplitudes of the admixed states, excited states of the core should be due to excitation of a single proton to a higher orbit. These states should have even parity and spin J = 2 and the orbital angular momenta of the excited proton should not differ by more than 2. For the sake of simplicity, the authors applied their approach to with one valence neutron and the Pb region with one or two valence neutron or neutron holes. General expressions for matrix elements were developed in terms of Racah coefficients for a Rosenfeld interaction with a Gaussian potential with a range of 1.9 fm. Radial integrals were computed for harmonic oscillator wave functions. The detailed calculation can be replaced in the case with the oscillator potential by a simpler calculation in which closure is used. In other cases it may be a useful approximation of the exact results. If energy denominators in the first order perturbation amplitudes of the admixtures are replaced by a constant energy difference, the calculation is greatly simplified. The matrix element of the quadrupole operator Q between states with J and J’, becomes equal to

where V is the two nucleon interaction and E is the constant energy difference. Evaluation of this expression is rather simple since it involves only the zero order shell model states. Actual computation yields for the quadrupole moment of the ground state of the value

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For the E2 transition from the level to the ground state, the calculated reduced matrix element has the value barn. The value fits well both experimental data. If E, the difference between two oscillator shells with the same parity, is taken to be about 12 MeV, becomes fairly close to 40 MeV which was adopted by Elliott and Flowers (1955). In the Pb region, the authors make use of closure and manage to replace the wave functions of the core protons by the proton density of Relying on the measured charge density, they obtained quadrupole moments and E2 transition probabilities in fair agreement with experimental data. It is interesting that about the same time, Raz (1957) published a short note on “Collective Effects in He considered the quadrupole moment and the E2 transition between the two lowest states and concluded that “in the straightforward application of the weak-coupling collective model gives consistent results”. What he actually showed is similar to results of Amado and Blin-Stoyle that almost the same effective charge of the valence neutron accounts for both the quadrupole moment and the E2 transition rate. In September of 1957 the International Conference on Nuclear Structure took place in Rehovot, Israel. Some of the lectures given in that conference are mentioned elsewhere in this review. A. de-Shalit (1957) gave a talk on “Effects of Configuration Mixing on Electromagnetic Transitions”. He later published a more detailed version in the Physical Review (de-Shalit 1958). In these papers, de-Shalit presented a general formulation of the mechanism used by previous authors. In it, first order corrections to simple shell model wave functions give rise to first order corrections of matrix elements of various single nucleon operators. His derivation demonstrated that these corrections are simply proportional to the matrix element of the operator between the unperturbed states. Thus, they lead to renormalization of the operator which can be expressed as giving it an effective charge. The effective charges depend on the nature of the single nucleon operator and on the states between which the matrix element is evaluated. They may still be the same for all states of the valence nucleons and hence, the concept of effective charge may by more useful. The derivation is simple and may be exact for semi-magic nuclei with protons forming closed shells, and the neutrons in states with in the configuration. Admixtures of some higher configurations to such simple states are given to first order by

In this expression the magnetic quantum numbers were suppressed and

are

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additional quantum numbers needed to specify the excited states of the protons. The energy denominators are the differences between the diagonal elements of the Hamiltonian in the simple shell model state and the excited state

The reduced matrix element of an irreducible tensor operator which is a single proton operator, between simple states with and is equal in first order to

Since acts only on proton states, its matrix elements between the simple states with and vanish. This also shows that non vanishing matrix elements must have and the states must be obtained from the state by raising one proton into a higher orbit. To further evaluate the matrix element, de-Shalit introduced the expansion of the interaction V between protons and neutrons as a linear combination of scalar products of irreducible tensor operators. If V is given as a function of nucleon coordinates and momenta, the radial integrations may be carried out. Summation over magnetic quantum numbers gives the most general expressions

The are single nucleon irreducible tensor operators whose reduced matrix elements between two given single nucleon states are equal to 1 and vanish otherwise. These expressions were obtained by using simple formulae of tensor algebra which may be used also to obtain simple expressions for the matrix elements of the operator In the latter, the neutron states cannot

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be different and hence, in the first summation and in the second one. Since the non-vanishing matrix elements of V are those in which Introducing the expressions for matrix elements of V into the expression for the matrix element yields the result

Due to the Wigner-Eckart theorem, matrix elements between given states of any two irreducible tensor operators are proportional. Thus, it is possible to define

Recalling that both

and

belong to the

configuration, it follows that

and Combining these expressions yields the result that the matrix element of between the states and to first order is equal to the matrix element of between neutron states and multiplied by

The physical meaning of the last expression is an effective charge of the for the matrix element under consideration. It still depends explicitly on and through the energy denominators. Amos de-Shalit explained that the energy differences between the neutron states and are within the ground state configuration. Hence, they are expected to be small compared with energy differences due to excitations of the proton core from its ground state to the states. This is expected to be true also for the interaction between neutrons in the and protons in states. Under these assumptions the effective charge of the is independent of and It is the same for all electromagnetic transitions between states of the neutron configuration as well as for moments of these states. The author argued that his results are applicable also to more general cases where protons are in a state, not necessarily in closed orbits, and valence neutrons are not limited to a configuration. He made the important

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observation that “the possibility of introducing the concept of effective charge depends on the operator we are considering and on the character of the interaction between protons and neutrons”. The author considered, in particular, the special case of the electric quadrupole operator which is proportional to the spherical harmonic Wigner forces contain in their expansion, spherical harmonics and contribute in this way to effective charges for electric multipole moments. As pointed out repeatedly, the proton-neutron interaction has a strong and attractive quadrupole-quadrupole term. Hence, the first order contributions to the electric quadrupole, act coherently to yield large values of neutron effective charges. The analogous argument for large effective charges for protons will be mentioned in subsequent sections.

3.4. Seniority and the The years described above, may be called the “naive period” but this is only due to the choice of the two nucleon interaction. Far from naive, however, were the methods used in those years. The powerful theoretical methods developed by Racah for atomic spectroscopy were immediately applied to nuclei. These methods were adopted and extended by Jahn in England where he inspired the work of students and colleagues in nuclear structure. The algebra of tensor operators introduced by Racah (1942) was independently developed in the US by Wigner, a few years earlier. However, he did not publish it because colleagues and referees found it uninteresting! With the re-emergence of the shell model, Wigner’s unpublished manuscript was circulated and used by nuclear physicists (Wigner (1940)). In Japan, work on nuclear spectroscopy and use of group theory, were due to the influence of Yamanouchi. A very important step in the work of Racah was the use of group theory in (atomic) spectroscopy. In the thirties, atomic spectroscopists had no use for group theory as clearly demonstrated in the famous book of Condon and Shortley (1935). Racah in his first papers, follows this tradition and avoids using it. In fact, he points out that he obtained an expression for ClebschGordan coefficients “without the use of the theory of groups” which is more convenient than the one obtained by Wigner who did use it. Racah (1943) also introduced the seniority scheme without group theory. He realized only after several years how useful group theory could be to spectroscopy and then made extensive use of it (Racah (1949)). The concept of seniority was introduced by Racah in his search for additional quantum numbers to distinguish between states of a given electron configuration with the same values of S, L and J. It is based on the idea of pairing of particles into L = 0 states (J = 0 in ). In atoms, due to the repulsive interaction between electrons, states with maximum pairing lie at

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high energies and are of little physical interest. In nuclei, however, pairing energy is large and attractive and, hence, seniority turned out to be a very useful concept. In the Amsterdam Conference which took place in September 1952, a paper by Racah and Talmi (1952) was presented dealing with seniority and short range potentials. In a paper written earlier, Racah (1952) proved analytically the expressions for ground state energies found by Maria Mayer, also for He also derived very similar expressions for LS-coupling. In the case of higher there are in configurations two or more states with J = 0 for even and two or more states for odd Racah showed that Mayer’s formulae hold for states with lowest seniorities ( and ) if the is diagonal in the seniority scheme. In the paper by Racah and Talmi (1952) this fact was proved in a simple way and also a simple derivation of Mayer’s formulae was given. A rather large class of interactions including tensor forces, was shown to yield such formulae. The original definition of seniority by Racah (1943) was in LS-coupling for configurations of identical particles (like electrons). The generalization to and isospin was first published by Flowers (1952c) and then independently by Racah (1952). Racah introduced a quantum number (in Hebrew, seniority is “vetek”) which is, loosely speaking, the number of unpaired particles. A state of the configuration with no J = 0 couplings is defined to have seniority Other states with seniority may be obtained by multiplying it by a product of states of pairs coupled to J = 0 and antisymmetrizing. The resulting state in the configuration, has the same spin J as the state. To prove the consistency of this scheme of states, Racah introduced an operator which later became known as the “pairing interaction”. In the configuration it is a two-nucleon operator whose only non-vanishing eigenvalue is in the state with J = 0, i.e.

Racah then calculated (without the use of group theory...) the eigenvalues of this operator in states of the configuration and obtained them in the form

The seniority scheme was defined as the (complete) set of eigenstates of this hermitean operator. States with different seniorities have different eigenvalues and are orthogonal. When Racah finally embraced group theory as a useful tool for spectroscopy, he recognized that seniority and the operator Q are simply related to a certain

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group. In the case of the SU(4) scheme, described above, it was explained how fully antisymmetric functions can be constructed. Spin-isospin states which form bases of irreducible representations of SU(4), have a certain symmetry under permutations or SU(4) transformations. They should be multiplied by spatial functions with the dual (opposite) symmetry. In the case of identical particles it is sufficient to consider spin functions only (instead of the SU(4) group, the SU(2) group is relevant) whose symmetry is fully determined by the value of spin S. The spatial functions are characterized by the orbital angular momentum L (and M).This is due to the shell model Hamiltonian being invariant under 3-dimensional rotations which are the elements of the O(3) group. All states with various L (and M) values which have the same symmetry under permutations transform irreducibly among themselves under transformations of the group of unitary transformations in a space of dimensions. These transformations are applied to the states, with various of every particle. The many particle wave functions transform linearly under such transformations. For example, in the configuration, the symmetric functions with L = 0, L = 2,..., transform irreducibly among themselves and all components of the antisymmetric ones, with odd values of L, transform irreducibly among themselves. The following description is based on lectures delivered in 1951 by Racah at the Institute for Advanced Study in Princeton. Notes taken by E. Merzbacher and D. Park were mimeographed several times and widely distributed. Only much later, they were published by Racah (1965). Classification of states can be viewed as starting from irreducible representations of a certain group and going to a subgroup for a finer distinction between states. The labels which characterize the irreducible representations can be used as quantum numbers which characterize the states. In the case of the configuration, starting with the its irreducible representations determine the symmetry of the states. They can be defined by the SU(4) labels and in the case of identical particles, by the value of the spin S. There is a subgroup of whose members are transformations among the components of a single particle states induced by rotations in the familiar 3dimensional space. Going to this O(3) subgroup, the irreducible representations are reduced to those of O(3). As well known, the 2L + 1 components with definite L, transform irreducibly among themselves under these rotations. It is worth while to point out that the shell model Hamiltonian is, in general, not invariant under transformations. Had it commuted with them, its eigenvalues would have been the same for all states in a given irreducible representation, independent of L. This is the case for a two body interaction which is a constant,

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like in the Wigner approximation of SU(4), described above. Still, any two body interaction can be expressed as a linear combination of scalar products of irreducible tensor operators which are generators of the group. Therefore, operating with the shell model Hamiltonian on a state which belongs to a basis of an irreducible representation of yields a state in the same basis. Hence, the eigenstates can be classified according to these representations. So far, the discussion above has been just a fancy way of introducing the quantum numbers S and L. Racah (1949), however, realized that the seniority classification is related to an intermediate group between and O(3). It is a subgroup of whose transformations leave unchanged the scalar product in the dimensional space. Its elements leave invariant the scalar product, which is simply the state (but not other states). It has therefore, O(3) as a subgroup under whose transformations the components of each L state transform among themselves. The group introduced by Racah, is the orthogonal group in the dimensional space, The state, with seniority is invariant under transformations. All components of other states of identical nucleons, with transform irreducibly among themselves under those transformations and have seniority There is an advantage in going from first to the subgroup and only then to O(3). If several states with the same value of L (and of S, i.e., symmetry) occur in the configuration, they may belong to different irreducible representations of Hence, such states may have additional labels for a more detailed classification, like the seniority in the case of identical nucleons. The situation in is analogous to the preceding one. All states of the configuration of identical particles are fully antisymmetric. Hence, they all belong to the fully irreducible representation of the group of unitary transformations in the dimensional space spanned by the states of a single In the case of protons and neutrons and isospin fully antisymmetric states are linear combinations of space and spin functions with a certain symmetry multiplied by isospin states with the dual (opposite) symmetry. The irreducible representations of the space and spin states of a nucleon are thus determined by the symmetry of the isospin states, i.e., by T. The generators (infinitesimal elements) of this group may be taken to be the components of the unit irreducible tensor operators in the dimensional space whose reduced matrix elements are equal to 1 between and zero otherwise. Components of these tensors are linearly independent and their number, is equal to the number of real parameters necessary for defining such unitary transformations.

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Among these, the generators of the subgroup of transformations induced by 3 dimensional rotations are the 3 components with Seniority may be introduced by considering a subgroup of whose transformations leave invariant the antisymmetric bilinear form They should satisfy the condition

The state has and is symmetric if is odd and antisymmetric if is even. Since is antisymmetric and is symmetric, the state must vanish if is odd. Thus, odd tensor operators are generators of the group whose transformations leave the state invariant. This group is the symplectic group in dimensions. These considerations, based on symmetry properties of the J = 0 state, are not limited to the case of identical nuclcons. They are valid also for states of protons and neutrons with The pairing interaction is related to the quadratic Casimir operator of which can be expressed as a linear combination of scalar products of odd tensor operators. Its eigenvalues determine the irreducible representations of which can be used as labels for states of configurations. In the case of identical particles they are fully determined by the seniority For states of nucleons with isospin, another quantum number is necessary. A convenient one is the reduced isospin introduced by Flowers (1952c), which is the isospin of the states with seniority in the configuration. In the case of identical nucleons, the reduced isospin is equal to Thus, the state with has The single state has The states with J > 0, with have if J is even and if J is odd. Transformations of leave invariant the state with seniority (and ). All states of two nucleons with and various M values, with transform irreducibly under these transformations. This is also the case for two-nucleon states with with In terms of and tion are given by

as well as

and T, eigenvalues of the pairing interac-

For the case of maximum isospin, the reduced isospin of all seniority states is equal to Substituting these values of T and the previous expression is obtained. There are many two-body interactions which

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are diagonal in the seniority scheme. They can all be expanded in terms of scalar products of odd tensor operators, a tensor operator with and for a constant multiplying the charge exchange operator. Such interactions have very interesting properties, some of which will be described in the following. The as shown by Racah and Talmi (1952), can be expressed as a linear combination of scalar products of odd tensors only. The absence of the monopole, term leads to the results of Mayer. As explained above, a potential interaction can be expanded as a linear combination of scalar products of spherical harmonics. It was mentioned that matrix elements of between single states vanish if is odd. This applies also to configurations, where all nucleons have the same orbital angular momentum Odd tensor operators, playing an important role in the seniority scheme, appear in spin dependent interactions, like As before, is expanded in a linear combination of of which only those with even should be kept. To evaluate matrix elements of each term in it is convenient to recouple into a linear combination of scalar products Matrix elements of the tensor product between states vanish unless is even. This follows from the symmetry properties of to which the matrix elements are proportional. Thus, for even values of the rank must be odd and this interaction contains only odd tensors in its expansion. Due to the Wigner-Eckart theorem, matrix elements of the tensors are proportional to matrix elements of the unit tensors In addition to the central (or scalar) interaction considered here, also the expansion of tensor forces, in configurations, contains scalar products of only odd tensor operators.

3.5. Applications of the clei

to Even-Even and Odd-Odd Nu-

Two approaches to the problem of the interaction between nucleons in the shell model may be recognized. In one of these, the interaction was supposed to be determined by nucleon-nucleon scattering experiments and the state of the deuteron. The resulting interaction was a central one with an exchange operator whose composition was determined by some saturation conditions. Historically, or for convenience, a Gaussian form for the (central) potential was usually adopted. In the other approach, a zero range potential, introduced by Mayer, was used. It was not considered as a realistic interaction but it seemed to reproduce well some features of nuclear spectra. Its use facilitated calculations and it was meant to yield general results, at least qualitatively. The general feeling was that soon the nucleon-nucleon interaction would be completely determined. Some authors thought that meanwhile it was safer to carry out cal-

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culations for various possible interactions. In particular, for mutual spin-orbit interactions and tensor forces whose existence was implied by the quadrupole moment of the deuteron. Also, it seemed that the Yukawa radial dependence of the potential should be used rather than the Gaussian shape. This is the attitude in the paper of Talmi (1952). In a long paper by Elliott (1953), he used a very general interaction with a Yukawa potential, including tensor forces and a mutual spin-orbit interaction suggested by Case and Pais (1950). Considering energy levels in and he attempts to determine the parameters of his interaction. He used LS-coupling wave functions which makes it difficult to compare his results with the model. A rather early applications of the was carried out by Pryce (1952). He considered nuclei in the vicinity of the doubly magic nucleus He explains that the approximation of a delta potential is probably justified for this rather heavy nucleus. He tried to calculate energy levels of nuclei with two valence nucleons or holes. He took single nucleon (or single hole) energies from experiment, used a delta potential and derived explicit expressions for its matrix elements. He introduced different strengths for T = 1 states (singlet states) and for T = 0 proton-neutron states (triplet states) and used these strengths as essentially adjustable parameters. At that time there were only a few known levels with spin assignments which made his work difficult. Pryce was well aware of these difficulties. “The intention of this work was frankly exploratory, and for that reason the methods of calculation were as simple as possible. This was achieved by assuming that the forces were of very short range, ignoring non-central forces and assuming rigorous ”. “The apparent success of much of the resulting theory ...must not be taken too seriously as support for the hypotheses. Other assignments may do just as well or better, and fatal discrepancies may yet be found. Further analysis, both theoretical and experimental, is now required.” These last words appeared in many papers published in those years. Odd-odd nuclei were considered interesting since their levels are determined by the proton-neutron interaction. Nordheim, in 1951, staled phenomenological rules for the ground state spins of these nuclei: I. If the valence protons are in a orbit and the neutrons in a orbit, or vice versa, the ground state spin is equal to II. If both valence protons and valence neutrons are in orbits respectively, the ground state spin is equal to or close to it.

and

An attempt to derive Nordheim rules by using a zero range potential was made by de-Shalit (1953b) who considered states of one valence and one

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valence The orbit is completely filled as is the case in heavier nuclei. He used a zero range interaction with spin dependent terms

The author defines Nordheim numbers by so Nordheim’s rules hold for N even and N odd respectively. In his paper he is made use of tensor algebra a la Wigner, in great detail, and perhaps for the first time, replaced the earlier conventional W of Racah coefficients. He calculated matrix elements of the independently of earlier results of Pryce and Talmi. The analytical formulae which he derived are used to plot, for certain values of and energy levels for and as functions of Looking at the plots for Nordheim Rule I seems to hold for any value of between 0 and 1. It is shown to be a “strong rule” in the sense that levels with spins higher than lie considerably higher than the ground state. In configurations where Rule II should apply, the situation is not that clear. For and the lowest state has also spin but it rises steeply as is increased and then the state with becomes the lowest one. This state, however, is not isolated. States with spins etc. are not much higher. Indeed, “this simple model can explain Nordheim’s rule and its decomposition into ’strong’ and ‘weak’ rules. It goes a little bit beyond the original formulation of the rule, in as much as it predicts that in the ’weak’ case the ground state will have an even or odd spin according to whether is even or odd.” It is interesting to note that ground state spins of several self conjugate odd-odd nuclei, measured later, turned out to be J = 0. This does not agree with the weak rule but is in agreement with de-Shalit’s calculations for small values of A continuation of de-Shalit’s calculations was carried out by Schwartz (1954). He used the same interaction but considered several and several (outside a closed orbit). To calculate energies of various states of this configuration, he made a drastic simplification. Schwartz assumed that the couple to a state with and seniority and the neutrons to a state with The states he considered are only those obtained by coupling of and He then used recursion relations, due to Racah, for odd (spin-dependent) and even tensors which appear in the expansion of the interaction, to calculate energies of the various states. Schwartz found values of for which the interaction in his configurations reproduces the measured ground state spins of several odd odd nuclei. His results for proton-hole neutron-hole systems are, of course, identical to those for the corresponding proton-neutron cases. But in other cases, his

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results agree with experiment whereas Nordheim’s rules fail. Of particular interest are the results for particle-hole configurations where his assumptions are exact. He could reproduce the J = 4 ground state spin of with his interaction, as Kurath before him, which is in contradiction with Nordheim rules. In other cases, as it turned out later, his approximation is far too drastic. Nevertheless, his results demonstrated that odd-odd nuclei are not very simple and could not be described by simple configurations as he notes that, “in general, the situation is more complicated”. The experimental information on spins of odd-odd nuclei is summarized by Brennan and Bernstein (1960). They compared the data with predictions of the Nordheim Rules and with calculations of Schwartz (1954) which are in better agreement with experiment. Another application of the zero range potential was made to explain relation between spins and parities for excited states in even-even nuclei. Glaubman (1953) made the observation that such excited levels with odd spins have odd parities and that most low lying even spins states have even parity. The order of levels with spins J = 0 , 2 , 4 . . . within a configuration was reproduced by the use of a Still, Glaubman quoted cases where states with even parity belonged to excited configurations. Talmi (1953) checked whether this relation between spin and parity could be reproduced in the simple configuration of identical nucleons with one hole in the orbit and one nucleon in the orbit. Unaware of the paper of Pryce, he calculated matrix elements of the in two nucleon configurations. The results are very simple in the LS-coupling scheme. The interaction energy in a state with and L, vanishes in all triplet states, with S = 1. Such states are symmetric in spin variables and hence, fully antisymmetric in space coordinates, vanishing for In singlet states, with S = 0, the interaction energy is proportional to the square of the with and L and Thus, it vanishes unless

The situation is not much more complicated in The state of nucleons with and coupled to spin J can be expanded in terms of LScoupling components. The triplet components have vanishing interaction energies of the The only contribution is due to the singlet component if the above condition is satisfied. This contribution is equal to the energy of the singlet state multiplied by a known geometrical factor. Thus, the only non vanishing energy in the configuration is in the state which satisfies the condition stated above. That condition determines that the spin of the lowest state in any excited configuration is odd if the parity is odd and it is even if the parity is even. Since, as shown by Racah and Talmi (1952), the

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can be expressed as an odd tensor interaction, this relation holds also for the nucleon-hole configuration. Looking at the order of levels with non-vanishing interaction energies, the lowest ones are those whose spins maximize the coefficient of the singlet term. If then the maximum value, and the lowest term, will be for the minimum spin, It is odd if the parity of the state is odd and even if that parity is even. In the other case, where and the spin of the lowest state will have the maximum value, That value also satisfies the spin-parity condition stated above. These rules, however, are considered to be less robust than the relation between spin and parity which is independent of the orientation of intrinsic spin with orbital angular momentum. “These rules are in a sense the opposite of Nordheim rules concerning the coupling of unlike nucleons (in that case the coupling has to produce maximum triplet component but in the case considered here the coupling produces maximum singlet component as the triplet states have space-antisymmetric wave functions).”

3.6. Simple Potential Interactions. Configuration Mixing Although the was widely used, many authors tried to use the interaction determined from nucleon-nucleon scattering and properties of the deuteron. Such interactions were considered to be more realistic than the schematic A nice example of the approach, in which a central interaction with a Gaussian potential is used, was offered by a long paper of Kurath (1953). It was based on his Ph.D thesis and thus contained also some previous results. It dealt with nuclei in which valence nucleons occupy the and orbits. The central potential of his two nucleon interaction is Gaussian and single nucleon wave functions are of a harmonic oscillator. He considered several exchange mixtures and particularly the operator . The results presented for 3 identical nucleons are the same as obtained by himself and by others earlier. Kurath also calculates binding energies in the shell. For isospins smaller than there are several states with odd, and J = 0, even. In such cases, he used some approximation for ground states which he believed is close to the and states of Flowers. Without stating whether it is obtained theoretically or numerically, he wrote an expression for the interaction energy. It effectively contains a linear term, a quadratic term in as well as a pairing term and a symmetry energy term. It has 3 coefficients “which depend on the interaction potential used and the nuclear size”. Together with single nucleon energies, Coulomb energies and the binding energy, that expression should be equal to binding energies

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in the shell. The values of the coefficients calculated from the assumed interaction do not reproduce the measured binding energies. The short range potential seems to be better than one with a longer range. If, however, a smooth function of mass number, roughly proportional to is added, good agreement is obtained. This term is taken to be the sum of single nucleon energies. The comparison of binding energies of a series of isobars agrees with the symmetry energy term T(T+1) much better than the T(T+4) term of the SU(4) model. In his paper, Kurath calculated ground state spins for various exchange mixtures and the dependence on the range of the interaction. He finds great sensitivity to these parameters. An interesting result is obtained by comparing particle-particle to particle-hole states. Acceptable interactions give different ground state spins in the two cases. This contradicts the rather schematic Nordheim rules. In the cases he considered, Kurath found that the ground state of a particle hole system has spin for any combination of intrinsic spin and orbital angular momentum. “This seems to be in good agreement with experiment and explains an outstanding exception to the Nordheim rule”. The spin 4 of the ground state should be compared with the spin 2 of where the Nordheim rule I agrees with the experiment. Meanwhile, more work was carried out on nuclei in the French, Halbert and Pandya (1955) carried out an intermediate coupling calculation for Although this is a special nucleus, where the shell model may not be applicable, they adopted for it the configuration. They used the two particle interaction of Rosenfeld

They did not specify the potential nor radial functions since they enter only through the radial integrals L and K. They fixed L by assuming L = 6K and left K as a parameter. In addition, they considered a spin-orbit term and the other parameter which they have is They found values of these coefficients which reproduce the positions of some levels but there was no complete information on spins and parities. It turned out later that some predicted levels were not observed even if the values of the parameters are appreciably changed. On the other hand, some levels are reasonably well accounted for. For instance, the ground state spin is for any value of Also levels with spins and are reasonably reproduced. This feature must be attributed to the available free parameters. As in many such calculations which were carried out later, the form of the interaction is taken from some fit to two nucleon data but its strength is varied as well as the strength of the spin-orbit interaction. The authors also calculated the magnetic and quadrupole moments and cross sections of some direct reactions.

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The J = 1 ground state spin of still presented a challenge. Tauber and Ta-You Wu (1954) wrote a paper on “Some and Nuclei in Intermediate Coupling.” To simplify the calculation they considered configurations with two valence nucleons and two hole configurations. They used harmonic oscillator wave functions which enabled them to examine several forms of interaction potentials with various ranges. They considered an exponential, a Yukawa and a Gaussian potentials. The exchange operator multiplying the potential was a combination of Majorana and Bartlett operators. They explained their choice of interaction by stating that “in the absence of sufficient knowledge, we shall assume that it is comparable with that between two nucleons unbound in a nucleus. On this assumption, the constants and and the depth of the potential have been obtained from the data on slow neutron scattering at low energies”. They included one-body spin-orbit interaction, which they vary over a wide range, and diagonalize the matrices for some levels of the and configurations. They looked for values of the various parameters for which the order and spacings of low lying levels agreed with experiment. They also calculated the ground state magnetic moments of and They concluded from the “somewhat exploratory nature” of their calculations that the order of levels depends on the choice of interaction and thus “it is necessary to study the situation for various types of nucleon-nucleon interactions.” They drew the conclusion that their “work also seems to suggest that a nucleus having only two nucleons outside closed shells is much closer to LS -coupling than to This is definitely demonstrated in the case of ”. Of course, the case of is not typical, all previous authors agreed that it is rather close to LS-coupling. Explicit wave functions were used by Redlich (1954) in his paper on “Interactions between Some Two-Nucleon Configurations”. He is considering the shell and some of his work could be called intermediate coupling in the Still, he calculated matrix elements between states with different numbers of and and also between states of the and configurations. He used harmonic oscillator wave functions and a Gaussian potential with various exchange operators. The depth and range of the potential “were taken from the data for the neutron-proton system in the triplet state”. The equivalent Yukawa potential is compared with the one given by Sachs. The range of the latter is close to the one in the paper but its depth is larger by factor two. Redlich believed that this may be due to his omission of tensor forces which are included in the potential in the book by Sachs. Redlich explained that “the results of the present calculations have at best qualitative significance.” Hence, he examined how robust were his results by varying the parameters of the interaction as well as that of the oscillator potential. He did not find significant changes. Redlich took the differences of

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single nucleon energies within the shell from experiment. The energy difference between the and orbits was not known and he adopted the value of about 10 MeV. With this value and his interaction he found that the largest admixture of state into the state occurs for the J = 0 state. There, its weight amounts to 3.4% which is rather small. On the other hand, Redlich found rather large admixtures within the shell, in particular for states with T = 0. The eigenstate for the ground state of is calculated to be

whereas the

ground state is

He used these wave functions to calculate the beta decay of to and found better agreement with the experimental rate than the one calculated within the configuration. The main features of these ground states are interesting and have been verified by subsequent detailed calculations. Redlich’s interaction may have been too strong but any mutual interaction within and configurations modifies the wave functions towards LS-coupling. This behaviour is more pronounced in states with lower isospin values. The overlap of the T = 1, J = 0 state with the main configuration is .895 and its weight is 80%. The weight of the state is less than 6%. If we look at the component of the wave function, these weights are 93% and 7% respectively whereas in the state they should be 60% and 40%. The overlap between the component of the ground state wave function and the wave function is .913 which should be compared with its overlap of .965 with the state of The situation is rather different in the T = 0 state. The overlap of the T = 0, J = 1 state with the main configuration is only .732 and the weight of the latter is less than 54%. Still, the state is different also from the state of LS-coupling. This is evident from the negligible weight of the configuration (less than 2% compared to 8% in the state). The overlap of the state with the state is .913, much higher than the overlap with the state. The same region was considered by Elliott and Flowers (1955). They wrote a long and detailed paper on “the structure of nuclei of mass 18 and 19”. They considered states of and nucleons and mixing of these configurations. They used harmonic oscillator wave functions whose extension is determined by the nuclear radius, and their interaction is central and given by

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This exchange character was suggested by Rosenfeld. The Yukawa potential seemed to be “the most reasonable” and its range is fixed by the pion mass of 276 times the electron mass. Still, the authors considered this interaction “as an effective force between particles in the nucleus which need not be that operating in the deuteron”. Thus, the strength of the force was chosen as a free parameter. Such explicit description of the interaction as an effective one is certainly a new development in the naive period. Still, a simple potential interaction is used which, as it turned out, too restrictive. The single nucleon energies of the and states were taken from experimental data of A = 17 nuclei. The detailed calculation was carried out in LS-coupling. The A = 18 nuclei were considered first, and it was found that there is considerable mixing of configurations with and nucleons. The authors state that there are large admixtures of states “in both the LS-coupling and the schemes.” This last statement is too sweeping since there are clear differences between the T = 0 and T = 1 states. For example, the ground state with T = 0, J = 1, has about 30% weight of the configuration. Within the configuration, however, it is almost a pure (96%) LS-coupling state with L = 0, S = 1 This state is rather different from the lowest state, the one with the configuration. Within the configuration the latter has the weight of only 48% while the state of the configuration has close to 47%. On the other hand, the T = 1 ground state of is much closer to the state of the which has the weight of 93% within the space of states (the state has 15% admixture of the configuration). As in the earlier calculation of Redlich, the higher T states are closer to and have smaller admixtures of other configurations. Elliott and Flowers considered next, nuclei with A = 19. In the value of yields a 1/2 ground state, a 5/2 level at 200 keV above it and a 3/2 level at 1.5 MeV excitation energy “in remarkable agreement with experiment”. The next level with even parity, is a 9/2 level, calculated at about 2 MeV. They identified it with the observed level at 2.8 MeV which turned out later to be correct. These results were very impressive and encouraging. In the situation was less impressive. The ground state was calculated to have spin 5/2 for any strength of the potential and two close levels with spins 1/2 and 3/2 should lie about .5 MeV above it. Experimentally, a level was found at 100 keV excitation energy and a level at 1.47 MeV. “This result is particularly disturbing in view of the excellent agreement obtained for The authors tried to explain this fact by pointing out that “the T = 3/2 levels of are much more dependent upon the exchange properties”. If the latter are varied, agreement with may probably be obtained with little change in

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The composition of the 3 nucleon configurations is interesting. The authors pointed out “that in the T = 1/2 levels of there is a large impurity in the lowest configuration. For T = 3/2, however, the impurity is much less and one might expect to get substantially the same results for in the scheme using and levels only.” This statement is more precise than the statement in the introduction that “the mode of coupling is found to be closer to LS-coupling than to ” Indeed, in the T = 1/2, J = 5/2 state, the dominant LS-coupling state has the weight of 55% which is 84% of the component of that state. The weight of admixed configurations is 35%. It is farther removed from the limit. Within the component of the state, the configuration has the weight of only 60% with 40% of the configuration. On the other hand, the T = 3/2, J = 5/2 state has only 11 % admixture of higher configurations. Within the component, the ground configuration of has the weight of 92% as compared to 8% of the configuration. Elliott and Flowers calculated in that paper also beta decay rates and magnetic moments of A = 18 and A = 19 nuclei. These nuclei were considered about the same time by Redlich (1955a, 1955b). His work follows his calculations on A = 18 nuclei mentioned above. He used harmonic oscillator wave functions and his interaction between nucleons is

The potential is a Gaussian and the interaction “accounts for all properties of the neutron-proton system in the triplet state at low energies except for the deuteron’s quadrupole moment”. Single nucleon energies of the and are taken from A = 17 nuclei and the scheme is used in the calculations. The comparison of calculated energies with experiment is more conservative and limited than that of Elliott and Flowers (1955). Redlich looked only at levels whose spins, or at least parities were definitely established. He obtained the correct ground states spins of and but the energy difference between them was off by 2 Mev, “as has recently been emphasized” (by Elliott and Flowers). The only other state that he considered is the first excited state of whose spin 2 agrees with experiment. Its calculated position was about 4 MeV above the ground state whereas the measured value is 2 MeV. For A = 19 nuclei he calculated only energies of the two lowest states. For Redlich found the correct spins 1/2 for the ground state and 5/2 for the state above it. The calculated energy difference is 170 keV, in excellent agreement

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with the experimental value of 200 keV. For however, he encountered difficulties, like Elliott and Flowers. The spin 5/2 for the ground state was correctly obtained but the 1/2 level was calculated to lie .6 Mev as compared to the experimental value of 1.47 MeV. Redlich ignored the excited level at .1 MeV since its spin and parity were not definitely determined (it has spin ). In any case, he expected that the calculated 3/2 level should lie above the one with spin 1/2. The composition of eigenstates obtained in Redlich’s paper is similar to that obtained by Elliott and Flowers. Like them, he concluded “that is a better approximation than LS-coupling for the larger T values, here for T = 3/2 and 1.” He offered the explanation that “this is easily understood in terms of the greater space-symmetry of the states with lower T” which leads to larger matrix elements of the interaction. In that paper, Redlich also calculated magnetic moments and ft-values of beta decay.

3.7. The Beta Decay of In a long and detailed review article, Inglis (1953) describes “The Energy Levels and the Structure of Light Nuclei”. Most of the discussion is on nuclei but he also considered light nuclei like up to He also considered excitations from the into that shell as seen in In his discussion of the spectrum of the latter nucleus, he included a description of it in terms of “the alpha [particle] model as an alternate possibility and as an occasional admixture”. Inglis described the experimental situation and related theories. In the he presented results of intermediate coupling calculations which seem to give fair agreement with experiment. A special section is devoted to “beta-decay and the surprising longevity of Inglis considered in detail the interaction matrices of the T = 1, J = 0 ground state of and the ground state with T = 0, J = 1. He included central interactions as well as single nucleon spin-orbit terms. He demonstrated that the beta decay matrix-element cannot become vanishingly small for any values of the interaction parameters. There is a factor 10 of retardation since the two hole wave function is 90% a state (it is 74% in ) “which is only a very small part of the factor needed”. Inglis then considered the effect of tensor forces on the wave functions and matrix elements. Tensor forces are given by a potential function multiplied by the scalar product of two rank irreducible tensor operators. One tensor is obtained by coupling the spins of two interacting nucleons. The other tensor is formed from the relative coordinates of the two nucleons. A

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simple expression of tensor forces is

The only non-vanishing non-diagonal matrix element of tensor forces is between the and of the T = 0, J = 1 state. Inglis argued that since the beta decay matrix element depends only on the amplitudes of the and in that state, “this tensor matrix element does not affect this result”. It affects the detailed curves “but not the lack of cancellation in the ground configuration .” Inglis concluded that the source of cancellation of the matrix element must be some configuration mixing. He was not sure, however, how it could be calculated. He describes the decay as “what has probably been the most persistent puzzle in the study of beta-decay”. A possible solution of the puzzle within the configuration was suggested by Jancovici and Talmi (1954). They demonstrated that tensor forces may give rise to an “accidental cancellation” of the beta decay matrix element. That matrix element depends indeed only on the and amplitudes in the ground state but these are determined by diagonalization of the interaction matrix which includes the contribution of tensor forces. Within the the T = 0, J = 0 state can be written as

The

ground state wave function can be expressed as

The Gamow-Teller matrix element connecting these states is proportional to

Inglis pointed out that within the configuration, central forces and spinorbit interaction lead to equal signs of and The matrix element can vanish only if and have the same sign but with any central forces and spin-orbit interaction, and have opposite signs as proved by Inglis. This is the case even if a mutual spin-orbit interaction is included, since its elements vanish in the matrices of the states considered here. Jancovici and Talmi calculated the matrix element of the tensor interaction between the and states as well as the other non-vanishing element - the expectation value in the state. They used a central plus tensor force which was adjusted to fit the ground state data of the deuteron. Jancovici and Talmi used harmonic oscillator wave functions with which matrix elements of tensor forces can be simply evaluated.

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For a reasonable value of the spin-orbit interaction the Gamow-Teller matrix element vanishes. Values of the coefficients for which there is exact cancellation are

The overlap of the T = 1, J = 0 state with the wave function, is 0.996 whereas the corresponding number for the T = 0, J = 1 state is only 0.84. Using these coefficients, Jancovici and Talmi calculated the magnetic moment of ground state to be .38 nm, in good agreement with the measured value of .40 nm. The calculated quadrupole moment of is .011 barns in fair agreement with the measured value of .007 barns (at that time the quoted value was .01). An indication that in the decay there is an accidental cancellation is provided by the mirror decay of Also this transition is strongly attenuated but its matrix element is about 7 times bigger than that of the decay. Jancovici and Talmi attributed the difference to the Coulomb interaction of the protons in derived a simple expression for the matrix element and obtained a fair estimate of the experimental value. The tensor force used by Jancovici and Talmi was extremely large and leads to order of T = 0 levels, in contradiction to observation. Therefore, they point out that their calculation “should not be considered as giving a satisfactory explanation. It is, however, interesting to see that tensor forces can play an important role in a possible cancellation of the [Gamow-Teller] matrix element”. Indeed, this possibility was picked up by several authors who managed to obtain better agreement with experiment without those difficulties. Similar difficulties were encountered by Adkins and Brennan (1955) who used a potential of a form suggested by analysis of the two-body problem. The radial dependence of that potential was a Yukawa shape and the radial functions of the nucleons were taken to be due to a harmonic oscillator potential. They considered two nucleons and two nucleon holes in the i.e., and To obtain a good fit to the data, they needed a strong tensor force. In however, imposing the conditions of vanishing matrix element of the beta decay of led to severe difficulties. They imposed other conditions which follow from measured observables and concluded that “for reasonable values of the ranges and depth of the potential, the numerical values of the spinorbit parameter and of the tensor parameter which have been found necessary ... are so large as to make difficult any physical interpretation of the results.” Sherr, Gerhart, Horie, and Hornyak (1955) made careful measurements of the beta decay and the life time of the J = 1, T= 0 level of That level is the analog of the ground state and its M1 strength can be expressed in terms of the coefficients of the ground state wave functions of and They found empirically a set of coefficients for the and ground state

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wave functions, within the configuration, consistent with the experimental data, including the magnetic moment and quadrupole moment of Later, these coefficients were calculated by using reasonable strengths of the tensor forces. In a long and detailed paper, Elliott (1956) tried to reconcile the observed spectrum with the cancellation of the beta decay matrix element. He explains “that the trouble which Jancovici and Talmi experienced with the spectrum arose from the ratio of two tensor force radial integrals which they calculated. This ratio turned out to be very sensitive to the radial dependence of the interaction”. He treated this ratio as a parameter to be varied to give results consistent with the spectrum and cancellation. Elliott related the radial integrals of the central interaction by L = 6K and the spin-orbit coefficient as The exchange mixture of the central interaction was restricted to and in which case the exchange operator has the factor The tensor interaction is also multiplied by this factor. By imposing conditions on the spectrum and on cancellation, Elliott found an interaction which sufficiently satisfies them. To estimate the relative strengths of the central and tensor interactions, he adopted a Yukawa potential for both. With harmonic oscillator wave functions whose parameter he equated to the range of the potential, he found the coefficients of the potentials to be and The range of these potentials is rather large, 1.64 fm as compared to the one adopted by Jancovici and Talmi, 1.185 fm. The strengths may be compared with those deduced from low energy two-nucleon data and He explained that “although these are based on ranges different from those used here, the general agreement in sign and magnitude is encouraging”. Elliott then calculated the magnetic and quadrupole moments of as well as the M1 transitions from the two lowest excited states of In view of the uncertainties, the agreement with experiment was fair. Elliott concluded by “we have shown how the long of can be reconciled with the spectrum of by using a tensor force which turns out to be small compared with the the central and spin-orbit forces and in general agreement with that deduced from deuteron and triton data”. At about the same time of Elliott’s work, Visscher and Ferrell (1957) carried out similar calculations (they presented a contribution to an APS meeting in 1955 but their manuscript was revised and published in 1957). The introduction to their paper makes the impression that theirs is the first attempt to solve the problem. They discuss at great length the wave functions of deriving the expression for the Gamow-Teller matrix element. They repeat the argument of Inglis about the impossibility to obtain cancellation with one-body spin-orbit interaction and generalize it to other forms of such interaction. They

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mentioned an earlier paper by Schulten and Ferrell where a complicated spinorbit interaction was used and some cancellation obtained, obviously due to some “relatively small error”. In the present paper, the authors used oscillator wave functions and, in addition to a central and tensor interactions also an “interparticle spin-orbit force”. The latter interaction led to an effective one-body force, where the coefficient is a linear combination of harmonic oscillator integrals of the mutual spin-orbit potential. They took the value of from experiment to be and from it determined the strength of the potential. It follows that the matrix elements of the mutual spin-orbit interaction, apart from their contribution to turn out to be small and are neglected. The tensor interaction which is obtained from one pion exchange is equal to –11.4 MeV, multiplying the product of the operator and the potential

with fm. The critical matrix element which connects the and states of the configuration is calculated to be –2.58 MeV. The authors were well aware of the hard core in the central potential which was discovered in high energy nucleon nucleon scattering. They were also aware of the Brueckner approach to this problem. “Thus the effective nucleon-nucleon interaction in the shell model bears not a priori simple relationship to the basic meson-theoretic interaction potential”. The authors then assumed that the effective central interaction is approximated by a Gaussian potential. The effective central strength for singlet states is determined by results of low energy scattering, it is and the range is 1.73 fm. From approximate calculation of binding energy they obtained and With these parameters they calculated L = -7.05 MeV and L /K = 6.29. Visscher and Ferrell used this effective interaction to construct the Hamiltonian sub-matrices for the various J and T states. The lowest J = 1, T = 0 eigenvalue agrees roughly with the energy of the ground state. About the other states they wrote: “all of the calculated levels fall within about a MeV of the experimental levels, which is a good agreement as can be expected from the present rather crude “first principles” approach”. The J = 1, T = 1 level was predicted about 7.5 MeV above the ground state and has not been found (there are no levels below 10 MeV). Also the 5.11 MeV level of which is assigned spin in the paper, turned out to be a one (a level was found at 7.03 MeV). The wave functions of the J = 1, T = 0 and J = 0, T = 1 states lead to some cancellation of the beta decay matrix element which is calculated to be 400 times the experimental value. The authors now adopt an “empirical approach” in which they modified some of the matrix elements. The diagonal matrix element is no longer

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zero but is equal to –4.91 MeV whereas the diagonal element is changed from –1.65 MeV to –3.83 MeV. The diagonal matrix element of the state is also changed from –9.23 MeV to –6.60 MeV. The resulting eigenstates now lead to the desired cancellation and the ground has the same eigenvalue as before. The calculations described above, show that the longevity of can be reproduced with shell model wave functions within the Difficulties encountered in those calculations may be attributed to insufficient knowledge of the two-nucleon interaction. A consistent shell model description of the complete which includes a highly retarded beta decay of is presented in Sect. 4.10.

3.8. Simple Potential Interactions. Ignoring the Hard Core As we saw, shell model theorists kept using in their calculations interactions which were determined from low energy scattering of free nucleons, including information from the bound state of the deuteron. It is very interesting that results of these calculations gave rough agreement with experimental data. By that time, there was ample evidence that at somewhat higher energies, the free nucleon interaction is rather different from the smooth Gaussian or Yukawa potentials. The scattering data indicated that at a short range, it has a strongly repulsive core. Matrix elements of such an interaction between shell model wave functions are meaningless. In low energy scattering, nucleon wave functions are adjusted to avoid the region of the hard core. Shell model wave functions, however, describe independent motion of nucleons and hence, must be modified if such an interaction is adopted. Many-body theorists like Brueckner, Eden and Jastrow made serious efforts to keep the single nucleon picture by suggesting that the free nucleon interaction should be renormalized into an effective interaction which may be sufficiently tame to be treated as a perturbation. The effective interaction could bear little resemblance to the one between free nucleons. Concrete attempts to calculate it were carried out in subsequent years. In 1955, however, the problem was not even mentioned by most authors. Perhaps it was hinted by Elliott and Flowers (1955) stating that they regard the interaction which they use “as an effective force between particles in the nucleus which need not at all be that operating in the deuteron”. A notable exception among shell model theorists were Levinson and Ford who were in 1955 in Bloomington in the company of Brueckner and Eden. In fact, in one of the seminal papers, Levinson collaborated with Brueckner on “approximate reduction of the many-body problem for strongly interacting particles to a problem of self-consistent fields”. Levinson and Ford wrote a series of 3 papers in which they applied the shell model to and In

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the first paper, Levinson and Ford (1955a) outlined the theory and developed methods of calculation. They summarized the approach that had been used by previous authors, stressing the role of configuration mixings. Among the questions that they would like to answer is: “Are the ‘interparticle forces’ that act between particles in the nucleus equivalent to the nuclear forces between two nucleons in the vacuum?”. This they claim has not been hitherto determined since “previous applications of such a model have utilized specific assumptions about the two-body forces which in general do not accord with current ideas about the nature of nuclear force as deduced from theory or from nucleon nucleon scattering experiments”. In view of the last statement it is rather surprising that while they realized that with repulsive cores “perturbation theory with independent particle wave functions is not appropriate”, the authors state that this “difficulty we do not strictly avoid, but rather ignore. The effect of a repulsive core on the radial parts of the wave function and short range correlations of nucleon positions was not taken into account.” They quote some “theoretical evidence that the independent particle picture is approximately correct at low energies in spite of these correlations”. In any case, Ford and Levinson believed that “the final test of this approximation must come through the success or failure of calculations which start with the independent particle assumption.” In the second paper, Ford and Levinson (1955) considered effects of coupling of individual nucleons to surface vibrations of the core. These vibrations are described by the collective model of Bohr and Motttelson and were studied by Ken Ford in his Ph.D thesis. In their third paper, Levinson and Ford (1955b) pointed out that some effects of this kind are included in single nucleon energies which are taken from experiment. Other effects which change the two nucleon interaction are also included since the potentials are fitted to yield the level spacings. In that paper levels of and were considered. The experimental information was then far from complete and the authors had to make guesses. It is instructive to see how justified their assignments turned out to be. First come the single nucleon levels of where the ground state is due to the neutron orbit. The 1.95 MeV state is taken to be the orbit and the level at 2.47 MeV to be the orbit (it turned out to have also spin ). “These levels bracket the 2.01 MeV level which from shell model considerations we take to be the state” (it turned out to be a level). “As a guess the 2.89 level is taken to be the state” (it has probable spin 7/2, a level was identified at 3.20 MeV). The position of the state was determined by various considerations to be at 5.89 MeV. Other single nucleon orbits were not considered. Next come the assignments of the levels of which the spin 0 ground state and the first excited level with spin 2 at 1.51 MeV were known. The next

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higher levels, measured at 1.95 and 2.29 MeV were not assigned spins. The authors argued that “any reasonable assumption on the interparticle forces together with any strength of surface coupling will give the level order 0,2,4,6. The third and fourth excited states of were accordingly assumed to have spins 4 and 6, The third and fourth levels in at energies 1.84 and 2.42 MeV, turned out later to have spins and Considering Levinson and Ford quote an experimental paper in which there are a 5/2 level .37 MeV above the 7/2 ground state, a 3/2 level at .6 MeV and a level at .81 MeV assigned spin 9/2 (probably the level at .99 MeV). These are the levels whose energies the authors aimed to calculate. They suggest a simple “test of the presence of configuration interaction”. For any two body interaction, “the relations between the levels of and should be simply given in terms of fractional parentage coefficients”. Assuming pure configurations they express the levels calculated from their empirical levels of and found that level spacings in the resulting spectrum are considerably bigger than measured ones although the order of levels is correct. “This proves that there exists no two-body potential which can predict the observed level spacings of both and and give negligible configuration interaction”. This lends support to their efforts to find the configuration interaction which is so needed. In calculating matrix elements, the authors made use of the similarity between the nodeless radial functions of and nucleons in an infinite square well potential, and assume that their radial integrals are equal. In the interaction between and nucleons, Slater radial integrals appear with odd values of The authors argue that the are smooth functions of and hence they “take the values of for to be given by linear interpolation or extrapolation of the for ”. With these assumptions, all matrix elements are given in terms of the four diagonal elements of the interaction in the J = 0, 2, 4, 6 of the configuration with S = 0 and L = J. They found that the main effect of mixing the and states was to lower the J = 0 ground state of These admixed states amount to 12% of the J = 0 state. In other states, their effect as well as their mixing amplitudes are negligible. There are, however, like in the J = 0 state, mixing of states with nucleons. Levinson and Ford considered next the effects of configurations which included nucleons. They argued that these effects are small and hence only estimated their influence on the calculated results. They use the experimental energies of states, according to their assignments, to determine these four diagonal elements mentioned above. They tried to obtain them from potentials of the form which have been used by Kurath. They

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“conclude that the effective nuclear potential in is indeed not well represented by a Gaussian constructed to fit the low-energy scattering data but is better represented by a Gaussian which is about one-half as deep and two times longer-ranged”. They investigate the shape dependence of matrix elements and conclude that it is “sufficient to raise the hope that study of nuclear energy levels may yield information on the shape of the potential”. The two nucleon matrix elements are then used to calculate the energy levels of They used an approximate way of diagonalizing the matrices for the various J values. The calculated binding energy of if single nucleon energies are taken from , is about .6 MeV smaller than the experimental one which the authors explained as due to a slight A dependence. Yet, “Spacings and spin assignments agree remarkably well with the experimental results”. Indeed, their calculated excitation energies of J = 5/2, J = 3/2 and J = 9/2 states are .38, .61 and .80 MeV above the J = 7/2 ground state as compared to their adopted experimental values of .37, .63 and .81 MeV. The authors modestly explained that “the almost exact agreement between experiment and theory is probably fortuitous” due to certain approximations and uncertainties in their work. They tried to analyze some of the latter and conclude “that the neglect of triplet forces in the calculation is a good approximation”. To gild the lily, Levinson and Ford calculated the magnetic moment of the ground state. In addition to the effect of the various admixed states, they considered also the (small) effect of the center of mass motion. The calculated value of the magnetic moment is –1.305 nm as compared with the measured value of –1.315 n.m. Indeed, “the agreement of the predicted and observed magnetic moment is nearly perfect”. In their conclusion the authors stated: “A remarkably accurate prediction of level spacings and magnetic moment of has been made on the basis of the shell model perturbed by central two-body forces”. This remarkable agreement was hailed as a great success of the shell model. It did not take long, however, before new experimental information, as pointed out above, demolished the basis of these calculations. In particular the levels which Ford and Levinson took to be J = 4 and J = 6 turned out to have spins 0 and 2. What is truly remarkable is how they managed to obtain such good agreement in spite of the wrong input. They made many assumptions and approximations but at that time it was difficult to see in their papers any parameter which could be tuned to yield the desired results. It seems that by spending many hours on a problem, where there are many unknowns, one knows instinctively which approximations should be made to obtain agreement with given data. A different and very interesting approach to states of Ca isotopes was presented by French and Raz (1956). They considered results of stripping reactions and tried to determine from them the shell model composition of

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various states. An important ingredient in their work was to determine relative cross sections for various from experimental data obtained for “a nearby ’closed shell plus one’ nucleus”. Their “results demonstrate that the coupling scheme is rather close to more precisely, they give small value for the amplitudes of certain minor components of the wave functions for low-lying odd-parity states” in and The authors then proceeded to determine an interaction which had such wave functions as eigenstates. This is unlike Levinson and Ford (1955b), “who focus their attention on level structures and magnetic moments but do not consider the deuteron reactions”. “As is customary” the authors “consider a central two-nucleon interaction for identical particles”. They studied various exchange admixtures and ranges for the potential which they took, following Kurath (1953) to be Gaussian. This potential was used with harmonic oscillator wave functions. Their search for the best interaction was hampered by the absence of experimental information on the splittings between the and orbits as well as between the and the ones. French and Raz find the interaction which gave the best fit to the spectrum to be

with fm. With this interaction and spin-orbit splittings which they assumed, wave functions of the lower states in and turned out to be rather pure states of configurations, The J = 0 ground state had the weight of 96% and the J = 2 state was even purer with 99.4%. In the J = 7/2 ground state, the J = 5/2 state at .37 MeV and the J = 3/2, .59 MeV state had the weights of 96.6%, 98.2% and 95.5% respectively, of the configuration. The calculated positions of the 5/2 and 3/2 levels agree with experiment. The authors predicted levels with spins 9/2, 11/2 and 15/2, below 2 MeV. These predictions are lower by about .5-.7 Mev than the experimental positions which were measured later. “The agreement with the the spectrum is mediocre”. The J = 2 level was calculated to be around 1 MeV, .5 MeV too low. The calculated positions of the J = 4 and J = 6 levels are almost degenerate at about 1.6 MeV which is lower than the 1.8 and 2.4 MeV levels which were expected to have spins 2 and 4. The authors mentioned that in the cases of and “the ground state, in particular, may be depressed by the inclusion of zero-coupled pairs from higher singleparticle levels”. They referred at this point to Levinson and Ford (1955b)... The new information about nucleon-nucleon scattering was not taken into account by nuclear theorists for some time. A comprehensive paper by Kurath (1956) deals with “Intermediate Coupling in the ”. He was still using a Gaussian central potential with a “mixture of 0.8 space exchange and

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0.2 spin exchange”. In addition, he included a one-body spin-orbit term. Kurath was, however, careful not to mention low energy nucleon scattering as justification for his choice of interaction. He was using harmonic oscillator wave functions for the and studied the dependence of the radial integrals L and K as a function of the ratio of the oscillator constant to the range of the potential. Whereas Inglis used in several cases interpolation between LS -coupling and limits, Kurath used a computer to diagonalize the various energy matrices. In this paper the complete was reviewed which is a very formidable task. In view of the rather meagre available measurements of levels and spinparity assignments, no definite conclusions on the success of this attempt could be reached. Still, it was very impressive that the calculated levels agreed in many cases with the experimental ones. In particular, T = 0 and T = 1 calculated levels in A = 10 nuclei, with reasonable values of the interaction parameters, agreed rather well with low lying experimental levels. There was also fair agreement between calculated and experimental magnetic moments. Kurath remarked that in view of the complexity of 6 nucleon configurations “it is encouraging, though somewhat surprising that it provides the best agreement of all the cases”. As in previous calculations, Kurath found that the best values of the spin-orbit interaction vary with mass number. At the end of the shell they are about twice their values at the beginning. In the conclusions he wrote: “The over-all picture for the shows that intermediate coupling model gives considerable improvement over the models of extremely weak or strong spin-orbit coupling. One can even begin to make quantitative comparisons with experiment using what is conceptually a very simple model.” In a later paper, Kurath (1957) used the wave functions from this calculation to obtain electromagnetic transitions between levels of nuclei. He explained that transition probabilities offer a more stringent test of the model than energies. Kurath included in his analysis in great detail, all the experimental information available at that time. He calculated strengths of M1 and E2 transitions and found reasonable agreement with experiment is several cases, specially for “transitions for which the M1 mode is the only one possible”. On the other hand, “it appears that the E2 transition strengths given by the model are of the right order of magnitude, but too weak”. Kurath believed “that some collective motion which would enhance the E2 transition strengths was needed, even for such light nuclei as these in the ”. Still, “one does not need nearly as much enhancement as is found for the strongly deformed nuclei”. He concludes that the “problem will be to see whether one can add collective effects sufficient to explain the E2 transitions without seriously disturbing the energy level schemes.” The relation between the shell model and the collective model descriptions

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of nuclear states has been a rather permanent topic of nuclear theory. There were some attempts to compare shell model wave functions obtained by diagonalization of the Hamiltonian sub-matrix, with states obtained by projection from Nilsson like single nucleon orbits. The latter were states with definite projection K on an axis fixed in space whose expansion in spherical orbits contained only those considered in the shell model calculation. An extreme example is offered by putting several nucleons in states of the single orbit and projecting states with definite T and J values. Lawson (1961) compared such wave functions with those obtained from shell model calculations and found striking similarities. A rather simple case for such analyses was offered by the shell model calculations of Kurath of mixed configurations of and nucleons. Kurath and Picman (1959) considered such wave functions in the where the deformed orbits were linear combinations of and orbits, like the state

In addition to normalization, there is only one parameter which fixes the coefficients of this state, the ratio between deformation and the coefficient of the spin-orbit interaction. There is one parameter which determines the intermediate coupling wave functions, the ratio of to one radial integral of the potential. The authors put several nucleons in the lowest Nilsson like orbits and project states with definite values of T and J which they compare to shell model wave functions. They found that for certain values of the two relevant parameters, these wave functions are very similar. “Wave functions generated from the lowest-energy Nilsson states have a remarkable similarity to the wave functions of the low energy states in the intermediate-coupling interaction approach”. The overlaps in some cases “are .995 or better”. The authors then went ahead and tried to see whether projected states might form rotational bands but the shell model space within the was too small for this study. In their conclusions, the authors made an interesting remark about their work. “The results demonstrate that the agreement between the two theoretical approaches goes beyond the explanation in terms of the SU(3) group, because the similarity in wave functions persists into regions where the super-multiplet classification breaks down and the partitions are well-mixed”. An earlier paper, barely mentioned by Kurath and Picman, was published by Redlich (1958) who considered wave functions of nuclei with A = 18 and A = 19. This is a more complicated case where the deformed orbits are linear combinations of single nucleon and orbits with given In his shell model calculations, Redlich used the interaction which he used in his earlier papers on these nuclei. It is a “central Gaussian Serber two-nucleon interaction” to which he added a single nucleon

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term “which raises the and levels by amounts observed in ”. For the deformed orbits, Redlich chose the expansion coefficients of the lowest K = 1/ 2 and K = 3 / 2 states to yield better similarity with the shell model wave functions. The chosen linear combinations are rather similar to those obtained from a deformed harmonic oscillator. “It is surprising that the” modified wave function for K = 1 / 2 “resembles most the” deformed oscillator function “for the lowest state of a prolate (cigar-shaped) spheroid, while an oblate spheroid would be expected at the beginning of a new shell”. The results of Redlich provide, in this case, a shell model basis of the Nilsson scheme. The agreement between wave functions projected from a deformed state and those calculated with a potential interaction is very good in general. The comparison with experiment is similar to the one in Redlich’s former calculations. The agreement with experiment for T = 1 / 2 states of is much better than for T = 3 / 2 states. As late as 1957 some nuclear theorists continued to use the “Rosenfeld mixture” with a Yukawa potential in calculations of nuclear levels. Elliott and Flowers (1957) wrote a long and detailed paper on “The odd-parity states of and ”. They quoted “the general success of detailed shell-model calculations in intermediate coupling” and “undertake a more ambitious program of calculations” involving excitations to a higher configuration in the next major shell. They used harmonic oscillator wave functions and the exchange mixture which according to Rosenfeld, should “automatically give saturation of nuclear binding energies”. The value of the coefficient of the Yukawa potential gives the correct binding energy of the deuteron but it is used as a parameter which is varied to give best results. They considered excited configurations obtained by raising one nucleon from the closed which is the ground slate of into the next All levels of these configurations have odd parity. The T = 1 states are the lowest ones in and whereas they, as well as the T = 0 states, are excited states in They determined the strength of the spin-orbit interaction, as well as the relative position of the orbit, from observed single nucleon states in and single hole states in (or ). Elliott and Flowers did not discuss explicitly the relation between their interaction and the one between free nucleons as well known already at that time. They used the convenient old prescription but they “have continued to treat the strength of the central interaction as a variable parameter, bearing in mind that the values of interest are likely to lie in the region of 30 to 50 MeV as they did in the earlier work”. The authors explained that using given wave functions and potential, all Slater integrals, 8 of which are needed in the present work, could be calculated. They explained, however, that their “rough procedure is probably adequate until such time as experimental data may be present

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in sufficient wealth to permit one to treat all Slater integrals as parameters to be adjusted by a least squares in the familiar manner of atomic spectroscopy. In the meantime one must rest content if the positions of nuclear levels can be calculated to within several hundred kilovolts of their observed positions”. The authors seemed to ignore, or to be unaware of a successful attempt, which was made about a year earlier, to determine, in a simple case, shell model matrix elements of the nuclear interaction from experimental data. This approach will be described in detail in the following. In their paper, Elliott and Flowers take into account the motion of the center of mass. If particles are bound in a potential fixed in space, their center of mass cannot be in a state with given linear momentum as it should for a free nucleus. If the central potential is that of a harmonic oscillator, the center of mass motion is decoupled from the relative motions of particles. Bethe and Rose (1937) showed that if the particles occupy the lowest oscillator shells then the center of mass is in its lowest bound state in all states of the configuration considered. Hence, as long as configurations in the valence major shell are considered, the center of mass motion contributes a constant term to the total calculated energy of the nucleus. Elliott and Skyrme (1955) had shown that some linear combinations of states of excited configurations may be due to excitations of the center of mass motion from its lowest state to higher states like Such spurious states, which do not describe real nuclear excitations, must be removed from the calculation. Elliott and Flowers identified the spurious state which is obtained by applying the center of mass coordinate to the ground state of It has S = 0, L = 1, T = 0 and the authors removed it from their calculation. The agreement with experiment was only semi-quantitative. “So far as the T = 0 are concerned, therefore, we can be sure only that the order of the lowest few odd parity states has been given correctly for all values of of interest”. The situation of the T = 1 levels was in better shape, “the calculations predict unequivocally a group of four close states of spins and ”. The order of those levels was not reproduced but the authors argued that they were “so close that their calculated order is not significant”. These levels span only 400 keV and the calculated value is 700 keV. The order of these T = 1 levels in is different which the authors correctly attribute to Coulomb effects. Other T = 1 levels were predicted to lie several MeV higher. This structure seems to provide evidence for where the lower levels are due to a hole coupled to a or nucleon. The authors do not list wave functions but state that “the actual situation is somewhat removed from strict ”. In addition to energies, Elliott and Flowers calculated the beta decay of electromagnetic transitions in and reduced widths for deuteron stripping reactions and even the photodisintegration of

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In the case of electric quadrupole transitions they found a certain enhancement which they attributed to polarization of the core by valence nucleons and holes (they considered it to be due to weak coupling to surface vibrations). This enhancement was treated semi-empirically by looking at single neutron transitions in They assumed for the neutrons an effective charge (only for quadrupole transitions) of about .5 of the proton charge and similarly an enhanced effective charge for the protons of about 1.5. The authors also discussed the possible “interpretation of the spectrum by means of the model”. It seemed that in spite of the great success of the shell model, the model did not fade away completely. The authors considered photodisintegration of and explained that it is dominated by a giant resonance which exhausts a large fraction of the dipole sum rule. In their model, a single nucleon dipole excitation of can reach five states in the excited configurations. They found that “much of the dipole sum is exhausted in the highest two states at 22.6 and 25.1 MeV. It is, therefore, very satisfying that” experimentalists “find the peak of the giant resonance in at 23.5 MeV”. This is certainly a very interesting result of the work of Elliott and Flowers. In their conclusion they write: “It appears to be possible to give a satisfactory account of the giant resonance phenomenon”. They explain that “Of the 12 odd-parity stales of below 13.5 MeV only one, the second state at 9.58 MeV, has been found definitely inconsistent with” the configurations adopted. The giant dipole resonance of so nicely reproduced by Elliott and Flowers, attracted the attention of Brown and Bolsterli (1959). They considered such giant resonances observed in many nuclei, and tried to explain their existence within the shell model. Like Elliott and Flowers, they took them to be linear combinations of states obtained by nucleons excited from the orbit into the orbit, the particle and hole coupled to J = 1 and negative parity. Brown and Bolsterli first considered proton excitations but then they also included isospin. The case of proton excitations was simpler and is outlined below. The authors constructed a schematic model which should explain the main features observed. First, the fact that the energy of the giant resonance is much higher than that of the individual particle-hole states, and second, that it contains most of the dipole strength. These features arise from diagonalization of the matrix of the interaction between the various particle-hole states. They took this interaction to be a zero range force and further assumed that the radial integrals, diagonal and non-diagonal, are all equal. Ignoring spin, they found in that case, the matrix elements

The interaction matrix in this case is separable (rank 1). If the various

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particle-hole states are assumed to have the same excitation energy E, there is only one non-zero eigenvalue, equal to the trace of the matrix given by

If the interaction is repulsive and there are several particle-hole states involved, the energy of this state could be rather high. The authors pointed out that this happens for the Rosenfeld exchange mixture. Thus, the uppermost state is pushed up by the coherent contributions of all particle-hole states. The situation is opposite to the case of the pairing interaction and indeed, the model was motivated by the degenerate case of the former. The corresponding eigenstate is a linear combination of all those states with coefficients proportional to The matrix elements of the electric dipole operator between the ground state and a particle-hole state are also proportional to All lower eigenstates are orthogonal to the uppermost one and hence, the matrix elements of the dipole operator between them and the ground state vanish. Thus, “through the particle-hole interaction, the lower levels have been denuded of their dipole transition strength and this has been transferred into the uppermost level”. In a later publication, Brown, Castilejo and Evans (1960) considered in detail the giant dipole states in nuclei. They included spins and used wave functions to calculate the dipole state in and Their approach “here is still somewhat simplified, and cannot be expected to produce quantitative results”. They were using zero-range interactions which, however, compare favourably with the results of Elliott and Flowers (1957) for They looked at several exchange mixtures and concluded that in all of them the particle-hole interactions in T = 1 states was repulsive. Like Elliott and Flowers, they found that in the dipole strength is concentrated in the upper two states and in it is in the upper 2 or 3 states, depending on the exchange mixture. The strength distribution among the upper states is rather sensitive to the exchange character of the interaction. During the years, other giant resonances have been found by experiment. Very extensive theoretical work followed these discoveries. There were attempts to use properties of these resonances to determine some quantitative features of the effective interaction in nuclei. These resonances are collective in the sense that they involve a large number of shell model states. It seems extremely difficult to extract from the experimental information about them, detailed knowledge on the effective interaction to be used in shell model calculations. Usually, a certain simple model with very few parameters was assumed and some of these parameters were determined. Thus, these giant resonances will not be further considered in this review.

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The same configurations were considered by Halbert and French (1957) who looked at the positive parity states of These they took as due to configurations in which there is a single or nucleon coupled to various states of the configuration. They also considered the configuration, but found that its contribution to low lying states is negligible. The nucleon-nucleon interaction which they used has the Rosenfeld exchange mixture and a Yukawa potential The overall coefficient of the potential is 37 MeV. The coefficients of the spin-orbit interaction were taken to be -4.22 MeV for the and -2.03 MeV for the The authors used harmonic oscillator wave functions and the LS-coupling scheme so, they could simply remove spurious states. When this interaction was used to calculate energies of single and nucleons outside the closed and orbits, the orbit was lower than the orbit by about 1 MeV. This is contrary to the experimental situation in where the position of the orbit, obtained from the ground state and the state, is higher by about 1 MeV from the .87 MeV state. Hence, the authors added to their Hamiltonian a single nucleon term which lowers the orbit by 2 MeV relative to the orbit. The authors seem to feel uneasy about this technique which “is identical with that, familiar in the theories, of assigning single-particle level differences. In any case, this is a very crude way of compensating for some of the inadequacies of the model”. The experimental situation was not completely established at the time and the authors had to go into a thorough analysis of the data to obtain tentative spin (and parity) assignments. To compare their results with the data, Halbert and French fixed the position of their lowest state at 5.28 MeV. They “do not calculate either the absolute binding energy or the energy separation between + and – levels”. For the levels below 9 MeV the “agreement in level positions is tolerable; in the worst case there is a 1.4 MeV discrepancy”. The agreement in widths of stripping reactions is “entirely satisfactory”. “Of particular interest is the nature of the 5.31 MeV level, which shows no stripping”. “Lane has suggested that it is the state” but the authors find that such states should be very much higher. They assign it to the configuration and the very small calculated “ width is due to a partial cancellation among contributing terms”. It is unfortunate that the authors used the LS-coupling scheme and hence, it was not clear what are the contributing terms in the simpler scheme. The situation of higher levels was less satisfactory, certainly in view of levels which were discovered subsequently by experiment. Should some of these be identified with calculated ones, the discrepancies in level positions could reach almost 4 MeV. A nice exception is the lowest J = 1/2 level with T = 3/2. Its calculated position is at 12.3 MeV as compared with the experimental value of 11.615 MeV. There

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was also good agreement between calculated and experimental width of the resonance in proton scattering experiments on There were some indications for independence of the interaction used in the shell model from the one which fits the deuteron data. Pinkston and Brennan (1958) tried to extract matrix elements of the interaction in the configuration from experimental data of They used the measured magnetic and quadrupole moments of the ground state to obtain “approximate values to the wave function coefficients”. They used energies of excited levels and assuming a central two body interaction and single nucleon spin-orbit term, could express the relevant matrix elements in terms of two of them. For the latter, they obtained only approximate values. The authors then tried to find some Yukawa type or Gaussian potentials which reproduce the matrix elements. Extremely weak tensor forces were needed to obtain the quadrupole moment which has a negligible effect on level energies. The interaction which they derived “has a central force strength which is larger than that expected from the two-nucleon analysis and has a different spin dependence”.

3.9. The Pb Region I Although much work was carried out on light and medium nuclei, nuclei near the doubly magic were the subject of intensive research. In 1954 excited states of the doubly magic nucleus were studied experimentally. The first few levels turned out to have odd parity and spins equal to 3, 5, 4 and 5. Tauber (1955) considered configurations in which a nucleon is excited from the core. He used the interaction

with a delta function potential a la Pryce but also a Gaussian potential with harmonic oscillator wave functions. He varied the ratio as well as the range of the Gaussian potential, to find the best fit to the observed levels. He correctly pointed out that nucleon-hole configurations have different order of levels than that two nucleon configurations. He found “that it is possible to obtain the energy levels of for certain configurations and interactions” but that there are too many unknown parameters to be determined. He did not consider configuration mixing which may be present. Pryce continued his interest in that region and Alburger and Pryce (1954) carried out a shell model study of the two hole configurations in Most of the paper is devoted to the experimental results and their analysis. The theoretical part is rather semi-quantitative. The zero order energies of the various configurations are determined by single neutron energies taken from the

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levels. Some of the states have zero interaction energy if the interaction a la Pryce is adopted. The interaction energies of other levels are calculated for the They are proportional to radial integrals whose values depend on the configuration. The latter integrals were not calculated since “the theory is at best approximate”. “Instead, rough values... have been estimated semi-empirically”. With these assumptions, positions of several low lying levels were determined. No configuration mixing was attempted and the wave functions with pure configurations were used to calculate various transitions. The authors concluded that “the agreement between the observed levels of and the prediction of the shell model could be considered very satisfactory”. It is amusing to see that “the shell model” in this case was restricted not only to the but also to other drastic approximations. The situation was not good for transition probabilities, specially E2 and E3. “Although the shell model appears to give reliable predictions concerning the energies of the levels and the selection rules for electron capture, it it unreliable about radiation intensities”. Strong electric transitions due to valence neutrons indicate “some kind of collective motion”. They did not yet use the notion of effective charge of neutrons due to polarization of the core. Some time later, True (1956) looked carefully at transitions between levels of with one valence hole, and whose levels he calculated a la Alburger and Pryce. In his paper, he tried to explain the strong electric transitions by coupling the valence neutron holes to surface vibrations of the core. He wrote: “If the surface coupling is sufficiently weak, it makes itself felt primarily only in electric quadrupole effects”. “Energy level spacings might, for example, be only slightly changed while E2 transition rates are greatly changed”. In the transition between the level to the level can be obtained by coupling to surface vibrations with the surface tension parameter C equal to 1100 MeV. In the transition between the level and the ground state implies the value 520 MeV for the surface tension parameter. The author stated that the difference in C for the two isotopes is real and “does not arise from the approximations made in the limiting case of weak coupling” with surface vibrations. In calculating the levels of True made the same approximations as Alburger and Pryce. He also ignored configuration mixings and adopted for the interaction the and the strengths adopted by those authors. Positions of some levels were calculated fairly well, in particular, an isomeric state with spin was calculated at 2.05 MeV above the ground state. Experimentally such a level was found at 2.19 MeV. The rather slow transition between the lowest and levels is attributed to their configurations differing by the orbits of 2 nucleons. The author concluded that “short-range two-body forces describe the interaction between the external nucleons fairly well”.

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A similar approach was used by Pryce (1956/57) to calculate the levels of He took single neutron hole energies from the observed levels of and used a zero range interaction. The two nucleon matrix elements were calculated by him in his first paper on He was aware of the need to include configuration mixing but, to take care of it, he was “making a rough estimate of the energy shifts... for a few of the more important levels and the rest have been corrected by guess-work and intuition”. Looking at his predictions, it is clear that his intuition was very successful. He predicted a level very close to the ground state which was later found at .002 MeV. Another predicted level with spin at about 1 MeV was found at 1.01 MeV above the ground state. A level predicted at about 2 MeV was later found at 2.2 MeV and he identified the .70 level to be the state rather than the which was experimentally favoured. He also suggested that the state should be isomeric which was later verified. The restriction to a zero range potential for Pb isotopes was removed by Kearsley (1957). She used instead, a Yukawa potential with the Rosenfeld exchange mixture

where

fm. The best fit to the level spacings was obtained with The single neutron hole energies were taken from the experimental level of and she used harmonic oscillator wave functions. The author must have been unaware of the many papers written about harmonic oscillator wave functions. Calculating matrix elements she carried out the transformation to center-of-mass and relative coordinates and arrived at the simple oscillator integrals of the potential. Kearsley is calculated effects of mixing of configurations due to the 5 lowest orbits by diagonalization of the Hamiltonian submatrices for each J value for values of up to 100 MeV. She obtained very good agreement with experimental level energies known then. Calculated energies of other levels agreed very well with later measurements. Some examples were excited levels predicted at 1.12 and 2.12 Mev, found at 1.165 and level calculated to be at 1.699 MeV was discovered at 1.704 MeV, and a second level predicted at 2.91 MeV and found at 2.940 MeV. The author found important effects of configuration mixing. In particular, the unperturbed state was shifted by .66 MeV due to interaction with higher configurations. This result is somewhat surprising since from binding energies of and it follows that the interaction energy in the ground state is only .63 MeV. This implies that the interaction energy of the two neutron holes is next to zero which is disturbing. This is perhaps due to the Rosenfeld exchange mixture which has a rather large repulsion in T = 1, S = 1 states.

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A finite-range potential was adopted also by True and Ford (1958). Their paper is long and detailed, and was based on the Ph.D thesis of True whose advisor was Ford. Much of it was devoted to discussion of the experimental data, level energies and transition rates. Instead of the highly simplified calculations with a zero range force, which “yielded qualitative agreement with experimental energy levels”, they used a finite-range potential. The authors were well aware of the fact that the “introduction of a repulsive core in the two-body force leads to practical difficulties in a shell-model calculation which have not yet been surmounted”. Still, they believed that for nucleons within the nucleus, where energies are low enough, “the repulsive core is not playing a dominant role”. Hence, they thought that using a potential without a hard core “is probably not a serious source of error in the calculation”. True and Ford adopted an interaction which vanishes for triplet-odd (space antisymmetric) states (“Serber force”). They examined the effects of adding a triplet-odd interaction and found them very small. They used harmonic oscillator wave functions and a Gaussian potential

This potential had “the same effective range and strength as for the lowenergy two-body system”. Configuration mixing was carried out for states up to 3.4 MeV. The authors found that the “agreement between theory and experiment for energy level positions is generally excellent”. They added that “the potential strength which fits two body scattering and the potential strength acting between extra nucleons in the shell model (in ) are nearly equal”. Indeed, the agreement was very impressive even for higher levels which were not measured then. A nice example is the and levels which were not reported by Kearsley. True and Ford predict their positions at 2.98 and 2.63 MeV respectively, and they were later discovered at 2.955 and 2.658 MeV above the ground state. There are some exceptions where the agreement is not excellent but it is still fair. The authors compared some of their results with those obtained by Kearsley (1957) and found them to be very close in spite the different exchange mixture and radial dependence of the potential. They attributed the similar results to the fact “that the singlet forces in the two calculations are nearly equal in strength”. True and Ford used the wave functions obtained by the diagonalization to calculated rates of various transitions, which “are a much more sensitive test of the eigenfunctions”. They calculated gamma transition rates and cross sections for stripping reaction. Results of the latter indicate that in the ground state, the configuration has a larger amplitude than the calculated one. To obtain agreement for transition rates, the authors invoked surface vibrations which are weakly coupled to the motion of valence

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nucleons (one phonon formalism). “From the observed rate of the E2 transition between the first excited state and ground state in one may deduce a strength of particle-hole coupling or, approximately, a nuclear surface tension”. In the earlier paper of True (1956) an error of factor was made and only pure configurations were considered. With the wave functions which the authors determined, the surface tension parameter determined from turns out to be C=1140 MeV and the neutron effective charge These values agreed very well with those obtained from C=1160 MeV and with possible errors of 12%, Weak coupling to surface vibrations gave rise to an additional interaction between valence nucleons. A quadrupole phonon emitted by one nucleon which is absorbed by another nucleon, gives rise to a quadrupole-quadrupole interaction between the two. The strength of this interaction was determined by the strength of the coupling which, in this case, was determined by the observed transition rates. Adding this force, makes visible changes in calculated energies which may be corrected by reducing the strength of the original interaction. The authors reduced the strength to 75% of its original value and the agreement with experiment was even somewhat improved. The authors concluded by stating that “there is no evidence for manybody forces”.

4. EFFECTIVE INTERACTIONS FROM EXPERIMENTAL NUCLEAR ENERGIES As mentioned above, new information about the interaction of free nucleons shattered the basis of using interactions extracted from low energy scattering experiments. It turned out that the free nucleon interaction contains a very repulsive, or even a hard core at very short distances. Beyond it, there is very strong attractive potential. The whole idea of nuclear saturation being due to exchange forces had to be strongly revised if not completely discarded. The nature of the free nucleon interaction emphasized the problem of reconciling it with the success of the shell model. It was clear that such an interaction cannot be treated in the usual perturbation theory. Some of its matrix elements between shell model wave functions are very large if not infinite. Brueckner, Eden and others looked into this problem and showed how under certain conditions, shell model wave functions may be used for calculation of energies and other observables, provided the interaction and other operators are strongly renormalized. The effect of the short range correlations which must exist in the real wave functions should be transformed onto the Hamiltonian. The renormalized nuclear interaction may become this way rather tame and could be used with shell model wave functions describing the motion of independent

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nucleons. The trouble was and still is, that there is no simple way to derive the renormalized or effective interaction from the one between free nucleons. It is indeed a very complicated many body problem. There has been a very large effort to solve it but so far no reliable method has been developed. Moreover, the very existence of the shell model as a good approximation has not been demonstrated.

These facts emphasized the exploratory nature of shell model calculations which were carried out using simple potentials. Those potentials may have been a reasonable approximation of the renormalized interaction. Yet, disagreement between calculations and experiment, need no longer be blamed on configuration mixing nor deviations from pure The disagreement could be due to the use of inadequate interactions. The problem facing nuclear theorists was how to use in a reliable manner the shell model for calculation of energies, wave functions and other observables. How to understand nuclear structure before a reliable solution of the nuclear many body problem will be reached. A possible way was found by a simple example which demonstrated how it is possible to proceed. It took sometime until this method became generally accepted and finally it became the standard approach to shell model calculations. This method is based on determination of matrix elements of the effective interaction from experimental data in a consistent way.

The first successful example will be described below but it is interesting that about the same time Ford and Levinson tried this approach in their work on Ca isotopes described above. They took two body matrix elements (or rather their differences) from the levels which they considered to be due to the configuration and calculated the resulting spectrum. They found disagreement with the levels and deduced the presence of large configuration mixing. As will be shown later, had they adopted the correct levels, the spectrum would have agreed with the calculation. At around the same time, Meshkov and Ufford (1956) tried to use the observed levels in order to calculate the spectrum. Since they considered the complete wave functions, there were many matrix elements of the interaction and not enough experimental data to determine them. Thus, they could not reach any conclusions. Actually, this method has been the standard approach in atomic spectroscopy where the electrostatic interaction between electrons is well known. The lack of detailed knowledge of the electron radial functions and possible effects of configuration mixing, do not allow reliable calculation of matrix elements.

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4.1. Simple Configurations The simple case where this method turned out to be successful is due to the measured spectrum of There are 4 low lying levels with odd parity and spins J = 4 (g.s.), J = 3 (at .03 MeV), J = 2 (.80 MeV), and J = 5 (.89 MeV). The higher levels are above 1.6 MeV. Goldstein and Talmi believed that these data indicated a simple configuration with one neutron in the orbit and one proton missing from the closed orbit. Also the magnetic moment of the ground state, fairly agreed with this configuration assignment, as mentioned above. Starting from these energies (or rather energy differences), energies of other states of configurations involving these orbits could be calculated and compared with experiment. Among those states, levels of were easy to calculate since there are only 4 states, with spins 2,3,4,5, in the configuration. Their energies, calculated simply by using coefficients of fractional parentage, are linear combinations of the energies and vice versa. The calculated levels were J = 2 (g.s), J = 5 (.70 MeV), J = 3 (.75 MeV), and J = 4 (1.32 MeV). The ground state spin of was known to be J = 2 which agreed with the calculation. The excited states quoted at that time, however, at 1.00 and 1.92 MeV, did not match the calculated ones. This did not come as a surprise to Goldstein and Talmi since such disagreement was rather common in simple calculations. If there was another level, still unobserved, at around 1 MeV, the quoted levels would turn out to be about 50% higher than the calculated ones. Several reasons could be suggested for this disagreement. In the simple calculation it was assumed that the interaction is a two body operator and three-body, four-body, ... interactions have negligible effect. This has not been clear from many body theory where such terms arise even if one starts from a two-body interaction. It is also assumed that matrix elements of the interaction do not change appreciably when going from to This point was not clear and also Meshkov and Ufford (1956) were worried about it. Most drastic, however, was the assumption of pure configurations and no configuration mixings. Naturally, those calculations were not published. Several months later, new experimental results on the levels were published and the correct levels agreed remarkably well with the calculated ones. At .67 MeV a J = 5, odd parity level was found and two other levels were measured at .76 and 1.31 MeV. The good agreement meant that the description of and was a very good approximation. This was particularly true of the J = 3/2 ground state of which should be described as a single proton outside the closed and proton orbits and the closed neutron shell. It also indicated that two-body interaction was adequate and that its matrix elements do not change appreciably when going

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from one nucleus to the next one. Some indication that radial functions may be constant for a given orbit was found in the calculation of Coulomb energies by Carlson and Talmi (1954). A short time after the Goldstein and Talmi (1956) paper, Pandya (1956) independently calculated in the same way the spectrum. He discovered a formula relating the particle-hole levels to those of the particle-particle configuration

Pandya applied his formula to the case. Being without much experience, it seems that he was not at all surprised by the good agreement he obtained. It is worth while to point out that in this calculation no assumptions were made on the two body rotationally invariant interaction nor on the radial functions of the nucleons. The interaction might be central or non-central with mutual spin-orbit interactions and tensor forces. It could be a complicated nonlocal interaction, like the one which would be obtained by solving the nuclear many body problem. Only a small number of matrix elements were determined in this way. They should be considered as targets for calculation by many body theory of the effective interaction from the one between free nucleons. The agreement between calculated and experimental energy levels of demonstrated the consistency of the procedure adopted. Six level spacings in and were determined by three spacings. It lent confidence in the matrix elements of the effective interaction between a proton and a neutron. The quantitative agreement was a severe test for these matrix elements and the consistency of the configuration assignment. These matrix elements could be, and have been, used in calculating levels of other nuclei where there are configurations with protons and neutrons. Several important conclusions could be drawn from the success of the shell model in the case described above. It was demonstrated that energy levels could be consistently calculated by using matrix elements of the effective interaction determined from experimental data. These matrix elements did not seem to change appreciably when going from one nucleus to nearby ones. It was clearly shown that may be a very good scheme in certain cases. Most important, the good agreement was obtained by adopting a twobody effective interaction. This was a natural assumption as long as the interaction was believed to be the one between free nucleons. That interaction, however, after renormalization according to many body theory, could contain

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significant terms involving 3 nucleons, 4 nucleons, etc. There was no assurance that these terms are small and serious calculations even indicated that they may be important. Restriction to two-body effective interactions was the key to the success of this approach. The results quoted above and many others, some of which will be described below, supply strong evidence for the validity of this restriction. Talmi and Thieberger (1956) were encouraged by the successful calculation described above, and tried to apply this method to binding energies of nuclei. They looked at binding energies of nuclei where protons and neutrons occupy a certain They adopted the orbits prescribed by the shell model, up to the shell, and completely ignored configuration mixing. They considered the most general two body interaction which is rotationally invariant and also charge independent. They took the radial parts of the single nucleon wave functions to be the same for all nuclei where a is being filled. With these assumptions matrix elements of all states in configurations could be expressed as linear combinations of energies of states in the configuration. To calculate these matrix elements, coefficients of fractional parentage could be used. Talmi and Thieberger chose a different approach which is “an elegant method due Racah” (1952). If there are two body operators, with known eigenvalues, such that a linear combination of them could express the energies, that linear combination could be used to calculate all energies. It turns out that averages of energies of groups of states with the same seniority quantum numbers, are linear combinations of such averages of energies. These are with seniority and reduced isospin the average of states with even J > 0 values, and average of states with odd J values, Explicitly,

There are 3 simple two body operators which could reproduce these averages and hence, be used for calculating averages in configurations. If the seniority scheme is adopted, ground states of even-even nuclei are expected to have and those of odd-even nuclei to have These are the only states in the configuration with these seniority labels and hence, the linear combination of the 3 operators is exactly equal to their energies. The operators are a constant 1, and the pairing interaction. A linear combination of them, with coefficients and summed over all nucleon pairs in a stale of the configuration is given by

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The coefficients are linear combinations of and which can be determined by equating this expression with to with to and with to The binding energy of a nucleus with the configuration is equal to the binding energy of the nucleus with (the nucleus with closed shells only), to which times the single energy should be added. The latter is the expectation value of the kinetic energy of a and its interaction with nucleons in closed shells. With the values of and in ground states, the resulting binding energy is given by

where and is the largest integer not exceeding This is a pairing term which is equal to for even and to for odd, thus reproducing the odd-even variation in binding energies. The term with the coefficient is the symmetry energy term. It includes a linear term in T, whose coefficient is smaller than the one in the SU(4) scheme. The coefficients and are linear combinations of two body matrix elements expressed by

The authors used a simple expression, a la Carlson and Talmi (1954) for the electrostatic interaction of the protons which gave a good fit to measured binding energy differences between mirror nuclei where data were available. The parameters and C for each were determined by a best fit to the experimental binding energies. It was the first attempt to calculate binding energies within the shell model. In spite of the drastic simplifying assumptions, the agreement between calculated and experimental binding energies turned out to be fair. In the case “the agreement is poor for the first nuclei up to A = 7, indicating breakdown of the shell model (or only of )”. Binding energies are rather large and the fact that reasonable results were obtained, was of great interest. Some of the deviations were rather large, about 1 up to 2 MeV but the average deviations turned out to be only a fraction of an MeV. The authors comment that “in the middle of the shell all the experimental binding energies are bigger than those calculated whereas both in the beginning and the end the situation is reversed. This may be associated with the effects of deformation (or possibly of configuration interaction)”. With the coefficients

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which gave the best fit, the pairing terms turned out to be large and attractive. The coefficient of the symmetry energy term is also large and repulsive as should follow from general features of binding energies. The quadratic term is attractive but its coefficient is rather small (for ). Goldstein and Talmi (1957) considered nuclei with protons in the orbit and neutrons in the orbit. They used the differences of matrix elements determined by the excellent agreement between calculated and experimental levels to calculate binding energies and also levels of They also calculated binding energies in nuclei with protons and neutrons. The agreement between experimental and calculated binding energies was good, in spite of the simple configurations adopted. More interesting were the results on the spectrum. The authors included in the calculation not only the lowest state (J = 7 / 2) of the configuration but also the known excited levels, J = 5 / 2 at excitation of .37 MeV and J = 3 / 2 at .59 MeV. If only diagonal matrix elements of the interaction between the protons and neutrons in the J = 7 / 2 state are calculated, the lowest state of has J = 3. The state with J = 4 lies above it and only the next state has J = 2, contrary to the experimental situation. There are non-vanishing matrix elements of the proton-neutron interaction between the various states of obtained by coupling the various states of the neutron to the proton state. The non-diagonal elements of the interaction matrices are linear combinations of differences of matrix elements which were determined so well. Of the excited states, only the lowest J = 5 / 2 and J = 3 / 2 whose energies were taken from the spectrum, were included in the calculation. The ground state of emerging from diagonalization of the matrices turned out to have J = 2 as experimentally measured. In its wave function the probability of the neutrons to be in the J = 7 / 2 state is only 55% while it is 42% to be in the J = 5 / 2 state (3% in the J = 3 / 2 state). The other calculated low lying levels have spins J = 3, 4, 3 whose positions were in fair agreement with subsequent measurements. It is perhaps worth while to point out that this calculation does not involve configuration mixing. Diagonalization of the shell model Hamiltonian was carried out within states of the proton neutron configuration. The ground state spin of was reproduced without any adjustable parameters. It stressed again the importance of proton-neutron interaction and the consequence, that states of odd-odd nuclei cannot be considered as due to protons and neutrons in their lowest states. Shell model wave functions obtained from such calculations, were used by Oquidam and Jancovici (1959) to calculate rates of certain beta decays. These are first forbidden transitions (with change of parity) where spins of initial and

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final states differ by 2 units. They are characterized by a unique shape of the intensity of emitted electrons their kinetic energy. The squared matrix elements of the corresponding operator are inversely proportional to another function of the energy and lifetime, In the configurations considered, transitions in which a neutron turns into a protons must be of this kind. The rates of these first forbidden unique transitions are considerably smaller than those of other first forbidden decays (where spins change by 1 or 0). Hence, they can be observed only if the nuclear states involved have spins which differ by 2. In such cases, small amplitudes of admixed states with neutrons or protons are not expected to give large contributions to transition matrix elements. In 1959, the values of these unique transitions were not very well determined by experiment. Oquidam and Jancovici studied the experimental papers and had to calculate several of the values from the raw data. The considered by the authors, exhibit large fluctuations. The authors considered 7 such transitions of which only the one between the ground state of and the ground state of was a single nucleon transition. The other ones involved more complicated wave functions which they obtained by diagonalization of the Hamiltonian submatrix following Goldstein and Talmi (1957). Their calculations gave the squared matrix element of each transition as the matrix element squared of the single nucleon transition multiplied by a factor. They divided the squares of matrix elements, deduced from experiment, by the corresponding factors. The results should be equal and yield the square of the matrix element of the single nucleon transition. Before using their calculated wave functions, the authors tried to use simplified wave functions which had been used by various authors before. In those, the protons and neutrons are coupled to states of their lowest seniorities ( J = 0 for the even group and for the odd group). This yielded values which had also wild fluctuations. On the other hand, the authors found “that the constancy of” the result “is improved by using the more detailed wave functions” which they calculated. The authors noted that it “must also be remembered that the experimental values are sometimes rather imprecise, and on the other hand that the beta matrix elements are sensitive to small changes in the wave functions”. The first part of this note is no longer valid. In fact, quoted in the literature bear little resemblance to those listed in the paper. They still exhibit large fluctuations and using simplified wave functions gives little improvement. Use of the wave functions calculated by Oquidam and Jancovici leads to better agreement. The squares of single nucleon matrix elements obtained from the various transitions are roughly equal, the r.m.s. deviation is about 30%. The deviations could be attributed to the second part of the note - the sensitivity of transition rates to small changes in the wave functions. The authors tried to compare the single nucleon matrix element obtained

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from their analysis with the one calculated by using wave functions of a harmonic oscillator or infinite square well. They found that the effective matrix element is much smaller than the calculated one. It is not clear to what extent this could be due to a smaller overlap between the radial functions of and nucleons or to an essential renormalization of the original operator. The wave functions calculated by Goldstein and Talmi (1957) were used also to calculate magnetic moments of several K isotopes. Talmi and Unna (1960a) used “effective g-factors for protons and neutrons”, one for a proton and a second one for neutrons. These were determined by a least square fit to the measured ground states magnetic moments of 5 isotopes, to The agreement between calculated moments and experimental ones was very good. The effective moment of the proton turned out to be .335 nm as compared with the Schmidt value of .124 nm. It is rather close to the measured moment of .392 nm. “Similarly, the effective neutron magnetic moment is –1.569 nm, whereas the Schmidt value is –1.913 nm”. This predicted value was rather close to the magnetic moment of which was measured later,–1.594 nm. The authors did not mention the fact that the magnetic moment of the ground state is –1.318 nm, farther removed from the Schmidt value. This deviation could well be due to an Arima-Horie type small admixture of a configuration with a neutron. In the calculation, however, only pure configurations were included. Such admixtures would not change the calculated moment of and perhaps also the moment of They would change the calculated value of the ground state where a pure was assumed. Perhaps the change would be small and may even further improve the good agreement already obtained. In the fall of 1956 Talmi gave a talk in Berkeley on determination of the effective interaction from measured energies of nuclear states. A few months later, Lawson and Uretsky (1957) “have used this technique to make spin assignments for the excited states of nuclei with protons and 28 (closed shell) neutrons”. They took the matrix elements (actually, differences of these) from the measured levels of by assuming “spin assignments and for the second and fourth excited states”. The adopted (“theoretical”) energies of the levels with J = 2, 4, 6, were taken to be at 1.41, 2.54 and 3.16 MeV. Using these, they calculated the levels of configurations and also the low lying, levels of configurations. Spacings of the latter should be simply equal to those in the configuration. They compared their predictions with available data in as well as in and The agreement between calculated and experimental levels was good. In particular, the prediction of a low lying J = 5/2 state in configuration from the two nucleon spectrum solved a long standing problem of

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the early days. These levels are the exceptions to Mayer’s coupling rules and as described above, were considered by several authors without much success. Here, their positions were predicted by measured two nucleon matrix elements. In the level at .32 MeV is rather close to the calculated position of .24 MeV. The positions of levels with and at .93, 1.61, 1.84, and 3.11 MeV, agree very well with with the calculated ones at .98, 1.74, 1.82, and 3.16 MeV respectively. The two-nucleon matrix elements of the effective interaction determined by Lawson and Uretzky were rather different from those of a short range potential. The zero range leads to spacings of the J = 2, 4, 6 levels much smaller than the 0-2 spacing. As a result, the J = 5/2 level is not sufficiently close to the J = 7/2 ground state. Use of the pairing interaction leads to even worse disagreement with experiment. The J = 2,4,... levels of the configuration (and levels in any configuration with even ) are all degenerate. The spacing between them and the J = 0 ground state is called the “energy gap”. As a result, in configurations, with odd of identical nucleons, the state and all other states are degenerate. Their position above the J = 0 ground state is equal to the energy gap multiplied by This factor is only 3/4 for and closer to 1 for higher values. Thus, a gap similar to the one in even-even nuclei should be also observed in odd-mass nuclei, contrary to experiment. The description by configurations is consistent and the actual values of the two-body matrix elements should eventually be derived by the nuclear many body theory. Talmi (1957) recognized the analogy between configurations of protons and those of neutrons in Ca isotopes. From this analogy followed that “in the levels should be the J = 0 ground state, the 2+ level at 1.53 MeV, the level at 2.75 (or 2.42) MeV with J = 4, and the level at 3.25 (or thereabout) with J = 6”. Corresponding levels were identified also in In and the situation was very similar to that in the corresponding proton configurations. The author then checked how binding energies of nuclei with these proton and neutron configurations agreed with this picture. Since the configurations in this case contain identical nucleons only, the formula for binding energies is a special case of the one used by Talmi and Thieberger (1956). It is also exact, since for identical nucleons, seniority is an exact quantum number. Binding energies, from which the binding energy of the nucleus with closed shells and no nucleons was subtracted, are given by

This is a remarkably simple expression which contains exactly a quadratic term in the number of particles and a pairing term. It shows the power and elegance of the seniority scheme. The odd-even variation of nuclear binding

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energies has a natural description in this expression. The single nucleon energy is C and the coefficient of the pairing term is as before. The coefficient is related to and by Using the available binding energies, values of these coefficients were determined by a least squares fit. In the binding energy formula these coefficients reproduced very well the measured energies. The predictions for nuclei not yet measured turned out to be in very good agreement with later measurements. The values of the coefficients which best reproduce the binding energies were determined to be

“The values of C are different since the closed shells are different in the two cases and C for the protons contains some Coulomb energy.” Binding energies were taken to be positive quantities and hence, the coefficient of the quadratic term is repulsive. It is larger in absolute value for the protons because “the proton’s two-body force contains the Coulomb force” and “also because of the probable different radial functions for protons and neutrons”. This comment is also true for the pairing term. Not quoted as another possible reason, is the possible difference in effective interaction due to the different closed shells. It is significant that the quadratic term is repulsive. This is consistent with the values obtained by Talmi and Thieberger (1956) since there, a is attractive but rather small whereas is large and repulsive, hence, is repulsive. A simple, perhaps surprising result is that, unlike some common statements, magic nuclei are not more strongly bound than their preceding even-even neighbours. They are more stable, simply because beyond them a new shell begins whose nucleons occupy higher, less bound, orbits. Indeed, beyond a magic number there is a large drop of separation energies. The and coefficients are linear combinations of two nucleon matrix elements. The values of however, can be determined only from binding energies. The values of can be compared with those obtained from energy level spacings of configurations. The comparison is rather good and specially with the value obtained from levels which have been measured right then. The value of in the measured position of the center-of-mass of the J = 2, 4, 6 levels above the ground state, is equal to 2.73 MeV, which should be compared to 8 × 3.12/9=2.77 MeV obtained from binding energies. Another important point which was not made in the paper, concerns possible effects due to polarization of the core. If the binding energy of the core, the closed shells nucleus, the single nucleon energies and matrix elements of the two nucleon interaction are all constant throughout the shell, then the binding energy formula with constant coefficients follows. Polarization of the core

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by valence nucleons, however, may lead to linear and quadratic changes of its binding energy. Such changes, as well as linear changes in single nucleon energies, will be absorbed into the C and coefficients and also into the pairing term. As long as these corrections are small perturbations, it is not possible to determine them from binding energies. About that time, Thieberger and de-Shalit (1957) applied the same approach and essentially the same formula, to binding energies of proton and neutron configurations in heavier nuclei. Some of the nuclei which they considered were real semi-magic nuclei, where there are identical valence nucleons outside closed shells. A nice example is offered by neutron configurations outside the closed shells of In this case, it seems that neutrons occupy the orbit and use of the formula is justified for two nucleon interactions which are diagonal in the seniority scheme. The agreement which they obtained in this case is indeed perfect but there were only 5 measured binding energies which were fitted by 3 parameters. Other semi-magic nuclei are Sn isotopes where in odd-even cases, ground state spins indicate filling of several The authors made some simplifying assumptions and managed to obtain also in this case, good agreement with the experimental energies. In the other cases, Thieberger and de-Shalit assume that in ground states, both protons and neutrons are in states with lowest seniorities. The results in these cases did not agree with experiment as in the case of semi-magic nuclei. The results were still impressive and the agreement between calculations and experiment seemed even better than in lighter nuclei where protons and neutrons occupy the same The approach described in this section was presented for the first time in an international conference in 1957. In September, 1957 the International Conference on Nuclear Structure was held at the Weizmann Institute of Science in Rehovot (1957), Israel. It was a forum in which many aspects of nuclear structure were presented during one week. There were traditional methods presented along with new approaches, some of which were still in embryonic form. Kurath and Flowers presented results of calculations with regular potentials and exchange operators. On the other hand, Eden talked about “Foundations of the Nuclear Shell Model”, presenting the many body aspects of the problem. In the same spirit, Brueckner talked about various topics in the nuclear many body problem. Talmi talked about “Shell Model Analysis of Nuclear Energies” introducing the determination of the effective interaction from measured binding and level energies. “Effects of Configuration Mixing on Electromagnetic Transitions” were described in a talk by de-Shalit. Skyrme described a “nuclear pseudo-potential”, a schematic interaction which, after a few years, was embraced by practitioners of Hartree-Fock theory. Pines talked about the analogy between nuclei and superconductors which he with Bohr and Mottelson were

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soon to introduce in detail. Also Mottelson, who talked about “The Theoretical Description of Nuclear Spectra” from the point of view of the collective model, referred to superconductivity in his talk. He also mentioned the model of Elliott which “clearly exhibits collective rotational spectra and is simple enough to handle”. Elliott’s SU (3) model was briefly presented in a session on “Group Theoretical Methods in Nuclear Spectroscopy”. Racah gave a talk on seniority and its applications, while Flowers’ talk was on “A Shell Model Interpretation of Collective Motion”. The detailed papers of Elliott on his SU(3) model were published in 1958. The analysis of binding energies of nuclei where the and orbits are filled, by Talmi and Thieberger (1956) and Goldstein and Talmi (1957), was taken up in a very detailed paper by Arima (1958). In addition to binding energies, he also looked at some excited states. His results on binding energies are essentially the same as the results of those authors. “The agreement is remarkably good”. He explained that binding energies did not determine all matrix elements and these could be only determined from excited states. Arima, however, “instead of using the excited states, these undetermined parameters are calculated from the two-body force required to give the best fit with the parameters already determined”. He used harmonic oscillator wave functions and Gaussian potentials for all interactions. These are a central interaction with the various exchange operators, isospin dependent tensor forces, as well as an isospin dependent mutual spin-orbit interaction. The “phenomenological spinorbit interaction” is defined by

In principle, it should reproduce the spin-orbit splitting observed in nuclei. The author found that his interaction was similar to the potentials used for the interaction between free nucleons. The matrix elements he obtained from it are in some contradiction to those which fit certain excited states. He believed that “in order to obtain the values of more experimental data are much needed”. The agreement between calculated level spacings and experimental ones was fair in some cases and less so in others. He concluded by pointing out that the agreement obtained for binding energies “indicates the adequacy of the shell model in describing the ground states of these nuclei. However, when the energies of the excited states are calculated, it seems frequently to be necessary to take into account the effect of configuration mixing.” Arima tried to explain this difference by noting that “states having the same parity and spin as the ground states usually have quite large excitation energies, so that in most ground states the effect of the configuration mixing on the binding energies can be neglected”. In other cases “there are states with the same parity and spin which compete and interact strongly with

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each other”. Another conclusion is that the determination of the phenomenological interaction “shows the necessity of the spin-orbit interaction and it has the right sign and intensity to produce the one-body spin-orbit forces needed in the shell model”.

4.2. Protons and Neutrons in Different Orbits. I. Unna and I. Talmi (1958) applied the approach described above to nuclei where according to the simple shell model, the orbit is being filled. The adoption of extreme to this region was motivated by the success of this description of nuclei. Extracting the effective interaction from experimental data in a consistent fashion, depends on having the number of matrix elements to be determined as small as possible. To achieve this goal it is necessary to start with the simplest possible shell model configuration. Since the energy is stationary at the correct eigenstates, the hope is to gain in this way good approximations to them. Binding energies of nuclei with configurations were rather well reproduced by Talmi and Thieberger (1956). The authors took to be a closed shells nucleus and valence nucleons to be in the or orbits. They considered first configurations with one nucleon, starting with and up to and Single neutron energy and single proton energy, as well as matrix elements between a and nucleons, are determined by a least squares fit. The latter were found to be equal to

With these values, calculated energies agreed rather well with the measured ones. It is interesting to see that the T = 0 matrix elements are attractive (they increase the binding energy and hence, taken to be positive) and large, whereas those with T = 1 are weak and repulsive on the average ((–.77+3×.01)/4=–.19). This average repulsion between identical nucleons in different orbits turned out to be a global phenomenon. Calculated energies of and with T = 1 in and as well as the T = 0 in

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agree well with experimental values. There is good agreement between calculated and measured energies of T = 1 levels with spins and in These same levels were considered by Elliott and Flowers (1957) but here they have a rather simple description. The same T = 1 levels in should lie at and above the ground state. They correspond very well to measured levels at 12.8 and 13.09 MeV respectively. This is no surprise since their positions in are simply determined by those in and the Coulomb energy. More interesting are the T = 0 levels in They are calculated to lie at and A T = 0, level was found experimentally at 10.96 MeV and the authors believed it to correspond to the calculated level at 11.89 MeV. This was also the conclusion of Flowers and Elliott who predicted it to lie at 1115 MeV excitation. The agreement is somewhat better for the predicted level at 10.08 MeV if it corresponds to the experimental level at 9.59 MeV. There is, however, another level at 7.12 MeV which Unna and Talmi “conclude that this low state of is due to another mode of excitation”. On the other hand, Elliott and Flowers believed that the 7.12 MeV level was the one due to excitation of a into the shell and “that the 9.59 MeV state comes from a more complicated configuration, such as...of triple excitation”. Elliott and Flowers, using LS- coupling states and harmonic oscillator wave functions were able to eliminate the spurious states. Unna and Talmi being unable to do it, calculated the amount of spurious states which might occur in their wave functions. They found it, where it appears, to be only a few percent which they hoped does not affect too much their results. A similar situation occurs also in and for levels, obtained by raising one nucleon into the orbit. To obtain their energies, a 2 by 2 matrix was diagonalized by the authors, and its eigenvalues yield states at 7.25 and 8.83 MeV above the ground state, 7.55 and 9.11 MeV in There is also a state which belongs to the same configuration whose calculated position is at 7.46 MeV in and 8.33 MeV in At that time, the experimental situation was not clear. There are known levels with these spins in and but there are large differences between calculated an measured energies. The most unsure assumption in the calculation was a pure configuration of the ground state (T = 0, J = 1) in contrast with the T = 1, J = 0 state. The authors were carried away and tried to calculate energies of states with several nucleons. To determine their mutual interaction, the authors had to assign configurations with these nucleons to certain states. Some of these assignments are oversimplified and invalidate the predictions. One interesting result which seems to be correct, is that the first excited state in a level at 6.05 MeV, could not be due to a two nucleon excitation and must have a large fraction of a 4 nucleon excitation. This observation was made several

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years earlier by Christy and Fowler (1954). Along with these calculations, Talmi and Unna (1960a) considered configurations in which a nucleon is raised into the orbit. In all these states there is no contribution from spurious states. The results, however, are very similar to those presented above. The results for these configurations were published in Annual Review of Nuclear Science. The matrix elements of the effective interaction between a and a nucleons were determined (with statistical errors)to be given, in MeV, by

These values show the same features as of the latter were quoted by Talmi and Unna (1960a), in MeV, as

interaction. The

These values differ somewhat from the ones given above, but the agreement with experimental data is practically the same. Also in the present case, the T = 0 interaction is attractive and large whereas the T = 1 one, between identical nucleons, is rather small and repulsive on the average (note that attraction has here a plus sign like the usual way for binding energies). Using the matrix elements above, the authors obtained very good agreement between calculated and experimental energies of states in oddeven nuclei and levels in even-even nuclei. Calculated positions of and with T = 0 in agreed very well with the experimental values 5.11 and 5.83 MeV. This is similar to the agreement between calculated and experimental levels with T = 0 and spins and The quoted values are 4.92 and 5.69 MeV experimentally as compared to the calculated 4.80 and 5.75 MeV. In the case of the T = 1 levels with spins and in and there was very good agreement between calculated and measured energies. The positions of these levels above the 2.31 MeV T = 1, J = 0 state in were lower than their positions in This effect is due to the Coulomb interaction. The corresponding shifts of the and levels were bigger which is

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consistent with their interpretation as due to a single nucleon. The same behaviour is found in the T = 1 quartet of states in and Their calculated energies were in very good agreement with measured ones and in they are shifted relative to Also here, the shift was larger for the and states in which one nucleon is in the orbit. The order of each pair of levels in and was the opposite to the order in and as follows from the matrix elements above. The interaction of a proton and a neutron in different orbits was equal to the sum of the T = 0 and T = 1 matrix elements divided by 2. These levels in and were considered by Elliott and Flowers (1957) who calculated their energies by using simple potential interactions. They had some difficulty with these levels with T = 0 in Also Talmi and Unna (1960a) did not get for these levels agreement with observed energy levels. In their review article, Talmi and Unna (1960a) describe in detail how matrix elements of the effective interaction can be consistently determined from experimental energies. In addition to the examples described above, they considered nuclei between and assuming that valence nucleons occupy the orbit. There are only 4 allowed states in the configuration, T = 1 states with J = 0, 2 and T = 0 states with J = 1, 3. Hence, the method due to Racah (1952) could be applied. Four independent simple operators, whose matrix elements can reproduce the energies of the 4 states, are used to calculate all energies. Thus the linear combination

where is the pairing interaction, may be used as the effective interaction for nucleons in the orbit. The eigenvalues of this interaction are directly obtained as

The coefficients are uniquely determined if this expression for is equated, with the appropriate quantum numbers to A direct consequence is that any two body interaction is diagonal in the seniority scheme for Energies of states are given by the expression above, to which single nucleon energies and the Coulomb interaction are added, i.e.

where and For binding energies, there are only 3 independent operators, as shown above. The formula in this case is exact

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since seniority and reduced isospin are exact quantum numbers. The agreement obtained for binding energies was very good. The situation for excited states, however, was not satisfactory. There were clear indications that some of the excited states could not belong to pure configurations. Perhaps ground states are less affected by mixing since states with the same spin and parity are usually higher than spacings between excited states. It is perhaps significant that in level is only .8 MeV above the ground state whereas in which was successfully taken to have a single proton, the is much higher, at 1.7 MeV. The is briefly and qualitatively described in the Talmi-Unna review article. It was explained that in the beginning of the shell, LS-coupling seems to be a good scheme whereas in heavier nuclei, becomes valid. This is particularly true for proton and neutron numbers beyond 6 where some nuclei, specially with higher isospins, seem to be well described by configurations. Nevertheless, they mentioned a very simplified calculation by Talmi and Unna intended to determine the ground state spin of In an ” it was “concluded that the asexperimental paper on “Beta Decay of signment for as expected from the shell model is possible but cannot be established firmly on the basis of the present evidence”. In a figure in that paper, the ground state is assigned spin in parentheses. In the summary, it is stated that “at present it seems at least possible that may be as expected by the shell model and also by the full individual-particle model in intermediate coupling”. The beta decays to the ground state and to the state at 2.14 MeV of are very weak, logft=6.77 and logft=6.63 respectively. If, however, the assignment was adopted, then “the transitions to the two lowest states of are allowed but strongly discouraged and it would be most interesting to know the prediction of the individual-particle model on this point”. The rates of these weak beta decays are typical of first forbidden transitions. They are consistent with spin of the ground state, a possibility which was somehow ruled out in the experimental paper. Talmi and Unna (1960b) tried to calculate the spin of the ground state by making use of measured energies in nearby nuclei. In the ground state has spin and a state lies at 3.09 MeV excitation energy. To simplify the configurations, thereby reducing the number of necessary matrix elements, they assumed that in the ground state there is one neutron outside the closed orbit and in the state, the valence neutron is in the orbit. To reach two protons should be removed leaving the other two protons in a J = 0 state which could couple to the valence neutron to form a state with spin 1/2. It was expected that the interaction of protons with a neutron was stronger than their interaction with a neutron. Hence,

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removal of two protons will lower the energy of the

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to the energy of the neutron. To calculate the magnitude of this effect it was necessary to determine the interaction of a proton with a neutron and with a neutron. The necessary information may be obtained from the energy levels of which are obtained from by removing one proton. Indeed, the four lowest states of have spins (g.s.), (.95 MeV), and These were interpreted as due to coupling of the proton hole with neutrons in the and orbits. The interaction between a and two coupled to J = 0 is given by

Hence, to obtain the difference of this expression for and we need to have only the difference in centers-of-mass of the matrix elements. The difference between centers-of-mass of the and pairs of levels is equal to the difference in single nucleon energies (3.09 MeV) minus one half of the difference of the expression above for the two competing orbits. Thus, we subtract from 3.09 MeV the spacing between the centers of mass in {(3×2.62+5×1.67)/8–(3× 0+5×.95)/8=1.43} and obtain 3.09–1.43=1.66 MeV as the difference between the two average interactions. When going to twice this energy should be subtracted from 3.09 MeV to obtain the difference between positions of the and the levels. Thus, we find 3.09–2×1.66=–.23 which means that the ground state has spin and a level should lie .23 MeV above it. The result was indeed very surprising, since it meant a change in order of orbits in different major shells. This change is due to properties of the proton-neutron effective interaction. Later, the spin was experimentally verified for the ground state and a level was found .32 MeV above it in very good agreement with the simple calculation. Using the same approach, Talmi and Unna (1960b) calculated the separation energy of the neutron in and obtained for it the value of .83 MeV, fairly close to the experimental value measured later of .50 MeV. An interesting point missed by the authors, was that a single neutron, without Coulomb and centrifugal barriers, with such a small separation energy has a highly extended radial function. could have been called a “halo nucleus” many years before it was found experimentally to be one. The negative parity states of mentioned above, as well as some positive parity states, were discussed later by True (1963). He followed Talmi and Unna (1960a) by assigning these states simple pure

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configurations. On the other hand, he followed Elliott and Flowers (1957) by carrying out “a conventional two-particle shell model calculation of taking to be “an inert core” with closed and orbits. He compared his results with those of Warburton and Pinkston (1960) who determined, on the basis of various experimental data, wave functions of these states. The two nucleons were assumed to occupy the and orbits. The single nucleon energies were taken from measured levels of and The two-nucleon interaction is “assumed to be a central interaction”. “It is desirable to choose the parameters used in any calculation from first principles as much as possible” but also “one must be guided by the parameters which have worked well in the past”. In view of the uncertainties, “the final justification is the comparison of the theoretical results with the experimental data”. The author used a Gaussian potential and harmonic oscillator wave functions for the nucleons. Thus, the interaction is

The actual values used were and which makes this force have a singlet-even strength the same as the one “used successfully in ” by True and Ford (1958). True tried several possible values for the parameter a and the oscillator constant. The best results were obtained for an oscillator constant” of about and an of about 1.6”. The agreement between calculated and experimental level positions was rather poor. Except for two levels which “are predominantly configurations” (the level at 3.95 MeV is not included since it is due to “core excitation”), “the calculated levels are in general about 1 MeV too low”. The author did not dare to change the effective interaction accordingly but tried to use the same interaction with different oscillator constants for and nucleons. “However, the improvement is not as good as one might expect”. The improvement was indeed barely perceptible. The calculated wave functions contain some configuration mixings but the dominant configurations, usually with weights above 90%, agree with results of Warburton and Pinkston (1960) and with the assignments of Talmi and Unna (1960a). In view of the results, the conclusion of the author that “it is heartening that the agreement between the calculated energies, spins, and parity, and those of the observed levels is so good”, seems somewhat surprizing. As mentioned above, Goldstein and Talmi (1957) calculated energies of nuclei with protons in the orbit and neutrons in the orbit. Matrix elements of the mutual interaction were taken from the striking success of the relation between the spectra of and These proton-neutron matrix elements are averages of the T = 0 and T = 1 matrix elements. Several years later, Erne (1966) considered “an arbitrary number of nucleons in the

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shell” and only one nucleon in the orbit. As long as nucleons are in the lowest orbits in the harmonic oscillator shells, no spurious states can arise. Once configurations excited across major shells are considered, some of the states may include components of spurious states. These are due to center-of-mass excitation from 1s to the 1p oscillator shell. This problem does not arise here, since the spurious states in this case, arise by acting on states of the ground configuration by the center-of-mass coordinate. Operating with cannot raise a nucleon to the orbit with To calculate energies and wave functions, 12 two-body matrix elements and 2 single nucleon energies are needed. “These parameters were obtained from a least-squares fit to 60 nuclear levels, including even-parity levels belonging to configurations in the shell only”. The matrix elements obtained in this way are listed with their statistical errors which seem reasonable. The T = 0 matrix elements of the effective interaction between and nucleons are all attractive. The corresponding T = 1 matrix elements are repulsive in J = 2, 4, and 5 states and the J = 3 matrix element is only marginally attractive, – .08±.20 MeV. The agreement with experiment was sporadic, good in some cases and very poor in others, including levels which were measured later. “The root mean square deviation of the 60 theoretical level energies with respect to the experimental energies amounts to 0.5 MeV”. It seems that the simple configurations considered by the author, were reasonable approximations for states with maximum isospin. Whenever there are both valence protons and neutrons, more orbits must be included in the calculation.

4.3.

The Zr Region I

The very good agreement obtained for binding energies of calcium isotopes and N = 28 nuclei, does not prove that the T = 1 effective interaction is diagonal in the seniority scheme. This is due to the fact that any two nucleon interaction in configurations is diagonal in the seniority scheme for To find out whether the effective interaction is diagonal in the seniority scheme, it is necessary to look at with The lowest of these is the orbit expected to be filled between nucleon numbers 40 and 50. Ford (1955) looked at levels of the nucleus and nearby nuclei. In also with a closed neutron shell, the ground state is consistent with a valence nucleon and the level at .91 MeV with a one. Ford realized that in the configuration is the lowest and the J = 0 of the one is 1.8 MeV above it before the two nucleon interaction is added. The diagonal matrix elements bring down both J = 0 states and a non-diagonal matrix element will mix them, thereby pushing them apart.

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Some qualitative arguments made him believe that the first excited state of should be a state. He predicted that it should be lower than the state at 2.32 MeV which he assigned to the configuration. A level was known at 2.19 MeV which Ford suggested to be the unperturbed J = 2 state of the configuration. Ford attributed the rather small spacing between it and the predicted state ( .5 MeV) to the upward shift of the state due to configuration mixing. Ford’s intuitive prediction of a first excited state in was firmly established by the experimental discovery of a level, 1.76 MeV above the ground state. This simple and yet exciting work was the beginning of a long effort to study the proton orbit in nuclei with N = 50. Bayman, Reiner and Sheline (1959) made a very detailed study of the energy levels and transitions between them for which there were experimental data. They followed Ford (1955) in taking the levels as due to the and configurations. All possible levels of these configurations were found with the exception of the level. They analyzed beta decays into the two states and also gamma transitions from the level leading to them. They used these data to determine the amplitudes of the two configurations in the ground state and the level at 1.76 MeV. The authors tried to reproduce the data by “using finite-range, spin-dependent, central forces”. Their two body interaction, between the two valence protons, is given in their notation by

The and are projection operators to singlet and triplet states and is a parameter which was adjusted (it is equal to 1/3 in the Rosenfeld mixture). They tried Yukawa and Gaussian potentials and obtained the best fit for a Gaussian potential with range of 2.4 fm. The value of the parameter was between 0 and –1/3 making the force in triplet states (space antisymmetric) vanish or weakly attractive. This potential also reproduced well the configuration mixing in the states which they took to be around 64% and 36% The agreement between calculated and experimental level energies was not very good, the constraints imposed by the assumed interaction were very clear. Some transition probabilities between various levels could not be reproduced within the assumed configurations. Effects of core polarization are sufficient to explain some of those but not all. This was a clear demonstration of the fact, met so often, that level energies which are stationary at the correct eigenstates may be well reproduced by an approximate calculation, while transitions are much more sensitive and may strongly depend on small admixtures to the wave functions.

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The opportunity to study the spectrum of the configuration attracted the attention of Talmi and Unna (1960c). They tried to calculate the spectrum of by using matrix elements of the effective interaction between valence protons determined from experimental data of and nearby nuclei. In the absence of sufficient data of proton configurations, they considered also energies of neutron hole levels assumed to have the same configurations. The latter are levels in Sr nuclei with one or two neutrons missing from the closed shells of N = 50. With this input, there are 9 experimental energies which should determine 6 matrix elements. The J = 2,4,6,8 levels in are taken to be the unperturbed ones of the configuration. The single proton energy difference is taken as a parameter to be determined and so is the corresponding difference between neutron hole states. The data include the experimental energy differences between the and levels in and The effective two body interaction was assumed to be the same for protons and for neutrons in the configurations considered. Differences due to different cores and the Coulomb interaction were ignored. Since only differences of level energies were considered, perhaps this is a fair approximation. An immediate conclusion was drawn from the energy differences of the levels with J = 2,4,6,8 and positive parity. The effective interaction is diagonal in the seniority scheme in configurations. The non-diagonal matrix element between the J = 9/2 states with seniorities and turned out to be only .03 MeV. All matrix elments which are non-diagonal in seniority are proportional to it. This value is very small compared to spacings between states with different seniorities which are of order 1 MeV. The probability of higher seniority states is then about and the shift due to such admixtures is about .001 MeV. As a result, spacings between the lowest J = 2, 4, 6, 8 states of the mixed and configurations, expected in should be equal to spacings of the levels in the configuration In the calculation, for the first time, a matrix element which gives rise to configuration mixing was determined from experimental energies. This is the one between the J = 0 states of the and configurations. In order to obtain energies of states of two and three valence nucleons it was necessary to diagonalize 2 × 2 matrices. The agreement between calculated and known experimental energies was very good, among them the level spacings in and The probabilities in the ground state of the and configurations were calculated to be 75% and 25% respectively. The best fit to the available data yielded also the position of the unperturbed levels of the configuration. These energies when used in the configuration, yielded a very low lying level with J = 7/2 whereas J = 5/2 and J = 3/2 lie considerably

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higher. These results should be compared with those obtained in the early days from a potential interaction. The situation is similar to the case with low lying J = 5/2 levels. The Talmi-Unna calculation yields a level in at .27 MeV compared to the .23 MeV experimental excitation energy. In a level was predicted at 1.70 MeV, slightly above an excited level at 1.48 MeV (the configuration lies above the one). Those levels were found later at 1.64 and 1.58 MeV respectively. Binding energies of unperturbed configurations, calculated from levels, exhibit the same features as those of configurations of identical nucleons. This paper was the first of a long series of papers, by several authors, on the structure of nuclei in the zirconium region. The nucleus has a closed neutron shell and a rather stable proton configuration, the lowest excited state lies 1.76 MeV above the ground state. The ground state of has spin taken to be due to a single neutron orbit. Other orbits lie considerably higher, at 1.21 MeV, at 1.88 MeV, at 2.04 MeV, and at 2.17 MeV. There is another state at 1.47 MeV excitation, which could be due to a neutron coupled to the first excited state of Results of some stripping reactions on Zr isotopes, with A between 91 and 96, indicated that ground states and some excited levels seem to be rather pure states. Talmi (1962a) looked at the available data which seemed to be sufficient to check the consistency of the simple description by configurations. In the three lowest excited states have spins (.93 MeV), and If the and states belong to the configuration, there should be such levels with identical spacings in the configuration expected in Such levels were really found in at .92 MeV and 1.47 MeV respectively. In the expected configuration, has only the ground state with J = 5/2, which agrees with the experimental spectrum. The orbit should be closed in and, indeed, it has a level at 1.59 MeV and the next is a level at 1.76 MeV, much higher than in its preceding even Zr isotopes. All interaction energies are linear combinations of the three interactions in states with J = 0, 2, 4. Hence, any linear combination of simple operators which will reproduce them may be used for calculating all interaction energies. A simple choice is which for given J and is given by

where the

single neutron energies C were added. Using the spacings of levels

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from or the spacings in the configuration in could be calculated. A level was predicted at .26 MeV above the ground state. A 3/2 level was found experimentally at .267 MeV excitation in The other level of the configuration has spin and is predicted to lie at 1.1 Mev but was not then, nor now, found experimentally. The other consistency checks concern just the binding energies for which the expression above is simplified, like in the case, to The best fit of this expression to 6 binding energies yielded the values of the 3 parameters. The agreement between calculated and experimental binding energies was very good. The coefficient of the quadratic term is small and repulsive, whereas the coefficient of the pairing term is large and attractive, Only the latter coefficient is related to level spacings. Multiplied by 6/7 it should be equal to the position of the center-ofmass of the J = 2 and J = 4 levels above the J = 0 ground state. In fact, (6/7)1.50=1.29 which agrees perfectly with the values 1.29 MeV from and 1.28 from the levels. A paper, on “Effective Interactions and Coupling Schemes in Nuclei”, was published by Talmi (1962b) in an issue of Reviews of Modern Physics in honor of Wigner on his 60th birthday. It contains an explanation for the need to determine matrix elements from experimental nuclear energies and how it can be carried our in a consistent way. It described some of the results obtained by using matrix elements of the effective interaction. It also contained description of some general features of the T = 1 and T = 0 parts of these effective interactions and the structure of nuclear states which follow from them. The T = 1 part of the effective interaction can be studied in configurations with valence identical nucleons. In a pure configuration, the information gleaned from binding energies and from energy levels, shows that the interaction in J = 0 states is strong and attractive. The average interaction in J >0 levels of the two nucleon interaction is rather small and repulsive. States in which the T = 1 interaction is diagonal are those with maximum pairing which are states of the seniority scheme. The pairing interaction, or the is a poor approximation of the detailed T = 1 effective interaction. The J > 0 levels are actually widely spaced which is consistent with the results from binding energies. The only condition is that the center-of-mass of these levels should be much higher than the J = 0 state. The observed spacings of these levels lead to the seniority state with to be rather close to the J = 0 ground state in odd configurations. Also the average interaction between identical nucleons in different orbits determined from experiment is repulsive. The repulsive part of the T = 1

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interaction follows directly from known properties of nuclear energies. Nuclei to which more and more identical nucleons are added, become less and less stable. This feature implies also that the average interaction between protons and neutrons should be attractive. This is verified in the shell model by the appearance of the symmetry energy term in binding energy formulae due to the seniority scheme, and its value where it is applicable. Also it was found that the average interaction between protons and neutrons in different orbits is attractive. The proton-neutron interaction creates the central potential in which nucleons move and has a strong influence on its shape. This was demonstrated by the calculation of the ground state of described above. The fact that the attractive T = 0 effective interaction also breaks seniority in a major way was not mentioned. The fact that it leads to special kind of configuration mixing which cannot be absorbed into the two nucleon interaction was described and demonstrated by a simple example. The possible admixtures into the J = 0 state, with are two nucleon excitations like the J = 0 state with The unperturbed position of such a state is twice the single nucleon spin-orbit splitting which is rather high compared with the non-diagonal matrix element. Effects of such, rather small admixtures may be absorbed into the effective interaction in the configuration. On the other hand, the T = 0, J = 1 state of the configuration may be strongly admixed with the T = 0, J = 1 state. Its unperturbed energy is only one single nucleon spin-orbit splitting and, with an S state interaction, it has much more interaction energy than the the lower state. Also the non-diagonal matrix element of such an interaction between these two J = 0 states is rather large. Effects due to such admixtures of states, obtained by single nucleon excitations, cannot be absorbed into the two nucleon effective interaction. The resulting wave functions may be obtained by putting nucleons in deformed single nucleon orbits, like States with definite values of T and J may be projected from such states where several nucleons occupy deformed orbits. Such states in the and their comparison with states calculated from a simple potential interaction were previously discussed by Kurath and Picman (1959). Even earlier, Redlich (1958) compared wave functions in the obtained from a potential interaction with states projected from deformed states which include not only and orbits but also the orbit. Talmi explained that as long as the admixtured are small “the deformed potential well is only a mathematical device for building wave functions”. If, however, the admixtures are large and more single nucleon (spherical) orbits are added, “some of the projected wave functions may be described as due to one definite intrinsic function in the deformed potential well”. Then “the deformed potential has a physical meaning and the eigenstates of the nucleus [may] form rotational

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bands”. Such single nucleon interactions cannot occur in states with maximum isospin, with J = 0 and indeed nuclei with identical valence nucleons are known to be spherical. It is the T = 0 part of the proton-neutron interaction which may lead to nuclear deformations. In the example given in the paper, there is an attractive interaction only in S states which is the same in singlet (T = 1) and triplet (T = 0) states. In states with maximum T it acts like the pairing interaction and leads to the seniority scheme. In states with lower isospins the very same interaction leads to deformed states. The coupling schemes of nuclear states may thus strongly depend on the value of the isospin. Another topic in this review concerned unfavoured beta decays. Blin-Stoyle and Caine (1957) showed how small deviation from pure could account for the observed rates of unfavoured beta transitions in configurations. Talmi attempted to show how the difference between favoured and unfavoured transitions in the SU (4) limit, could survive a strong spin-orbit interaction that leads to T = 3/2 states which are rather close to states of the configuration. Without spin-orbit interaction, the J = 5/2 ground state of for instance, would belong to the (3/2,1/2,–1/2) supermultiplet. The spin-orbit interaction admixes it with states of the (3/2,3/2,3/2) supermultiplet but cannot admix it with states of the (1/2,1/2,1/2) supermultiplet to which the J = 5/2 state belongs (it has only T = 1/2 states). Thus, the transition from the ground state to the latter should be still forbidden. The T = 1/2, J = 5/2 state could be expanded in wave functions

The decay considered, being a single nucleon transition, can proceed only to the first two states in the expansion. Thus, the transition rate vanishes even if the spin-orbit interaction makes the coefficients and of two nucleon and three nucleon excitations, much smaller than in the SU (4) scheme. If the admixture of the state obtained by single nucleon excitation is smaller than in the pure SU (4) case, the transition is expected to be allowed but still unfavoured.

4.4.

The

Shell

In 1962 the level structure of was studied experimentally. Two levels were discovered at 1.434 MeV and at 2.965 MeV. In addition to the level at 2.369 MeV, another level at 2.766 MeV was found. The level at 3.112 MeV was verified and several other levels were found at higher energies. Electromagnetic transitions between various levels were measured which offered the opportunity to check various selection rules which follow from seniority. In a paper dealing with these issues, Talmi (1962c) pointed

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out that if seniority states have energies taken from or from itself (assuming for the 1.434 MeV level and 2.766 MeV level), the calculated position of the level with seniority is 2.32 MeV, very close to the measured 2.369 MeV. In configuration of identical nucleons, any two nucleon interaction is diagonal in the seniority scheme. In the seniority scheme, matrix elements between states with the same seniority of an even rank tensor, like the quadrupole operator, vanish in the middle of the shell. Hence, no E2 transitions should take place between states with the same seniority in the configuration. In actual nuclei, the description of states in terms of pure configuration is not very accurate. Also three nucleon interactions could mix the two states in Measured transition rates may supply information on these points. The higher levels may be identified but there are many levels at those energies and comparison with experiment is doubtful. The experimental rates, if accurate, indicate considerable seniority admixtures in the two levels. These are based, however, on some small branching whose ratios may be difficult to measure accurately. Admixtures with other configurations, particularly with and orbits may well contribute to the various transitions.

A systematic study of the

with both protons and neutrons was

taken up by McCullen, Bayman and Zamick. They took two nucleon matrix elements from the configuration in wrote down the interaction matrices for the various states and diagonalized them. They compared the eigenvalues with the experimental data and used the resulting wave functions to calculate magnetic moments and beta decay probabilities. The results of the latter calculations, in which improvements over the single nucleon values was obtained, were published first by Bayman, McCullen and Zamick (1963). A long paper was published later by McCullen, Bayman and Zamick (1964). The authors realized that, unlike the proton or neutron configurations, the protonneutron could not be very pure. One test which they suggested was comparison of nuclei with and nuclei with Such “cross conjugate nuclei” should have identical level spacings but experimentally, this condition was not very well satisfied. The authors derived formulae to be used with their wave functions. Their effective interaction is charge independent but their states are constructed by coupling proton states with neutron states. There is no need to use the isospin formalism for the calculation of energies, magnetic moments or spectroscopic factors for stripping reactions. For beta decay, however, explicit isospin wave functions are necessary and to calculate ft-values the authors antisymmetrized their wave functions. The experimental situation in was not very clear at that time and some level assignments made by the authors were not accurate.

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The T = 1 levels were correctly identified by their analogues in Among the T = 0 levels, the only firm assignment was J = 7 for the .617 MeV level. The authors adopted J = 1 for a level which was observed at 1.035 MeV since they wanted “the spin 1 level as low as possible”. The 1.035 MeV level has meanwhile disappeared but the real J = 1 level was discovered even lower, at .611 MeV. The levels with spins 3 and 5 were taken by the authors to be at 2.25 and 1.96 MeV respectively. Such levels were later discovered at 1.491 and 1.511 MeV. The T = 1 levels with spins 2 and 4 are now identified at 1.586 and 2.82 MeV (the authors use 1.509 and 2.998 MeV). For the J = 6 level they assumed 3.4 MeV which is higher than in and but not much. McCullen, Bayman and Zamick (MBZ) noticed an interesting symmetry of their Hamiltonian submatrix. The two nucleon interaction is invariant under cross conjugation described above, where proton holes are replaced by neutrons and neutron holes by protons. The nucleus is self cross conjugate and hence, its eigenstates are either invariant under this operation or change their sign. Matrix elements of the interaction vanish between such even and odd states and “pairs of very close eigenvalues can occur, for example, E = 3.55 (odd) and 3.485(even) are the energies of the two lowest I = 6 states of Two J = 6 states were experimentally verified at 3.333 and 3.508 MeV above the ground state. The authors made a systematic comparison of their calculated energies with experiment. For nuclei with only valence protons or valence neutrons, the agreement was fair and similar to earlier results, since in them only T = 1 matrix elements occur. For other nuclei, “the theoretical predictions are less successful in the odd-even nuclei than in either the odd-odd or even-even” nuclei. It was generally better for high spin states than for or levels which may be affected by nearby configurations. In for example, the and levels are predicted to lie higher than 1.6 MeV but were found at .38 and .72 MeV respectively. The 9/2, 11/2, 15/2 levels predicted at 1.70, 1.28, 2.35 MeV were found (later) to be at 1.66, 1.24, 2.11 MeV respectively. In the 3/2 and 9/2 are shifted down more than the 5/2 and 11/2 levels, whereas in the 3/2 and 5/2 levels were much lower than the calculated ones and the 9/2, 11/2 levels are well reproduced. In a low lying 5/2 level is found and it is actually predicted to be the ground state with a low lying 7/2 level. There is, however, also a low lying 3/2 level (.037 MeV) which is predicted at about 1.7 MeV. In even-even nuclei, the agreement between theory and experiment is better. Calculated positions of the lowest J = 2 and J = 4 above the J = 0 ground state were higher than the experimental ones by about .2 MeV. In whose two J = 6 states were mentioned above, MBZ predicted a J = 3 level at 3.01 MeV. They wrote that “the chief failure of the theory in the even-

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even isotopes is the predicted level...for which there is no experimental evidence”. They gave several possible reasons why it may have escaped detection and indeed it was discovered later at 3.224 MeV. The odd-odd nuclei which MBZ consider have one odd proton (or hole) or one odd neutron hole, i.e., Sc isotopes and N = 27 nuclei. The ground state of is correctly predicted to have J = 2 and the low lying states with J = 6, 4, 3, 1 are fairly well reproduced, as are the higher 7 and 5 levels. The situation is similar in where J = 6 is predicted to be the ground state with a J = 4 level close to it. The experimental order is reversed, J = 4 is the ground state spin but a J = 6 level is only .05 MeV above it. No such inversion is observed in the cross conjugate with J = 6 ground state spin. Also other predicted levels were in fair agreement with the observed spectrum. In the cross conjugate to the agreement was worse. The ground state has spin 6 and the 2-6 spacing changes by .65 MeV, between the two nuclei. MBZ remark that “the magnetic moment of the state is in violent disagreement with calculation”. In calculating magnetic moments MBZ determined an effective moment for protons to be 5.25 nm and for neutrons –1.35 nm. With these moments they obtained good agreement with most measured moments. The calculated magnetic moment of the J = 6 ground state of 3.05 nm was in excellent agreement with the measured value of 3.06 nm. The calculated value for the J = 2 state, .01 nm seemed to be in “violent disagreement” with the quoted measured value of 1.02 nm. In later measurements, however, the value .0076 nm was obtained. In with the proton particle neutron hole configuration, the authors use the Pandya transformation from the levels to predict the spectrum. The statement “the energy agreement is excellent” is based on incorrect experimental levels. Actually it is not very good which is not surprizing in view of the wrong spin assignments in The authors presented calculated ft-values of several Gamow-Teller beta decays and compare them with measured ones. They determine the single nucleon transition strength from the decay of the J = 7 state in to the J = 6 state of They assumed that these levels were relatively pure. The ft-values calculated this way were in rough agreement with measured values. The authors point out that transition rates are rather sensitive to small changes in the wave functions. Several measured “hole-hole conjugate” transitions had rather different ft-values. An extreme example was offered by the transition between the J = 7/2 ground states of and It has logft value of 8.5 whereas the hole-hole conjugate transition between and has logft value of 5.7. The calculated value is 5.85 but a slight change in the wave functions may yield the experimental value, “the overlap between the energy eigenfunction and the wave function necessary to fit the experimental logft value is

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0.998”. McCullen, Bayman and Zamick presented in an appendix, wave functions of ground states. It is clear that the even group of identical nucleons is not in its lowest seniority J = 0 state, nor is the odd group in its J = 7/2 state. For instance, in the ground state wave function, the weight of the state is only 53%, of the is 29% while the total weight of all states with various amounts to 40%. The rest of the wave function is a linear combination of states with and various MBZ compared their wave functions to states with lowest seniority quantum numbers, for even-even and for oddeven nuclei. They mentioned that for identical 7/2 nucleons, any two body interaction is diagonal in the seniority scheme but this is no longer the case for protons and neutrons. They calculated the overlaps of their ground state wave functions with those of lowest seniority and found it to be considerably smaller than 1. The probabilities of lowest and states in their calculated wave functions ranged between 84% for through 88% in and up to 93% in and The seniority scheme is indeed broken by the effective T = 0 part of the effective interaction. MBZ made another comparison of their wave functions, to those obtained from a quadrupole-quadrupole interaction. It is equal to the term with in the expansion of the interaction given above. They normalized it to the spectrum and repeated the calculations. They found that “although there are some deficiencies, it is quite clear that the quadrupole force spectrum agrees fairly well with experiment”. The probabilities of the resulting ground state wave functions in those of MBZ are rather large. They are larger in cases where seniority states have smaller probabilities and vice versa. They range from 97% in and through 98% in up to 99.3% in and The importance of the quadrupole interaction has been emphasized in the collective model and it will be further discussed in the following. It was made clear by the work of MBZ that there are many states where the configuration is strongly perturbed by near lying configurations. Still, the main features of many nuclei in the region have been fairly reproduced by the work of McCullen, Bayman and Zamick who used an effective charge independent two nucleon interaction and configurations. An interesting example is offered by the levels with T = 1/2 of the configuration. Some of the calculated levels are shown in the MBZ paper but the authors were not sufficiently brave to present some higher levels, which are all listed in a Princeton preprint. Looking at them, the existence of a long lived isomeric state followed directly. The calculated position of the J = 19/ 2 state is at 3.64 MeV whereas the J = 17/2,15/2,13/2 states are predicted to lie

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above it (at 4.62,3.71,3.85 MeV respectively). Hence, there is a “spin gap” between the J = 19/2 state and the state with J = 11/2, calculated to lie at 2.44 MeV, to which it can decay by E4 gamma transition. A long lived isomer with J = 19/2 was later discovered in by experimentalists who were unaware of the unpublished MBZ prediction. It lies at 3.04 MeV and states with J = 15/2 and J = 13/2 lie above it at 3.46 and 3.18 MeV respectively (no J = 17/2 state was yet detected). The J = 11/2 state lies at 2.34 MeV and a J = 9/2 state calculated to lie at 1.68 MeV was discovered at 1.33 MeV. The experimental situation in with the equivalent configuration, is not clear. After the MBZ letter was published and before their long paper appeared, a short note was published by Ginocchio and French (1963). They had been working on the shell making the same assumptions as MBZ. They were not sure about the spin assignment of the levels and chose the “one which has already been used by Bayman et al”. In their case “the basic representation used is one in which isobaric spin and symplectic symmetry are diagonal”. They did not write to what extent seniority (“symplectic symmetry”) is broken in the eigenstates which they obtain. They gave a very short description of the spectra which are the same as those obtained by MBZ. They presented calculations of binding energies which were not considered by MBZ. To obtain them they had to use the interaction energy of the J = 0 state of two nucleons which could be simply set to zero if only level spacings are calculated. The neutron-neutron interaction was taken from binding energies of and The desired value is equal to the binding energy difference from which two single neutron energies should be subtracted. The corresponding neutron-proton value was similarly determined by binding energies of and to be –3.20 MeV, somewhat different from the neutron-neutron value of -3.11 MeV. In the determination of the proton-proton interaction in the J = 0 state, the authors used binding energies of and The result, –2.69 MeV, includes the Coulomb repulsion which is also the reason for the rather large difference between single proton and single neutron energies. The Coulomb interaction was considered as a perturbation on the nuclear one and was taken to contribute equal amounts to all states of the proton-proton configuration. Its inclusion added to the charge independent Hamiltonian, constructed from differences of level energies, a constant and a term proportional to These terms depend on the number of nucleons but have the isospin T as a good quantum number. Binding energies due to valence nucleons are thus given by adding to the eigenvalues of the charge independent Hamiltonian the terms

where

and

are the numbers of

neutrons and pro-

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tons respectively. The last two terms are single neutron and single proton energies. Note that binding energies calculated in this way are negative numbers. The authors calculate binding energies whose deviations from the experimental ones are rather large, 12 MeV (from 70 MeV) at the end of the shell. The authors achieved much better agreement by adding to all two nucleon energies (“moving up their centre-of-gravity”) a small constant term, .087 MeV. This adds to the binding energies the term which could be justified as due to changes in the core due to the valence nucleons. The deviations were considerably reduced, reaching at the end of the shell the value of only 2.8 MeV. The r.m.s. deviation was reduced by addition of this term, from 4 to 1 MeV.

4.5. Mixed Configurations in the

Shell

Hamamoto and Arima (1962) considered “Properties of and the ProtonNeutron Interaction” in order to understand the fact that the “ground state of has an anomalous spin of and a very small magnetic moment”. These facts “could not be expected from the conventional shell model”. They considered the valence protons to be in the orbit and the valence neutrons in the and orbits. Single neutron energies are taken from the levels and their mutual interaction is taken to be a potential and a quadrupole interaction, i.e., a term proportional to

Harmonic oscillator wave functions were used and the strength of the zero range potential “is fixed so that the pairing energy of is 0.9 MeV”. The strength of the force “is varied to find the best fit for the spectra of and The best fit was not very good. Nevertheless, the authors couple the lowest 4 states of the 3 neutron configuration, with spins to the J = 0, J = 2 and J = 4 states of the proton configuration. The energies of the latter states are taken from levels of and The matrix elements of the proton-neutron interaction were calculated from a quadrupole interaction whose strength “is varied to give the best fit for the spectrum of In the best fit, the 3 lowest levels were very well reproduced “with the correct level order”. These levels lie within .14 MeV experimentally and .09 MeV in the fit. The higher levels, however, were not in good agreement with the calculation. There are and levels at .36 and .71 MeV whereas levels with these spins are calculated to lie at .90 and 1.30 MeV. The authors explained that the lowering of the state is because the state “is

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heavily mixed with the state” J = 1/2 (the weights are 35% and 65% respectively). The magnetic moment of the ground state wave function was calculated to be .43 nm and with some other corrections it is .35 nm. This value is indeed smaller than the Schmidt moment, .64 nm, but still far bigger than the measured moment, –0.091 nm. The calculated moment of the first excited state was –1.17 nm and with further corrections, –.67 nm compared to the Schmidt value of –1.91 nm and the experimental moment of –0.155 nm. The calculated quadrupole moment of the state, with effective neutron charge 1 and proton charge 2, as well as rates of E2 transitions were in fair agreement with experiment. Vervier (1963) looked at “Effective Proton-Neutron Interaction for He considered nuclei in which there are valence protons and one or two valence neutrons beyond the N = 28 closed shell. The latter he took to be in the orbit. To calculate energies, the author took excitation energies of the proton configuration from experiment. Differences of matrix elements of the effective interaction between protons and neutrons were determined from the known hole particle levels of and some information on levels of The latter is assumed to have the hole configuration. Vervier then calculated binding energies of nuclei with one neutron by diagonalizing the matrices obtained by coupling the valence neutron to various states of the protons. He obtained the best fit to 7 binding energies with two parameters. One is the separation energy of the neutron from the core. The other is the full value of one of the matrix elements of the interaction. The agreement with experiment was good, the r.m.s. deviation is .136 MeV. Vervier predicted for states J = 2 (g.s.), J = 5 (.025 MeV), J = 3 (.45 MeV), J = 4 (.793 MeV). These predictions were in fair agreement with the levels measured later, J = 5 (g.s), J = 2 (.257 MeV), J = 3 (.329 MeV), and J = 4 (.76 MeV). The author then went to nuclei with two neutrons. To calculate their energies he needed also the excitation energy of the J = 2 state of the configuration which he took from He used binding energies of 5 nuclei to determine the interaction energy of the J = 0 state. The agreement with experiment was still fair but less good than for N = 29 nuclei. Using the resulting wave functions, the author calculated magnetic moments of several nuclei with effective moments. The agreement with experimental values was “satisfactory”. He made an interesting observation about the interaction. He obtained good agreement between the matrix elements he determined and those obtained from a zero range interaction

with

Vervier made use of this interaction in a subsequent paper.

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Maxwell and Parkinson (1964) carried out less extensive but more detailed calculations for N = 29 nuclei, apparently unaware of the paper by Vervier. Their “paper concerns the properties of energy levels in which have been studied experimentally by means of the reaction and the nuclear structure of and In the first part of the paper the authors described their experimental techniques and results followed by the shell model calculation. They took the protons to be in configurations and their interaction energies were taken from “the experimentally observed level splittings in the neighbouring even nuclei”. The valence neutron was assumed to occupy either the or the orbits. The single neutron energies were taken from the observed levels of The number of diagonal and nondiagonal matrix elements of the proton neutron interaction is 20 in the present case. The authors must have been aware of the impossibility of determining them from the existing data since they chose the interaction

with the range The authors used harmonic oscillator wave functions with The parameters of the exchange operators were chosen to be those of a Serber force, W = M = 0.5, B = H = 0 with a well depth of or those of a “Serber-like exchange mixture, W = M = 0.4, B = 0.2, H = 0 with Both choices “were found to give quite reasonable results”. No attempts were made to obtain the best fit. The authors used these elements to construct the Hamiltonian sub-matrices for J= 1/2,3/2,5/2 and diagonalized them (the largest order was 12). The agreement between calculated and experimental level positions was fair. It is better for low lying levels than for higher ones. It is rather good for and less so for the heavier ones. The authors used the wave functions obtained from the diagonalization to calculate stripping strengths. The agreement of those with experimental data was also reasonable. The authors concluded that their shell model description “seems to account, at least in a qualitative way, for the properties of the low lying levels in and The energies and reduced widths compare rather well with the experimentally observed values”. The Energy Levels of were considered by Wells (1965). Some of these levels were not yet assigned spins and parities and the author used various reactions in an attempt to obtain the correct assignments. Some of his results turned out to agree with later measurements but some others were incorrect. He took the 7 protons to be in the orbit (one hole) and the valence neutron was assumed to occupy the or the orbits. The difference of single neutron energies was fixed by the best fit to the data at values between

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.7 and .8 MeV. The proton-neutron interaction was assumed to be

The range of the Yukawa potential was taken to be

fm and harmonic

oscillator wave functions were used to evaluate its matrix elements. The best value for the spin-dependent part was The agreement between calculated energy levels and experimental ones was rather poor. The author found the dominant configurations for certain levels to be and for some higher ones, He used the wave functions to obtain rates of electromagnetic transitions but incorrect spin assignments invalidate various results and the discussion. In a long and detailed paper, Vervier (1996a) returned to the “Effective Nucleon-Nucleon Interaction in the Nuclei with 29 and 30 Neutrons”. In his paper, Vervier presented first a calculation for protons and neutrons. This was similar to his 1963 paper but it included “recent experimental data and more complete calculations and corrects some errors”. In these calculations “qualitative agreement with experiment is generally achieved”. Following this calculation, Vervier took into account also the and orbits for the valence neutrons. To calculate the many matrix elements he used zero range forces “whose strengths are deduced both from the results obtained on the configuration and from the motion of the and neutron orbits with respect to between and The author then found that “very satisfactory quantitative agreement with experiment is generally obtained for energy level positions, spectroscopic factors of the reaction, magnetic moments and M 1 transition probabilities, and only qualitatively, for beta-decay data”. As in the earlier paper, where only the is considered, calculated bind-

ing energies of nuclei with 29 neutrons were in good agreement with the experimental ones. Level spacings, however, were only in qualitative agreement with the data. This is particularly true of and levels in even odd nuclei. In the case of N = 30 nuclei, the situation was similar and even somewhat better, since the level schemes seem to be primarily determined by the configurations. Magnetic moments of some nuclei, calculated with effective proton and neutron moments were in fair agreement with the measured ones. Also rates of E2 transitions, calculated with effective quadrupole operators were in reasonable agreement with measured rates. The conclusion of this simplified calculation was that “some successes, mainly of qualitative nature”, were obtained, as described above and also for “magnetic and quadrupole moments, M1 and E2 transition probabilities and spectroscopic factors for nucleon transfer reactions”. There were indications for the need of other neutron orbits which are considered in the second part of the paper.

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The same nuclei were considered by Vervier assuming that “their 29th neutron can be in the and orbits”. As before, “the effective proton-proton integration in the orbit will be taken directly from the experimental spectra of the nuclei with the same Z but N = 28”. The single neutron energies were taken from the levels. The proton-neutron interaction obtained in the simple analysis, could be derived from the zero range potential

with

and multiplied by the radial integral I given by The same form of interaction was adopted also for the other matrix elements, keeping the same ratio of The changes in the single neutron energy spacings from to were attributed to the interaction between the neutrons and the protons. Using them, the values of for the and neutrons interactions with the protons were determined. The value of for the interaction turned out to be 0.806 MeV and for the case, 1.080 MeV. The larger value of the latter is attributed to the larger overlap between the and radial functions. For the nondiagonal matrix elements, the strengths of the interactions were determined by some interpolation between those strengths for the diagonal elements. No assumptions were made on the neutron-neutron interaction and the calculation was limited to one valence neutron. The calculations and experiment for and “displays quite satisfactory agreement, which is at least qualitative and, in most cases, quantitative”. In particular, as could be expected, there was good agreement for the lowest and states. The situation was similar in the odd-odd nuclei and there is considerable improvement on the results of the simpler picture. Vervier concludes by stating that the “agreement with experiment is generally quite satisfactory and in most cases, quantitative”. Vervier made a detailed comparison of his results with those of Maxwell and Parkinson (1964) and with earlier results of Ramavataram (1963). The latter, according to Vervier, took positive parity levels in the cores of and to be quadrupole vibrations, up to 3 phonons. The cores need not be the actual and nuclei and the phonon energy is taken as a free parameter. The valence neutron in the or are coupled to the core. Two spacings of these orbits, the strength of the core-particle coupling, as well as are free parameters for each nucleus. These “four parameters of the calculations change in peculiar ways from to and “no explanation is offered for these features”. “Qualitative agreement with experiment is claimed”. Nevertheless, Vervier found out that the “over-all similarity between

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the three theoretical predictions, which adopt rather different approaches especially” Ramavataram, “and their general agreement with experiment is striking”. About a year later, McGrory (1967) published a paper on “Shell-Model Calculations for N = 30 Nuclei”. He actually extended the work of Vervier (1966) to nuclei with two valence neutrons in the and orbits. The protons in the nuclei he considered were assumed to occupy the orbit and their interactions were taken from the experimental spectrum of Single neutron energies were taken from measured level of and matrix elements of their mutual interaction were taken from the analysis of Ni isotopes by Cohen, Lawson, Macfarlane, Pandya, and Soga (1967). The many matrix elements of the proton-neutron interaction were taken from the paper of Vervier (1966a). Using these interactions, McGrory constructed the Hamiltonian sub-matrices for the various J-values (with orders up to 70) and diagonalized them. Some preliminary results of the energy-level calculations were reported previously (McGrory (1966)). Only spacings between levels were calculated and the predicted levels compared with the experimental ones. The author explained that the “agreement for levels which lie in the energy region from 0-2 MeV is generally satisfactory, with the striking exception of The level schemes he was referring to are of to and the agreement was rather good. The author explained that his calculations were an extension of those of Vervier (1966a) for some N = 30 nuclei. Thus, he found, for and that “the inclusion of configuration mixing leads to distinct improvement in the quality of the agreement between theory and experiment”. There were, however, some states not accounted for which could be intruder states, probably arising from excitations of the configuration of protons and neutrons. Such states could be a state in a state in and and two states in McGrory was disappointed that in “there is no similarity between the experimental and theoretical spectrum”. Subsequent measurements show that the agreement with experiment, apart from the intruder levels, was as good in as in the other nuclei. The author made use of the wave functions obtained by diagonalization for N = 30 nuclei as well as for N = 29 ones, to calculate spectroscopic factors for stripping reactions and pick-up reactions. He found that the “predicted strengths” of nucleon transfer reactions “are not in as good agreement with experiment as the excitation energies”. He argued that “those factors extracted from experimental data” are “meaningful only to the extent that the procedure for extracting the numbers from experiment is accurate”. It may be significant that the “predicted strengths for the two-nucleon-transfer reaction are in bet-

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ter agreement with experiment than those found for the one-nucleon-transfer reactions”. In spite of the progress made in understanding the specially for only valence neutrons or valence protons, it was not accepted by all. Raz and Soga (1965) explained that in spite of expectations to observe in calcium isotopes, this “is not the case at all, as several theorists have discovered”. The references include MBZ as well as Ford and Levinson... The authors describe their work as “the first successful shell-model result for the calcium isotopes”. They included in their calculations only the and neutron orbits and considered the nuclei to The singlet (S = 0) part of the nucleon-nucleon interaction was a potential whose matrix elements were used for relative L = 0 states. The four harmonic oscillator integrals were taken as free parameters in the fit. To correct for the approximation, a Gaussian potential was added to it, with a given range, whose strength is another parameter. Two more parameters were provided by the strengths of the triplet (S = 1) potential and the tensor Gaussian potentials whose ranges are fixed at 1.2 times the range of the extra S = 0 potential. The authors tried 3 fits to all levels and also to only some of them. They claim that the “overall agreement is very gratifying”. It is clear, however, that their interaction is much too restrictive. For example, they predicted a level in at .72 or .91 MeV excitation which was later discovered at 2.09 MeV. The level in predicted at .44 or .71 MeV excitation was found later to be at 1.99 MeV. A simpler calculation of calcium isotopes was carried out by Engeland and Osnes (1966). They considered in addition to configurations, also states in which one neutron was raised to the orbit. The single neutron energies were taken from the levels and the 10 two-body matrix elements were taken as free parameters to be fitted by 20 experimental energies of to The authors explained that including more than one neutron in the calculation would introduce 5 more parameters “and the experimental information might then become too scarce”. They diagonalized the Hamiltonian sub-matrices and with the best values of the matrix elements obtained good agreement with experiment. The “model represents an appreciable improvement over the pure configuration model”, the r.m.s. deviation between the 20 calculated energies and experimental ones is only .087 MeV. An interesting result of the paper was the low position of the level in compared to the pure configuration prediction of about 1.3 MeV and to its position in (1.43 MeV). Also the lower position of the state in compared to was nicely reproduced. As could be expected, ground states of even isotopes turned out to belong to almost pure configurations (99-100%). Also the odd isotopes ground state configurations were not strongly perturbed (95-99% pure). The mixings were larger in excited states.

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The first excited state in has only 62% of the configuration. It is a “somewhat surprising fact” that the weight of the state is only 4% whereas 34% are due to the This agreed well with stripping data. The authors explained that they did not attempt to fit the first excited J = 0 and the second excited J = 2 states in “which have recently been described nicely as deformed states” by Federman (1966). Federman(1966) noticed the similarity between the and spectra. In each of them there are two low lying extra states with J = 0 and J = 2 which cannot be due to the two valence neutrons. In both cases, these states seem to belong to a rotational band which is due to some deformed state arising from excitation of the core. Federman carried out a calculation in analogy to the one in He constructed a deformed state by raising two protons from the orbit into the orbit. The two protons and the two valence neutrons occupy the K = 1/2 and K = –1/2 states which contain, for simplicity, the and orbits only. The shell model states which admix with the J=0,2,4,6 state projected from the deformed state, are states whose energy differences were taken from the two proton configuration in The states projected from the deformed state were assumed to form a rotational band, similar to the one observed in The moment of inertia of the rotational band in was taken from the band. The non-diagonal matrix elements between shell model and deformed states were calculated from a Gaussian potential with the Rosenfeld exchange mixture and were found to be rather small. The only parameter to be determined was the unperturbed position of the J = 0 deformed state. It was estimated to be roughly 2 MeV above the ground state and was slightly adjusted to 1.7 MeV to obtain better agreement. The agreement with the measured energies of was very good indeed. The mixing between the two sets of states is up to 20% with energy shifts up to .2 MeV. Yet differences between those shifts are much smaller, yielding level spacings almost identical to those of A more complete analysis of calcium isotopes was carried out by Federman and Talmi (1966) who included one and two neutron excitations into the orbit as well as deformed states. As in the paper by Federman (1966), they assumed a rotational band for the deformed states whose moment of inertia is taken from the similar band in The unperturbed position of the J = 0 band head was taken to be 2 MeV. Single neutron energies were taken from the levels. There are 15 diagonal and non-diagonal matrix elements of the two-nucleon interaction. Two of them, the interaction energies of the J = 0 and J = 2 of the configuration, were taken from the work of Auerbach (1966a) on nickel isotopes. Thus, they were left with 14 parameters, including a non-diagonal matrix element between one shell model and

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the corresponding deformed state. The other such non-diagonal elements are proportional to it. These parameters were determined to fit 30 experimental energies. The statistical errors on some of the matrix elements were rather large. One of them could not be determined well and a fixed value was assigned to it. With the matrix elements thus determined, very good agreement was obtained between calculated and experimental energies, the r.m.s. deviation was only .13 MeV. The spectra of both even and odd isotopes were very well reproduced. Also calculated binding energies of and agreed very well with the experimental values. In good agreement was obtained for the binding energy as well as positions of the first excited J = 2 and J = 0 states. The wave functions obtained by diagonalization yielded strengths for stripping reactions which were consistent with experimental data. A simple demonstration that the effective interaction depends on the configuration adopted, can be found in that paper. The matrix elements in the configuration, extracted from the fit are considerably smaller, in absolute values, from those obtained by considering only configurations. The agreement with experiment, obtained by Federman and Talmi (1966) was due to explicit mixing with configurations containing also nucleons. For example, the interaction energy in the state J = 0 is determined by the fit to be –2.64±.30 MeV. This should be compared with the value = –3.10 MeV determined from binding energies by using pure configurations. The J = 2 interaction energy is similarly affected. The change in the J = 4 and J = 6 states is smaller. In a long paper by Vervier (1966a) to be described below, was not discussed, possibly due to lack of experimental information. With the publication of relevant measurements, Vervier (1966b) carried out a shell model calculation. In this case there are no protons and their interaction with the neutrons need not be known. He considered neutrons in the and orbits. As before, single nucleon energies were taken from the experimental data on binding energies and levels of The matrix elements of the effective interaction between the two valence neutrons, were taken by Vervier from the work of Auerbach (1966a) on Ni isotopes. The results for the binding energy and positions of levels up to 5 MeV were in very good agreement with the measured ones. There were some experimental levels which have no corresponding ones in the configurations considered. The author points out that those are probably due to excitation of the core. Indeed, such levels with spins and were found in the nucleus. Using the wave function he obtained, Vervier calculated the strengths of the to the three states. He found that the calculation showed that in this reaction, the ground state is strongly favoured over the higher states. Federman and Talmi (1966) concluded that first excited J = 0 states in

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heavier calcium isotopes are mainly due to neutrons and are not strongly mixed with core excited states. This conclusion was challenged by McGrory and Wildenthal (1968). In their paper “Nature of Levels in Calcium Isotopes”, they carried out “conventional shell model calculations” using several effective interactions. Using them they could not reproduce the positions of the first excited J = 0 states in and Their measured positions above ground states are 1.84, 1.88 and 2.42 MeV respectively, whereas the calculated positions were typically 5, 4.5 and 3.8 MeV. The interactions used by the authors included: a)the one derived by Kuo and Brown (1968) from the Hamada-Johnston free nucleon interaction (KB), b)the modified surface delta interaction c)a central interaction parametrized in terms of radial integrals, and d)the KB interaction in which certain “matrix elements were adjusted to better fit experimental binding energies”. The configurations were limited to, and those excited from it by raising one or two neutrons into the and orbits and those in which 1 to 4 neutrons are allowed to occupy the orbit. The authors concluded that in “contradiction to the conclusions of Federman and Talmi, our results strongly suggest that the core-excited state persists as the first excited state through McGrory, Wildenthal and Halbert (1970) considered the “Shell-Model Structure of In these nuclei, the 20 protons were assumed to form closed shells. On the other hand, the authors took the neutrons to occupy not only the orbit but also the and orbits. They also carried out calculations where also the neutron orbit was included. Thus, only was assumed to be an inert core. To limit the order of the Hamiltonian submatrices, the authors restricted the number of neutrons in the orbit (and also in the orbit) to 2 at most. The authors tried to use the interaction between these valence neutrons which was derived from the Hamada-Johnston interaction between free nucleons. Kuo and Brown (1966) carried out an approximate renormalization of that strong interaction and obtained a tame version which was appropriate for using with shell model wave functions in shell nuclei. Kuo and Brown (1968) extended their work to the shell and McGrory et al., tried to use their interaction for calcium isotopes. The results of Kuo and Brown were later extended and amplified by Kuo and co-workers. The work of Kuo and Brown has been a big step towards deriving the effective interaction from the one between free nucleons. The Kuo-Brown (KB) interaction has been used in many shell model calculations, more or less successfully. These calculations, as well as others based on such “realistic” effective interactions, are not included in the present review since some important ingredients are still missing from most of them. Only recently, some progress has been made in deriving effective interactions which seem to yield results in good agreement with experiment.

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This point is clearly exemplified by the results of McGrory et al., as may be illustrated by looking at ground state binding energies. As explained above, binding energies of calcium isotopes, N = 20 to N = 28, are nicely reproduced by the simple formula of the seniority scheme. Single nucleon separation energies as a function of N, lie on two straight and parallel lines, for odd and even nucleon numbers. The measured separation energies of and lie considerably lower than these lines, which help characterizing 28 as a magic number. The behaviour of separation energies calculated with the KB interaction is very different. They lie on two straight and parallel lines up to N = 28, but the separation energy of is higher than the line. The calculated value is 7.00 MeV as compared to the experimental 5.15 MeV. The calculated separation energy is 8.81 MeV, much larger than the experimental value of 6.35 MeV. The magicity of N = 28 is not reproduced by the KB interaction. The authors find from the calculated spectra that “the interactions of neutrons with and neutrons are too strong”, i.e., too attractive. The authors modified the KB interaction and obtained much better agreement with experiment. This procedure was adopted later in large scale shell model calculations. McGrory et al., could not determine the 94 matrix elements needed to calculate energies and wave functions, from the 25 experimental energies available. They singled out the 8 most important matrix elements, 4 energies of states and 4 energies of states of the configuration. All other matrix elements were taken from the KB interaction. Still “the four matrix elements could not all be accurately determined by the set of observed levels included in the search”. The values of the 3 energies of states with spins 2,3 and 4 were fixed at a certain stage in the search for the best fit. Thus, five parameters were used in the final fit. Comparing these 8 matrix elements of the modified KB (KB’) interaction to the original elements, there are clear differences. The diagonal matrix elements in the states with “J = 0 and J = 2 were made more attractive by 0.3 MeV” whereas those with J = 4 and J = 6 were hardly changed. All diagonal elements of the states “were made more repulsive by 0.3 MeV”. This change is significant. It was emphasized above that the effective interaction between identical nucleons in different orbits determined from experimental data is repulsive on the average. This average is defined by the center-of-mass of the diagonal matrix elements

In the KB interaction, this average for and is attractive, –.096 MeV. In the modified KB’ interaction this average is repulsive, .214 MeV.

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The agreement between calculated and experimental spectra was good with the exception of some intruder (core-excited) states and and in The latter discrepancies have been removed by another change of the KB interaction. The authors have added a .25 MeV repulsion to all diagonal matrix elements of and states (KB”). Binding energies calculated with the KB’ interaction reproduce very well the experimental energies. The r.m.s. deviation is only .16 MeV and the sharp drop of separation energy beyond N = 28 is clearly evident. “The calculated spectroscopic factors for transfers between the various calcium isotopes are in good agreement with experiment”. The mixing of other configurations into the states is rather large if the KB interaction is used. It is smaller for the KB’ interaction (and probably smaller for the KB” one). The weight of the configuration in the ground state is 92% and it goes down to 86% in The weight of the in the ground state of is also high, 84%. The mixing becomes appreciable in ground state where the weight of the state is only 46%. The authors did not give the other configurations which could be either and or those in which neutrons are excited into higher orbits. Auerbach (1967a) considered proton configurations in nuclei with N = 28 neutrons. He pointed out some irregularities in level positions which indicated that the configurations cannot be pure. In particular, E2 transitions from the J = 6 level in to the lower J = 4 level should be forbidden if it has seniority Experimentally, its B(E2) is smaller by only a factor 2 than the one to the higher, J = 4 level. As remarked earlier, such a situation may indicate that the two states with different seniorities are mixed which could not occur in a pure configurations of identical nucleons and two nucleon interactions. Stripping reactions also indicate considerable mixing between the J = 4 states with seniorities and as well as mixing with configurations which contain also protons. Thus, the author considered such configuration mixings. He adopts, along with proton configurations outside the core, configurations obtained by raising one proton into the orbit. Such excited configurations do not have definite isospins but the author argued that isospin impurities are “in most cases very small and cannot change appreciably the general picture”. There are 10 two-nucleon matrix elements of the effective interaction to be determined along with the single proton energy. The single proton energy of the orbit was then determined from the spectrum. These 11 parameters were determined by a least squares fit to 25 experimental energies. With the best values of the matrix elements, the calculated energies were in very good agreement with the experimental ones. The r.m.s. deviation was

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only .09 MeV. The ground state wave functions had very small admixtures of the configurations. “Most of the other states, however, are considerably admixed”. The E2 transitions and stripping widths calculated from them were in much better agreement with experiment than results with pure configurations. In particular, configuration mixing leads also to strong mixing of different seniorities in the two lowest J = 4 levels of The upper J = 4 level had only 56% of the state with and 32% of the state, the other 12% belong to the excited configuration. In the lower J = 4 level, the corresponding weights were 53% of the state, 34% of the state and 13% states of the configuration.

4.6. The Nickel Isotopes Neutron orbits beyond the magic number 28 are the and They are supposed to be filled in Ni isotopes between and whose spectra indicate rather strong mixing of configurations based on these orbits. Hsu and French (1965) used Ni isotopes “to study the relationship between various microscopic models for low-lying states of vibrational nuclei”. They compared various approximation methods which aim at describing collective excitations, with exact shell model calculations. For the latter, they took single nucleon energies from levels and a special two-nucleon interaction. It acts only in relative L=0 states “produced, via a least squares analysis to energy levels, by Pandya and Soga”. This interaction is defined by the values of the 4 harmonic oscillator integrals ( with ). The agreement between calculated level energies in even Ni isotopes from to and 27 experimental energies was “reasonable” (r.m.s. deviation is .19 MeV). Adding a constant (monopole) interaction, –.84 MeV, to all diagonal matrix elements “gives a satisfactory treatment for the binding energies”. With a neutron effective charge of 1.55e, B(E2) values for the first states to the ground states and E2, M1 mixing ratios between the lowest two states “agree well with experiment”. A very interesting result was that “almost all the electric quadrupole strength from the ground state goes to the first state, which is almost identical with Q(e.m.)|0 >, where Q(e.m.) is the E2 operator and |0 > the ground state wave function”. This point was elaborated several years later in the framework of generalized seniority. The promise of a shell model description of “vibrational states” in even Ni isotopes attracted the attention of Auerbach (1966a). He tried to obtain a shell model description of Ni isotopes by using matrix elments of the effective interaction determined from experimental data. In his paper, he explained that it is difficult to determine all matrix elements from experiment because of their

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large number. The number of diagonal elements and specially the number of non-diagonal ones, increases as the number of available orbits increases. The author made a simplifying assumption for which he quotes calculations by Cohen, Lawson, Macfarlane, and Soga who were probably motivated by the results of Hsu and French. Auerbach considered ground states of even Ni isotopes which are mixtures of states where in each orbit there is an even number of neutrons which are coupled to J=0 states (also but it is superfluous for the orbits involved). He made the same assumption also on the first excited state (which “implies that the [first] excited state is not a two-phonon state”). Similarly, he considered the lowest states in odd nuclei to be mixtures of states where states of -neutrons with to be in J = 0 states and the odd group to be in the state The author took differences of single neutron energies from levels of and was left with one single neutron energy and 11 diagonal and non-diagonal matrix elements. He determined their values by fitting 21 experimental binding energies and level positions. “The agreement between the calculated and experimental energies is very good”, the r.m.s. deviation is .11 MeV. In addition to binding energies, positions of first excited states in and were in very good agreement with experimental data. In odd Ni isotopes, spacings between lowest states with spins 3/2, 5/2, and 1/2 were very well reproduced, including the crossing of these levels. Not all matrix elements could be determined accurately, in particular, the interaction in the state was obtained to be .89±1.05 MeV. When this matrix element was fixed at .76 MeV and the fit repeated, there was a slight improvement in the agreement with experiment. The values of the other matrix elements were hardly changed but the statistical errors on them were appreciably reduced. The author tried to fit the data by using the schematic pairing interaction with one free parameter, its strength, and one single neutron energy. A fit to 21 energies gave “results in poor agreement with experimental data”. The r.m.s. deviation was 1.1 MeV which was a typical deviation. An important ingredient missing from the pairing interaction was the average repulsion between identical nucleons in different orbit. When the 3 average repulsions were introduced as free parameters the agreement with experiment was much improved, the r.m.s. deviation was reduced to .19 MeV. The values of the average repulsion were equal, within the statistical errors, to those of the more detailed interaction which is in better agreement with experiment. Not long after that paper, Auerbach (1966b) published a short paper in which “spectra of and were calculated without the restriction to states of lowest seniorities”. Eleven of the required matrix elements for the complete calculation were taken from the previous paper but 19 more

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were needed and they could not be determined from the small number of measured energies. The 11 matrix elements determined earlier included centersof-mass of levels with J > 0. Auerbach fixed the individual energies by using the matrix elements calculated by Kallio and Kolltveit (1964). Those authors used the Moszkowski-Scott separation method to get rid of the hard core in the interaction between free nucleons. Also for the non-diagonal matrix elements, between J > 0 states, the Kallio-Kolltveit (1964) interaction was used. The author found that these matrix elements were very similar to those calculated from a Yukawa potential with the Rosenfeld exchange mixture. The procedure adopted by Auerbach became widely used in many extensive shell model calculations in coming years. In cases where there are too many matrix elements, only a few important ones are determined from experimental data. The others, with less effect on the results are taken from some theoretical paper. Their exact values may have more effect on high lying states, but these are rarely observed experimentally. The agreement between calculations and experiment “is very gratifying” and the resulting wave functions are interesting. Auerbach checked how justified is the approximation he made before. He assumed that in even nuclei, the ground state and the first excited state had only seniority zero components which means that neutrons in each orbit are coupled to J = 0 states. Similarly, he assumed that in the lowest levels in odd Ni isotopes, neutrons in 2 of the orbits are coupled to J = 0 states and in the third, they are coupled to a state with where or 1/2 (“seniority one”). He found that admixtures into these “seniority ground and first excited states are very small. In the and ground states these admixtures did not exceed 2% and the resulting shifts less than .05 MeV. In the first excited state of the admixtures are 5% with a shift of .07 MeV. In these numbers were even smaller. The admixtures in the lowest and states of were less than 5% and the shifts were about .05 MeV. They were larger for the lowest state of about 9%. “Thus the present calculation confirms the approximations adopted in” the earlier paper. The “first levels in as well as in are almost pure state”, the impurity is only .5%. “Seniority” is defined here as the sum of seniorities of states of the various mixed configurations. These impurities in the lowest states in and are also small, 3% and 9% respectively. On the other hand, rather large admixtures were found in the second excited states. The admixtures in was 27% and in it was even larger, 48%. Lawson, Macfarlane and Kuo (1966) in their calculation of Ni isotopes, adopted a similar method to determine the multitude of matrix elements. They determined single neutron energies from levels of and its binding energy, minus binding energy. They determined from experiment only the rela-

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tive L = 0 part of the interaction, a la Hsu and French (1965), by leaving the 4 oscillator integrals as free parameters. The other parts of the two-nucleon interaction were calculated from a “fixed range potential with singlet-central, triplet-central, two-body spin-orbit and tensor components”. “The resulting interaction matrix elements agree closely with those determined by Auerbach”. Like him, they found good agreement between calculated and experimental level spacings and binding energies. The main point of their paper, however, was to check how well energies of Ni isotopes can be obtained by using matrix elements calculated by Kuo and Brown from the Hamada-Johnston interaction between free nucleons. The calculated matrix elements of the reaction matrix yielded poor agreement with observed level spacings. Corrections due to single nucleon or two nucleon excitations from the core changed the matrix elements but did not improve the agreement with the data. The inadequacy of these matrix elements was clearly seen in the calculated binding energies. The effective interaction determined from experiment contained repulsion between neutrons in different orbits and thus, fitted very well the measured binding energies. On the other hand, the reaction matrix was attractive in those cases and failed to follow the binding energy curve (the difference in is about 18 MeV). Corrections due to core polarization were repulsive but not sufficiently, reducing the difference in to about 4 MeV. A long paper with detailed discussion of the “Shell Model of the Nickel Isotopes” was published by Cohen, Lawson, Macfarlane, Pandya, and Soga (1967). In this paper, the authors discussed the various approximations involved in arriving at a feasible shell model calculation. They considered neutrons in the and orbits whose energies were taken from (and the binding energy). To determine the 30 matrix elements of the twonucleon interaction, the authors first tried a simple local interaction

where and is the relative orbital angular momentum. The were taken to be Gaussian potentials with fixed (unequal) ranges. The 4 strength coefficients were considered as free parameters to be determined by a least squares fit to the experimental energies. Since the agreement with experiment obtained with this interaction was not satisfactory, the authors tried various changes but “conclude that the poor agreement between theory and experiment is probably due to the inflexibility of the assumed form of interaction potential”. To obtain a better interaction they added, a la Hsu and French (1965), an

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interaction which acts only in relative states which was defined by 4 harmonic oscillator integrals The authors considered these 4 integrals as additional free parameters which, together with the four are determined by a fit to 24 observed energies in to With the values of these parameters thus determined, the resulting “interaction provides satisfactory description of the low-lying spectra of the Ni isotopes”, the r.m.s. deviation is about. 15 MeV. Calculated energies for to also agreed well with experimental ones which had not been included in the fit. The authors compared the matrix elements which they determined with those calculated by Kuo and Brown from the Hamada-Johnston interaction. Whereas the reaction matrix results were rather different from them, those corrected by core polarization were in many cases “in fair agreement with the phenomenological matrix elements” of the paper. The latter, although they “cannot be said to have been derived rigorously from the force between free nucleons, it is clear that a rough quantitative relationship has been established”. Matrix elements determined in this paper agreed rather well with those determined by Auerbach (1966a) in his approximate calculation. The authors attributed the similarity to the structure of low-lying states, which Auerbach adopted. They found in ground states of even nuclei, large percentages of states with only even groups of various coupled to J = 0. In J = 2, 4 states, one even group is coupled to J or in two of the orbits, neutrons are coupled to states (with J = 3/2, 5/2 or 1/2). In odd nuclei, only even groups coupled to J = 0 and one odd group with (“seniority 0”, “seniority 2” and “seniority 1” wave functions). In the ground state they found only .3% of admixtures to such a wave function, in the lowest state they find .5% admixtures. The admixtures in the lowest J = 4 state are higher, 7%, in the first excited J = 0 state they amount to 13% and in the second excited J = 2 state to 11%. In the lowest J = 3/2 and J = 5/2 states of the admixtures are about 5% whereas they are 9% in the lowest J = 1/2 state. Similar results were obtained by Auerbach (1966b) in his more detailed calculation. Using their wave functions, the authors calculated electromagnetic transition probabilities and single nucleon spectroscopic factors. The measured E2 transition rates between the lowest J = 0 and J = 2 states in and could be reproduced by using an overall effective charge of about 1.7e to the valence neutrons. The small ratio of B(E2) values between J = 0 ground states and second J = 2 levels and the B(E2) values between the two lowest J = 2 states observed in and was nicely reproduced and given a simple explanation. Like Hsu and French, the authors found that most of the B(E2) strength from the J = 0 ground states goes to the lowest J = 2 level, which may well be approximated by the quadrupole operator acting on the ground state. This effect “has little to do with the de-

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tailed structure of the second excited state” and hence, does not “indicate a vibrational characteristic in the even-mass Ni isotopes”. Sum rules for single nucleon spectroscopic factors may be used to determine the occupation numbers of neutrons in the various orbits, The authors found good agreement between experimental values for J = 0 ground states of to and occupation numbers calculated from their wave functions. They described it as “a modest, though satisfying, achievement of the theory”. Three months after the paper by Cohen et al., appeared, a long and detailed paper on Ni isotopes was published by Auerbach (1967). He adopted in his analysis the matrix elements which he determined in his previous two publications. He used them to calculate energies of Ni isotopes, from to as well as wave functions, E2 transition rates and spectroscopic factors. His matrix elements can be compared with those of Cohen et al. Although these elements were determined in different ways, most of them are very close, within the statistical errors. There are some exceptions but they do not seem to change much the predictions for low lying levels. Auerbach also found that the “agreement between the calculated and experimental energy levels (including binding energies) was very gratifying”. In fact, he had a better fit to binding energies. He discussed in detail the experimental level schemes of the various isotopes and compared them to his calculated ones, Auerbach calculated from his wave functions spectroscopic factors for stripping and pick-up reactions on even Ni isotopes into various levels of odd isotopes. “The spectroscopic factors for the individual levels (from the pick-up as well as from the stripping reactions) were in good agreement [with experiment] for the lighter isotopes and were in somewhat poorer agreement for the heavier isotopes In spite of the strong configuration mixing, most of the strength of these reactions, 90% to 95%, is contained in the lowest J= 1/2,3/2,5/2 states. The author attributed this feature to the fact that ground states are almost pure “lowest seniority states” as explained above. Less experimental information was available on pick-up and stripping reactions on odd Ni isotopes. The calculated spectroscopic factors of reactions leading to individual levels of and were in agreement with the observed ones. The B(E2) values for transitions between J = 0 ground states and lowest J = 2 states, calculated with a neutron effective charge of 1.1e, agreed fairly well with measured ones. The relatively weak transition from the second excited J = 2 state to the ground state compared with the transition from the first excited J = 2 state was semiquantitatively reproduced. Five years later the structure of Ni isotopes was taken up by Glaudemans, De Voigt and Steffens (1972). In their shell model calculations they considered valence neutrons in the and outside the core. The two-

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nucleon interaction which they used was the modified surface delta interaction. This interaction is described in Sect. 5.2 below. The modification is by having a different strength in T = 0 and T = 1 states and by adding to twobody matrix elements, a constant, state independent, term B[2T(T + 1) – 3]. For neutron configurations, T = 1, and B are taken as free parameters to be determined by a fit to experimental energies. In addition, 3 single neutron energies are taken as free parameters. The matrix elements of the modified surface delta interaction between antisymmetrized two neutron states are given by

The values of the 5 parameters were determined by a least squares fit to 36 excitation energies and 9 binding energies. They are given in MeV by

Very good agreement was obtained for binding energies which were used in the fit, up to but a “rather strong deviation occurs for The calculated binding energies of and were not given but it seemed that the deviations for them will be much bigger. The single neutron energies listed above agreed well with the observed spectrum and binding energy of The first term in the two nucleon matrix element may lead to attractive interaction between neutrons in certain different orbits, unlike the situation in preceding calculations. The constant term leads to the repulsive interaction in states of n neutrons which more than compensates for the attractive term. “Only three parameters are involved in the calculation of excitation energies”, and 2 single neutron energy differences. It is indeed remarkable how good the agreement was with experimental 36 level spacings. The “average absolute deviation” was only . 11 MeV. “In fact, the agreement with experiment is even better than in” the papers of Cohen et al., and Auerbach “although in their calculations more parameters are used to obtain values of two-body matrix elements”. Much of the paper dealt with electromagnetic properties. For E2 transitions, the authors obtained by fitting 22 transition strengths, an effective charge for the neutron of 1.7e. The agreement with the data was only fair and was not

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improved much by a fit with 5 different coefficients of the effective quadrupole operator. From 17 M1 transitions and 2 magnetic moments, an effective single neutron M1 operator was obtained. It has 5 independent reduced matrix elements whose values turned out to have the same signs as the values obtained from the Schmidt moments. The ratios between bare and effective matrix elements vary between .5 and 2. An effective non-vanishing matrix element between and states was obtained by the fit. The agreement between calculated and experimental M1 transition rates was only fair. The calculated magnetic moments in odd isotopes were in good agreement with measured ones, also those which were measured later. If the lowest states were really “seniority 1” states, the magnetic moments would not change along the shell. In fact the calculated moments changed very little and so were, to some extent the measured ones. The situation may be very different for the lowest J = 2 states in even isotopes. Whereas the predicted moments vary by only 15% from to and are negative, the experimental values change from –.12 nm in change sign and reach .92 nm in If these experimental numbers will turn out to be correct, the structure of Ni isotopes may have to be revised. The rather constant positions of the first excited J = 2 levels in the nickel isotopes was interpreted in the collective model as an indication of quadrupole vibrations. Shell model calculations attempted to reproduce these states and a consistent description was obtained. There is nothing in the resulting wave functions which can be viewed as a vibration. For example, if the J = 2 states are one phonon states two phonon states, with J=0,2,4, are expected. Such a description was not reproduced by shell model calculations. Still, the approximate constancy of 0-2 spacings in Ni isotopes and in other semi-magic nuclei suggested that a simple description should be found. Such a description was later offered by generalized seniority which will be described in the following.

4.7.

The Pb Region II

An isomeric state of was discovered by Perlman et al., (1962) at 2.91 MeV above the ground state. It has a half-life of 45 seconds and its only observed de-excitation is by Glendenning (1962) carried out a shell model calculation of the lowest configurations in this nucleus to see whether a high spin state lay below states which have spins close to it. He adopted a central potential interaction

where and are projection operators into singlet even and triplet even states. This interaction is the even part of the Ferrell-Visscher interaction.

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As late as 1962, and even later, some nuclear theorists continued the use of (Gaussian) potential interactions for the effective interaction between nucleons. Glendenning was not considering configuration mixing since the calculated non-diagonal matrix element between the J = 0 states of the two lowest configurations is only .23 MeV, shifting the levels by only . 1 MeV. The lowest proton orbit is and the author considered only states with the proton configuration. The lowest neutron orbit is and the author considered states with two neutrons in this orbit. He considered, however, also states of two neutrons one in each of the next orbits, and These single neutron orbits were observed in at .78 and 1.42 MeV but Glendenning was using values taken from a theoretical paper by Nilsson and Mottelson (1955), .41 and .62 MeV, which choice had considerable effect on his results. For each two proton - two neutron configuration, the author found “that the level ordering of the lowest levels is normal for a short-range force, i.e., J=0,2,4,... . However, the higher spin states appear at lower energy than some of the intermediate spin states. This happens because the interaction energy between like nucleons becomes very small with increasing spin while the interaction energy between neutron and proton increases as the spin approaches (in steps of two) either of the extreme values from intermediate ones”. This led indeed to an isomeric state but the author found the states of the configuration to be lower by about 1 MeV than the states. This was probably due to the larger overlap of the radial parts of the nodeless and orbitals and very strong interaction. The state of maximum spin 8+10=18 of the lowest configuration was lower than all states with spins 17-11 and only the J = 10 lay below it. Its excitation energy is 1.7 MeV which “is not a serious discrepancy, however, since the residual interaction is not well known”. Band, Kharitonov and Sliv (1962) studied the levels with spins up to 4 and excitation energy up to 2 MeV, since other levels were not known experimentally. In two earlier papers they, with others (Guman, Kharitonov, Sliv, and Sogonomova (1961) and Kharitonov, Sliv and Sogonomova (1961)) considered a model in which a Wood-Saxon single nucleon potential and a spinorbit term proportional to its derivative were used to determine single nucleon wave functions. They also introduced weak coupling to surface vibrations of the core and included in their wave functions also components with one or two phonons. Their two-nucleon interaction, which they called “pairing interaction”, was a Gaussian potential (range 2 fm) with different strengths for spin singlet and triplet states which were determined empirically. They included configuration mixing, which they called “pair correlations”, but in some cases, the diagonalization was approximate. In the two papers, the authors de-

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veloped the theory of the model and tried to determine its parameters from spectra of two nucleons outside and two holes in it In for instance, they obtained in the J = 0 ground state wave function only less than 3% states with one phonon whereas in the ground state, such components with one or two phonons, have a 14% weight. Band, Sliv and Kharitonov (1962) in their paper used the same model for the low lying levels. For the J = 0 ground state they obtained 18% components with one and two phonons. The agreement with some low-lying, low-spin levels was fair. When the isomeric state was discovered, Sliv and Kharitonov (1963a) looked at high spin states in They used the nucleon-nucleon interaction with values of the parameters determined in the previous paper on low-lying levels. They seemed to ignore completely the complications due to coupling to phonons and considered only configurations with two valence protons and two valence neutrons. Spin 16 is the maximum spin of the configuration which they adopted, and they found that states with higher spins, of other configurations, “lie higher than the I = 16 level because their singleparticle coupling energy is less”. The authors plotted diagonal matrix elements of the proton-proton and neutron-neutron interaction as well as diagonal elements of the proton-neutron interaction. Due to these general features, the J = 16 level turns out to be isomeric. Glendenning (1962) calculated diagonal and non-diagonal matrix elements between all states J > and diagonalized the matrices for given total J. It seems that Sliv and Kharitonov (1963a) did not go through this necessary procedure. “In the determination of the relative level positions it is possible to avoid calculating the off-diagonal matrix elements, since the latter cause an almost identical energy shift of the first levels with a given spin I”. To calculate the position of the J = 16 level “it is necessary to calculate the coupling energy, calculating the contribution not only of the diagonal, but also of the off-diagonal matrix elements”. Presumably for calculating the energy of the ground state. The J = 16 state was calculated to lie at 2.4 MeV and the only two levels with either or which lie below it are In a subsequent paper, Sliv and Kharitonov (1963b) considered general features of the “residual interaction in heavy nuclei” which follow from using a Wigner force and forces which depend on the intrinsic spins. They calculated matrix elements for a 9/2 proton - 9/2 neutron configuration. For spin dependent forces they found different behaviour for odd and even Nordheim numbers. They used these properties to predict isomeric levels in several nuclei in the Pb region. Band, Kharitonov and Sliv (1964) extended their calculations in the Pb re-

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gion. In a long paper, they developed the necessary formulae to be used in many nucleon configurations. They used the same potential interaction as in their paper on which they described as “pairing interaction”. They applied their method to the the spectrum of with 4 valence protons and 6 valence neutrons. They constructed the matrices for the various J values but if all configurations were included, the orders were too large. They truncated the matrices keeping the lowest configurations and those with large nondiagonal elements. There was still not much experimental information and the agreement with experiment was limited to the calculated position of the lowest J = 2 level, .35 MeV compared to the experimental position at .324 MeV. The authors concluded that they confirmed that the residual interaction was “mainly a two-particle interaction”. “A conclusive result in this respect” was the calculated binding energy of 52 MeV in addition to the binding energy of compared to the measured value of 50.6 MeV. Auerbach and Talmi (1964a) heard from I. Perlman about the isomeric state in which he and his collaborators discovered. They tried to see whether there was sufficient experimental information to carry out a shell model calculation for that nucleus. Another isomeric transition had been known for several years in the neighbour nucleus Both isomers were believed to be due to a “spin gap” - a large difference in spins between the isomeric state and states below it (later called “Yrast trap”). Glendenning (1964a) in his calculations found such a spin gap by using a potential interaction. Auerbach and Talmi (1964a), unaware of the work of Sliv and Kharitonov (1963), tried to determine the relevant matrix elements from experimental data. The lowest proton configuration in and was taken to be Its energy levels could be obtained from the spectrum whose lowest known levels were J = 0(g.s.), J = 2(1.18 MeV), J = 4(1.43 MeV) and J = 6(1.47 MeV). The J = 8 level was not known at the time and the authors put it at 1.53 MeV (it was later found at 1.56 MeV). There was no way to find out whether these levels belong to a pure configuration, the restriction to a single proton configuration was adopted to make the calculations possible. In contrast to lighter nuclei, the spectrum of this and other configurations in these heavy nuclei, is rather similar to the one calculated from a zero range potential. The lowest neutron orbit in is the orbit and the spin of the isomeric state at 1.46 MeV, is equal to the maximum spin in the configuration. To calculate the states of this configuration, the matrix elements of the neutron were needed. These could be obtained from the lowest levels of where levels with spins ranging from 0 to 9 were observed and interpreted as due to the configuration. The authors expected to find a spin gap in since the interaction in the J = 9 state

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(at .27 MeV) was considerably stronger than in the states with J=8,7,6,5,4 (at .58,.43,.55,.44,.50 MeV). “Due to the proximity of the and levels in the configuration, the proton-neutron interaction plays a dominant part”. The calculated position of the J = 25/2 state was 1.45 MeV compared to the experimental value of 1.462 MeV. The calculated positions of all levels with J = 23/2, 21/2, 19/2 lie above it. The lowest J = 17/2 level is calculated to lie below the isomeric state but rather close to it, at 1.41 MeV (1.428 experimentally) As a result the J = 25/2 state decays by with a lifetime of 25 seconds. To calculate the spectrum of matrix elements of the configuration were needed. The lowest levels in have spins 2,4,6 at .80,1.10, 1.20 MeV. The spin 8 level was not yet known and the authors took it to be at 1.2 MeV, slightly above 1.17 MeV, the position of the J = 6 level quoted at that time. “Unlike the situation in the existence of the isomeric state is sensitive to the 6-8 spacings” in the proton and neutron configurations. The calculated position of the J = 16 state was 2.68 MeV, “in fair agreement” with the experimental value of 2.922 MeV. The J = 15 and J = 13 levels had higher calculated energies while the lowest J = 14 level had the same calculated energy (it was later found at 2.886 MeV). In fact, the spin of the isomeric state in may be 18 and not 16, in which case it would belong to a different configuration. Auerbach and Talmi (1964a) predicted a spin gap in which would be a close analog of the spin gap. They calculated the energy of the J = 25/2 state at 1.16 MeV. Much later, a state with possible spin 25/2 was found at 1.257 MeV above the ground state. A level with possible spin 21/2 was found at 1.227 MeV but the decay scheme is not yet clear.

4.8. The Zr region II The zirconium region became active again due to Sweet, Bhatt and Ball (1964) who considered the levels of which were recently measured. They explained that in that nucleus, the expected states are due to a proton and a neutron. Levels with spins between 2 and 7 should be found in the low part of the spectrum. To explain the intensities of stripping reactions to the various states, it is necessary to know the ground state wave function of It should be a linear combination of the states whose amplitudes are determined by the energies of the states and the proton neutron interaction. The former were taken from the spectrum and matrix elements of the latter were taken from the levels. The results obtained by diagonalizalion gave, for a cer-

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tain choice of spins for those levels, intensity ratios in good agreement with experiment. Pandya (1964) tried to calculate the energies of the proton by using a “conventional potential model”. Using the information from the levels, he arrived at the interaction

with a Gaussian potential and harmonic oscillator wave functions. He varied the range of the potential and for a certain value, obtained fair agreement with the observed level spacings. He made an interesting observation, “a surprising feature of the calculations is that in all cases the level is predicted below the state”. This was contrary to the assignments made in the experimental paper and the author pointed out the need for more accurate determination of spins and parities. It is indeed remarkable that such a simple theory predicted the correct assignments in this case. Auerbach and Talmi (1964b) made use of the experimental information on the levels to study the spectrum of An isomeric level was known in that nucleus for a long time and was assigned spin 21/2. It decays to the 5/2 ground state via levels with spins 13/2 and 9/2. It was believed that these levels were due to the neutron coupled to the proton levels with spins 0,2,4, and 8. There are, however, 24 levels which belong to this configuration and it was not clear why the cascade of gamma rays goes through these “stretched” states and why there was a spin gap below the 21/2 level. Knowledge of the proton neutron interaction was necessary to find out the answers. The authors were aware of the fact that in the proton state is not a single proton but a linear combination of and states. Still, for simplicity, they adopted the simple configuration with proton neutron interaction taken from The energies of levels, with were taken from the lowest levels with these spins of In spite of the simplifications and the inverted spin assignments of the J = 3 and J = 5 levels in the results were encouraging. The 21/2 level was calculated at 2.40 MeV excitation compared with the experimental value of 2.425 MeV. There was a spin gap between it and the 13/2 state at 2.16 MeV (calculated at 2.18 MeV). That level would decay preferentially to the 9/2 level at 1.477 MeV (calculated at 1.49 MeV). The spin gap was attributed to the fact that the strongest proton neutron interaction is in the state with maximum spin 9/2+5/2=7. In particular, it is much stronger than in the state with the next spin value 6. The maximum couplings in J = 7 proton neutron interaction is in states which are as aligned as possible. Although the 21/2 state is based on the state, the latter is not much higher than the state. The successful

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treatment of isomerism in the nucleus encouraged Auerbach and Talmi to tackle the isomerism in and nuclei. When more experimental data in the Zr region became available, better theoretical analyses followed. Cohen, Lawson, Macfarlane, and Soga (1964) considered the N = 50 nuclei from to They adopted configurations with and protons, taking the difference between single nucleon energies from the measured levels in The 9 independent two-nucleon matrix elements in these configurations and one single nucleon energy were determined by fitting to 20 measured binding energies and level positions. The overall agreement of energies calculated with these matrix elements with experimental values was very good. At least 10 predicted levels were in good agreement with those later discovered experimentally (in Table 1 of their paper several calculated levels of were omitted). The authors found that the effective interaction in configurations is diagonal in the seniority scheme. Like Talmi and Unna (1960c), they found that the non-diagonal matrix element between the J = 9/2 states with seniorities and is only .03 MeV. From this, follows the result that the mixing with states cannot cause mixings of states with different seniorities. Another explicit feature of the effective interaction which they obtained is a rather strong average repulsion between and protons. This feature was obtained in all following papers on the Zr region. The repulsive nuclear interaction between identical nucleons in different orbits, was noted earlier in lighter nuclei. In the case of protons, it includes the Coulomb interaction which is also repulsive. The authors concluded by stating that “a severely truncated shell model is in excellent agreement with the known experimental facts about the lowest levels of nuclei with 50 neutrons”. About the same time, Auerbach and Talmi (1965) considered the same nuclei and in addition, some nuclei with protons in the and orbits and neutrons in the one. The authors confirmed that the effective interaction in configurations was diagonal in the seniority scheme. From this follows that expected spacings between the lowest J=2,4,6,8 levels in should be equal to those of Since this fairly agrees with experimental data, they took these spacings to be the spacings of the unperturbed configuration. This leaves two single nucleon energies (outside ) and 5 matrix elements, the non-diagonal and the average interaction

These 7 theoretical parameters were determined by fitting calculated energies to 11 experimental ones. The latter included binding energies and excited states

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in N = 50 nuclei from to The agreement between calculated and experimental energies was very good. Several predicted level energies agreed well with later measurements. From the value of and the observed position of the level of the position of the level was calculated to be at 2.66 MeV in good agreement with the experimental 2.74 MeV. The calculated positions of the and levels in were 2.64 and 2.98 MeV in good agreement with the observed positions at 2.53 and 3.06 MeV. To calculate energies of nuclei with neutrons beyond N = 50, the interaction between and protons with neutrons was needed. It can be obtained from the and levels, as well as from the positive parity levels with J = 2 to J = 7 levels in Of the latter, only the J = 7 and J = 2 were known then experimentally. In addition, all these levels were observed in where they should belong to the configuration which is strongly mixed with the higher one. Using the information from both nuclei, as well as the proton proton effective interaction, matrix elements of the proton neutron interaction were extracted. They were used to recalculate the spectrum which turned out to be similar to the one from the approximate calculation, and the existence of the isomeric state was verified. The proton neutron interaction was used to calculate energy levels of nuclei with several neutrons up to and The agreement with the meagre experimental data was fair. Several years later, many more experimental data were obtained and more theoretical work was carried out on this region. Bhatt and Ball (1965), who explored the experimental energy levels of the proton - neutron configuration, used their results to calculate spectra of Nb, Mo and Tc. They simplified the proton states to be due to a configuration, ignoring the admixtures of pairs. They argued that the “effect of this pair excitation on the calculated energy spectra is expected to be partially compensated for by the use of experimentally determined two-body matrix elements”. The valence neutrons are taken to be in the orbit. The authors, avoiding configuration mixing, used only diagonal matrix elements which they determined from level energies in nuclei with simple configurations. They took the neutron energies from the levels, the proton neutron energies from the correct levels of and the proton energies from the spectrum. As previous work showed, the effect of admixtures is already rather small in and heavier N = 50 isotones. It is well known that the best way to determine matrix element from experimental data is to try to obtain the best fit to all of them. Small deviations in the two-nucleon matrix elements may be amplified in many nucleon configurations. Since the authors started from fixed values of matrix elements, the agreement with experiment was very good for certain states but only semi-

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quantitative in others. Many of the measured levels did not have then spin and parity assignments and only later was it possible to compare their positions with calculated ones. In general, the low-lying level positions in odd-even and odd-odd nuclei are well reproduced. For example, in the lowest calculated levels with spins 6,3,4,7,5 are very close, and even in the correct order, to the experimental ones, For higher levels the difference increase to . 1 and .2 Mev. In the situation is similar, levels with those spins are the lowest, are in the right order (different from the one in ) and in good agreement with experiment. In calculated levels about 1.5 MeV above the ground state, correspond to measured levels but some differences amount to .8 MeV. In the nucleus, up to 2 MeV there is good agreement between calculated and measured levels. There is, however, a state just above 2 MeV with no corresponding one in the calculation. This situation is even more disturbing in where a measured level is lower by about 1 MeV from the next calculated level. In that nucleus, a level was measured at 1.15 MeV above the ground state whereas no calculated level lies below 2.8 MeV. In there are no neutrons and the calculated levels belong to the configuration. Like the situation in previous papers, the agreement with experiment was very good. Only the 15/2,17/2 and 21/2 calculated levels were too low by about .2 MeV. In the lowest calculated states with spins 7,2,3,6,5,4 are in very good agreement with measured levels. Bhatt and Ball calculated also binding energies relative to the core for which they need 5 more parameters. These are the J = 0, the J = 0 and the J = 7 energies as well as the and single nucleon energies. In this calculation, they determined the values of these parameters by a least squares fit to 20 experimental binding energies. The authors obtained excellent agreement between calculated and experimental energies. The binding energies under consideration ranged between 5 and 47 MeV but the r.m.s. deviation between calculated and measured energies was only .115 MeV. Bhatt and Ball calculated B(E2) values for the transitions from the ground state to the first state in and The effective quadrupole moments of a proton and a neutron were taken from the corresponding B(E2) measured in and The “calculated values are smaller then the experimental ones by about 30-50%”. The measurements were not very accurate so the agreement seems to be satisfactory. In their summary, the authors suggested various experiments which could clarify the extent of configuration mixing in these nuclei. The same nuclei were considered in detail in a long paper by Vervier (1966). In the first part, the author carried out an independent calculation which was identical with that of Bhatt and Ball (1965). He considered valence protons in

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the orbit and valence neutrons in neutrons with effective interactions taken from the and levels. He used the spin assignments made in the experimental paper but also the assignments suggested by Pandya (1964). The latter, also justified by more recent experiments, led to somewhat better agreement with the data. The author concluded this part by stating that the calculations “account rather successfully for the low-lying energy levels of the Nb, Mo and Tc isotopes with The agreement is better when the neutron number is close to 50 and becomes worse when it increases”. This trend is clearly evident in the spectrum of where the calculated excited states lie higher by more than .75 MeV above the measured ones. The author correctly attributed this trend to the influence of other neutron orbits, which “explanation is further substantiated by the results of and reaction studies on the Mo isotopes”. Indeed, it seemed that the attraction of protons and neutrons lowers the orbit and gives rise to configuration mixings. In fact, it turned out that the ground state of as well as those of several Pd and Cd nuclei with N >57, have spin indicating that the orbit is not closed at N = 56. In the second part, Vervier included in his calculation also the proton orbit and mixing between proton configurations. For the proton configurations considered, there are two single proton energies and 9 two-body matrix elements to be determined from the experimental energies. The number of these parameters can be reduced to 5 if only level spacings are calculated and spacings of levels with seniority are taken from the Zr levels. The values of these parameters were calculated to give the best fit to 7 measured energies. With these values, the author calculated the spectra of nuclei with Z=39,40,41,42, and 43 and N = 50. He found that these calculations “are able to account successfully for most experimental energy levels in these nuclei and for some of their properties: spectroscopic factors for direct reactions... and E2 transition probabilities”. In the introduction, Vervier mentioned that when his work was completed, similar studies were published. He explained that the first part was similar to the work of Bhatt and Ball (1965). The part with only proton configurations, he found in good agreement with the papers of Cohen, Lawson, Macfarlane, and Soga (1964) and Auerbach and Talmi (1965). Vervier then determined the interaction between protons and neutrons by using the and levels and neutron energies from levels. For the latter, he used the correct spin assignments. He used these matrix elements to calculate spectra of nuclei with Z=39,40 and 41 with The agreement between calculated and experimental level positions was “quite satisfactory”. It is also in fair agreement with more recent experiments. As in the previous analysis, it becomes worse as the neutron number increases. In particular, the results for and indicate that, unlike in the Zr case, for

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N = 46 the neutron orbit is not closed. The author remarked that some of these nuclei were considered by Auerbach and Talmi (1965) but his results were more extensive. He states that “the results of the two analyses are very similar where they” overlap “and both reproduce satisfactorily the experimental data”. Vervier noticed that “numerous first-forbidden unique are observed in the Y and Zr isotopes, corresponding to a neutron to proton transformation”. He was aware that their rates were “generally smaller than the single-particle estimate”. Like Oquidam and Jancovici (1959), of whose work he was not aware, the author used an effective matrix element for the transition which he determines from the of the of to the ground state of Using the wave functions which he obtained in his analysis, Vervier calculated the for 12 more known transitions. The agreement between rates calculated in this way and experimental rates quoted in the paper, was only fair as could have been expected from transitions which are very sensitive to small admixtures in the wave functions. The current experimental are rather different from the quoted values. According to them, the weight of the state in the Zr ground state is about 75% as compared to the calculated percentage of 62. The Zr region will occupy nuclear theorists for many more years. Ball, McGrory, Auble, and Bhatt (1969) extended the calculations described above, to including also the higher-lying odd-parity levels, and to The interaction they used is the one in the paper by Cohen, Lawson, Macfarlane, and Soga (1964). They obtained good agreement between the calculated spectrum of and the measured one, including the high-lying and levels. The measured positions of the latter are 3.62 and 4.25 MeV whereas the calculated values are 3.6 and 4.4 MeV respectively. Levels with these spins are predicted also in at 3.70 and 4.39 MeV which agree very well with measured levels, tentatively assigned spins of and at 3.65 and 4.35 MeV respectively. There is good agreement also for the other levels. The authors were impressed by the “ability of the calculation to reproduce the observed changes in the spectra with the addition of two protons”. Equally impressive was the fact that this addition keeps the spacings between with J=2,4,6,8 practically unchanged, as follows from the seniority scheme. The authors calculated also the binding energy of relative to the core, to be 38.81 MeV. Only later relevant binding energy were measured and the experimental difference was found to be 38.38 MeV. After a few years, Ball, McGrory and Larsen (1972) published calculations for several N = 50 nuclei. Starting with and going up to they could use “a much larger body of experimental data” than what was available to previous authors. First they tried to use the modified surface delta interaction in which, for T = 1 interactions, there were two parameters. They fitted them, as

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well as two single proton energies, to 6 binding energies and 25 excited states. They found that “while the fit to binding energies is acceptable, the excited states are not well reproduced by such a simple assumption for those nuclei”. Then the authors carried out a search for the values of the 11 parameters which give the best fit to 6 binding energies and 38 excitation energies. “Thus, a total of 44 pieces of experimental data are fit with an average deviation of 53 keV”. The authors pointed out that there are also low-lying levels which seem to be outside the model space. In “even nuclei, this is the octupole vibration which occurs near 3 MeV of excitation”. The authors found that “an interesting feature” of the resulting interaction “is a significantly increased mixing of higher seniority components in the lowlying states. For example, the probabilities in the predominantly states of spin and in are...almost an order of magnitude larger than predicted by using the interaction of” Cohen, Lawson, Macfarlane and Soga (1964). The actual weights of the mixed states were, however, negligibly small, “0.1 to 0.2%”. Even if such admixtures are accepted, seniority is still an excellent quantum number in these configurations. The authors calculated also the binding energies of and relative to the binding energy of the core. Their calculated values are 41.43 and 46.50 MeV respectively. Later measurement are in excellent agreement with these predictions, 41.435 and 46.565 MeV. A detailed study of N = 50 nuclei was performed by Gloeckner and Serduke (1974). They continued earlier work but tried to fit the data with 4 different effective interactions, all determined from experimental data. The 11 matrix elements needed for describing configurations with and protons outside the inert core, were determined first, by the best fit to 45 energies. Next, the authors imposed the extra condition that seniority be an exact quantum number in configuration. The spacings were restricted by the condition that their linear combination which is the matrix element between the J = 9/2 states with and should vanish. In a third fit, the authors “considered contributions from excitation energies and from binding energies separately”. Wave functions calculated from the effective interaction determined in either of these 3 methods were used to calculate E2 transition rates. They agreed fairly with measured rates if an effective charge of 1.7e was adopted. One glaring discrepancy, noted before by Ball, McGrory, Auble, and Bhatt (1969), was the rate of the to transition in The calculated rate was faster by a factor 20 to 25 than the measured one. Also the calculated to transition rate is faster by factor 2 than the experimental rate. The authors tried to obtain an effective interaction which would lead to better agreement with E2 transition rates. The resulting interaction had indeed the desired effect but the r.m.s. deviation obtained with it for energies is

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.08 MeV. This should be compared with .0519, .0522 and .0594 obtained with the first 3 interactions respectively. A “preliminary report on this work” dealing mainly with the “inhibited transitions” was published by Gloeckner, Macfarlane, Lawson, and Serduke (1972). Gloeckner and Serduke used their matrix elements to calculate levels of and which were not known experimentally. The agreement with their predictions and subsequent measured level spacings was also very good. Another interesting results of the paper was the determination of the sign of the non-diagonal matrix element between the J = 0 states of the and configurations. “The energy of all states in the N = 50 nuclei is independent of this sign”. They fixed it by calculating electromagnetic M4 rates due to transitions between a proton in the and orbits. “Agreement with experiment is far superior if” it “is chosen positive”. They noted “that this choice agrees with the sign predicted by a surface delta interaction”. A major problem seemed to be posed by the high spin states, at 4.257 MeV in tentatively assigned and a level in which was thought to be at 4.597 MeV but is actually at 5.858 MeV. Both in the paper and in the preliminary report, the authors stated that they “cannot be reproduced within our simple model”. The calculated position of the level is nearly 1 MeV higher in all four sets of matrix elements. The effective interactions determined by Gloeckner and Serduke in the various ways have very similar features. The authors found that the “most significant result that emerged here is that a seniority conserving effective interaction is entirely consistent with the known experimental levels in the N = 50 isotones”. Other features of this T = 1 interaction were consistent with those found in simpler configurations. The interaction between protons in the and orbits was repulsive in both J = 4 and J = 5 states. The average repulsion was .43 or .44 MeV. The coefficient of the pairing term in the seniority scheme binding energy formula, was large and attractive, 2.21 to 2.24 MeV in absolute value. The coefficient of the quadratic term was repulsive and considerably smaller, .51 to .52 MeV. The rather strong repulsion was due to the Coulomb interaction which was included in the effective interaction between the protons. Its contribution to diagonal matrix elements could be estimated as a repulsion of about .3 MeV. After several years, many more levels were measured in and nuclei. Amusa and Lawson (1982) made use of one of the effective interactions determined by Gloeckner and Serduke (1974) to calculate energies of those levels. The agreement between their calculated level positions and binding energies relative to and measured values, was very good for low-lying levels. It was fair for higher levels up to the level close to 5 MeV and the

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level at about 5.5 MeV. This was the case for more than 40 levels in both nuclei. Amusa and Lawson also found good agreement between experimental and calculated E2 transition rates. The latter were calculated with an effective proton charge of 1.72e and An interesting feature of the calculation was the occurrence of two closely lying levels in They are due to coupling of a proton to two states with J = 17/2 of the configuration. One was the state with seniority and the other is the state with Hence, the state, due to the J = 21/2, would have zero E2 matrix element to the lower state as dictated by the seniority scheme for states with the same in an orbit half-filled. Experimentally, a large rate was found. The authors explained that due to the proximity of the calculated states, small admixtures from outside the model space could cause mixing of the two states and, hence, give rise to the observed rate. The predicted positions were at 4.254 and 4.319 MeV. The level to which the level decays was observed at 4.198 MeV. The higher level has not been observed. An experimental paper on magnetic moments of N = 50 isotones was published by Haeusser et al., (1977). Magnetic moments were measured for states in several nuclei and compared with the predictions of simple configurations. The authors got into complicated calculations to account for the slight differences between the simple picture and experimental data. These involve core polarization Arima and Horie (1954) as well as meson exchange corrections to magnetic moments. Even without core polarization, it seemed that simple wave functions, with effective of the protons, led to a reasonable description of the experimental situation. If the for and protons were taken to be 1.365 and –.31 nm respectively, there is reasonable agreement with the data. The of the states in and and the and states in should be equal to that of a single proton. The measured values are 1.356, 1.413, 1.387, and 1.354 nm respectively. The g-factors of the states in and should be 1.266 nm as compared to the measured values 1.277 and 1.231 nm respectively. Finally, the g-factor of the state in was calculated to be 1.289 nm whereas the measured value is 1.262nm. Gloeckner (1972) considered new experimental results on proton occupation numbers in Zr isotopes. According to Vervier (1966) these should decrease when going from contrary to the experimental data. He then concluded that the neutron orbit considered by Vervier was not adequate and the orbit should be included. His main argument was that from the spectrum follows that the average attraction of neutrons and protons was somewhat weaker than their interaction with protons. Actually, slight variation in the matrix elements could lead to agreement with the occupation numbers without the orbit and without significant change in

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calculated energies. This would be also consistent with the 1.088 MeV gap between the and levels in which is bigger than the corresponding gap of .909 MeV in Detailed description of the calculation and results were published later by Gloeckner (1975). There, he looked at nuclei with by considering the and valence neutrons and valence protons occupying the and orbits. The single proton energies and their effective interaction were taken from Gloeckner and Serduke (1974). The neutron-neutron interaction (8 matrix elements and two single neutron energies), as well as the protonneutron interaction (14 matrix elements), were determined by a least square fit to 91 experimental energies. Energies calculated with the matrix elements thus determined, were in good agreement with observed levels in and This was also the case for Zr isotopes from to although there was clear deterioration towards higher excitations and heavier isotopes. The author calculated strengths of various stripping and pickup reactions involving Zr isotopes and found fair agreement with experiment. He noted that in neutron stripping into even Zr isotopes, “the strength to the lowest states decreases from to to this state looks less and less like a neutron coupled to an even Zr core”. The author seemed to be unaware that this is precisely the seniority scheme dependence on for configurations. The dependence leads here to the ratios 1 to 2/3 to 1/3. These values are very close to the calculated values .98 to .67 to .36 and agree well with the measured ones. This should be contrasted with the neutron pickup reactions from these isotopes whose strengths should be proportional to This behaviour seems to agree with the rather widely scattered experimental data. Early data on these reactions were quoted by Talmi (1962a) as evidence for rather pure configurations in Zr isotopes. Gloeckner criticized that paper where “good agreement for the binding energies of Zr isotopes in a pure model” was found. He explained that “this suggests that good agreement with energy level data (binding energies in this case) in simple models should not be used to confirm pure configuration assignments”. While this statement is true, the author seems to be unaware of the fact that in the case of Zr isotopes, not only binding energies are in agreement with data. Relations between all known levels which may be due to configurations also agree with experiment and with information from binding energies. It seems that if admixtures of neutron configurations are present, their effect is a small perturbation and can be absorbed into the effective interaction between neutrons. The neutron-neutron effective interaction determined by Gloeckner probably over emphasizes the role of neutrons. It exhibits a small but attractive average interaction between and neutrons, contrary to the situation in all other cases. Indeed, the author found

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that “the most poorly determined combination of matrix elements involve the neutron-neutron diagonal and off-diagonal and matrix elements”. The calculated binding energies of Zr isotopes, relative to are in good agreement with measured ones. Still, in the case of the discrepancy jumps to .35-.46 MeV. Rather large deviations of level spacings between calculation and experiment are obtained for the heavier Nb isotopes. The author commented that “there exist many states in the region considered that are outside the model space”. The situation of heavier Zr isotopes is rather different. Up to lowlying levels are consistent with the valence neutrons outside occupying the orbit. In the ground state spin, can be viewed as due to a neutron in the orbit. Excited states of with spins and lie higher than 1 MeV above the ground state. The spectrum of is consistent with a ground state. There is a level at .853 MeV and a level at 1.223 MeV excitation. The measured spectrum of was a big surprise. The measured J = 2 level is only .213 MeV above the ground state and the ratio between the energies of the J = 4 and J = 2, E4/E2, is 2.655. The nature of this “phase transition” is even more pronounced in the position of the level is only at .152 MeV and E4/E2=3.149. These levels seem to be the lowest members of a rotational band (in a perfect J(J+1) rotational band, E4/E2=20/6=.333...). This phenomenon is even more pronounced in Sr isotopes, with the same neutron numbers. In the position is at .145 MeV and E4/E2=3.001, and in the excitation energy is only .129 MeV and E4/E2=3.232 is very close to the value of a perfect rotor. In Mo isotopes, the rotational limit is approached only in with 64 neutrons, with the excitation energy of .172 MeV and E4/E2=3.044. It was clear that such spectra could not arise from a normal filling of orbits. Adding more neutrons, in particular in orbits, adds attraction to protons in the orbit. They may be raised into this orbit from the and orbits, in which case, collective spectra may naturally arise. This mechanism was introduced and discussed by Federman and Pittel (1977). In their first paper, they presented only qualitative arguments. They stressed the importance of the strong attraction between protons and neutrons as proposed by de-Shalit and Goldhaber (1953). And attribute it, Talmi (1962b), to strong interaction in L = 0 states like in the state, with T = 0, of the configuration. In a detailed paper, Federman and Pittel (1979) presented shell model calculations which reproduce the situation in the Zr region. To simplify the calculation, they took the neutron orbit to be closed. “The valence neutrons were restricted to the and

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orbitals and the valence protons to the and orbitals”. “The single-particle splittings” were taken from the observed levels of and The two-body interaction was taken from a paper by Auerbach and Vary (1976) where those authors compare results of using several interactions to calculate the spectrum. This interaction is a Yukawa potential “with Rosenfeld admixture with: and Federman and Pittel calculated level spacings and wave functions of and the ground state of They show only the level scheme. The agreement between calculated and measured level positions was good for the first excited and levels. At higher energies, there are more experimental levels than predicted ones and the situation was unclear at best. The authors’ main interest was in the composition of the J = 0 states which may indicate the effect which they explored. This effect is clearly demonstrated by the occupation probabilities of the various orbits. In the ground state, the occupation probabilities are the expected normal ones. They are given for the neutron and orbits, by 1.81, .12 and .08 and for the proton and orbits, they are 1.63 and .34. In the first excited state, however, the situation is different, those numbers are .33, .28 and 1.39 for the neutrons and .25 and 1.67 for the protons. In the situation is reversed, the ground state has more neutrons and protons. The corresponding occupation numbers are 1.82, .80 and 1.39 for the and neutron orbits whereas for the proton and orbits they are given by .42 and 1.48. It is clear that this calculation is only an approximation, but it certainly points to the correct reason for this behaviour of nuclei in the Zr region. Measurement of energies of nearby nuclei, enabled the consideration of configurations with neutron holes in the orbit. Nilsson and Grecescu (1973) reported high-spin states in and which all have N = 49. Levels were interpreted as due to proton states in corresponding N = 50 nuclei, coupled to a hole state. The authors explained that the matrix elements of the proton neutron hole interaction could be taken from levels in They also quote unpublished calculations of Vervier for the positive parity states in There was good agreement between calculated and measured levels with spins 9/2(g.s.), 13/2,17/2,21/2,23/2,25/2. Gross and Frenkel (1974) took up the theoretical study of these configurations. Protonproton interactions were taken from previous analyses of N = 50 nuclei. The authors used 26 excited states in N = 49 nuclei to obtain the matrix elements of the effective interaction of and protons with neutrons (holes). They did not consider binding energies and determined only spacings between various energy levels. The values of these 10 spacings (the energy of the J = 0 state does not contribute to the energies considered) were determined with rather large errors. Still, they used them to calculate spectra of

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odd-odd nuclei, and as well as, odd-mass nuclei, Mo and The agreement between calculation and experiment is good, the r.m.s. deviation is 0.1 MeV. The authors checked the charge independence of their interaction by using the Pandya particle-hole transformation. Differences of the transformed energies were compared to differences of the energies (of states with even J values) used in the calculation. The deviations turned out to be “within 50 keV”. Gross and Frenkel explained their plan to include also the neutron orbit in a forthcoming calculation. Meanwhile, Serduke, Lawson and Gloeckner (1976) published a detailed paper on the “Shell-Model Study of the N = 49 Isotones”. They followed earlier work by adopting as an inert core and by taking the proton-proton interaction from the analysis of N=50 nuclei. They improved on former calculations by letting the neutron (hole) be in both and orbits. Energies of single-hole energies were taken from binding energy and the level spacing in Without invoking various symmetries, there are 20 matrix elements which define the interaction between a proton and a neutron hole. These were determined by a least squares fit to 63 energy levels in N = 49 nuclei, to The authors tried four ways to determine these matrix elements. In the first, they took all of them to be free parameters. In the second, they imposed one condition on the matrix elements with T = 1, obtained from the matrix elements by the Pandya transformation. These should satisfy the condition that seniority is an exact quantum number in configurations with maximum isospin, In the third way, the authors “impose the requirement that the nuclear part of the model effective interaction conserve isospin”. They imposed conditions on the nucleon-nucleon T = 1 matrix elements, obtained from the nucleonhole matrix elements by the Pandya transformation. They should be equal to those determined from N = 50 nuclei from which the Coulomb energy was subtracted. The requirement of charge independence of matrix elements, implies also the equality of matrix elements of nucleon hole and hole nucleon states with J = 4 and J = 5. The remaining free parameters are the 9 matrix elements with T = 0 in the configurations considered. In the fourth way, the authors simply used only level spacings to be fitted by all matrix elements “subject to the constraint that the effective interaction conserve seniority”. As in the second way, there are here 19 free parameters. The matrix elements determined in these ways were used to calculate energies and wave functions of states in the N = 49 nuclei considered. The agreement between calculated and experimental energies was assessed by the root mean square deviation for the four different choices of effective interactions. The authors refer to it as “a figure of merit” and is given by the

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usual definition of the r.m.s. deviation

where is the number of energies considered and is the number of free parameters. This is the meaningful definition since if is omitted, perfect agreement could be obtained by using free parameters. The authors found this deviation to be .0833 MeV for the first way, .0830 MeV for the second and .0886 MeV for the fourth. The deviation for the third way, in which charge independence of matrix elements was imposed, is twice as big, .166 MeV. The authors “interpret this to be strong evidence that the N = 49 energy level data favour a small isospin breaking in the model space residual twobody effective interaction”. This “small isospin breaking” seemed to be small indeed. The “isospin impurities” in the “lowest isobaric analog states are several hundredths of one percent for the T = 0 [third] fit and roughly a factor of ten larger for the” seniority conserving interaction. Calculations of spectra were carried out by the authors using the set of the “seniority-conserving proton-neutron hole matrix elements”. The agreement between calculated and experimental energies “is very good, especially for yrast levels”. Still, there were many measured levels which seem to be outside the model space. This is probably due to the proximity of and hole states, .5 and .85 MeV above the hole state in Many levels are well reproduced. In the calculated positions of the and levels agreed well with the measured ones. Since then, high-energy levels with spins and have been discovered and their positions agree well with the predicted ones. Binding energies of the nuclei considered in the paper were also calculated and agreed very well with the measured ones. The “overall good agreement” between calculation and experiment, obtained by Serduke, Lawson and Gloeckner included also “electromagnetic decay characteristics and spectroscopic factors”. In particular, M4 transition rates were well reproduced by using effective operators for the protons and neutrons. In a subsequent paper, Gloeckner, Lawson and Serduke (1976) used the wave functions obtained by Serduke, Lawson and Gloeckner (1976) to calculate “allowed beta decay of the N = 49 isotones”. Years earlier, de-Shalit and Goldhaber (1953) noticed some highly retarded allowed beta decays in several nuclei, including decays from They attributed such “abnormal” decays to a “stabilization” of proton configurations by the neutrons and vice versa. This effect should be due to the stronger interaction between protons and neutrons in the same orbit. Hence, in a state with a neutron hole, the two valence protons in strongly interacting with neutrons, would occupy the orbit. Similarly, in a state with a neutron hole, there is a smaller at-

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traction with protons and the neutrons would make the valence protons occupy the orbit. If such configurations were pure enough, the beta transitions between the yrast states in and as well as between the yrast states of these nuclei, were strongly attenuated. Goldhaber and de-Shalit believed that such pure configurations could exist near closed shells which would explain the large logft values of 7.1 and 6.2 measured for those transitions respectively. They wrote on the basis of several such cases, “the assumption of “purified” states in these cases, therefore, seems to be quite plausible”. It was even then doubtful that such pure configurations could exist. With the knowledge of the effective interaction, the amount of mixing in the proton configurations in the and states in was determined to be appreciable. Serduke, Lawson and Gloeckner looked for the explanation of these retarded beta decays. They noticed that in these, as well as in other cases, the Gamow-Teller matrix elements contain contributions from transitions between a proton and neutron, as well as between a proton and neutron. The bare single nucleon matrix elements for a nucleon and a nucleon have opposite signs. By using effective matrix elements, with opposite signs, they were able to obtain large cancellations in the matrix elements between calculated and wave functions. They calculated beta decay rates of several states of N = 49 nuclei and obtained fair agreement with measured rates. A few months later, a similar calculation was published by Gross and Frenkel (1976). They also took as an inert core and consider valence protons in and orbits and neutron holes in the closed and orbits. The authors used in their analysis also nuclei with N = 50 neutrons and aimed at “using two-body effective interaction consistent with charge independence”. They took the neutron-neutron interaction to be equal to the proton-proton interaction from which the Coulomb energy was removed. This is also taken as the T = 1 part of the interaction between protons and neutrons. Matrix elements of the latter were obtained by the Pandya particle-hole transformation from the matrix elements of the proton-neutron hole system. The authors took the Coulomb interaction to have the same value in all states of a given two proton configuration and hence, obtain 6 conditions on the matrix elements. Since there are 9 matrix elements with T = 1 in this shell model sub-space, 3 more conditions were needed. Two conditions were obtained by equating the diagonal matrix element in the proton neutron hole states with J = 4, 5 to the ones in the proton neutron hole state. The authors did not calculate the single holes energies from the two-nucleon interaction and the single proton energies. They could determine this way the single protons energy in the nucleus but the estimated Coulomb energy difference between and could not be trusted. Instead, the authors im-

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pose the third condition which makes the T = 6 state of (the isobaric analog state) equal to the state obtained by acting on the ground state of by the isospin lowering operator. This condition is a linear relation between the and proton energies and the corresponding hole energies, the energies in the two J = 0 particle hole states as well as the non-diagonal element between them. “Since the parameters of the effective interaction” that the authors “fit have statistical errors”, they “have decided not to use the above constraints for reducing the number of independent matrix elements, but rather to include them in the fit, together with the energy levels”. The authors included in the fit differences of the two sides of the constraints obtained from the conditions on charge independence. They gave them 3 times the weight of the differences between calculated and experimental energies. Thus, they have 33 free parameters which they determined by the best fit to 95 experimental energies. Gross and Frenkel use the effective interaction determined in this way to calculate nuclear spectra. The r.m.s. deviation of calculated from experimental energies which they obtained, is .07 MeV. They explained that without the conditions imposed by charge independence, they could obtain a better fit, with a .04 MeV r.m.s.. The resulting effective interaction in that case, however, would “strongly violate charge independence”. Their “calculated spectra of N = 50 nuclei were similar to those” obtained by previous authors. Most of the calculations for N = 49 nuclei are similar to the results of Serduke, Lawson and Gloeckner (1976). This is the case also for the levels which were not included in the fit and some of them were measured later. The level of the higher multiplet in predicted at .957 MeV was later tentatively quoted at .96 MeV excitation. The authors excluded from the fit the states in and since “their inclusion badly violates the additional conditions” imposed. The calculated binding energies quoted in the paper were in very good agreement with measured ones. For example, the single hole energy was determined to be 11.137 MeV as compared to the single neutron separation energy of which is 11.113 MeV. Yet, in the same nucleus, the calculated position of the state was . 134 above the ground state whereas the measured position in is .388 MeV. Since the constraints were not strict, the effective interaction obtained in the analysis exhibits slight deviations from charge independence which are, however, within the statistical uncertainties. For example, the energy in the J = 4 state of a proton neutron hole is .438 ±.077 MeV, whereas the energy of the J = 4 state of a proton neutron hole is .427± .072 MeV. Nevertheless, the overlap of the state obtained by acting with the isospin lowering operator on the ground state of and state obtained by diagonalization, deviates from 1 by less than This is also the case for states in other N = 49 nuclei where the overlaps are

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very close to 1, the largest deviations are less than The T = 1 matrix elements within the configurations lead to seniority as a good quantum number. The non-diagonal element between T = 3/2, states with seniorities and is only of the difference between th diagonal elements. On the other hand, the T = 0 interaction leads to strong mixing between states with different seniorities. The lowest J = 9/2 state with T = 1/2 has only 73% of the component, the rest are states. The authors tried to calculate spectra of N = 48 nuclei by using their charge independent effective interaction. They found rather large disagreement with observed low spin levels, indicating possible admixtures of excited configurations. Another nucleus with N = 48, became the subject of detailed experiments and following that, a theoretical analysis by Amusa and Lawson (1983). They followed the work of Serduke, Lawson and Gloeckner (1976), taking as a closed shells nucleus and considering protons and neutron holes. They used one effective interaction from that paper and took two-hole matrix elements from the proton-proton matrix elements, removing from them the Coulomb repulsion. With these matrix elements, they calculated level positions up to 8 MeV obtaining good agreement with experiment. The highest positive parity level, compared with available data, has spin 37/2, is at about 6.5 MeV. The highest negative parity level has spin 41/2 and is at about 7.5 MeV. The authors predicted “that the state lies 183 keV above the ground state. It was later discovered at 140 keV excitation. The calculated binding energy, relative to is 8.29 MeV compared with the experimental value of 8.37 MeV. The authors considered in detail E2 and M1 transitions and succeed in obtaining fair agreement with experimental rates by using effective charges and They pointed out that observed B(E2) values vary “from state to state ranging from about 1/10 to 20 Weisskopf units”. In a couple of cases where discrepancies were found, they “found it possible to bring theory and experiment into agreement by a small mixing” of wave functions. An experimental paper by Oxorn, Mark, Kitching, and Wong (1985) included “a shell-model description of the N = 48 and N = 47 nuclei”. These are and and as well as and The authors used the effective interaction derived by Gross and Frenkel (1976) and the same shell model space. The best agreement with observed level spacings yields a r.m.s. of 140 keV in and 100 keV in This was obtained, however, by excluding in both nuclei, the lowest levels with spins 0,2,0,2,4 and several negative parity states where the deviations are considerably larger. The situation was similar in and good agreement was obtained for many levels but others were not well reproduced. In the N = 47 nuclei, the shell model

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space does not seem to account for all observed low lying levels. If these are not included, and the lowest and 1/2– are also omitted, the agreement between calculations and experimental level spacings is fair, with a r.m.s. of 170 keV. In only few levels were measured and their spacings are well reproduced by the shell model calculation. The authors concluded that “in spite of the presence of vibrational modes of excitation in the same region, there are states which have a clear shell-model structure”. Blomqvist and Rydstrom (1985) repeated the shell model analysis of N = 50 isotones between and With more experimental data available, they expected a better determination of the effective interaction in the space of and proton orbits. They explained that is not a very good closed shells nucleus. Hence, by using more data from heavier nuclei they hoped “to obtain matrix elements that are more relevant for the nuclei close to They determined the 9 two-body matrix elements and 2 single proton energies from a fit to 63 experimental energies from the nuclei to Energies calculated with these matrix elements were in very good agreement with experimental energies, the r.m.s. deviation is 64 keV. The values of these matrix elements were rather close to those obtained by Gloeckner and Serduke (1974). The average differences were between 31 and 44 keV for the various fits of those authors. “The largest single difference is 85 keV”. Seniority is a very good quantum number for the wave functions which they obtain. The matrix element between the and states of the configuration was calculated to be only 7.4 keV. The authors concluded that the model could be improved by incorporating particle hole excitations from proton orbits. This improvement was taken up by Xi and Wildenthal (1988). They considered N = 50 isotones where valence protons occupied the and orbits. They considered about 170 energy levels due to these configurations in nuclei from through There are 65 two-body matrix elements and 4 single proton energies which were determined by a fit to the 170 energies. The authors used a truncation scheme “to reduce the size of the Hamiltonian matrix to dimensions of 1500 or smaller. The truncation scheme allows only four particles to be excited across the Fermi surface”. They excluded from the analysis “states with very high energies” and “low-lying states of some even-mass nuclei”. Not all of the 69 parameters were well determined by the fit to the selected set of energy levels. Only “about 35 combinations of these parameters were well determined”. “These 35 were fitted to the data and the remaining combinations were held constant” during the four final iterations. The agreement between calculated and experimental energies was good. “The average deviation in excitation energies between experiment and theory

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finally obtained was about 150 keV”. The “binding energies are well reproduced by the present calculation. The r.m.s. deviation between experiment and theory is about 100 keV”. The authors explained that their two-body matrix elements in the space were almost identical to those of Blomqvist and Rydstrom (1985). Actually, the differences were . 1-.2 MeV. A glaring difference is in the (J = 0) energy which was attractive in the former paper (–.465 MeV) and repulsive in the present one (.688 MeV). The matrix elements contain contributions from the repulsive Coulomb energy but those could not account for a repulsive pairing energy. Coulomb energy contributions should make the theoretical average repulsion between protons in different orbits more repulsive. These are indeed repulsive between and protons, as well as for and protons. The average interaction between and protons is also repulsive but for it is barely repulsive (.038 MeV) and for it is attractive (–.038). The authors observed that the predictions “are better for nuclei above than below it”. For those nuclei, as well as for “the wave function structures are much more complicated, and thus very sensitive to many less well determined interaction parameters”. In fact, “the off-diagonal matrix elements which connect the and subspaces are important in determining the structure. These matrix elements are least well determined from our fitting procedure”. The great improvements in experimental nuclear physics following the introduction of multidetector arrays, offered new challenges to theorists. High spin states in some N = 50 nuclei are reported by Roth, Arnell, Foltescu, Skeppstedt, Blomqvist, Nilsson, Kuroyanagi, Mitarai, and Nyberg (1994). States with spins up to 39/2 and higher, were observed in and semi-magic, N = 50 nuclei. The authors assigned shell model configurations to some of these states. The highest spins cannot arise from the ground configuration in which the protons occupy the and orbits and the neutrons form closed shells. The authors considered configurations in which one nucleon is raised to higher orbits. They found that such high spin states are not contained in configurations in which a proton is raised from the or orbits into the or orbits. Such states could be obtained if a neutron is raised from the closed orbit into the orbit. Since these states belong to a definite configuration, their energies may be calculated without the knowledge of all diagonal, as well as non-diagonal matrix elements. Also, no large matrices need be diagonalized. In the energies of the unique and states, which are the maximumspin members of “the and configurations, have been calculated by using the exper-

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imental energy of the state in adding the proton-neutron interaction energies derived from levels in N = 49 and N = 51 nuclei”. The calculated level positions were 7.85, 9.41 and 10.06 MeV respectively, in very good agreement with the measured energies, 7.811, 9.419 and 10.271 MeV. In it was necessary to diagonalize a 2×2 matrix, since there are two states with J = 19, due to the and proton configurations. The nondiagonal matrix element between them was well determined from the extensive analyses of low-lying levels in this region. The calculated positions of the two states and the state, were 9.96, 11.30 and 11.01 MeV, in very good agreement with the measured values, 9.920, 11.451 and 11.042 MeV. Similar results to those of were obtained for the high-spin levels of with spins and The agreement between calculated positions, 8.91 and 9.30 MeV, and measured ones, 8.874 and 9.346 MeV, was striking. Several years later, the zirconium region became a subject of intensive experimental research. Many of the experimental papers included shell model calculations of the observed data. In these, the shell model space was taken to include the and orbits of protons and neutrons. The effective interaction determined by Gross and Frenkel (1976) was widely used in those papers. In the following, several examples of such calculations will be presented. Weiszflog et al., (1992) published an experimental study of The paper included also a shell model calculation of this N = 46 nucleus. The authors adopted the interaction of Gross and Frenkel and obtained fair agreement of level spacings, with the exception of the lowest J = 0, 2, 4 for which there were considerable deviations from measured values. They also needed a shift of 524 keV of the binding energy. The authors concluded that their calculation “appears to account for the measured high spin spectrum up to more than 11 MeV and spin 23”. A more complicated case, was the subject of an experimental paper by Rudolph et al., (1992) where a shell model calculation was included. The authors use the Gross-Frenkel model space and effective interaction. Fair agreement with measured level spacings was obtained for levels up to the level at 4.2 MeV. To obtain the best agreement, the calculated binding energy was shifted by 366 keV. Another nucleus studied around that time was Weiszflog et al., (1993) who carried out the experiments, performed also a shell model calculation of that nucleus. They were surprised by “the success of the shell model for up to the highest spin observed and within the small configuration space (only and single particle orbits)”, described above. The authors used within this shell model space, the interaction of Gross and Frenkel and obtained fair agreement with the measured level spacings (they needed “a 368 keV shift of the whole spectrum”). With this shift, they obtained a r.m.s. deviation of 220 keV. The spin of the highest positive parity level, at about 6.7 MeV, is

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J = 37/2 and the negative parity one, at about 7.6 MeV was 39/2. Both calculated levels are rather close to the experimental positions. The authors discussed in detail, wave functions of various levels and list the configurations of their main components. The nucleus, considered by Amusa and Lawson (1983) was studied by more detailed experiments by Rudolph et al., (1994a). The authors also carried out shell model calculations using several interactions, and compared their results with experimental energies and transitions. They used the effective interaction of Gross and Frenkel (1976) and compared the results with those obtained with the modified surface delta interaction a la Glaudemans, Brussaard and Wildenthal (1967). The agreement between level spacings calculated with the Gross-Frenkel effective interaction with measured ones was good for positive parity levels up to 6.5 MeV and for negative parity levels up to 8 MeV. The r.m.s. deviation for levels above 2 MeV and below those energies, was quoted as 120 keV. The agreement was poorer for higher levels. Calculations with the modified surface delta interaction (MSDI) were carried out in the space (SD-1) and also in a larger space (SD-2). In the latter, up to two protons were allowed to be raised from the and into the and orbits. The interaction strengths were “chosen as and 0.44 for SD-1 and SD-2 respectively”. The agreement with experiment of calculated levels with both SD-1 and SD-2 was worse than with the Gross-Frenkel interaction. The authors calculated branching ratios of electromagnetic transitions and found that “the overall best agreement with the experiment was obtained in the calculation GF-2”. In this calculation, the wave functions were determined by the Gross-Frenkel interaction but the branching ratios were calculated with the experimental energy spacings. The agreement with experiment was good in some cases and fair in others. The authors also calculated transition strengths and mixing ratios of E2 and M1 transitions. For these, they used effective charges for the protons of 1.72e and 1.44e for the neutrons, as well as taken from simple states in nearby nuclei. The agreement between calculations and experiment was fair. The authors explained that “the predictions concerning branching ratios and transition probabilities change dramatically if only minor changes in the components of some wave functions are proposed”. A little earlier, Boedecker et al., (1993) carried out an experimental study of another N = 48 nucleus, and found “seniority levels in its spectrum. They calculated the shell model energy levels using the “singleparticle energies and effective two-body matrix elements” deduced by Gross and Frenkel (1976). To compare the theoretical energies with experimental ones, the former have “been shifted by 153 keV in order to adjust the experimental and theoretical yrast states”. This could be interpreted simply

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as just calculating level spacings. The authors explained that all positive parity states with J >25/2 must be due to three protons and two neutron holes and must have Similarly, all negative parity states with J >19/2 can be obtained by coupling of a nucleon and four protons and neutron holes. The latter must be coupled to states with and hence and the authors referred also to these states as seniority states. They predicted a level whose main component is of this kind at 3.194 MeV. Another state, a state, due to proton and neutron hole coupled to J = 9, as its main component, is predicted at 3.312 MeV. So far, only one level around 3 MeV has been identified. In general, the agreement between calculated and experimental energies was fair for the quoted levels. Position of other predicted levels were not given. The authors calculate branching ratios and E2/M1 mixing ratios by using effective charges 1.72e for protons and 1.44e for neutrons. They “used the calculated transition energies and slightly quenched single-particle magnetic moments”. The agreement with experiment is fair in many cases and rather poor in others. The authors concluded that the “overall success of the shell model with this very much restricted model space is surprising, indeed”. Weiszflog et al., (1994) measured some magnetic moments in which they describe as “a discriminating test of the shell model”. They calculated wave functions by using the Gross-Frenkel (1976) interaction and compared the results with those obtained using MSDI with They used effective “taken from the corresponding single or two particle states”. These are, in nm, and The experimental of the state at 2.549 MeV, is 1.10± .28 nm, in spite of its large error, “favours the stronger proton component (76%) of the GF calculation with gGF=1.15 over the SDI(52%) with gSDI=.63”. The measured of the state at 4.842 MeV, is .42 ± .13 nm, which is smaller than the calculated values, .80 (GF) and 1.18 (SDI). It still “gives a clear preference for the GF wavefunction over the SDI”. For the state at 4.556 MeV, the predictions of both calculations are similar, .59 (GF) and .64 (SDI). Both are close to the measured value of .50 ±.06 nm. Although it seems that MSDI is too schematic an interaction for shell model calculations, it has attractive features. In any case, it should be kept in mind that calculation of magnetic moments can be rather sensitive to small admixtures of wave functions outside the model space used. Heese et al., (1994) studied experimentally and also carried out a theoretical analysis, of the “shell model description of the neutron deficient nuclei and They used the shell model space and effective interaction of Gross and Frenkel (1976) to calculate energy levels of these N = 48 and N = 47 nuclei. They compared calculated level spacings with experiment

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since the deviations of binding energies from measured values are .50 MeV in and .28 MeV in There are only a few measured levels in observed in a cascade from a level at 6.388 MeV. The agreement between calculated level spacings and experiment was semi-quantitative and the order of these levels was correctly predicted. The quoted r.m.s. deviation was 196 keV but there are deviations which are twice that value. Many more levels were measured in and their energies calculated in configurations with 3 neutron holes. “The calculations excellently reproduce” the positive parity states, the r.m.s. deviation is 73 keV. The situation was rather different for negative parity states where the r.m.s. deviation was 410 keV. The authors pointed out that the “agreement between experiment and calculation is better at higher spins”. The authors calculated rates of E2 and M1 transitions, assuming effective charges for protons 1.72e and for neutrons 1.44e. They took from moments of and states in nuclei with N ~50. Fair agreement was obtained between calculated and experimental branching ratios in and for positive parity states in There were discrepancies for negative parity states which could be attributed to some extent, to proximity of levels with identical spins. Mixing of such states may give rise to discrepancies. The authors felt, however, that also “the main properties of the level scheme are still reproduced within the model space”.

Another N = 47 nucleus, was studied experimentally by Rudolph et al., (1993). The authors also carried out shell model calculations, using the space and effective interaction of Gross and Frenkel (1976). They found that “the level scheme can be reproduced very well up to spin J = 21”. The r.m.s. deviation of calculated level spacings from experimental ones is only 100 keV. The authors described in detail the structure of the resulting wave functions and compared them with those of neighbouring nuclei. In a subsequent paper, Rudolph et al., (1994b), described measurements of branching ratios and transition probabilities of some high spin state in They compared the experimental data with shell model calculations. They found that “in most cases the agreement with the calculations using the Gross-Frenkel parameters is striking but the results for several yrare states, very sensitively depend on the residual interaction used”. In addition to the original Gross-Frenkel interaction (GF), they used also the modified surface delta (MSDI) interaction with As in other cases, the latter “proved less successful”. The authors also tried several modifications of GF which do not change much the spectrum, but help in reducing some discrepancies. This occurs specially when levels with the same spin and parity are close in energy.

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The

Shell

A region in which configuration mixing was expected is the shell beyond Talmi and Unna (1962) thought that such mixings might be less important for states with high isospins and considered only isotopes of oxygen. They included in their analysis, binding energies and levels to which they assigned configurations. They explained that there were not enough data to include mixing with configurations including neutrons in the way they treated nuclei in the Zr region. The results for binding energies agreed well with experiment (also the predictions for and agree with subsequent measurements). The binding energies show the same general features as in the and cases with identical nucleons. These are a small and repulsive quadratic term and a large attractive pairing term. From the latter, by fixing the position of the level at 2 MeV, the excitation energy of the J = 4 level is calculated to be about 4 MeV. This is higher than the 3.55 MeV observed in but agrees better with levels in and in (and in which was measured later) with similar proton configurations. The J = 3/2 state in is then calculated to be very low above the J = 5/2 ground state as experimentally observed. As in the shell, the position of this level is due to the rather large separation between the J = 2 and J = 4 in which is very different from results for the The authors tried to calculate levels of configurations but not enough energies were available and the results were not well established. These nuclei will be considered in many subsequent papers including configuration mixing with expected as well as unexpected configurations. In spite of the meagre experimental data on oxygen isotopes, Pandya (1963) attempted to determine the effective interaction for configurations with and neutrons. In this case, there are J=0,2,4 states of the configuration, states and a J = 0 state of the configuration. Hence, there are 8 diagonal and non-diagonal matrix elements of a two-nucleon interaction acting in these states. These 8 matrix elements should be determined by 10 experimental data - energies of 10 states in and which include binding energies. The single neutron energies of the and orbits were taken from observed levels in The Hamiltonian sub-matrices are at most of order 4 and the author found “sets of two-particle matrix elements which give fairly good values for the energy levels of the three oxygen isotopes”. This is not surprising in view of the small number of energies considered and the large number of free parameters. The author found “a large amount of configuration mixing in the wave functions not inconsistent with the results of the stripping experiments”. Pandya compared the resulting matrix elements with those of Talmi and Unna (1962) and with matrix elements

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approximately calculated from the interaction between free nucleons. Using harmonic oscillator wave functions, he assumed a central interaction and deduced the values of the harmonic oscillator integrals He tried various potentials and concluded that “the structure of the radial integrals in even angular momentum states can be satisfactorily understood if we have not only a reasonably short range attractive potential, but also a repulsive central region”. He even constructed a combination of two Gaussian potentials, a long range attractive potential and a short range repulsive one from which he “can obtain reasonable values of the Talmi integrals”. A similar study, including configuration mixing, of oxygen isotopes was carried out by Cohen, Lawson, Macfarlane, and Soga (1964). They considered the neutron and orbits and determined matrix elements of the effective interaction from experimental data. Theirs was an extension of the analysis of Talmi and Unna (1962) who omitted configuration mixing and of Pandya (1963) who included it. The authors tried to fit by a least squares fit, 13 experimental energies, including binding energies, by 8 two body matrix elements. Single neutron energies were taken from the levels. The agreement between calculated and experimental energies was very good and was a substantial improvement on previous calculations. The authors listed the matrix elements which they obtained and compared them with those used in earlier calculations. The values determined in this paper were similar, in many cases, with those quoted by Talmi and Unna and by Pandya. They were rather different from those calculated by Elliott and Flowers (1955) from a potential interaction. New experimental information about levels posed a problem for the simple shell model description of these levels. In addition to levels expected from two neutrons in and orbits, a level at 5.26 and a level at 5.34 MeV were found. States with one or two neutrons in the orbit were expected at considerably higher energies. These new states were discovered by stripping reactions indicating sizable components of simple shell model configurations. There were two questions to be answered, first, what is the nature of these states and second, what is their effect on shell model calculations. Engeland (1965) and Brown (1964) suggested that such states were collective excitations similar to first excited J = 0 and J = 2 in The latter are possible analogs of the ground state rotational band where there are 4 nucleons in the shell. The analog states in may well have large components of such states. Brown, in his talk, reported calculations in which the deformed states in had rather low energies and rather large non-diagonal matrix elements with shell model states. The resulting strong mixing seemed to contradict not only the simple shell model description but also results of stripping reactions.

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Federman and Talmi (1965a) tried to calculate the matrix elements between shell model states and deformed states by using various interactions. Their prescription for the deformed state was that the 2 protons in the orbit are raised into the shell and with the 2 neutrons in form states like those suggested for The latter were assumed to arise from a K = 1/2 Nilsson orbit which contains contributions from and orbits. Talman (1962) used such states as variational wave functions for with the two nucleon interaction used by Elliott and Flowers (1955). Unna (1963) carried out a similar analysis using effective interactions determined from experiment. Federman and Talmi took from these papers the coefficients of the and orbits in the expansion of the K = 1/2 state. Using these coefficients they calculated matrix elements between states in which there are no protons and 4 nucleons are in the K = 1/2 orbit and shell model states in which the orbit is full and two neutrons occupy the and orbits. Several choices were made of the interaction whose matrix elements were calculated. One was the interaction used by Elliott and Flowers (1955) as well as by Talman (1962). Another one was the Kallio-Kolltveit (1964) interaction which they derived from the interaction between free nucleons, eliminating the hard core by an approximate separation point method. The third interaction was obtained by Dawson, Talmi and Walecka (1962) by an approximate solution of the Bethe-Goldstone equation. With all choices, the matrix elements thus calculated, were much smaller than those reported by Brown. To obtain wave functions of the J = 0 and J = 2 states, 3 × 3 matrices were diagonalized. Diagonal and non-diagonal matrix elements between shell model states were taken from the papers quoted above. The unperturbed positions of the deformed states were estimated from the binding energy and single nucleon states outside the core. The eigenvalues of these matrices were in very good agreement with the observed energies. The wave functions of the lower states were mainly due to neutrons in the and orbits. In the ground state, the weight of the deformed state was only 15%, in the 3.63 MeV state it is 8.4% while the deformed state was the main component of the state at 5.34 MeV, its weight is 76%. The situation was similar in the J = 2 states. A more comprehensive study of oxygen isotopes, including deformed states, was carried out later by Federman and Talmi (1965b). The various matrix elements were determined by a least squares fit to the experimental energies. The parameters to be fitted include 8 matrix elements between shell model states of and neutrons. In addition, the moment of inertia of the rotational band built on the deformed state and a common factor of all matrix elements between shell model and deformed state

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In are the coefficients of and orbits in the deformed state. The unperturbed positions of the deformed states were estimated from binding energies and single nucleon energies outside the and cores. The spacing between the and single nucleon energies was taken from These 10 parameters were fitted to reproduce 19 experimental energies in and The agreement between calculated and experimental energies was very good. According to the results of this calculation, levels of and were less affected by the deformed states due to their higher unperturbed positions. In the states are similar to those in earlier calculation. The weight of the deformed state in the states of was 12%, 14% and 73% (using another set of these numbers are 10, 6 and 83). The corresponding weights in J = 2 states were 7%, 12% and 81% (or 7, 3 and 90). Such wave functions give also good agreement with results of stripping reactions. The conclusion is that “coexistence of shell model and deformed states in oxygen isotopes” was demonstrated. A detailed description of the work described above, was presented in a long paper by Federman (1967). It included also results of some further calculations and comparison with experimental data. The various matrix elements were determined by a least squares fit to the experimental energies. The parameters to be fitted included 8 matrix elements of the two-body interaction between shell model states of and neutrons. In addition, the moment of inertia of the rotational band built on the deformed state and a common factor of all matrix elements between shell model and deformed states is

In are the coefficients of and orbits in the expansion of the deformed state. The unperturbed positions of the deformed states were estimated from binding energies and single nucleon energies outside the and cores. The spacing between the and single nucleon energies was taken from These 10 parameters were fitted to reproduce 19 experimental energies in and The agreement between calculated and experimental energies was very good. According to the results of this calculation, levels of and are less affected by the deformed states due to their higher unperturbed positions. In the states are similar to those in the earlier calculation. The weight of the deformed state in the states was 12%, 41% and 47%. The corresponding weights in J = 2 states were 7%, 13% and 80%. Such wave functions give also good agreement with results of stripping reactions. Also in this paper the conclusion was that “coexistence of shell model and deformed states in oxygen isotopes” was demonstrated. A more detailed study of the shell was published by Arima, Cohen, Lawson, and Macfarlane (1968). They tried to describe nuclei between

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and by states of protons and neutrons occupying the and orbits. They were aware of the core excited states in but they believed “that a large body of data can be correlated by a model in which the core is inert”. The authors had in their model 16 matrix elements of two-nucleon interactions with T = 1 and T = 0, to be determined. The single neutron energy was taken from the difference between the and binding energies and similarly, the single proton energy from and The difference in single nucleon energies of the and orbits was taken from the levels (.871 MeV). This difference for a proton in is only .495 MeV, but perhaps the authors attributed it to the small separation energy and thought that this difference was smaller in nuclei where the separation energies are larger. They also seemed to ignore the Coulomb energy contribution to T = 1 matrix elements between states of two protons. Such states occur only in and the effect of the Coulomb contribution may be less important. The 16 matrix elements were determined by a fit to 36 energies, including binding energies. It seems that the number of experimental energies was not sufficient for a good fit. If the level at 1.70 MeV in was included, the calculation predicted this state at 2.34 MeV and also for all other states the agreement was not good. The authors suggested that this state was due to excitation of the core in which two nucleons are raised to the shell. If it was excluded, the agreement between calculated and experimental energies was very good. This includes, in addition to the ground state band with J=0,2,4,6,8, of also a second state, a second state and two excited states. The largest deviations occured for levels. For the agreement was very good, not only for the lowest and states but also for the lowest level. The predictions of levels with T = 0 and the second and states, as well as of a state agreed very well with results of subsequent measurements. The agreement between calculation and experiment in oxygen isotopes was also very good. This includes the lowest two states and lowest two states in Possible effects of core excited states seemed to be somehow absorbed into the effective interaction. In the case of the and were nicely reproduced. The second J = 1 state, however, was predicted to lie at 8.75 MeV and the second J = 3 state at 8.35 MeV. The authors explained that the matrix elements between the T = 0, J = 1,3 states “are the least accurately determined in the least-squares fit”. Indeed, J = 1 and J = 3 states were later discovered at excitation energies of 3.72 and 3.36 MeV. The authors found that some of their matrix elements were very sensitive to the choice of data. Even if a specific experimental energy was omitted from the fit, wild changes occurred in some of the matrix elements. They diagonalized the error matrix and found that “certain linear combinations of the two-body matrix elements

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are much more precisely determined by the data than are others”. They also found that “the least well determined linear combinations heavily weight contributions from the T = 0 diagonal energies and the off-diagonal J = 1 and J = 3 interaction matrix elements”. The authors calculated E2 transition rates from their wave functions by assigning the proton an effective charge of 1.7e and the neutron .7e. The agreement in the ground state band in was very good and in the other nuclei the predictions agree “to within a factor two of their observed values”. A notable exception is the transition from the first excited J = 0 level of to the lowest J = 2 level, where the observed rate is about 50 times the predicted one. This is perhaps a manifestation of admixtures of core excited deformed states. The agreement of predicted M1 transitions and beta decays was less impressive which is explained by the large sensitivity of their matrix elements to small admixtures of configurations with nucleons. The authors explained that in the M1 transition between the J = 3/2 state to the J = 5/2 ground state “would be seniority-forbidden” if both states belong to the neutron configuration. Actually, as explained by Talmi and Unna (1962), the M1 operator is proportional in any configuration of identical particles to the total spin J and hence, irrespective of seniority, only its diagonal elements need not vanish. A region with valence nucleons in the shell, which seems to be free of low-lying intruder states, is of nuclei between and It was described in terms of and orbits by Goldstein and Talmi (1957), by Arima (1958) and by Talmi and Unna (1960a). These calculations, however, did not include mixing of configurations of nucleons in the two orbits. Configuration mixing of nucleons in these orbits was carried out by Glaudemans, Wiechers and Brussaard (1964a) and (1964b). In the first paper they presented in detail the method of calculation. In the second, “calculations on nuclei in the region are performed with the assumption of an inert core”. There are 17 parameters which define the subspace of the Hamiltonian in this case. These are the and single nucleon energies and 15 two-body matrix elements. The authors determined their values by the best fit to energies of 50 positive parity states with definitely established spins. The effective interaction obtained in this way, was used to calculate energies and wave functions of all 377 states which belong to the shell model sub-space considered. The agreement between calculated binding and level energies and experimental values was good in many cases and fair in other ones. This was the case also for levels whose energies were not included in the fit, as well as for some levels whose spins were measured in later years. In some nuclei, however, levels which seem to be outside the model space

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lie rather low. This feature may indicate that a larger space is needed for description of observed spectra. The wave functions obtained by diagonalization of the Hamiltonian sub-matrices, contain rather large configuration mixing. In the lowest J = 0 state of two nucleons, the component has the weight of 73%, whereas the rest is due to the component. In taken as an inert core by previous authors, in the J = 0 ground state, the closed orbit is only 50%. Almost 40% are due to the state. Towards the end of the shell, states become purer. In the J = 3/2 ground state of for instance, the weight of the state is 93%. Similarly, the weight of the in the J = 0 ground state of (or is 95%. These wave functions were used by the authors to calculate the rates (spectroscopic factors) of various and stripping reactions. They compared the calculated values with the meagre experimental data. “The experimental relative values for each reaction have been normalized to the computed reduced width of the strongest component in the ground-state transitions”. They concluded that in view of the large uncertainties in the experimental data and the extraction of the reduced widths from them, “one observes that rather good agreement is obtained between the predicted values” and measured ones. These wave functions were also used by Wiechers and Brussaard (1965) to calculate magnetic moments and M1 transition rates of the nuclei considered. They refer to the papers of Blin-Stoyle (1953), Blin-Stoyle and Perks (1954) and Arima and Horie (1954) where rather small configuration mixing led to “a considerable effect on the value of the magnetic moment. As it turns out this is not so in our case. Although admixtures range from 50% for nuclei consisting of 32 particles to some 10% for nuclei with nearly complete shells, the effect of these admixtures is generally small”. The authors did not explain the difference between their results and the quoted ones, where the corrections to magnetic moments are linear in the amplitude of the admixed configurations. The latter corrections can take place only if the difference between the mixed configurations is by the state of one nucleon raised to an orbit with the same value. To obtain better agreement with experiment, the authors use effective for the intrinsic spins of protons and neutrons in and orbits. They pointed out that due to the limited shell model space “it would not seem unreasonable to replace the single-particle factors by values which produce the best agreement with experimental results”. Using the 4 effective they obtained fair agreement with 8 measured magnetic moments. “A notable exception is the magnetic moment of the ground state”. They could not obtain a proper fit to the measured moment. They wrote that “here one may question the configuration that was obtained from” the paper by Glaudemans,

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Wiechers and Brussaard (1964b). Indeed, reasonable agreement with the measured moment can be obtained from the pure configuration, but its weight in the calculated wave function is only 57%. A couple of years later, Glaudemans, Wildenthal and McGrory (1966) repeated the analysis of Wiechers and Brussaard (1964b) and determined several sets of matrix elements of the effective interaction. The first set was determined by a fit to 60 energies of states, including binding energies. The matrix elements were close to those obtained earlier and the differences indicate how well they were determined by the data. The agreement with measured energies was as before, the r.m.s. deviation is .16 MeV. A fit to the same set of data but putting to zero non-diagonal elements, gave worse agreement, the r.m.s. deviation was found to be .37 MeV. The non-vanishing matrix elements, however, were rather similar to those of the other set. Another fit avoided binding energies and used only 35 level spacings to determine the values of 15 two-body matrix elements and the difference between the single nucleon and energies. The resulting matrix elements differ in some cases appreciably from those of the other sets. The level spacings calculated using them were also in fair agreement with the measured spacings. The somewhat larger .20 MeV r.m.s. deviation was probably due to the smaller number of measured energies.

4.10.

The Complete

and Beyond

An attempt to carry out a complete calculation with effective interactions determined from experiment was made by Amit and Katz (1964). They calculated also binding energies starting from down. The single hole energies as well as two nucleon matrix elements were determined from experimental energies. They used wave functions to construct the Hamiltonian sub-matrices for all states with the same values of J and T of The number of free parameters to be determined by a least squares fit was 15+2=17. The authors noticed “that the exclusion of the energy levels for improved the agreement between the calculated and experimental levels to a great extent”. They excluded them from the fit and were left with 36 energies to determine the 17 parameters. Not all parameters were determined with the same accuracy. In fact, the error bars on two of them exceeded the values of the matrix elements. The authors set for them the value zero, for convenience. The other 15 parameters were determined with rather large error bars. Energies calculated with these matrix elements were in fair agreement with the experimental ones. The authors pointed out that there was no need to change the single nucleon spin-orbit splitting from A = 8 to A = 16. The actual change observed along the shell must be due to a mutual spin-orbit interaction which is included in their matrix elements of the effec-

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tive interaction. Amit and Katz concluded that intermediate coupling in the seems to give a good description of nuclei beyond A = 7. They were not sure, however, about “possible inclusion of wrongly identified levels into the calculation”. In the paper of Amit and Katz (1964), a rather large number of nuclei was considered and for the first time several non-diagonal matrix elements were determined from experimental energies by a least squares fit. Their paper was followed by a paper of Cohen and Kurath (1965) on “Effective interactions in the In that paper, a more careful choice of energy levels was made and the authors succeeded in determining all matrix elements of the effective interaction. There are 15 matrix elements of the two-nucleon interaction and 2 single nucleon energies which were determined from 35 experimental energies of nuclei from A = 8 up to A = 16. Cohen and Kurath also tried to fit the data by using in the LS-coupling scheme, only 11 two-nucleon matrix elements which would arise from a potential interaction. In that scheme there are 10 diagonal matrix elements but only one non-diagonal element connecting two space symmetric states, and Results of fitting the data with the smaller number of parameters were also in good agreement with experiment. The agreement between experimental energies and those calculated from matrix elements obtained by the authors was very good. To check the wave functions obtained by diagonalization, the authors calculated magnetic moments and found them in good agreement with measured ones. Magnetic moments calculated with a potential interaction were not very different. The authors remarked “that the wave functions of the low-lying states are not appreciably different from those which resulted from the old calculation despite the fact that the interaction is considerably different”. They noticed, however, improvements on the old results. For the beta decay, the calculated Gamow-Teller matrix element was reduced by factor 5 from its value in the old calculation. If a minute change of the ground state wave function was made to one with .998 overlap, nearly complete cancellation of the element occurred. This was evidence for the effective matrix element containing contributions from tensor forces. The authors analyzed the resulting matrix elements of the effective interaction to obtain the contributions of central, two-body spin-orbit and tensor interactions. Not all matrix elements were accurately determined. “The best determined integrals are those involving the even state central forces” which make the major contributions. “The signs and magnitudes appear to be determined for the integrals with the tensor force, but some particular changes by as much as 50% still leave reasonable fits”. The authors criticized Amit and Katz (1964) for their choice of data. They also compared their matrix elements with those of Amit and Katz which led to rather strange interactions. This criticism, however, is not fair since Amit and

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Katz put some very poorly determined matrix elements to zero rather arbitrarily. The zero elements were not meant to be due the correct effective interaction and they could have been assigned very different values. Amit and Katz also had a positive sign of a non-diagonal matrix element because the sign is irrelevant for energy calculations. Any sensible interaction would give it a negative sign. In their conclusion, Cohen and Kurath explain that since there were not many experimental data, the choice of levels was important. In particular, they omitted from the fit the T = 0 level at 7.65 MeV in Their calculated position was 5-6 MeV higher and they thought that the observed level had a large component of excitations into the shell. The overall picture was indeed impressive and the authors conclude that “it is perhaps surprising that the pure picture works as well as it does”. The work of Cohen and Kurath has been a landmark in the use of effective interactions determined from experimental data. It also contains a thorough and careful analysis of the experimental information, on energies as well as on moments and transitions. The success of the shell model calculations in the complete by Cohen and Kurath (1965), encouraged Millener and Kurath (1975) to go beyond it. They were motivated by new experiments on beta decay of to states in In according to the level order of the shell model, there should be one neutron in the shell. To calculate rates of beta decay, the wave functions of states with valence nucleons in the and shells must be calculated. The authors adopted the interaction of Cohen and Kurath for the nucleons. Then they looked for an effective interaction between nucleons and nucleons. They derived this effective interaction from a potential interaction with central, mutual spin-orbit and tensor force components. They adjusted the parameters of that interaction to obtain better agreement with various energy levels. Their central interaction had the exchange combination

where are projection operators onto states with given T and S. In terms of the familiar exchange operators, these are given by and The exchange operator (93) is multiplied by an attractive Yukawa potential. The two-body spin-orbit interaction is equal to

multiplied by an attractive Yukawa potential. The tensor forces were similarly defined by

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which is also multiplied by an attractive Yukawa potential. The last two interactions vanish when acting on a spin singlet S = 1 state. This interaction is denoted by MK. The authors compared their interaction with other potential interaction, as well as to a G-matrix calculation by Kuo (1967). They observed that the MK interaction “is closely related to, and not very different in character from, the bare G-matrix interaction”. In fact, the “only essential change which has been made to the G-matrix interaction is to add a repulsive component to the triplet-odd central force, a feature which seems to be a general requirement for empirically derived effective interactions”. The MK interaction was applied to states in which one nucleon is raised from the shell into the shell. For certain states, it was necessary to go beyond this one-hole one-particle configuration and include also 3 hole 3 particle states. Spurious states, due to excitation of the center-of-mass may arise in this case and their number may reach 10% or so of all states. The authors explained that the “elimination of spurious states is not, in fact, difficult to perform, regardless of the classification scheme employed, since the complete basis of a given unperturbed (harmonic oscillator) energy is to be used”. The calculated level spacings were compared to experimental ones in several nuclei with A = 11 to A = 16. The calculated position of the state above the ground state of was 1.49 MeV rather close to the measured position of 1.78 MeV. “For ground state is predicted with a state at 510 keV”, this was verified later experimentally, the spacing is 740 keV. The agreement with positive parity experimental levels of was good, with deviations smaller than .4 MeV. The agreement is worse for negative parity levels of and For some levels there was fair agreement with experiment but there are several large deviations of 1 MeV or bigger. This was also the situation in whereas it was better for the negative parity levels of Millener and Kurath used the wave functions obtained by diagonalization, to calculate rates of beta decay of the ground state of to levels. The calculated logft value for the decays into the and states of are 4.13, 5.33 and 4.83 which were in good agreement with the experimental values of 4.22±.05, 5.10+.30–.08 and >4.8 respectively. Hsieh and Horie (1970) considered “non-normal parity states in A = 15, 14 and 13 nuclei”. These states were assumed to be due to raising one nucleon from the orbit into the or raising one into the The two-body interactions between were taken from Cohen and Kurath (1965). “The position of the and relative to the levels are obtained from the experimental values of The orbit was put 2 MeV below the center-of-mass of the and levels. “The spacing between the and levels is assumed to be the same as that between the

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and levels”. For the interactions between nucleons in different orbits, the authors “adopt the same interactions employed by Elliott and Flowers” (1957)

They used harmonic oscillator wave functions and Before diagonalizing the energy matrices, the authors eliminated the spurious states. Level diagrams are presented for T = 0 and T = 1 levels in and levels of (and ). The authors calculated also proton and neutron reduced widths for some levels, and also E1 transition rates. The agreement between calculated and experimental level positions in was fair only for the lowest and levels. For higher levels, it was only qualitative, the deviations amounted to several hundreds keV and even to more than 1 MeV. The situation for was similar. The agreement was particularly poor for the yrast states whose main components should be states obtained by coupling a hole to a and nucleon. In the numerical calculation of the levels, some restriction was imposed on T = 1/2 states. “Also the positions of the and levels relative to the level are chosen to be larger than that obtained from the experimental value of for 0.8 MeV, and the position of the level is chosen to be 2.84 MeV below the centre of gravity of the and levels”. With these modifications, much better agreement with experiment was obtained for the spectrum, up to levels at excitation of about 8 MeV.

4.11.

“Pseudonium Nuclei”

Severe doubts were expressed on the validity of shell model calculations using matrix elements of the effective interaction determined from experimental nuclear energies. Cohen, Lawson and Soper (1966) presented a case where the interpretation in terms of pure configurations is totally wrong. “If the consequences of assuming a simple, pure configuration are quantitatively borne out by experiment, it is tempting to say that this confirms the hypothesis that the shell is fairly pure”. The authors show “that this conclusion is not warranted” with “consequences both for the shell-model itself and for such things as the shell-model effective interaction”. The case presented is a fictitious one, neutrons in degenerate and orbits interacting by a central interaction with a Yukawa potential, which exhibits strong configuration mixing. They considered neutron numbers between 2 and 12 but their interest is in numbers between 4 and 12. They diagonalized the Hamiltonian in this space and took the results as “pseudo experimental data” which are used to determine matrix elements of an effective interaction for pure configurations.

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For obvious reasons their pseudo nuclei were called pseudonium isotopes with having 4 neutrons and with 12 ones. The very notion of effective interaction implies important configuration mixing induced by the singular parts of the interaction between free nucleons. In the shell model, it is assumed that the mixing is of highly lying configurations whose contributions to energies could be replaced by renormalization of the interaction. The admixtures presented by the authors, however, were of configurations whose energies are rather close, and their weight is very large. The ground state, for instance, contains only 9% of the closed orbit, the state. In with the probability that the lowest state is due to the configuration is only 17%. These probabilities in ground states, increase monotonically with reaching 88% in and, of course, 100% in and in Still, the authors found for all cases with even a spin 0 ground state. For odd values of the ground state spin is For “the spins of the lowest states of appropriate parity are just those that would arise from the pure configuration. There were 4 matrix elements which, together with the single energy relative to taken as a core, determined all energies in all configurations. Cohen, Lawson and Soper tried to determine them from 31 pseudo energies which they have calculated. “The least squares fit is excellent. Both the binding energies and the spectra are very well reproduced”. The authors calculated magnetic moments and M1 transitions and found that the “pseudo-experimental magnetic moments are found to lie within 1% of the Schmidt value”. All M1 transitions, forbidden in a pure configuration, were very small. The E2 transition probabilities in a pure configuration were also in remarkable agreement with the pseudo-data. There are some 60 E2 pseudodata and the only disagreement was found in 3 cases in The authors explained that these features were due to the dominant mode of mixing, which includes states where “one or more zero-coupled pairs excited out of the closed shell” which is They attributed this feature to the fact that “the interaction tends to favour zero-coupled pairs”.

More pseudonium cases were presented also by Lawson and Soper (1967) and by Soper (1970). This pseudonium case came as a big surprise and may have shaken the faith of some theorists in the determination of effective interaction from experiment. Only several years later, irrelevance of this case for real nuclei was demonstrated and shown to be a special case of generalized seniority (see 6.2). This case, impressive as it may be, is only of academic interest and will not be further described here.

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SOME SCHEMATIC INTERACTIONS AND APPLICATIONS

5.1. The Pairing Interaction In 1957 the long standing problem of superconductivity was finally resolved. Bardeen, Cooper and Schrieffer (1957) (BCS) suggested a mechanism for an attractive interaction between electrons near the Fermi surface. That interaction had non-vanishing matrix elements between S = 0 pairs of electrons which are in single electron states with opposite momenta. Such “Cooper pairs” could in some way condense into a many electron superconducting state. BCS have shown how an approximation of such a state can be constructed rather simply. In their theory, the number of electrons is constant only on the average which is a good approximation if is very large. The BCS theory turned out to be very successful and it was not surprising that nuclear physicists tried to apply it also to nuclei. David Pines while visiting Copenhagen talked to Bohr and Mottelson about it and Bohr, Mottelson and Pines (1958) wrote a paper on “Possible analogy between the excitation spectra of nuclei and those of the superconducting metallic state”. For nuclei, the BCS interaction was replaced by the pairing interaction defined above and it was realized that Racah’s seniority scheme is a set of eigenstates of such an interaction. While it can be used in energy calculations, it was explained above that the pairing interaction certainly cannot replace the effective interaction which yields good agreement with experiment. It was also clear that in metals, the main interaction between electrons is the Coulomb repulsion and the BCS correlations take place only at the Fermi surface. In spherical nuclei there are large degeneracies and hence, any two nucleon interaction leads to a correlated state. In particular, interactions with large attractive eigenvalue in the state lead in the configurations with even to ground states with seniority In the case of several the pairing interaction is defined by

If the single nucleon orbits are degenerate, an exact simple expression for the lowest eigenstates and their eigenvalues can be obtained as will be described below. The differences between nucleons in nuclei and conduction electrons in metals did not deter nuclear theorists from using the pairing interaction. Moreover, some of them applied the BCS formalism to nuclei where the number of valence nucleons is rather low. In such applications, a distinction was made between the degenerate and non-degenerate situations. In the former, essentially the results of seniority were reproduced whereas the later situation was applied

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to deformed nuclei. In those, single nucleon levels in a deformed potential well are not degenerate. This way another schematic interaction was introduced into nuclear structure physics along with the whose spectrum has similar features. Among the effects associated with the pairing interaction is an energy gap between the ground state and excited states in even-even-nuclei. Another one is the variation in binding energies between odd-even and eveneven nuclei. Unlike in the general case of seniority, these two observables, as well as the energy due to the pairing interaction, should be obtained from the same parameter, the coefficient of the pairing interaction. This is a severe constraint which cannot be satisfied by the experimental data. This is definitely the case for spherical nuclei but some of the difficulties appear also in nuclei which exhibit rotational spectra. The pairing interaction contains one important ingredient of the effective interaction between identical nucleons. This is the strong attraction in two nucleon states with J = 0. It is also diagonal in the seniority scheme in configurations. Still, ignoring the interaction in J > 0 states leads, as shown above, to level spacings which are in disagreement with experimental ones. In spite of this, the pairing interaction has been widely applied to strongly deformed nuclei described by the collective model. The introduction of any two nucleon interaction into a single nucleon Hamiltonian with a deformed potential well, leads to a more accurate physical picture. Since the pairing interaction is easy to use in the BCS approximation, it became very popular among nuclear physicists. Bardeen Cooper and Schrieffer (1957) introduced a variational wave function for the ground state of the system

where is a (linear) momentum and create particles with momenta and respectively. The physical meaning of the coefficients is clear. The probability of the pair state to be occupied in the ground state wave function is whereas is the probability of this state to be unoccupied. Clearly, this wave function does not have a definite number of particles. To have a given average number the number operator is added to the pairing Hamiltonian with a Lagrange multiplier which is determined by The pairing Hamiltonian is thus expressed by

where are single particle energies. In the non-degenerate case, the coefficients

and

for which the expec-

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where

is defined by

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is the lowest, are given by

and

Hence

from which follows that if the lowest state is the normal ground slate of particles occupying the lowest states. If is not zero, then the following condition should be satisfied

which is called the gap equation. The value of which plays the role of the chemical potential, is then determined by equating the expectation value of to the given number of particles

The expectation value of the Hamiltonian is then given by

In its application to nuclei, is replaced by and the summation is carried out over states with In the degenerate case, where all are equal, and for a single the resulting expectation value of the pairing interaction in the BCS approximation, is equal to the exact result up to order It was recognized (Mottelson (1959)) that the part of the BCS variational wave function with a given number of particles is equal to

In the case of a single

the coefficients may be taken to be given by Then the state is just the one in which N pair states with J = 0 are multiplied and antisymmetrized, i.e., the state with seniority

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Since the BCS ground state wave function does not have a definite number of particles, it can be viewed as the vacuum state of “quasi-particles”. The creation and annihilation operators of these quasi-particles are defined by the Bogolyubov-Valatin transformation from creation and annihilation of particles, to be

The creation and annihilation operators of quasi-particles satisfy the usual anticommutation relations of fermions. This is guaranteed by putting and in addition to The BCS ground state wave function is indeed the quasi-particle vacuum, it is annihilated by all quasi-particle annihilation operators. These are creation operators of anti quasi-particles which implies that the BCS ground state is proportional to the state in which all anti quasi-particle states are filled. Lowest states of systems with odd numbers of particles are one quasiparticle states with momentum obtained by acting on the BCS wave function by quasi-particle creation operators. In nuclei, these states correspond to single nucleon states and their spins are equal to the values of the various orbits. With the usual approximations of the BCS theory, the single quasi-particle energy is given by

This energy, for state near the Fermi surface, is roughly equal to and thus, it is positive which accounts for the odd-even nuclear binding energy differences. Excited states of a system with even number of particles, may be obtained from the ground state by operating on the latter by two quasi-particle creation operators. The energies of two quasi-particle states are equal to The gap between the ground state and such states is roughly equal to The states obtained by acting on the BCS ground state wave function with and for any contain some spurious components due to not having a definite value in the BCS ground state. Here, is not proportional to but it should not be another physically meaningful state. Hence, the state which is orthogonal to the ground state is a spurious state. Its components should be removed from the two quasi-particle states In nuclei, if several are considered, there is one spurious state with J = 0 which is a linear combination of states of two quasiparticles with spin Other states, orthogonal to it, correspond to excited J = 0 states.

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The BCS theory in the non-degenerate situation has been applied to actual nuclei. Most applications have been made to strongly deformed, axially symmetric nuclei having single nucleon levels which are only two fold degenerate, and states. These cases are sufficiently removed from degeneracy. The odd-even variation in binding energies turns out to be approximately equal to in this formalism. It varies slowly with mass number and in the region of strongly deformed nuclei, is approximately 1 MeV. This is also the energy of what is considered intrinsic excitations in even even nuclei which should also be given by The total amount of pairing energy as given above, with constant values of and G, or values which change slowly with A, is fairly constant. It can be evaluated by using the gap equation and the result is 1 to 2 MeV. Thus, “the total pairing energy is extremely small compared to the total binding energy”. Indeed, it “is surprisingly small compared with the total binding energy”. The situation in such cases is very different from that in spherical nuclei. Semi-magic nuclei are the natural places to see effects of pairing. Binding energies in the seniority scheme include the pairing term, which gives rise to the odd-even variation in binding energies. This term, for even is proportional to In the case of semi-magic nuclei there is also a repulsive term which is quadratic in Still, the binding energy due to the mutual interactions is not constant but increases with The dependence on which is rather exact for semi-magic nuclei, is observed also in other nuclei, including those with rotational spectra. There is no real gap in spectra of even-even semi-magic nuclei. Had the effective interaction been like the pairing one, all levels should have been degenerate. This is far from the actual situation. The lower of these levels is the one with J = 2 and it seems to lie “within the gap”. No special reasons should be looked for the lower positions of J = 2 levels. The actual positions of levels in even-even nuclei determine the positions of levels in odd-even nuclei. In the latter, the states are far from degenerate and in particular, the level is rather close to the ground state. Nevertheless, attempts were made to apply the BCS theory, in the non-degenerate case, also to spherical nuclei. In Copenhagen, the BCS theory was applied to actual nuclei by Belayev (1959) and in great detail by Kisslinger and Sorensen (1960). The importance of the quadrupole degree of freedom in nuclei was recognized by its dominant role in the collective model. Therefore, in their calculations, the authors included a quadrupole-quadrupole interaction which was described as a long range force. In fact, the title of the paper by Kisslinger and Sorensen is “Pairing Plus Long Range Force for Single Closed Shell Nuclei”. They explained that using such a simple force, the pairing interaction and the quadrupole “ force”, they “cannot expect to derive the detailed quanti-

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tative properties of these nuclei, but rather attempt to find the main systematic features and to identify the main parts of the nuclear wave functions”. This is just a general excuse to be used, if necessary, since the authors claimed to obtain quantitative agreement with the experimental data. The authors used the pairing interaction and the BCS ground state wave functions. They applied to the Hamiltonian the Bogolyubov-Valatin transformation which defines creation and annihilation operators of “quasi-particles” by

The coefficients and replace here the and defined above and are independent of due to the spherical symmetry of the central potential. The non-degeneracy of single nucleon orbits is due to the inclusion of several As explained above, the BCS ground state wave function is the quasiparticle vacuum, it is annihilated by their annihilation operators. Lowest states of an odd-even nucleus are one quasi-particle states obtained by acting on the BCS wave function by quasi-particle creation operators. These states correspond to single nucleon states and their spins are equal to the values of the various orbits. The single quasi-particle energy is given by

This energy, roughly equal to is positive which accounts for the odd-even binding energy differences. Excited states of even-even nuclei with various spins may be obtained from the ground J = 0 states by operating on the latter by two quasi-particle creation operators. The energies of two quasi-particle states are equal to The “gap” between the ground state and such states is roughly equal to “For the long range part of the shell model particle interaction” the authors use the two-body force. The reason for this choice is that “for nuclei with or near closed shells” this force “can provide an explanation for the observed quadrupole vibrational spectra”. They considered the action of the force only between valence nucleons, including effects of the core by renormalizing the qudrupole force and the effective charge of the valence nucleons. Kisslinger and Sorensen argued that the force cannot be considered in perturbation theory. One argument is that E2 transition rates are faster than those calculated with effective charges. The other argument is that “the lowest state is well below the two quasi-particle states produced by a pairing force of such strength as to be consistent with other data”. “Since this state must be constructed from the two quasi-particle states, and is far separated

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from them in energy and of a different character from them, a non-perturbation treatment is necessary for this state”. They followed Belayev, expressing the quadrupole interaction by quasi-particle creation and annihilation operators and keeping what they consider the important terms. Then, like Belayev, they defined a “collective parameter, Q, the quadrupole field or the total nuclear quadrupole moment”. The authors used “a method suggested by A.Bohr which is equivalent, within the approximation used, to that of Belayev”. They arrived at a collective Hamiltonian which, apart from a constant, is given by

where the collective variables are essentially linear combinations of products of two quasi-particles creation operators coupled to and of two annihilation operators coupled to This is the Hamiltonian of a five-dimensional harmonic oscillator where the restoring force coefficient C and the inertial parameter B are functions of the and matrix elements of the quadrupole operator between single nucleon states. When quantized, this Hamiltonian “will lead to the spectrum associated with the harmonic surface oscillations, the quanta being phonons of spin 2. With this description of the lowest state, its properties can be easily obtained”. The level spacing is given by

In this way, the authors made the connection between levels of odd-even nuclei and their even-even neighbours very complicated. In view of the shell model results in the case of and orbits, it is clear that, at least in those cases, the formalism used by Kisslinger and Sorensen will not work. Indeed, in the case of nuclei with N = 28 neutrons, and proton number between 20 and 28, the authors found that although “the state is fit pretty well as a collective state, the levels in the odd-A isotopes are not well fit”. The low lying and in and lie 3 and 2 MeV below the lowest calculated states. The authors “also performed exact diagonalization of the pairing force plus force” and found “that it is possible to fit the experimental data for the even-A isotopes and the low-lying states”. Still, the calculated J =4 and J =6 levels were degenerate and the calculated level was much higher than the experimental one. It lay also above the calculated level. This should be compared with the very good agreement obtained in the calculations of Lawson and Uretsky (1957). The poor agreement in this case cannot be blamed on “the small degeneracy of the the levels in this region, i.e., which “leads to poor Bardeen solutions”. The authors did not refer to the state at .68 MeV above the

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ground state of which, in their model must be a 3 quasi-particle slate and lie at around 3 MeV. Kisslinger and Sorensen considered the Pb isotopes with N < 126, the Sn isotopes, the Ni isotopes, and nuclei with N = 82, N = 50 and N = 28. They have two interaction parameters, G of the pairing interaction and X, the strength of the force. The values of G which give the best agreement with the data ranged between 20/A for lighter nuclei and 25/A MeV for the Pb region. The corresponding values of X ranged between 100/A and 150/A MeV. In addition, the single nucleon energies of the various orbits had to be determined. The authors mentioned several ways to determine them, none very reliable. In some cases they were “working back from the known experimental levels to find the values of the single-particle levels which are consistent with the experimental data”. The agreement between calculated level positions and experimental ones was semi-quantitative at best. In even-even nuclei the position of states constrains the values of the interaction parameters and hence, the fit is reasonable, apart from a few exceptions. In odd-even nuclei the situation is similar. Positions of one quasi-particle states are calculated from the single nucleon energies and are also modified by interaction with surface oscillations. Most ground state spins are reproduced correctly and so are positions of the lower single nucleon levels. The authors tried to calculate magnetic moments of nuclei but concluded that “it does not seem to be possible to understand the shift of the quasi-particle magnetic moment of single closed shell odd-A nuclei from the single-particle value on the basis of the coupling of the quasi-particle to the collective oscillations”. The authors explained that configuration mixing leading to changes of magnetic moments a la Blin-Stoyle and Arima and Horie did not take place with the pairing interaction. They believed that if they used instead a delta potential, magnetic moments would be correctly reproduced. Kisslinger and Sorensen calculated also electromagnetic transition probabilities. The most consistent ones were found for E2 transitions between first excited states and ground states. The agreement between calculated and experimental rates were good when the effective neutron charge was taken to be 1 and that of the proton to be 2. In the summary, Kisslinger and Sorensen explained that the “deformed field approximation is used to calculate the effect of the relatively long range part of the nuclear force, and in particular to determine the position of the collective states”. These states were “lying in the gap between the ground state and the intrinsic excited states”. They claimed to have found “a simple explanation for this state”. These states are “always the first excited quadrupole vibrational level”. The authors concluded by stating “that the simple model which we have tried has been successful in deriving the observed low lying systematic features

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of single closed shell nuclei, and that our results might serve as a good basis for a more detailed quantitative investigation”. In a later paper, Sorensen (1961) calculated spectra of odd-even single closed shell nuclei. He used the same Hamiltonian which “is diagonalized to include all states containing up to two phonons of quadrupole vibration”. In the Kisslinger Sorensen (1960) paper, coupling of quasi-particles was limited to only one phonon. Sorensen found that amplitudes of such two phonon components “in nearly all cases they are less than 0.1”. His conclusion was “that the improved treatment of the Hamiltonian maintains the agreement (or disagreement) with experiment concerning the low-lying levels of odd-mass spherical nuclei investigated in” the earlier paper. It led to predictions of energies and wave functions of higher levels. “Where experimental data is available, no obvious gross discrepancies with these predictions occur”. A strong argument against the use of the quadrupole interaction for identical valence nucleons is due to seniority considerations. Experimental levels of configurations and of configurations, of identical nucleons show that the effective two-nucleon interaction is diagonal in the seniority scheme (see 6.1). Thus, the T = 1 interaction can be expressed in terms of odd tensors plus a monopole, term only. Adding a quadrupole interaction will break seniority in a major way. It should be pointed out that any two-body interaction may be expressed in such configurations in terms of even tensors only. A quadrupole term may well be present there but its seniority breaking effects will be compensated by other even tensors with Only in special cases may the quadrupole term contribute without effects from other even rank tensors. This is the case with first order contributions to the E2 effective charge considered above, where the even tensor expansion is useful. Up to any two body interaction, which may include the force, is diagonal in the seniority scheme. Also in configurations with the effective two body interaction, is diagonal in the seniority scheme. In cases of configuration mixing, like those considered by Kisslinger and Sorensen, seniority-like spectra are observed which will be discussed in the following sections. These are characterized by fairly constant spacings between J =0 ground states and J =2 first excited states. A quadrupole interaction which has non-vanishing matrix elements between different orbits will cause a strong reduction of those spacings, even if some of the orbits have The positions of J =2 states, as well as the J =4,6,... states, in semi-magic nuclei are due to the nature of the odd tensor interactions between nucleons. It is unnecessary and wrong to attribute the lower position of the J =2 states to the presence of a quadrupole interaction (without other even tensor interactions). Invoking a special mechanism for lowering the states makes it necessary to look for another, independent, mechanism to explain the low-lying

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states in odd-even nuclei with and It is interesting to note that the quadrupole interaction used by Kisslinger and Sorensen led to rather constant 0-2 spacings. This is probably due to their presenting the lowest J =2 state as a surface vibration rather than diagonalizing the Hamiltonian. This is an example where the approximation fits the experiment better than the exact calculation. This trend seems to be contrary to the expectation of Mottelson (1960) in his talk at the 1960 Kingston International Conference on Nuclear Structure. The talk was on “Nuclear Coupling Schemes and the Microscopic Description of Collective States”. In it Mottelson described “two essentially different types of correlation between the nucleons in the nucleus”. In ground states of the aligned coupling scheme, the wave function is a product of single particle states which best fit into the average (non-spherical) potential which the particles themselves generate”. Such a wave function “takes into account the part of the nuclear forces which can be described in terms of an average field; this field producing part of the nuclear force is associated with the part of the interaction that is of longest range”. The other coupling scheme “is associated with the short range part of the inter-nucleon interaction”. This is the seniority coupling scheme or the pairing scheme. “Which of the two coupling effects will dominate depends, of course, partly on the nuclear forces and partly on the nuclear configuration. Roughly we can say that the binding energy of the pairing scheme goes as (

maximum number of nucleons in the shell)

where is the binding energy of a single pair and is the number of particles outside of closed shells. For the aligned scheme (

maximum number of particles in the shell)

where F is the strength of the non-spherical average field produced by a single nucleon. Thus, the aligned coupling scheme will dominate for larger values of while for small values of we should expect the pairing to be more important”. Note that this argument deals just with “nucleons” or “particles”. The fact that there are two kinds of nucleons is not even mentioned, nor is the possibility that the coupling scheme may strongly depend on isospin. The binding energy in the seniority scheme is not just linear in and it even goes down as increases. There is, however, no region of nuclei where binding energies increase quadratically with as is implied in the case of the aligned coupling scheme. The Pauli principle may not allow the number of nucleons which can “best fit into the average (non-spherical) potential which the particles themselves generate”, to be sufficiently large, but then the argument loses its validity.

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There is indeed a competition between two coupling schemes, the seniority like scheme and the one leading to deformation. The actual type of ground states strongly depend on the isospin of the valence nucleons. The T = 1 part of the nuclear interaction, apart from the strong pairing term, is repulsive. This is seen from the quadratic repulsive term in the interaction within a configuration and from the average repulsion between identical nucleons in different orbits. Hence, it does not contribute to the central potential well of the shell model. It is diagonal in seniority and hence, does not lead to the reduction of 0-2 spacings which is the signature of transition to rotational spectra and deformation. In fact, semi-magic nuclei are spherical and have a rather constant 0 - 2 spacings. The T = 0 part of the interaction which acts between protons and neutrons is attractive on the average and is responsible for the average central potential. It determines also the shape of the spherical potential and positions of single nucleon levels as exhibited by the case of If there are several valence protons and neutrons outside closed shells, The T = 0 interaction may lead to deformed nuclei and rotational spectra. It has a quadrupole component which breaks seniority and leads to strong reduction of 0-2 spacings. The number of valence nucleons by itself does not determine the coupling scheme. Semimagic nuclei have fairly constant 0-2 spacings, independent of which, for Sn isotopes ranges up to 16 in the middle of the major shell. Once there are both valence protons and neutrons, the 0-2 spacings go down appreciably with In a subsequent long and detailed paper, “Spherical Nuclei with Simple Residual Forces”, Kisslinger and Sorensen (1963) considered also other nuclei. These are nuclei in which there are valence protons and neutrons but no rotational spectra. They made similar assumptions as in the previous papers and carried out the Bogolyubov-Valatin transformation separately for protons and neutrons (usually in different shells). In addition, they considered a quadrupole interaction between proton and neutron quasi-particles with the coefficient This is the “only neutron-proton interaction which occurs explicitly” in the Hamiltonian. The authors explained that in the “systematic study of even-even, odd-odd and odd-mass nuclei for the spherical region one is generally concerned with quite different aspects of nuclear structure”. Among these, those with new features are the even-even nuclei. Also in the regions considered in this paper, their lowest excited states “are not properly described as two quasi-particle or other simple-particle states, but more nearly as quadrupole vibrational states”. In the absence of a proton-neutron interaction, “two states are lowered into the energy gap. One is a linear combination of neutron states and the other of proton states”. The proton-neutron quadrupole interaction lowers one linear combination of the two states and raises the orthogonal combination.

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“The experimental observation of only one low (and not a doublet) shows furthermore that must at least be a sizable fraction of and The authors actually used which is equal to the strengths of the proton and neutron quadrupole interactions, and which are taken to be equal. They also took the coefficients of the pairing interaction for protons and for neutrons to be equal. In addition to these two parameters, single nucleon energies in the various regions should be known. These are determined from experimental data and by various theoretical considerations (“judicious choice”). The pairing interaction strength is determined from semi-magic nuclei. The “value of X used in the calculation of other properties was chosen to fit the experimental level energy”. Apparently, there were some difficulties since the authors found “evidence for the need of a neutron-proton interaction in addition to the quadrupole interaction. In the even isotopes this is suggested by the fact that the phonon energies for the single-closed-shell isotopes cannot be fitted with the same quadrupole parameters as apply for the cases with both neutrons and protons”. Then “the quasi-particle random phase or dilute quasi-particle approximation has been used to introduce the phonons, which approximately account for the interaction between the quasi-particles due to the quadrupole interaction”. Using this formalism, the authors determined the region of spherical nuclei. For semi-magic nuclei they reproduced the results of their earlier paper. They presented results of detailed calculations for many regions of nuclei. The agreement with experimental energies was usually only semi-quantitative. Also in the present paper they were baffled by the occurrence of low lying levels. “For the Tc, Rh and Ag isotopes it seems almost certain that our coupling scheme is breaking down. The occurrence of the low lying 7/2 and perhaps 5/2 positive parity states would have to be explained in our method by a coupling of the quasi-particle to the phonon. However, we are never able to bring that level nearly as low as required”. Including 3 quasi-particle states, “important corrections would probably be introduced”. The authors calculated magnetic moments of odd-even nuclei and found that “the qualitative results of” the previous paper “are unchanged, i.e., that the phonons themselves do not contribute very much to these moments” and “that the predicted deviations from the single-particle values are too small to account for the experimental results”. They presented calculated moments of the “phonon” state for nuclei in the 50-82 shell. Some of them fairly agreed with later measurements whereas some others do not but the predicted signs agreed with experiment. The authors calculated also quadrupole moments of odd-even nuclei. They included contributions from “certain configurations admixed by a force” which contributed to quadrupole moments, comparable to those of single particles. Still, the major contribution came from the phonon component in

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the wave function. The effective proton charge is taken to be 2e whereas that of the neutron to be e. The agreement with experimental data was fair for certain nuclei but was rather poor for other ones. There were no measured moments of states so there is no calculation of the phonon quadrupole moments. The situation is similar for E2 transitions. The authors calculated rates of various allowed beta decays. Since all of these are unfavoured, the meaning of such calculations is not clear. It is interesting that as late as 1963, physicists carried out detailed and extensive calculations using rather schematic interactions. As pointed out above, these interactions have little resemblance to the effective interaction in nuclei.

5.2.

The Surface Delta Interaction, the Quasi-Spin Scheme and Extensions

It was explained above, that in the case of configuration of identical particles, the state with J =0, is equal to the the part with particles in the BCS ground state wave function. This state, however, was expressed in terms of fermion creation operators. To make the connection between this formalism and seniority, let us return to the seniority scheme in a single Products of two creation operators, and have the properties of spin components. This was first realized by Wada, Takano and Fukuda (1958) and independently by Kerman (1961) who quotes Anderson (1958). If we define the hermitean conjugate is and their commutation relation is

The operator

defined by this relation satisfies the following conditions

These commutation relations are the same as those of components of angular momentum, and Hence, they are called quasi-spin operators and are generators of a SU(2) Lie algebra. As shown by Kerman, the seniority scheme may be defined and developed by using these operators. The pair creation operator its hermitean conjugate and obey the same commutation relations and are generators of SU(2). The pairing interaction introduced by Racah is

with eigenvalues where is either integer or half integer. Comparing this expression with the one in terms of

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we obtain the relation States with seniority are simply and states are given by Kerman, Lawson and Macfarlane (1961) and Lawson and Macfarlane (1965) showed how to use this SU(2) algebra to derive all properties of the seniority scheme. As seen from this description, the pairing theory of BCS is not necessary in the case of a single In that case, exact solutions for ground states exist not only for the pairing interaction but also for all interactions which are diagonal in the seniority scheme. In situations where there are several with equal single particle energies, a seniority scheme may be obtained if matrix elements of the pairing interaction are independent of i.e., if it is equal to Then the operators and are generators of SU(2) and the seniority scheme can be defined and used as in the case of a single The same expressions are obtained for matrix elements and eigenvalues provided is replaced by In particular, this holds for binding energies given by the simple expression and level spacings being independent of particle number. This is a rather simple generalization of seniority in a single which may be called the quasispin scheme. As will be shown, however, it cannot give a good description of nuclear slates and energies. Properties of the quasi-spin scheme were derived in detail, by Arima and Kawarada (1964) and by Arima and Ichimura (1966). The pairing interaction is diagonal in the seniority scheme in configurations of identical nucleons. The zero range potential, the delta interaction is also diagonal in this scheme. In the case of identical nucleons occupying several as shown above, the pairing interaction is diagonal in the quasispin scheme which is a direct generalization of seniority. A special zero range interaction which was introduced by Moszkowski and Green turned out to be also diagonal in the quasi-spin scheme. In a paper based on Green’s Ph.D thesis, Green and Moszkowski (1965) defined the surface delta interaction (SDI) as a delta potential whose radial integrals are equal, irrespective of the orbits. The authors applied the SDI to some simple configurations and compare the results to those obtained with an ordinary delta potential. In the case of the configuration they found that for SDI the low lying levels, with J = 0,2,4, were more evenly spaced than for an ordinary delta potential. Going to the configuration and T = 2, they “find that the states are characterized by definite seniority”. Levels which appear in the two nucleon configuration appeared with the same spacings in the four nucleon case. The situation for T = 0 was rather different. The authors explained that the delta interaction was no longer diagonal in the seniority scheme. To see it in a single the authors calculated the levels of the configuration using a delta potential (which is in this case equivalent to SDI). Whereas in T = 2 states the and level spacings were the same as in the

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configuration, the T = 0, J = 2 level lay lower. Still, the T = 0 “spectrum was only slightly different from the T = 2 case for the lower levels”. In the case of mixed orbits, in the configuration with T = 0, “the ordinary delta function interaction gives a spectrum similar to that for the two-particle case. On the other hand, for a surface delta interaction, the lowest L = 0,2,4 states form a band which has nearly a rotational spacing of levels”. Green and Moszkowski concluded that “it is possible to obtain rotational spectra even with a short range interaction. It is necessary, however, to have mixed configurations and both neutrons and protons participating. Indeed, it is well known that low-lying rotational spectra occur only in nuclei with both protons and neutrons outside closed shells”. In a subsequent paper, Arvieu and Moszkowski (1966), discussed the relation between “Generalized Seniority and the Surface Delta Interaction”. They demonstrated that, for identical nucleons, the SDI is diagonal in the quasispin scheme which is a simple generalization of seniority in a single They used rather than LS-coupling which was used in the paper by Green and Moszkowski (1965). Within configurations, seniority is introduced by using the operator creating a state with S = 0, L = 0 which is equal to

The state on the r.h.s. is created by the operator and hence, seniority in the configuration is equivalent to the quasi-scheme in the mixed configurations with and orbits. The authors defined the pair creation operator as the sum of the operators each of which is multiplied by the phase where is the orbital angular momentum of the corresponding This operator, together with its hermitean conjugate and their commutator, are generators of the SU(2) algebra, like the operators defined above. In fact, any operator may be multiplied by any phase factor. It amounts to using in that space, a frame of reference rotated by 180° around the “z-axis”. Arvieu and Moszkowski proved that the SDI two-body interaction is the sum of a quasi-spin scalar and a term proportional to the number of nucleons. Hence, if all single nucleon energies of participating orbits are equal, the SDI Hamiltonian is diagonal in the quasi-spin scheme. In a following paper, Plastino, Arvieu and Moszkowski (1966) applied the SDI interaction to some “single-closed-shell nuclei, and primarily to the spectra of even nuclei”. The authors realized that the SDI had only one parameter - the strength of the interaction - whereas interactions which were applied to

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these nuclei have several parameters. Therefore, they did not expect to obtain as good an agreement as in more detailed calculations. “It is expected, of course, that the surface delta interaction gives only a first approximation and cannot account for all the effects which are given by such complicated forces”. The authors took single nucleon energies from experiment. They carried out exact diagonalizations for all two nucleon configurations and also in some more complicated cases. Since they had non-degenerate single nucleon states, where the seniority scheme does not apply, the authors used in some cases an approximation based on pairing theory. For odd nuclei, they calculated energies of one quasi-particle states which roughly correspond to seniority slates. For even nuclei they diagonalized the Hamiltonian with SDI and single nucleon energies, in the space of two quasi-particle states. These roughly correspond to states with seniority Comparison between exact and approximate results showed “that the approximation method worked quite well for the two-particle case. However, for more than two particles outside closed shells, the approximation works well for the or the but not so well for the other states”. The authors calculated spectra of even mass nuclei to to to and and N =82 nuclei and Levels of odd mass nuclei were calculated for and levels as well as odd-even binding energy differences, for to and N =82 nuclei, and The strength of the SDI was determined in some cases by the odd-even mass differences and in others by fit to first excited levels. The agreement between calculated and experimental level spacings and odd-even-mass differences was fair in the case of Ni isotopes and only semi-quantitative in the other cases. This is not surprising in view of the limited number of adjustable parameters which the authors use. Still they may be right in their conclusion that “in the nuclei which are well described by a few configurations of the shell model the SDI is able to give a similar (or even better) agreement with experiment than other interactions”. The surface delta interaction became a favourable schematic interaction among shell model theoreticians. There are no radial integrals to calculate and there are analytical expressions for its matrix elements. Very soon after its appearance, Glaudemans, Wildenthal and McGrory (1966) made use of it in a shell model calculation. They considered nuclei with taking into account the orbits, following Glaudemans, Wiechers and Brussaard (1964b). They took the matrix elements determined from experimental data and compared them with those derived from a surface delta interaction. The authors calculated matrix elements of SDI by using 2 interaction strengths, for T = 1 (and S = 0) states and for T =0 (and S = 1) states. They varied these factors, as well as the difference of and sin-

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gle nucleon energies, to obtain the best fit to the matrix elements determined from level spacings. They found some similarities between these two sets, for MeV, but level spacings calculated with these parameters did not agree well with the data. The r.m.s. deviation was .92 MeV and the single nucleon spacing determined this way, was 2.2 MeV, compared to the measured value of 1.27 MeV in The authors then tried to fit all level spacings by varying and the single nucleon spacing. The parameters determined were and the spacing was better, 1.8 MeV. The agreement with measured values was improved, the r.m.s. deviation is .35 MeV which “compares favourably to the results achieved with 16 free parameters”. The diagonal matrix elements calculated from SDI were all attractive. This feature is in contradiction with properties of the T = 1 part of the effective nuclear interaction determined from experimental data, in this region and in others. The average interaction between identical nucleons in different orbits was repulsive. Also the quadratic term in the binding energy formula should be repulsive whereas here it was small but attractive. These problems were simply solved by using the modified surface delta interaction (MSDI) where a constant term was added for T = 1 states and another one for T =0 states. The MSDI was introduced by Glaudemans, Brussaard and Wildenthal (1967) by “the addition of a T-dependent but J-independent term to the surface delta interaction”. This “greatly improves the agreement between the values of the two-body matrix elements calculated from this interaction and the corresponding values obtained both from empirical fits to level energies and from a realistic (Hamada-Johnston) interaction”. In addition to the two strengths, coefficient for T =1 states and for T =0 ones, the authors introduced two constants and These should be added to the SDI two-body matrix elements with T =1 and T =0 respectively. The operator to be added to the SDI is

and its contribution to any state of

nucleons with isospin T is accordingly

The authors determined the coefficients which gave the best fit to energies of 23 ground states and 35 excited states in the upper half of the shell. They included nuclei with assuming the and orbits for the nucleons. These values were and The additive constants made the T =1 average interaction less attractive and increased the average attraction in T =0 states, in agreement

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with experiment. Indeed, the authors obtained fair agreement with measured binding energies and “the agreement for the excitation energies is of the same quality”. The authors compared matrix elements determined by Glaudemans, Wildenthal and McGrory (1966) with those calculated from MSDI. Some matrix elements were well reproduced while others show large deviations. The authors used the same coefficients of MSDI to calculate diagonal and non-diagonal matrix elements of states of and nucleons. The comparison of these with matrix elements determined from experiment by Erne (1966) was even more impressive. They next determined the T =1 parameters of MSDI by a fit to energies of 8 ground states and 20 excited states of Ca isotopes, taking valence neutrons to be in and orbits. The authors then compared matrix elements calculated with and with matrix elements determined from experimental data. Values of the latter were taken from Engeland and Osnes (1966) and Federman and Talmi (1966). The agreement between empirical and MSDI matrix elements was striking, there were only a couple of exceptions. The authors used the same parameters, and to calculate matrix elements of configurations with and neutrons. They compared them with matrix elements of the effective interaction derived from the Hamada-Johnston potential, calculated, using many-body theory, by Lawson, Macfarlane and Kuo (1966) in their analysis of Ni isotopes. The agreement between the two sets was good for some matrix elements and less so for others. The authors wrote that the two-body matrix elements in the complete determined by Cohen and Kurath (1965), “could also be fairly well described with the MSDI parameters” and They made the same claim about the T =1 MSDI parameters, and matrix elements determined from experiment by Cohen, Lawson, Macfarlane, and Soga (1964). An application of MSDI was carried out by Wildenthal (1969) for N =82 nuclei. He took the valence protons to be mostly in the and orbits, but included also excitations of one proton into the higher or orbits. Three differences of single proton energies and the parameter were determined by a fit to 40 level spacings. The single energy and the value of B(= B1) were then determined by “the best fit to the known binding energies of N = 82 ground states”. The values so determined were A =.38 MeV, B=.597 MeV and the and single proton energies were –10.14, –9.62, –7.02, –7.19 MeV. The agreement between calculated spectra and experiment which is presented in the paper was only fair. Single proton energy spacings, measured later in are .96 MeV and 2.71 MeV of the and levels above the ground state. The calculated levels were at .52 MeV and 3.12 MeV respectively. The agreement between the calculated

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and experimental spectrum of was considerably better. Although some deviations are about .2 MeV, the general features of the spectrum were nicely reproduced. Even the J =5 level, the only one “which lacks a probable experimentally observed counterpart” was found later, very close to the predicted position. The same region was studied later by Wildenthal and Larson (1971) using the same shell model space and MSDI. They determined the parameters only from lighter N = 82 nuclei with A = 136 – 140. They obtained a different set of parameters, in which “the significant change ... is an increase, from 0.48 MeV to 0.88 MeV” of the splitting between the and single proton energies. This value was in much better agreement with the observed splitting of .96 MeV in The agreement between calculated level spacings and the 9 experimental ones was very good. This should not be attributed just to the small number of available energies. With 3 exceptions the authors did not publish predictions of unknown levels. Two of them, the and levels in were measured later at 2.126 MeV and 2.262 MeV in very good agreement with the predicted positions of 2.20 MeV and 2.33 MeV, respectively. In there are two low-lying levels, at .604 and .870 MeV, well reproduced by the calculated positions of .65 and .82 MeV. The calculated wave functions of these states were admixtures of mainly and configurations. These wave functions are in reasonable agreement with data on pick-up reactions from to these states. The authors calculated also various E2 transition rates. The only measured rates were of the transitions between the lowest and levels. The B(E2) values were in and only in If an effective proton charge is taken to be 1.47e, to fit the value, the calculated value for the rate was small, The authors attributed the marked reduction in B(E2) value to the well known selection rule which forbids E2 transitions in a half filled orbit between states with the same seniority. Another application of MSDI to nuclei is the work of Glaudemans, De Voigt and Steffens (1972) on the structure of Ni isotopes, described above. There were also other applications but they will not be mentioned here. Some of them may be found in the book of Brussaard and Glaudemans (1977). Many years later, Rejmund et al., (1997) measured high spin states in Yrast levels up to at about 4 MeV were found. The authors carried out also a shell model calculation for valence neutrons occupying the orbits. Single neutron energies were taken from experiment. The authors used for the two-body interaction the SDI with the strength parameter equal to With this value of the

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binding energy of relative to was calculated to be –9.219 MeV compared to the measured value of –9.122 MeV. Very good agreement between calculated and measured energies were obtained for the J =2,4,6,8 levels whose main components are states. The weights of these components, and weights of other main components in the corresponding eigenstates was The agreement was also very good for the higher J =10 state of which the main component is the state. It was a little worse for the negative parity and with the main components and respectively. There was very good agreement for the level whose main component is the J =14 state of the configuration. Attempts were made to deduce a global interaction which would yield the correct matrix elements of the effective interaction. Molinari, Johnson, Bethe, and Alberico (1976) looked at nuclei with two valence nucleons or holes, and deduced from the observed low-lying levels, two-body matrix elements. They tried to reproduce them by using a short range delta force to which, a “long range core-mediated component” was added. They tried to determine that component by using various diagrams of many-body theory. The authors concluded that “very little evidence has emerged from our analysis for a monopole force”. An “important result of our analysis concerns the T = 1 channel for nonequivalent particles: the semiclassical approach fails due to the very large repulsive interaction”. The authors explained that, in their analysis they use a method due to Schiffer (1971). They write: “In conclusion, the Schiffer method proves once more the validity of the old “pairing plus quadrupole” interaction of the Copenhagen school”. In 5.1 above, a severe criticism of this interaction was presented and there is no need to repeat it here. Also Schiffer, who applied his method to nuclei, seemed not to concur with their conclusion. A very serious attempt to obtain “the effective interaction between nucleons deduced from nuclear spectra”, was made by Schiffer and True (1976). They described in detail two methods of obtaining matrix elements of the effective interaction. One was the reaction matrix approach and another one “is to use a phenomenological force for the residual interaction”. Little attention was paid to a “third approach, that is applicable in a few restricted regions of the periodic table, is to do a least-squares fitting to a selected set of experimental data”. Then “the resultant wave functions can be checked for “further” consistency” by comparing with experiment predictions of moments and transition rates. Schiffer and True describe their approach: “In selected nuclei throughout the periodic table, it is possible to determine reasonably well from experimental data the value of the diagonal matrix elements of the residual interaction”. They tried to find cases with a pure configuration”. In cases in which the state is “distributed among two or more actual levels”, it may be pos-

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sible to obtain help from data of direct reactions. It may then be “possible to deduce the energy centroid of the unperturbed two-particle configuration and thus one is still able to determine the value of the two-body matrix element”. From single nucleon energies, the unperturbed position of a configuration was determined and then, “provided that one has a pure configuration, the difference is a measure of the two-body matrix element”. Here, is the measured energy of the It was obtained from the centroid of levels, with the same J, weighted according to the fraction of the pure which they contain according to direct reactions. This determination depended on the accuracy of the results which various direct reactions claim. It is rather difficult to apply to cases where several configurations are mixed. Still the authors went very methodically and carefully over all regions of the periodic table and deduced values of many diagonal matrix elements. They tried to find various interactions, including tensor forces and mutual spin orbit interactions, which fitted best the matrix elements which they determined. They compared their approach to that of Molinari et al., and pointed out the differences. Schiffer and True used for the central interaction, a combination of two Yukawa potentials with different ranges and strengths. The authors concluded that “a reasonably satisfactory over-all fit is obtained to well over one hundred experimental matrix elements from nuclei throughout the periodic table. The interaction has 12 parameters, though all but the last 32% in is accomplished by only five parameters in the central interaction. The need for a two-range interaction, with a shorter range attraction and a longer range repulsion is necessary for T = 1”. In this way, the average repulsion between identical nucleons was achieved. In the MSDI, the attractive potential has zero range and the repulsive one has infinite-range. “For the T = 0 interaction the need for a second range is much less clear. The tensor term in the interaction improves the fit for rather few specific matrix elements, while the L – S term seems to improve the level of fit in a more general way”. The authors were aware of the fact that “this investigation has only considered diagonal matrix elements; there is not enough reliable data available from transfer reactions to extract off-diagonal ones”. In spite of the admiration that one can have for the extensive and intensive effort invested, it is justified to ask what kind of effective interaction was determined. The effective interaction for the shell model must include also non-diagonal matrix elements. Even to calculate spectra of the two-nucleon configurations which the authors considered, non-diagonal matrix elements are required. They are certainly needed for calculating energies and wave functions of more complicated configurations. There is no reason to believe that the potentials which may be used to calculate diagonal matrix elements will yield also reliable non-diagonal ones. The interactions described above are intended to be applicable to nuclei in

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the whole periodic table. The effective interactions which they represent are to be used in shell model calculations in which only valence nucleons are active. There are global interactions which are intended for use in calculations in which all A nucleons participate, like in the Hartree-Fock approach. Among them, a very popular one is the Skyrme interaction which contains several parameters. Various determinations of them give rise to many versions of this interaction. It is based on the and its derivatives, and contains explicit three-body interactions. Another global interaction is the Migdal force which is a multiplied by a linear combination of exchange operators. A global interaction with a finite-range, instead of a zero range, was introduced by Gogny. These interactions will not be described here since they are not intended for use in shell model calculations with valence nucleons. In fact, using some of them in this way, leads to absurd results. They are described in various text books, such as Ring and Schuck (1980).

5.3.

The SU (3) Scheme

In the supermultiplet theory it is assumed that the interaction between nucleons depends only on space coordinates and momenta. Eigenstates can then be classified according to the symmetry properties of the spatial parts of the wave functions. The latter are determined by the symmetry properties of the spin-isospin part which are characterized by the irreducible representations of the SU(4) group. A more detailed specification of the states, with given values of T and S, which belong to a given supermultiplet, can be obtained only by a more detailed study of the spatial parts. In the configuration, due to the antisymmetry of the complete wave functions, their spatial parts have the symmetry which is opposite (dual) to the symmetry of the spin-isospin SU(4) states. The former are bases of irreducible representations of the unitary group in dimensions. If the interaction fulfils certain conditions, states can be characterized by the seniority quantum numbers. Such states form bases of irreducible representations of the group which is a subgroup of U(2l+1) and has O(3) as a subgroup. If several are admixed in the wave function, there are usually no simple prescriptions for a detailed characterization of states. In some special cases, however, there is a simple scheme which is of great theoretical interest. It is the SU(3) scheme introduced by Elliott (1958a, 1958b). Elliott introduced the SU(3) scheme in a series of papers. The title of the first is “Collective motion in the nuclear shell model I. Classification schemes for states of mixed configurations” and it was followed by II and III. To obtain the SU(3) scheme, it is necessary to start with a single nucleon Hamiltonian which is invariant under transformations of the SU(3) group.

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Such a Hamiltonian is offered by the harmonic oscillator potential well, and may be expressed as

where the vector operators

and A are defined by

The oscillator Hamiltonian is invariant under a large group of transformations in the 3-dimensional space of and Elements of the U(3) group of unitary transformations of the (complex) vectors leave invariant the scalar product Thus, these transformations commute with the scalar product and hence, with the oscillator Hamiltonian. This fact implies that the eigenvalues of this Hamiltonian which belong to states in the same basis of an irreducible representation of U(3) are degenerate. This is indeed the case for single nucleon orbits which occupy an oscillator major shell, like the the etc. More interesting are two-nucleon interactions which are diagonal in the SU(3) scheme. Two-nucleon interactions may be obtained by forming scalar operators from products of generators of the group. The generators (infinitesimal transformations) of a unitary group in dimensions can be chosen as the set of irreducible tensor operators with ranks to Here, tensors with may be constructed from and A by taking their various tensor products. The scalar product is equal to up to an additive constant. The vector product is equal to the vector product of and multiplied by and thus proportional to l. The trace of the symmetric product of and A is equal to which is proportional to the scalar product. There are thus 5 independent components of the traceless symmetric product. Linear combinations of them may be formed which are components of a rank irreducible (spherical) tensor. The 9 component of these irreducible tensor operators are generators of the unitary group U(3) in the 3 dimensional space. When going from the full U(3) group to its subgroup SU(3) of elements whose determinants are equal to 1, the irreducible representations are not reduced. Certain U(3) irreducible representations become equivalent SU(3) representations. They differ only in the number of particles and their bases contain the same states. Thus, it is the classification scheme of SU(3) which is of physical interest. The SU(3) generators are the infinitesimal elements of U(3) with vanishing traces. Hence, the generators of SU(3) are the components of the orbital angular momentum l and the quadrupole operator

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The generators of SU(3) in the general case, with several nucleons in the major oscillator shell, are the 3+5=8 components of

Scalar products of these tensors, which contain some single nucleon terms, provide the only two-nucleon interactions whose eigenstates belong to bases of irreducible representations of SU(3). The irreducible representations of SU(3) are characterized by two integers, and µ which satisfy the relation All single nucleon state in the N-th oscillator shell form a basis of an irreducible SU(3) representation with Their single nucleon energies due to the oscillator Hamiltonian H0, are all equal to (note that here the shell has N = 1 and the lower shell has N = 0). The only two nucleon interactions which are diagonal in the SU(3) scheme are thus given by an L.L interaction and a certain quadrupole-quadrupole interaction. Such interaction has only two independent coefficients. The Hamiltonian is given by

The single nucleon energies in H are given by those in to which single nucleon contributions from the interaction terms should be added. The eigenvalues of H may be evaluated by using the Casimir operator of SU(3). The quadratic function of the generators which commutes with all SU(3) transformations is

Its eigenvalues are given by

The eigenvalues of the Hamiltonian for

We can express H by

nucleons are then given by

For a given ( µ) irreducible representation, states are grouped into rotational like bands with excitation energies proportional to L(L+1). Such a term arises from an attractive quadrupole interaction alone and has the right sign, i.e., a positive moment of inertia. If the electromagnetic quadrupole operator is taken to be proportional to the generator of SU(3), then E2 transitions may take place only within the band. The shell model SU(3) scheme yields features of the collective model.

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The similarity between the SU(3) scheme and the collective model goes even further. In the latter, spins of even-even nuclei, in the ground state rotational band, are equal to J=0,2,4,... These are the quantum states of a symmetric rotor. A rotation around the axis of symmetry leaves the state invariant and hence, the component of the angular momentum along the axis must vanish. This component is denoted by K and may have a definite value even though and have definite values. That component is the scalar product of J with a unit vector directed along an axis defined by the dynamical variables of the system. Therefore, it commutes with the components and defined along axes fixed in space. The Hamiltonian of the nucleus is invariant under space rotations in any model. Hence, the eigenvalues of the 2J+1 states, with the various eigenvalues M of the component, are equal. This is not the case for the projection K on a body-fixed axis. Unlike the geometrical quantum number M, the value of K is determined by the internal state of the system. From the meaning of this quantum number it follows that If K = 0 than the spins in the rotational band are given by J=0,2,4,... If then the band contains states with J = K, K+1, K+2,... In the SU(3) scheme, if there are several states with the same value of L in a given irreducible representation, an additional quantum number, K is needed to distinguish between them. According to the definition of Elliott, this K plays a similar role to the K in the collective model. In an SU(3) irreducible representation ( µ) with the values of K are K = µ,µ –2,...,1 or 0. The values of L of states included in the basis of this representation for K = 0 are or 0. For they are given by K, K + 1, K + 2,..., For any values of and µ, in these formulae, K should be replaced by min( µ) and by max( µ). Unlike the bands in the collective model, the SU(3) rotational bands have a maximum value of L determined by the shell model (“microscopic”) structure. The energy of states with the same ( µ) and L are equal, independent of K. The relative positions of the bands are determined by the quantum numbers and µ. The SU(3) scheme provides a shell model description of collective states which is is very elegant and yet simple and transparent. Its introduction into nuclear structure physics caused great excitement among nuclear theorists. The fact that it is possible to obtain by using rather simple shell model wave functions, states which have features of rotational states, was appreciated by all. It attracted the attention of many theorists who tried to apply it to actual nuclei. Clearly, the schematic interaction of SU(3) with two parameters, is much too restrictive. Hence, attempts were made to see whether various potential interactions lead to eigenstates which can be simply described in terms of SU(3). The question was whether a given eigenstate belongs to a basis of an irreducible representation of SU(3) or is a linear combination of states which be-

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long to only few such bases. Most of the work was carried out in the shell. Interactions which must be expressed by spin operators mix different SU(3) states. In particular, the spin-orbit interaction strongly breaks the description in terms of SU(3) irreducible representations. The large spin-orbit splitting between and (about 5 MeV) drastically reduces the mixing of nucleon states which is implied by SU(3) symmetry. As mentioned above, this is particularly true for states with higher isospins. The SU(3) description was applied to the rotational-like band in whereas does not exhibit any rotational features. A summary of applications of SU(3) to shell nuclei can be found in a review paper by Harvey (1968). As explained above, the strong spin-orbit interaction which determines the orbits in the central potential well of the shell model, breaks SU(3) symmetry in a major way. This fact prevented the successful application of the SU(3) scheme to nuclei beyond the shell. In fact, the shell is the highest one which contains all the orbits in an oscillator shell. In heavier nuclei, the orbit with the highest is lowered into the lower shell and the orbit with the highest in the next oscillator shell, with different parity, is included. In some cases, however, the formalism of the SU(3) scheme could be applied to actual nuclei. A simple example is offered by the neutron orbits in Ni isotopes. The orbit is closer to the orbit than its spin-orbit partner, the orbit. There are more, and better examples of such bunching which was found by Ginocchio (1997) to follow from the relativistic mean field theory. It was suggested to consider the and as due to coupling of a pseudo-spin to a pseudo-orbit with Such pairs of orbits are not separated by the strong spin-orbit interaction and hence, may be approximated by degenerate orbits and states of several nucleons may be represented by a pseudo LS-coupling scheme. Moreover, since the energy of the orbit is also rather close, it could be included as an orbit with pseudo-orbit The SU(3) scheme of the shell could then be applied to these orbits. This is how the pseudo SU(3) scheme was introduced by Arima, Harvey and Shimizu (1969) and by Adler and Hecht (1969). An important fact which enabled the use of quadrupole-quadrupole interactions for nucleons in such orbits is a property of spherical harmonics with even rank Their matrix elements between and states depend only on and not on provided both orbits have the same parity. Such a scheme cannot include intruder orbits with different parity, like the in the present example. No successful applications of this approach have been made to actual nuclei. A revival of SU(3) symmetry in nuclear structure physics came with the introduction of the interacting boson model by Arima and Iachello (1975). An equivalent model was introduced earlier by Janssen, Jolos and Donau (1974).

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In the first version, IBM-1, the dynamics of even-even nuclei was described by that of an assembly of N bosons. These have either ( bosons) or ( bosons). It was customary to associate these bosons with nucleon pairs coupled to J = 0 or J = 2, but the latter are either proton or neutron pairs. This distinction is made in the IBM-2 version and, with explicit inclusion of isospin, in the IBM-3 version. The IBM-2 introduced two kinds of and bosons, proton bosons and neutron and bosons. The IBM-1 Hamiltonian contained single and boson energies and boson-boson interactions. The space part of the bosons was not specified and hence, the natural expression of operators was by boson creation and annihilation operators. Each term of the boson-boson interaction is a product of two creation and two annihilation operators. Hence, the total number of bosons

commutes with the IBM-1 Hamiltonian and is conserved. The following products of creation and annihilation operators

may be taken as the 1+5+5+25=36 generators of U(6), the group of unitary transformations in the 6-dimensional space, spanned by the 1+5=6 components of the and bosons. The IBM-1 Hamiltonian may be constructed as a linear combination of scalar operators which are products of these generators. Hence, all its symmetric eigenfunctions, for a given value of N, transform irreducibly under U(6) transformations. They all, irrespective of L and M, form the basis of the fully symmetric irreducible representation of U(6), uniquely determined by N. Before proceeding with all symmetries which may occur in IBM-1, let us make the connection with IBM-2 (Arima, Otsuka, Iachello and Talmi (1977)). The Hamiltonian of the latter, includes two-body interactions between proton bosons, between neutron bosons and interactions between proton bosons and neutron bosons. The IBM-2 Hamiltonian conserves electric charge and the number of proton bosons is conserved, as well as the number of neutron bosons. The eigenfunctions of this Hamiltonian are thus, linear combinations of products of proton boson wave functions and neutron boson wave functions. The proton states form a basis of the fully symmetric irreducible representation of characterized by Similarly, the neutron boson wave functions form an irreducible representation of determined by The complete boson wave function is fully symmetric with respect to interchanges of proton bosons with neutron bosons. It may be expressed, however, in a form where the proton or neutron nature of the bosons is separated from their other variables. In fact, a quantum number,

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F-spin was introduced to distinguish between a proton boson (F = 1/2) and a neutron boson (F = –1/2). The F-spin plays a similar role to ordinary spin in LS-coupling or to isospin, but is clearly different from the latter. It is possible to construct F-spin states with given value of the total F-spin, from which is fully symmetric in proton and neutron bosons, down to with decreasing symmetries. The projection of the total F-spin vector on the “3-axis” is Fully symmetric states in proton bosons and neutron bosons can be constructed, in principle, as linear combinations of products of F-spin states by states of the other boson variables which have the same symmetry as the F-spin states. In particular, the F-spin state with maximum symmetry, F = N/2, should be just multiplied by fully symmetric states in the other boson variables. This procedure may be expressed as going from the irreducible representations of the group characterized by to those of the subgroup characterized by N,F. If the IBM-2 Hamiltonian is fully symmetric under interchanges of proton bosons and neutron bosons, it commutes with the square of the total F-spin vector. Then the eigenstates are characterized by definite F-spin values and the eigenvalues are equal for all states with different projections of F, i.e., different values of and satisfying The states with maximum symmetry have F = N/2 and their eigenvalues are independent of and satisfying Hence, the eigenvalues are equal to the case in which or In such cases, the IBM-2 states are equivalent to states of a IBM-1 Hamiltonian. Thus, IBM-2 states with maximum symmetry, F = N/2 correspond exactly to IBM-1 states. The following discussion of symmetries will be based on IBM-1 without considering the validity of the assumptions leading to it. In this review, the differences between T = 1 and T = 0 interactions are emphasized at every opportunity. It is rather difficult to see how Hamiltonians in which those differences are ignored, can represent correctly the physical situation. In this respect, as well as in others, IBM-1 is similar to the collective model where no distinction is made between proton and neutron excitations. Eigenstates and eigenvalues of the general IBM-1 Hamiltonian can be obtained by writing its sub-matrix for a given N in a basis of U(6) and diagonalizing. Arima and Iachello realized that if the Hamiltonian is restricted in certain ways, very simple and elegant symmetries emerge. The simplest way is to express the Hamiltonian without the use of and operators. In that case, it is expressed by the operators, i.e., by generators of U(5), the group of unitary transformations in the 5 dimensional space spanned by the states of a single boson. Eigenvalues of such U(5) Hamiltonians can be expressed by

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a closed formula. There are only 3 independent scalar operators constructed by products of 2 creation and 2 annihilation operators of bosons. Hence, any U(5) Hamiltonian can be expressed as a linear combination of 3 quadratic Casimir operators of a chain of groups. These are, U(5), the subgroup O(5) of U(5) which is the group of orthogonal transformations in the 5 dimensional space, and the subgroup O(3) of it. The latter is the group of orthogonal transformations in the 5 dimensional space induced by 3 dimensional rotations. The generators of O(3) are the components of the angular momentum which satisfy the usual commutation relations. The angular momentum is carried by bosons only and its components are given by

The quadratic Casimir operator of O(3) is simply with eigenvalues L(L+1). There are also simple expressions for the eigenvalues of the other Casimir operators which furnish quantum numbers to characterize the states. Since these Casimir operators commute, by definition, the eigenvalues of the Hamiltonian are a linear combination of their eigenvalues. A situation like this is called dynamical symmetry. The U(5) symmetry is considered to be relevant for the description of vibrational nuclei of the collective model. There are two other chains of subgroups of U(6) which exhibit dynamical symmetries. These symmetries may be also realized in certain shell model spaces. The generators of SU(3) described above, are the components of a special quadrupole operator and the 3 angular momentum components. Certain linear combinations of the U(6) generators form the components of a quadrupole operator with special properties. The commutation relations between its components are equal to linear combinations of the angular momentum components. The commutation relations between angular momentum components and components of any irreducible tensor operator are proportional to the latter. Hence, SU(3) defined by these generators is a subgroup of U(6). The generators of O(3) are a subset of those of SU(3) and hence, O(3) is a subgroup of SU(3). The most general U(6) Hamiltonian, with single boson terms and bosonboson interactions, which has SU(3) symmetry, is rather simple. It is a linear combination of the scalar products of the only two tensor operators which may be constructed from SU(3) generators. These are the angular momentum and the quadrupole operator whose components are given by

This linear combination of scalar products can be conveniently expressed as a

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linear combination of the Casimir operators of SU(3) and O(3),

The eigenvalues of are quadratic functions of and µ which characterize the irreducible representations of SU(3), as presented above,

Thus, in the SU(3) symmetry, the energy eigenvalues are given by a simple closed formula. The spectrum of the SU(3) Hamiltonian is very similar to that of the collective model for strongly deformed (rotational) nuclei. The similarity is even more pronounced than in the shell model. For N bosons, an attractive quadrupole interaction leads to the lowest level having µ=0. With the positive (repulsive) sign of the (L.L) coefficient, which is obtained from an attractive quadrupole interaction, a ground state band emerges with energies proportional to L(L+1) up to L =2N. The next higher irreducible representation is characterized by µ=2. There are two rotational bands in this case. One is for K = 0 and it contains states with L=0,2,...,2N–4, a “beta band”. The other is a “gamma band” with K = 2 and its levels have spins L=2,3,...,2N–2. Levels with the same L in both bands have the same excitation energy. This feature does not arise in the collective model and is due to the simplicity of the SU(3) Hamiltonian. The use of the boson SU(3) Hamiltonian for the description of rotational nuclei, does not depend on the particular bunching of single nucleon orbits. Its use is possible in any region of nuclei but the use of the boson model itself should be first justified. This discussion will not be presented nor discussion of the extensive use of the boson model and comparison with experiment. Before going back to the shell model, let us look at the last group chain with dynamical symmetry in IBM-1. A subgroup of U(6) is the group of orthogonal transformations in the 6 dimensional space spanned by states of a boson and boson. Generators of this group, O(6), are the components of tensor operators constructed from the U(6) generators. These are, in addition to L, a tensor of rank 3, similarly constructed from creation and annihilation operators of bosons, and a quadrupole operator. The total number of components is 3+7+5=15 which is equal to 6×5/2 required for an orthogonal group in a 6 dimensional space. The O(6) quadrupole operator is defined by

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O(6) has a subgroup of orthogonal transformations in the 5 dimensional space of the boson components. The generators of this O(5) group are those of O(6) which are constructed from only boson operators, i.e., the rank 3 tensor and the rank 1 angular momentum vector (7+3=10=5x4/2). The components of the latter are the generators of O(3) the group of orthogonal transformations among those of O(5) which are induced by 3 dimensional rotations. The general U(6) Hamiltonian with O(6) symmetry is a linear combination of scalar products of these tensor operators. Thus, it can be simply expressed by a linear combination of the quadratic Casimir operators of O(6), O(5) and O(3). Each of these is a linear combination of scalar products of tensors whose components are generators. The eigenvalues of which belong to fully symmetric irreducible representations, are given by a quantum number as The possible values of are or 0. The eigenvalues of are similarly given by where The quantum number is actually the seniority which can be defined for bosons. The O(6) spectrum has rotational like bands, due to the L(L+1) term in the Hamiltonian. It is more complex than the SU(3) spectrum. It corresponds to nuclei in the collective model whose deformation potential has a sharp minimum at a fixed value of but is completely independent of This is different from the case of SU(3) symmetry where the deformation is axially symmetric and the minimum is at Ginocchio (1980) constructed shell model states which correspond exactly to the SU(3) and O(6) symmetries of the boson model. He constructed operators of two (identical) nucleons which create states with J = 0 and J = 2, S and D pairs, and single nucleon tensor operators with ranks Components of the latter obey the same commutation relations as the boson generators of O(6) or SU(3). Hence, they generate the same groups in the fermion case. The commutators of those components with and pair creation operators, are also the same as the commutators of the boson generators with the boson creation operators and Thus, the irreducible representations of O(6) and SU(3) in the shell model space are created by operating with and on the vacuum state as was the case with bosons. The reason why the commutators with and do not produce other pairs, with or without the same angular momenta, is simple. The individual spins of orbits in the shell model sub-space considered, are obtained by coupling a spin to a spin To obtain O(6) symmetry, we put and is any integer, the shell model space includes 4 orbits (for ) with values of For SU(3), is any half integer and and 3 orbits (for ) are included with equal to and The tensor operators act only on components of in the first case, and on those of in the other. In the pairs which create the basis for O(6) states, the total K vanishes

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and hence, only unique J = 0 and J = 2 antisymmetric pairs of the configuration exist. Similarly, in the SU(3) pairs, the total I vanishes and there are only unique J = 0 and J = 2 symmetric states. The auxiliary and spins need not appear explicitly in the various definitions. The transformation from a scheme defined by to the scheme of is a change of coupling transformation,

Hence, the state created by the pair creation operator in which I = 0, K = 0, can be expressed by using standard angular momentum algebra, as

The operator which creates this state may be expressed as and hence, can be used to create a quasi-spin scheme in the model space considered. This SU(2) symmetry is rather simple compared to the O(6) and SU(3) cases and its applicability to nuclei was discussed above. The expression of is the same for both latter symmetries, whereas the operator and the irreducible single nucleon operators are different as will be shown below. Tensor operators which act only on the variables can be expressed also in terms of creation and annihilation operators of The creation operators transform irreducibly under rotations like the Annihilation operators with the same transformation properties are Both tensorial sets are used to express those tensor operators as

The commutator of the L = 2 quadrupole operator with operator

yields the

The tensor operators with L= 1,2,3 generate the O(6) group. Shell model states which form irreducible O(6) representations may be created by and in the same way that boson O(6) representations are created by and To every nucleon representation there is

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a corresponding boson one. The inverse holds only up to the middle of the shell, For nucleon states are determined by the number of holes, whereas the number of bosons is unlimited. Shell model Hamiltonians constructed from scalar products of the tensor operators with L =1,2,3 are diagonal in the O(6) scheme. They may be expressed by a linear combination of the quadratic Casimir operators of O(6), O(5), and O(3), like in the boson case. The eigenstates form bases of irreducible representations of O(6) and can be characterized by the eigenvalues of the Casimir operators. The energy eigenvalues are given by a linear combinations of those eigenvalues. To obtain the SU(3) symmetry, a quadrupole operator and an angular momentum vector are needed. These operators should act only on the variables, for In the scheme, they can be expressed as

These operators generate in the shell model space considered, transformations induced by unitary transformations in the 3 dimensional space of the variables. Hence, they generate SU(3) transformations in the space of with The operators which correspond to those in the boson model are and Thus, the operator in the SU(3) case is

This operator and create states which are bases of irreducible representations of SU(3), exactly like and in the boson case. Shell model Hamiltonians which are linear combinations of scalar products of Q and L, possess the SU(3) dynamical symmetry. They can be expressed as linear combinations of the quadratic Casimir operators of SU(3) and O(3). Energy eigenvalues are linear combinations of their eigenvalues, which serve as quantum numbers which characterize eigenstates. To every state of nucleons there is a corresponding boson state. There are, however, boson states which do not have matching nucleon states. This occurs not only in beyond the middle of the shell, as in the case of the O(6) symmetry. Beyond one third of the shell, for certain SU(3) irreducible representations are simply not realized for nucleons. The boson representation which contains the ground state band has µ=0. Its highest level has J = 2N and its M = 2N state is cre-

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ated by

In the case of nucleons, the corresponding state should be but it vanishes if Indeed, since I = 0 and K = 2 in the operator, in such a state, all of the nucleons have projection + 1 on the According to the Pauli principle, their states should be different. There are such possible states and this is the highest number of (identical) nucleons which can form this J =2N state. Since such states with vanish, along with all other states in the same irreducible representation. The Ginocchio models are very interesting. They demonstrate that it is possible, in principle, to map nucleon states and operators on boson states and operators. This mapping is exact in spite of the Pauli principle and leads to very different commutation relations of the corresponding operators. The relevance of these models to description of nuclear states is less clear. First, the models exhibit SU(3) rotational or O(6) features for identical valence nucleons in clearly defined orbits. As emphasized above, these features appear only when there are both protons and neutrons outside closed shells, irrespective of the orbits occupied. Another difficulty, called by Ginocchio a “fatal flaw” of the model, is the absence of the ground state band around the middle of the shell where maximum collectivity is expected. It is not difficult to include valence protons and neutrons in the SU(3) description (the same considerations apply to O(6)). A shell model Hamiltonian which is a linear combination of scalar products and has SU(3) dynamical symmetry. This is like the boson IBA-2 model with similar consequences. Eigenstates transform under the irreducible representations of group which is a subgroup of the group. This approach is the basis of the Fermion Dynamical Symmetry Model (FDSM) introduced by Feng et al. In each major shell, they considered the available orbits and imposed accordingly a symmetry group for the protons and the neutrons. In certain regions they adopted in others, etc. Since the orbits included in the Ginocchio models have the same parity, the orbit with highest with opposite parity, is treated differently. In FDSM, the only symmetry of those nucleons is SU(2) and they are coupled to pairs with J = 0 in low lying levels. Their presence also helps to cure the “fatal flaw” in the case of SU(3) symmetry, since the number of nucleons in “normal parity” orbits need not exceed The FDSM was explained by Wu, Feng, Chen, Chen, and Guidry (1987) and the actual applications were summarized by Wu, Feng and Guidry (1994) where references to earlier papers are given.

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SENIORITY AND GENERALIZED SENIORITY IN SEMI-MAGIC NUCLEI

6.1. The Seniority Scheme and Applications As pointed out above, there is evidence that the T = 1 effective interaction, the nuclear interaction between identical valence nucleons, is diagonal in the seniority scheme. It was explained that any two body interaction is diagonal in seniority in configurations of identical nucleons, for any Thus, the first occasion to find out whether the effective T = 1 interaction is indeed diagonal in the seniority scheme was by looking at proton configurations in nuclei with N = 50. These configurations are mixed with ones. The configuration has only one state, with J =0. Hence, if the interaction is diagonal in seniority, spacings between levels, with J=2,4,..., as well as levels, are constant, independent of They are equal to the spacings of unperturbed states of the configuration. The same is true for levels in odd configurations. Spacings of levels which are independent of the number of valence nucleons, thus provide the first test of validity of the seniority scheme. Matrix elements which are non-diagonal in seniority are proportional to matrix elements of the interaction between the state with and states of the configuration. The seniority scheme maximizes the amount of interactions, the maximum is for the states for even values of and states for odd The matrix elements which are non-diagonal in seniority, are linear combinations of only spacings between levels of the configuration with As mentioned above, measured level spacings of nuclei with N = 50, clearly show that the effective interaction in proton configurations is diagonal in the seniority scheme to a very good approximation, No experimental information is available on neutron configurations. There is also little experimental information on neutron configurations from lead nuclei (Z = 82) with N >126. The ground state of has spin indicating a valence neutron, whereas the first excited state, with spin lies .78 MeV above it. Spacings of J=0,2,4,6,8 levels in and are fairly equal, consistent with states of configurations. The higher excited states lie about .5 MeV above these states. Ground states of and have spins but only some levels with no spin assignments are known in If the levels are used, the calculated position of the level is .446 MeV above the ground state. The first excited state known in is at .439 MeV which could support the configuration assignment. In any case, the calculated matrix element between the J = 9/2 states with and in the configuration is negligible. Its value is .016 MeV

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calculated from the levels and –.026 MeV from those of There is more experimental information on proton configurations in N = 126 nuclei. The overall agreement with the predictions for such configurations is good but spectra of both even and odd nuclei show some slight but systematic spreading with A. The computed non-diagonal matrix elements between states with and range between .04 MeV and –.05 MeV. Rather pure proton configurations, seem to occur in an unexpected region of nuclei. Kleinheinz and co-workers looked at which has closed neutron shells (N = 82). They were interested in it since the and orbits could have been completely filled. They expected to have some features of a doubly magic nucleus. Indeed, they established by experiment, that its first excited state is a state, lower than the one, contrary to the situation in non-magic nuclei. The first paper was published by Kleinheinz, Lunardi, Ogawa, and Maier (1978) and it was followed by many experimental papers over several years, which also showed that apparently, two or more, valence protons in Z >64, N = 82, nuclei go into the orbit. Some of these papers were written with the participation of theorists, J.Blomqvist and R.D.Lawson, and contain relevant shell model calculations. Some levels were considered as due to coupling of valence proton(s) to the excited state of the core. In several of the papers, nuclei with N = 83 neutrons were considered. In those nuclei, certain levels were taken to be due to coupling of the valence neutron to states in which a proton is coupled to the core excitation. The octupole state was assumed to be a collective excitation of the core. Such excitations were not discussed in this review and this applies also to the interesting theoretical work related to those levels in which a neutron is coupled to them. References to the experimental work are given in a review article by Blomqvist (1984). In this review, the author presented the discussion of configurations and possible mixings of other ones. Since his review was based on lectures he gave at a Nordic Winter School, the author must have felt obliged to include a discussion based on BCS theory... Needless to mention that this is in sharp contrast to the simple and successful shell model description of the experimental data in this region. Seniority was used but not discussed in that review, it plays an important role in a paper by Lawson (1981). He concluded that the data show the conservation of “seniority to a high degree”. Looking at level schemes of nearby nuclei, it is not easy to understand why protons occupy the orbit. Much talk is devoted to some “shell gaps” which are supposed to make the orbit far removed from others in the major shell. Irrespective of the method by which those are obtained, we can

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look at the spectrum of That spectrum, obtained by adding one valence proton to the core, was not fully known in those years. The ground state has spin the state is .05 MeV above it and the state is at .253 MeV excitation. These states are due to the valence proton occupying the the and the orbits. These orbits are assumed to be far removed in energy from the and orbits. In however, there are and (hole) states at .355 and .719 MeV respectively. Even if the and proton orbits are completely filled, the apparent proximity of the and orbits may give rise to considerable configuration mixing into configurations. One possible reason for the apparent purity of states may be the rather strong mutual interaction between nucleons. Another reason was considered by Lawson (1981). He explained that because of the different parities, only configurations with two (or four) protons occupying the and orbits could be mixed into ones. Their effects on energies, if taken in second order perturbation theory, would simply renormalize the effective interaction of the protons. In fact, if the two nucleons are coupled to J = 0 (as it must be for protons), the contributions in second order perturbation theory are equivalent to the addition of a pairing interaction to the original interaction. In that case, the levels in the configuration are not affected. Lawson used their values, taken from to calculate the non-diagonal matrix element between the J = 11/2 states, with and of the configuration. This value, to which all non-diagonal elements are proportional, is equal to –.006 MeV. It is indeed very small compared to differences of diagonal elements which are usually more than 1 MeV. Thus, to a very good approximation, the effective interaction in configurations is diagonal in the seniority scheme. It was already mentioned by Talmi and Unna (1960c), that only states with the same seniority of the and the configurations could be mixed and spacings of states with the same seniority remain unchanged by the mixing, even if it is not a small perturbation. In his paper, Lawson (1981) used the level spacings of to calculate the spectra of configurations in heavier N = 82 nuclei. The calculated level schemes follow the predictions of the seniority scheme to a high degree of accuracy. Calculated spacings of and levels in even-even nuclei are almost exactly equal to those of the configuration in The calculated spacings of and levels for the and cases are also almost equal. Lawson calculated the spectrum and compared it with levels in The calculated levels were in good agreement with the measured ones which have spins 27/2,23/2,19/2,15/2 lying above the 11/2 ground state. Lawson considered also electromagnetic transi-

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tions and pointed out that the seniority scheme “number dependence of the E2 transition probabilities” could be used. The dependence of B(E2) on of transitions between states with a given seniority is This dependence provides a “more stringent test of the model” and the author used it to predict various transition rates. Using the measured rate of the to transition in the author calculated the rate of the transition between the and the levels in the configuration in The calculated value is which “is in excellent agreement with the” measured value of Using appropriate radial functions, these transition rates could be calculated from an effective proton charge of 1.5 e. Lawson also calculated positions of levels obtained by raising or protons into the orbit. Matrix elements “characterizing the proton interaction with a or nucleon, were calculated using the central Schiffer-True interaction”. He used the interaction introduced by Schiffer and True (1976) with and Positions of several levels were predicted.

Further intensive and impressive experimental work added many pieces of data on N = 82 nuclei. It is summarized in the lecture notes of Blomqvist (1984) where references are listed. Blomqvist presented the comparison between experimental level spacings of and with calculations of and configurations. The two nucleon matrix elements are taken from the spectrum. The agreement was only fair but this was due to his fixing the calculated position of the J = 16 level at the experimental one. He argues that the J = 16 state, with seniority is the least likely to be perturbed. Had he fixed in the same way the position of any level, there would have been excellent agreement between the measured levels, with J=2,6,8,10 and very good one with the J = 12 level. The predicted J = 0 ground state would have been a bit low and even the calculated J = 14 level a little too high. The calculated position of the J = 16 level would then be about .2 MeV too high but this level is higher than 5 MeV above the ground state. The same situation occurs also in The agreement with experiment could certainly be improved by introducing explicitly mixing with other configurations. Unfortunately, there are not enough experimental data to carry out a meaningful calculation. On the other hand, the simple assignment gives an adequate description of the data mentioned above, as well as those measured later. In fact, looking at recently measured energies, a two nucleon interaction could be simply chosen which fits nicely the experimental data. The calculated position of the J = 16 level of the configuration is only .067 MeV above the observed position of this level in (Talmi (1993)).

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Blomqvist presented results of calculated B(E2) transition rates based on the description and compared them to experiment. These rates were measured for the to transitions in even nuclei and the to ones in odd nuclei. There was very good agreement with measured lifetimes for and 5, the effective charge is about 1.5 e. According to the seniority scheme, the rate should vanish if the number is equal to which is 6 here. Indeed, the rate of the to transition measured in is very small, rather unusual for E2 transitions. It is .76(10) as compared to the analogous values 11.3(7) in and 43(3) in Blomqvist tried to speculate about the effect of possible admixtures to the wave functions. In a later paper, McNeill et al., (1989) presented experimental results on “exotic N = 82 nuclei and The B(E2) values of the corresponding transitions in these nuclei were measured and found to be in (n=7) and in (they also quote a more accurate value for It seemed that the trend was according to the seniority formula but the actual values were much smaller than expected for the and cases. The authors tried to determine where the orbit is half filled. They plotted the square roots of the measured B(E2) values which should be proportional to the matrix elements. These should lie on straight lines, one for odd and the other for even changing sign at The experimental values lie fairly well on two lines which cross zero at a point between and (the negative square roots are used for and The authors conclude that these “results, together with those for lighter N = 82 isotones, provide an outstanding illustration of the dependence of E2 transition rates between states on the subshell occupation, and demonstrate that half-filling of the subshell in the N = 82 series occurs just below Z = 71 This analysis seems to be an over simplification. The picture which emerges from the data is that there are some admixtures to the states whose effect on energies may well be absorbed into an effective interaction. The E2 matrix element due to this perturbation, seems to have a constant sign which adds to the matrix elements for At its effect is rather small. For the matrix elements change sign and hence, the admixtures act to reduce the total matrix elements and the B(E2) values. In a paper by Nolte, Korschinek and Setzensack (1982), the authors presented their experimental studies of and nearby nuclei. Their results on energy levels and E2 transitions were mentioned by Blomqvist (1984). In their paper they consider another feature of seniority - “favoured transitions The beta decays considered are from the J = 0 ground state of a N = 82 nucleus to a J = 1 state, with positive parity, in the odd-odd nucleus with N = 83. These states are assumed to be due to coupling of

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the J = 11/2 state to a single neutron in the orbit. The matrix elements of these Gamow-Teller transitions have a simple form in the seniority scheme. They can be viewed as the product of the matrix element for removing one proton from the J = 0 state, leaving the protons in the J = 11/2 state and the matrix element for putting a single neutron in the empty orbit. The former is proportional to the square root of and the latter is a constant. Hence, the transition rates are proportional to and the ft-values are proportional to This is the behaviour which the authors find for the ft-values of the decays of and They used for the rates an expression derived from “general pairing-model wave functions”. Then they made simplifying assumptions about the and coefficients. “Since the neutron orbital is practically completely empty, equals about 1”. “The factors equal roughly where is the number of protons outside the core”. This way, they obtained approximately the simple exact result of seniority in configurations. There was an overall reduction of the observed rates from those expected from favoured transitions in light nuclei. This reduction by about a factor 6, could be due to poor overlap between proton and neutron radial functions. It could also be due to some renormalization of the single nucleon operator due to many body effects.

6.2. Generalized Seniority The occurrence of fairly pure configurations in some semi-magic nuclei, presented the opportunity to study the T = 1 effective interaction in rather simple cases. It is possible to conclude from such cases that the interaction between identical nucleons is diagonal in the seniority scheme to a high degree of accuracy. In other semi-magic nuclei the situation is not the same, there is clear evidence for configuration mixing. Still, some of the features of seniority seem to survive. Binding energies of even-even nuclei are well described by the simple formula of the seniority scheme. There are in semi-magic nuclei, fairly constant spacings between the J = 0 ground states and J = 2 first excited states. In the collective model, these J = 2 states were described as quadrupole vibrations of spherical nuclei. This description could not account for some experimental data. It turned out that configuration mixing in semi-magic nuclei found a good description in terms of a generalization of seniority. Talmi (1971) addressed this problem, starting from the nickel isotopes. Auerbach (1996a) and (1967) and Cohen, Lawson, Macfarlane, Pandya, and Soga (1967) found “that the ground states of even Ni isotopes are almost exclusively built from

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pairs coupled to J = 0 in the various orbits. It is therefore interesting to see whether a simpler description of these states can be found which will be a natural generalization of the single seniority scheme”. The generalization of seniority to be described, did not follow directly from the quasi-spin scheme. It was reached via consideration of the part with a fixed number of particles in the BCS ground state wave function. In the case of a spherically symmetric potential, it has the form

Using the conventional phase convention and the definition of the pair creation operator this state could be expressed as

Gambhir et al., (1969) adopted these expressions as variational wave functions. They tried to determine the values of the which yield the lowest expectation value of some Hamiltonian. They also considered excited states, like the J = 2 states in even-even nuclei, by changing the coupling in one J = 0 pair state. The creation operator of the J >0 pair was multiplied by N – 1 creation operators of pairs coupled to J = 0 (“broken pair approximation”). Talmi (1971) considered these wave functions from a rather opposite approach, not starting from a Hamiltonian and checking whether these are good approximations of its eigenstates. Instead, he took them to be exact eigenstates of the shell model Hamiltonian and examined the consequences of this assumption. Those turned out to be very interesting. The pair creation operator is taken to be

As mentioned above, if all are equal (or their absolute values are equal), the quasi-spin scheme follows. The scheme of simple seniority, due to the SU(2) group, is the set of eigenstates of the (hermitean) pairing interaction Here, no such assumption is made. Ground states of semi-magic nuclei are assumed to have the form

which is analogous to states with seniority the shell model Hamiltonian H, then

If these are eigenstates of

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In the normalization of H, the binding energy of the closed shells is omitted so that In the case of one pair, N = 1, this relation becomes

This means that the eigenstate with eigenvalue V, is a pair state with correlations determined by the values of the coefficients. Next, the N = 2 case yields after some commutations

If this state should be an eigenstate, the second term on the r.h.s. should be proportional to the state. Hence, the following condition is obtained

If the shell model Hamiltonian contains single nucleon energies and two-body interactions, its double commutator with contains only terms with 4 creation operators. Hence, the condition above is equivalent to an operator equation

These conditions, which follow from (144) for N = 1 and N = 2, are necessary ones. They are also sufficient conditions for the state (143) to be an eigenstate for any value of N. The coefficients are constant throughout the shell and do not change with N. The author proved that from these conditions follows

Thus, the state (143) is an eigenstate for any N with simple eigenvalues. The structure of the eigenstates guarantees the simple binding energy formula. It has terms which are linear and quadratic in N and is the analog of the formula of the seniority scheme for states. Here, however, the contribution of single nucleon energies is included in the linear and quadratic terms. It is surprising that this important feature of seniority survives the generalization described here. In the seniority scheme, the binding energy formula is derived by using eigenvalues of the Casimir operator of SU(2). This formula is obtained here, for even-even nuclei, even though there is no group which could be applied. If the coefficient are not all equal, is no longer a generator of the SU(2) group. An important feature of this binding energy formula is

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the absence of discontinuities at possible closures of sub-shells. The mixing of configurations of several with non-vanishing coefficients, implies smooth filling of all of them. The participation of nucleons in an orbit whose coefficient is small may be considered a small perturbation for small values of N. But as N increases, orbits with large coefficients are filled first and then the amplitudes of states with smaller are no longer small. The author checked whether ground states of even Ni isotopes have the structure (143). It was remarked earlier that some authors found those ground states to be mostly linear combinations of products of various states with The states (143) have this feature but there are many different states of this kind which do not follow the special prescription (143). He found that binding energies, both experimental and calculated in the shell model, were well reproduced by the formula (149). The coefficients which give a good fit were V=22.33 MeV and W=–1.88 MeV (repulsive). The (unnormalized) ground state of the two neutron configuration in as calculated by Auerbach (1966a), is obtained by acting on the vacuum state by the pair creation operator

In order to check the accuracy of generalized seniority it is not necessary to evaluate the double commutator condition. Since it is equivalent to (144) for N = 2, it is sufficient to check whether the eigenstate in the 4 nucleon configuration is equal to (143) for N = 2. The author found that the overlap of the state and the calculated ground state of “turns out to be higher than 99%”. It is not explained whether this is the square of the overlap or that the overlap is just .99. Talmi derived from the condition on the double commutator, specific conditions involving matrix elements of the two-body interaction and the coefficients. In the case of a single these conditions imply that the two-body interaction is an odd tensor one plus a constant term. This is no surprise since it was known for a long time (de-Shalit and Talmi (1963)) that any twobody interaction which is diagonal in the seniority scheme must have this form. More interesting is a general condition to be satisfied in the case of several orbits. The Hamiltonian sub-matrices in the various configurations should be diagonal in their (SU(2)) seniority schemes. Moreover, the quadratic terms in the expressions for energies of their states should be all equal. In this case, the value of W is equal to the common quadratic term multiplied by 4. In the case of Ni isotopes, due to all values being smaller than 9/2, the interaction is diagonal in the seniority scheme. The coefficients of the quadratic term in the energy formula for states, taken from Auerbach (1966a) are –.52±. 15 MeV for the orbit and –.35±.10 MeV for the orbit. Both

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values agree, within the quoted errors, with the value –.47 MeV, which is the value of W/4. The paper dealt with an interesting question, why do shell model Hamiltonians satisfy these conditions? Although single nucleon energies do not appear explicitly in the double commutator, the coefficients do. The latter are obtained by diagonalization of the sub-matrix of the Hamiltonian in the two nucleon configuration. Its elements are determined by both single nucleon energies and two-body matrix elements of the effective interaction. Single nucleon energies are determined also by the effective interaction of the valence nucleon with nucleons in the core. “It is, however, very difficult to imagine that condition” (148) “with the various considered, has in it so much information and restrictions on the nuclear interaction”. The author wondered whether the conditions were dictated by the behaviour of binding energies. The structure of ground states led to a linear and quadratic dependence of binding energies on N. Is the inverse true? If binding energies are given by

where does it follow that ground states are given by If all are equal the answer is positive. The proof, however, cannot be extended to unequal In fact, counter examples were found. Those, however, show that if binding energies have the dependence (149) on N, a shell model Hamiltonian which will reproduce them is most likely to obey the double commutator condition and have eigenstates in which J = 0 pairing is highly favoured. The states (143) may be referred to as generalized seniority states. Yet it is important to be aware of the price for having unequal coefficients. The seniority scheme is an orthogonal and complete set of wave functions. If two states and with the same values of N, J, M, are orthogonal, then also the states and are orthogonal. This feature of the seniority scheme, may not hold in the case of generalized seniority, with unequal coefficients. It is still possible to define one state with a given even J > 0, which may be called a state with generalized seniority The main interest is in J = 2 states so the author defines a creation operator of two nucleons coupled to spin J = 2. If denoted by its creating an eigenstate of H is expressed by

The author then studied the implications on H, satisfying the conditions above, that has also eigenstates which have the form

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Such a state is the analog of seniority in the case of SU(2) symmetry. Applying H to these states leads to the expression

Using the relations ((145) and (148)) this expression simplifies into

Hence, the state (153) is an eigenstate of H if the double commutator in (155) is proportional to By looking at a part in both expressions, the author deduces that the proportionality factor must be W. Hence the condition

is necessary and sufficient for the state (153) to be an eigenstate. The eigenvalue is then equal to

The energy of the J = 2 state is equal to the energy in (149) of the J = 0 ground state plus a constant term which is independent of N. The suggested structure of the J = 2 states guarantees constant 0-2 spacings throughout the shell. Also this important feature of seniority survives the generalization presented here. This generalization reproduced very well the prominent features of eveneven semi-magic nuclei. These are the binding energy dependence on the nucleon number and the constant position, independent of of the first excited J = 2 level above the J = 0 ground state. This generalization of seniority, however, was limited to even-even nuclei. Talmi tried to see whether this approach could be applied to odd-even nuclei. He considered the states which are the analogs of seniority states. He then checked the conditions that these are eigenstates of the shell model Hamiltonian with single nucleon energies and two-body interactions. The conditions are

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for all with non-vanishing coefficients. These conditions may be satisfied only if all values are equal. In this quasi-spin limit, all level spacings do not change with nucleon number. The author points out that this applies to such states in odd-even nuclei which need not be degenerate. This means that for equal these levels do not cross as the nucleon number is changed. The ground state spin would be constant throughout the major shell, in sharp contrast with experiment. Talmi looked at the pseudonium isotopes in the papers of Cohen, Lawson and Soper (1966) and Soper (1970). Using the interaction of those papers he found that the case was rather close to generalized seniority. Calculating binding energies of Ps isotopes, neutron pair separation energies of to lie on a straight line as follows from (149). He calculated, however, binding energies of lighter Ps isotopes and found that separation energies of and lie on the same straight line. “There is no break at the “pseudo-magic” shell of 20 in the pair separation energies”. Considering also these “pseudo data”, the immediate conclusion would be that there is rather strong mixing between configurations with and neutrons, probably described by (143). The author presented a plot of experimental separation energies in Ca isotopes which exhibited large breaks at neutron numbers 40 and 48. He explained that this did not exclude mixing into configurations. This possible mixing, however, could not be as strong as implied by (143). In the paper described above, very little was said about the quasi-spin scheme. It seems as if the author insisted on dissociating generalized seniority from the simple, unphysical SU(2) case. In the next paper (Shlomo and Talmi (1972)), this special case was explicitly considered. The authors proved in detail that if all are equal, then acting on any eigenstate with operators yields an eigenstate. Spacings between levels are not changed by this operation in even as well as in odd nuclei. The authors defined a single nucleon operator which, acting on a generalized seniority state, yields a state with generalized seniority The latter states in the two nucleon case are obtained explicitly, for any J > 0 even, by acting on the vacuum by the pair creation operator

The operator for J = 2 is The single nucleon operator is defined as an irreducible tensor operator of rank J, by the commutation relation

The authors showed how a hermitean single nucleon operator could be constructed in terms of the and coefficients. It follows that the only

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state which is connected to the state (143) by non-vanishing matrix elements of the operator is the JM state with generalized seniority This follows from the relation

and that the resulting eigenstate on the r.h.s. is orthogonal to all other eigenstates, In the seniority scheme, a single nucleon operator has non-vanishing matrix elements only between states whose seniorities differ by at most 2. The special selection rule derived above, is a survivor of the generalization and may still have important consequences. If the quadrupole operator “is proportional to the effective E2 operator..., no E2 transitions will take place between” other J = 2 “states and the ground state. As is well known, this is indeed the case in semi-magic nuclei”. The constant 0 - 2 spacings may be emphasized by using the operator. Using (162), the following relation is obtained

An equation like defines harmonic vibrations, but this is not the case here. The relation (162) is not an operator equation, it holds only if the commutator is applied to the ground state with generalized seniority The authors derive detailed conditions on shell model Hamiltonians which have generalized seniority and eigenstates. They construct some non-trivial examples of such Hamiltonians, first those satisfying (145) and (148), in addition to the example of Ni isotopes considered by Talmi (1971). “Non-trivial” here means cases in which not all coefficients are equal. They note that it “is more difficult to find simple yet non-trivial examples in which both and conditions are satisfied”. One “example which attempts to deal with actual nuclei” is offered by the shell model calculation of Wildenthal and Larson (1971). Those authors considered nuclei with 82 neutrons and valence protons beyond Z = 50. In the J = 0 state of two protons, there are only and pairs and the state is created by the operator

The coefficients are far from equal and the overlap of the J = 0 state created by it, with the qausi-spin state is only .963. The J = 0 of the 4 proton state is

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whereas the state created by

Igal Talmi

is

The overlap between these two states is .9991 (it is .9996 if the is ignored. This should be compared with the overlap of the state (164) with the quasi-spin state, .925. The creation operator creating the lowest J = 2 state of two protons, contains contributions from and configurations. Still, the overlap between the calculated J = 2 state in the 4 proton case, with the state created by is .967. The overlap of the latter with the part with lowest seniority of the calculated state is .988. The overlap of the calculated state with the corresponding quasi-spin state is only .896. The overlaps for the lowest J = 4 state are almost the same. It seems that small modifications in the Wildenthal-Larson matrix elements would yield a Hamiltonian satisfying the conditions (145) and (148). The states created by the S and D pairs of generalized seniority were mapped on states of, proton or neutron, and bosons in IBM-2. This mapping was introduced by Arima, Otsuka, Iachello, and Talmi (1977) and further developed by Otsuka, Arima, Iachello, and Talmi (1978). Its introduction motivated some shell model calculations using generalized seniority. Their purpose was to learn about the parameters which define the effective Hamiltonian of the bosons. Scholten and Kruse (1983) and Scholten (1983)in more detail, presented a “calculation for the N = 82 isotones in the generalized seniority scheme” and compared it with unpublished calculations by Kruse and Wildenthal. They concluded that “it has been shown that the generalized seniority concept generally provides a valid truncation scheme of the full shellmodel basis, in the sense that a calculation in a much smaller basis using the same interaction gives essentially the same excitation energies for the first few levels of each spin”. They noted that in “a pure generalized seniority scheme one expects that the coefficients do not vary from isotope to isotope”. It is clear that the authors used the states as variational wave functions for the Kruse-Wildenthal interaction. Thus, they obtained coefficients which vary with N. They noted that “for the were indeed rather constant” but for higher values of Z they changed appreciably. In fact, level spacings change with proton number even for nuclei with Also the experimental level spacings change appreciably in this region which indicates that generalized seniority is not the correct description of these semi-magic nuclei. The authors correctly pointed out that generalized seniority provides a very powerful truncation scheme, this aspect will be discussed in a subsequent section. Many years later, N = 82 nuclei were considered by Holt, Engeland, Osnes, Hjorth-Jensen, and Suhonen (1997). They were trying to derive an effective interaction from an interaction between free nucleons by using many body

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theory. They used “the so-called folded-diagram expansion method” of Kuo and Osnes (1990) and the free nucleon interaction was the Bonn A one-bosonexchange model. The shell model space which they considered consists of all proton orbits in the major shell Z = 50 – 82. The calculations reproduced measured level spacings in even nuclei up to very reasonably. The “results are rather good”, the deviation of “calculated energy levels from the experimental ones is generally within 0.1-0.3 MeV. Up to five states, three states and two states are well reproduced throughout the sequence of isotones”. The authors also calculated transition probabilities and found that “with one exception the known E2 transitions were well reproduced”. They used an effective proton charge of 1.4 MeV. The case is too difficult so the authors just calculated the 0-2 spacing which “is found to be 1.864 MeV and compares well to the experimental value of 1.972 MeV”. It is not clear whether the model reproduced a sub-shell closure. Although it is not clear to what extent the data support the generalized seniority picture in this case, the authors tried to check how well this feature exists in their calculations. They determined coefficients of the and creation operators from the two proton states in They next calculated the overlaps of states created by and and corresponding 4 proton calculated states. They found overlaps of .99 for J = 0 states and .97 for J = 2. The overlaps of calculated states and generalized seniority states went down with increasing number of valence protons. It was clear, however, that a very good approximation to the effective interaction used in this paper, could have J = 0 and J = 2 generalized seniority eigenstates. In a talk at a conference, Broda, Fornal, Krolas, Lach, Pawlat, Wrzesinski, Bazzaco, Lunardi, Rossi-Alvarez, de Angelis, Daly, Zhang, Grabowski, Cocks, Butler, Maier, and Blomqvist (1998), presented various new experimental results. Along with the data, possible configurations were assigned to many levels. Results of shell model calculations were compared with measured energies, mostly of high spin states of with N = 82. The authors mentioned that Wildenthal constructed the Hamiltonian for protons occupying the and orbits. His Hamiltonian described “in a consistent way many features observed experimentally in all series of N = 82 isotones”. The authors explained that new measurements “gave additional experimental input to improve some of the parameters. J. Blomqvist introduced those changes and performed calculations with the new Hamiltonian prior to our analysis of experimental data”. The levels considered were the ground state of positive parity states with J=11/2,15/2,17/2,21/2,23/2, and higher negative parity states with J=23/2,27/2,29/2, and 31/2. Positions of these levels were calculated also with the Wildenthal Hamiltonian and the mean deviation from measured positions was 159 keV. The calculated energies

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of these states, including the binding energy, with the Blomqvist interaction, were in much better agreement with experiment. The mean deviation is only 23 keV. “The yrast structure of isotope predicted quantitatively in these calculations was confirmed by experiment with spectacular accuracy”. Many years later, Sandulescu et al., (1997) wrote a paper on “Generalized seniority scheme in light Sn isotopes”. They had a much more sophisticated effective interaction, derived by theoretical many-body methods, from one of the Bonn versions of the interaction between free nucleons. They included all orbits in the major shell and hence, had to diagonalize large matrices. They explained that in looking for a truncation scheme, they considered also the scheme of generalized seniority. In for instance, there are only 9 states with J = 2 in that scheme, compared to the order 86,990 of the full shell model matrix which should be diagonalized. Still they were able to calculate the J=0,2,4,6 in and compare their results for to to results of generalized seniority. They tried to construct states in that scheme in two ways. In the first they used fixed coefficients determined from the calculated wave function of the 2 neutron system in In the other, they determined the coefficients by minimizing the energy of the The coefficients of the operator were determined by “diagonalizing the given interaction in the space of all possible seniority two basis states” for each nucleus. The authors knew that “a constant pair structure is within the philosophy of the original generalised seniority scheme”. They also “stress that the validity of a truncation scheme depends on the effective interaction employed”. Still, the authors concluded that “a fixed pair structure description is not meaningful for the heavy Sn isotopes”. This conclusion was based on the poor overlaps of the calculated eigenstates of their effective interaction with generalized seniority states. The overlaps for J = 0 went down from .975 in with fixed (.983 for adjusted ) to .876 (.953) in For J = 2 states, the overlaps went down even faster, from .965 (.963) in to .648 (.881) in For J = 4 and J = 6 the overlaps are even smaller. In another place the authors staled “that a truncation scheme based on seniority zero and two states is inadequate when the number of valence particles gets large”. In the abstract, however, the authors expressed the correct conclusion that “with the realistic effective interaction applied here”, this truncation scheme “is inadequate when the number of valence particles gets large”. It can be argued that, in view of the fair overlaps in small modifications in the interaction which will not affect agreement with experiment, could make the overlaps equal to 1. In that case, a constant pair structure would hold throughout the shell and lead to constant 0-2 spacings. It is worthwhile to stress that the “realistic effective interaction” was derived from a highly approximate free nucleon

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interaction. The derivation of the G-matrix was also only approximate, as were harmonic oscillator wave functions. The derivation was certainly a great theoretical achievement but the outcome is far from perfect. The agreement with experiment was not spectacular. The calculated positions of the J = 2 states in to were 1.45, 1.42, 1.57, 1.63, 1.65 MeV. The experimental values are 1.216, 1.207, 1.206, 1.212, 1.257 MeV respectively. The scheme of generalized seniority was introduced in order to understand important experimental features of semi-magic nuclei. A good effective interaction, which will reproduce these features, will most probably will have eigenstates prescribed by generalized seniority. About a year later, Holt, Engeland, Hjorth-Jensen, and Osnes (1998), published a paper on similar calculations in heavy Sn isotopes. They derived in the same way, the effective interaction for neutron holes in the major shell 50-82. The authors used this interaction to calculate spectra of isotopes down to The calculations for lighter isotopes, apparently involved great computational difficulties and the range to was not considered. The number of states of the shell model space considered in the for is about 6 million, whereas it grows to more than 12 million for and it is about 16 million for The authors obtained good to fair agreement of calculated level spacings of odd and even Sn isotopes down to with 12 holes. This calculation was an example of a large scale shell model calculation. It is presented in this section, since the authors tried to see whether their results were consistent with generalized seniority. They determined the coefficients of the and creation operators from the calculated wave functions of the two hole configuration in They computed the overlap of their calculated states of 4 holes with those created by two pair creation operators. The authors found the overlap for the ground J = 0 state to be .98 and for the lowest J = 2 state, to be .96. These overlaps decrease with the number of holes and reach for the values .89 and .86, respectively. This decrease is not surprising since small deviations in the two hole case, are magnified as more holes are considered. A more interesting question is whether small modifications of the matrix elements, which would make the overlaps in equal to 1, could still keep the good agreement obtained by the authors. This is of special interest since they found good agreement with experimental 0-2 spacings (which exhibit, however, more constant behaviour). Possible modifications of the effective interaction were necessary in any case. The authors “conclude that the relative location of the states is satisfactory, but it ought to be mentioned that the absolute values of the binding energies are far off. “For example, the ten-particle system is overbound by more than 7 MeV with the present effective interaction”. This seems to be a general malady of effective interactions derived in this way and will be discussed in

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the next part. It may be argued that the positions of J = 2 levels above the J = 0 ground states are not strictly constant. The changes in their positions for various numbers of valence neutrons, should be contrasted with the situation in other nuclei. The 0-2 spacings in nuclei from to deviate from their mean value, 1.217 MeV by less than .085 MeV (7% of the mean value). This variation should be compared with the change in level positions in odd Sn isotopes. The spacings between the yrast and levels change by more than 2 MeV between ( ground state) and This change is a clear proof that the coefficients in the pair creation operator cannot be all equal. The relative constancy of the 0-2 spacings in Sn nuclei could be contrasted with the situation where both protons and neutrons occupy valence orbits. In the semi-magic the J = 2 level lies 1.436 MeV above the ground state. Adding more neutrons leads to a drastic reduction of the 0-2 spacing, down to .181 MeV in When high precision calculations of nuclear energies will be available, the slight variation in 0-2 spacings in Sn isotopes will be reproduced. At the moment, we can be satisfied with the approximation in which these spacings are constant which is reproduced by generalized seniority.

7. LARGE SCALE SHELL MODEL CALCULATIONS Success of various shell model calculations, some of which were described above, increased the confidence of physicists in using the shell model for obtaining quantitative results and predictions. Advanced computing facilities and techniques, enabled removing the need for truncations which could not be justified on physical considerations. A new era of large scale shell model calculations began and has not yet reached its peak. These developments occured on time to match an important experimental effort to reach nuclei at the limits of stability. In the following, a rather sketchy description of some of this work will be presented. The essential difficulties in carrying out such calculations will become evident.

7.1.

The

and Beyond

Several shell model calculations were described above in which parts of the shell were considered. They may have been reasonable approximations but it was not possible to check their accuracy in comparison with a complete calculation. A shell model calculation for all nuclei with valence protons and neutrons in the complete shell was finally launched by Wildenthal and his collaborators. It took about ten years of work of several physicists before it

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developed into a complete scheme in which the same set of matrix elements of the effective interaction was used for the full shell. The first paper in the series was published by Wildenthal (1969) but even as late as 1976, nuclear energies were fitted by two sets of matrix elements, one set for the beginning of the shell (A=17-28) and another for its end (A=28-39). The situation at that time was presented by Wildenthal (1977), with earlier references, in his lectures at the 1976 Varenna summer school on nuclear physics. Only later, did Wildenthal (1984) find a way to match the two regions by letting the matrix elements of the two nucleon interaction change as a function of A. Starting with the values of these matrix elements for A = 18, their values at higher values of A are obtained by multiplying by This change is supposed to take care of the very large change in nuclear masses, from A = 16 to A = 40. The single nucleon energies of the and orbits were kept constant throughout the shell. There are altogether 63 matrix elements of the two nucleon interaction between nucleons in the three orbits. In the final version of the calculation, the 66 parameters were determined by a least squares fit to 447 binding and excitation energies. The two-nucleon Hamiltonian is charge independent and the resulting eigenstates have definite isospins. The Coulomb energy of the protons in a given state should be added before comparison with experiment is performed. This Coulomb energy is determined by comparing the measured energies of analog states in neighbouring nuclei. The agreement between energies calculated with the matrix element thus determined, with experiment, was very good (for 447 energies!), the r.m.s. deviation was only .185 MeV. Even the overwhelming number of experimental energies turned out to be insufficient for accurate determination of all matrix elements. The relevant parameters were 47 linear combinations of matrix elements which were best determined. The others were fixed to agree with some values calculated from the interaction between free nucleons. These matrix elements are referred to as the W, or USD effective interaction. Some of the matrices which were diagonalized in this calculation are very large. In the middle of the shell, for the levels, they are the largest. If the is used in setting up the Hamiltonian sub-matrices, the matrix with M = 0 contains all states of 6 protons and 6 neutrons with all possible values of T (and ), and J values (M = 0). This matrix is of order 93,710 but also in the J, T scheme the situation is still complex. To obtain the energies of T = 1, J = 3 states of a matrix of order 6,706 had been diagonalized. The two nucleon matrix elements which were determined from experiment should be used with all states in the full shell. Any truncation of the shell model space would require a different, probably stronger interaction. In fact, if the simplest configuration were chosen for the ground

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state and the same matrix elements used, its binding energy would be about 10 MeV less than the one calculated in the full shell model space. The calculated binding energies are in good agreement with measured energies. Rather large deviations are found only in the case of Na and Mg isotopes with large neutron numbers (N = 19, 20). These are attributed to the observed fact that for these nuclei, N = 20 seems to lose its magicity and apparently higher neutron orbits must be included in the shell model description. Some features of the W matrix elements which were determined are similar to those obtained in earlier, more restricted, calculations. Some of these involve the average interaction between nucleons in different orbits

It was found previously to be small and repulsive for T = 1 and large and attractive for T = 0. From the list of W matrix elements follows that which is indeed strong and attractive, whereas which is rather weak and repulsive. The situation is not the same in the other cases. The average interactions, and are large and attractive. The average T = 1 interactions, however, and are attractive. Their absolute values are rather small and hence, slight changes in the matrix elements, consistent with the statistical errors, could change their signs. Within the configurations, the coefficient of the pairing term, calculated from the matrix elements, was –2.749 MeV which is large and attractive. The coefficient of the quadratic term however, turned out to be –.071 MeV which is indeed small but attractive. Since its absolute value is small, slight changes in the matrix elements could give it the proper sign. The calculated coefficient of T(T+1) in the symmetry energy term was rather large and repulsive, equal to 1.635 MeV. The corresponding calculated values for the orbit were (repulsive) and the coefficient of the symmetry energy term, 1.718 MeV. In a summary of the extensive and intensive work on the shell, Brown and Wildenthal (1988) presented results obtained for other observables by using the calculated wave functions. The authors demonstrated the agreement with experiment on two-dimensional plots. The coordinate of each point is the experimental value and the coordinate is the calculated value. They carried out two sets of calculations of magnetic moments, M1 dipole gamma transitions and Gamow-Teller beta decay matrix elements. In one set, they used the operators of free nucleons and in the other set they used effective operators, with renormalized coefficients. The latter are single nucleon operators which are determined by the best fit to the measured observables. The coefficients of the effective operators change with A according to The effective

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M1 operator can be expressed in terms of corrections to the free nucleon operator which involve 3 rank 1 operators s, l and The points on the plot of Ml transitions were around the line When calculated with the effective M1 operator they were much closer to that line. The magnetic moments calculated with the free nucleon operator were in very good agreement with the measured ones. Use of an effective operator made the agreement even better. The proton effective was reduced from 2 × 2.79 to 2 × 2.23 nm while this factor of the neutron was changed from –2 × 1.91 to –2× 1.68 nm. The factor was less well determined and its effective value for the proton was 1.16 and for the neutron .07 nm. The addition of the term was the least certain, its coefficient, for either protons or neutrons, was roughly determined as about 0.4. The electric quadrupole transitions are stronger than those calculated with the electric charges of the free nucleons. Using an effective charge of 1.3e for the proton and .5e for the neutron, the situation was greatly improved. “With a few exceptions, the agreement between experiment and effective-operator predictions is excellent”. Mass measurements of neutron rich Na isotopes up to carried out in the late 1970’s, indicated some irregularities. Single neutron separation energies did not exhibit the drop expected at the magic number 20. It was found that the 21st neutron is more tightly bound than the 19th one. The behaviour of neutron rich Mg and Ne isotopes is similar but less pronounced. Excitations of J = 2 states in Mg isotopes were measured. The position of this state in is at 1.484 MeV but in where the neutron shell is expected to be closed, the position of this state was measured to be only .886 MeV above the ground state. The disappearance of the magic number N = 20 was clearly demonstrated by Wildenthal and Chung (1980). They used their best available effective interactions in the shell to calculate binding energies and excited states in Ne, Na, and Mg nuclei with neutron number up to 20. Their calculated binding energy of was several MeV too low. Their calculated position of the J = 2 state of was higher than 2 MeV. They concluded that these features “are not to be explained within the configuration space”. “The new challenge for shell model theory is to determine an effective Hamiltonian which, while operative in a space expanded beyond the confines of the shell, maintains structure intact for the appropriate N, Z systems and then induces the radical changes currently observed at Z = 11 – 12, N = 20”. This challenge was taken up by Watt, Singhal, Storm, and Whitehead (1981) who added to the shell model space also the orbit. They explained that the discrepancy in binding energies “can be explained by allowing neutron excitations from the shell into the shell”. For nuclei they used the Preedom-Wildenthal (1972) interaction and the “remaining matrix el-

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ements required for the calculation are taken without modification from the G matrix of Kuo and Brown (1966). The single nucleon energies are taken from the spectrum and the orbit is assumed to lie 10.85 MeV above the orbit. To limit the size of the matrices to be diagonalized, the authors impose some restrictions on occupation of orbits. The protons are confined to the orbit and the neutron and orbits are taken to be completely filled. Valence neutrons may occupy the orbit and two of them at most, may be excited into the orbit. “An additional virtue of the present truncation scheme is that spurious centre-of-mass excitations cannot occur”. Model states cannot have spurious components since the center-of-mass coordinate has vanishing matrix elements between and single nucleon states. Binding energies calculated within the shell and those calculated in the enlarged space were compared to the experimental values. For and with N = 18, results of both calculations were very close and close to the measured energies. For and excitations add some energy but the experimental binding is stronger. “The largest effects are seen in and Their calculated ground states are “dominated by configurations”, their weight being more than 90%. The binding energies in the enlarged space are larger than the results by 7.4 MeV for and by 5.8 MeV for The authors commented that, “although the results are still 2 MeV short of the experimental values, the improvement is obvious”. The authors explained that their “calculations were intended to be exploratory and a more detailed investigation of this region is required”. In a subsequent paper, Storm, Watt and Whitehead (1983) considered “crossing of single-particle energy levels resulting from neutron excess in the shell”. The authors used the same interaction and single neutron spacings as in the previous paper. They considered, however, only neutrons in orbits and in the orbit. Removing all protons is an extreme case in which the effects associated with shell closure at N = 20 may be best studied. They looked at single nucleon levels and gave a rather vague definition of them. Their examples were only of the separation energy of a single nucleon outside a closed orbit and of the separation energy of a single nucleon from a closed orbit. The latter energy may be called a single hole energy but is not very useful. It does not make much sense to define single nucleon energies of interacting nucleons in the same orbit. In any case, the authors find that the orbit is below the orbit for valence neutrons but is above it for This meant that one “can no longer ignore the orbital when we are dealing with isotopes with significantly more neutrons than protons”. The authors “do not expect this behaviour to be confided to the region of the periodic table which we have studied and we believe that it may be of importance elsewhere as work progresses in the study of exotic nuclei”.

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The authors were unaware that a similar behaviour was observed almost 20 years earlier in where the neutron orbit lies below the orbit. The interaction between identical nucleons, apart from the attractive pairing term is repulsive on the average. It is more repulsive between nucleons in the same orbit than between them and nucleons in other orbits. Hence, when more neutrons are added to or to orbits in the higher shell become lower. It is only when protons are added, their attractive interaction with neutrons favours orbits in the same shell. Thus, from to the neutron orbit lies above the neutron orbits. The same occurs for the neutron orbit which lies below the orbits in and other nuclei with more Several years later, Poves and Retamosa (1987) considered nuclei with N ~20 and various proton numbers. They considered protons in the orbit and neutrons in the shell along with “configurations with two neutrons promoted from the orbit to the orbits”. For the shell, they adopted an “average of the Chung-Wildenthal interactions” as in Wildenthal (1977). For “the and matrix elements” they took the Kahana, Lee and Scott (1969) G-matrix interaction, whose monopole term was adjusted according to Pasquini and Zuker (1977). In such calculations, some of the states with one or two neutrons may have components of spurious states. Such states, due to center-of-mass excitations, were not mentioned in the paper. The authors diagonalized the Hamiltonian sub-matrices, which “in spite of these severe truncations, dimensions reach 800×800”. They presented calculated ground state spins and the percentages of configurations in their wave functions. For negative parity states in N = 21 nuclei, these percentages referred to configurations with 12 nucleons and one or nucleon (these do not contain spurious components). For N = 18 and N = 19 they calculated those percentages for Ne, Na and Al, and found values between 91 and 95. For N = 20, these percentages were calculated to be for Z = 8, 68, for Z = 9, 42 but for Z = 10, it was only 20 and for Z = 11 even smaller, only 8. For Z = 12, it was 23 but for Z = 13 it increased to 50 and reaches 71 for Z = 14. The situation was very similar for N = 21 nuclei. The rather low percentages of sd configurations for Z = 14, cast some doubt on the interactions adopted. The reason for the effect becoming less important for Z = 8, 9 was explained by the authors. They plotted “the differences among the centroids of the” two types of configurations for N = 20 and found, apparently in contrast to Storm, Watt and Whitehead (1983), that it went down with Z. Its decrease was rather slight and it did not go down much below 3 MeV. The reason for the dominance of configurations in Z = 10 – 12 cases, was attributed by the authors to their larger “correlation energy”. This latter quantity, defined by the authors in a subsequent paper, was small for Z = 8 and Z = 14 and reached it highest value,

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about –4.5 MeV, for Z = 11. The structure of these shell model states corresponds to deformed states. The calculated spectra of these configurations for and look like rotational bands with J(J+l) dependence on energy. Also calculated B(E2) values between these states, support this picture. In a subsequent paper, Poves and Retamosa (1994) “developed and improved” their model. “The new mass measurements have reduced the discrepancies with the results for standard calculations”. Measurements of excited states, however, “confirm that the region of deformation actually exists” around N = 20. Whereas the first excited J = 2 in is very high, about 3.5 MeV, indicating shell closure, that state in is only at .881 MeV, much lower than in lighter Mg isotopes. In there are 5 levels below 1 MeV, unlike in any lighter odd-mass isotope. Thus, the authors “claim that far from stability the ordering of configurations may be different from the one dictated by the single-particle energies near stability”. The shell model space which they used is the same as in their preceding paper. Within the configurations, they used the USD (or W) interaction of Wildenthal (1984). All other matrix elements were those of Kahana, Lee and Scott (1969)(LKS)of which some T = 1 elements were modified. To the diagonal T = 1 matrix elements in the configuration –.50 MeV is added. Also the diagonal T = 1 matrix elements and those in states, were made more attractive by .35 MeV. Single nucleon energies were taken from the measured spectrum. Those of the and were taken from spectra of “neutron-rich nuclei with A = 31”. Both these orbits were placed 3 MeV above the orbit. The authors calculated the “correlation energy” for the “intruder states” defined by the difference between the lowest eigenvalue of the Hamiltonian sub-matrix and its lowest diagonal element. For N = 20 and also for N = 19, they found a sharp minimum of the correlation energy at Z = 11, whose value is close to –5 MeV. The authors “plotted the intruder yrast bands for and Except in the case of they look like rotational bands”. They also found that the calculated “intraband E2 transitions are large compared with interband transitions”. The authors “conclude that in this zone the intruder states are deformed structures when valence protons are available”. To take into account the mixing between “standard” and intruder configurations, the authors modified the monopole part of the Hamiltonian. They referred to “the modified LKS+USD interaction as PR” and explained that its main point “is a shift in the neutron number where the transition takes place(for Z <13), from N = 20 to halfway between N = 19 and 20”. The “centroids of the PR interaction” of T = 1 states, were repulsive for (.415 MeV), (.272 MeV) states and even for states, in spite of the attraction in the J = 0 state. On the other hand, the centroid en-

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ergy of states was only mildly repulsive, .075 MeV. The centroid energies of and T = 1 states were attractive, –.643 and –.646 MeV respectively. With this interaction, and some further adjustments, calculated binding energies were in fair agreement with measured ones. The authors presented the calculated “percentage of normal configurations in the ground states of all nuclei studied as a function of Z and N. Nuclei with N >19 and Z <13 belong to the region of intruder dominance with a single exception, which is an even mixture of intruder and normal components”. A detailed study of this region was taken up by Warburton, Becker and Brown (1990). They proceeded to diagonalize the Hamiltonian sub-matrices built of states in which nucleons occupy the lowest orbits, the and the higher orbits, according to the conventional scheme. The sub-matrices of states in which two nucleons are excited into orbits, were separately diagonalized. No interaction was considered between states in these two sub-spaces due to some reasons explained in detail. In the excited configurations, in the computer program for diagonalization, “spuriosity is investigated and often eliminated using the approximate method of Gloeckner and Lawson” (1974). “The interaction - designated WBMB - was developed by Warburton, Becker, Millener, and Brown” (1987, unpublished). The interaction used by the authors within the shell was the USD interaction of Wildenthal (1984). The interaction between nucleons was taken from McGrory (1973). For the interaction between nucleons and nucleons, the authors modified the Millener and Kurath (1975) interaction. They used the same MK potential interaction and calculated the 510 two-body matrix elements needed. The authors varied the eight T = 0 and T = 1 diagonal matrix elements of states with and nucleons and the corresponding matrix elements. The average of T = 1 matrix elements in the former case was very mildly repulsive, .017 MeV, and of the latter only slightly more repulsive, .08 MeV. On the other hand, the T = 0 matrix elements seemed too attractive when compared with some experimental data. The authors calculated binding energies of nuclei for configurations in which orbits are filled according to the conventional scheme They recalled that energies calculated with the USD interaction in nuclei were in very good agreement with experiment. The r.m.s. deviation for 440 energies were 185 keV as shown by Brown and Wildenthal (1988). Using the WBMB interaction for ground states of nuclei with very good agreement was obtained with experiment. The r.m.s. deviation for the 18 binding energies was 195 keV. If also seven nuclei with N > 20, Z = 13 – 15 were included, the r.m.s. deviation rose to 305 keV. Although larger, it was still small compared to the deviations of calculated binding energies, within

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the space, of Na and Mg nuclei, with experiment. In these cases, the calculations underbind and by more than 4 MeV. The authors mentioned that Wildenthal and Chung (1980) showed that the USD interaction in the space underbinds and The present results showed that and were also included in the “island of inversion”, as was found by Poves and Retamosa (1987). Warburton, Becker and Brown tried to use the WBMB interaction to calculate energies of lowest states in the space of configurations in which two nucleons are excited into the shell. The order of matrices which should be diagonalized in this case was, in most cases, far beyond the computing ability of the authors. The only exception was the N = 20 case for which the matrices are relatively small for the lowest states of Z = 8 – 10 nuclei. The authors looked for truncation schemes and found that if only neutrons were excited, the results were very close to results of the exact calculation. The calculated lowest energies, either exact or after certain truncations, were compared to the results. The latter are larger for most nuclei with the exception of nuclei in the “island of inversion” where there is more binding in states. These nuclei are Ne, Na and Mg isotopes with neutron numbers N = 20, 21 and 22. The authors calculated the spectrum of in the space. They found levels with J=2,4,...,14 with energies following roughly a J( J+l) dependence and large intraband B(E2) values. The calculated excitation energy of the J = 2 state was 1.055 MeV which is considerably higher than the .3 MeV value found by Poves and Retamosa (1987). The authors tried to answer the question “why is there an island of inversion”? They calculated “effective neutron single-particle energies” of the various orbits in the and shells. Some of these were calculated exactly while the derivation of others involved some truncation. The authors found that “the gap between and shells shows a moderate decrease from E=7230 keV in to 5115 keV in This is in contrast to Storm”, Watt and Whitehead (1983) who “predict that the gap actually becomes negative for Hence, the authors quoted “three important mechanisms which combine to give the inversion of relative to the (small) reduction in the single-particle gap, the increase in the pairing energy and the increase in the proton-neutron interaction energy This latter interaction, led to collective-like states with increased interaction energy. A case for the actual lowering of the neutron orbit relative to the one, could be made on the basis of a simplified description. In the orbit is unbound by .95 MeV. The separation energy of a neutron from the ground state of is 8.64 MeV. This large increase could be roughly attributed to the attraction between and protons and a neutron. If several of these protons were removed, the orbit became much less bound

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and the orbit might compete with it. This was similar to the mechanism shown by Talmi and Unna (1964) to account for the ground state of Warburton, Becker and Brown expressed the hope “that an effective interaction for the full model space could be derived” which will be used in this region. Indeed, large scale shell model of this kind were carried out later and will be mentioned in the next section.

7.2.

The

Nuclei with protons and neutrons have been described in terms of configurations. Although binding energies show a definite break at N = 28 (Z = 20) or Z = 28 (N = 28), excited states do not fully correspond to the simple picture. There is evidence for mixing of configurations with nucleons in higher orbits, specially for states with both valence protons and neutrons. Attempts were mentioned above, to include in certain cases, mixing with configurations in which one or two nucleons are excited into the orbit. A better description could be obtained by considering the complete shell. It includes in addition to the orbit also the and the orbits. The number of states of several nucleons occupying these orbits is very very high. Also the number of matrix elements of the effective interaction is rather large. There are 4 single nucleon energies and 195 matrix elements of the effective two-nucleon interaction, 94 T = 1 matrix elements and 101 T = 0 ones. It is out of the question to determine all of them from measured energies. The various groups engaged in this work had to find ways to overcome this difficulty. Zuker and his student Pasquini, tried to determine matrix elements of the effective interaction by starting from a “realistic interaction”. Pasquini and Zuker (1977) started with the Kuo-Brown (1968) interaction (KB) and modified it to be useful. They noticed that using it without modifications led to absurd results. In for instance, the closed sub-shell configuration was calculated to lie several MeV above a four nucleon - four hole configuration. The authors had a rather simple cure for the “very severe flaw” which the KB interaction seemed to have in its “monopole behaviour”. They used a “minimal tampering” approach and modified only the monopole parts of the KB interaction. They defined the interaction between nucleons and those in higher orbits by adding to each matrix element a constant term MeV. It added an attraction between unlike nucleons and added a repulsive term to T = 1 matrix elements. The authors introduced further modifications to energies obtaining the KB” interaction which they used to calculate energy levels and wave functions of several nuclei using some truncation of the shell model space.

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In a detailed report, Poves and Zuker (1981) described this approach and included improvements of previous calculations. They defined a modified KB interaction by first adding to all energies of T = 0 states –.35 MeV and to those with T = 1, –. 11 MeV. They then added 300 keV monopole term to the interactions of nucleons with those in higher orbits. This modified interaction is denoted by KB1. For better agreement with experiment some further modifications were made to the matrix elements. These had, in the KB3 interaction, the following values in MeV For J =

0 –1.92

1 –1.77

2 –1.09

3 –0.86

4 –0.19

5 –0.71

6 0.18

7 –2.45

An important aim of the authors was to apply the idea of a renormalized interaction to a pure configuration. They wanted to show how many states with protons and neutron numbers between 20 and 28 exhibit fairly well, features of configurations. They made use of the interaction in the full space to modify the effective interaction within configurations. Hence, they diagonalized exactly the KB3 interaction only for states of and T = 0 states of There was fair agreement between calculated energy levels obtained by exact diagonalization and experimental levels of and In the “low lying triplet is obviously of intruder nature”. In the “ground state band” with J=0,2,4,...,12 was fairly reproduced but there were many more experimental levels. In these nuclei, there is good agreement between exact calculations and those within “quasi-configurations”, where the renormalized interaction is used. Levels of other nuclei were calculated only by using that interaction for configurations. In future publications, advanced diagonalization methods were developed and there was no longer the need of reducing the shell model space. It seems that effective interactions, derived from the interaction between free nucleons, do not reproduce correctly the monopole part. In a series of papers, Cole (1990a, 1990b, 1991) tried to determine from experiment, the correct monopole part. He explained about those “realistic interactions” that “the problems can be traced to incorrect centroids of the interaction between particles in different orbitals”. Determination of those from experiment “permits accurate empirical modification to be made to realistic interactions calculated from bare nucleon-nucleon potentials”. These centroids may be used as input for determination of the effective interaction from measured energies. It is “advantageous to use as input an interaction with accurate centroids, since this will greatly reduce the number of iterations required”. Cole explained that “the aim of the present work is somewhat more limited than a full fit - namely, to extract directly from experimental data information on the centroids themselves”.

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These centroids are equal to averages of two-nucleon interaction energies, or to the monopole part of that interaction. They are determined from measured binding energies in cases that those are linear combinations of only average interactions. Within configurations, Cole was using the formulae of the seniority scheme, applicable to states of identical nucleons up to the orbit. For the average interaction energies of nucleons in different orbits, he was looking at separation energies of single nucleons. The single nucleon is either outside closed shells or “weakly” coupled to nucleons occupying a lower orbit which are coupled to a state. To apply this approach, the author had to assume the “normal” order of filling of shell model orbits. Cole listed the centroids of various two-nucleon interactions, determined for several mass regions. He compared them with those derived from realistic interactions and noted that the average T = 1 interactions between nucleons in different orbits, were more repulsive. Indeed, all such average T = 1 interactions determined by Cole, were definitely repulsive which feature agrees with results of earlier analyses. A “new effective interaction for the shell” was published by Richter, van der Merwe, Julies, and Brown (1991). They used 63 binding and excitation energies to determine the effective interaction. The data were taken from calcium and scandium isotopes in the mass range 41-49. Suspected intruder states were excluded from the fit. The authors determined two sets of matrix elements of the effective interaction. Dependence of the two nucleon matrix elements on A was introduced by multiplying the values for It was necessary to reduce drastically the number of matrix elements which together with 4 single nucleon energies amounts to 199. One way was to vary only 60 of the 199 parameters and choose among them the 12 linear combinations with the smallest statistical errors. All other linear combinations, not well determined by the experimental data, as well as the other 139 matrix elements not in the selected set, were held fixed at values obtained from the Kuo-Brown (1968) interaction. The other effective interaction was determined in a way which was used before in the shell by Brown, Richter, Julies, and Wildenthal (1988). This interaction is a sum of potential interactions where the central, mutual spinorbit and tensor interactions have the form obtained from one boson exchange. The potentials, which depend on the relative coordinates of the nucleons, were multiplied by a function of the magnitude of their center of mass coordinate. This is supposed to take care of the density dependence of the effective interaction. Another term in this interaction is a monopole term which depends on the isospin T and spin S. To reduce the number of independent parameters, the authors used several of them taken from the work in the shell, like the spin-orbit and tensor

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interactions. They were left with 10 parameters, four of the central interaction (“for the four channels”), two monopole terms and four single nucleon energies. The parameters of both sets were determined by a fit to 61 experimental energies. The two effective interactions thus determined, the direct one with 12 free parameters, and the other one, FPD6, “give an excellent reproduction of the energies of the fitted levels”. The r.m.s. deviations are .16 and .18 MeV respectively. Also levels of some nuclei not included in the fit were nicely reproduced. Magnetic moments were in fair agreement with measured ones when calculated with of free nucleons. The agreement was greatly improved if effective were used. Quadrupole moments calculated with proton and neutron effective charges 1.33e and .64e, were also in fair agreement with measured values. The results using the two interactions are comparable in general. A notable exception is in the spectrum where FPD6 predictions are in good agreement with the data. The other interaction, however, predicts the ground state to be J = 1/2 in sharp disagreement with its experimental position, 2 MeV above the J = 7/2 ground state. Examination of the matrix elements of that interaction shows that in 6 of the average interactions between identical nucleons in different orbits, only 2 are repulsive, the others are attractive (one of which is close to zero), In the case of the FPD6 interaction, 5 are repulsive and the only attractive one is rather small. Perhaps this is the reason for the situation in the calculated levels. This demonstrates the difficulty of determination of the effective interaction with a large number of matrix elements from experimental energies. The two interactions, FPD6 and KB3 of the Madrid-Strasbourg group, became the standard ones for such calculations. In a paper by Caurier, Zuker, Poves, and Martinez-Pinedo (1994), “Full shell model study of A = 48 nuclei”, the authors managed to tackle more complex spectra. The authors pointed out that the number of states as a function of the number of valence nucleons “increases so fast that three generations of computers and computer codes have been necessary to move from to in the shell”. In a configuration with given the total number of states with given T, is larger for lower values of T. It was convenient to write the matrices in the and diagonalize them by using the Lanczos method. The authors quoted the number of states with M = 0, which is the total number of states, with the lowest isospin and up to T = 4, in several A = 48 nuclei. It starts with 12,022 for and rapidly rises to 1,489,168 for and to 1,963,461 for Using the J and T quantum numbers reduces the size of the matrices which should be diagonalized (it is, however, more complicated to construct them). For T = 4, J = 4, the order of the matrix is 1,755 but for T = 2, J = 4, it is 63,757 and for T = 1, J = 5, it is already 102,225.

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Later, the Strasbourg-Madrid group, as well as other groups, managed to diagonalize much larger matrices. The authors used the KB3 interaction, introduced by Poves and Zuker (1981). This interaction, however, did not seem to improve appreciably the agreement with experiment obtained by using KB1. The calculated binding energies were too large by the same amount. Introducing a “constant shift of 780 keV, whose origin may be related to residual monopole defects” led to good agreement with measured binding energies. The authors obtained “detailed agreement with measured level schemes and electromagnetic transitions” in A = 48 nuclei. “Gamow-Teller strength functions are systematically calculated and reproduce the data to within the standard quenching factor”. The authors paid special attention to the “yrast band” of It is not a true rotational band but exhibits “back bending” indicating possible band crossing. This behaviour was very nicely reproduced by the shell model calculations, the agreement with measured spacings was very good, up to J = 16 which was measured later. Similar behaviour was obtained also for as described by Martinez-Pinedo, Poves, Robledo, Caurier, Nowacki, Retamosa, and Zuker (1996). The yrast band was observed up to J = 14 and was very well reproduced by the calculation using the KB3 interaction. A “backbending is predicted at corresponding to a collective to non-collective transition”. The authors computed the occupation of the various orbits and concluded that the collectivity is due mainly to the and orbits. They described the situation as a “quasi-SU 3” symmetry. A systematic calculation of A = 47 and A = 49 isotopes of Ca, Sc, Ti, V, Cr, and Mn is presented in a paper by Martinez-Pinedo, Zuker, Poves, and Caurier (1997). The authors used the KB3 interaction and diagonalize the Hamiltonian sub-matrices in the The order of these matrices range from 7,531 for to 6,004,205 for the nucleus. Calculated level spacings were in general, in good agreement with observed spacings. The agreement extends to several MeV above ground states. It was noted by Caurier, Zuker, Poves, and Martinez-Pinedo (1994) that KB3 overbinds A = 48 nuclei. The monopole correction keV used there, leads to good agreement between calculated and measured binding energies also for A = 47 and A = 49 nuclei. The authors calculated magnetic and quadrupole moments as well as rates of E2 and M1 transitions. They used for quadrupole moments and transitions, an effective charge of 1.5e for protons and .5e for neutrons. The agreement with experiment was good in some cases and fair in others. The authors calculated also strengths of stripping reactions with a fair amount of success. A certain amount of collective behaviour was predicted in and in the mirror pairs and The authors concluded correctly that “com-

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plete diagonalization in the major shell lead to very good agreement with the experimental data”. Results and prospects of large scale diagonalizations in the shell were presented by Andre Zuker in the International Conference on Nuclear Physics, Paris 1998 (Caurier, Nowacki, Zuker, Martinez-Pinedo, Poves, and Retamosa (1999)). States of such configurations were experimentally observed as members of a positive parity band in by Bednarczyk et al., (1997). The band starts with a state, .012 MeV above the ground state, and extends up to J = 35/2. The authors attributed these states to one proton raised from the orbit into the shell and attempt to calculate their energies. They explained that “negative parity states in and are well reproduced by shell-model calculations peformed assuming “inert” ore with the major shell acting as the valence space and the FPD6 interaction”. For the positive parity states the authors adopted the interaction of Warburton, Becker and Brown (1990), “replacing their two-body matrix elements with those of the FPD6 interaction”. They still needed some truncation, only one particle is raised from the or and “the occupation of the and orbits was limited to no more than three particles”. The authors found that “for the overall structure of the entire positive- parity band is rather well reproduced”. Relative to the ground state, however, “the calculated excitation energies for the positive-parity states in are systematically higher than the experimental values by approximately 1 MeV”. For the authors could calculate energies without truncation and excitation of more particles from the shell and into the and orbits is allowed. Energies obtained by the full calculation were lower by about .5 MeV than those in the truncated calculation. The authors believed that part of the discrepancy was due to truncation. They calculated B(E2) values within the band, using effective charge 1.33e for protons and .64e for neutrons. The agreement with experiment was fair. The authors concluded by explaining that “the positive-parity bands in odd-A nuclei around are among the most spectacular examples of shape coexistence”. The collectivity is evident in the large intraband B(E2) values. “Large-scale shell-model calculations are remarkably successful in reproducing on the same footing both nearly spherical and collective structures”. In a later experimental paper on high spin structure of some odd-even nuclei, Bednarczyk et al., (1998), presented calculations of states. They considered negative parity levels in and and calculated level spacings using the FPD6 interaction in the full shell model space. The agreement between calculated and experimental level positions was very good for and up to J = 23/2 around 6 MeV. It was worse for higher states. The highest spin observed in is 27/2 which is the maximum spin possible

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in the configuration. For the agreement was good for the levels up to J = 15/2 at about 3 MeV which is the highest observed state. The exception was the J = 3/2 state for which the deviation between theory and experiment was larger. This work is an example of experiment and theory working together to obtain a better understanding of nuclear spectra. The shell model description of “positive-parity rotational bands in oddA nuclei” was considered by Poves and Solano (1998). They took them to be due to “the coupling of a hole in the orbit to the more deformed even-even configuration”. They adopted the full shell and the KB3 as the interaction for its nucleons. The matrix elements of the interaction between the hole and the nucleons were taken from the interaction of Retamosa, Caurier, Nowacki, and Poves (1997). The calculated yrast negative parity levels of were in good agreement with experiment up to the level at about 8 MeV, and in fair agreement for higher levels. The situation in the positive parity band of starting from a state and going up to a state at about 12 MeV, was similar. The calculated position of the state above the ground state was “1.5 MeV higher than the experimental one”. The authors believed that this “must be due to the overly large monopole gap between the and shells given by” the effective interaction. There was no such discrepancy in the case of the positive parity band in The band starts with a state, at .012 MeV excitation and the “calculation places it 70 keV below the which represents a very good agreement. The agreement between calculated level spacings and experimental one was “quite good. The larger differences, that occurred in the high-spin states, was of the order of 500 keV”. These states are at energies of about 12 MeV. The agreement with experiment was similar to that obtained by Bednarczyk et al., (1997). The authors calculated E2 transition rates using effective charge 1.5e for protons and .5e for neutrons. The calculated values agreed well with some rates but failed in others. The B(M1) values, calculated with “bare electromagnetic factors, “compare well with the experimental data”. Around 1995, Otsuka and his group at the University of Tokyo, developed a novel method for diagonalization of large shell model Hamiltonian matrices. They described it as a Quantum Monte Carlo Diagonalization (QMCD) and tested its convergence in complicated cases where exact diagonalizations were available. In later publications they refer to its use for actual nuclei as the Monte Carlo Shell Model (MCSM). A review of this method can be found in a paper by Otsuka, Mizusaki and Honma (1999). One of the first application of this method was the complete shell calculation of the spectrum. First results were reported by Otsuka, Honma and Mizusaki (1998) for the yrast levels. Fair agreement was obtained with experimental level spacings by using the FPD6 interaction which, in this case, works better than KB3. “The

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M = 0 Hilbert space has a dimension of for in the full shell”. The authors found that the weight of the component in the ground state wave function was only 49%. “This is rather small compared to what would be expected for a closed shell nucleus”. The occupation number of the orbit was only 14.6. In comparison, the weight of the component in was calculated to be 86%. Perhaps the situation was not so bad for the shell. In a conference talk, Poves (1998) reported some results on shell model calculations for The only full calculations which the Strasbourg-Madrid could carry out at that time, were for J = 0 states. They “use the KBF interaction” that “keeps all the good features of KB3 and corrects some of its defects at, and beyond N = Z = 28”. Their results for the ground state are interesting. “The doubly magic configuration dominates the ground state of in the full calculation with a 50% - 60% probability, depending on the effective interaction used. Every other component has very small amplitude. Thus, we can still consider as a very correlated - doubly magic nucleus”. In a subsequent paper, Mizusaki, Otsuka, Utsuno, Honma, and Sebe (1999) presented results of QMCD also for non-yrast levels of These seem to be members of a rotational band whose level spacings were very well reproduced by the calculation and large E2 transition rates were predicted. The authors described the situation as coexistence of this band with the spherical, non-collective yrast states. Both level schemes emerged from their shell model calculation in the spherical basis. There are several nice examples of experimental papers in which new data have been well reproduced by large scale shell model calculations. Ur et al., (1998) presented new results on excited states in Its structure has been analyzed in the full shell, using the KB3 interaction with single nucleon energies taken from the observed spectrum of “The dimension, 109,954,620, is the largest attained so far”. Ten calculated levels, with spins J=0,2,4,6,8,10,12, up to 8 MeV, were in “fairly good agreement” with experiment. The “root-mean-square deviation for the yrast levels” was 175 keV. There were larger deviations for non-yrast levels. Of particular interest was the “energy inversion of the isomeric state with the yrast” level. It yielded an yrast trap, due to the spin gap to the lower states. The authors calculated B(E2) values by using effective charge 1.5e for protons and .5e for neutrons. The agreement between calculated and experimentally observed transition rates was fair. Precise measurements of magnetic moments and E2 transition rates, in J = 2 and J = 4 states of Ti and Cr nuclei were reported by Ernst et al.,

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(2000). The new data were compared with magnetic moments and B(E2) values, calculated from wave functions of the full space. The latter were calculated by using the KB3 interaction “in which the gap around is reduced and a density dependence is included”. The agreement between calculated and measured B(E2) values was fair but not impressive. The agreement between calculated and measured of the J = 4 state of the J = 2,4 states of and the J = 4 state of was excellent. The calculated of the J = 2 state of and the J = 2,4 states of followed the trend of the measured but was lower by almost factor 2. The authors attributed these deviations to possible mixing of states with excitations of the core. The authors emphasized the role of excitations from the configuration to higher orbits. They quoted results of Zamick and Zheng (1996) for of states which agreed with the new measured values only if the number of excited nucleons was at least 2. A very nice example of a fruitful collaboration between experimentalists and theorists from the Tokyo group is in a paper on the N=Z, odd-odd nucleus In the paper of Schmidt et al., (2000), 21 levels are reported, 16 for the first time, up to 3.6 MeV. These are rather low-spin levels, J <= 7, and their number at low excitations is enhanced by the proximity of T = 0 and T = 1 states. Two effective interaction were used in the full shell model space. One is KB3 and the other is FPD6 and exact diagonalization is carried out without any truncation. Definite level assignments could be made only levels up to 2.5 MeV and these are compared to calculated levels. The two “calculations lead almost to similar results and to reasonable agreement with experiment”. Absolute rates of E2 and M1 transitions are difficult to measure but intensity ratios can be compared to the shell model predictions. The authors noted that “the good agreement between theory and experiment for the branching ratios and the E2/M1 multipole mixing ratios shows that the relative strengths of M1 and E2 transitions are well accounted for in the shell model”.

7.3.

Nuclei in which N = 20 Loses Its Magicity

“Large-scale state-of-the-art shell model calculations” were carried out by Fukunishi, Otsuka and Sebe (1992) on nuclei with N = 20 and Z = 10 – 20. They considered the full shell as well as configurations where two nucleons are excited into the and orbits. For they extended the space to include excitations up to four nucleons from orbits. The matrix, for M = 0, has in this case the order 910,878. The authors used the USD interaction in the shell and the Kuo Brown (1968) G-matrix interaction “for matrix elements within the shell and for matrix elements involving

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both the orbits and the orbit. The remaining part gives only a very minor contribution” and the authors used for it the MK interaction. All matrix elements were scaled, like those of USD, by the function Another refinement was due to the and orbits being included explicitly. Their implicit contribution to the effective USD interaction “are subtracted by using the second order perturbation so as to avoid double counting”. “The spurious center-of-mass motion was removed by using the approximate method of Gloeckner and Lawson (1974). Spectra of several even-even nuclei were compared with experiment. The calculated position of the J = 2 level was very high in and whereas it was low in and For nuclei between and the occupation numbers of protons and neutrons remained small, .6. For nuclei with lower Z values, there was a continuous and large increase in occupation probability of neutrons (the proton went to zero). The authors calculated E2 transition rates using effective charges 1.3e and .5e for protons and neutrons. Their calculated B(E2) for the J = 0 to J = 2 transitions agreed very well with the rather small experimental values for and There was a sharp rise of the calculated B(E2)values for and The authors plotted effective single particle energies of the and orbits. These were defined as “the sum of the original s.p.e. and the expectation value of the two-body interaction with respect to the relevant proton closed subshell plus one neutron particle in the or one neutron hole in “The effective s.p.e. is introduced for schematic demonstration” only. What it demonstrated was that “as Z decreases, the difference between the and effective s.p.e.’s decreases gradually”. At Z = 10, the effective became lower than the one of This behaviour implies “the vanishing of the N = 20 shell gap towards Z << N = 20”. The Tokyo group returned later to this region, equipped with the QMCD method of diagonalization. In their paper, Utsuno, Otsuka, Mizusaki, and Honma (1999) considered the complete shell and the and orbits as the shell model space. For nucleons they used the USD interaction and for the ones they used the Kuo-Brown (1968) “interaction obtained from the renormalized G matrix”. The cross shell part of the interaction was taken from Warburton, Alburger, Becker, Brown, and Raman (1986) which was based on the MK interaction. Warburton et al., modified the 8 diagonal matrix elements between states with T = 0 and T = 1 and J = 2 – 5. All two-body matrix elements were scaled by the factor. The authors further modified the interaction to make it agree better with established features. The T = 1 diagonal matrix elements were made more repulsive by .3 MeV, whereas the T = 0 ones were made more attractive by .7 MeV. The corresponding changes in matrix elements

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were .16 MeV and –.5 MeV. The modified average T = 1 interactions are then only slightly attractive, –.01 MeV for states and –.08 MeV for states. Since the states were included explicitly, the authors removed their effects which were included in USD. This was accomplished by reducing the attraction in matrix elements of J = 0 states. The authors added to all matrix elements . MeV. A crucial step in the calculation was adopting single particle energies (SPE). They “are determined so as to reproduce the neutron separation energies and the one particle spectra of and The adopted SPE for the and orbits lay 1.45 MeV above the SPE in the orbit. (The observed yrast level in lies only .612 MeV above the yrast level and the yrast level, only .293 MeV above it). The authors calculated what they describe as effective SPE. They found that as “the proton number increases from Z = 8, the effective SPE’s go down as a whole”. They “notice also that the gap between the and shell becomes larger”. This was interpreted as due to the strong attractive T = 0 monopole interaction, which “is more attractive within the same major shell”. The role of the varying gap of SPE’s in forming the island of inversion was not discussed explicitly. The authors calculated binding energies and J = 2 and J = 4 excitation energies and found good agreement with experimental data. They reproduced the rather smooth behaviour of two-neutron separation energies across N = 20 for Ne, Mg and Si. They reproduced the strong reduction of the positions of those levels of Mg at N = 20. They calculated B(E2) values for transitions, with effective charges 1.3e for protons and .5e for neutrons. They found good agreement with measured B(E2) values in Mg and Si isotopes. They found considerable mixing with configurations in which neutrons were raised into orbits. They presented the average number of such excited neutrons. It started to deviate from 0 for N = 18, and for N = 20 reached 2 in Ne and Mg, whereas for Si it was only .3, and stayed small also for higher N values. It was still 2 at N = 22 for Ne and Mg and decreased to 1-1.5 at N = 24 for Ne and Mg. These results indicated that “the intruder configuration cannot be negligible even at N = 24”, contrary to previous calculations in which “N = 24 does not belong to the island” of inversion. Even earlier, the Strasbourg-Madrid group considered nuclei in which nucleons may occupy orbits in the and shells. In the first paper, Retamosa, Caurier, Nowacki, and Poves (1997) considered nuclei with and In particular, they investigated whether for “N = 28 isotones far from stability”, there is “a collapse” of the N = 28 shell closure, analogous to the situation at N = 20. Their “valence space consists of the full shell for Z – 8 protons and the full shell for N – 20 neutrons”. They did

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not include states obtained by exciting nucleons from the to the shell, since they believed that these would be less important near N = 28. For the interaction of nucleons they used USD. For the matrix elements of nucleons they used the interaction which “is better suited to this valence space than the KB3 interaction”. “The cross-shell interaction is the G matrix of” Kahana, Lee and Scott (1969) which the authors “have proceeded to modify phenomenologically”. The mass dependence of the matrix elements was given by The authors calculated the spectra of and The agreement with the rather meagre experimental information was good in general. Of particular interest were the positions of yrast J = 2 levels in nuclei from S to Cr where the agreement with experiment was very good. The 02 calculated and experimental spacings were the lowest in and The authors took these small spacings together with the large quadrupole moments and B(E2) values, to indicate that these nuclei are deformed. The calculated occupancy of the orbit in nuclei with N = 28, was lowest for Z = 15 and 16. The authors thought that this indication of large mixing with neutrons gave rise to the observed deformation in and The authors asked “Does the N = 28 shell gap persist with large neutron excess?”. They calculated binding energies and compared two neutron separation energies with measured data. The agreement was good where measurements are available. The calculated values of showed a sharp decrease at N = 28 for Ca, K and and Si isotopes. “For the other isotopes, there are not sharp changes in the slopes but still decreases” contrary to “mass systematic extrapolations, which lead to almost constant around N = 28, interpreted as a signature of a new region of deformation”. The authors concluded “that the neutron configuration at N = 28, although somewhat eroded by the large neutron excess, is still dominant”. “Therefore, the vanishing of the shell closure far from stability that happens at N = 20, does not seem to occur at N = 28. In a subsequent paper, the same authors extended their calculations to neutron rich isotopes from O to Si. Caurier, Nowacki, Poves, and Retamosa (1999) used the same interaction as in the preceding paper, to study the same model space. They extended it also to intruder configurations in which two neutrons were excited from the to the and orbits. They calculated level spacings and B(E2) values for “normal configurations” and found for Ne isotopes with N = 22 – 26, that the calculated positions of J = 2 states were rather low lying, and found corresponding large B(E2) values. No measured data were available but the situation for was better. The measured 0-2 spacing was only .89 MeV compared to 1.48 MeV in The calculation yielded a larger spacing for 1.71 MeV, indicating breakdown of

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the N = 20 shell. The calculated positions of J = 2 levels in heavier Mg isotopes was rather low up to The authors explained that “according to our calculation N = 28 is not a closed shell at Z = 12”. All these nuclei were considered to be rather strongly deformed. They calculated binding energies for normal configurations and compared single neutron and two neutron separation energies with experiment. For Ne, Na and Mg, there were significant deviations between calculated and measured around N = 20. These deviations were “the fingerprint of the breaking of the shell closure”. They calculated the binding energy of the intruder configurations for nuclei with and The authors found inversion of the lowest configuration for and They argued that even if the intruders were a few hundreds keV lower, the “anomalous region is still constrained to

8. WHAT HAVE WE LEARNED? WHERE DO WE STAND? Looking back at the last fifty years, we may ask what have we learned about the effective interaction in the shell model. As explained above, this question is about the interaction between valence nucleons to be used with shell model wave functions. The most important conclusion is that energies and shell model wave functions can be accurately calculated by using an effective two-body interaction. Shell model wave functions form a complete set of states and hence, any wave function of A bound nucleons can be expressed as a linear combination of them. The calculations considered here, however, involve only valence nucleons occupying one or at most two major shells. The success of these calculations is obtained if we do not adopt some simplified interaction. At various stages, physicists tried to use interactions based on a Yukawa or even a Gaussian potential. Results obtained in those calculations were sometimes good but in most cases, the agreement with experiment was only qualitative. Originally, it was believed that such potentials were acting between free nucleons and that more accurate scattering experiments would reveal their exact depth, range and exchange character. This hope was not fulfilled. The interaction between free nucleons which was determined by experiment, was too strong at short distances and must be sufficiently renormalized to allow nucleons to move independently. There are many-body theories which aim at such renormalization. Effective interactions derived so far, however, gave rather poor results in shell model calculations. Some of them seemed to be unable to reproduce the very existence of magic numbers. Hence, good agreement with experiment has been obtained by determination of matrix elements of the effective interaction from experimental energies.

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The possibility of obtaining in this way, quantitatively significant agreement with experiment was first demonstrated for rather simple configurations. For most nuclei, particularly in cases where collective spectra appear, several configurations are involved and in such cases, only the first steps were taken, like in the shell. The success in these cases, lends confidence in the belief that also more collective spectra, like rotational ones, will be reproduced by sufficiently large shell model calculations. Rotational spectra follow simply from the SU(3) scheme. This scheme, however, is due to a schematic interaction acting in special shell model spaces. The realistic effective interaction is much more complicated and major shells are rather different from the simple spaces required for SU(3) to be valid. Still, the observed spectra exhibit, for many nuclei, very simple rotational bands. To reproduce them is a challenge to large scale shell model calculations. Shell model wave functions yield not only energies but also other observables. Good agreement is obtained between calculated and experimental rates of E2 and M1 transitions, as well as quadrupole and magnetic moments, and rates of allowed beta decays. Also these observables may be reliably calculated provided effective operators are used for their determination. It seems as if the effects of short range correlations, imposed by the strong interaction between free nucleons at short distances, may be transformed onto the Hamiltonian and single nucleon operators. This transformation is expected to be achieved by many-body theory, but so far, its results have not turned out to be sufficiently reliable. Still, there is progress in obtaining reliable matrix elements of the effective interaction from the interaction between free nucleons. In fact, it was described above how such matrix elements may be modified and then used in shell model calculations. It seems that single nucleon wave functions are not just model states to be used with effective operators. This follows from results of high energy scattering on nuclei, direct nuclear reactions and even from detailed analysis of charge distributions. This information indicates that the real wave functions of single nucleons can be approximated by shell model wave functions. The actual occupation of shell model orbits is not well determined. Still, even lower estimates are “that only ~2/3 of the time a nucleon acts as an independent particle bound in an average potential”. Discussion of this point, however, is beyond the scope of this review. Determination of the effective interaction from experiment, yields a set of matrix elements for the various orbits considered. Their values depend on the particular shell model orbits which the valence nucleons are assumed to occupy. If good agreement with experiment is obtained, effects due to other orbits are absorbed into the effective interaction in the limited space. If some of the other orbits are included in the shell model space considered, the effects

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due to them, will not be included in the effective interaction. For these reasons it is difficult to find a general formula for the effective interaction. Still, some of its general features can be deduced from cases in which good agreement with experiment was obtained. The quantitative agreement with experiment is the basis of these deduced general features. The most important conclusion is that the effective interaction between valence nucleons can be expressed as a twobody interaction. In fact, the restriction to two-nucleon effective interaction enabled its determination from experimental energies. When a large region of nuclei is considered, some dependence on A is required. In calculations of binding energies, introducing a small term, cubic in the number of valence nucleons, yields better agreement with experiment. Such corrections, however, may be attributed to the change in volume of the nucleus and no explicit three-body interactions were found to be necessary. In many-body theory of the effective interaction, three-body terms appear in the expansions, even if the original interaction is only a two-body one. The conclusion from shell model calculations is not in contradiction with possible effects of three-body forces on energies and wave functions of three and four nucleon systems. Nor is it in apparent contradiction with possible effects of three-body terms on the calculations of total binding energies of nuclei. Hopefully, this point will be cleared when the shell model for finite nuclei will be reliably based on many-body theory. At this stage, only matrix elements determined from experimental energies are available. Features of the effective interaction which were determined in simple cases, seem to be present also in more complicated ones. Some of these features are consistent with general properties of nuclear energies which were known from the very early days of nuclear physics. The T=1 part of the effective interaction has a strong pairing term but apart from it, is repulsive on the average. It is the only interaction between identical nucleons and it can be studied best in semimagic nuclei. The contribution to the binding energy of identical valence nucleons in the can be expressed by a formula of the seniority scheme. It contains a term linear in due to single nucleon energies, a quadratic term and a pairing term. The coefficient of the pairing term is large and attractive but the coefficient of the quadratic term, although small, is repulsive. The average interaction energy of two identical nucleons in different orbits is also repulsive. This feature is consistent with the saturation properties of nuclear binding energies. For equal numbers of protons and neutrons, apart from the Coulomb energy, binding energies are fairly linear in A = Z+N. If N > Z, for instance, the symmetry energy, which is proportional to T(T+1) is repulsive. If N = the ground state isospin is and the symmetry energy is equal to a repulsive coefficient multiplying Hence, there is a repulsive term quadratic in the number of valence identical

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nucleons. If the is completely filled, additional valence nucleons go into a higher The quadratic repulsion, due to the symmetry energy term, should not stop at that point. Hence, it implies an average repulsion between the nucleons in the full and nucleons in the In fact, the quadratic part of the symmetry energy term is given in this case, by

From this follows that the average repulsion between identical nucleons in different orbits should be twice the value of the average quadratic term in configurations. The interaction between free neutrons, or the interaction in a T = 1 state of a proton and a neutron, is attractive. It is weaker than the T=0 interaction and does not lead to a bound state of two nucleons. These features do not seem to lead directly to the features of the T = l effective interaction in nuclei. This is possibly due to the fact that nucleons are bound in the nucleus and their interaction may be strongly renormalized by effects of the many-body system. This feature leads to the conclusion that the T=1 interaction cannot give rise to nuclear binding. Nuclei are held together by the T= 0 part of the nuclear interaction which acts between protons and neutrons. The effective interaction between protons and neutrons in different orbits is equal to the average of the T= l and T= 0 interactions. In all simple cases we find that it is attractive on the average. The interaction of a proton with neutrons in closed orbits, or a neutron with such protons, is attractive. It means that the T=0 part leads to the binding of nucleons in the attractive potential well of the shell model. When valence protons and neutrons occupy the same in states with low values of T, the repulsion due to the symmetry energy term is absent. In fact, if there are valence protons and valence neutrons outside closed shells with then

The coefficient of the symmetry energy term is repulsive, consistent with repulsive and terms. Hence, the coefficient of the average interaction energy between valence protons and neutrons, is attractive. The fact that the proton-neutron interaction creates the potential well in which nucleons move, has important implications. The depth of this well, as well as its shape for neutrons, depend on the occupation of proton orbits and vice versa. This means that the spacings and even the order, of single proton orbits may depend on the occupation of neutron orbits and vice versa. A simple example is offered by the ground state spin of which was discussed in detail above. Whereas in the neutron orbit lies below

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the orbit, the order is reversed in The shell model explanation is simple. The attraction between protons and a neutron is stronger than their attraction to a neutron. Removing a pair of protons from affects more strongly the position of the orbit. This could be described as disappearance of shell closure at N=8. This seems to be also the reason for the situation for nuclei with N=20 and Z ~11. In nuclei with a sufficient number of their attraction makes the neutron considerably lower than the neutron If are removed, the neutron is more affected than the neutron As a result, when more are removed, the gap between these neutron orbits decreases. If a sufficient number of is removed, N=20 is no longer a magic number. Configurations including orbits are strongly admixed into configurations. This effect was described above and it has been reproduced both in truncated as well as in large scale shell model calculations. Another example is offered by the situation in heavier Zr isotopes, discussed above. When neutrons occupy the orbit, the proton orbit is lowered and protons move into it. The nucleus with Z=40 protons is no longer an inert core and collective spectra emerge. The question was raised whether the shell model may be used for exotic nuclei, where the proton or neutron drip line is approached. These examples show that single nucleon orbits may change their positions as a result of changes in occupation numbers. There seems to be nothing peculiar about those changes and it seems that the shell model is able to reproduce them and hence, is useful also in such cases. There are other, more specific differences between the T=1 and T=0 parts of the effective interaction in nuclei. The interaction between identical nucleons in configurations is diagonal in the seniority scheme. In semi-magic nuclei where configuration mixing is evident, generalized seniority has a simple prescription for J=0 ground states and J=2 first excited states. Simple features of the seniority scheme are preserved in this generalization and are displayed by these states. An important feature is the constant spacings between these J=0 and J=2 states, which are independent of the number of valence identical nucleons. This feature characterizes semi-magic nuclei as spherical ones. When there are both valence protons and neutrons, their interaction includes also the T=0 part of the effective interaction. If they are in the same orbit, in the configuration and the two-body interaction need not be diagonal in the seniority scheme. We find that seniority is badly broken by the T=0 effective interaction. Had this not been the case, 0-2 spacings would have been constant, independent of the number of valence nucle-

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ons. We see, however, that the 0-2 spacings decrease considerably as the number of nucleons, or holes, is increased in nuclei where both valence protons and neutrons are present. This decrease is a clear signature of the transition towards rotational spectra. The attractive proton-neutron interaction which creates the potential well of the shell model, is responsible for its depth and its shape as mentioned above. This interaction leads also to deviation of the well from its spherical shape into a deformed one. In a configuration with scalar products of even rank tensors break seniority. A typical such interaction is the quadrupole-quadrupole one. If scalar products of other even rank tensors, with are added to it with certain coefficients, the resulting interaction may become diagonal in seniority. The quadrupole interaction by itself breaks seniority in a major way. Indeed, from the early days of the collective model, the importance of the quadrupole degree of freedom has been recognized and emphasized. A recent calculation demonstrated these features of the effective interaction. The Tokyo group considered the “Transition from Spherical to Deformed Shapes of Nuclei”. They use a rather schematic interaction which incorporates the features of T=1 and T=0 mentioned above. In their paper, Shimizu, Otsuka, Mizusaki, and Honma (2001) used their MCSM to carry out a large scale shell model calculation of Ba isotopes. They chose to calculate spectra of Ba isotopes with since experimentally they exhibit the transition between “ a semi-magic spherical nucleus, and a rotational deformed nucleus”. The authors took the protons to occupy the and The neutrons occupy the and orbits. “The single particle energies are adopted from the experimental energy levels of for neutrons and from those of for protons”. “The proton-neutron interaction is assumed to consist of the quadrupolequadrupole interaction”. “The two-body effective interaction between identical nucleons is assumed to be comprised of the monopole pairing, quadrupole pairing and quadrupole-quadrupole interactions”. The coefficients which determine the strengths of these interactions were given for neutrons (protons) by .13 (.21) MeV, .14 (.22) MeV and respectively. A word of explanation should be added about the quadrupole interaction between identical nucleons. As emphasized on several occasions above, the interaction between identical nucleons is diagonal in the seniority scheme and this feature is badly broken by a quadrupole interaction. Here, however, quadrupole pairing in T=1 states is introduced explicitly. The expansion of an interaction whose two-body matrix elements are proportional to within a single configuration, in terms of scalar products of even tensor operators was derived by de-Shalit and Talmi (1963). The same derivation can be used to show that an attractive interaction can be expanded in terms of odd ten-

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sors and a repulsive quadrupole-quadrupole term. The odd tensor interaction is diagonal in the seniority scheme but the quadrupole term is not. Hence, to obtain an interaction which is diagonal in seniority, a quadrupole term with the same sign should be added. The calculation described here, involves many configurations but perhaps the added attractive T=1 quadrupole interaction compensates the seniority breaking effect of the quadrupole term which is due to the quadrupole pairing interaction. The T=1 quadrupole term added by the authors, is much smaller than the proton-neutron attractive quadrupole interaction whose strength is determined by the The Hamiltonian with single nucleon levels and two-body interaction remains constant in the calculation of spectra of the various isotopes. The number of neutrons is varied between 82 and 94. Calculated positions of yrast levels with J=2,4,6,8,10 are in good agreement with all measured positions. The J=2 levels, both experimental and calculated, show a strong and regular drop from 1.436 MeV at N=82, to .142 MeV at N=92. The decrease of J=4 and J=6 level positions is also fairly well reproduced. The calculated positions of the J=8 are considerably higher than the measured ones for A=138 and 140 but drop closer to the measured positions for A=144 and A=146. The calculated B(E2) values for the transitions agree well with the measured ones in Ba isotopes with A=138 and A=142-146. The rise of measured E2 transition rates is followed closely by rates calculated with effective charges 1.6e and .6e of protons and neutrons. The calculated positions of J > 2 levels show a break in the slope at N=88. The experimental positions of J=6 and J=8 levels also show clearly this break. Its presence is less pronounced in the measured positions of J=4 levels. These results are encouraging. They indicate that monopole and quadrupole pairing in T=1 states and a quadrupole-quadrupole interaction between protons and neutrons are the most important ingredients of the effective nuclear interaction. Still, it should be kept in mind that the effective interaction is far more detailed and complicated. It would be very interesting to see the transition from simple shell model spectra to rotational ones, obtained with a more realistic effective interaction. The good agreement between results of shell model calculations and experimental energies is limited to states of valence nucleons. Features of the effective interaction described above, were gleaned from such calculations. Configuration mixing, however, may be important not only for orbits in the same major shell. Most binding energy calculations which accurately reproduce the data, involve valence nucleons in their lowest major shells. Still, some examples of intruder states which affect ground states were described above. At higher excitations, such effects are much more pronounced. The energy differences between states of nucleons in two major shells is not very large. Hence, there is a rather large overlap between energy regions spanned by lev-

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els of configurations due to nucleons in different major shells. In many nuclei there is coexistence between sets of states, one predominantly due to the lower major shell and the other to the higher one. Mixing with states of such configurations is usually not included in large scale shell model calculations and hence, it may give rise to deviations from measured energies. This effect, however, is probably less important for yrast states and, in particular, to high-spin ones. An example is offered by the spectra of Sn isotopes whose low lying levels are clearly due to only valence neutrons. The yrast J=2 levels are around 1.3 MeV above the J=0 ground states. As explained above, properties of these levels are consistent with their description as generalized seniority eigenstates. Hardly any other levels are observed up to 2 MeV excitation, but right above it, many levels were experimentally found. As more neutrons are added, the nature of some of these states seems to be revealed. In and in an excited J=0 state is observed at 1.757 (and 1.758) MeV above the ground states. These J=0 states are the ground states of bands whose highest measured states, at 5.4 MeV, have spins J=12. These bands have the properties of rotational bands, even though the energies do not strictly follow the J(J+1) dependence. Actually, energies of the J=6,8,10,12 levels, in both nuclei, agree rather well with this dependence, much better than the J=2,4 levels which seem to be more perturbed. From what we learned about the effective interaction, rotational levels cannot be due to valence neutrons only. They must be due to configurations in which protons are excited from the Z=50 closed shells. Another example of coexistence is in the domain of deformed nuclei. In many nuclei, in addition to “normal” rotational bands, super-deformed bands are observed. The normal bands may be due to valence protons and neutrons which occupy the lowest major shells available to them. The more collective and higher energy super-deformed bands may be due to configurations in which protons are excited from normal closed shells to higher major shells. There is hope that a shell model description of rotational spectra, due to a deformed potential well, will be found. It may be also possible to hope for a shell model description of super-deformed rotational bands, presumably due to a second minimum of the potential well. Three main difficulties have been mentioned in reference to large scale shell model calculations, since they involve construction of very large matrices which then should be diagonalized. The first difficulty is our lack of information how to construct them. The number of diagonal and non-diagonal matrix elements increases very rapidly with the number of orbits considered. There is no reliable way to calculate them and their number is so high that it is not possible to determine them from experimental energies. The second difficulty is due to the very large order of the matrices. It is very difficult to diagonalize

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them and even to determine their lowest eigenstates and eigenvalues. The third difficulty is conceptual, rather than practical. The eigenstates in the computer output have billions of components. Low lying levels of most nuclei exhibit rather simple features. It would be very difficult to extract from wave functions with so many components, real understanding of how simple structures of the spectrum arise. An actual case may illustrate this point. Semi-magic nuclei are simpler than others and the problems mentioned above are less difficult. The nucleus has Z=50 and the 12 valence neutrons may occupy the and orbits. All possible neutron configurations contain: 56,907 states with J=0 and even parity 267,720 states with J=2 and even parity and 426,558 states with J=4 and even parity. To find the eigenvalues and eigenstates of the shell model Hamiltonian, matrices of these orders should be constructed and diagonalized. Diagonalization of such matrices is no real problem for present day computers and computer programs. For constructing these matrices, 160 two-body matrix elements plus 5 single neutron energies are required. It would be a difficult task to determine all of them from experimental data. It seems, however, that theorists have found ways to overcome this difficulty by modifying matrix elements obtained from many-body theory. This procedure seems to be sufficiently accurate for low-lying levels. The third difficulty mentioned above, would be still there. If energies of J=0 and J=2 states would be calculated for Sn isotopes and found to have a constant separation, in agreement with experiment, it would be a success of the shell model. It would still be difficult to understand by looking at eigenstates with so many components why these spacings are constant, independent of neutron number. A very simple prescription for the lowest J=0 and J=2 states is offered by generalized seniority. As explained above, it provides a very simple explanation of the constant 0 - 2 spacings. Even if it is only an approximation, it provides a simple prescription for a simple regularity. If the shell model Hamiltonian has J=0 and J=2 generalized seniority eigenstates, its sub-matrices for for instance, have a special structure. The lowest J=0 eigenstate is completely decoupled from the other 56,906 states with J=0. The lowest J=2 eigenstate is completely decoupled from the other 267,719 J=2 states which are orthogonal to it. If this is the case, generalized seniority can be viewed as an efficient truncation scheme. It may hold only for yrast states but these are the interesting states in semi-magic nuclei which show characteristic regularities. Many millions of the other states are less interesting, they lie at higher energies where they may be mixed with states of excited configurations as actually observed in tin isotopes. Many of the higher levels do not correspond to

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bound states. All matrix elements needed for determination of spectra of Sn isotopes are between identical nucleons. The interaction between valence identical nucleons, in all ground configurations of semi-magic nuclei, is between two-nucleon states with T=1. It seems that calculation of these matrix elements from the interaction between free nucleons, is more reliable than the derivation of the T=0 matrix elements. Even crude approximations, like the one made by Dawson, Talmi and Walecka (1962), led to reasonable T=1 matrix elements for An example of the advances made in deriving the T=1 effective interaction from the one between free nucleons, is offered by the following papers. Andreozzi, Coraggio, Covello, Gargano, Kuo, Li, and Porrino (1996) presented “realistic shell-model calculations” for to They compared their results for two versions of the interaction between free nucleons, the Bonn A and the Paris potentials. They considered the and neutron orbits. The two-body matrix elements were calculated by the method of approximation developed by Tom Kuo and his collaborators. Due to the absence of experimental information on single nucleon energies were determined in a complicated way from data on neighboring nuclei. The calculated level spacings were compared to the observed ones in and The results obtained with matrix elements derived from the Bonn A potential were in better agreement with experiment and the authors adopted it for further calculations. In the agreement with experiment was good, “the main point of disagreement being the position of the state which lies about 250 keV above the observed one”. The authors quoted the square root of the sum of squared deviations divided by the number of levels, as equal to .129 MeV. In the calculated levels are “slightly overestimated”, the “maximum discrepancy with experiment being 350 keV for the excitation energy of the” second excited state above 2 MeV. The average deviation, as defined above, in the case of is .241 MeV. The same approach resulted in even better agreement with experiment for N=82 isotones and Andreozzi, Coraggio, Covello, Gargano, Kuo, and Porrino (1997) presented the calculated level schemes of those nuclei and compare them with experiment. The agreement between theoretical and measured, rather complex spectrum, of was good up to known levels at about 4.5 MeV. The mean deviation was .145 MeV and most actual deviations were less than .1 MeV. Up to 3.5 MeV, calculated levels corresponded to all measured ones, with the exception of a and a levels, predicted at 2.51 and 2.65 MeV. The agreement with experiment was even better for where the mean deviation was only .077 MeV. Up to 1.5 MeV, there was complete correspondence between calculated and experimental levels. The correspondence of levels above 1.5 MeV up to about 4 MeV, was shown only for levels with

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firm spin assignments. Binding energies were calculated but comparison with experiment was hampered by lack of detailed information on the Coulomb energy of the valence protons. However, even a simple minded estimate indicated very reasonable agreement with measured energies. A subset of the authors of the papers described above, with the addition of another author, used the same approach to calculate spectra of certain Pb isotopes. Coraggio, Covello, Gargano, Itaco, and Kuo considered and in which there are neutron holes in the and orbits. Energies of single hole states were taken from the observed spectrum of Good agreement was obtained between calculated and measured level spacings. The mean deviations are “207 and 216 keV for and respectively. The agreement with experiment was even better for with a mean deviation of only .074 MeV. Up to 2.5 MeV in the measured levels all had corresponding calculated levels, with the exception of two levels without assignment around 2.2 MeV excitation. They may correspond to calculated and levels. The situation in was similar up to 1.5 MeV. Above this excitation, correspondence was established only for high spin levels up to around 5 MeV. In the spectrum there was good agreement between calculated and measured positions of a level at about 7.5 MeV. The authors explained that above 4.3 MeV, the agreement is rather worse, “the largest discrepancy being about 400 keV for the state”. The authors found good agreement between the calculated binding energies relative to The calculated values for and are–14.240, –22.147 and –28.927 MeV, compared to the measured energies, –14.106(6), –22.194(6) and –28.925(6) MeV respectively. The agreement between theory and experiment in these papers is very impressive. The approximations made in deriving the effective interaction may have been somehow adjusted to obtain better agreement with experiment. Even if it is so, this method seems to yield matrix elements which may be used in successful shell model calculations. The small discrepancies observed in the case of few valence nucleons, may be magnified when a larger number of nucleons is considered. Should this happen, the calculated two-body matrix elements could be considered as good parameters to start from. Slight modifications of their values could lead to better agreement with experiment. The order of the Hamiltonian sub-matrices which should be diagonalized, increases tremendously when both valence protons and valence neutrons are present. A typical rotational spectrum is observed in with 12 protons occupying the and orbits, and 10 neutrons occupying the and orbits. Coupling of all allowed proton states with all allowed neutron states gives rise to:

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41,654,193,516,797 states with J=0 and positive parity 346,132,052,934,889 states with J=2 and positive parity and 530,897,397,260,575 states with J=4 and positive parity. Determination of the 1307 two-body matrix elements and 11 single nucleon energies needed to construct the Hamiltonian sub-matrices is a formidable task. Hopefully, they may be reliably derived by many-body theory from the interaction between free nucleons. Diagonalization of matrices of these orders are most probably beyond present computing capability. Clearly, a truncation method is necessary for a clear shell model description of such nuclei. It should be simple enough to allow physical understanding and yet, include the important ingredients. It should yield a rotational like spectrum in and but a vibrational like spectrum in and The successful developments of large scale shell model calculations were developed on time to match important developments in experimental nuclear physics. New facilities are exploring regions of the nuclear chart which were not accessible before. The results of new experiments will offer new challenges to shell model theorists and their results may provide guidance to experimental physicists. There are already some examples of fruitful cooperation between physicists performing new experiments and those deriving new theoretical results. Many more of these will be seen in the future. Looking back at the last 50 years, reveals the tremendous progress in nuclear physics which was made possible by the shell model. The shell model provided guidance to experimental physicists whose earlier work laid its foundations. It gave theorists a concrete scheme for carrying out quantitative calculations of nuclear energies and other observables. It is possible to look back with satisfaction at the progress made in shell model calculations. It is amazing how some simple potential interactions could yield a semi-quantitative description of nuclear spectra. Some theorists could not part with such potentials even many years after the true facts about the interaction between free nucleons were discovered. It has been a difficult challenge to start from the interaction between free nucleons and, using many-body theory, to derive the effective interaction. Only in the last few years, the many attempts in this direction, seem to yield matrix elements which may be reliably used in shell model calculations. Meanwhile, quantitatively significant calculations of nuclear energies have been carried out by using matrix elements extracted from energy levels of nuclei. The restriction to two-body effective interaction leads to a consistent way to carry out this procedure. Matrix elements between many nucleon states are linear combinations of those between two-nucleon states. The calculation is successful if the experimental energies considered, satisfy the relations which

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follow from this fact. These relations depend on the choice of the shell model space adopted. If these consistency checks are satisfied, they provide evidence for the validity of the choice of configurations. The two-body matrix elements may then be reliably used in calculations of energies and other observables of nuclei in which these configurations are present. In large scale shell model calculations, there are not enough experimental data for determination of all matrix elements. Theorists had to resort to using matrix elements derived by many-body theory from the interaction between free nucleons. Some of these matrix elements were modified by requiring that they should reproduce certain measured nuclear energies. Those are the elements which are most important for the calculation of yrast stales. Nuclear structure theorists have been using two different approaches to obtain the effective nuclear interaction for the shell model. One approach is described in this review. The other approach starts from some version of the free nucleon interaction. In large scale shell model calculations, good agreement with experiment was achieved by a combination of the two approaches. There is an ever growing number of matrix elements needed for such calculations. There are indications that the reliability of matrix elements, derived from the interaction between free nucleons is increasing. Hopefully, the efforts in the two approaches will merge, which will lead to results, even more impressive than those obtained so far.

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Chapter 2

INTRODUCTION TO CHIRAL PERTURBATION THEORY Stefan Scherer Institut für Kernphysik Johannes Gutenberg- Universät Mainz J.J. Becher Weg 45, D-55099 Mainz, Germany

1.

Introduction

278

2.

QCD and Chiral Symmetry

290

3.

Spontaneous Symmetry Breaking and the Goldstone Theorem

323

4.

Chiral Perturbation Theory for Mesons

339

5.

Chiral Perturbation Theory for Baryons

417

6.

Summary and Concluding Remarks

496

Appendix

501

References

529

277

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ABSTRACT This article provides a pedagogical introduction to the basic concepts of chiral perturbation theory and is designed as a text for a two-semester course on that topic. Section 1 serves as a general introduction to the empirical and theoretical foundations which led to the development of chiral perturbation theory. Section 2 deals with QCD and its global symmetries in the chiral limit; the concept of Green functions and Ward identities reflecting the underlying chiral symmetry is elaborated. In Section 3 the idea of a spontaneous breakdown of a global symmetry is discussed and its consequences in terms of the Goldstone theorem are demonstrated. Section 4 deals with mesonic chiral perturbation theory and the principles entering the construction of the chiral Lagrangian are outlined. Various examples with increasing chiral orders and complexity are given. Finally, in Sect. 5 the methods are extended to include the interaction between Goldstone bosons and baryons in the single-baryon sector, with the main emphasis put on the heavy-baryon formulation. At the end, the method of infrared regularization in the relativistic framework is discussed.

1.

INTRODUCTION

1.1. Scope and Aim of the Review The present review has evolved from two courses I have taught several times at the Johannes Gutenberg-Universität, Mainz. The first course was an introduction to chiral perturbation theory (ChPT) which only covered the purely mesonic sector of the theory. In the second course the methods were extended to also include baryons. I have tried to preserve the spirit of those lectures in this article in the sense that it is meant to be a pedagogical introduction to the basic concepts of chiral perturbation theory. By this I do not mean that the material covered is trivial, but that rather I have deliberately also worked out those pieces which by the “experts” are considered as well known. In particular, I have often included intermediate steps in derivations in order to facilitate the understanding of the origin of the final results. My intention was to keep a balance between mathematical rigor and illustrations by means of (numerous) simple examples. This article addresses both experimentalists and theorists! Ideally, it would help a graduate student interested in theoretical physics getting started in the field of chiral perturbation theory. However, it is also written for an experimental graduate student with the purpose of conveying some ideas why the experiment she/he is performing is important for our theoretical understand-

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ing of the strong interactions. My precedent in this context is the review by A.W. Thomas [304] which appeared in this series many years ago and served for me as an introduction to the cloudy bag model. Finally, this article clearly is not intended to be a comprehensive overview of the numerous results which have been obtained over the past two decades. For obvious reasons, I would, right at the beginning, like to apologize to all the researchers who have made important contributions to the field that have not been mentioned in this work. Readers interested in the present status of applications are referred to lecture notes and review articles [230,59,253,232, 44, 286, 274, 118, 246, 275, 83] as well as conference proceedings [55, 49, 52]. The present article is organized as follows. Section 1 contains a general introduction to the empirical and theoretical foundations which led to the development of chiral perturbation theory. Many of the technical aspects mentioned in the introduction will be treated in great detail later on. Section 2 deals with QCD and its global symmetries in the chiral limit and the concept of Green functions and Ward identities reflecting the underlying chiral symmetry is elaborated. In Section 3 the idea of a spontaneous breakdown of a global symmetry is discussed and its consequences in terms of the Goldstone theorem are demonstrated. Section 4 deals with mesonic chiral perturbation theory and the principles entering the construction of the chiral Lagrangian are outlined. Various examples with increasing chiral orders and complexity are given. Finally, in Sect. 5 the methods are extended to include the interaction between Goldstone bosons and baryons in the single-baryon sector with the main emphasis put on the heavy-baryon formulation. At the end, the method of infrared regularization in the relativistic framework is discussed. Some technical details and simple illustrations are relegated to the Appendices.

1.2. Introduction to Chiral Symmetry and Its Application to Mesons and Single Baryons In the 1950’s a description of the strong interactions in the framework of quantum field theory seemed to fail due to an ever increasing number of observed hadrons as well as a coupling constant which was too large to allow for a sensible application of perturbation theory [174]. The rich spectrum of hadrons together with their finite sizes (i.e., non-point-like behaviour showing up, e.g., in elastic electron-proton scattering through the existence of form factors) were the first hints pointing to a substructure in terms of more fundamental constituents. A calculation of the anomalous magnetic moments of protons and neutrons in the framework of a pseudoscalar pion-nucleon interaction gave rise to values which were far off the empirical ones (see, e.g., [54]). On the other hand, a simple quark model analysis [31, 258] gave a prediction –3/2

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for the ratio which is very close to the empirical value of —1.46. Nevertheless, the existence of quarks was hotly debated for a long time, since these elementary building blocks, in contrast to the constituents of atomic or nuclear physics, could not be isolated as free particles, no matter what amount of energy was supplied to, say, the proton. Until the early 1970’s it was common to talk about “fictitious” constituents allowing for a simplified group-theoretical classification of the hadron spectrum [148, 333], which, however, could not be interpreted as dynamical degrees of freedom in the context of quantum field theory. In our present understanding, hadrons are highly complex objects built from more fundamental degrees of freedom. These are on the one hand matter fields with spin 1/2 (quarks) and on the other hand massless spin-1 fields (gluons) mediating the strong interactions. Many empirical results of medium- and high-energy physics [11] such as, e.g., deep-inelastic lepton-hadron scattering, hadron production in electron-positron annihilation, and lepton-pair production in Drell-Yan processes may successfully be described using perturbative methods in the framework of an SU(3) gauge theory, which is referred to as quantum chromodynamics (QCD) [181, 321, 140]. Of particular importance in this context is the concept of asymptotic freedom [181, 182, 277], referring to the fact that the coupling strength decreases for increasing momentum transfer providing an explanation of (approximate) Bjorken scaling in deep inelastic scattering and allowing more generally for a perturbative approach at high energies. Sometimes perturbative QCD is used as a synonym for asymptotic freedom. In Refs. [86, 330] it was shown that Yang-Mills theories, i.e., gauge theories based on non-Abelian Lie groups, provide the only possibility for asymptotically free theories in four dimensions. At present, QCD is compatible with all empirical phenomena of the strong interactions in the asymptotic region. However, one should also keep in mind that many phenomena cannot be treated by perturbation theory. For example, simple (static) properties of hadrons cannot yet be described by ab initio calculations from QCD and this remains one of the largest challenges in theoretical particle physics [174]. In this context it is interesting to note that, of the three open problems of QCD at the quantum level, namely, the “gap problem,” “quark confinement,” and (spontaneous) “chiral symmetry breaking” [208], the Yang-Mills existence of a mass gap has been chosen as one of the Millennium Prize Problems [92] of the Clay Mathematics Institute. From a physical point of view this problem relates to the fact that nuclear forces are strong and short-ranged. One distinguishes among six quark flavours (up), (down), (strange), (charm), (bottom), and (top), each of which coming in three different colour degrees of freedom and transforming as a triplet under the fundamental representation of colour SU(3). The interaction between the quarks and

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the eight gauge bosons does not depend on flavour, i.e., gluons themselves are flavour neutral. On the other hand, due to the non-Abelian character of the group SU(3), also gluons carry “colour charges” such that the QCD Lagrangian contains gluon self interactions involving vertices with three and four gluon fields. As a result, the structure of QCD is much more complicated than that of Quantum Electrodynamics (QED) which is based on a local, Abelian U(l) invariance. However, it is exactly the non-Abelian nature of the theory which provides an anti-screening due to gluons that prevails over the screening due to pairs, leading to an asymptotically free theory [266], Since neither quarks nor gluons have been observed as free, asymptotic slates, one assumes that any observable hadron must be in a so-called colour singlet state, i.e., a physically observable state is invariant under SU(3) colour transformations. The strong increase of the running coupling for large distances possibly provides a mechanism for colour confinement [182]. In the framework of lattice QCD this can be shown in the so-called strong coupling limit [326]. However, one has to keep in mind that the continuum limit of lattice gauge theory is approached for a weak coupling and a mathematical proof for colour confinement is still missing [208]. There still exists no analytical method for the description of QCD at large distances, i.e., at low energies. For example, how the asymptotically observed hadrons, including their rich resonance spectrum, are created from QCD dynamics is still insufficiently understood.* This is one of the reasons why, for many practical purposes, one makes use of phenomenological, more-or-less QCD-inspired, models of hadrons (see, e.g., [304, 196, 262, 329, 183, 56, 259, 157, 314, 109, 307]). Besides the local SU(3) colour symmetry, QCD exhibits further global symmetries. For example, in a strong interaction process, a given quark cannot change its flavour, and if quark-antiquark pairs are created or annihilated during the interaction, these pairs must be flavour neutral. In other words, for each flavour the difference in the number of quarks and antiquarks (flavour number) is a constant of the motion. This symmetry originates in a global invariance under a direct product of U(l) transformations for each quark flavour and is an exact symmetry of QCD independent of the value of the quark masses. Other symmetries are more or less satisfied. It is well known that the hadron spectrum may be organized in terms of (approximately) degenerate basis states carrying irreducible representations of isospin SU(2). Neglecting electromagnetic effects, such a symmetry in QCD results from equal and (current) masses. The extension including the quark leads to the famous flavour SU(3) symmetry [167] which, however, is already significantly broken due to the * For a prediction of hadron masses in the framework of lattice QCD see, e.g., Refs. [84, 9],

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larger mass. The masses of the three light quarks and are small in comparison with the masses of “typical” light hadrons such as, e.g., the meson (770 MeV) or the proton (938 MeV). On the other hand, the eight lightest pseudoscalar mesons are distinguished by their comparatively small masses.† Within the pseudoscalar octet, the isospin triplet of pions has a significantly smaller mass (135 MeV) than the mesons containing strange quarks. One finds a relatively large mass gap of about 630 MeV between the isospin triplets of the pseudoscalar and the vector mesons, with the gap between the corresponding multiplets involving strange mesons being somewhat smaller. In the limit in which the masses of the light quarks go to zero, the lefthanded and right-handed quark fields are decoupled from each other in the QCD Lagrangian. At the “classical” level QCD then exhibits a global symmetry. However, at the quantum level (including loops) the singlet axial-vector current develops an anomaly [5, 2, 24, 70, 6] such that the difference in left-handed and right-handed quark numbers is not a constant of the motion. In other words, in the so-called chiral limit, the QCD-Hamiltonian has a symmetry. Naturally the question arises, whether the hadron spectrum is, at least approximately, in accordance with such a symmetry of the Hamiltonian. The symmetry is connected to baryon number conservation, where quarks and antiquarks are assigned the baryon numbers B = 1/3 and B = –1/3, respectively. Mesons and baryons differ by their respective baryon numbers B = 0 and B = 1. Since baryon number is additive, a nucleus containing A nucleons has baryon number B = A. On the other hand, the symmetry is not even approximately realized by the low-energy spectrum. If one constructs from the 16 generators of the group the linear combinations and _ the generators form a Lie algebra corresponding to a subgroup H of G. It was shown in Ref. [313] that, in the chiral limit, the ground state is necessarily invariant under the group H, i.e., the eight generators annihilate the ground state. The symmetry with respect to H is said to be realized in the so-called WignerWeyl mode. As a consequence of Coleman’s theorem [93], the symmetry pattern of the spectrum follows the symmetry of the ground state. Applying one of the axial generators to an arbitrary state of a given multiplet of welldefined parity, one would obtain a degenerate state of opposite parity, since

has negative parity and, by definition, commutes with the Hamiltonian in †

They are not considered as “typical” hadrons due to their special role as the (approximate) Goldstone bosons of spontaneous chiral symmetry breaking.

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the chiral limit. However, due to Coleman’s theorem such a conclusion tacitly assumes that the ground state is annihilated by the Since such a parity doubling is not observed in the spectrum one reaches the conclusion that the do not annihilate the ground state. In other words, the ground state is not invariant under the full symmetry group of the Hamiltonian, a situation which is referred to as spontaneous symmetry breaking or the Nambu-Goldstone realization of a symmetry [264, 267, 268]. As a consequence of the Goldstone theorem [168,169], each generator which commutes with the Hamiltonian but does not annihilate the ground state is associated with a massless Goldstone boson, whose properties are tightly connected with the generator in question. The eight generators have negative parity, baryon number zero, and transform as an octet under the subgroup leaving the vacuum invariant. Thus one expects the same properties of the Goldstone bosons, and the light pseudoscalar octet qualifies as candidates for these Goldstone bosons. The finite masses of the physical multiplet are interpreted as a consequence of the explicit symmetry breaking due to the finite and masses in the QCD Lagrangian [162]. Of course, the above (global) symmetry considerations were long known before the formulation of QCD, In the 1960’s they were the cornerstones of the description of low-energy interactions of hadrons in the framework of various techniques, such as the current-algebra approach in combination with the hypothesis of a partially conserved axial-vector current (PCAC) [149, 3, 306, 8], the application of phenomenological Lagrangians [319, 295, 320, 95, 85, 155], and perturbation theory about a symmetry realized in the Nambu-Goldstone mode [102,113, 240, 273]. All these methods were equivalent in the sense that they produced the same results for “soft” pions [113]. Although QCD is widely accepted as the fundamental gauge theory underlying the strong interactions, we still lack the analytical tools for ab initio descriptions of low-energy properties and processes. However, new techniques have been developed to extend the results of the current-algebra days and systematically explore corrections to the soft-pion predictions based on symmetry properties of QCD Green functions. The method is called chiral perturbation theory (ChPT) [322, 163, 164] and describes the dynamics of Goldstone bosons in the framework of an effective field theory. Although one returns to a field theory in terms of non-elementary hadrons, there is an important distinction between the early quantum field theories of the strong interactions and the new approach in the sense that, now, one is dealing with a so-called effective field theory. Such a theory allows for a perturbative treatment in terms of a momentum—as opposed to a coupling-constant—expansion. The starting point is a theorem of Weinberg stating that a perturbative description in terms of the most general effective Lagrangian containing all

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possible terms compatible with assumed symmetry principles yields the most general S matrix consistent with the fundamental principles of quantum field theory and the assumed symmetry principles [322]. The proof of the theorem relies on Lorentz invariance and the absence of anomalies [231, 199] and starts from the observation that the Ward identities satisfied by the Green functions of the symmetry currents are equivalent to an invariance of the generating functional under local transformations [231]. For that reason, one considers a locally invariant, effective Lagrangian although the symmetries of the underlying theory may originate in a global symmetry. If the Ward identities contain anomalies, they show up as a modification of the generating functional, which can explicitly be incorporated through the Wess-Zumino-Witten construction [328, 327]. All other terms of the effective Lagrangian remain locally invariant. In the present case, the assumed symmetry is the symmetry of the QCD Hamilton operator in the chiral limit, in combination with a restricted symmetry of the ground state. For center-ofmass energies below the mass, the only asymptotic states which can explicitly be produced are the Goldstone bosons. For the description of processes in this energy regime one organizes the most general, chirally invariant Lagrangian for the pseudoscalar meson octet in an expansion in terms of momenta and quark masses. Such an ansatz is naturally suggested by the fact that the interactions of Goldstone bosons are known to vanish in the zero-energy limit. Since the effective Lagrangian by construction contains an infinite number of interaction terms, one needs for any practical purpose an organization scheme allowing one to compare the importance of, say, two given diagrams. To that end, for a given diagram, one analyzes its behaviour under a linear rescaling of the external momenta, and a quadratic rescaling of the light quark masses, Applying Weinberg’s power counting scheme [322], one finds that any given diagram behaves as where is determined by the structure of the vertices and the topology of the diagram in question. For a given D, Weinberg’s formula unambiguously determines to which order in the momentum and quark mass expansion the Lagrangian needs to be known. Furthermore, the number of loops is restricted to be smaller than or equal to D/2 – 1, i.e., Weinberg’s power counting establishes a relation between the momentum expansion and the loop expansion.‡ Effective field theories are not renormalizable in the usual sense. However, this is no longer regarded as a serious problem, since by means of Weinberg’s counting scheme the infinities arising from loops may be identified order by ‡

The counting refers to ordinary chiral perturbation theory in the mesonic sector, where D is an even number.

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order in the momentum expansion and then absorbed in a renormalization of the coefficients of the most general Lagrangian. Thus, in any arbitrary order the results are finite. Of course, there is a price to pay: the rapid increase in the number of possible terms as the order increases. Practical applications will hence be restricted to low orders. The lowest-order mesonic Lagrangian, is given by the nonlinear model coupled to external fields [163, 164]. It contains two free parameters: the pion-decay constant and the scalar quark condensate, both in the chiral limit. The specific values are not determined by chiral symmetry and must, ultimately, be explained from QCD dynamics. When calculating processes in the phenomenological approximation to i.e., considering only tree-level diagrams, one reproduces the results of current algebra [322]. Since tree-level diagrams involving vertices derived from a Hermitian Lagrangian are always real, one has to go beyond the tree level in order not to violate the unitarity of the S matrix. A calculation of one-loop diagrams with on the one hand, leads to infinities which are not of the original type, but also contributes to a perturbative restoration of unitarity. Due to Weinberg’s power counting, the divergent terms are of order and can thus be compensated by means of a renormalization of the most general Lagrangian at

The most general, effective Lagrangian at

was first constructed by

Gasser and Leutwyler [164] and contains 10 physical low-energy constants as well as two additional terms containing only external fields. Out of the ten physically relevant structures, eight are required for the renormalization of the infinities due to the one-loop diagrams involving The finite parts of the constants represent free parameters, reflecting our ignorance regarding the underlying theory, namely QCD, in this order of the momentum expansion. These parameters may be fixed phenomenologically by comparison with experimental data [163, 164, 61]. There are also theoretical approaches for estimating the low-energy constants in the framework of QCD-inspired models [124, 125, 116, 58], meson-resonance saturation [119, 120, 108, 216, 235] and lattice QCD [252, 170]. Without external fields (i.e., pure QCD) or including electromagnetic processes only, the effective Lagrangians and have an additional symmetry: they contain interaction terms involving exclusively an even number of Goldstone bosons. This property is often referred to as normal or even intrinsic parity, but is obviously not a symmetry of QCD, because it would exclude reactions of the type or In Ref. [327], Witten discussed how to remove this symmetry from the effective Lagrangian and essentially re-derived the Wess-Zumino anomalous effective action which describes the chiral anomaly [328], The corresponding Lagrangian, which is of cannot be written as a standard local effective Lagrangian in terms of

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the usual chiral matrix U but can be expressed directly in terms of the Goldstone boson fields. In particular, for the above case, by construction it contains interaction terms with an odd number of Goldstone bosons (odd intrinsic parity). In contrast to the Lagrangian of Gasser and Leutwyler, the Wess-ZuminoWitten (WZW) effective action does not contain any free parameter apart from the number of colours. The excellent description of the neutral pion decay for is regarded as a key evidence for the existence of three colour degrees of freedom. Chiral perturbation theory to has become a well-established method for describing the low-energy interactions of the pseudoscalar octet. For an overview of its many successful applications the interested reader is referred to Refs. [230, 59, 253, 232, 55, 44, 286, 274, 118, 246, 49, 275, 83, 52]. In general, due to the relatively large mass of the quark, the convergence in the SU(3) sector is somewhat slower as compared with the SU(2) version. Nevertheless, ChPT in the SU(3) sector has significantly contributed to our understanding of previously open questions. A prime example is the decay rate of which current algebra predicts to be much too small. In Ref. [166] it was shown that one-loop corrections substantially increase the theoretical value and remove the previous discrepancy between theory and experiment. For obvious reasons, the question of convergence of the method is of utmost importance. The so-called chiral symmetry breaking scale is the dimensional parameter which characterizes the convergence of the momentum expansion [254, 153]. A “naive” dimensional analysis of loop diagrams suggests that this scale is given by where denotes the pion-decay constant in the chiral limit and the factor originates from a geometric factor in the calculation of loop diagrams in four dimensions. A second dimensional scale is provided by the masses of the lightest excitations which have been “integrated out” as explicit dynamical degrees of freedom of the theory—in the present case, typically the lightest vector mesons. In a phenomenological approach the exchange of such particles leads to a propagator of the type where the expansion only converges for The corresponding scale is approximately of the same size as Assuming a reasonable behaviour of the coefficients of the momentum expansion leads to the expectation that ChPT converges for center-of-mass energies sufficiently below the mass. Of course, the validity of such a statement depends on the specific process under consideration and the quantum numbers of the intermediate states. Clearly, for a given process, it would be desirable to have an idea about the size of the next-to-leading-order corrections. In the odd-intrinsic-parity sector such a calculation is at least of order because the WZW action itself is already of order Thus, according to Weinberg’s power counting,

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one-loop diagrams involving exactly one WZW vertex and an arbitrary number of vertices result in corrections of Several authors have shown that quantum corrections to the Wess-Zumino-Witten classical action do not renormalize the coefficient of the Wess-Zumino-Witten term [114, 200, 57, 1, 115, 68]. Furthermore, the one-loop counter terms lead to conventional chirally invariant structures at [114, 200, 57, 1, 115, 68]. There have been several attempts to construct the most general odd-intrinsic-parity Lagrangian at and only recently two independent calculations have found the same number of 23 independent structures in the SU(3) sector [117, 68]. For an overview of the application of ChPT to anomalous processes, the interested reader is referred to Ref. [59]. In general, next-to-leading-order corrections to processes in the evenintrinsic-parity sector are of However, there are also processes which receive their leading-order contributions at In particular, the reactions [261, 105, 32, 217, 25, 34] and [16, 25, 220, 34, 205] have received considerable attention, because the predictions at [26, 107] and [16], respectively, were in disagreement with experimental results ([248] and [175], respectively). In the case of loop corrections at lead to a considerably improved description, with the result only little sensitive to the tree-level diagrams at [32]. The opposite picture emerges for the decay where the tree-level diagrams at play an important role. A second class of calculations includes processes which already receive contributions at such as scattering [63] or [82], Here, calculations may be viewed as precision tests of ChPT. The first process is of fundamental importance because it provides information on the mechanism of spontaneous symmetry breaking in QCD [63]. The second reaction is of particular interest because an old current-algebra low-energy theorem [303] relates the electromagnetic polarizabilities and of the charged pionat to radiative pion decay Corrections at were shown to be 12% and 24% of the values for and respectively [82]. On the other hand, experimental results for the polarizabilities scatter substantially and still have large uncertainties (see, e.g., Ref. [309]) and new experimental data are clearly needed to test the accuracy of the chiral predictions. In the SU(3) sector, the first construction of the most general even-intrinsic parity Lagrangian at was performed in Ref. [ 141 ]. Although it was later shown that the original list of terms contained redundant structures [66], even the final number of 90 + 4 free parameters is very large, such that, in contrast to the Lagrangian of Gasser and Leutwyler, it seems unlikely that all parameters can be fixed through comparison with experimental data. However, chiral symmetry relates different processes to each other,

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such that groups of interaction terms may be connected with each other and through comparison with experiment the consistency conditions of chiral symmetry may be tested. Furthermore, the same theoretical methods which have been applied to predict the coefficients of may be extended to the next order [33] which, however, involves much more work. Chiral perturbation theory has proven to be highly successful in the mesonic sector and, for obvious reasons, one would like to have a generalization including the interaction of Goldstone bosons with baryons. The group-theoretical foundations for a nonlinear realization of chiral symmetry were developed in Refs. [320, 95, 85], which also included the coupling of Goldstone bosons to other isospin or, for the more general case, SU(N)-flavour multiplets. Numerous low-energy theorems involving the pion-nucleon interaction and its SU(3) extension were derived in the 1960’s by use of current-algebra methods and PCAC. However, a systematic study of chiral corrections to the low-energy theorems has only become possible when the methods of mesonic ChPT were extended to processes with one external nucleon line [144]. The situation turned out to be more involved than in the pure mesonic sector because the loops have a more complicated structure due to the nucleon mass which, in contrast to the Goldstone boson masses, does not vanish in the chiral limit. This introduces a third scale into the problem beyond the pion decay constant and the scalar quark condensate. In particular, it was shown that the relativistic formulation, at first sight, does not provide such a simple connection between the chiral expansion and the loop expansion as in the mesonic sector [144], i.e., higher-order loop diagrams also contribute to lower orders in the chiral expansion of a physical quantity. This observation was taken as evidence for a breakdown of power counting in the relativistic formulation. Subsequently, techniques borrowed from heavy-quark physics were applied to the baryon sector [206, 37], providing a heavy-baryon formulation of ChPT (HBChPT), where the Lagrangian is not only expanded in the number of derivatives and quark masses but also in powers of inverse nucleon masses. The technique is very similar to the Foldy-Wouthuysen method [143], There have been many successful applications of HBChPT to “traditional” current-algebra processes such as pion photoproduction [36, 46] and radiative pion capture [131], pion electroproduction [38, 42, 43, 51], pion-nucleon scattering [45, 256, 132, 137], to name just a few (for an extensive overview, see Ref. [44]). In all these cases, ChPT has allowed one to either systematically calculate corrections to the old currentalgebra results or to obtain new predictions which are beyond the scope of the old techniques. Other applications include the calculation of static properties such as masses [204, 40, 238, 73, 250] and various form factors of baryons [38, 47, 130, 225]. The role of the pionic degrees of freedom has been ex-

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tensively discussed for real Compton scattering off the nucleon in terms of the electromagnetic polarizabilities [39, 41, 152, 251, 226, 156]. The new frontier of virtual Compton scattering off the nucleon [179, 111, 288] has also been addressed in the framework of ChPT [187, 188, 191, 242]. As in the mesonic sector, the most general chiral Lagrangian in the single-baryon sector is needed which, due to the spin degree of freedom, is more complicated [144, 224, 121, 133]. In the baryonic sector, the resonance plays a prominent role because its excitation energy is only about two times the pion mass and its (almost) 100% branching ratio to the decay mode In Ref. [189], the formalism of the so-called small scale expansion was developed, which also treats the nucleon-delta mass splitting as a “small” quantity like the pion mass. Subsequently, the formalism was applied to Compton scattering [190], baryon form factors [50], the transition [151] and virtual Compton scattering [191]. While the heavy-baryon formulation provided a useful low-energy expansion scheme, it was realized in the context of the isovector spectral function entering the calculation of the nucleon electromagnetic form factor that the corresponding perturbation series fails to converge in part of the low-energy region [47]. Various methods have been suggested to generate a power counting which is also valid for the relativistic approach and which respects the singularity structure of Green functions [302, 126, 71, 158, 146, 241, 237]. The so-called “infrared regularization” of Ref. [71 ] decomposes one-loop diagrams into singular and regular parts. The singular parts satisfy power counting, whereas the regular parts can be absorbed into local counter terms of the Lagrangian. This technique solves the power counting problem of relativistic baryon chiral perturbation theory at the one-loop level and has already been applied to the calculation of baryon masses in SU(3) ChPT [127], of form factors [214, 331, ?], pion-nucleon scattering [72] as well as the generalized Gerasimov-Drell-Hearn sum rule [53]. At present, the procedure has not yet been generalized to higher-order loop diagrams. In Ref. [146] another approach, based on choosing appropriate renormalization conditions, was proposed, leading to the correct analyticity structure and a consistent power counting, which can also be extended to higher loops. Finally, the techniques of effective field theory have also been applied to the nucleon-nucleon interaction (see, e.g., Refs. [323, 272, 209, 222, 123, 29, 135]). Clearly this is a very important topic in its own right but is beyond the scope of the present work.

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QCD AND CHIRAL SYMMETRY

Chiral perturbation theory (ChPT) provides a systematic framework for investigating strong-interaction processes at low energies, as opposed to a perturbative treatment of quantum chromodynamics (QCD) at high momentum transfers in terms of the “running coupling constant.” The basis of ChPT is the global symmetry of the QCD Lagrangian in the limit of massless and quarks. This symmetry is assumed to be spontaneously broken down to giving rise to eight massless Goldstone bosons. In this chapter we will describe in detail one of the foundations of ChPT, namely the symmetries of QCD and their consequences in terms of QCD Green functions.

2.1.

Some Remarks on SU(3)

The group SU(3) plays an important role in the context of strong interactions, because on the one hand it is the gauge group of QCD and, on the other hand, flavour SU(3) is approximately realized as a global symmetry of the hadron spectrum (Eightfold Way [265, 147, 167]), so that the observed (low-mass) hadrons can be organized in approximately degenerate multiplets fitting the dimensionalities of irreducible representations of SU(3). Finally, as will be discussed later in this chapter, the direct product is the chiral-symmetry group of QCD for vanishing and masses. Thus, it is appropriate to first recall a few basic properties of SU(3) and its Lie algebra su(3) [79, 271, 207]. The group SU(3) is defined as the set of all unitary, unimodular, 3 x 3 matrices U, i.e., U †U = 1,§ and det(U) = 1. In mathematical terms, SU(3) is an eight-parameter, simply connected, compact Lie group. This implies that any group element can be parameterized by a set of eight independent real parameters varying over a continuous range. The Lie-group property refers to the fact that the group multiplication of two elements and is expressed in terms of eight analytic functions i.e., where It is simply connected because every element can be connected to the identity by a continuous path in the parameter space and compactness requires the parameters to be confined in a finite volume. Finally, for compact Lie groups, every finite-dimensional representation is equivalent to a unitary one and can be decomposed into a direct sum of irreducible representations (Clebsch-Gordan series). § In this report we often adopt the convention that 1 stands for the unit matrix in dimensions. It should be clear from the respective context which dimensionality actually applies.

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Elements of SU(3) are conveniently written in terms of the exponential representation¶

with real numbers, and where the eight linearly independent matrices are the so-called Gell-Mann matrices, satisfying

An explicit representation of the Gell-Mann matrices is given by

The set constitutes a basis of the Lie algebra su(3) of SU(3), i.e., the set of all complex traceless skew Hermitian 3×3 matrices. The Lie product is then defined in terms of ordinary matrix multiplication as the commutator of two elements of su(3). Such a definition naturally satisfies the Lie properties of anti-commutativity as well as the Jacobi identity

¶

In our notation, the indices denoting group parameters and generators will appear as subscripts or superscripts depending on what is notationally convenient. We do not distinguish between upper and lower indices, i.e., we abandon the methods of tensor analysis.

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In accordance with Eqs. (2.1) and (2.2), elements of su(3) can be interpreted as tangent vectors in the identity of SU(3). The structure of the Lie group is encoded in the commutation relations of the Gell-Mann matrices,

where the totally antisymmetric real structure constants Eq. (2.4) as

are obtained from

The independent non-vanishing values are explicitly summarized in the scheme of Table 2.1. Roughly speaking, these structure constants are a measure of the non-commutativity of the group SU(3).

The anticommutation relations read

where the totally symmetric

are given by

and are summarized in Table 2.2. Clearly, the anticommutator of two GellMann matrices is not necessarily a Gell-Mann matrix. For example, the square of a (nontrivial) skew-Hermitian matrix is not skew Hermitian. Moreover, it is convenient to introduce as a ninth matrix

such that Eqs, (2.3) and (2.4) are still satisfied by the nine matrices In particular, the set constitutes a basis of the Lie algebra u(3) of U(3), i.e., the set of all complex skew Hermitian 3×3 matrices. Finally, an arbitrary 3×3 matrix M can then be written as

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where

2.2.

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are complex numbers given by

The QCD Lagrangian

The gauge principle has proven to be a tremendously successful method in elementary particle physics to generate interactions between matter fields through the exchange of massless gauge bosons (for a detailed account see, e.g., [7, 271]). The best-known example is, of course, quantum electrodynamics (QED) which is obtained from promoting the global U(l) symmetry of the Lagrangian describing a free electron,||

to a local symmetry. In this process the parameter describing an element of U(l) is allowed to vary smoothly in space-time, which is referred to as gauging the U(l) group. To keep the invariance of the Lagrangian under local transformations one introduces a four-potential into the theory which transforms under the gauge transformation The method is referred to as gauging the Lagrangian with respect to U(l): where

** The covariant derivative of

is defined such that under a so-called gauge transformation of the second kind

||

We use the standard representation for the Dirac matrices (see, e.g., Ref. [27]). ** We use natural units, i.e., and

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it transforms in the same way as

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itself:

In Eq. (2.15), the term containing the squared field strength makes the gauge potential a dynamical degree of freedom as opposed to a pure external field. A mass term is not included since it would violate gauge invariance and thus the gauge principle requires massless gauge bosons.†† In the present case we identify the with the electromagnetic four-potential and with the field strength tensor containing the electric and magnetic fields. The gauge principle has (naturally) generated the interaction of the electromagnetic field with matter. If the underlying gauge group is non-Abelian, the gauge principle associates an independent gauge field with each independent continuous parameter of the gauge group. QCD is the gauge theory of the strong interactions [181, 321, 140] with colour SU(3) as the underlying gauge group.‡‡ The matter fields of QCD are the so-called quarks which are spin-1/2 fermions, with six different flavours in addition to their three possible colours (see Table 2.3). Since quarks have not been observed as asymptotically free states, the meaning of quark masses and their numerical values are tightly connected with the method by which they are extracted from hadronic properties (see Ref. [247] for a thorough discussion). Regarding the so-called current-quark-mass values of the light quarks, one should view the quark mass terms merely as symmetry breaking parameters with their magnitude providing a measure for the extent to which chiral symmetry is broken [296]. For example, ratios of the light quark masses can be inferred from the masses of the light pseudoscalar octet (see Ref. [233]). Comparing the proton mass, with the sum of two up and one down current-quark masses (see Table 2.3),

shows that an interpretation of the proton mass in terms of current-quark mass parameters must be very different from, say, the situation in the hydrogen atom, where the mass is essentially given by the sum of the electron and proton masses, corrected by a small amount of binding energy. The QCD Lagrangian obtained from the gauge principle reads [260, 10]

††

Masses of gauge fields can be induced through a spontaneous breakdown of the gauge symmetry. ‡‡ Historically, the colour degree of freedom was introduced into the quark model to account for the Pauli principle in the description of baryons as three-quark states [171, 193].

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For each quark flavour the quark field consists of a colour triplet (subscripts and standing for “red,” “green,” and “blue”),

which transforms under a gauge transformation described by the set of parameters according to*

Technically speaking, each quark field transforms according to the fundamental representation of colour SU(3). Because SU(3) is an eight-parameter group, the covariant derivative of Eq. (2.19) contains eight independent gauge potentials

We note that the interaction between quarks and gluons is independent of the quark flavours. Demanding gauge invariance of imposes the following * For the sake of clarity, the Gell-Mann matrices contain a superscript C, indicating the action in colour space.

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transformation property of the gauge fields

Again, with this requirement the covariant derivative transforms as i.e., Under a gauge transformation of the first kind, i.e., a global SU(3) transformation, the second term on the right-hand side of Eq. (2.23) would vanish and the gauge fields would transform according to the adjoint representation. So far we have only considered the matter-field part of including its interaction with the gauge fields. Equation (2.19) also contains the generalization of the field strength tensor to the non-Abelian case,

with the SU(3) structure constants given in Table 2.1 and a summation over repeated indices implied. Given Eq. (2.23) the field strength tensor transforms under SU(3) as

Using Eq. (2.4) the purely gluonic part of

can be written as

which, using the cyclic property of traces, Tr(AB) = Tr(BA), together with UU † = 1, is easily seen to be invariant under the transformation of Eq. (2.25). In contradistinction to the Abelian case of QED, the squared field strength tensor gives rise to gauge-field self interactions involving vertices with three and four gauge fields of strength and respectively. Such interaction terms are characteristic of non-Abelian gauge theories and make them much more complicated than Abelian theories. From the point of view of gauge invariance the strong-interaction Lagrangian could also involve a term of the type

where

denotes the totally antisymmetric Levi-Civita tensor.

†

is an even permutation of {0,1,2,3} is an odd permutation of {0,1,2,3}

†

The so-

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called term of Eq. (2.26) implies an explicit P and CP violation of the strong interactions which, for example, would give rise to an electric dipole moment of the neutron (for an upper limit, see Ref. [186]). The present empirical information indicates that the term is small and, in the following, we will omit Eq. (2.26) from our discussion and refer the interested reader to Refs. [279, 77, 212].

2.3. Accidental, Global Symmetries of

2.3.1. Light and Heavy Quarks The six quark flavours are commonly divided into the three light quarks and and the three heavy flavours and

where the scale of 1 GeV is associated with the masses of the lightest hadrons containing light quarks, e.g., which are not Goldstone bosons resulting from spontaneous symmetry breaking. The scale associated with spon taneous symmetry breaking, is of the same order of magnitude [273, 254, 153]. The masses of the lightest meson and baryon containing a charmed quark, and are (1869.4 ± 0.5) MeV and (2284.9 ± 0.6) MeV, respectively [175]. The threshold center-of-mass energy to produce, say, a pair in collisions is approximately 3.74 GeV, and thus way beyond the low-energy regime which we are interested in. In the following, we will approximate the full QCD Lagrangian by its light-flavour version, i.e., we will ignore effects due to (virtual) heavy quark-antiquark pairs In particular, Eq. (2.18) suggests that the Lagrangian containing only the lightflavour quarks in the so-called chiral limit might be a good starting point in the discussion of low-energy QCD:

We repeat that the covariant derivative only, but is independent of flavour.

acts on colour and Dirac indices

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2.3.2. Left-Handed and Right-Handed Quark Fields In order to fully exhibit the global symmetries of Eq. (2.28), we consider ‡ the chirality matrix and introduce projection operators

where the indices R and L refer to right-handed and left-handed, respectively, as will become more clear below. Obviously, the 4 x 4 matrices and satisfy a completeness relation,

are idempotent, i.e., and respect the orthogonality relations

The combined properties of Eqs. (2.30) - (2.32) guarantee that and are indeed projection operators which project from the Dirac field variable to its chiral components and

We recall in this context that a chiral (field) variable is one which under parity is transformed into neither the original variable nor its negative [110].§ Under parity, the quark field is transformed into its parity conjugate,

and hence

¶ and similarly for The terminology right-handed and left-handed fields can easily be visualized in terms of the solution to the free Dirac equation. For that purpose, let us ‡ Unless stated otherwise, we use the convention of Ref. [27]. § In case of fields, a transformation of the argument is implied. ¶ Note that in the above sense, also is a chiral variable. However, the assignment of handedness does not have such an intuitive meaning as in the case of and

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consider an extreme relativistic positive-energy solution with three-momentum

where we assume that the spin in the rest frame is either parallel or antiparallel to the direction of momentum

In the standard representation of Dirac matrices we find

such that

and similarly

In the extreme relativistic limit (or better, in the zero-mass limit), the operators and project to the positive and negative helicity eigenstates, i.e., in this limit chirality equals helicity. Our goal is to analyze the symmetry of the QCD Lagrangian with respect to independent global transformations of the left- and right-handed fields. In order to decompose the 16 quadratic forms into their respective projections to right- and left-handed fields, we make use of [145]

where and Equation (2.34) is easily proven by inserting the completeness relation of Eq. (2.30) both to the left and the right of

and by noting and relations of Eq. (2.32) we then obtain ||

Here we adopt a covariant normalization of the spinors,

Together with the orthogonality

etc.

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and similarly

We stress that the validity of Eq. (2.34) is general and does not refer to “massless” quark fields. We now apply Eq. (2.34) to the term containing the contraction of the covariant derivative with This quadratic quark form decouples into the sum of two terms which connect only left-handed with left-handed and right-handed with right-handed quark fields. The QCD Lagrangian in the chiral limit can then be written as

Due to the flavour independence of the covariant derivative under

is invariant

where and are independent unitary 3×3 matrices. Note that the GellMann matrices act in flavour space. is said to have a classical global symmetry. Applying Noether’s theorem (see, for example, [192, 8]) from such an invariance one would expect a total of 2 × (8 + 1) = 18 conserved currents.

2.3.3.

Noether’s Theorem

In order to identify the conserved currents associated with this invariance, we briefly recall the method of Ref. [161] and consider the variation of Eq. (2.35) under a local infinitesimal transformation.** For simplicity we consider only internal symmetries. To that end we start with a Lagrangian depending on independent fields and their first partial derivatives,

** By exponentiating elements of the Lie algebra u(N) any element of U(N) can be obtained.

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from which one obtains

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equations of motion:

For each of the generators of infinitesimal transformations representing the underlying symmetry group, we consider a local infinitesimal transformation of the fields [161],††

and obtain, neglecting terms of order

as the variation of the Lagrangian,

According to this equation we define for each infinitesimal transformation a four-current density as

By calculating the divergence

of Eq. (2.41)

where we made use of the equations of motion, Eq. (2.38), we explicitly verify the consistency with the definition of according to Eq. (2.40). From Eq. (2.40) it is straightforward to obtain the four-currents as well as their divergences as

††

Note that the transformation need not be realized linearly on the fields.

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For a conserved current,

the charge

is time independent, i.e., a constant of the motion, which is shown in the standard fashion by applying the divergence theorem for an infinite volume with appropriate boundary conditions for So far we have discussed Noether’s theorem on the classical level, implying that the charges can have any continuous real value. However, we also need to discuss the implications of a transition to a quantum theory. After canonical quantization, the fields and their conjugate momenta are considered as linear operators acting on a Hilbert space which, in the Heisenberg picture, are subject to the equal-time commutation relations

As a special case of Eq. (2.39) let us consider infinitesimal transformations which are linear in the fields,

where the are constants generating a mixing of the fields. From Eq. (2.41) we then obtain‡‡

where and are now operators. In order to interpret the charge operators let us make use of the equal-time commutation relations, Eqs. (2.45), and calculate their commutators with the field operators,

‡‡

Normal ordering symbols are suppressed.

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Note that we did not require the charge operators to be time independent. On the other hand, for the transformation behaviour of the Hilbert space associated with a global infinitesimal transformation, we make an ansatz in terms of an infinitesimal unitary transformation*

with Hermitian operators

Demanding

in combination with Eq. (2,46) yields the condition

By comparing the terms linear in

on both sides,

we see that the infinitesimal generators acting on the states of Hilbert space which are associated with the transformation of the fields are identical with the charge operators of Eq. (2.48). Finally, evaluating the commutation relations for the case of several generators,

we find the right-hand side of Eq. (2.53) to be again proportional to a charge operator, if i.e., in that case the charge operators

form a Lie algebra

with structure constants The quantization of the charges (as opposed to continuous values in the classical case) can be understood in analogy to the algebraic construction of the angular momentum eigenvalues in quantum mechanics starting from the su(2) algebra. Of course, for conserved currents, the * We have chosen to have the fields (field operators) rotate actively and thus must transform the states of Hilbert space in the opposite direction.

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charge operators are time independent, i.e., they commute with the Hamilton operator of the system. From now on we assume the validity of Eq. (2.54) and interpret the constants as the entries in the row and column of an matrix

Because of Eq. (2.54), these matrices form an a Lie algebra, The infinitesimal, linear transformations of the fields in a compact form,

representation of

may then be written

In general, through an appropriate unitary transformation, the matrices may be decomposed into their irreducible components, i.e., brought into blockdiagonal form, such that only fields belonging to the same multiplet transform into each other under the symmetry group.

2.3.4.

Global Symmetry Currents of the Light Quark Sector

The method of Ref. [161] can now easily be applied to the QCD Lagrangian by calculating the variation under the infinitesimal, local form of Eqs. (2.36),

from which, by virtue of Eqs. (2.42) and (2.43), one obtains the currents associated with the transformations of the left-handed or right-handed quarks

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The eight currents transform under as an (8, 1) multiplet, i.e., as octet and singlet under transformations of the left- and righthanded fields, respectively. Similarly, the right-handed currents transform as a (1, 8) multiplet under Instead of these chiral currents one often uses linear combinations,

transforming under parity as vector and axial-vector current densities, respectively,

From Eqs. (2.42) and (2.43) one also obtains a conserved singlet vector current resulting from a transformation of all left-handed and right-handed quark fields by the same phase,

The singlet axial-vector current,

originates from a transformation of all left-handed quark fields with one phase and all right-handed with the opposite phase. However, such a singlet axialvector current is only conserved on the classical level. This symmetry is not preserved by quantization and there will be extra terms, referred to as anomalies [5, 2, 24, 70, 6], resulting in†

where the factor of 3 originates from the number of flavours. †

In the large (number of colours) limit of Ref. [195] the singlet axial-vector current is conserved, because the strong coupling constant behaves as

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2.3.5.

The Chiral Algebra

The invariance of under global transformations implies that also the QCD Hamilton operator in the chiral limit, exhibits a global symmetry. As usual, the “charge operators” are defined as the space integrals of the charge densities,

For conserved symmetry currents, these operators are time independent, i.e., they commute with the Hamiltonian,

The commutation relations of the charge operators with each other are obtained by using the equal-time commutation relations of the quark fields in the Heisenberg picture,

where and are Dirac indices and and flavour indices, respectively.‡ The equal-time commutator of two quadratic quark forms is of the type

where and tively. Using

‡

are 4 × 4 Dirac matrices and 3×3 flavour matrices, respec-

Strictly speaking, we should also include the colour indices. However, since we are only discussing colour-neutral quadratic forms a summation over such indices is always implied, with the net effect that one can completely omit them from the discussion.

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we express the commutator of Fermi fields in terms of anticommutators and make use of the equal-time commutation relations of Eqs. (2.70) – (2.72) to obtain

With this result Eq. (2.73) reads

After inserting appropriate projectors Eq. (2.75) is easily applied to the charge operators of Eqs. (2.66), (2.67), and (2.68), showing that these operators indeed satisfy the commutation relations corresponding to the Lie algebra of

It should be stressed that, even without being able to explicitly solve the equation of motion of the quark fields entering the charge operators of Eqs. (2.66) (2.68), we know from the equal-time commutation relations and the symmetry of the Lagrangian that these charge operators are the generators of infinitesimal transformations of the Hilbert space associated with Furthermore, their commutation relations with a given operator specify the transformation behaviour of the operator in question under the group

2.3.6.

Chiral Symmetry Breaking Due to Quark Masses

The finite and masses in the QCD Lagrangian result in explicit divergences of the symmetry currents. As a consequence, the charge operators are, in general, no longer time independent. However, as first pointed out by Gell-Mann, the equal-time-commutation relations still play an important role even if the symmetry is explicitly broken [147]. As will be discussed later on in more detail, the symmetry currents will give rise to chiral Ward identities relating various QCD Green functions to each other. Equation (2.43) allows one to discuss the divergences in the presence of quark masses. To that

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end, let us consider the quark-mass matrix of the three light quarks and project it on the nine matrices of Eq. (2.13),

In particular, applying Eq. (2.34) we see that the quark mass term mixes leftand right-handed fields,

The symmetry-breaking term transforms under of a (3, 3*) + (3*, 3) representation, i.e.,

as a member

where Such symmetry-breaking patterns were already discussed in the pre-QCD era in Refs. [180, 150]. From one obtains as the variation under the transformations of Eqs. (2.36),

which results in the following divergences,

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The anomaly has not yet been considered. Applying Eq. (2.34) to the case of the vector currents and inserting projection operators as in the derivation of Eq. (2.64) for the axial-vector current, the corresponding divergences read

where the axial anomaly has also been taken into account. We are now in the position to summarize the various (approximate) symmetries of the strong interactions in combination with the corresponding currents and their divergences. In the limit of massless quarks, the sixteen currents and or, alternatively, and are conserved. The same is true for the singlet vector current whereas the singlet axial-vector current has an anomaly. For any value of quark masses, the individual flavour currents and are always conserved in strong interactions reflecting the flavour independence of the strong coupling and the diagonality of the quark mass matrix. Of course, the singlet vector current being the sum of the three flavour currents, is always conserved. In addition to the anomaly, the singlet axial-vector current has an explicit divergence due to the quark masses. For equal quark masses, the eight vector currents are conserved, because Such a scenario is the origin of the SU(3) symmetry originally proposed by Gell-Mann and Ne’eman [167]. The eight axial currents are not conserved. The divergences of the octet axial-vector currents of Eq. (2.84) are proportional to pseudoscalar quadratic forms. This can be interpreted as the microscopic origin of the PCAC relation (partially conserved axial-vector current) which states that the divergences of the axial-vector currents are proportional to renormalized field operators representing the lowest lying pseudoscalar octet (for a comprehensive discussion of the meaning of PCAC see Refs. [149, 3, 306, 8]).

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Various symmetry-breaking patterns are discussed in great detail in Ref. [273].

2.4.

Green Functions and Chiral Ward Identities

2.4.1. Chiral Green Functions For conserved currents, the spatial integrals of the charge densities are time independent, i.e., in a quantized theory the corresponding charge operators commute with the Hamilton operator. These operators are generators of infinitesimal transformations on the Hilbert space of the theory. The mass eigenstates should organize themselves in degenerate multiplets with dimensionalities corresponding to irreducible representations of the Lie group in question.§ Which irreducible representations ultimately appear, and what the actual energy eigenvalues are, is determined by the dynamics of the Hamiltonian. For example, SU(2) isospin symmetry of the strong interactions reflects itself in degenerate SU(2) multiplets such as the nucleon doublet, the pion triplet and so on. Ultimately, the actual masses of the nucleon and the pion should follow from QCD (for a prediction of hadron masses in lattice QCD, see e.g., Refs. [84, 9]). It is also well-known that symmetries imply relations between S-matrix elements. For example, applying the Wigner-Eckart theorem to pion-nucleon scattering, assuming the strong-interaction Hamiltonian to be an isoscalar, it is sufficient to consider two isospin amplitudes describing transitions between states of total isospin I = 1/2 or I = 3/2 (see, for example, [128]). All the dynamical information is contained in these isospin amplitudes and the results for physical processes can be expressed in terms of these amplitudes together with geometrical coefficients, namely, the Clebsch-Gordan coefficients. In quantum field theory, the objects of interest are the Green functions which are vacuum expectation values of time-ordered products.¶ Pictorially, these Green functions can be understood as vertices and are related to physical scattering amplitudes through the Lehmann-Symanzik-Zimmermann(LSZ) reduction formalism [228]. Symmetries provide strong constraints not only for scattering amplitudes, i.e., their transformation behaviour, but, more generally speaking, also for Green functions and, in particular, among Green functions. The famous example in this context is, of course, the Ward identity of QED §

¶

Here we assume that the dynamical system described by the Hamiltonian does not lead to a spontaneous symmetry breakdown. We will come back to this point later. Later on, we will also refer to matrix elements of time-ordered products between states other than the vacuum as Green functions.

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associated with U(1) gauge invariance [315],

which relates the electromagnetic vertex of an electron at zero momentum transfer, to the electron self energy, Such symmetry relations can be extended to non-vanishing momentum transfer and also to more complicated groups and are referred to as WardFradkin-Takahashi identities [315, 139, 301] (or Ward identities for short). Furthermore, even if a symmetry is broken, i.e., the infinitesimal generators are time dependent, conditions related to the symmetry breaking terms can still be obtained using equal-time commutation relations [147]. At first, we are interested in time-ordered products of colour-neutral, Hermitian quadratic forms involving the light quark fields evaluated between the vacuum of QCD. Using the LSZ reduction formalism [228, 201] such Green functions can be related to physical processes involving mesons as well as their interactions with the electroweak gauge fields of the Standard Model. The interpretation depends on the transformation properties and quantum numbers of the quadratic forms, determining for which mesons they may serve as an interpolating field. In addition to the vector and axial-vector currents of Eqs. (2.59), (2.60), and (2.63) we want to investigate scalar and pseudoscalar densities, ||

which enter, for example, in Eqs. (2.84) as the divergences of the vector and axial-vector currents for nonzero quark masses. Whenever it is more convenient, we will also use

instead of and Later on, we will also consider similar time-ordered products evaluated between a single nucleon in the initial and final states in addition to the vacuum Green functions. This will allow us to discuss properties of the nucleon as well as dynamical processes involving a single nucleon. Generally speaking, a chiral Ward identity relates the divergence of a Green function containing at least one factor of [see Eqs. (2.59) and (2.60)] to some linear combination of other Green functions. The terminology ||

The singlet axial-vector current involves an anomaly such that the Green functions involving this current operator are related to Green functions containing the contraction of the gluon fieldstrength tensor with its dual.

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chiral refers to the underlying group. To make this statement more precise, let us consider as a simple example the two-point Green function involving an axial-vector current and a pseudoscalar density,**

and evaluate the divergence

where we made use of This simple example already shows the main features of (chiral) Ward identities. From the differentiation of the theta functions one obtains equal-time commutators between a charge density and the remaining quadratic forms. The results of such commutators are a reflection of the underlying symmetry, as will be shown below. As a second term, one obtains the divergence of the current operator in question. If the symmetry is perfect, such terms vanish identically. For example, this is always true for the electromagnetic case with its U(1) symmetry. If the symmetry is only approximate, an additional term involving the symmetry breaking appears. For a soft breaking such a divergence can be treated as a perturbation. Via induction, the generalization of the above simple example to an point Green function is symbolically of the form

where

stands generically for any of the Noether currents.

** The time ordering of points products of theta functions.

gives rise to

distinct orderings, each involving

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2.4.2. The Algebra of Currents In the above example, we have seen that chiral Ward identities depend on the equal-time commutation relations of the charge densities of the symmetry currents with the relevant quadratic quark forms. Unfortunately, a naive application of Eq. (2.75) may lead to erroneous results. Let us illustrate this by means of a simplified example, the equal-time commutator of the time and space components of the ordinary electromagnetic current in QED. A naive use of the canonical commutation relations leads to

where we made use of the delta function to evaluate the fields at It was noticed a long time ago by Schwinger that this result cannot be true [294]. In order to see this, consider the commutator

where we made use of current conservation, one would necessarily also have

which we evaluate for

If Eq. (2.90) were true,

between the ground state,

Here, we inserted a complete set of states and made use of

Since every individual term in the sum is non-negative, one would need for any intermediate state which is obviously unphysical. The solution is that the starting point, Eq. (2.90), is not true. The corrected version of Eq. (2.90) picks up an additional, so-called Schwinger term containing a derivative of the delta function.

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Quite generally, by evaluating commutation relations with the component of the energy-momentum tensor one can show that the equal-time commutation relation between a charge density and a current density can be determined up to one derivative of the function [202],

where the Schwinger term possesses the symmetry

and

denote the structure constants of the group in question. However, in our above derivation of the chiral Ward identity, we also made use of the naive time-ordered product (T) as opposed to the covariant one (T*) which, typically, differ by another non-covariant term which is called a seagull. Feynman’s conjecture [202] states that there is a cancelation between Schwinger terms and seagull terms such that a Ward identity obtained by using the naive T product and by simultaneously omitting Schwinger terms ultimately yields the correct result to be satisfied by the Green function (involving the covariant T* product). Although this will not be true in general, a sufficient condition for it to happen is that the time component algebra of the full theory remains the same as the one derived canonically and does not possess a Schwinger term. For a detailed discussion, the interested reader is referred to Ref. [202]. Keeping the above discussion in mind, the complete list of equal-time commutation relations, omitting Schwinger terms, reads

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2.4.3. Two Simple Examples We now return to our specific example, namely, the divergence of Eq. (2.88). Inserting the results of Eqs. (2.84) and (2.92) one obtains

The second term on the right-hand side of Eq. (2.93) can be re-expressed using Eq. (2.80) and the anti-commutation relations of Eq. (2.11) in combination with the coefficients of Table 2.2 (no summation over a implied),

Equation (2.93) serves to illustrate two distinct features of chiral Ward identities. The first term of Eq. (2.93) originates in the algebra of currents and thus represents a consequence of the transformation properties of the quadratic quark forms entering the Green function. In general, depending on whether the appropriate equal-time commutation relation of Eq. (2.92) vanishes or not, the resulting term in the divergence of an Green function vanishes or is proportional to an Green function. In our specific example, the divergence of the Green function involving the axial-vector current and the pseudoscalar density is related to the so-called scalar quark condensate which will be discussed in more detail in Sect. 4.1.2. The second term of Eq. (2.93) is due to an explicit symmetry breaking resulting from the quark masses. This shows the second property of chiral Ward identities, namely, symmetry breaking terms give rise to another Green function. To summarize, chiral Ward identities incorporate both transformation properties of quadratic quark forms as well as symmetry breaking patterns.

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As another well-known and simple example, let us briefly consider, for the two-flavour case, the nucleon matrix element of the axial-vector current operator††

This matrix element serves as an illustration of chiral Ward identities which are taken between one-nucleon states instead of the vacuum. According to Eq. (2.84), the divergence of Eq. (2.94) is related to the pseudoscalar density evaluated between one-nucleon states. Of course, in the chiral limit M = 0 and the axial-vector current is conserved.

2.4.4.

QCD in the Presence of External Fields and the Generating Functional

Here, we want to consider the consequences of Eqs. (2.92) for the Green functions of QCD (in particular, at low energies). In principle, using the techniques of the last section, for each Green function one can explicitly work out the chiral Ward identity which, however, becomes more and more tedious as the number of quark quadratic forms increases. However, there exists an elegant way of formally combining all Green functions in a generating functional. The (infinite) set of all chiral Ward identities is encoded as an invariance property of that functional. To see this, one has to consider a coupling to external c-number fields such that through functional methods one can, in principle, obtain all Green functions from a generating functional. The rationale behind this approach is that, in the absence of anomalies, the Ward identities obeyed by the Green functions are equivalent to an invariance of the generating functional under a local transformation of the external fields [231]. The use of local transformations allows one to also consider divergences of Green functions. For an illustration of this statement, the reader is referred to Appendix A.1. Following the procedure of Gasser and Leutwyler [163, 164], we introduce into the Lagrangian of QCD the couplings of the nine vector currents and the eight axial-vector currents as well as the scalar and pseudoscalar quark densities to external c-number fields and

The external fields are colour-neutral, Hermitian 3 × 3 matrices, where the matrix character, with respect to the (suppressed) flavour indices and of ††

This matrix element will be dealt with in Sect. 5.3.1.

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the quark fields, is‡‡

The ordinary three flavour QCD Lagrangian is recovered by setting and in Eq. (2.95). If one defines the generating functional*

then any Green function consisting of the time-ordered product of colourneutral, Hermitian quadratic forms can be obtained from Eq. (2.97) through a functional derivative with respect to the external fields. The quark fields are operators in the Heisenberg picture and have to satisfy the equation of motion and the canonical anti-commutation relations. The actual value of the generating functional for a given configuration of external fields and reflects the dynamics generated by the QCD Lagrangian. The generating functional is related to the vacuum-to-vacuum transition amplitude in the presence of external fields [163, 164],

where the dynamics is determined by the Lagrangian of Eq. (2.95). For example,† the component of the scalar quark condensate in the chiral limit, is given by

‡‡ As in Refs. [163, 164], we omit the coupling to the singlet axial-vector current which has an

anomaly, but include a singlet vector current which is of some physical relevance in the two-flavour sector. * Many books on quantum field theory such as Refs. [201, 94, 292, 287] reserve the symbol for the generating functional of all Green functions as opposed to the argument of the exponential which denotes the generating functional of connected Green functions. † In order to obtain Green functions from the generating functional the simple rule

is extremely useful. Furthermore, the functional derivative satisfies properties similar to the ordinary differentiation, namely linearity, the product and chain rules.

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where we made use of Eq. (2.13). Note that both the quark field operators and the ground state are considered in the chiral limit, which is denoted by the subscript 0. As another example, let us consider the two-point function of the axialvector currents of Eq. (2.60) of the “real world,” i.e., for and the “true vacuum”

Requiring the total Lagrangian of Eq. (2.95) to be Hermitian and invariant under P, C, and T leads to constraints on the transformation behaviour of the external fields. In fact, it is sufficient to consider P and C, only, because T is then automatically incorporated owing to the CPT theorem. Under parity, the quark fields transform as

and the requirement of parity conservation,

leads, using the results of Table 2.4, to the following constraints for the external fields,

In Eq. (2.103) it is understood that the arguments change from

to

Similarly, under charge conjugation the quark fields transform as

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where the subscripts and are Dirac spinor indices, is the usual charge conjugation matrix in the convention of Ref. [27] and refers to flavour. Using Eq. (2.104) in combination with Table 2.5 it is straightforward to show that invariance of under charge conjugation requires the transformation properties‡

where the transposition refers to the flavour space.

Finally, we need to discuss the requirements to be met by the external fields under local transformations. In a first step, we write Eq. (2.95) in terms of the left- and right-handed quark fields. Besides the properties of Eqs. (2.30) - (2.32) we make use of the auxiliary formulae

and

to obtain

where Similarly, we rewrite the second part containing the external scalar and pseudoscalar fields, ‡

In deriving these results we need to make use of are anti-commuting field operators.

since the quark fields

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yielding for the Lagrangian of Eq. (2.95)

Equation (2.107) remains invariant under local transformations

where and are independent space-time-dependent SU(3) matrices, provided the external fields are subject to the transformations

The derivative terms in Eq. (2.109) serve the same purpose as in the construction of gauge theories, i.e., they cancel analogous terms originating from the kinetic part of the quark Lagrangian. There is another, yet, more practical aspect of the local invariance, namely: such a procedure allows one to also discuss a coupling to external gauge fields in the transition to the effective theory to be discussed later. For example, we have seen in Sect. 2.2 that a coupling of the electromagnetic field to point-like fundamental particles results from gauging a U(l) symmetry. Here, the corresponding U(l) group is to be understood as a subgroup of a local Another example deals with the interaction of the light quarks with the charged and neutral gauge bosons of the weak interactions. Let us consider both examples explicitly. The coupling of quarks to an external electromagnetic field is given by

where Q = diag(2/3, –1/3, –1/3) is the quark charge matrix:

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On the other hand, if one considers only the two-flavour version of ChPT one has to insert for the external fields

In the description of semileptonic interactions such as or neutron decay one needs the interaction of quarks with the massive charged weak bosons

where

refers to the Hermitian conjugate and

Here, denote the elements of the Cabibbo-Kobayashi-Maskawa quarkmixing matrix describing the transformation between the mass eigenstates of QCD and the weak eigenstates [175],

At lowest order in perturbation theory, the Fermi constant is related to the gauge coupling and the W mass as

Making use of

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we see that inserting Eq. (2.112) into Eq. (2.107) leads to the standard chargedcurrent weak interaction in the light quark sector,

The situation is slightly different for the neutral weak interaction. Here, the SU(3) version requires a coupling to the singlet axial-vector current which, because of the anomaly of Eq. (2.65), we have dropped from our discussion. On the other hand, in the SU(2) version the axial-vector current part is traceless and we have

where is the weak angle. With these external fields, we obtain the standard weak neutral-current interaction [175]

where we made use of

2.4.5. PCAC in the Presence of an External Electromagnetic Field Finally, the technique of coupling the QCD Lagrangian to external fields also allows us to determine the current divergences for rigid external fields, i.e., which are not simultaneously transformed. For the sake of simplicity we restrict ourselves to the SU(2) sector. (The generalization to the SU(3) case is straightforward.) If the external fields are not simultaneously transformed and one considers a global chiral transformation only, the divergences of the currents read [see Eq. (2.43)]

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As an example, let us consider the QCD Lagrangian for a finite light quark mass in combination with a coupling to an external electromagnetic field In this case the expressions for the divergence of the vector and axial-vector currents, respectively, read

where we have introduced the isovector pseudoscalar density

and is the electromagnetic field strength tensor. The third component of the axial-vector current, has an anomaly [5, 2, 24, 70, 6] which is related to the decay We emphasize the formal similarity of Eq. (2.117) to the (pre-QCD) PCAC relation obtained by Adler through the inclusion of the electromagnetic interactions with minimal electromagnetic coupling (see the Appendix of Ref. [4]).§ Since in QCD the quarks are taken as truly elementary, their interaction with an (external) electromagnetic field is of such a minimal type.

3.

SPONTANEOUS SYMMETRY BREAKING AND THE GOLDSTONE THEOREM

So far we have concentrated on the chiral symmetry of the QCD Hamiltonian and the explicit symmetry breaking through the quark masses. We have discussed the importance of chiral symmetry for the properties of Green functions with particular emphasis on the relations among different Green functions as expressed through the chiral Ward identities. Now it is time to address a second aspect which, for the low-energy structure of QCD, is equally important, namely, the concept of spontaneous symmetry breaking. A (continuous) symmetry is said to be spontaneously broken or hidden, if the ground state of the system is no longer invariant under the full symmetry group of the Hamiltonian. In this chapter we will first illustrate this by means of a discrete symmetry and then turn to the case of a spontaneously broken continuous global symmetry. § In Adler’s version, the right-hand side of Eq. (2.117) contains a renormalized field operator

creating and destroying pions instead of From a modern point of view, the combination serves as an interpolating pion field (see Sect. 4.6.2.) Furthermore, the anomaly term is not yet present in Ref. [4].

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3.1.

Degenerate Ground States

Before discussing the case of a continuous symmetry, we will first have a look at a field theory with a discrete internal symmetry. This will allow us to distinguish between two possibilities: a dynamical system with a unique ground state or a system with a finite number of distinct degenerate ground states. In particular, we will see how, for the second case, an infinitesimal perturbation selects a particular vacuum state. To that end we consider the Lagrangian of a real scalar field [153]

which is invariant under the discrete transformation sponding classical energy density reads

The corre-

where one chooses so that is bounded from below. The field which minimizes the Hamilton density must be constant and uniform since in that case the first two terms take everywhere their minimum values of zero. It must also minimize the potential since (see Fig. 3.1), from which we obtain the condition

We now distinguish two different cases: (see Fig. 3.2): In this case the potential has its minimum In the quantized theory we associate a unique ground state with this minimum. Later on, in the case of a continuous symmetry, this situation will be referred to as the Wigner-Weyl realization of the symmetry.

for

(see Fig. 3.3): Now the potential exhibits two distinct minima. (In the continuous symmetry case this will be referred to as the NambuGoldstone realization of the symmetry.) We will concentrate on the second situation, because this is the one which we would like to generalize to a continuous symmetry and which ultimately

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leads to the appearance of Goldstone bosons. In the present case, maximum for and two minima for

has a

As will be explained below, the quantized theory develops two degenerate vacua and which are distinguished through their vacuum expec¶ tation values of the field

We made use of translational invariance, and the fact that the ground state is an eigenstate of energy and momentum. We associate with the transformation a unitary operator acting on the Hilbert space of our model, with the properties

In accord with Eq. (3.4) the action of the operator given by

on the ground states is

For the moment we select one of the two expectation values and expand || the field with respect to

A short calculation yields

such that the Lagrangian in terms of the shifted dynamical variable reads

¶ The case of a quantum field theory with an infinite volume V has to be distinguished from,

say, a nonrelativistic particle in a one-dimensional potential of a shape similar to the function of Fig. 3.3. For example, in the case of a symmetric double-well potential, the solutions with positive parity have always lower energy eigenvalues than those with negative parity (see, e.g., Ref. [172]). || The field instead of is assumed to vanish at infinity.

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In terms of the new dynamical variable the symmetry is no longer manifest, i.e., it is hidden. Selecting one of the ground states has led to a spontaneous symmetry breaking which is always related to the existence of several degenerate vacua. At this stage it is not clear why the quantum mechanical ground state should be one or the other of and not a superposition of both. For example, the linear combination

is invariant under as is the original Lagrangian of Eq. (3.1). However, this superposition is not stable against any infinitesimal external perturbation which is odd in (see Fig. 3.4),

Any such perturbation will drive the ground state into the vicinity of either rather than

This can easily be seen in the framework of perturbation theory for degenerate states. Consider

such that The condition for the energy eigenvalues of the ground state, to first order in results from

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Due to the symmetry properties of Eq. (3.5), we obtain

and similarly Setting which can always be achieved by multiplication of one of the two states by an appropriate phase, one finds

resulting in

In other words, the degeneracy has been lifted and we get for the energy eigenvalues The corresponding eigenstates of zeroth order in are and respectively. We thus conclude that an arbitrarily small external perturbation which is odd with respect to will push the ground state to either or In the above discussion, we have tacitly assumed that the Hamiltonian and the field can simultaneously be diagonalized in the vacuum sector, i.e., Following Ref. [324], we will justify this assumption which will also be crucial for the continuous case to be discussed later. For an infinite volume, a general vacuum state is defined as a state with momentum eigenvalue where is a discrete eigenvalue as opposed to an eigenvalue of single- or manyparticle states for which is an element of a continuous spectrum (see Fig. 3.5). We deal with the situation of several degenerate ground states which will be denoted by etc.** and start from the identity

from which we obtain for t = 0

** For continuous symmetry groups one may have a non-countably infinite number of ground states.

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Let us consider the left-hand side,

where we inserted a complete set of states which we split into the vacuum contribution and the rest, and made use of translational invariance. We now define and assume to be reasonably behaved such that one can apply the lemma of Riemann and Lebesgue,

At this point the assumption of an infinite volume, is crucial. Repeating the argument for the right-hand side and taking the limit only the vacuum contributions survive in Eq. (3.10) and we obtain

for arbitrary ground states and In other words, the matrices and commute and can be diagonalized simultaneously. Choosing an appropriate basis, one can write

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where denotes the expectation value of in the state In the above example, the ground states and with vacuum expectation values are thus indeed orthogonal and satisfy

3.2.

Spontaneous Breakdown of a Global, Continuous, Non-Abelian Symmetry

We now extend the discussion to a system with a continuous, non-Abelian symmetry such as SO(3). To that end, we consider the Lagrangian

where with Hermitian fields The Lagrangian of Eq. (3.11) is invariant under a global “isospin” rotation,††

For the to also be Hermitian, the Hermitian must be purely imaginary and thus antisymmetric. The provide the basis of a representation of the so(3) Lie algebra and satisfy the commutation relations We will use the representation with the matrix elements given by As in Sect. 3.1, we now look for a minimum of the potential which does not depend on and find

Since can point in any direction in isospin space we now have a noncountably infinite number of degenerate vacua. In analogy to the discussion of the last section, any infinitesimal external perturbation which is not invariant under SO(3) will select a particular direction which, by an appropriate orientation of the internal coordinate frame, we denote as the 3 direction,

†† Of course, the Lagrangian is invariant under the full group O(3) which can be decomposed

into its two components: the proper rotations connected to the identity, SO(3), and the rotationreflections. For our purposes it is sufficient to discuss SO(3).

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Clearly, of Eq. (3.14) is not invariant under the full group G = SO(3) ‡‡ since rotations about the 1 and 2 axis change To be specific, if

we obtain

Note that the set of transformations which do not leave invariant does not form a group, because it does not contain the identity. On the other hand, is invariant under a subgroup H of G, namely, the rotations about the 3 axis:

In analogy to Eq. (3.6), we expand

where is a new field replacing the potential

with respect to

and obtain the new expression for

Upon inspection of the terms quadratic in the fields, one finds after spontaneous symmetry breaking two massless Goldstone bosons and one massive boson:

The model-independent feature of the above example is given by the fact that for each of the two generators and which do not annihilate the ground state one obtains a massless Goldstone boson. By means of a two-dimensional simplification (see the “Mexican hat” potential shown in Fig. 3.6) the mechanism at hand can easily be visualized. ‡‡ We say, somewhat loosely, that

finite group elements generated by become clearer later on.

and and

do not annihilate the ground state or, equivalently, do not leave the ground state invariant. This should

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Infinitesimal variations orthogonal to the circle of the minimum of the potential generate quadratic terms, i.e., “restoring forces linear in the displacement,” whereas tangential variations experience restoring forces only of higher orders. Now let us generalize the model to the case of an arbitrary compact Lie group G of order resulting in infinitesimal generators.* Once again, we start from a Lagrangian of the form [169]

where is a multiplet of scalar (or pseudoscalar) Hermitian fields. The Lagrangian and thus also are supposed to be globally invariant under G, where the infinitesimal transformations of the fields are given by

The Hermitian representation matrices are again antisymmetric and purely imaginary. We now assume that, by choosing an appropriate form of the Lagrangian generates a spontaneous symmetry breaking resulting in a ground state with a vacuum expectation value which is invariant under a continuous subgroup H of G. We expand with respect to * The restriction to compact groups allows for a complete decomposition into finite-dimensional irreducible unitary representations.

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The matrix must be symmetric and, since one is expanding around a minimum, positive semidefinite, i.e.,

In that case, all eigenvalues of are nonnegative. Making use of the invariance of under the symmetry group G,

one obtains, by comparing coefficients,

Differentiating Eq. (3.25) with respect to in the matrix equation

and using

results

Inserting the variations of Eq. (3.21) for arbitrary we conclude The solutions of Eq. (3.27) can be classified into two categories: 1.

is a representation of an element of the Lie algebra belonging to the subgroup H of G, leaving the selected ground state invariant. In that case one has

such that Eq. (3.27) is automatically satisfied without any knowledge of

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is not a representation of an element of the Lie algebra belonging to the subgroup H. In that case and is an eigenvector of with eigenvalue 0. To each such eigenvector corresponds a massless Goldstone boson. In particular, the different are linearly independent, resulting in independent Goldstone bosons. (If they were not linearly independent, there would exist a nontrivial linear combination

such that T is an element of the Lie algebra of H in contradiction to our assumption.) Let us check these results by reconsidering the example of Eq. (3.11). In that case and generating 2 Goldstone bosons [see Eq. (3.19)]. We conclude this section with two remarks. First, the number of Goldstone bosons is determined by the structure of the symmetry groups. Let G denote the symmetry group of the Lagrangian, with generators and H the subgroup with generators which leaves the ground state after spontaneous symmetry breaking invariant. For each generator which does not annihilate the vacuum one obtains a massless Goldstone boson, i.e., the total number of Goldstone bosons equals Second, the Lagrangians used in motivating the phenomenon of a spontaneous symmetry breakdown are typically constructed in such a fashion that the degeneracy of the ground states is built into the potential at the classical level (the prototype being the “Mexican hat” potential of Fig. 3.6). As in the above case, it is then argued that an elementary Hermitian field of a multiplet transforming non-trivially under the symmetry group G acquires a vacuum expectation value signaling a spontaneous symmetry breakdown. However, there also exist theories such as QCD where one cannot infer from inspection of the Lagrangian whether the theory exhibits spontaneous symmetry breaking. Rather, the criterion for spontaneous symmetry breaking is a non-vanishing vacuum expectation value of some Hermitian operator, not an elementary field, which is generated through the dynamics of the underlying theory. In particular, we will see that the quantities developing a vacuum expectation value may also be local Hermitian operators composed of more fundamental degrees of freedom of the theory. Such a possibility was already emphasized in the derivation of Goldstone’s theorem in Ref. [169].

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3.3. Goldstone’s Theorem By means of the above example, we motivate another approach to Goldstone’s theorem without delving into all the subtleties of a quantum fieldtheoretical approach [35]. Given a Hamilton operator with a global symmetry group G = SO(3), let denote a triplet of local Hermitian operators transforming as a vector under G,

where the are the generators of the SO(3) transformations on the Hilbert space satisfying and the are the matrices of the three dimensional representation satisfying We assume that one component of the multiplet acquires a non-vanishing vacuum expectation value:

Then the two generators and do not annihilate the ground state, and to each such generator corresponds a massless Goldstone boson. In order to prove these two statements let us expand Eq. (3.28) to first order in the

Comparing the terms linear in the

and noting that all three

can be chosen independently, we obtain

which, of course, simply expresses the fact that the field operators as a vector. we find

In particular,

transform

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with cyclic permutations for the other two cases. In order to prove that and do not annihilate the ground state, let us consider Eq. (3.28) for

From the first row we obtain

Taking the vacuum expectation value

and using Eq. (3.29) clearly since otherwise the exponential operator could be replaced by unity and the right-hand side would vanish. A similar argument shows At this point let us make two remarks. The “states” cannot be normalized. In a more rigorous derivation one makes use of integrals of the form

and first determines the commutator before evaluating the integral [35]. Some derivations of Goldstone’s theorem right away start by assuming However, for the discussion of spontaneous symmetry breaking in the framework of QCD it is advantageous to establish the connection between the existence of Goldstone bosons and a non-vanishing expectation value. Let us now turn to the existence of Goldstone bosons, taking the vacuum expectation value of Eq. (3.30)

We will first show A = –B. To that end we perform a rotation of the fields as well as the generators by about the 3 axis [see Eq. (3.28) with

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and analogously for the charge operators

We thus obtain

where we made use of i.e., the vacuum is invariant under rotations about the 3 axis. In other words, the non-vanishing vacuum expectation value can also be written as

We insert a complete set of states

into the commutator†

and make use of translational invariance

Integration with respect to the momentum of the inserted intermediate states yields an expression of the form

† The abbreviation

includes an integral over the total momentum quantum numbers necessary to fully specify the states.

as well as all other

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where the prime indicates that only states with need to be considered. Due to the Hermiticity of the symmetry current operators as well as the we have

such that

From Eq. (3.32) we draw the following conclusions. 1. Due to our assumption of a non-vanishing vacuum expectation value there must exist states for which both and do not vanish. The vacuum itself cannot contribute to Eq. (3.32) because 2. States with

contribute

to the sum. However, states with

is the phase of

is time-independent and therefore the sum over must vanish.

3. The right-hand side of Eq. (3.32) must therefore contain the contribution from states with zero energy as well as zero momentum thus zero mass. These zero-mass states are the Goldstone bosons.

3.4.

Explicit Symmetry Breaking: A First Look

Finally, let us illustrate the consequences of adding to our Lagrangian of Eq. (3.11) a small perturbation which explicitly breaks the symmetry. To that end, we modify the potential of Eq. (3.11) by adding a term

where and with Hermitian fields Clearly, the potential no longer has the original O(3) symmetry but is only invariant under O(2). The conditions for the new minimum, obtained from read

Introduction to Chiral Perturbation Theory

Let us solve the cubic equation for

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using the perturbative ansatz

from which we obtain

Of course,

corresponds to our result without explicit perturbation. The

condition for a minimum [see Eq. (3.23)] excludes Expanding the potential with we obtain, after a short calculation, for the masses

The important feature here is that the original Goldstone bosons of Eq. (3.19) are now massive. The squared masses are proportional to the symmetry breaking parameter Calculating quantum corrections to observables in terms of Goldstone-boson loop diagrams will generate corrections which are nonanalytic in the symmetry breaking parameter such as In [240]. Such socalled chiral logarithms originate from the mass terms in the Goldstone boson propagators entering the calculation of loop integrals. We will come back to this point in Chapter 4 when we discuss the masses of the pseudoscalar octet in terms of the quark masses which, in QCD, represent the analogue to the parameter in the above example.

4.

CHIRAL PERTURBATION THEORY FOR MESONS

Chiral perturbation theory provides a systematic method for discussing the consequences of the global flavour symmetries of QCD at low energies by means of an effective field theory. The effective Lagrangian is expressed in terms of those hadronic degrees of freedom which, at low energies, show up as observable asymptotic states. At very low energies these are just the members of the pseudoscalar octet which are regarded as the Goldstone bosons of the spontaneous breaking of the chiral symmetry down to The non-vanishing masses of the light pseudoscalars in the “real”

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world are related to the explicit symmetry breaking in QCD due to the light quark masses. We will first consider the indications for a spontaneous breakdown of chiral symmetry in QCD and then, in quite general terms, discuss the transformation properties of Goldstone bosons under the symmetry groups of the Lagrangian and the ground state, respectively. This will lead us to the concept of a nonlinear realization of a symmetry. After introducing the lowest-order effective Lagrangian relevant to the spontaneous breakdown from to we will illustrate how Weinberg’s power counting scheme allows for a systematic classification of Feynman diagrams in the so-called momentum expansion. We will then outline the principles entering the construction of the effective Lagrangian and discuss how, at lowest order, the results of current algebra are reproduced. After presenting the Lagrangian of Gasser and Leutwyler and the Wess-Zumino-Witten action we will discuss some applications at chiral order We will conclude the presentation of the mesonic sector with referring to some selected examples at

4.1. Spontaneous Symmetry Breaking in QCD While the toy model of Sect. 3.2., by construction led to a spontaneous symmetry breaking, it is not fully understood theoretically why QCD should exhibit this phenomenon [208]. We will first motivate why experimental input, the hadron spectrum of the “real” world, indicates that spontaneous symmetry breaking happens in QCD. Secondly, we will show that a non-vanishing singlet scalar quark condensate is a sufficient condition for a spontaneous symmetry breaking in QCD.

4.1.1.

The Hadron Spectrum

We saw in Sect. 2.3., that the QCD Lagrangian possesses a symmetry in the chiral limit in which the light quark masses vanish. From symmetry considerations involving the Hamiltonian only, one would naively expect that hadrons organize themselves into approximately degenerate multiplets fitting the dimensionalities of irreducible representations of the group The symmetry results in baryon number conservation‡ and leads to a classification of hadrons into mesons (B = 0) and baryons (B = 1). The linear combinations and of the left- and right-handed charge operators commute ‡

See Ref. [175] for empirical limits on nucleon decay as well as baryon-number violating Z and decays.

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with have opposite parity, and thus for any state of positive parity one would expect the existence of a degenerate state of negative parity (parity doubling) which can be seen as follows. Let denote an eigenstate of with eigenvalue having positive parity, such as, e.g., a member of the ground state baryon octet (in the chiral limit). Defining because of we have

i.e, the new state is also an eigenstate of but of opposite parity:

with the same eigenvalue

The state can be expanded in terms of the members of the multiplet with negative parity, However, the low-energy spectrum of baryons does not contain a degenerate baryon octet of negative parity. Naturally the question arises whether the above chain of arguments is incomplete. Indeed, we have tacitly assumed that the ground state of QCD is annihilated by Let symbolically denote an operator which creates quanta with the quantum numbers of the state whereas creates degenerate quanta of opposite parity. Let us assume the states and to be members of a basis of an irreducible representation of In analogy to Eq. (2.49), we assume that under the creation operators are related by

The usual chain of arguments then works as

However, if the ground state is not annihilated by the reasoning of Eq. (4.1) does no longer apply. Two empirical facts about the hadron spectrum suggest that a spontaneous symmetry breaking happens in the chiral limit of QCD. First, SU(3) instead

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of is approximately realized as a symmetry of the hadrons. Second, the octet of the pseudoscalar mesons is special in the sense that the masses of its members are small in comparison with the corresponding 1– vector mesons. They are candidates for the Goldstone bosons of a spontaneous symmetry breaking. In order to understand the origin of the SU(3) symmetry let us consider the vector charges [see Eq. (2.59)]. They satisfy the commutation relations of an SU(3) Lie algebra [see Eqs. (2.76)-(2.78)],

In Ref. [313] it was shown that, in the chiral limit, the ground state is necessarily invariant under i.e., the eight vector charges as well as the baryon number operator§ annihilate the ground state,

If the vacuum is invariant under then so is the Hamiltonian [93] (but not vice versa). Moreover, the invariance of the ground state and the Hamiltonian implies that the physical states of the spectrum of. can be organized according to irreducible representations of The index V (for vector) indicates that the generators result from integrals of the zeroth component of vector current operators and thus transform with a positive sign under parity. Let us now turn to the linear combinations satisfying the commutation relations [see Eqs. (2.76) - (2.78)]

Note that these charge operators do not form a closed algebra, i.e., the commutator of two axial charge operators is not again an axial charge operator. Since the parity doubling is not observed for the low-lying states, one assumes that the do not annihilate the ground state,

§ Recall that each quark is assigned a baryon number 1/3.

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i.e., the ground state of QCD is not invariant under “axial” transformations. According to Goldstone’s theorem [264, 267, 268, 168, 169], to each axial generator which does not annihilate the ground state, corresponds a massless Goldstone boson field with spin 0, whose symmetry properties are tightly connected to the generator in question. The Goldstone bosons have the same transformation behaviour under parity,

i.e., they are pseudoscalars, and transform under the subgroup which leaves the vacuum invariant, as an octet [see Eq. (4.4)]:

In the present case, with with and we expect eight Goldstone bosons.

4.1.2.

and

The Scalar Quark Condensate

In the following, we will show that a non-vanishing scalar quark condensate in the chiral limit is a sufficient (but not a necessary) condition for a spontaneous symmetry breaking in QCD.¶ The subsequent discussion will parallel that of the toy model in Sect. 3.3., after replacement of the elementary fields by appropriate composite Hermitian operators of QCD. Let us first recall the definition of the nine scalar and pseudoscalar quark densities:

The equal-time commutation relation of two quark operators of the form where symbolically denotes Dirac- and flavour matrices and a summation over colour indices is implied, can compactly be written as [see Eq. (2.75)]

With the definition

¶ In this section all physical quantities such as the ground state, the quark operators etc., are

considered in the chiral limit.

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and using

we see, after integration of Eq. (4.10) over that the scalar quark densities of Eq. (4.8) transform under as a singlet and as an octet, respectively,

with analogous results for the pseudoscalar quark densities. In the limit and, of course, also in the even more restrictive chiral limit, the charge operators in Eqs. (4.11) and (4.12) are actually time independent.|| Using the relation

for the structure constants of SU(3), we re-express the octet components of the scalar quark densities as

which represents the analogue of Eq. (3.30) in the discussion of Goldstone’s theorem. In the chiral limit the ground state is necessarily invariant under [313], i.e., and we obtain from Eq. (4.14)

where we made use of translational invariance of the ground state. In other words, the octet components of the scalar quark condensate must vanish in the chiral limit. From Eq. (4.15), we obtain for

||

The commutation relations also remain valid for equal times if the symmetry is explicitly broken.

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i.e.,

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and for

i.e., Because of Eq. (4.11) a similar argument cannot be used for the singlet condensate, and if we assume a non-vanishing singlet scalar quark condensate in the chiral limit, we thus find using Eq. (4.15)

Finally, we make use of (no summation implied!)

in combination with

to obtain

where we have suppressed the dependence on the right-hand side. We evaluate Eq. (4.17) for a ground state which is invariant under assuming a non-vanishing singlet scalar quark condensate,

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where, because of translational invariance, the right-hand side is independent of Inserting a complete set of states into the commutator of Eq. (4.18) yields, in complete analogy to Sect. 3.3., [see the discussion following Eq. (3.31)] that both the pseudoscalar density as well as the axial charge operators must have a non-vanishing matrix element between the vacuum and massless one particle states In particular, because of Lorentz covariance, the matrix element of the axial-vector current operator between the vacuum and these massless states, appropriately normalized, can be written as

where MeV denotes the “decay” constant of the Goldstone bosons in the chiral limit. Assuming a non-zero value of is a necessary and sufficient criterion for spontaneous chiral symmetry breaking. On the other hand, because of Eq. (4.18) a non-vanishing scalar quark condensate is a sufficient (but not a necessary) condition for a spontaneous symmetry breakdown in QCD. Table 4.1 contains a summary of the patterns of spontaneous symmetry breaking as discussed in Sect. 3.3., the generalization of Sect. 3.2., to the socalled O(N) linear sigma model, and QCD.

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4.2. Transformation Properties of the Goldstone Bosons The purpose of this section is to discuss the transformation properties of the field variables describing the Goldstone bosons [320, 95, 85, 22, 230]. We will need the concept of a nonlinear realization of a group in addition to a representation of a group which one usually encounters in Physics. We will first discuss a few general group-theoretical properties before specializing to QCD.

4.2.1. General Considerations Let us consider a physical system with a Hamilton operator which is invariant under a compact Lie group G. Furthermore we assume the ground state of the system to be invariant under only a subgroup H of G, giving rise to Goldstone bosons. Each of these Goldstone bosons will be described by an independent field which is a continuous real function on Minkowski space ** We collect these fields in an n-component vector and define the vector space

Our aim is to find a mapping an element

which uniquely associates with each pair with the following properties:

Such a mapping defines an operation of the group G on The second condition is the so-called group-homomorphism property [79, 271, 207]. The mapping will, in general, not define a representation of the group G, because we do not require the mapping to be linear, i.e., Let denote the “origin” of [230] which, in a theory containing Goldstone bosons only, loosely speaking corresponds to the ground state configuration. Since the ground state is supposed to be invariant under the subgroup H we require the mapping to be such that all elements map the origin onto itself. In this context the subgroup H is also known as the little group of Given that such a mapping indeed exists, we need to verify for infinite groups that (see Sect. 2.4 of [207]): 1. H is not empty, because the identity

maps the origin onto itself.

** Depending on the equations of motion, we will require more restrictive properties of the functions

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2. If

and

are elements satisfying

so does i.e., because of the

homomorphism property also the product 3. For

we have

i.e.,

Following Ref. [230] we will establish a connection between the Goldstone boson fields and the set of all left cosets which is also referred to as the quotient G/H. For a subgroup H of G the set defines the left coset of (with an analogous definition for the right coset) which is one element of G/H. † † For our purposes we need the property that cosets either completely overlap or are completely disjoint (see, e.g., [207]), i.e., the quotient is a set whose elements themselves are sets of group elements, and these sets are completely disjoint. Let us first show that for all elements of a given coset, maps the origin onto the same vector in

Secondly, the mapping is injective with respect to the cosets, which can be proven as follows. Consider two elements and of G where We need to show Let us assume

However, this implies or in contradiction to the assumption. Thus cannot be true. In other words, the mapping can be inverted on the image of The conclusion is that there exists an isomorphic mapping between the quotient G/H and the Goldstone boson fields.‡‡ Now let us discuss the transformation behaviour of the Goldstone boson fields under an arbitrary in terms of the isomorphism established above. †† An invariant subgroup has the additional property that the left and right cosets coincide for

each which allows for a definition of the factor group G/H in terms of the complex product. However, here we do not need this property. ‡‡ Of course, the Goldstone boson fields are not constant vectors in but functions on Minkowski space [see Eq. (4.20)]. This is accomplished by allowing the cosets to also depend on

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To each corresponds a coset with appropriate denote a representative of this coset such that

Now apply the mapping

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Let

to

In other words, in order to obtain the transformed from a given we simply need to multiply the left coset representing by in order to obtain the new left coset representing This procedure uniquely determines the transformation behaviour of the Goldstone bosons up to an appropriate choice of variables parameterizing the elements of the quotient G/H.

4.2.2. Application to QCD Now let us apply the above general considerations to the specific case relevant to QCD and consider the group SU(N), and which is isomorphic to SU(N). Let We may uniquely characterize the left coset of through the SU(N) matrix [22],

if we follow the convention that we choose the representative of the coset such that the unit matrix stands in its first argument. According to the above derivation, U is isomorphic to a The transformation behaviour of U under is obtained by multiplication in the left coset:

i.e. As mentioned above, we finally need to introduce an

dependence so that

Let us now restrict ourselves to the physically relevant cases of N = 2 and N = 3 and define

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Furthermore let matrices,

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denote the set of all Hermitian and traceless N × N

which under addition of matrices defines a real vector space. We define a second set continuous}, where the entries are continuous functions. For N = 2 the elements of and are related to each other according to

where the are the usual Pauli matrices and gously for N = 3,

with the Gell-Mann matrices and a real vector space. Let us finally define

Analo-

Again,

forms

At this point it is important to note that does not define a vector space because the sum of two SU(N) matrices is not an SU(N) matrix. We are now in the position to discuss the so-called nonlinear realization of SU(N) × SU(N) on The homomorphism

defines an operation of G on

1. 2.

since

because and

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3. Let

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and thus

The mapping is called a nonlinear realization, because is not a vector space. The origin i.e., denotes the ground state of the system. Under transformations of the subgroup corresponding to rotating both left- and right-handed quark fields in QCD by the same V, the ground state remains invariant,

On the other hand, under “axial transformations,” i.e., rotating the left-handed quarks by A and the right-handed quarks by the ground state does not remain invariant,

which, of course, is consistent with the assumed spontaneous symmetry breakdown. Let us finally discuss the transformation behaviour of under the subgroup Expanding

we immediately see that the realization restricted to the subgroup H,

defines a linear representation on

Let us consider the SU(3) case and parameterize

because

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from which we obtain, by comparing both sides of Eq. (4.25),

However, this corresponds exactly to the adjoint representation, i.e., in SU(3) the fields transforms as an octet which is also consistent with the transformation behaviour we discussed in Eq, (4.7):

For group elements of G of the form one may proceed in a completely analogous fashion. However, one finds that the fields do not have a simple transformation behaviour under these group elements. In other words, the commutation relations of the fields with the axial charges are complicated nonlinear functions of the fields [320].

4.3.

The Lowest-Order Effective Lagrangian

Our goal is the construction of the most general theory describing the dynamics of the Goldstone bosons associated with the spontaneous symmetry breakdown in QCD, In the chiral limit, we want the effective Lagrangian to be invariant under It should contain exactly eight pseudoscalar degrees of freedom transforming as an octet under the subgroup Moreover, taking account of spontaneous symmetry breaking, the ground state should only be invariant under Following the discussion of Sect. 4.2.2., we collect the dynamical variables in the SU(3) matrix

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The most general, chirally invariant, effective Lagrangian density with the minimal number of derivatives reads

where MeV is a free parameter which later on will be related to the pion decay (see Sect. 4.6.1.) First of all, the Lagrangian is invariant under the global transformations of Eq. (4.23):

because

where we made use of the trace property Tr(AB) = Tr(BA). The global invariance is trivially satisfied, because the Goldstone bosons have baryon number zero, thus transforming as under which also implies The substitution or, equivalently, provides a simple method of testing, whether an expression is of so-called even or odd intrinsic parity,* i.e., even or odd in the number of Goldstone boson fields. For example, it is easy to show, using the trace property, that the Lagrangian of Eq. (4.29) is even. The purpose of the multiplicative constant in Eq. (4.29) is to generate the standard form of the kinetic term which can be seen by expanding the exponential resulting in

* Since the Goldstone bosons are pseudoscalars, a true parity transformation is given by

or, equivalently,

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where we made use of In particular, since there are no other terms containing only two fields starts with interaction terms containing at least four Goldstone bosons) the eight fields describe eight independent massless particles.† A term of the type may be re-expressed as‡

i.e., up to a total derivative it is proportional to the Lagrangian of Eq. (4.29). However, in the present context, total derivatives do not have a dynamical significance, i.e., they leave the equations of motion unchanged and can thus be dropped. The product of two invariant traces is excluded at lowest order, because Let us prove the general SU(N) case by considering an SU(N)-valued field

with

Hermitian, traceless matrices we expand the exponential

and real fields

Defining

and consider the derivative§

We then find

†

‡ §

At this stage, this is only a tree-level argument. We will see in Sect. 4.9.1., that the Goldstone bosons remain massless in the chiral limit even when loop corrections have been included. In the present case and are matrices which, in general, do not commute.

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where we made use of Let us turn to the vector and axial-vector currents associated with the global symmetry of the effective Lagrangian of Eq. (4.29). To that end, we parameterize

In order to construct Then, to first order in

set

and choose

(see Sect. 2.3.3.)

from which we obtain for

(In the last step we made use of

which follows from differentiating currents

and, completely analogously, choosing

We thus obtain for the left

and

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for the right currents. Combining Eqs. (4.35) and (4.36) the vector and axialvector currents read

Furthermore, because of the symmetry of under both vector and axial-vector currents are conserved. The vector current densities of Eq. (4.37) contain only terms with an even number of Goldstone bosons,

On the other hand, the expression for the axial-vector currents is odd in the number of Goldstone bosons,

To find the leading term let us expand Eq. (4.38) in the fields,

from which we conclude that the axial-vector current has a non-vanishing matrix element when evaluated between the vacuum and a one-Goldstone boson state [see Eq. (4.19)]:

In Sect. 4.6.1.,

will be related to the pion-decay constant entering

So far we have assumed a perfect symmetry. However, in Sect. 3.4., we saw, by means of a simple example, how an explicit symmetry

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breaking may lead to finite masses of the Goldstone bosons. As has been discussed in Sect. 2.3.6., the quark mass term of QCD results in such an explicit symmetry breaking,

In order to incorporate the consequences of Eq. (4.39) into the effectiveLagrangian framework, one makes use of the following argument [153]: Although M is in reality just a constant matrix and does not transform along with the quark fields, of Eq. (4.39) would be invariant if M transformed as

One then constructs the most general Lagrangian which is invariant under Eqs. (4.24) and (4.40) and expands this function in powers of M. At lowest order in M one obtains

where the subscript s.b. refers to symmetry breaking. In order to interpret the new parameter let us consider the energy density of the ground state (U =

and compare its derivative with respect to (any of) the light quark masses with the corresponding quantity in QCD,

where is the chiral quark condensate of Eq. (4.16). Within the framework is thus related to the of the lowest order effective Lagrangian, the constant chiral quark condensate as

Let us add a few remarks. 1. A term Tr(M) by itself is not invariant.

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2. The combination ity

because

has the wrong behaviour under par-

3. Because

contains only terms even in

In order to determine the masses of the Goldstone bosons, we identify the terms of second order in the fields in

Using Eq. (4.28) we find

For the sake of simplicity we consider the isospin-symmetric limit so that the term vanishes and there is no mixing. We then obtain for the masses of the Goldstone bosons, to lowest order in the quark masses,

These results, in combination with Eq. (4.43), correspond relations obtained in Ref. [150] and are referred to as the Gell-Mann, Oakes, and Renner relations. Furthermore, the masses of Eqs. (4.45)–(4.47) satisfy the Gell-Mann-Okubo relation

independent of the value of Without additional input regarding the numerical value of Eqs. (4.45) - (4.47) do not allow for an extraction of the

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absolute values of the quark masses and because rescaling in combination with leaves the relations invariant. For the ratio of the quark masses one obtains, using the empirical values of the pseudoscalar octet,

Let us conclude this section with the following remark. We saw in Sect. 4.1.2., that a non-vanishing quark condensate in the chiral limit is a sufficient but not a necessary condition for a spontaneous chiral symmetry breaking. The effective Lagrangian term of Eq. (4.41) not only results in a shift of the vacuum energy but also in finite Goldstone boson masses.¶ These are related via the parameter and we recall that it was a symmetry argument which excluded a term Tr(M) which, at leading order in M, would decouple the vacuum energy shift from the Goldstone boson masses. The scenario underlying of Eq. (4.41) is similar to that of a Heisenberg ferromagnet [12, 230] which exhibits a spontaneous magnetization breaking the O(3) symmetry of the Heisenberg Hamiltonian down to O(2). In the present case the analogue of the order parameter is the quark condensate In the case of the ferromagnet, the interaction with an external magnetic field is given by which corresponds to Eq. (4.42), with the quark masses playing the role of the external field However, in principle, it is also possible that vanishes or is rather small. In such a case the quadratic masses of the Goldstone bosons might be dominated by terms which are nonlinear in the quark masses, i.e., by higher-order terms in the expansion of Such a scenario is the origin of the so-called generalized chiral perturbation theory [218, 219, 299]. The analogue would be an antiferromagnet which shows a spontaneous symmetry breaking but with The analysis of recent data on in terms of the isoscalar scattering length supports the conjecture that the quark condensate is indeed the leading order parameter of the spontaneously broken chiral symmetry. For a recent discussion on the relation between the quark condensate and scattering the interested reader is referred to Ref. [234]. ¶ Later on we will also see that the

scattering amplitude is effected by

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4.4. Effective Lagrangians and Weinberg’s Power Counting Scheme An essential prerequisite for the construction of effective field theories is a “theorem” of Weinberg stating that a perturbative description in terms of the most general effective Lagrangian containing all possible terms compatible with assumed symmetry principles yields the most general S matrix consistent with the fundamental principles of quantum field theory and the assumed symmetry principles [322]. The corresponding effective Lagrangian will contain an infinite number of terms with an infinite number of free parameters. Turning Weinberg’s theorem into a practical tool requires two steps: one needs some scheme to organize the effective Lagrangian and a systematic method of assessing the importance of diagrams generated by the interaction terms of this Lagrangian when calculating a physical matrix element. In the framework of mesonic chiral perturbation theory, the most general chiral Lagrangian describing the dynamics of the Goldstone bosons is organized as a string of terms with an increasing number of derivatives and quark mass terms, where the subscripts refer to the order in the momentum and quark mass expansion. The index 2, for example, denotes either two derivatives or one quark mass term. In the context of Feynman rules, derivatives generate fourmomenta, whereas the convention of counting quark mass terms as being of the same order as two derivatives originates from Eqs. (4.45) - (4.47) in conjunction with the on-shell condition In an analogous fashion, and denote more complicated terms of so-called chiral orders and with corresponding numbers of derivatives and quark mass terms. With such a counting scheme, the chiral orders in the mesonic sector are always even because Lorentz indices of derivatives always have to be contracted with either the metric tensor or the Levi-Civita tensor to generate scalars, and the quark mass terms are counted as Weinberg’s power counting scheme [322] analyzes the behaviour of a given diagram under a linear rescaling of all the external momenta, and a quadratic rescaling of the light quark masses, which, in terms of the Goldstone boson masses, corresponds to The chiral dimension D of a given diagram with amplitude is defined by

and thus

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where denotes the number of vertices originating from and is the number of independent loops. Clearly, for small enough momenta and masses diagrams with small D, such as D = 2 or D = 4, should dominate. Of course, the rescaling of Eq. (4.51) must be viewed as a mathematical tool. While external three-momenta can, to a certain extent, be made arbitrarily small, the rescaling of the quark masses is a theoretical instrument only. Note that loop diagrams are always suppressed due to the term in Eq. (4.52). It may happen, though, that the leading-order tree diagrams vanish and therefore that the lowest-order contribution to a certain process is a one-loop diagram. An example is the reaction [26]. In order to prove Eq. (4.52) we start from the usual Feynman rules for evaluating an S-matrix element (see, e.g., Appendix A.4 of Ref. [201]). Each internal meson line contributes a factor

For each vertex, originating from we obtain symbolically a factor together with a four-momentum conserving delta function resulting in for the vertex factor and for the delta function. At this point one has to take into account the fact that, although Eq. (4.51) refers to a rescaling of external momenta, a substitution for internal momenta as in Eq. (4.53) acts in exactly the same way as a rescaling of external momenta:

where and denote external and internal momenta, respectively. So far we have discussed the rules for determining the power referring to the S-matrix element which is related to the invariant amplitude through a four-momentum conserving delta function,

The delta function contains external momenta only, and thus re-scales under so

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We thus find as an intermediate result

where denotes the number of internal lines. The number of independent loops, total number of vertices, and number of internal lines are related by||

because each of the vertices generates a delta function. After extracting one overall delta function this yields conditions for the internal momenta. Using we finally obtain from Eq. (4.54)

By means of a simple example we will illustrate how the mechanism of rescaling actually works. To that end we consider as a toy model of an effective field theory the self interaction of a scalar field,

** The Feynman where the coupling constant has the dimension of rules give the amplitude corresponding to the simple tree diagram of Fig. 4.1 for the scattering of two particles,

As expected, the behaviour under rescaling is in agreement with Eq. (4.52) for and for all remaining Now let us consider a typical loop diagram of Fig. 4.2 contributing to the same process, where the 2 in the interaction blob indicates the term in the Lagrangian containing two derivatives. || Note that the number of independent momenta is not the number of faces or closed circuits that

may be drawn on the internal lines of a diagram. This may, for example, be seen using a diagram with the topology of a tetrahedron which has four faces but (see, e.g., Sect. 6.2 of Ref. [201]). ** Recall that the dimensions of a Lagrangian density and a field are energy4 and energy, respectively.

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Applying the usual Feynman rules, with the vertex of Eq. (4.56), we obtain

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This agrees with the value D = 4 given by Eq. (4.52) for and For the sake of completeness, let us comment on the symmetry factor 1/2 in Eq. (4.57). When deriving the Feynman rule of Eq. (4.56), we took account of 4! = 24 distinct combinations of contracting four field operators with four external lines. The “product” of two such vertices thus contains 24 × 24 combinations. However, from each vertex two lines have to be selected as internal lines and there exist 6 possibilities to choose one pair out of 4 field operators to form internal lines. For the two remaining operators one has two possibilities of contracting them with external lines. Finally, the respective pairs of internal lines of the first and second vertices may be contracted in two ways with each other, leaving us with 12 × 12 × 2 = (24 × 24)/2 combinations. In the discussion of the loop integral we did not address the question of convergence. This needs to be addressed since applying the substitution in Eq. (4.57) is well-defined only for convergent integrals. Later on we will regularize the integrals by use of the method of dimensional regularization, introducing a renormalization scale which also has to be rescaled linearly. However, at a given chiral order, the sum of all diagrams will, by construction, not depend on the renormalization scale. Finally, the proof of Weinberg’s theorem [231, 199] for chiral perturbation

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theory is rather technical and lengthy and beyond the scope of this review. In Ref. [231] it was shown that global symmetry constraints alone do not suffice to fully determine the low-energy structure of the effective Lagrangian. In fact, a determination of the (low-energy) Green functions of QCD off the mass shell, i.e., for momenta which do not correspond to the mass-shell conditions for Goldstone bosons, one needs to study the Ward identities, and therefore the symmetries have to be extended to the local level. One thus considers a locally invariant, effective Lagrangian although the symmetries of the underlying theory originate in a global symmetry. If the Ward identities contain anomalies, they show up as a modification of the generating functional, which can explicitly be incorporated through the Wess-Zumino-Witten construction [328, 327].

4.5. Construction of the Effective Lagrangian In Sect. 4.3., we have derived the lowest-order effective Lagrangian for a global symmetry. On the other hand, the Ward identities originating in the global symmetry of QCD are obtained from a locally invariant generating functional involving a coupling to external fields (see Sect. 2.4.4., and Appendix A.1). Our goal is to approximate the “true” generating functional of Eq. (2.97) by a sequence where the effective generating functionals are obtained using the effective field theory. Therefore, we need to promote the global symmetry of the effective Lagrangian to a local one and introduce a coupling to the same external fields and as in QCD. In the following we will outline the principles entering the construction of the effective Lagrangian for a local symmetry (see Refs, [141, 66, 117] for details).†† The matrix U transforms as where and are independent space-lime-dependent SU(3) matrices. As in the case of gauge theories, we need external fields and [see Eqs. (2.96), (2.106), and (2.109) and Table 4.2] corresponding to the parameters and of and respectively. For any object A transforming as such as, e.g., U we define the covariant derivative as

††

In principle, we could also “gauge” the symmetry. However, this is primarily of relevance to the SU(2) sector in order to fully incorporate the coupling to the electromagnetic field [see Eq. (2.111)]. Since in SU(3), the quark-charge matrix is traceless, this important case is included in our considerations. For further discussions, see Ref. [117].

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where we made use of Again, the defining property for the covariant derivative is that it should transform in the same way as the object it acts on.‡‡ Since the effective Lagrangian will ultimately contain arbitrarily high powers of derivatives we also need the field strength tensors and corresponding to the gauge fields,

The field strength tensors are traceless,

because and the trace of any commutator vanishes. Finally, following the convention of Gasser and Leutwyler we introduce the linear combination with the scalar and pseudoscalar external fields of Eq. (2.96), where is defined in Eq. (4.43). Table 4.2 contains the transformation properties of all building blocks under the group (G), charge conjugation (C), and parity (P), In the chiral counting scheme of chiral perturbation theory the elements are counted as:

The external fields and count as to match and is of because of Eqs. (4.45) - (4.47). Any additional covariant derivative counts as The construction of the effective Lagrangian in terms of the building blocks of Eq. (4.62) proceeds as follows.* Given objects A, B,…, all of which transform as one can form invariants by taking ‡‡ Under certain circumstances it is advantageous to introduce for each object with a well-defined

transformation behaviour a separate covariant derivative. One may then use a product rule similar to the one of ordinary differentiation [see Eqs. (18) and (19) of Ref. [141]]. * There is a certain freedom in the choice of the elementary building blocks. For example, by a suitable multiplication with U or any building block can be made to transform as without changing its chiral order [141]. The present approach most naturally leads to the Lagrangian of Gasser and Leutwyler [164].

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the trace of products of the type

The generalization to more terms is obvious and, of course, the product of invariant traces is invariant:

The complete list of elements up to and including order reads

For the invariants up to

we then obtain

transforming as

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In * we made use of two important properties of the covariant derivative

The first relation results from the unitarily of U in combination with the definition of the covariant derivative, Eq. (4.58). Equation (4.67) is shown using together with Eq. (4.30),

The relations ** can either be verified by explicit calculation or, more elegantly, using the product rule of Ref. [141] for the covariant derivatives. Finally, we impose Lorentz invariance, i.e., Lorentz indices have to be contracted, resulting in three candidate terms:

The term in Eq. (4.69) with the minus sign is excluded because it has the wrong sign under parity (see Table 4.2), and we end up with the most general, locally invariant, effective Lagrangian at lowest chiral order, †

Note that contains two free parameters: the pion-decay constant and of Eq. (4.43) (hidden in the definition of ). Let us finally derive the equations of motion associated with the lowestorder Lagrangian. These are important because they can be used to eliminate so-called equation-of-motion terms in the construction of the higher-order Lagrangians [154, 230, 173, 20, 298] by applying field transformations [89, 210]. To that end we need to consider an infinitesimal change of the SU(3) matrix Since the set of SU(3) matrices forms a group, to each pair of elements U and corresponds a unique element connecting the two via Let us parameterize by means of the Gell-Mann matrices,

† At

invariance under C does not provide any additional constraints.

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and consider small variations of the SU(3) matrix as

where the are now real functions. With such an ansatz, the matrix satisfies both conditions

‡ up to and including the terms linear in Given the fields at and the dynamics is determined by the principle of stationary action. We obtain for the variation of the action

In the second equation we made use of the standard boundary conditions the divergence theorem, and the definition of the covariant derivative of Eq. (4.58). The third equality results from and the invariance of the trace with respect to cyclic permutations. The functions may be chosen arbitrarily, and we obtain eight EulerLagrange equations

Since any 3 × 3 matrix A can be written as

‡

Some derivations in the literature neglect the second condition of Eq. (4.73) and thus obtain the wrong equations of motion.

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the eight equations of motion of Eq. (4.75) may compactly be written in matrix form§

The additional term involving the trace is included to guarantee that the component proportional to the identity matrix vanishes identically and thus one does not erroneously generate a ninth equation of motion.

4.6.

Applications at Lowest Order

Let us consider two simple examples at lowest order D = 2. According to Eq. (4.52) we only need to consider tree-level diagrams with vertices of

4.6.1. Pion Decay Our first example deals with the weak decay which will allow us to relate the free parameter of to the pion-decay constant. At the level of the degrees of freedom of the Standard Model, pion decay is described by the annihilation of a quark and a antiquark, forming the into a boson, propagation of the intermediate and creation of the leptons and in the final state (see Fig. 4.3).

The coupling of the W bosons to the leptons is given by

whereas their interaction with the quarks forming the Goldstone bosons is effectively taken into account by inserting Eq. (2.112) into the Lagrangian of Eq. (4.70). Let us consider the first term of Eq. (4.70) and set with, at §

Applying Eq. (4.65) one finds

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this point, still arbitrary

Using

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we find

where only the term linear in

is shown. If we parameterize

the interaction term linear in

reads

where we made use of Eq. (4.35) defining using Eq. (4.28) to first order in

Again, we expand

by

from which we obtain the matrix element

Inserting of Eq. (2.112), we find for the interaction term of a single Goldstone boson with a W

Thus, we need to calculate¶

¶ Recall that the entries

and

of the Cabibbo-Kobayashi-Maskawa matrix are real.

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We then obtain for the interaction term

In combination with the Feynman propagator for W bosons,

the Feynman rule for the invariant amplitude for the weak pion decay reads

where

denotes the four-momentum of the pion and

is the Fermi constant. The evaluation of the decay rate is a standard textbook exercise and we only quote the final result ||

The constant is referred to as the pion-decay constant in the chiral limit.** It measures the strength of the matrix element of the axial-vector current operator between a one-Goldstone-boson state and the vacuum [see Eq. (4.19)]. || See Sect. 10.14 of Ref. [27] with the substitution in Eq. (10.140). ** Of course, in the chiral limit, the pion is massless and, in such a world, the massive leptons would decay into Goldstone bosons, e.g., However, at the symmetry breaking term of Eq. (4.41) gives rise to Goldstone-boson masses, whereas the decay constant is not modified at

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Since the interaction of the W boson with the quarks is of the type [see Eq. (2.112)] and the vector current operator does not contribute to the matrix element between a single pion and the vacuum, pion decay is completely determined by the axial-vector current. The degeneracy of a single constant in Eq. (4.19) is lifted at [164] once SU(3) symmetry breaking is taken into account. The empirical numbers for and are 92.4 MeV and 113 MeV, respectively.††

4.6.2. Pion-Pion Scattering Our second example deals with the prototype of a Goldstone boson reaction: scattering. For the sake of simplicity we will restrict ourselves to the SU(2)×SU(2) version of Eq. (4.70). We will contrast two different methods of calculating the scattering amplitude: the “direct” calculation in terms of the Goldstone boson fields of the effective Lagrangian versus the calculation of the QCD Green function in combination with the LSZ reduction formalism. Loosely speaking, the “direct” calculation is somewhat more along the spirit of Weinberg’s original paper [322]: one considers the most general Lagrangian satisfying the general symmetry constraints and calculates S-matrix elements with that Lagrangian. The second method will allow one to also consider QCD Green functions “off shell,” i.e., for arbitrary squared invariant momenta. We will discuss under which circumstances the two methods are equivalent and also work out the more general scope of the Green function approach. For the “direct” calculation we set to zero all external fields except for the quark mass term, [see Eq. (4.45)],

In our general discussion of the transformation behaviour of Goldstone bosons at the end of Sect. 4.2.1., we argued that we still have a choice how to represent the variables parameterizing the elements of the set of cosets G/H. In the present case these are elements of SU(2) and we will illustrate this freedom by making use of two different parameterizations of the matrix U [142],‡‡

†† In the analysis of Ref. [175] is used. ‡‡ The first parameterization is popular, because the pion field appears only linearly in the term

proportional to the Pauli matrices, leading to a substantial simplification when deriving Feynman rules. It is specific to SU(2) because, in contrast to the general case of SU(N), in SU(2) the totally symmetric symbols vanish [see Eq. (2.12)], On the other hand, the exponential parameterization can be used for any N.

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where in both cases the three Hermitian fields and describe pion fields transforming as isovectors under The fields in the two parameterizations are non-linearly related,

This can be interpreted in terms of a change of variables which leaves the freefield part of the Lagrangian unchanged [89, 210]. As a consequence of the equivalence theorem of field theory [89, 210] the result for a physical observable should not depend on the choice of variables. The substitution corresponding, respectively, to and tells us that generates only interaction terms containing an even number of pion fields. Since there exists no vertex involving 3 Goldstone bosons, scattering must be described by a contact interaction at By inserting the expressions for U of Eqs. (4.87) and (4.88) into Eq. (4.86) and collecting those terms containing four pion fields we obtain the interaction Lagrangians

Observe that the two interaction Lagrangians depend differently on the respective pion fields. The corresponding Feynman rules are obtained in the usual fashion by considering all possible ways of contracting pion fields of with initial and final pion lines, with the derivatives generating for an initial (final) line. For Cartesian isospin indices the Feynman rules for the scattering process as obtained from Eqs. (4.90) and (4.91) read, respectively,

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where we introduced

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and the usual Mandelstam variables

which are related by If the initial and final pions are all on the mass shell, i.e., the scattering amplitudes are the same, in agreement with the equivalence theorem [89, 210],* The on-shell result also agrees with the current-algebra prediction for low-energy scattering [318]. We will come back to scattering in Sect. 4.10.2., when we also discuss corrections of higher order [63]. On the other hand, if one of the momenta of the external lines is off mass shell, the amplitudes of Eqs. (4.92) and (4.93) differ. In other words, a “direct” calculation gives a unique result independent of the parameterization of U only for the on-shell matrix element. The second method, developed by Gasser and Leutwyler [163], deals with the Green functions of QCD and their interrelations as expressed in the Ward identities. In particular, these Green functions can, in principle, be calculated for any value of squared momenta even though ChPT is set up only for a lowenergy description. For the discussion of scattering one considers the fourpoint function [163]

with the pseudoscalar quark densities of Eq. (4.9). In order so see that Eq. (4.94) can indeed be related to scattering, let us first investigate the matrix element of the pseudoscalar density evaluated between a single-pion state and the vacuum, which is defined in terms of the coupling constant [163]:

At we determine the coupling of an external pseudoscalar source to the Goldstone bosons by inserting into the Lagrangian of Eq. (4.70) (see Fig. 4.4),

* For a general proof of the equivalence of S-matrix elements evaluated at tree level (phenomeno-

logical approximation), see Sect. 2 of Ref. [95].

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where the first and second lines refer to the parameterizations of Eqs. (4.87) and (4.88), respectively. From Eq. (4.96) we obtain independent of the parameterization used which, since the pion is on-shell, is a consequence of the equivalence theorem [89, 210]. As a consistency check, let us verify the PCAC relation of Eq. (2.117) (without an external electromagnetic field) evaluated between a single-pion state and the vacuum. For the axial-vector current matrix element, we found at

Taking the divergence we obtain

where we made use of Eq. (4.45) for the pion mass. Multiplying Eq. (4.95) by and using we explicitly verify the PCAC relation. Every field which satisfies the relation

can serve as a so-called interpolating pion field [75] in the LSZ reduction formulas [228, 201]. For the case of the reduction formula relates the S-matrix element to the Green function of the interpolating field as

After partial integrations, the Klein-Gordon operators convert into inverse free propagators

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In the present context, we will use

which then relates the S-matrix element of scattering to the QCD Green function involving four pseudoscalar densities

Using translational invariance, let us define the momentum space Green function as

where we define all momenta as incoming. The usual relation between the S matrix and the T matrix, implies for the T-matrix element and, finally, for

We will now determine the Green function using the parameterizations of Eqs. (4.87) and (4.88) for U. In the first parameterization we only obtain a linear coupling between the external pseudoscalar field and the pion field [see Eq. (4.96)] so that only the Feynman diagram of Fig. 4.5 contributes

where is given in Eq. (4.92). The Green function depends on six independent Lorentz scalars which can be chosen as the squared invariant momenta

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and the three Mandelstam variables

and

satisfying the constraint

Using the second parameterization we will obtain a contribution which is of the same form as Fig. 4.5 but with replaced by of Eq. (4.93). Clearly, this is not yet the same result as Eq. (4.102) because of the terms proportional to in Eq. (4.93). However, in this parameterization the external pseudoscalar field also couples to three pion fields [see Eq. (4.96)], resulting in four additional diagrams of the type shown in Fig. 4.6. For example, the contribution shown in Fig. 4.6 reads

where

In combination with the contribution of the remaining

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three diagrams, we find a complete cancelation with those terms proportional to of Fig. 4.5 (in the second parameterization) and the end result is identical with Eq. (4.102). Finally, using and inserting the result of Eq. (4.102) into Eq. (4.101) we obtain the same scattering amplitude as in the “direct” calculation of Eqs. (4.92) and (4.93) evaluated for on-shell pions.

This example serves as an illustration that the method of Gasser and Leutwyler generates unique results for the Green functions of QCD for arbitrary four-momenta. There is no ambiguity resulting from the choice of variables used to parameterize the matrix U in the effective Lagrangian. These Green functions can be evaluated for arbitrary (but small) four-momenta. Using the reduction formalism, on-shell matrix elements such as the scattering amplitude can be calculated from the QCD Green functions. The result for the scattering amplitude as derived from Eq. (4.101) agrees with the “direct” calculation of the on-shell matrix elements of Eqs. (4.92) and (4.93). On the other hand, the Feynman rules of Eqs. (4.92) and (4.93), when taken off shell, have to be considered as intermediate building blocks only and thus need not be unique.

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4.7. The Chiral Lagrangian at Order Applying the ideas outlined in Sect. 4.5., it is possible to construct the most general Lagrangian at Here we only quote the result of Ref. [164]:

The numerical values of the low-energy coupling constants are not determined by chiral symmetry. In analogy to and of they are parameters containing information on the underlying dynamics and should, in principle, be calculable in terms of the (remaining) parameters of QCD, namely, the heavyquark masses and the QCD scale In practice, they parameterize our inability to solve the dynamics of QCD in the non-perturbative regime. So far they have either been fixed using empirical input [163, 164, 61] or theoretically using QCD-inspired models [124, 125, 116, 58], meson-resonance saturation [119, 120, 108, 216, 235], and lattice QCD [252, 170]. From a practical point of view the coefficients are also required for another purpose. When calculating one-loop graphs, using vertices from of Eq. (4.70), one generates infinities which, according to Weinberg’s power counting of Eq. (4.52), are of i.e., which cannot be absorbed by a renormalization of the coefficients and In the framework of dimensional regularization (see Appendix A.2) these divergences appear as poles at space-time dimension In Refs. [163, 164] the poles, together with the relevant counter terms, were given in closed form. To that end, Gasser and Leutwyler made use of the so-called saddle-point method which, in the path-integral approach, allows one to identify the one-loop contribution to the generating functional. The action is expanded around the classical solution and the path integral is performed with respect to the terms quadratic in the fluctuations about the classical solution. The resulting one-loop piece of the generating functionalis treated within the dimensional-regularization procedure and the poles are

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isolated by applying the so-called heat-kernel technique.† Except for and the low-energy coupling constants and the “contact terms”—i.e., pure external field and are required in the renormalization of the one-loop graphs [164]. Since and contain only external fields, they are of no physical relevance [164]. By construction Eq. (4.104) represents the most general Lagrangian at and it is thus possible to absorb the one-loop divergences by an appropriate renormalization of the coefficients and [164]:

where R is defined as (see Appendix A.2)

with denoting the number of space-time dimensions and being Euler’s constant. The constants and are given in Table 4.3. The renormalized coefficients depend on the scale µ introduced by dimensional regularization [see Eq. (A.37)] and their values at two different scales and are related by

We will see that the scale dependence of the coefficients and the finite part of the loop-diagrams compensate each other in such a way that physical observables are scale independent. We finally discuss the method of using field transformations to eliminate redundant terms in the most general effective Lagrangian [154, 230, 173, 20, 298]. From a “naive” point of view the two structures

would qualify as independent terms of order Loosely speaking, by using the classical equation of motion of Eq. (4.77) these terms can be shown to be redundant. We will justify this statement in terms of field transformations. † Since the whole procedure is rather technical, we will restrict ourselves, by means of the example

to be discussed in Sect. 4.9.1., to an explicit verification that the renormalization procedure indeed leads to finite predictions for physical observables.

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To that end let us consider another SU(3) matrix by a field transformation of the form

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which is related to

Since both U and are SU(3) matrices, must be a Hermitian traceless 3 × 3 matrix. We demand that satisfies the same symmetry properties as (see Table 4.2),

from which we obtain the following conditions for S:

The most general transformation is constructed iteratively in the momentum and quark-mass expansion,

where the matrices are of satisfy the properties of Eq. (4.112), and have to be constructed from the same building blocks as the effective Lagrangian. To be explicit, let us derive the most general matrix At the field strength tensors cannot contribute as building blocks because of their antisymmetry under interchange of the Lorentz indices. Imposing the transformation behaviour under the group we obtain a list of

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five terms

Parity eliminates three combinations and we are left with

Demanding Hermiticity and a vanishing trace, we end up with two terms at

where and are real numbers. At charge conjugation does not provide an additional constraint. What are the consequences of working with instead of In Sect. 4.6.2., we have already argued, by means of a simple example, that the results for the Green functions are independent of the parameterizations of of Eqs. (4.87) and (4.88). Expressing of Eq. (4.110) by using Eq. (4.116) and inserting the result into of Eq. (4.70), we obtain

where

to leading order in

is given by

The functional form of has been defined in Eq. (4.77). Note, however, that we do not assume We have dropped a total derivative, since it does not modify the dynamics. Both and are of order so that is of order Of course, higher powers of in Eq. (4.118) induce additional terms of higher orders in the momentum expansion which we will discuss in a moment. Through a suitable choice of the parameters and it is possible to eliminate two structures at order i.e., one generates a new Lagrangian with a different functional form which, however, according to the equivalence theorem leads to the same observables [89, 210]. Such a procedure is commonly referred to as using the classical equation of motion to eliminate terms. For example, it is straightforward but tedious to re-express the two structures of Eq. (4.109) through the terms of Gasser and Leutwyler, Eq. (4.104), and the

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following two terms

Choosing and in Eq. (4.116), the two terms of Eq. (4.119) and the modification of Eq. (4.118) precisely cancel and one is left with the canonical form of Gasser and Leutwyler. A field redefinition, of course, also leads to modifications of the functional form of the effective Lagrangians of higher orders. However, for such terms are at least of order as are the higher-order terms in Eq. (4.118). Thus one proceeds iteratively [298]. Using one generates the simplest form of Next one constructs inserts it again into to simplify etc. From a point of view of constructing the simplest Lagrangian at a given order it is sufficient to identify those terms proportional to the classical, i.e., lowest-order, equation of motion and drop them right from the beginning using the argument that, by choosing appropriate generators, they can be transformed away. A completely different situation arises if one tries to express the effective Lagrangian obtained within the framework of a specific model in the canonical form. In such a case it is necessary to explicitly perform the iteration process consistently to a given order and, in particular, take into account the modification of the higher-order coefficients due to the transformation. An explicit example is given in Appendix A.11 of Ref. [33].

4.8.

The Effective Wess-Zumino-Witten Action

The Lagrangians and discussed so far exhibit a larger symmetry than the “real world.” For example, if we consider the case of “pure” QCD, i.e., no external fields except for with the quark mass matrix M of Eq. (4.39), the two Lagrangians are invariant under the substitution As discussed in Sect. 4.3., they contain interaction terms with an even number of Goldstone bosons only, i.e., they are of even intrinsic parity, and it would not be possible to describe the reaction .‡ Analogously, the process cannot be described by and in the presence of external electromagnetic fields. These observations lead us to a discussion of the effective Wess-ZuminoWitten action [328, 327]. Whereas normal Ward identities are related to the invariance of the generating functional under local transformations of the external fields, the anomalous Ward identities [5, 2, 24, 70, 6], which were first obtained in the framework of renormalized perturbation theory, give a par‡

The ø meson can decay into both

and

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ticular form to the variation of the generating functional [328, 163]. Wess and Zumino derived consistency or integrability relations which are satisfied by the anomalous Ward identities and then explicitly constructed a functional involving the pseudoscalar octet which satisfies the anomalous Ward identities [328]. In particular, Wess and Zumino emphasized that their interaction Lagrangians cannot be obtained as part of a chiral invariant Lagrangian. In the construction of Witten [327] the simplest term possible which breaks the symmetry of having only an even number of Goldstone bosons at the Lagrangian level is added to the equation of motion of Eq. (4.77) for the case of massless Goldstone bosons without any external fields,§

where is a (purely imaginary) constant. Substituting in Eq. (4.120) and subsequently multiplying from the left by U and from the right by we verify that the two terms transform with opposite relative signs. Recall that a term which is even (odd) in the Lagrangian leads to a term which, in the equation of motion, is odd (even). However, the action functional corresponding to the new term cannot be written as the four-dimensional integral of a Lagrangian expressed in terms of U and its derivatives. Rather, one has to extend the range of definition of the fields to a hypothetical fifth dimension,

where Minkowski space is defined as the surface of the five-dimensional space for Let us first quote the result of the effective Wess-Zumino-Witten action in the absence of external fields (denoted by a superscript 0):

where the indices run from 0 to 4, completely antisymmetric tensor with §

is the and

In order to conform with our previous convention of Eq. (4.23), we need to substitute Furthermore of Ref. [327] corresponds to Finally,

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By calculating the variation of the action functional as in Eq. (4.74) we find that the constant of Eq. (4.120) and of Eq. (4.122) are related by Using topological arguments Witten showed that the constant appearing in Eq. (4.122) must be an integer. Below, will be identified with the number of colours Expanding the SU(3) matrix in terms of the Goldstone boson fields, one obtains an infinite series of terms, each involving an odd number of Goldstone bosons, i.e., the WZW action is of odd intrinsic parity. For each individual term the integration can be performed explicitly resulting in an ordinary action in terms of a four-dimensional integral of a local Lagrangian. For example, the term with the smallest number of Goldstone bosons reads

In the last step we made use of the fact that exactly one index can take the value 4. The term involving has been integrated with respect to α whereas the other four possibilities cancel each other because the tensor in four dimensions is antisymmetric under a cyclic permutation of the indices whereas the trace is symmetric under a cyclic permutation. In particular, the WZW action without external fields involves at least five Goldstone bosons [328]. The connection to the number of colours is established by introducing a coupling to electromagnetism [328, 327]. In the presence of external fields there will be an additional term in the anomalous action,

given by [90, 278, 244, 59]

with

Introduction to Chiral Perturbation Theory

where we defined the abbreviations and commonly used expression [59], we performed the replacement

387

In the

in order to generate a manifestly C invariant and Hermitian action. Without this replacement charge-conjugation invariance and Hermiticity are satisfied up to a total derivative only. As a special case, let us consider a coupling to external electromagnetic fields by inserting

where is the quark charge matrix [see Eq. (2.110)]. The terms involving three and four electromagnetic four-potentials vanish upon contraction with the totally antisymmetric tensor because their contributions to are symmetric in at least two indices, and we obtain

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We note that the current

by itself is not gauge invariant and the additional terms of Eq. (4.128) are required to obtain a gauge-invariant action. The identification of the constant with the number of colours [327] results from finding in Eq. (4.128) the interaction Lagrangian which is relevant to the decay Since Eq. (4.128) contains a piece

where we made use of a partial integration to shift the derivative from the pion field onto the electromagnetic four-potential. The corresponding invariant amplitude reads

which agrees with a direct calculation of the anomaly term in terms of and quarks with colours (see, e.g., Ref. [311]). After summation over the final photon polarizations and integration over phase space, the decay rate reads

which is in good agreement with the experimental value (7.7 ± 0.6) eV for

4.9. Applications at Order

4.9.1. Masses of the Goldstone Bosons A discussion of the masses at will allow us to illustrate various properties typical of chiral perturbation theory:

1. The relation between the bare low-energy coupling constants and the renormalized coefficients in Eq. (4.105) is such that the divergences of one-loop diagrams are canceled. 2. Similarly, the scale dependence of the coefficients on the one hand and of the finite contributions of the one-loop diagrams on the other hand leads to scale-independent predictions for physical observables.

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3. A perturbation expansion in the explicit symmetry breaking with respect to a symmetry that is realized in the Nambu-Goldstone mode generates corrections which are non-analytic in the symmetry breaking parameter [240], here the quark masses. Let us consider for QCD with finite quark masses but in the absence of external fields. We restrict ourselves to the limit of isospin symmetry, i.e., In order to determine the masses we calculate the self energies of the Goldstone bosons. The propagator of a (pseudo-) scalar field is defined as the Fourier transform of the two-point Green function:

where the index 0 refers to the fact that we still deal with the bare unrenormalized field—not to be confused with a free field without interaction. At lowest order (D = 2) the propagator simply reads

where the lowest-order masses

are given in Eqs. (4.45) - (4.47):

The loop diagrams with and the contact diagrams with result in socalled proper self-energy insertions which may be summed using a geometric series (see Fig. 4.7):

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Note that consists of one-particle-irreducible diagrams only, i.e., diagrams which do not fall apart into two separate pieces when cutting an arbitrary internal line. The physical mass, including the interaction, is defined as the position of the pole of Eq. (4.135),

Let us assume that

can be expanded in a series around

where the remainder depends on the choice of We then obtain for the propagator

and satisfies

Taking in Eq. (4.138) and applying the condition of Eq. (4.136), the propagator may be written as

where we have introduced the wave function renormalization constant

Introducing renormalized fields as gator is given by

the renormalized propa-

In particular, since in the vicinity of the pole, the renormalized propagator behaves as a free propagator with physical mass Let us now turn to the calculation within the framework of ChPT (see, e.g., Ref. [291]). Since and without external fields generate vertices with an even number of Goldstone bosons only, the candidate terms at D = 4 contributing to the self energy are those shown in Fig. 4.8.

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For our particular application with exactly two external meson lines, the relevant interaction Lagrangians can be written as [291]

where

is given by

The terms of proportional to and do not contribute, because they either contain field-strength tensors or external fields only. Since the and terms of Eq. (4,104) are and need not be considered. The only candidates are the terms, of which we consider the term as an explicit example,¶

The remaining terms are treated analogously and we obtain for

¶

For pedagogical reasons, we make use of the physical fields. From a technical point of view, it is often advantageous to work with the Cartesian fields and, at the end of the calculation, express physical processes in terms of the Cartesian components.

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where the constants

At

and

are given by

the self energies are of the form

where the constants and receive a tree-level contribution from and a one-loop contribution with a vertex from (see Fig. 4.8). For the tree-level contribution of this is easily seen, because the Lagrangians of Eq. (4.141) contain either exactly two derivatives of the fields or no derivatives at all. For example, the contact contribution for the reads

where, as usual, generates and for initial and final lines, respectively, and the factor two takes account of two combinations of contracting the fields with external lines. For the one-loop contribution the argument is as follows. The Lagrangian contains either two derivatives or no derivatives at all which, symbolically, can be written as and respectively. The first term results in or depending on whether the or the are contracted with the external fields. The “mixed” situation vanishes upon integration. The second term, does not generate a momentum dependence. As a specific example, we evaluate the pion-loop contribution to the self energy (see Fig. 4.9) by applying the Feynman rule of Eq. (4.93) for

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and

393

||

where 1/2 is a symmetry factor, as explained in Sect. 4.4. The integral of Eq. (4.144) diverges and we thus consider its extension to dimensions in order to make use of the dimensional-regularization technique described in Appendix A.2. In addition to the loop-integral of Eq. (A.37),

||

Note that we work in SU(3) and thus with the exponential parameterization of U.

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where

is given in Eq. (4.107), we need

where we have added

in the numerator. We make use of

in dimensional regularizalion (see the discussion at the end of Appendix A.3.2.2) and obtain

with energy is thus

of Eq. (4.145). The pion-loop contribution to the

self

which is indeed of the type discussed in Eq. (4.143) and diverges as After analyzing all loop contributions and combining them with the contact contributions of Eqs. (4.142), the constants and of Eq. (4.143) are given by

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where, for simplicity, we have suppressed the dependence on the scale µ and the number of dimensions in the integrals [see Eq. (4.145)]. Furthermore, the squared masses appearing in the loop integrals of Eq. (4.146) are given by the predictions of lowest order, Eqs. (4.45)–(4.47). Finally, the integrals I as well as the bare coefficients (with the exception of ) have poles and finite pieces. In particular, the coefficients and are not finite as The masses at are determined by solving the general equation

with the predictions of Eq. (4.143) for the self energies,

where the lowest-order terms, obtain

are given in Eqs. (4.45)–(4.47). We then

because and Expressing the bare coefficients in Eq. (4.146) in terms of the renormalized coefficients by using Eq. (4.105), the results for the masses of the Goldstone bosons at read

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In Eqs. (4.148)–(4.150) we have included the subscripts 2 and 4 in order to indicate from which chiral order the predictions result. First of all, we note that the expressions for the masses are finite. The bare coefficients of the Lagrangian of Gasser and Leutwyler must be infinite in order to cancel the infinities resulting from the divergent loop integrals. Furthermore, at the masses of the Goldstone bosons vanish, if the quark masses are sent to zero. This is, of course, what we had expected from QCD in the chiral limit but it is comforting to see that the self interaction in (in the absence of quark masses) does not generate Goldstone boson masses at higher order. At the squared Goldstone boson masses contain terms which are analytic in the quark masses, namely, of the form multiplied by the renormalized lowenergy coupling constants However, there are also non-analytic terms of the type chiral logarithms—which do not involve new parameters. Such a behaviour is an illustration of the mechanism found by Li and Pagels [240], who noticed that a perturbation theory around a symmetry which is realized in the Nambu-Goldstone mode results in both analytic as well as non-analytic expressions in the perturbation. Finally, the scale dependence of the renormalized coefficients of Eq. (4.108) is by construction such that it cancels the scale dependence of the chiral logarithms. Thus, physical observables do not depend on the scale µ. Let us verify this statement by differentiating Eqs. (4.148)–(4.150) with respect to µ. Using Eq. (4.108),

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we obtain

and, analogously, for the chiral logarithms

As a specific example, let us differentiate the expression for the pion mass

where we made use of the

of Table 4.3.

4.9.2. The Electromagnetic Form Factor of the Pion As a second application at we discuss the electromagnetic (or vector) form factor of the pion in SU(2)×SU(2) chiral perturbation theory. We will work with two commonly used versions of the SU(2) × SU(2) mesonic Lagrangian [163, 144] which are related by a field transformation (see Appendix A.4.1). Furthermore, we will perform the calculation with the two parameterizations for U of Eqs. (4.87) and (4.88). We will thus be able to extend the observations of Sect. 4.6.2., regarding the invariance of physical results under a change of variables to the one-loop level. According to Eq. (2.111), in the two-flavour sector the coupling to the elec-

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tromagnetic field

contains both isoscalar and isovector terms:

i.e.

When evaluating the electromagnetic current operator

between and the isoscalar first term does not contribute,** and the matrix element of the electromagnetic current operator must be of the form††

In other words, we only need to consider Eq. (4.151) which corresponds to a coupling to the third component of the isovector current operator. In the calculations that follow we make use of the parameterization of Eq. (4.87) for the SU(2) matrix

but we will comment along the way on the features, which would differ when using the parameterization of Eq. (4.88). (The equivalence theorem guarantees that physical observables do not depend on the specific choice of parameterization of U [89, 210].) The covariant derivative of U with the external fields of Eq. (4.151) reads [see Eq. (4.58)]

and generates, when inserted into the lowest-order Lagrangian of Eq. (4.70), the interaction term **

The matrix element must be of the form which results in for the neutral pion On the other hand, under charge conjugation and and thus †† A second structure proportional to vanishes for on-mass-shell pions because of current conservation.

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At

399

therefore, the interaction is that of point-like pions with form factor resulting in the Feynman amplitude (see Fig. 4.10)

where denotes the polarization vector of the external real or virtual photon.‡‡ In particular, using the parameterization of Eq. (4.154), all interaction terms

containing one electromagnetic field and pions vanish for This is not the case for the exponential parameterization of Eq. (4.88) which generates the more complicated interaction Lagrangian

which is the same as Eq. (4.155) only at lowest order in the fields. At we need to consider a contact term of (Fig. 4.11) and one-loop diagrams with vertices from (Figs. 4.12 and 4.13). We will first work with the Lagrangian of Gasser and Leutwyler [163], Eq. (A.79) of Appendix A.4.1,*

which produces the contact interaction

‡‡

*

For example, in electron scattering reactions often the polarization vector is used, with four-momentum transfer The low-energy coupling constants of the SU(2)×SU(2) Lagrangian are denoted by tinction to the of the SU(3) × SU(3) Lagrangian.

in dis-

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resulting in the Feynman amplitude

which vanishes for real photons, The second term vanishes if both pions are on the mass shell, but can be of relevance if the vertex is used as an intermediate building block in a calculation such as virtual Compton scattering off the pion [308]. The Feynman amplitude resulting from Eq. (4.158) is the same for both parameterizations. When using the Lagrangian of Ref. [144], Eq. (A.90), one obtains an additional contact interaction,

where

For both parameterizations of U we obtain the addi-

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401

tional term

Let us now turn to the one-loop diagram of Fig. 4.12. The corresponding Feynman amplitude in the parameterization of Eq. (4.154) using the pion-pion vertex of Eq. (4.92) reads

where the 1/2 is a symmetry factor. The integral diverges and its extension to dimensions is given by

where the functions and are defined in Eqs. (A.47), (A.49), and (A.51) of Appendix A.3.1. Inserting the results of Eqs. (A.54) and (A.55) the one-loop contribution of Fig. 4.12 finally reads

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The (infinite) contribution of the first term will be precisely canceled by taking the pion wave function renormalization into account. The second structure is separately gauge invariant and also contains an infinite piece which will be canceled by a corresponding infinite piece of the bare coefficient [see Eq. (4.160)]. Finally, a calculation of the one-loop diagram of Fig. 4.12 with the exponential parameterization of Eq. (4.88) and the pion-pion vertex of Eq. (4.93) yields exactly the same result as Eq. (4.164). Using the exponential parameterization there is a vertex at [see Eq. (4.157)] resulting in the additional loop diagram of Fig. 4.13. The corresponding contribution in dimensional regularization,

also generates an infinite contribution to the vertex at zero four-momentum transfer. The renormalized vertex is obtained by adding the bare contributions and multiplying the result by a factor for each external pion line. The wave function renormalization constant is not an observable and depends on both the parameterization for U and the Lagrangian. The corresponding results are summarized in Table A.2 of Appendix A.4.2. We add the bare contributions which were obtained using the parameterization of Eq. (4.154) and the Lagrangian of Eq. (4.159), Eqs. (4.156), (4.160), and (4.164), and multiply the result by the appropriate wave function renormalization constant [see entry “GL, Eq. (A.92)” of Table A.2],

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The factor counts as because the external electromagnetic field, represented by the polarization vector counts as [see Eq. (4.62)]. It is multiplied by

Since the wave function renormalization constant is (see Appendix A.4.2), it is only the tree-level contribution derived from which gets modified. The product is such that the renormalized vertex is properly normalized to the charge at The factor is and thus the apparent infinity R cannot be canceled through the wave function renormalization. Here it is the connection between the bare parameter and the renormalized parameter which cancels the divergence [see Eq. (9.6) of Ref. [163]]. Moreover, the explicit dependence on the renormalization scale µ cancels with a corresponding scale dependence of the parameter Finally, using the exponential parameterization or the Lagrangian of Ref. [144] results in the same expression as Eq. (4.166). The additional contributions from Eqs. (4.161) and/or (4.165) to the unrenormalized vertex are precisely canceled by modified wave function renormalization constants, resulting in the same renormalized vertex. On the mass shell and we obtain for the electromagnetic form factor [163]

where we replaced the quantities and by their physical values, the error introduced being of order Given a spherically symmetric charge distribution normalized so that the form factor in a nonrelativistic framework is given by

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where denotes the mean square radius.† In analogy, the Lorentz-invariant form factor of Eq. (4.167) is expanded for small as‡

and the charge radius of the pion is defined as

where we made use of Following Ref. [163], we introduce a scale-independent quantity (see Appendix A.4.1)

which can be determined using the empirical information on the charge radius of the pion: In a two-loop calculation of the vector form factor [65], higher-order terms in the chiral expansion terms were also taken into account and a fit to several experimental data sets was performed with the result where the last error is of theoretical origin. Once the value of the parameter has been determined it can be used to predict other processes such as, e.g., virtual Compton scattering off the pion [308, 309]. The results for the electromagnetic form factors of the charged pion, and the charged and neutral kaons in SU(3) × SU(3) chiral perturbation theory at can be found in Refs. [165, 291]. The calculation is very similar to the SU(2)×SU(2) case and the mean square radii of the charged pions and kaons are dominated by the low-energy parameter whereas the one-loop diagrams generate a small contribution only:§

† ‡ §

For neutral particles such as the neutron or the one has Breit-frame kinematics, i.e., comes closest to the nonrelativistic situation. The numerical values in Ref. [165] were obtained with

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405

In Ref. [165] the empirical value of [101] was ¶ The result for the mean square radius of the charged kaon is used to fix then a prediction which has to be compared with the empirical values of [100] and of [14]. In Ref. [81] the empirical data on the charged pion and kaon form factors were analyzed at two-loop order and the low-energy constant including the terms was determined as At the form factor of the receives one-loop contributions only and thus is predicted in terms of the pion-decay constant and the Goldstone boson masses. The mean square radius is given by

which has to be compared with the empirical value [257]. For a two-loop analysis of the neutral-kaon form factor, see Refs. [281, 282]. Since the neutral pion and the eta are their own antiparticles, their electromagnetic vertices vanish because of charge conjugation symmetry as noted in footnote ** for the case of the

4.10. Chiral Perturbation Theory at Mesonic chiral perturbation theory at order has led to a host of successful applications and may be considered as a full-grown and mature area of low-energy particle physics. In this section we will briefly touch upon its extension to [200, 1, 141, 66, 115, 117, 68], which naturally divides into the even- and odd-intrinsic-parity sectors. Calculations in the evenintrinsic-parity sector start at and two-loop calculations at are thus of next-to-next-to-leading order (NNLO). NNLO calculations at have been performed for [32], vector two-point functions [159, 243, 106, 17] and axial-vector two-point functions [17], scattering [63], [96], [80], Sirlin’s combination of SU(3) form factors [281], scalar and electromagnetic form factors of the pion [65], the form factors [18], the electromagnetic form factor of the [282], the form factors [283], and the electromagnetic form factors of pions and kaons in SU(3)×SU(3) ChPT [81]. Further applications deal with more technical aspects such as the evaluation of specific two-loop integrals [284, 176] and the renormalization of the even-intrinsicparity Lagrangian at [67]. ¶

A more recent value is given by [15]. Also (model-dependent) results have been obtained from pion-electroproduction experiments [236, 312].

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The odd-intrinsic-parity sector starts at with the anomalous WZW action, as discussed in Sect. 4.8. In this sector next-to-leading-order (NLO), i.e., one-loop calculations, are of It has been known for some time that quantum corrections to the WZW classical action do not renormalize the coefficient of the WZW term [114, 200, 57, 1, 115, 68]. The counter terms needed to renormalize the one-loop singularities at are of a conventional chirally invariant structure. The inclusion of the photon as a dynamical degree of freedom in the odd-intrinsic-parity sector has been discussed in Ref. [13]. For an overview of applications in the odd-intrinsic-parity sector, we refer to Ref. [59]. A two-loop calculation at for was performed in Ref. [185]. Here, we will mainly be concerned with some aspects of the construction of the most general mesonic chiral Lagrangian at and discuss as an application a two-loop calculation for the scattering lengths [63].

4.10.1. The Mesonic Chiral Lagrangian at Order The rapid increase in the number of free parameters when going from to naturally leads to the expectation of a very large number of chirally invariant structures at One of the problems with the construction of effective Lagrangians at higher orders is that it is far too easy to think of terms satisfying the necessary criteria of Lorentz invariance and invariance under the discrete symmetries as well as chiral transformations. To our knowledge there is neither a general formula, even at for determining the number of independent structures to expect nor an algorithm to decide whether a set of given structures is independent or not. Experience has shown that for almost any sector of higher-order effective chiral Lagrangians the number of terms found to be independent has gone down with time (see Refs. [200, 1, 141, 117, 68] for the odd-intrinsic-parity sector, [141, 66] for the even-intrinsic-parity sector, and [121, 133] for the heavy-baryon Lagrangian, respectively). For that reason, it is important to define a strategy for obtaining all of the independent terms without generating a lot of extraneous terms which have to be eliminated by hand. In the following, we will outline the main principles entering the construction of the Lagrangian and refer the reader to Refs. [141, 66, 117, 68] for more details. The effective Lagrangian is constructed from the elements U, and the field strength tensors and (see Sect. 4.5., in particular, Table 4.2). The external fields and only appear in the field strength tensors or the covariant derivatives which we define as

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In other words, the covariant derivative knows about the transformation property under of the object it acts on and adjusts itself accordingly. With such a convention a product rule analogous to that for ordinary derivatives holds. Given the product Z = XY where X, Y, Z have, according to Eq. (4.172), well-defined but not necessarily the same transformation behaviour, the product rule

applies, which can be easily verified using the definitions of Eq. (4.172). This product rule is valuable as an intermediate step in a number of the derivations of various relations. In order to avoid unnecessary and tedious repetitions during the process of construction, one would like to perform as many manipulations as possible on a formal level without explicit reference to the specific building blocks. It is thus convenient to handle the external field terms and in the same way. To that end we define

and introduce

as a common abbreviation for any of the building blocks and With these definitions, we have only two basic building blocks covariant derivatives acting on them and the respective adjoints. Due to the product rule of Eq. (4.173) it is not necessary to consider derivatives acting on products of these basic terms. All building blocks then transform as U (or ). In terms of the momentum expansion, U is of order 1, of order and each covariant derivative of order Up to this point we have treated a building block and its adjoint on a different footing which we will now remedy by defining the Hermitian and antiHermitian combinations

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where A is taken as or or as some number of covariant deriva|| Here is defined as the square root of U, i.e., tives acting on or In order to discuss the transformation behaviour of Eq. (4.175), we define the SU(3)-valued function referred to as the compensator field [118], through [144]

from which one obtains

From a group-theoretical point of view, K defines a nonlinear realization of SU(3)×SU(3) [144], because**

It is important to note that the first K has the transformed its argument. With these definitions, the building blocks

as transform as

The corresponding covariant derivative is defined as††

where

||

is the so-called connection [118], and is given by

In Ref. [141] the building blocks transforming as were used. The notation of Eq. (4.175) has some advantages when implementing the total-derivative procedure to be discussed below. Moreover, it is more closely related to the conventions used in the baryonic sector (see Sect. 5.1.) ** K does not define an operation of SU(3)×SU(3) on SU(3), because (see Sect. 4.2.1.) †† From an aesthetical point of view it would have been more satisfactory to introduce the covariant derivative as to generate a closer formal correspondence to Eqs. (4.172). However, we follow the standard convention used in the literature.

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As usual, the covariant derivative transforms in the same way as the object it acts on, Invariants under objects, each transforming as consider the trace

are constructed by forming products of and then taking the trace. For example,

where we made use of and the invariance of the trace under cyclic permutations. Obviously, products of such traces are also invariant. Basically, the construction of the most general Lagrangian then proceeds by forming products of elements where A is either or or covariant derivatives of these objects, taking appropriate traces, and forming Lorentz scalars by contracting the Lorentz indices with the metric tensor or the totally antisymmetric tensor After having chosen the set of building blocks and their transformation behaviour, we define a strategy concerning the order of constructing invariants at As shown in Refs. [141, 117], by applying the product rule, it is sufficient to restrict oneself to and integer with because the other combinations and can either be expressed in terms of these or vanish. One then immediately finds that all possible terms at can either include no, one, two, or three blocks which naturally defines four distinct levels to be considered:

1. terms with six 2. terms with four

and one

3. terms with two

and two

4. terms with three We always try to get rid of terms as high in the hierarchy (with the most ) as possible. In particular, with this strategy one ensures that the number of terms is minimal also for the special case in which all external fields are set

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equal to zero.‡‡ The motivation for such an approach is that at each level there exist relations which allow one to eliminate structures, as long as one keeps all terms at the lower levels of the hierarchy. To be specific, when considering multiple covariant derivatives, just one general (i.e., non-contracted) index combination is actually independent in the sense of the hierarchy defined above. For double derivatives this statement reads

i.e., it is sufficient to only keep the block as long as one considers all terms lower in the hierarchy. The construction then proceeds as follows. First of all, write down all conceivable Lorentz-invariant structures satisfying P and C invariance, Hermiticity, and chiral order in terms of the basic building blocks defined above. Then collect as many relations as possible among these structures and use these to eliminate structures. The relations can follow from any of the following mechanisms: (1) partial integration; (2) equation-of-motion argument; (3) epsilon relations; (4) Bianchi identities; (5) trace relations.

Let us illustrate the meaning of each of the above items by selected examples. The partial-integration or total-derivative argument refers to the fact that a total derivative in the Lagrangian density does not change the equation of motion. One thus generates relations of the following type

‡‡

In that sense, the final set given in Ref. [66] for the even-intrinsic-parity sector is not minimal when setting all external fields to zero. In that case the structures and are not independent and can be eliminated. However, if one replaces these two terms by the terms (120) and (139) of Ref. [141], then the entire set will remain independent whether or not there are external fields and both structures, (120) and (139), vanish explicitly when the fields are set to zero.

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411

where we made use of Eq. (4.180) and the product rule. This derivative shifting procedure is also valid for multiple traces. At this stage we note the advantage of working with the basic building blocks of Eq. (4.175) in comparison with those of Ref. [141] due to the relatively simple connection between the covariant derivative outside the block brackets and the covariant derivative inside when are used:

From a technical point of view Eq. (4.185) is important because it helps avoid extremely tedious algebraic manipulations one had to perform in the old framework of Ref. [141]. The combination of shifting derivatives back and forth and interchanging indices of multiple derivatives is referred to as index exchange. In Ref. [141] not all total-derivative terms were properly identified which has led to subsequent reductions in the number of terms in both the even-intrinsicparity sector [66] and the odd-intrinsic-parity sector [117, 68]. The equation-of-motion argument makes use of the invariance of physical observables under field transformations, as discussed in Sect. 4.7. The aim is to collect as many terms as possible containing a factor Such terms can be supplemented by corresponding terms lower in the hierarchy to generate an equation-of-motion term which can be eliminated by a field redefinition. The epsilon relations refer to the odd-intrinsic-parity sector with the basic idea being as follows. Consider a structure with six Lorentz indices transforming under parity as a Lorentz pseudotensor, i.e., In order to form a Lorentz scalar, one needs to contract two indices pairwise and the remaining four with the totally antisymmetric tensor in four dimensions. Suppose is neither symmetric nor antisymmetric under the exchange of any pair of indices. Naively one would then expect 5 + 4 + · · ·+ 1 = 15 independent contractions. However, such a counting does not take the totally antisymmetric nature of the epsilon tensor into account [ 1 ], from which one obtains, for the above case of no symmetry in the indices, five additional conditions [1, 141]. These additional identities have not been considered in the pioneering construction of Ref. [200]. In general, has some symmetry in its indices, and not all five epsilon relations are independent. For example, using the transformation behaviour of Table 4.2 it is easy to verify that

is an example for a pseudotensor which is invariant under G (and Hermitian).

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Stefan Scherer

Its symmetries are given by

from which one would naively end up with a single combination

which, however, vanishes due to the epsilon relation. The Bianchi identities refer to certain relations among covariant derivatives of field-strength tensors. Starting from the Jacobi identity

we consider the linear combination

where use of the Schwarz theorem, relabeling of indices, and the Jacobi identity, Eq. (4.186), has been made. Observe that the cyclic permutations of the indices µ, and has been denoted by The same arguments hold for the independent field strength tensor and we can summarize the constraints as

which, because of their similarity to an analogous equation for the RiemannChristoffel curvature tensor in general relativity, are referred to as the Bianchi identities (see, e.g., Refs. [292, 324]). Equation (4.188) does not require that satisfy any equations of motion. In terms of the building blocks and the Bianchi identities read

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413

Given the definition of Eq. (4.174), the Bianchi identities can be used to generate relations among the building blocks in terms of structures kept in lower orders of the hierarchy defined above. In Ref. [141] each term of the sum on the left-hand side of Eqs. (4.189) and (4.190) was treated as an independent element so that the final list of supposedly independent structures contained redundant elements. Finally, the trace relations refer to the fact that the construction of invariants uses traces and products of traces. One is thus particularly interested in finding any relations among those traces. We know from the Cayley-Hamilton theorem that any matrix A is a solution of its associated characteristical polynomial For this statement reads

Setting in (4.191) and making use of one ends up with the matrix equation

which is the central piece of information needed to derive the trace relations in the SU(2)×SU(2) sector. The analogous equation is slightly more complex

We can now derive trace relations by simply multiplying Eq. (4.192) or Eq. (4.193) with another arbitrary matrix of the same dimension and finally

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taking the trace of the whole construction, i.e.,

Note that may be any matrix, even a string of our basic building blocks. For example, Eq. (A.84) of Appendix A.4.1 is identical to Eq. (4.194). In principle, the ideas developed above apply to the general case and only at the end it is necessary to specify the number of flavours The reduction to the cases and is achieved in terms of the trace relations summarized in Eqs. (4.194) and (4.195). Although we have never come across a trace relation that could not be obtained in the manner explained above, we are not aware of a general proof showing that any kind of trace relation must be related to the Cayley-Hamilton theorem. In the even-intrinsic-parity sector the Lagrangian has 112 in principle measurable + 3 contact terms for the general case, 90 + 4 for the case, and 53 + 4 for the case [66]. The contact terms refer to structures which can be expressed in terms of only external fields such as the and terms of the Lagrangian of Eq. (4.104). The reduction in the number of terms in comparison with the 111 structures of Ref. [141] is due to a more complete application of the partial-integration relations, the use of additional trace relations, and the use of four relations due to the Bianchi identities which were not taken into account in Ref. [141]. The odd-intrinsic-parity sector was reconsidered in Refs. [117, 68]. Both analyses found 24 and 5 terms. Moreover, 8 additional terms due to the extension of the chiral group to were found, which are of some relevance when considering the electromagnetic interaction for the two-flavour case. In comparison to Ref. [141], the new analysis of Ref. [117] could eliminate two structures via partial integration, 6 via Bianchi identities and one by a trace relation. It is unlikely that the coefficients of all the terms of will be determined from experiment. However, usually a much smaller subset actually contributes to most simple processes, and it is possible to get information on some of the corresponding coefficients.

4.10.2. Elastic Pion-Pion Scattering at Elastic pion-pion scattering represents a nice example of the success of mesonic chiral perturbation theory. A complete analytical calculation at twoloop order was performed in Ref. [63].

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415

Let us consider the T-matrix element of the scattering process

where and denote the usual Mandelstam variables, the indices refer to the Cartesian isospin components, and the function A satisfies [318]. Since the pions form an isospin triplet, the two isovectors of both the initial and final states may be coupled to I = 0,1,2. For the strong interactions are invariant under isospin transformations, implying that scattering matrix elements can be decomposed as

For the case of scattering the three isospin amplitudes are given in terms of the invariant amplitude A of Eq. (4.196) by [163]

For example, the physical scattering process is described by Other physical processes are obtained using the appropriate Clebsch-Gordan coefficients. Evaluating the T matrices at threshold, one obtains the lengths*

where the subscript 0 refers to wave and the superscript to the isospin. vanishes because of Bose symmetry). The current-algebra prediction of Ref. [318] is identical with the lowest-order result obtained from Eqs. (4.92) or (4.93),

where we replaced by and made use of the numerical values MeV and MeV of Ref. [63]. In order to obtain the results of Eq. (4.200), use has been made of and *

The definition differs by a factor of [163] from the conventional definition of scattering lengths in the effective range expansion (see, e.g., Ref. [285]).

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The predictions for the

scattering lengths at

read [63]

The corrections at consist of a dominant part from the chiral logarithms (L) of the one-loop diagrams and a less important analytical contribution (anal.) resulting from the one-loop diagrams as well as the tree graphs of The total corrections at amount to 28% and 21% of the predictions, respectively. At one obtains two-loop corrections, one-loop corrections, and tree-level contributions. Once again, the loop corrections ( involving double chiral logarithms, and L) are more important than the analytical contributions. The influence of was estimated via scalar- and vector-meson exchange and found to be very small. The empirical results for the scattering lengths are, so far, obtained from the decay and the reactions. In the former case, the connection with low-energy scattering stems from a partial-wave analysis of the form factors relevant for the decay in terms of angular momentum eigenstates. In the low-energy regime the phases of these form factors are related by (a generalization of) Watson’s theorem [316] to the corresponding phases of I = 0 and I = 1 elastic scattering [97]. Using a dispersion-theory approach in terms of the Roy equations [290, 19], the most recent analysis of [276] has obtained This result has to be compared with older determinations [289, 138, 263]

and the more recent one from

[211]

which makes use of an extrapolation to the pion pole to extract the amplitude. In particular, when analyzing the data of Ref. [276] in combination with the Roy equations, an upper limit was obtained in Ref. [97] for the

Introduction to Chiral Perturbation Theory

417

scale-independent low-energy coupling constant which is related to of the SU(2)×SU(2) Lagrangian of Gasser and Leutwyler (see Appendix A.4.1). The great interest generated by this result is to be understood in the context of the pion mass at [see Eq. (A.96) of Appendix A.4.2],

where Recall that the constant is related to the scalar quark condensate in the chiral limit [see Eq. (4.43)] and that a nonvanishing quark condensate is a sufficient criterion for spontaneous chiral symmetry breakdown in QCD (see Sect. 4.1.2.). If the expansion of in powers of the quark masses is dominated by the linear term in Eq. (4.204), the result is often referred to as the Gell-Mann-Oakes-Renner relation [150]. If the terms of order were comparable or even larger than the linear terms, a different power counting or bookkeeping in ChPT would be required [218, 219, 299]. The estimate implies that the Gell-Mann-Oakes-Renner relation [150] is indeed a decent starting point, because the contribution of the second term of Eq. (4.204) to the pion mass is approximately given by

i.e., more than 94 % of the pion mass must stem from the quark condensate [97].

5.

CHIRAL PERTURBATION THEORY FOR BARYONS

So far we have considered the purely mesonic sector involving the interaction of Goldstone bosons with each other and with the external fields. However, ChPT can be extended to also describe the dynamics of baryons at low energies. Here we will concentrate on matrix elements with a single baryon in the initial and final states. With such matrix elements we can, e.g, describe static properties such as masses or magnetic moments, form factors, or, finally, more complicated processes, such as pion-nucleon scattering, Compton scattering, pion photoproduction etc. Technically speaking, we are interested in the baryon-to-baryon transition amplitude in the presence of external fields (as opposed to the vacuum-to-vacuum transition amplitude of Sect. 2.4.4.) [144, 224],

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Stefan Scherer

determined by the Lagrangian of Eq. (2.95),

In Eq. (5.1) and denote asymptotic one-baryon in- and outstates, i.e., states which in the remote past and distant future behave as free oneparticle states of momentum and respectively. The functional consists of connected diagrams only (superscript c). For example, the matrix elements of the vector and axial-vector currents between one-baryon states are given by [224]

where

denotes the quark-mass matrix and

As in the mesonic sector the method of calculating the Green functions associated with the functional of Eq. (5.1) consists of an effective Lagrangianapproach in combination with an appropriate power counting. Specific matrix elements will be calculated applying the Gell-Mann and Low formula of perturbation theory [160]. The group-theoretical foundations of constructing phenomenological Lagrangians in the presence of spontaneous symmetry breaking have been developed in Refs. [320, 95, 85]. The fields entering the Lagrangian are assumed to transform under irreducible representations of the subgroup H which leaves the vacuum invariant whereas the symmetry group G of the Hamiltonian or Lagrangian is nonlinearly realized (for the transformation behaviour of the Goldstone bosons, see Sect. 4.2.)

5.1.

Transformation Properties of the Fields

Our aim is a description of the interaction of baryons with the Goldstone bosons as well as the external fields at low energies. To that end we need to specify the transformation properties of the dynamical fields entering the Lagrangian. Our discussion follows Refs. [153, 144]. To be specific, we consider the octet of the baryons (see Fig. 5.1). With each member of the octet we associate a complex, four-component Dirac field which we arrange in a traceless 3 × 3 matrix B,

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419

where we have suppressed the dependence on For later use, we have to keep in mind that each entry of Eq. (5.5) is a Dirac field, but for the purpose of discussing the transformation properties under global flavour SU(3) this can be ignored, because these transformations act on each of the four components in the same way. In contrast to the mesonic case of Eq. (4.28), where we collected the fields of the Goldstone octet in a Hermitian traceless matrix the of the spin 1/2-case are not real (Hermitian), i.e., Now let us define the set which under the addition of matrices is a complex vector space. The following homomorphism is a representation of the abstract group on the vector space M [see also Eq. (4.25)]:

420

First of all,

Stefan Scherer

is again an element of M, because Equation (5.7) satisfies the homomorphism

and is indeed a representation of SU(3), because

Equation (5.7) is just the familiar statement that B transforms as an octet under † (the adjoint representation of) Let us now turn to various representations and realizations of the group We consider two explicit examples and refer the interested reader to Ref. [153] for more details. In analogy to the discussion of the quark fields in QCD, we may introduce left- and right-handed components of the baryon fields [see Eq. (2.33)]:

We define the set which under the addition of matrices is a complex vector space. The following homomorphism is a representation of the abstract group on

where we have suppressed the dependence. The proof proceeds in complete analogy to that of Eq. (5.7). As a second example, consider the set with the homomorphism i.e., the transformation behaviour is independent of . The mapping defines a representation of the group although the transformation †

Technically speaking the adjoint representation is faithful (one-to-one) modulo the center Z of SU(3) which is defined as the set of all elements commuting with all elements of SU(3) and is given by

Introduction to Chiral Perturbation Theory

421

behaviour is drastically different from the first example. However, the important feature which both mappings have in common is that under the subgroup of G both fields transform as an octet:

We will now show how in a theory also containing Goldstone bosons the various realizations may be connected to each other using field redefinitions. The procedure is actually very similar to Sect. 4.10.1., where we discussed how, by an appropriate multiplication with U or all building blocks of the mesonic effective Lagrangian could be made to transform in the same way. Here we consider the second example, with the fields of Eq. (5.10) and U of Eq. (4.28) transforming as

and define new baryon fields by

so that the new pair

Note in particular that

transforms as

still transforms as an octet under the subgroup

Given that physical observable are invariant under field transformations we may choose a description of baryons that is maximally convenient for the construction of the effective Lagrangian [153] and which is commonly used in chiral perturbation theory. We start with and consider the case of later. Let

denote the nucleon field with two four-component Dirac fields for the proton and the neutron and U the SU(2) matrix containing the pion fields. We have already seen in Sect. 4.2.2., that the mapping defines a nonlinear realization of G. We denote the square root of U by and define the SU(2)-valued function K(L, R, U) by [see Eqs. (4.176) and (4.177)]

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Stefan Scherer

i.e.

The following homomorphism defines an operation of G on the set in terms of a nonlinear realization:

because the identity leaves

invariant and [see Sect. 4.2.2., and Eq. (4.178)]

Note that for a general group element the transformation behaviour of depends on U. For the special case of an isospin transformation, = L = V, one obtains because

Comparing with Eq. (5.12) yields or K(V, V, U) = V, i.e., transforms linearly as an isospin doublet under the isospin subgroup A general feature here is that the transformation behaviour under the subgroup which leaves the ground state invariant is independent of U. Moreover, as already discussed in Sect. 4.2.2., the Goldstone bosons transform according to the adjoint representation of i.e., as an isospin triplet. For the case one uses the nonlinear realization

where K is defined completely analogously to Eq. (5.12) after inserting the corresponding SU(3) matrices.

Introduction to Chiral Perturbation Theory

5.2.

423

Lowest-Order Effective Baryonic Lagrangian

Given the dynamical fields of Eqs. (5.13) and (5.14) and their transformation properties, we will now discuss the most general effective baryonic Lagrangian at lowest order. As in the vacuum sector, chiral symmetry provides constraints among the single-baryon Green functions contained in the functional of Eq. (5.1). Analogous to the mesonic sector, these Ward identities will be satisfied if the Green functions are calculated from the most general effective Lagrangian coupled to external fields with a local invariance under the chiral group (see Appendix A.1). Let us start with the construction of the effective Lagrangian which we demand to have a local symmetry. The transformation behaviour of the external fields is given in Eq. (2.109), whereas the nucleon doublet and U transform as

The local character of the transformation implies that we need to introduce a covariant derivative with the usual property that it transforms in the same way as [compare with Eq. (2.17) for the case of QED]:

Since K not only depends on covariant derivative to contain nection of Eq. (4.181) (recall

and but also on U, we may expect the and and their derivatives. In fact, the con-

is also an integral part of the covariant derivative of the nucleon doublet:

What needs to be shown is

To that end, we make use of the product rule,

Stefan Scherer

424

in Eq. (5.19) and multiply by

reducing it to

Using Eq. (5.12),

we find

i.e., the covariant derivative defined in Eq. (5.18) indeed satisfies the condition of Eq. (5.16). At there exists another Hermitian building block, the socalled vielbein [118],‡

which under parity transforms as an axial vector:

‡ The relation with the notation of Sect. 4.10.1., is given by

[117].

Introduction to Chiral Perturbation Theory

The transformation behaviour under

425

is given by

which is shown using [see Eq. (5.12)]

and the corresponding adjoints. We obtain

The most general effective Lagrangian describing processes with a single nucleon in the initial and final states is then of the type where is an operator acting in Dirac and flavour space, transforming under As in the mesonic sector, the Lagrangian must be a Hermitian Lorentz scalar which is even under the discrete symmetries C, P, and T. The most general such Lagrangian with the smallest number of derivatives is given by [144]§

It contains two parameters not determined by chiral symmetry: the nucleon

mass

and the axial-vector coupling constant

§ The power counting will be discussed below.

both taken in the chiral

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Stefan Scherer

limit (denoted by ). The overall normalization of the Lagrangian is chosen such that in the case of no external fields and no pion fields it reduces to that of a free nucleon of mass Since the nucleon mass does not vanish in the chiral limit, the zeroth component of the partial derivative acting on the nucleon field does not produce a “small” quantity. We thus have to address the new features of chiral power counting in the baryonic sector. The counting of the external fields as well as of covariant derivatives acting on the mesonic fields remains the same as in mesonic chiral perturbation theory [see Eq. (4.62) of Sect. 4.5.]. On the other hand, the counting of bilinears is probably easiest understood by investigating the matrix elements of positive-energy plane-wave solutions to the free Dirac equation in the Dirac representation:

where

denotes a two-component Pauli spinor and with In the low-energy limit, i.e., for nonrelativistic kinematics, the lower (small) component is suppressed as in comparison with the upper (large) component. For the analysis of the bilinears it is convenient to divide the 16 Dirac matrices into even and odd ones, and [143, 129], respectively, where odd matrices couple large and small components but not large with large, whereas even matrices do not. Finally, acting on the nucleon solution produces which we write symbolically as where we count the second term as i.e., as a small quantity. We are now in the position to summarize the chiral counting scheme for the (new) elements of baryon chiral perturbation theory [224]:

where the order given is the minimal one. For example, has both an piece, as well as an piece, A rigorous nonrelativistic reduction may be achieved in the framework of the Foldy-Wouthuysen method [143] or the heavy-baryon approach [206, 37] which will be discussed later (for a pedagogical introduction see Ref. [194]). The construction of the Lagrangian proceeds similarly except for the fact that the baryon fields are contained in the 3 × 3 matrix of Eq. (5.5) transforming as As in the mesonic sector, the building blocks are written as products transforming as with a trace taken at the end.

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427

The lowest-order Lagrangian reads [153, 224]

where denotes the mass of the baryon octet in the chiral limit. The covariant derivative of B is defined as

with of Eq. (5.17) [for ]. The constants D and F may be determined by fitting the semi-leptonic decays at tree level [76]:

5.3.

Applications at Tree Level

5.3.1. Goldberger-Treiman Relation and the Axial-Vector Current Matrix Element We have seen in Sect. 2.3.6., that the quark masses in QCD give rise to a non-vanishing divergence of the axial-vector current operator [see Eq. (2.84)]. Here we will discuss the implications for the matrix elements of the pseudoscalar density and of the axial-vector current evaluated between singlenucleon states in terms of the lowest-order Lagrangians of Eqs. (4.70) and (5.21). In particular, we will see that the Ward identity

where is satisfied. The nucleon matrix element of the pseudoscalar density can be parameterized as

where Equation (5.28) defines the form factor in terms of the QCD operator As we have seen in the discussion of scattering of Sect. 4.6.2., the operator serves as an interpolating pion field [see Eq. (4.99)], and thus is also referred to as the pion-nucleon form factor (for this specific choice of the interpolating pion field). The pionnucleon coupling constant is defined through evaluated at

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The Lagrangian of Eq. (5.21) does not generate a direct coupling of an external pseudoscalar field to the nucleon, i.e., it does not contain any terms involving or At lowest order in the chiral expansion, the matrix element of the pseudoscalar density is therefore given in terms of the diagram of Fig. 5.2, i.e., the pseudoscalar source produces a pion which propagates and is then absorbed by the nucleon. The coupling of a pseudoscalar field to the pion in the framework of has already been discussed in Eq. (4.96),

When working with the nonlinear realization of Eq. (5.13) it is convenient to use the exponential parameterization of Eq. (4.88),

because in that case the square root is simply given by

According to Fig. 5.2, we need to identify the interaction term of a nucleon with a single pion. In the absence of external fields the vielbein of Eq. (5.20) is odd in the pion fields,

Introduction to Chiral Perturbation Theory

Expanding

and

429

as

we obtain

which, when inserted into tion Lagrangian:

of Eq. (5.21), generates the following interac-

(Note that the sign is opposite to the conventionally used pseudovector pionnucleon coupling.¶) The Feynman rule for the vertex of an incoming pion with four-momentum and Cartesian isospin index is given by

On the other hand, the connection of Eq. (5.17) with the external fields set to zero is even in the pion fields,

i.e., it does not contribute to the single-pion vertex. We now put the individual pieces together and obtain for the diagram of Fig. 5.2

¶ In fact, also the definition of the pion-nucleon form factor of Eq. (5.28) contains a sign opposite

to the standard convention so that, in the end, the Goldberger-Treiman relation emerges with the conventional sign.

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where we used and the Dirac equation to show At so that, by comparison with Eq. (5.28), we can read off the lowest-order result

i.e., at this order the form factor does not depend on In general, the pionnucleon coupling constant is defined at which, in the present case, simply yields

Equation (5.37) represents the famous Goldberger-Treiman relation [177, 178, 264] which establishes a connection between quantities entering weak processes, and (to be discussed below), and a typical strong-interaction quantity, namely the pion-nucleon coupling constant The numerical violation of the Goldberger-Treiman relation, as expressed in the so-called Goldberger-Treiman discrepancy

is at the percent level,|| although one has to keep in mind that all four physical quantities move from their chiral-limit values etc., to the empirical ones etc. Using Lorentz covariance and isospin symmetry, the matrix element of the axial-vector current between initial and final nucleon states—excluding second-class currents [317]—can be parameterized as**

|| Using

and one obtains ** The terminology “first and second classes” refers to the transformation property of strangenessconserving semi-leptonic weak interactions under conjugation [317] which is the product of charge symmetry and charge conjugation A second-class contribution would show up in terms of a third form factor contributing as

Assuming a perfect

-conjugation symmetry, the form factor

vanishes.

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431

where and and are the axial and induced pseudoscalar form factors, respectively. At lowest order, an external axial-vector field couples directly to the nucleon as which is obtained from through coupling to the pions is obtained from

The with

which gives rise to a diagram similar to Fig. 5.2, with The matrix element is thus given by

replaced by

from which we obtain, by applying the Dirac equation,

At this order the axial form factor does not yet show dependence. The axialvector coupling constant is defined as which is simply given by We have thus identified the second new parameter of besides the nucleon mass The induced pseudoscalar form factor is determined by the pion exchange which is the simplest version of the so-called pion-pole dominance. The behaviour of is not in conflict with the book-keeping of a calculation at chiral order because, according to Eq. (4.62), the external axial-vector field counts as and the definition of the matrix element contains a momentum and the Dirac matrix [see Eq. (5.23)] so that the combined order of all elements is indeed It is straightforward to verify that the form factors of Eqs. (5.36), (5.42), and (5.43) satisfy the relation

which is required by the Ward identity of Eq. (5.27) with the parameterizations of Eqs. (5.28) and (5.39) for the matrix elements. In other words, only two

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of the three form factors and are independent. Note that this relation is not restricted to small values of but holds for any In the chiral limit, Eq. (5.27) implies

which also follows from Eq. (5.42) and Eq. (5.43) for

Equation (5.45)

for requires that in the chiral limit the induced pseudoscalar form factor has a pole in the limit The interpretation of this pole is, of course, given in terms of the exchange of a massless Goldstone pion. To understand this in more detail consider the most general contribution of the pion exchange to the axial-vector current matrix element:

where for the renormalized propagator [see Eq. (4.137)]. The functions and denote the most general parameterizations for the pion-decay vertex and the pion-nucleon vertex (note that we have not specified the interpolating pion field). For general their values depend on the interpolating field, but for they are identical with the pion-decay constant and the pion-nucleon coupling constant respectively. In the chiral limit, we obtain

where In other words, the most general contribution of a massless pion to the induced pseudoscalar form factor in the chiral limit is given by

We divide the pseudoscalar form factor into the pion contribution and the rest. Making use of Eq. (5.45), we consider the limit r

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433

from which we obtain the Goldberger-Treiman relation

Of course, we have assumed that there is no other massless particle in the theory which could produce a pole in the residual part

5.3.2.

Pion-Nucleon Scattering at Tree Level

As another example, we will consider pion-nucleon scattering and show how the effective Lagrangian of Eq. (5.21) reproduces the Weinberg-Tomozawa predictions for the scattering lengths [318, 305]. We will contrast the results with those of a tree-level calculation within pseudoscalar (PS) and pseudovector (PV) pion-nucleon couplings. Before calculating the scattering amplitude within ChPT we introduce a general parameterization of the invariant amplitude for the process

with the two independent scalar kinematical variables

where variables satisfying ††

and

are the usual Mandelstam From pion-crossing symmetry

One also finds the parameterization [72]

with

where, for simplicity, we have omitted the isospin indices.

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we obtain for the crossing behaviour of the amplitudes

As in scattering one often also finds the isospin decomposition as in Eq. (4.197), where the relation between the two sets is given by [128]

Let us turn to the tree-level approximation to the scattering amplitude as obtained from of Eq. (5.21). In order to derive the relevant interaction Lagrangians from Eq. (5.21), we reconsider the connection of Eq. (5.17) with the external fields set to zero and obtain

The linear pion-nucleon interaction term was already derived in Eq. (5.33) so that we end up with the following interaction Lagrangian:

The first term is the pseudovector pion-nucleon coupling and the second the contact interaction with two factors of the pion field interacting with the nucleon at a single point. The Feynman rules for the vertices derived from Eq. (5.51) read for an incoming pion with four-momentum and Cartesian isospin index

for an incoming pion with

and an outgoing pion with

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435

The latter gives the contact contribution to

We emphasize that such a term is not present in a conventional calculation with either a pseudoscalar or a pseudovector pion-nucleon interaction. For the and nucleon-pole diagrams the pseudovector vertex appears twice and we obtain

The and contributions are related to each other through pion crossing and In what follows we explicitly calculate only the channel and make use of pion-crossing symmetry at the end to obtain the result. Moreover, we perform the manipulations such that the result of pseudoscalar coupling may also be read off. Using the Dirac equation, we rewrite

and obtain

We repeat the above procedure

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yielding

where, for the identification of the PS-coupling result, one has to make use of the Goldberger-Treiman relation [177, 178, 264] (see Sect. 5.3.1.)

where ing

denotes the pion-nucleon coupling constant in the chiral limit. Us-

we find

where we again made use of the Dirac equation. We finally obtain for the contribution

As noted above, the expression for the and

We combine the

and

channel results from the substitution

contributions using

Introduction to Chiral Perturbation Theory

437

and

and summarize the contributions to the functions Table 5.1 [see also Eq. (A.26) of Ref. [144]].

and

of Eq. (5.46) in

In order to extract the scattering lengths, let us consider threshold kinematics

Since we only work at lowest-order tree level, we replace Together with‡‡

we find for the threshold matrix element

Using

‡‡Recall that we use the normalization

etc.

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we obtain

where we have indicated the results for the various coupling schemes. Let us discuss the scattering lengths resulting from Eq. (5.61). Using the above normalization for the Dirac spinors, the differential cross section in the center-of-mass frame is given by [128]

which, at threshold, reduces to

The

scattering lengths are defined as*

The subscript 0+ refers to the fact that the system is in an orbital wave with total angular momentum 1/2 = 0 + 1/2. Inserting the results of

* The threshold parameters are defined in terms of a multipole expansion of the

amplitude [87]. The sign convention for the the convention of the effective range expansion.

scattering parameters

scattering is opposite to

Introduction to Chiral Perturbation Theory

439

Table 5.1 we obtain‡ where we have also indicated the chiral order. Taking the linear combinations and [see Eq. (5.49)], we see that the results of Eqs. (5.65) and (5.66) indeed satisfy the WeinbergTomozawa relation [318, 305]:‡

As in scattering, the scattering lengths vanish in the chiral limit reflecting the fact that the interaction of Goldstone bosons vanishes in the zero-energy limit. The pseudoscalar pion-nucleon interaction produces a scattering length proportional to instead of and is clearly in conflict with the requirements of chiral symmetry. Moreover, the scattering length of the pseudoscalar coupling is too large by a factor in comparison with the twopion contact term of Eq. (5.54) (sometimes also referred to as the WeinbergTomozawa term) induced by the nonlinear realization of chiral symmetry. On the other hand, the pseudovector pion-nucleon interaction gives a totally wrong result for because it misses the two-pion contact term of Eq. (5.54). Using the values

the numerical results for the scattering lengths are given in Table 5.2. We have included the full results of Eqs. (5.65) and (5.66) and the consistent corresponding prediction at The results of heavy-baryon chiral perturbation theory (HBChPT) (see Sect. 5.5.) are taken from Ref. [256]. At the calculation involves nine low-energy constants of the chiral Lagrangian which have been fit to the extrapolated threshold parameters of the partial wave analysis of Ref. [223], the pion-nucleon term and the Goldberger-Treiman discrepancy. Up to and including the HBChPT calculation contains 14 free parameters [136]. In Ref. [136] the complete one-loop amplitude at was fit to the phase shifts provided by three different partial wave analyses [221] [I], [249] [II], and SP98 of [293] [III]. Table 5.2 includes the results for the ‡ We do not expand the fraction because the dependence is not of dynamical origin. ‡ The result, in principle, holds for a general target of isospin T (except for the pion) after replacing

3/4 by T(T + 1) and

by

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scattering lengths obtained from those fits in combination with the empirical values of the three analyses. Finally, the results of the recently proposed manifestly Lorentz-invariant form of baryon ChPT [R(elativistic)BChPT] [71] (see Sect. 5.6.) are included up to [72]. The first entries (a) refer to a dispersive representation of the function entering the threshold matrix element [see Eq. (5.64) and recall ] whereas the second entries (b) involve only the one-loop approximation. Whereas for there is no difference, the value for differs substantially which has been interpreted as the result of an insufficient approximation of the one-loop representation to allow for an extrapolation from the Cheng-Dashen point to the physical region [72]. The empirical results quoted have been taken from low-energy partial-wave analyses [221, 249] and recent precision X-ray experiments on pionic hydrogen and deuterium [297].

5.4.

Examples of Loop Diagrams

In Sect. 4.4., we saw that, in the purely mesonic sector, contributions of diagrams are at least of order i.e., they are suppressed by in comparison with tree-level diagrams. An important ingredient in deriving this result was the fact that we treated the squared pion mass as a small quantity of order Such an approach is motivated by the observation that the masses of the Goldstone bosons must vanish in the chiral limit. In the frame-

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441

work of ordinary chiral perturbation theory [see Eq. (4.45) and the discussion at the end of Sect. 4.10.2.] which translates into a momentum expansion of observables at fixed ratio On the other hand, there is no reason to believe that the masses of hadrons other than the Goldstone bosons should vanish or become small in the chiral limit. In other words, the nucleon mass entering the pion-nucleon Lagrangian of Eq. (5.21) should—as already anticipated in the discussion following Eq. (5.21)—not be treated as a small quantity of, say, order Naturally the question arises how all this affects the calculation of loop diagrams and the setup of a consistent power counting scheme. We will follow Ref. [144] and consider, for illustrative purposes, two examples: a one-loop contribution to the nucleon mass and a loop diagram contributing to scattering.

5.4.1.

First Example: One-Loop Correction to the Nucleon Mass

The discussion of the modification of the nucleon mass due to pion loops is very similar to that of Sect. 4.9.1., for the masses of the Goldstone bosons. The lowest-order Feynman propagator of the nucleon, corresponding to the free-field part of of Eq. (5.21),

is modified by the self energy (see for example the one-loop contribution of Fig. 5.3) in a way analogous to the modification of the meson propagator in Eq. (4.135),

resulting in the full (but still unrenormalized) propagator

In the absence of external fields (but including the quark mass term), the most general expression for the self energy can be written as

where and are as yet undetermined functions of the invariant We assume that and may be determined in a perturbative (momentum or loop)

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expansion which, symbolically, we denote by some indicator

When switching off the interaction, we would like to recover the lowest-order result of Eq. (5.69), i.e., implying The mass of the nucleon is defined through the position of the pole of the full propagator, i.e., for we require

from which we obtain

The perturbative result to first order in

[Note that the argument

reads

of the functions

and

also has to be expanded

in powers of The wave function renormalization constant is defined through the residue at

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443

i.e., the renormalized propagator, defined through has a pole at with residue 1. Using Eq. (5.73) we find that near the pole at

yielding for the wave function renormalization constant

With these definitions let us consider the contribution of Fig. 5.3 to the self energy, where, for the sake of simplicity, we perform the calculation in the chiral limit Using the vertex of Eq. (5.34) we obtain the contribution of the self energy

Counting powers we see that the integral has a cubic divergence. We make use of (normal) dimensional regularization [203], where the integrand is first simplified using§

In the standard fashion, we first insert

§ For a recent discussion of the problem with

in dimensional regularization, see Ref. [203]. Since we are neither dealing with matrix elements containing anomalies nor considering closed fermion loops, we can safely make use of normal dimensional regularization [144, 203].

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simplify the numerator using Eq. (5.78),

and obtain, with

Indeed, when discussing the contribution to the nucleon mass [see Eq. (5.74] we only need to consider the first integral of Eq. (5.79), because the second term does not contribute at and the third term vanishes in dimensional regularization because the integrand is odd in Using Eqs. (A.42) and (A.43) of Appendix A.3.1.1 with the replacement we obtain, in the language of Eq. (5.72),

Applying Eq. (5.74) we find for the nucleon mass including the one-loop contribution of Fig. 5.3 [see Eq. (4.1) of Ref. [144]]

where

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The pion loop of Fig. 5.3 generates an (infinite) contribution to the nucleon mass, even in the chiral limit, i.e., the parameter of needs to be renormalized. The same is true for the second parameter [144]. This situation is completely different from the mesonic sector, where the two parameters and of the lowest-order Lagrangian do not change due to higher-order corrections in the chiral limit. For example, in the SU(2)×SU(2) sector, the pion-decay constant at is given by [see Eq. (12.2) of Ref. [163]]

where and the scale-independent low-energy parameter is defined in Eq. (A.81). Since in the chiral limit the piondecay constant in the chiral limit is still given by of Similarly, in the chiral limit the Goldstone boson masses vanish, not only at but also at higher orders, as we have seen in Eqs. (4.148)–(4.150).

5.4.2. Second Example: One-Loop Correction to

Scattering

In the previous section we have seen that the parameters of the lowest-order Lagrangian must be renormalized in the chiral limit. As a second example, we will discuss the -scattering loop diagram of Fig. 5.4, which will allow us to draw some further conclusions regarding the differences between the mesonic and baryonic sectors of ChPT.

Given the Feynman rule of Eq. (5.53), the contribution of Fig. 5.4 to the

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invariant amplitude reads

where, counting powers, we expect the integral to have a cubic divergence. The isospin structure is given by

i.e., the diagram contributes to both

isospin amplitudes. We obtain

We will outline the evaluation of the integral using dimensional regularization. To do this, we first combine the denominators using Feynman’s trick, Eq. (A.46) of Appendix A.3.1.2, yielding

where Shifting the integration variables as the amplitude reads

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447

For the final conclusions, it is actually sufficient to consider the chiral limit, which simplifies the discussion of the loop integral. We define

and will discuss the properties of the function A in more detail below. Note that A is a real, but not necessarily positive, number. The numerator of our integral is of the form

generating integrals of the type

where the integrals with an odd power of integration momenta in the numerator vanish in dimensional regularization, because of an integration over a symmetric interval. (The denominator is even). Let us discuss the scalar integral (numerator 1) of Eq. (5.82).¶ After a Wick rotation [see Eq. (A.28)], one chooses spherical coordinates for the Euclidean integral, and the angular integration is carried out as in Eq. (A.31). The remaining one-dimensional integration can be done using Eq. (A.35), and the result is expanded for small

where is Euler’s constant. The infinity as must be canceled by some counter term of the effective Lagrangian. In order to perform the remaining integration over the Feynman parameter we make use of Eq. (A.40),

i.e., we need to discuss A as a function of (for, in principle, arbitrary ). It is easy to show that A can take negative values in the interval ¶

It is straightforward to also determine the second-rank tensor integral of Eq. (5.82) using the methods described in Appendices A.2 Regarding the analyticity properties we are interested in, one does not obtain any new information.

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only if in which case with Eq. (5.84) we obtain

for

In combination

The remaining integral can be evaluated using elementary methods, and the final expression is

At this point, we refrain from presenting the final expression of in detail, because Eq. (5.85) suffices to point out the difference between one-loop diagrams in the mesonic and the baryonic sectors. To do this, we expand for small pion four-momenta in the chiral limit about

[Note that is a small quantity of chiral order ] Taking into account that the two extracted Dirac structures (which we have not displayed) are (at least) of order [see Eq. (4.3) of Ref. [144]], one can draw the following conclusions [144]: The counter term needed to renormalize the contribution of Fig. 5.4 must contain terms which are of order and

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The finite part of the loop diagram has a logarithmic singularity of the form Expanding the finite part of the diagram in terms of small external momenta one obtains an infinite series with arbitrary powers of (small) momenta [see Eq. (5.86)]. In combination with the result of the previous section we see that a loop calculation with the relativistic Lagrangians and using dimensional regularization leads to rather different properties in the mesonic and baryonic sectors. The example of the nucleon mass shows that loop diagrams may contribute at the same order as the tree diagrams which has to be contrasted with the mesonic sector where, according to the power counting of Eq. (4.52), loops are always suppressed by a factor with denoting the number of independent loops. In particular, with each new order of the loop expansion one has to expect that the low-energy coefficients including those of the lowest-order Lagrangian have to be renormalized. On the other hand, in the mesonic sector a one-loop calculation in the even-intrinsic parity sector leads to a renormalization of the coefficients (and possibly higher-order coefficients if vertices of higher order are used), a two-loop calculation to a renormalization of the and so on. A second difference refers to the orders produced by a loop contribution. In the mesonic sector, a one-loop calculation involving vertices of produces exclusively an contribution. We have seen in the example above that in the baryonic sector all higher orders are generated, even though, in principle, there is nothing wrong with such a result as long as one can organize and predict the leading order of the corresponding contribution beforehand. In the next section we will discuss the so-called heavy-baryon formulation of ChPT [206, 37], which provides a framework allowing for a power counting scheme which is very similar to the mesonic sector. One trades the manifestly covariant formulation for the systematic power counting. Moreover, under certain circumstances, the results obtained in HBChPT do not converge in all of the low-energy region. This problem has recently been solved in the framework of the So-called infrared regularization [71] which will be discussed in Sect. 5.6.

5.5.

The Heavy-Baryon Formulation

We have already seen in Sect. 5.2., that the baryonic sector introduces another energy scale—the nucleon mass—which does not vanish in the chiral

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limit. Furthermore, the mass of the nucleon has about the same size as the scale which appears in the calculation of pion-loop contributions [see, for example, the discussion of scattering, where the tree-level contributions of Table 5.1 are whereas the one-loop diagram of Fig. 5.4 is The heavy-baryon formulation of ChPT [206, 37] consists in an expansion (of matrix elements) in terms of

and where represents a small external momentum. Clearly cannot simply be the four-momentum of the initial and final nucleons of Eq. (5.1), because the energy components and are not small. Instead, a method has been devised which separates an external nucleon four-momentum into a large piece of the order of the nucleon mass and a small residual component. The approach is similar to the nonrelativistic reduction of Foldy and Wouthuysen [143] which provides a systematic procedure to block-diagonalize a relativistic Hamiltonian in and produce a decoupling of positive- and negative energy states to any desired order in A criterion for the Foldy-Wouthuysen method to work is that the potentials in the Dirac Hamiltonian (corresponding to the interaction with external fields) are small in comparison with the nucleon mass. This may be considered as the analogue of treating external fields as small quantities of order or in ChPT. As in the previous cases we will discuss the lowest-order Lagrangian in quite some detail. For a discussion of the higher-order Lagrangians, the reader is referred to Refs. [121, 48, 133].

5.5.1.

Nonrelativistic Reduction

Before discussing the heavy-baryon framework let us start with the more familiar nonrelativistic limit of the Dirac equation for a charged particle interacting with an external electromagnetic field. Using this example, we will later be able to develop a better understanding of a peculiarity inherent in the heavy-baryon formulation regarding wave function (re)normalization. Our presentation will closely follow Refs. [270, 104]. Consider the Dirac equation of a point-particle of charge and mass interacting with an electromagnetic four-potential||

|| In order to facilitate the comparison with the Foldy-Wouthuysen result below, we make use of

the “non-covariant” form of the Dirac equation.

Introduction to Chiral Perturbation Theory

where

and

451

are the usual Dirac matrices

and is the momentum operator. For simplicity, we consider the interaction with a static external electric field,

Since we want to describe a nonrelativistic particle-like solution, it is convenient to separate a factor from the wave function,**

so that the Dirac equation (after multiplication with

results in

Note that both H and are Hermitian operators. In the spirit of the nonrelativistic reduction, we write in terms of a pair of two-component spinors and (L for large and S for small)

and obtain, after insertion into Eq. (5.88), a set of two coupled partial differential equations

The second equation can formally be solved for

where, for later use, we have introduced the abbreviation A for the operator We expand Eq. (5.92) in terms of up to and including order

** The (second) quantization of the relevant fields will be discussed in Sect. 5.5.4.

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452

where we made use of the commutation relation and of [see Eqs. (5.90) and (5.92)]. Inserting this result into the right-hand side of Eq. (5.90) and using we obtain the Schrödinger-type equation

As already noted by Okubo [270], the last term on the right-hand side of Eq. (5.94) is not Hermitian and, when written as represents the interaction of an electric field with a (momentum-dependent) imaginary electric dipole moment [104].†† As pointed out in Ref. [104], the non-Hermiticity of the Hamilton operator of Eq. (5.94) is a consequence of the procedure for eliminating the small-component spinors. The method can be thought of as applying the transformation

to the four-component spinor to generate a four-component spinor consisting exclusively of the upper component and then solving the corresponding transformed Dirac equation. Since S is not a unitary operator, i.e.,

the norm of the original spinor will not be the same

and the transformed spinor

in general,

Equation (5.96) suggests considering a field redefinition of the form [270, 104]

so that the new spinor has the same norm as nian of Eq. (5.88) we have

For the specific Hamilto-

†† The standard textbook treatment of the nonrelativistic reduction leading to the Pauli equation

considers only terms of Refs. [27, 201]).

and thus does not yet generate non-Hermitian terms (see, e.g.,

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453

so that we find‡‡

When inserting Eq. (5.98) into Eq. (5.94), we make use of

and, as above, tion for the two-component spinor order

yielding the Schrödinger equaincluding relativistic corrections up to

where the second term, for a central potential, corresponds to the usual spinorbit interaction and the last term is the so-called Darwin term [27, 201]. Note that the Hamiltonian here is Hermitian, i.e., the imaginary dipole moment has disappeared. Moreover, because of Eqs. (5.96) and (5.97), the spinors are normalized conventionally. The result of Eq. (5.99) is identical with a nonrelativistic reduction using the Foldy-Wouthuysen method [143] which uses a sequence of unitary transformations to block-diagonalize a relativistic Hamiltonian of the form

to any desired order in In Eq. (5.100) and denote the so-called odd and even operators of H, respectively, where odd operators couple large and small components whereas even operators do not. In the present case we have

and after three successive transformations one obtains the block-diagonal Hamiltonian (see, e.g., Refs. [27, 201, 129])

‡‡ In the framework of plane-wave solutions, Eq. (5.98) already provides a hint that one may

have to expect “unconventional normalization factors” when dealing with Feynman rules in the heavy-baryon approach.

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where

Restricting ourselves to the upper left block of Eq. (5.101) and noting that in Eq. (5.88) we have already separated the time dependence from we find that Eqs. (5.99) and (5.101) are indeed identical. In Ref. [104], the equivalence of the two approaches was explicitly shown to order We will see that the heavy-baryon approach proceeds along lines very similar to the non-relativistic reduction leading from Eq. (5.87) to (5.94). In analogy to Eqs. (5.96) and (5.97) we thus have to be alert to surprises related to the normalization of the relevant wave functions.

5.5.2.

Light and Heavy Components

As mentioned above, the idea of the heavy-baryon approach consists of separating the large nucleon mass from the external four-momenta of the nucleons in the initial and final states and, in a sense to be discussed in Sect. 5.5.3., below, eliminating it from the Lagrangian. The starting point is the relativistic Lagrangian of Eq. (5.21),

where the covariant derivative and are defined in Eqs. (5.18) and (5.20), respectively. The corresponding Euler-Lagrange equation for the nucleon field reads

(For notational convenience we replace and Sect. 5.5.3.). For a general four-vector we define the projection operators*

and in Sect. 5.5.2., with the properties and

and introduce the so-called velocity-dependent fields

* It may be worthwhile to remember that

matical sense, because they do not satisfy used in Eq. (5.89).

and

as

do not define orthogonal projectors in the mathewith the exception of the special case

Introduction to Chiral Perturbation Theory

so that

455

can be written as

The fields

and

satisfy the properties

For a particle with four-momentum the particular choice corresponds to its world velocity which is why is also referred to as a four-velocity. The fields and are often called the light and heavy components of the field which will become clearer below. In order to motivate the ansatz of Eq. (5.107) let us consider a positiveenergy plane wave solution to the free Dirac equation with three-momentum

where

are ordinary two-component Pauli spinors, and We can think of entering the calculation of, say, an S-matrix element through covariant perturbation theory in terms of the matrix element of an infield between the vacuum and a single-nucleon state:

where i.e.,

denotes the nucleon isospinor. For the special case

the components and are, up to the modified time dependence, equivalent to the large and small components of the “one-particle wave function”

(5.109)

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In other words, for this choice of the light and heavy components of the positive-energy solutions are closely related to the large and small components of the nonrelativistic reduction discussed in Sect. 5.5.1. Moreover, assuming varies slowly with time in comparison with of with the result that a time derivative generates a factor which is small in comparison with Another choice is in which case and correspond to the usual projection operators for positive- and negative-energy states

For this case we find

i.e., the dependence has completely disappeared in and, due to the projection property vanishes identically. In general, one decomposes the four-momentum of a low-energy nu† cleon into and a residual momentum

so that

For

in the vicinity of (1,0,0,0), a partial derivative acting on produces a small residual momentum and, in particular,

The actual choice of is, to some extent, a matter of convenience. For low-energy processes involving a single nucleon in the initial and final states, the four-momentum transferred in the reaction is defined as and is considered as a small quantity of chiral order For and where, say, is a small residual momentum in the sense of Eq. (5.112), also is a small four-momentum. The implications on a chiral power-counting scheme will be discussed in Sect. 5.5.8., below. † Of course, the decomposition of Eq. (5.111) alone is not a sufficient criterion for

Taking, for example,

one finds

for large

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5.5.3.

457

Lowest-Order Lagrangian

In order to proceed with the construction of the lowest-order heavy-baryon Lagrangian we insert Eq. (5.107) into the EOM of Eq. (5.104),‡

make use of Eq. (5.108), multiply by

and obtain

In the next step we would like to separate the and the part of the EOM of Eq. (5.113). To that end we make use of the algebra of the gamma matrices to derive

where

As an example, let us explicitly show the first relation of Eq. (5.114)

‡

For a derivation in the framework of the path-integral approach, see Ref. [245] and Appendix A.1 of Ref. [37].

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The remaining results of Eq. (5.114) follow analogously. Using Eqs. (5.105) and (5.114) we are now in the position to project onto the and parts of the EOM of Eq. (5.113),

which corresponds to Eqs. (5.90) and (5.91) of the nonrelativistic reduction of Sect. 5.5.1. We formally solve Eq. (5.116) for

which, similar, to the discussion of Sect. 5.5.1., shows that the heavy component is formally suppressed by powers of relative to the light compo§ nent Inserting Eq. (5.117) into Eq. (5.115), the EOM for the light component reads

which represents the analogue of Eq. (5.94). This EOM may be obtained from applying the variational principle to the effective Lagrangian¶

Note that the second term is suppressed by relative to the first term. Equation (5.119) corresponds to the leading-order result for Eq. (A. 10) of Ref. [37] § In fact, setting all external fields to zero and dropping the interaction term proportional to

it is easy to verify that the one-particle wave functions indeed satisfy the relation implied by Eq. (5.ll7). ¶ Replacing and omitting all terms containing the chiral vielbein and interpreting the covariant derivative as that of QCD, the result of Eq. (5.119) is identical with Eq. (7) of the discussion of heavy quark effective theory in Sect. 2 of Ref. [23].

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which was obtained in the framework of the path-integral approach, but does not yet represent the final form commonly used in HBChPT.|| Having the discussion following Eq. (5.95) in mind, in order for the two Lagrangians of Eqs. (5.103) and (5.119) to describe the same observables, we cannot expect both fields and to be normalized in the same way. We will come back to this question in Sect. 5.5.4. To obtain the heavy-baryon Lagrangian we define the spin matrix as**

which, in four dimensions, has the properties

Using the properties of Eq. (5.108) together with Eq. (5.121), the 16 combinations may be written as [see Eqs. (9)-(12) of Ref. [206]]

|| In order to be able to invert the operator

of Ref. [37], strictly speaking the projection operators should not be included in the definition of ** For the classification of the irreducible representations of the Poincaré group, one makes use of the so-called Pauli-Lubanski vector

where is the completely antisymmetric tensor in four indices, denotes the generalized angular momentum operator, and is the four-momentum operator (see, e.g, Refs. [201, 207]). Both and are Lorentz invariant and translationally invariant and are thus used as Casimir operators, where the eigenvalues are denoted by and . For the massive spin-1/2 case one obtains

Using (in four dimensions)

together with the special choice one easily finds that the spin matrix is, for this special case, proportional to the Pauli-Lubanski vector,

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For example,

where we made use of Eq. (5.108). The remaining relations are shown analogously. Eqs. (5.122) result in a nice simplification of the Dirac structures in the heavy-baryon approach, because one ultimately only deals with two groups of 4 × 4 matrices, the unit matrix and instead of the original six groups on the left-hand side of Eq. (5.122). Expanding Eq. (5.119) formally into a series in

and applying Eq. (5.122), the lowest-order term reads

where we made use of the fourth relation of Eq. (5.122) and the first relation of Eq. (5.121). (Recall that the second term of Eq. (5.119) is of order Equation (5.124) represents the lowest-order Lagrangian of heavy-baryon chiral perturbation theory (HBChPT), indicated by the symbol . When comparing with the relativistic Lagrangian of Eq. (5.103), we see that the nucleon mass has disappeared from the leading-order Lagrangian. It only shows up in higher orders as powers of as will be discussed in Sect. 5.5.7. In the power counting scheme counts as because the covariant derivative and the chiral vielbein both count as When calculating loop diagrams with the Lagrangian of Eq. (5.124) one will encounter divergences which are treated in the framework of (normal) dimensional regularization [see Eq. (5.78)]. Since the definition of the spin matrix contains and the commutator of two such spin matrices, in four dimensions, involves the epsilon tensor, one needs some convention for dealing with products of spin matrices when evaluating integrals in dimensions. To be on the safe side, we always reduce the gamma matrices using only the rules of Eq. (5.78). Let us consider the following example, which appears in the calculation of the pion-nucleon form factor at the one-loop level:††

††In evaluating Eq. (5.125), we made use of

in

dimensions.

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461

where we consistently made use of the anticommutation relations of Eq. (5.78). In contrast, using the anticommutator and commutator of Eq. (5.121), one ends up with which only coincides with Eq. (5.125) for However, the factor needs to be written as when it is multiplied with a singularity of the form in order to produce the constant nondivergent term in the product. Similarly, using the conventions of Eq. (5.78), the squared spin operator in dimensions reads

5.5.4.

Normalization of Fields and States

So far we have calculated matrix elements of the relativistic Lagrangian of Eq. (5.21) in the framework of covariant perturbation theory based on the formula of Gell-Mann and Low [160] in combination with Wick’s theorem [325]. Let us recall that, for a generic field described by the Lagrangian the “magic formula of covariant perturbation theory” [184] allows one to calculate the Green functions

in terms of the generating functional

While the Green functions of Eq. (5.127) involve the interacting field and the vacuum of the corresponding interacting theory, the formula of GellMann and Low, in principle, provides an explicit expression for the generating functional in terms of the quantities and defined in the free theory. Note that the Green functions of Eq. (5.127) are expressed in terms of the bare fields and, in the end, still have to be renormalized (see, e.g., Sect. 4.9.1.). Here we want to address the question of how to establish contact between matrix elements evaluated perturbatively using the relativistic Lagrangian of Eq. (5.21), on the one hand, and the heavy-baryon Lagrangian of Eq. (5.119) on the other hand. The presentation will make use of the ideas developed in Refs. [112, 23], where this issue was discussed for the case of a heavy-quark

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effective theory. A different route was taken in Refs. [122, 300, 213], where the path-integral approach to the generating functional was used to define the wave function renormalization constant (including interaction). Later on we will explicitly see that the two approaches yield identical results at For later comparison with the heavy-baryon approach, we first need to collect a few properties of the free Dirac field operator which we decompose in the standard fashion in terms of the solutions of the free Dirac equation‡‡

where erties

The Dirac spinors have the following prop-

The creation operators and (annihilation operators and ) of particles and antiparticles, respectively, satisfy the anticommutation relations

where all remaining anticommutators such as, e.g., With this convention, single-particle states ized as

vanish. are normal-

Let us now turn to the heavy-baryon formulation and consider the leadingorder term of Eq. (5.124) which we write as

for later use in the formula of Gell-Mann and Low. We decompose the solution to the free equation of motion ‡‡ For the sake of simplicity, we consider one generic fermion field.

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463

as

where at leading order, is defined through Eq. (5.134), i.e.,

in order to satisfy

The spinors are given by

with

ordinary two-component Pauli spinors. Note that, at lowest order in the spinors do not depend on the residual momentum Moreover, for an arbitrary choice of the are four-component objects which only for the special case effectively reduce to two-component spinors. As will be shown below, the operators and destroy and create a nucleon (isospin index suppressed) with residual three-momentum They satisfy the anticommutation relations

where, as usual, the anticommutator of two annihilation or two creation operators vanishes. Accordingly, the single-particle states are normalized as

Note that the normalization of the states of Eqs. (5.132) and (5.138) coincide only at leading order in (or as ). Using Eq. (5.137) it is straightforward to verify that the (free) theory has been quantized “canonically” [112], i.e.

where is the momentum conjugate to we made use of the completeness relation

and

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Constructing the energy-momentum tensor corresponding to

(we made use of the equation of motion) we obtain for the four-momentum operator

Using mutation relations

it is then straightforward to verify the com-

Eq. (5.141) implies that and destroy and create quanta with (residual) four-momentum and total four-momentum This can be seen by comparing with the four-momentum operator of the free relativistic theory

which, to leading order in is related to Eq. (5.140) by where is the number operator [112]. Using the orthogonality relations of Eq. (5.136) we may express the creation and annihilation operators in terms of the fields in the standard way

Equations (5.143) are the starting point for the LSZ reduction [228,28,201] in the framework of the heavy-baryon approach. We consider the matrix element of Eq. (5.1) for the transition in the presence of external fields and (we omit spin and isospin labels)

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The intermediate steps indicated by . . . proceed in complete analogy to the usual reduction formula as described in, e.g., Refs. [28, 201]. In Eq. (5.144), the factors of the type are related to the relative normalization of the states [see Eq. (5.132) vs. (5.138)], whereas refers to the wave function renormalization in the framework of the heavy-baryon Lagrangian. The Green function entering Eq. (5.144) will be calculated perturbatively using the formula of Gell-Mann and Low [160],

where, on the right-hand side, denotes the vacuum of the free theory, and * the external fields are part of the Lagrangian

5.5.5.

Propagator at Lowest Order

We will now discuss the propagator of the lowest-order Lagrangian both on the “classical level” as well as in the quantized theory of the last section. The lowest-order equation of motion corresponding to Eq. (5.124) reads

where the second relation implies [see Eq. (5.108)]. We define the propagator corresponding to Eq. (5.146) through

In order to solve Eq. (5.146) perturbatively, we re-write the equation of motion in the standard form as

* Strictly speaking we should also include the mesonic Lagrangian.

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where V denotes the interaction term, and search for the unperturbed Green function satisfying the properties

In terms of

the propagator

is then given by

Inserting the standard ansatz in terms of a Fourier decomposition

into Eq. (5.148),

we obtain by comparing both sides

The boundary condition of Eq. (5.150) may be incorporated by introducing an infinitesimally small imaginary part into the denominator:

That this is indeed the correct choice is easily seen by evaluating the integral

as a contour integral in the complex plane (see Fig. 5.5). For the contour is closed in the lower half plane and one makes use of the residue theorem. On the other hand, for the contour is closed in the upper half

Introduction to Chiral Perturbation Theory

467

plane and, since the contour does not contain a pole, the integral vanishes. We then obtain

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For the special choice of a static source

the propagator reduces to that

Finally, it is easy to show that a definition of the propagator in terms of the field operators and [112],

yields the same result as Eq. (5.153). To that end, one inserts for each of the two operators a sum according to Eq. (5.135), commutes the creation and annihilation operators using Eq. (5.137), applies the completeness relation of Eq. (5.139), and makes use of for the individual Fourier components. Performing the remaining integration over one ends up with Eq. (5.153), i.e., as expected the two methods yield the same result.

5.5.6. Example:

Scattering at Lowest Order

As a simple example, let us return to scattering, but now in the framework of the heavy-baryon Lagrangian of Eq. (5.124). The four-momenta of the initial and final nucleons are written as respectively, with to leading order in The relevant interaction Lagrangian is obtained in complete analogy to Eq. (5.51),

and the corresponding Feynman rules for the vertices derived from Eq. (5.155) read for a single incoming pion with four-momentum and Cartesian isospin index

for an incoming pion with

and an outgoing pion with

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As in the case of the relativistic calculation of Sect. 5.3.2., the latter gives rise to a contact contribution to

where the spinors are given in Eq. (5.136) and N and are the normalization factors appearing in the reduction formula of Eq. (5.144). The result for the direct-channel nucleon pole term reads

where, at leading order, we can make use of is obtained from Eq. (5.159) by the replacement crossing).

The crossed channel and (pion

The evaluation of the total matrix element particularly simple for the special choice have

is for which we

In that case, the calculation effectively reduces to that of a two-component theory as in the Foldy-Wouthuysen transformation, because the 4 × 4 matrices of the vertices are multiplied both from the left and the right by originating from either the propagator of Eq. (5.152) or the spinors of Eq. (5.136). To be specific, for a 4 × 4 matrix of the type

where each block A, B, C, and D is a 2 × 2 matrix, one has

and

Moreover, the spin matrix of Eq. (5.120) is very simple for

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470

where has been defined in Eq. (5.102). With this special choice of matrix in the center-of-mass frame reads

the T

Performing a non-relativistic reduction of Eq. (5.46) in the center-of-mass frame,

and using

in the expansion of A and B of Table 5.1, one verifies that, at leading order in the relativistic Lagrangian of Eq. (5.21) and the heavy-baryon Lagrangian of Eq. (5.124) indeed generate the same scattering amplitude. We emphasize that in order to obtain this equivalence of the two approaches an expansion of Eq. (5.162) to is mandatory, because the functions and contain terms of leading order These terms disappear through a cancellation in the final result.†

5.5.7. Corrections at First Order in So far we have concentrated on the leading-order, heavybaryon Lagrangian of Eq. (5.124). In comparison with Eq. (5.23), the chiral counting scheme of HBChPT is different, because an ordinary partial derivative acting on a heavy-baryon field produces a small residual four† The overall factor 2

Eq. (5.60)].

in Eq. (5.162) is a result of our normalization of the spinors [see

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471

momentum [see also Eq. (4.62) for the mesonic sector]:

In the heavy-baryon approach four-momenta are considered small if their components are small in comparison with either the nucleon mass or the chiral symmetry breaking scale both of which we denote by a common ‡ It is clear that the Lagrangian of Eq. (5.123) also genscale erates terms of higher order in and, in analogy to the mesonic sector, we also expect additional new chiral structures from the most general chiral Lagrangian at higher orders. Recall that in the baryonic sector the chiral orders increase in units of one, because of the additional possibility of forming Lorentz invariants by contracting (covariant) derivatives with gamma matrices (see Sect. 5.2.). (The relativistic Lagrangian at ) has (partially) been given in Ref. [144].) Let us first consider the correction resulting from Eq. (5.123)

We make use of Eqs. (5.122) to identify the relevant replacements in the heavybaryon bilinears:

‡

In reality, the excitation energy of the convergence of the expansion.

resonance very often provides the limit of

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where the expression for the commutator of the covariant derivative of Eq. (5.18) is obtained after straightforward algebra. The field-strength tensors are defined as

where and are given in Eqs. (4.59) and (4.60), respectively. Collecting all terms, we finally obtain as the contribution of Eq. (5.123) of order (returning to the notation in terms of expressions in the chiral limit)

Applying the counting rules of Eqs. (4.62) and (5.163), we see that Eq. (5.165) is indeed of where the suppression relative to (5.124) is of the form At the heavy-baryon Lagrangian contains another contribution which, in analogy to in Sect. 5.5.3., may be obtained as the projection of the relativistic Lagrangian of [144] onto the light components. Here we quote the result in the convention of Ref. [48](except for the and terms, where, following Ref. [121], we explicitly separate the traceless and isoscalar terms)§

where § The nomenclature of Refs. [144] and [37] differs from the (more or less) standard convention of

Eq. (5.166). The constants Eq. (5.166).

of Ecker and

[121] differ by a factor

from those of

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In the parameterization of Eq. (5.166) the constants carry the dimension of an inverse mass and should be of the order of in order to produce a reasonable convergence of the chiral expansion. (The details of convergence generally depend on the observable in question). The complete heavy-baryon Lagrangian at is then given by the sum of Eqs. (5.165) and (5.166),

It is worthwhile mentioning that the contribution of Eq. (5.165) to contains chirally invariant structures that are not part of Eq. (5.166). Unless such terms can be transformed away by a field transformation (see below) their coefficients are fixed in terms of the parameters of the lowest-order Lagrangian. As stressed by Ecker [118], these fixed coefficients are a consequence of the Lorentz covariance of the whole approach. A related issue is the so-called reparameterization invariance, i.e., if a heavy particle of physical four-momentum is described by, say, with and implying physical observables should not change under the replacement giving rise to an equivalent parameterization if satisfies [239]. As a result, some coefficients of terms in the effective Lagrangian which are of different order in the expansion are related. For a detailed discussion, the reader is referred to Refs. [239, 88, 134]. The seven low-energy constants are determined by comparison with experimental information. For example, if we consider the interaction with an external electromagnetic field [see Eq. (2.111)],

we obtain

so that the interaction with the field-strength tensor is given by

[We made use of Eq. (5.121).] For the special choice we find [see Eq. (5.160)]

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Stefan Scherer

and the interaction term reduces to¶

which describes the interaction Lagrangian of a magnetic field with the magnetic moment of the nucleon. We define the isospin decomposition of the magnetic moment (in units of the nuclear magneton ) as

where and denote the isoscalar and isovector anomalous magnetic moments of the nucleon, respectively, with empirical values and A comparison with Eq. (5.170) shows that the constants and are related to the anomalous magnetic moments of the nucleon in the chiral limit The results for

and

up to and including

[44,130]

are used to express the parameters and in terms of physical quantities. Note that the numerical correction of –1.96 [parameters of Eq. (5.68)] to the isovector anomalous magnetic moment is substantial. Differences by factors of about 1.5 were generally observed for the determination of the at and to one-loop accuracy [44,48]. The numerical values of the low-energy constants have been de-

termined in Ref. [48] by performing a best fit to a set of nine pion-nucleon scattering observables at which do not contain any new low-energy constants from the Lagrangian. Finally, was determined in terms of the strong contribution to the neutron-proton mass difference. The results in units of are given by (see also Ref. [74])

For a phenomenological interpretation of the low-energy constants in terms of (meson and ) resonance exchanges see Ref. [48]. ¶ Recall that

are two-component fields.

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475

We will see in the next section that the constants are not required to compensate divergences of one-loop integrals. Such infinities first appear at The Lagrangian of Eq. (5.165) still contains terms of the type D which appears in the lowest-order equation of motion of Eq. (5.146). As discussed in detail for the mesonic sector in Sect. 4.7., and Appendix A.4.1, such terms can be eliminated by appropriate field redefinitions. For example, the field transformation eliminating in Eq. (5.165) the term

is given by [121]

Inserting Eq. (5.172) into the lowest-order Lagrangian of Eq. (5.124) yields

The second term cancels the equation-of-motion term, whereas rewriting the last term by using Eq. (5.121),

we find that some of the coefficients at (and at higher orders) are modified. As in the case of the SU(2)×SU(2) mesonic Lagrangian at (see Appendix A.4.1) one finds equivalent parameterizations of (and also of the higher-order Lagrangians) in the baryonic sector. For the sake of completeness we quote the result of Ecker and [121],

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Stefan Scherer

where the relation to the coefficients

of Eq. (5.166) is given by

Of course, the Lagrangians of Eq. (5.168) and (5.174) yield the same results for physical observables, provided their parameters are related by Eq. (5.175). However, they will differ for intermediate mathematical quantities such as vertices or wave function renormalization constants as observed in Ref. [130] for the case of the nucleon wave function renormalization constant. We repeat that the coefficients of the first two terms of Eq. (5.174) are fixed in terms of and whereas the constants are free parameters which have to be determined by comparison with experimental information.

5.5.8.

The Power Counting Scheme

The power counting scheme of HBChPT may be formulated in close analogy to the mesonic sector (see Sect. 4.4.). On the scale of either the nucleon mass or we consider as small external momenta the four-momenta of pions, the four-momenta transferred by external sources, and the residual momenta of the nucleon appearing in the separation For a given Feynman diagram we introduce the number of independent loop momenta the number of internal pion lines the number of pion vertices

originating from

the total number of pion vertices the number of internal nucleon lines the number of baryonic vertices

originating from

and the total number of baryonic vertices As in the mesonic sector, the internal momenta appearing in the loop integration are not necessarily small. However, via the four-momentum conserving delta functions at the vertices and a substitution of integration variables,

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the rescaling of the external momenta is transferred to the internal momenta (see Sect. 4.4.). The chiral dimension D of a given diagram is then given by [323, 118]

We make use of the topological relation [see, e.g., Eq. (2.130) of Ref. [91]]

to eliminate

from Eq. (5.176)

For processes containing exactly one nucleon in the initial and final states we have|| and we thus obtain

The power counting is very similar to the mesonic sector. We first observe that Moreover, as already mentioned in Sect. 5.5.7., loops start contributing at D = 3. In other words, the low-energy coefficients of are not needed to renormalize infinities from one-loop calculations. Again, we have a connection between the number of loops and the chiral dimension D: Each additional loop adds two units to the chiral dimension. As an example, let us consider the two-loop contribution to the nucleon self energy of Fig. 5.6. First of all, the number of independent loops is in agreement with Eq. (5.177) for and 4. The counting of the chiral dimension is most intuitively performed in the framework of Eq. (5.176), because it associates with each building block a unique term which is easy to remember (+4 for each independent loop, –2 for each internal meson propagator, etc). For and we obtain D = 8 – 4 – 3 + 0 + 2 + 4 = 7. || In the heavy-baryon formulation one has no closed fermion loops. In other words, in the single-

nucleon sector exactly one fermion line runs through the diagram connecting the initial and final states.

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5.5.9. Application at

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One-Loop Correction to the Nucleon Mass

As a simple example, we will return to the modification of the nucleon mass through higher-order terms in the heavy-baryon approach. The calculation will proceed along the lines of Ref. [130], where use was made of the Lagrangian of Ecker and [121] [see Eq. (5.174)]. The determination of the physical nucleon mass and the discussion of the wave function renormalization factor will be very similar to Sect. 4.9.1., for the masses of the Goldstone bosons and Sect. 5.4.1., for the nucleon mass in the relativistic approach. Let us denote the four-momentum of the nucleon by where, since we are interested in the propagator, we must allow the four-momentum to be off the mass shell. The on-shell case is, of course, given by with denoting the physical nucleon mass. Let us stress that, due to the interaction, we must expect the physical mass to be different from the mass in the chiral limit. We start from the lowest-order propagator of Eq. (5.152),

and first determine its modification in terms of the tree-level contribution of Eq. (5.174) to the self energy.** Neglecting isospin-symmetry breaking effects proportional to we obtain at

where denotes the squared pion mass at comes from the term in proportional to

The term which involves no

** In the remaining part of this section, we adopt the common practice of leaving out the projector

in the propagator and (possibly) in vertices with the understanding that all operators act only in the projected subspace.

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479

pions. In the spirit of the reduction formula of Eq. (5.144), in combination with the formula of Gell-Mann and Low of Eq. (5.145), we choose to include this as part of the interaction rather than part of the free Lagrangian and reserve for the free Lagrangian the term from The term involving is a contact term coming ultimately from Eq. (5.166), where The heavy-baryon Lagrangian at [121] does not produce a contact contribution to the self energy, because all the structures contain at least one pion or an external field. Moreover, given the Feynman rule of Eq. (5.157), the second one-loop diagram of Fig. 5.7 (b) vanishes, because In other words, the only contribution at results from the one-loop diagram of Fig. 5.7 (a). Using the vertex of Eq. (5.156) and the propagator of Eq. (5.152) we obtain for the one-loop contribution at

As in the relativistic case of Eq. (5.77), counting powers, we expect the integral to have a cubic divergence. Extending the integral to dimensions, using performing the substitution and applying Eq. (A.71) of Appendix A.3.2.2, we obtain the intermediate result

480

Since and the first equality of Eq. (A.75),

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[see Eq. (5.126)], we obtain, applying

where the integrals and are given in Eqs. (A.67) and (A.42), respectively. Combining with Eq. (5.181) and using Eq. (A.67) we thus obtain for the (unrenormalized) nucleon self energy at [130]

for Clearly, the self-energy contribution generated by the loop diagram of Fig. 5.7 (a) contains a divergent piece proportional to of Eq. (A.38). We have chosen to express the self energy as a function of the four-momentum In the relativistic case of Eq. (5.71) we needed two scalar functions depending on to parameterize the self energy. In contrast to the relativistic case, the heavy-baryon self energy of Eq. (5.184) is given by one function depending on two scalar variables for which one can take, say, and or

Making use of

the two sets are related by

The choice of Eq. (5.185) is convenient for the determination of the physical nucleon mass and renormalization constant because, in view of Eq. (5.180), we want the full (but yet unrenormalized) propagator to have a pole at which includes both the mass-shell condition and In the vicinity of the pole at the second choice of variables corresponds to terms which are, respectively, first and second order in the

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481

(small) distance from the pole. Thus in the following discussion we will use both notations and for the self energy, where it should be clear from the context which expression applies. In analogy to the mesonic case discussed in Sect. 4.9.1., the full heavybaryon propagator is written as [see Eq. (4.138)]

where

In these equations respect to evaluated at

denotes the first partial derivative of

with

and

is at least of second order in the distance from the pole. For the evaluation of and we need to expand Eq. (5.184). To the order we are working the term contributes only to whereas the loop piece contributes to all three. In contrast to the mesonic sector at is not zero in this case. Using Eq. (5.186), we obtain for the term

The first term on the right-hand side contributes to but is since, as we will see, the difference is The second term is and will contribute to Finally the third term contributes only to

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Applying Eq. (5.189) we obtain for the physical mass

†† We can thus neglect the second term which implies on the right-hand side of Eq. (5.192). The loop contribution is only a function of and thus a function only of and, neglecting terms of higher order in we may replace by 0, yielding

We finally obtain for the physical nucleon mass

where, in the expression between the brackets, we have replaced all quantities in terms of the physical quantities, because the difference is of higher order in the chiral expansion. In the chiral limit, both the counter-term contribution and the pion-loop correction disappear. In other words, in the heavy-baryon framework the situation is again as in the mesonic sector, where the parameters of the lowest-order Lagrangian do not get modified due to higher-order corrections in the chiral limit. The same is actually true for the second parameter of Eq. (5.124) [see, e.g., Eq. (50) of Ref. [130]]. Using the parameters of Eqs. (5.68) and (5.171) one finds that the counter term and the pion loop generate contributions to the physical nucleon mass of 0.0733 and –0.0163 in units of respectively. The wave function renormalization constant is obtained from Eq. (5.190) as

To the order we are considering, we have from Eqs. (5.193) and (5.184), respectively,

††

Strictly speaking we should say result originates from the loop contribution.

where the second

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483

Finally, expressing all quantities in terms of physical quantities, the wave function renormalization constant reads

As in the pion case (see Table A.2 of Appendix A.4.2) contains the infinite constant entering through dimensional regularization, i.e., is not a finite quantity. However, this is not a problem, because the wave function renormalization constant is not a physical observable. Moreover, as we have seen explicitly for the pion case, and as discussed in Ref. [130] for the heavybaryon Lagrangian, will also depend on the specific parameterization of the Lagrangian. In Ref. [122] it was shown that the wave function renormalization “constant” is in fact a non-trivial differential operator and should, in momentum space, depend on the momentum of the initial or final nucleon. Here we argue that the findings of Ref. [122] and the method used above do not seem to be in conflict with each other. To that end, we first note that Ref. [122] made use of the external spinor Using relativistic spinors normalized as in Eq. (5.131) this corresponds to a normalization of the heavy baryon spinors to To facilitate the comparison, let us consider the special choice In the framework of the reduction formula of Eq. (5.144), we work with a factor for an external nucleon in the initial state, where Ecker and would have It is now straightforward to show that the normalization factor exactly produces the additional term which, in the approach of Ref. [122], results from the additional term in the wave function renormalization. This explains why the two approaches, at least up to order generate the same result. For further discussion on this topic, the reader is referred to [130, 122, 300, 213].

5.6.

The Method of Infrared Regularization

In the discussion of the one-loop corrections to the nucleon self energy and pion-nucleon scattering of Sect. 5.4., we saw that the relativistic framework for baryons did not naturally provide a simple power counting scheme as for mesons. One major difference in comparison with the mesonic sector is related to the fact that the nucleon remains massive in the chiral limit which also introduces another mass scale into the problem. Thus, because of

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the zeroth component one can no longer argue that a derivative acting on the baryon field results in a small four-momentum. This problem is avoided in the heavy-baryon approach discussed in Sect. 5.5., where, through a field redefinition, the mass dependence has been shifted into an (infinite) string of vertices which are suppressed by powers of Since the derivatives in the heavy-baryon Lagrangian produce small residual four-momenta in the lowenergy regime, a power counting scheme analogous to the mesonic sector can be formulated [see Eqs. (5.176) and (5.179)]. A vast majority of applications of chiral perturbation theory in the baryonic sector were performed in the framework of the heavy-baryon approach. However, it was realized some time ago that the heavy-baryon approach, under certain circumstances, may generate Green functions which do not satisfy the analytic properties resulting from a (fully) relativistic field theory [47]. Clearly, it would be desirable to have a method which combines the advantages of the relativistic and the heavy-baryon approaches and, at the same time, avoids their shortcomings—absence of a power counting scheme on the one hand and failure of convergence on the other hand. Such approaches have been proposed and developed by various authors [302, 126, 71, 146, 241, 30, 237] and here we will briefly outline the ideas of the so-called infrared regularization [71]. Our presentation will closely follow Refs. [71, 30] to which we refer the reader for technical details. Some recent applications of the new approach deal with the electromagnetic form factors of the nucleon [214] and the baryon octet [?], scattering [72], axial-vector current matrix elements [331], and the generalized Gerasimov-Drell-Hearn sum rule [53]. In order to understand the problems of the heavy-baryon approach regarding the analytic behaviour of invariant functions let us start with a simple example [30]. To that end we consider the channel of pion-nucleon scattering (see Sect. 5.3.2.). The invariant amplitudes of Table 5.1 develop poles for (the upper and lower signs correspond to and respectively). For example, the singularity due to the nucleon pole in the channel is understood in terms of the relativistic propagator

which, of course, has a pole at or, equivalently, (Analogously, a second pole results from the channel at ) We also note that the propagator of Eq. (5.196) counts as because it is part of a tree-level diagram so that the four-momentum is assumed to be small, i.e., of Although both poles are not in the physical region of pion-nucleon scattering, analyticity of the invariant amplitudes requires these poles to be present in the amplitudes. Let us compare the situation with a heavy-baryon

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485

type of expansion, where, for simplicity, we choose as the four-velocity

Clearly, to any finite order the heavy-baryon expansion produces poles at instead of a simple pole at and will thus not generate the (nucleon) pole structures of the functions As a second example, we consider the so-called triangle diagram of Fig. 5.8 which will serve to illustrate the different analytic properties of invariant functions obtained from loop diagrams in the relativistic and heavy-baryon approaches. A diagram of this type appears in many calculations such as the scalar or electromagnetic form factors of the nucleon, where represents an external scalar or electromagnetic field, or or Compton scattering, where stands for two pion or electromagnetic fields. In all of these cases a fourmomentum is transferred to the nucleon and the analytic properties of the Feynman diagram as a function of are determined by the pole structure of the propagators.

Thus we need to discuss some properties of the integral

where We assume the initial and final nucleons to be on the mass shell, which implies Counting powers we see that the integral converges. The function is analytic in except

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for a cut along the positive real axis starting at which expresses the fact that two on-shell pions can be produced for In the following discussion of the analytic properties of Eq. (5.198) we will concentrate on the imaginary part of which we will derive applying the Cutkosky (or cutting) rules [99, 227, 280]. The rules, as summarized in [280], read: In order to obtain first cut through the diagram in all possible ways such that the cut propagators can simultaneously be put on shell. Next, for each cut one need to replace each cut propagator Finally, sum the contributions of all possible cuts. In the present case, the two terms where the nucleon propagator is simultaneously cut with either the first or the second pion propagator do not contribute. The result from cutting the two pion propagators reads

In order to evaluate Eq. (5.199), we choose a frame where and Using

with

we find, as an intermediate result,

For the delta function in Eq. (5.200) always vanishes, showing that the cut starts, as anticipated, at Applying the mass-shell condition we write

Performing the integration using spherical coordinates, the result for the first case reads

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where

The second case,

is obtained analogously by the replacement

Equations (5.201) and (5.202) agree with the results given in Eq. (B.43) of Ref. [144]. In the low-energy region and Eq. (5.201) becomes

Taking the factors resulting from the vertices and the relevant tensor structures of the loop integral into account, the contribution of Fig. 5.8 to the imaginary part of the scalar form factor of the nucleon reads [144, 71]

where as

is defined in terms of the

and

scalar densities

and

where and We will now investigate two limiting procedures. First, we consider a fixed value and let In that case and one would use the expansion

Keeping only the leading order term, we find

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which corresponds exactly to the result of HBChPT at This result corresponds to the standard chiral expansion which treats the quantity of Eq. (5.203) as because and However, for a fixed we may also consider a small enough close to the threshold value so that In that case the expansion of the arctan reads

yielding

where we have neglected higher powers of sponding to is given by

The critical value of

corre-

where Clearly the behaviour of Eq. (5.206) is very different from the chiral expansion of Eq. (5.205) and, similar to the discussion of Eq. (5.197), a finite sum of terms in HBChPT cannot reproduce such a threshold behaviour [71]. The rapid variation of the imaginary part can be understood in terms of the analytic properties of the arctan which, as a function of the complex variable is analytic in the entire complex plane save for cuts along the positive and negative imaginary axis starting at These branch points corresponding to are obtained for which is just below the physical threshold For that reason an expansion around corresponding to has a small radius of convergence. Clearly, the heavy-baryon approach does not produce the correct analytic structure as generated by the relativistic loop diagram. Moreover the lowenergy behaviour of Eq. (5.203) cannot be accounted for in the standard chiral analysis because the argument is of order What is needed is a method which produces both the relevant analytic structure and a consistent power counting. Here we will illustrate the method of Ref. [71] by means of the nucleon self energy diagram of Fig. 5.3. For a—at this stage—qualitative discussion of its properties we focus on the scalar loop integral

where, as usual, the right-hand side is thought of as a Feynman integral which has to be analytically continued as a function of the space-time dimension

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Counting powers, we see that, for the integrand behaves for large values of the integration variable as producing a logarithmic ultraviolet divergence while, on the other hand, the integral converges for Let us now consider the limit In this case, for both the integral is infrared regular for because, for small momenta, the integrand behaves as and respectively. For the integral is infrared regular for but singular for For any smaller value of it is infrared singular for arbitrary The infrared singularity as originates in the region, where the integration variable is small, i.e., of the order Counting powers of momenta, we (naively) expect this part to be of order On the other hand, for loop momenta of the order of and larger than the nucleon mass we expect power counting to fail, because the momentum of the nucleon propagating in loop integral is not constrained to be small in contrast to the case of tree-level diagrams [see Eq. (5.196)]. In order to explain these qualitative statements let us discuss the integral in more detail. We first introduce the Feynman parameterization‡‡

with shift

perform the and obtain

where We then apply Eq. (A.39) of Appendix A.2,

The relevant properties can nicely be displayed at the threshold where is particularly simple. The small imaginary part can be dropped in this case, because is never negative. Splitting the integration interval into and with ‡‡ In order to make it easier for the interested reader to follow Ref. [71] we have used the nota-

tion there, omitting a factor previous sections.

and choosing the opposite overall sign in comparison with

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we have, for

yielding, through analytic continuation, for arbitrary

The first term, proportional to is defined as the so-called infrared singular part I which, as behaves as in the qualitative discussion above. Since implies this term is singular for The second term, proportional to is defined as the infrared regular part and can be thought of as originating from an integration region where is of order so that the tree-level counting rules no longer apply [see Eq. (5.196)]. Note that for non-integer the infrared singular part contains non-integer powers of while an expansion of the regular part always contains non-negative integer powers of only. Let us now turn to a formal definition of the infrared singular and regular parts [71] which makes use of the Feynman parameterization of Eq. (5.209). Introducing the dimensionless variables

counting as as

and

so that H is now given by

where

respectively, we rewrite

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491

The infrared singularity originates from small values of where the function goes to zero as In order to isolate the divergent part one scales the integration variable so that the upper limit in Eq. (5.212) corresponds to as An integral I having the same infrared singularity as H is then defined which is identical to H except that the upper limit is replaced by

where (The pion mass defined as

is not sent to zero.) Accordingly, the regular part of H is

so that Let us verify that the definitions of Eqs. (5.214) and (5.215) indeed reproduce the behaviour of Eq. (5.210). To that end we make use of yielding

which converges for write [71]

In order to continue the integral to

and make use of a partial integration

we

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For the first expression vanishes at the upper limit and, at the lower limit, yields Bringing the second expression to the lefthand side, we may then continue the integral analytically as

so that we obtain for

which agrees with the infrared singular part I of Eq. (5.210). The threshold value of the regular part of Eq. (5.215) is obtained by analytic continuation from

which is indeed the regular part of Eq. (5.210). What distinguishes I from is that, for non-integer values of the chiral expansion of I gives rise to non-integer powers of whereas the regular part may be expanded in an ordinary Taylor series. For the threshold integral, this can nicely be seen by expanding and in the pion mass counting as On the other hand, it is the regular part which does not satisfy the counting rules valid at tree level. The basic idea of the infrared regularization consists of replacing the general integral H of Eq. (5.207) by its infrared singular part I, defined in Eq. (5.214), and dropping the regular part defined in Eq. (5.215). In the low-energy region H and I have the same analytic properties whereas the contribution of which is of the type of an infinite series in the momenta, can be included by adjusting the coefficients of the most general effective Lagrangian. As discussed in detail in Ref. [71], the method can be generalized to an arbitrary one-loop graph. Using techniques similar to those of Appendices A.3.1.2 and A.3.2.2, it is first argued that tensor integrals involving an expression of the type in the numerator may always be reduced to scalar loop integrals of the form

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where and are inverse meson and nucleon propagators, respectively. Here, the refer to four-momenta of and the are four-momenta which are not far off the nucleon mass shell, i.e., Using the Feynman parameterization, all pion propagators and all nucleon propagators are separately combined, and the result is written in such a way that it is obtained by applying and partial derivatives with respect to and respectively, to a master formula. A simple illustration is given by

where Of course, the expressions become more complicated for larger numbers of propagators. The relevant property of the above procedure is that the result of combining the meson propagators is of the type 1/A with where is a linear combination of the momenta with an analogous expression 1/B for the nucleon propagators. Finally, the expression

may then be treated in complete analogy to H of Eq. (5.207), i.e., the denominators are combined as in Eq. (5.208), and the infrared singular and regular pieces are identified by writing A crucial question is whether the infrared regularization respects the constraints of chiral symmetry as expressed through the chiral Ward identities. The argument given in Ref. [71] that this is indeed the case is as follows. The total nucleon-to-nucleon transition amplitude of Eq. (5.1) is chirally symmetric, i.e., invariant under a simultaneous local transformation of the quark fields and the external fields (see Appendix A.1 for an illustration). In terms of the effective theory, the contribution from all the tree-level diagrams is chirally symmetric so that the loop contribution must also be chirally symmetric. Since we work in dimensional regularization this statement holds for an arbitrary However, as we have seen in the example of Eq. (5.210), the separation into infrared singular and regular parts amounts to distinguishing between contributions of non-integer and non-negative integer powers in the momentum expansion. Since these powers do not mix for arbitrary the infrared singular and regular parts must be separately chirally symmetric. Finally, the regular part can be expanded in powers of either momenta or quark masses, and thus may as well be absorbed in the (modified) tree-level contribution. Let us finally establish the connection between the infrared singular part I and the corresponding result in HBChPT. To that end, we first consider the

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relativistic propagator by expressing the (off-shell) four-momentum as

In the last step, we have assumed that is small enough to allow for an expansion in terms of a geometric series. The result of Eq. (5.221) is displayed in Fig. 5.9 and may be interpreted as an infinite series in terms of the heavy-baryon propagator and the self-energy insertion which has the form of a non-relativistic kinetic energy. (Note that the expression still involves the operator Let

us apply Eq. (5.221) to the loop integral H of Eq. (5.207) by first expanding the integrand and then performing the summation. This corresponds to the prescription proposed in Refs. [302, 126] for identifying the low-energy or, in the nomenclature of Ref. [302], soft contribution to a Feynman graph. In the case at hand we obtain

where

which is somewhat easier to handle if we perform the shift

and then

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the substitution

As above, we explicitly discuss the threshold into Eq. (5.224). Defining we have

The different

by inserting and

are related by a simple recursion relation

implying

Equation (5.226) is easily verified:

where we made use of the fact that the first term in the second line vanishes in dimensional regularization [see Eq. (A.77)]. We then obtain for the series, evaluated at threshold,

What remains to be determined is the threshold integral

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Performing a shift performing another shift

combining the denominators as in Eq. (A.57), and making use of Eq. (A.39), one finds

Finally, performing a substitution tion of Eq. (5.218) with we obtain

and using the analytic continua-

Inserting Eq. (5.228) into the series, the final result reads

which is the same as of Eq. (5.219). This example shows that the infrared regularized amplitude is related to an infinite sum of heavy-baryon amplitudes with self-energy insertions in the heavy-baryon propagator, as depicted in Fig. 5.9. The advantage of the relativistic approach is obvious, because for a general one-loop amplitude it may be very difficult, if not impossible, to obtain a closed expression for the sum of all insertions. To conclude this section, the method of infrared regularization provides a fully relativistic framework producing amplitudes having the relevant analytic properties and satisfying the chiral power-counting rules. At the moment, it is not yet clear, whether it can be generalized beyond the one-loop level.

6.

SUMMARY AND CONCLUDING REMARKS

As we have discussed in great detail, the chiral symmetry of QCD in the limit of vanishing and masses (Sect. 2.3.4.), together with the assumption of its spontaneous breakdown to in the ground state (Sect. 4.1.), is one of the keys to understanding the phenomenology of the strong interactions in the low-energy regime. The importance of chiral symmetry was realized long before the formulation of QCD and led to a host of predictions within the current-algebra and PCAC approaches of the 1960’s [3]. Some of the consequences of an explicit symmetry breaking, yielding non-analytic terms in the perturbation, were worked out in the early 1970’s, but the development came to a halt [273] because it was not clear how to systematically organize a perturbative expansion.

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From the present point of view, the explicit symmetry breaking is due to the finite and masses, leading to divergences of the symmetry currents (Sect. 2.3.6.). In 1979 Weinberg [322] laid the foundations for further progress with his observation that the constraints due to (chiral) symmetry may perturbatively be analyzed in terms of the most general effective field theory. A very important ingredient was the formulation of a consistent power-counting scheme (Sect. 4.4., and Sect. 5.5.8.) which allowed for a systematic perturbative analysis in contrast to various commonly used ad hoc phenomenological approaches to the strong interactions at low energies. In particular, the inclusion of loop diagrams allowed for a perturbative restoration of unitarity which would be violated if only tree-level diagrams were used. Subsequently Gasser and Leutwyler [163, 164] combined the ideas of Weinberg with other modern techniques of quantum field theory to analyze the Ward identities of QCD Green functions in terms of a local invariance of the generating functional under the chiral group (Sect. 2.4., and Appendix A.1). These papers were the starting point of what is nowadays called chiral perturbation theory. The mesonic sector has generated a host of successful applications, some of which have reached two-loop accuracy. Here, we have concentrated on a few elementary observables and processes, namely: masses of the Goldstone bosons (Sect. 4.3., and Sect. 4.9.1.), weak and electromagnetic decays (Sect. 4.6.1., and Sect. 4.8.), scattering (Sect. 4.6.2., and Sect. 4.10.2.), and electromagnetic form factors (Sect. 4.9.2.). Moreover, we have discussed in quite some detail how to construct the mesonic effective Lagrangian (Sect. 4.2., Sect. 4.7., and Sect. 4.10.1.). At first sight, it might appear that the large number of low-energy parameters at would make any quantitative prediction at the two-loop level impossible. However, there are several reasons why this is not the case. To start with, there exist observables which do not depend on any new parameters at i.e., which can be predicted in terms of the and low-energy constants only. An example is given by the correction to Sirlin’s theorem discussed in Ref. [281]. Clearly, such cases provide a natural testing ground for the convergence of the approach. Secondly, only a limited set of low-energy parameters contribute to any given process. It follows from the nature of the Ward identities that different physical processes are interrelated due to the underlying symmetries so that coefficients which have been fixed using one reaction can be used to predict another observable. In view of the ordinary implementation of symmetries, such as in the Wigner-Eckart theorem, this is not a surprise, because it is well-known that symmetries imply relations among elements. However, the Ward identities provide additional constraints among Green functions of a different type and allow one to also

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include an explicit symmetry breaking (Sect. 2.4.). It is this second case which can systematically be studied in the framework of ChPT and which provides interesting new insights into our understanding of both spontaneous and explicit symmetry breaking within QCD. Finally, different methods exist which allow one to estimate the value of the parameters and thus, in combination with the ChPT result, test our physical picture of the strong interactions. In this work we have only considered elementary processes at an introductory level, not the extensions to and combinations with other methods. We omitted, for example, the weak interactions of kaons which are mediated by the exchange of W bosons between the quark currents [118, 286, 274]. We also did not discuss the breaking of isospin symmetry which requires the inclusion of the electromagnetic interaction in terms of dynamical (virtual) photons [310, 269, 13]. Chiral symmetry also dictates the interaction of the Goldstone bosons with other hadrons (Sect. 5.1., and Sect. 5.2.). By studying the axial-vector current matrix element (Sect. 5.3.1.) and scattering (Sect. 5.3.2.) we verified that a tree-level calculation using the lowest-order Lagrangian reproduces the Goldberger-Treiman relation and the Weinberg-Tomozawa result for the scattering lengths, respectively. As we have seen, the first systematic study in the pion-nucleon sector [144] raised the question of a consistent power counting (Sect. 5.4.). This problem was subsequently overcome in the framework of the heavy-baryon approach [206] (Sect. 5.5.) and most of the numerous applications in this sector have been performed in HBChPT [44]. In the baryonic sector the chiral orders increase in units of one, because of the additional possibility of forming Lorentz invariants by contracting (covariant) derivatives with gamma matrices (Sect. 5.2.). As a result, in the SU(2) × SU(2) baryonic sector at the one-loop level, up to and including one has in total 2 + 7 + 23 + 118 = 150 [133] low-energy constants as opposed to the 2 + 7 = 9 free parameters of the corresponding mesonic sector [163]. Nevertheless, numerous results have been obtained in the baryonic sector because, at the same time, a large amount of very precise experimental data are available due to the existence of a stable proton target. (Neutron data can also be extracted, e.g, from experiments on the deuteron.) The availability of new high-precision data in combination with the techniques of chiral perturbation theory have led to a considerable improvement of our understanding of the strong interactions at low energies, in particular since systematic corrections to the old current-algebra predictions could be worked out and (successfully) tested. We have not discussed the approach of the so-called small scale expansion to include the resonance as an explicit degree of freedom [189]. Clearly this is an important issue in the baryonic sector because the first nu-

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cleon excitation is such a prominent feature of the low-energy spectrum as seen, e.g., in the total pion-nucleon scattering or the total photo absorption cross sections. Most recently, the method of infrared regularization [71] has opened the possibility of reconciling the relativistic approach with a consistent chiral power counting scheme (Sect. 5.6.). One may expect that this method will have a large impact insofar as many of the results obtained within the heavy-baryon framework will have to be checked with respect to relativistic corrections. The question regarding the radius of convergence in the baryonic sector remains a big challenge because, ultimately, a calculation at the two-loop level is, in general, required to quantitatively assess higher-order corrections [250]. In comparison to two-loop calculations in the SU(2)×SU(2) mesonic sector such an investigation in the (relativistic) nucleon sector is even more complicated for two reasons. First, due to the spin of the nucleon, the structure of vertices is richer than for spin-0 particles. Second, the nucleon mass introduces another mass in the propagators making the evaluation of the two-loop integrals more difficult than for a single mass. Finally, we would like to mention that a description of the nucleon-nucleon interaction within the framework of effective field theory has made tremendous progress and that a rigorous treatment of nuclei within field theory is no longer out of reach [323, 272, 209, 222, 123, 29, 135]. In conclusion, chiral perturbation theory has added a new and unprecedented level of systematics to the description of strong-interaction processes at low energies and continues to be a very fruitful and rich field with promising perspectives. If this introductory review encourages students and newcomers to chiral perturbation theory to participate in this field of research, it has served its purpose.

ACKNOWLEDGMENTS I am greatly indebted to David R. Harrington for carefully and critically reading the whole manuscript (!) and his uncountably numerous suggestions for improvement. He continuously forced me to explain the meaning of concepts instead of hiding behind the chiral jargon. I would like to thank my academic teachers Dieter Drechsel, Harold W. Fearing, and Justus H. Koch for sharing their deep insights into theoretical physics with me. Learning from them has been a pleasure! Numerous discussions with my collaborators and colleagues Thomas Ebertshäuser, Thomas Fuchs, Thomas R. Hemmert, Barry R. Holstein, Germar Knöchlein, Anatoly I. L’vov, Andreas Metz, Barbara Pasquini, and

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Christine Unkmeir are gratefully acknowledged. Special thanks go to Rolf Brockmann for many discussions on effective field theory, to Jambul Gegelia for his valuable discussions on the infrared regularization, to Martin Renter for his kind help in preparing Appendix A. 1 on Green functions and Ward identities, and to Thomas Walcher for challenging discussions on spontaneous symmetry breaking. Last but not least, I would like to thank Erich Vogt for his incredible enthusiasm and his continuous encouragement to finish the manuscript. I dedicate this work to my family. I hope it was worth it!

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APPENDIX A.1.

Green Functions and Ward Identities

In this appendix we will show how to derive Ward identities for Green functions in the framework of canonical quantization on the one hand, and quantization via the Feynman path integral on the other hand, by means of an explicit example. In order to keep the discussion transparent, we will concentrate on a simple scalar field theory with a global O(2) or U(l) invariance. To that end, let us consider the Lagrangian

where

with real scalar fields and Furthermore, we assume and so there is no spontaneous symmetry breaking and the energy is bounded from below. Equation (A.1) is invariant under the global (or rigid) transformations

or, equivalently,

where is an infinitesimal real parameter. Applying the method of Gell-Mann and Lévy [161], we obtain for a local parameter

from which, via Eqs. (2.42) and (2.43), we derive for the current corresponding to the global symmetry,

Recall that the identification of Eq. (2.43) as the divergence of the current is only true for fields satisfying the Euler-Lagrange equations of motion.

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We now extend the analysis to a quantum field theory. In the framework of canonical quantization, we first define conjugate momenta,

and interpret the fields and their conjugate momenta as operators which, in the Heisenberg picture, are subject to the equal-time commutation relations

and

The remaining equal-time commutation relations, involving fields or momenta only, vanish. For the quantized theory, the current operator then reads

where: : denotes normal or Wick ordering, i.e., annihilation operators appear to the right of creation operators. For a conserved current, the charge operator, i.e., the space integral of the charge density, is time independent and serves as the generator of infinitesimal transformations of the Hilbert space states,

Applying tion relations*

it is straightforward to calculate the equal-time commuta-

In particular, performing the space integrals in Eqs. (A. 12), one obtains

* The transition to normal ordering involves an (infinite) constant which does not contribute to the commutator.

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In order to illustrate the implications of Eqs. (A. 13), let us take an eigenstate of with eigenvalue and consider, for example, the action of on that state,

We conclude that the operators and and ] increase (decrease) the Noether charge of a system by one unit. We are now in the position to discuss the consequences of the U(1) symmetry of Eq. (A.1) for the Green functions of the theory. To that end, let us consider as our prototype the Green function

which describes the transition amplitude for the creation of a quantum of Noether charge +1 at propagation to interaction at via the current operator, propagation to with annihilation at First of all we observe that under the global infinitesimal transformations of Eq. (A.3), or in other words We thus obtain

the Green function remaining invariant under the U(1) transformation. (In general, the transformation behaviour of a Green function depends on the irreducible representations under which the fields transform. In particular, for more complicated groups such as SU(N), standard tensor methods of group theory may be applied to reduce the product representations into irreducible components [79, 271, 207]. We also note that for U(1), the symmetry current is charge neutral, i.e., invariant, which for more complicated groups, in general, is not the case.) Moreover, since is the Noether current of the underlying U( 1) there are further restrictions on the Green function beyond its transformation behaviour under the group. In order to see this, we consider the divergence of Eq. (A.14) and apply the equal-time commutation relations of Eqs. (A.12) to obtain (see Sect. 2.4.1.)

where we made use of Equation (A.16) is the analogue of the Ward identity of QED [see Eq. (2.85)] [315, 139, 301]. In other words, the underlying symmetry not only determines the transformation behaviour of Green

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functions under the group, but also relates Green functions containing a symmetry current to -point Green functions [see Eq. (2.89)]. In principle, calculations similar to those leading to Eqs. (A.15) and (A.16), can be performed for any Green function of the theory. However, we will now show that the symmetry constraints can be compactly summarized in terms of an invariance property of a generating functional. The generating functional is defined as the vacuum-to-vacuum transition amplitude in the presence of external fields,

where and are the field operators and is the Noether current. Note that the field operators and the conjugate momenta are subject to the equaltime commutation relations and, in addition, must satisfy the Heisenberg equations of motion. Via this second condition and implicitly through the ground state, the generating functional depends on the dynamics of the system which is determined by the Lagrangian of Eq. (A.1). The Green functions of the theory involving and are obtained through functional derivatives of Eq. (A. 17). For example, the Green function of Eq. (A.14) is given by

In order to discuss the constraints imposed on the generating functional via the underlying symmetry of the theory, let us consider its path integral representation [332, 103],†

where

†

Up to an irrelevant constant the measure considered as independent variables of integration.

is equivalent to

with

and

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denotes the action corresponding to the Lagrangian of Eq. (A.1) in combination with a coupling to the external sources. Let us now consider a local infinitesimal transformation of the fields [see Eqs. (A.3)] together with a simultaneous transformation of the external sources,

The action of Eq. (A.20) remains invariant under such a transformation,

We stress that the transformation of the external current is necessary to cancel a term resulting from the kinetic term in the Lagrangian. We can now verify the invariance of the generating functional as follows,

We made use of the fact that the Jacobi determinant is one and renamed the integration variables. In other words, given the global U(1) symmetry of the Lagrangian, Eq. (A.1), the generating functional is invariant under the local transformations of Eq. (A.21). It is this observation which, for the more general case of the chiral group SU(N)×SU(N), was used by Gasser and Leutwyler as the starting point of chiral perturbation theory. We still have to discuss, how this invariance allows us to collect the Ward identities in a compact formula. We start from Eq. (A.23),

Observe that

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and similarly for the other terms, resulting in

Finally we interchange the order of integration, make use of partial integration, and apply the divergence theorem:

Since Eq. (A.24) must hold for any deriving Ward identities,

we obtain as the master equation for

We note that Eqs. (A.23) and (A.25) are equivalent. As a final illustration let us re-derive the Ward identity of Eq. (A. 16) using Eq. (A.25). For that purpose we start from Eq. (A. 18),

apply Eq. (A.25),

make use of functional derivatives,

and

for the

and, finally, use the definition of Eq. (A.17),

which is the same as Eq. (A. 16). In principle, any Ward identity can be obtained by taking appropriate higher functional derivatives of W and then using Eq. (A.25).

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Dimensional Regularization: Basics

For the sake of completeness we provide a simple illustration of the method of dimensional regularization. For a detailed account the interested reader is referred to Refs. [197, 229, 198, 91, 94, 311]. Let us consider the integral

which appears in the calculation of the masses of the Goldstone bosons [see Eq. (4.144)]. We introduce

so that

and define

In order to determine as part of the calculation of I, we consider in the complex plane and make use of Cauchy’s theorem

for functions which are differentiable in every point inside the closed contour C. We choose the contour as shown in Fig. A.1,

and make use of

to obtain for the individual integrals

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In combination with Eq. (A.27) we obtain the so-called Wick rotation

As an intermediate result the integral of Eq. (A.26) reads

where denotes a Euclidian scalar product. In this special case, the integrand does not have a pole and we can thus omit the

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which gave the positions of the poles in the original integral consistent with the boundary conditions. The degree of divergence can be estimated by simply counting the powers of momenta [311]. If the integral behaves asymptotically as the integral is said to diverge quadratically, linearly, and logarithmically, respectively. Thus, our example I diverges quadratically. Various methods have been devised to regularize divergent integrals. We will make use of dimensional regularization, because it preserves algebraic relations between Green functions (Ward identities) if the underlying symmetries do not depend on the number of dimensions of space-time. In dimensional regularization, we generalize the integral from 4 to dimensions and introduce polar coordinates

where is then symbolically of the form

A general integral

If the integrand does not depend on the angles, the angular integration can explicitly be carried out. To that end one makes use of

which can be shown by induction. We then obtain for the angular integration

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We define the integral for

dimensions ( integer) as

where for convenience we have introduced the renormalization scale µ so that the integral has the same dimension for arbitrary (The integral of Eq. (A.32) is convergent only for ) After the Wick rotation of Eq. (A.28) and the angular integration of Eq. (A.31) the integral formally reads

For later use, we investigate the (more general) integral

where we made use of the substitution Beta function

We then make use of the

where the integral converges for and diverges if or For non-positive values of or we make use of the analytic continuation in terms of the Gamma function to define the Beta function and thus the integral of Eq. (A.33).‡ Putting and our (intermediate) integral reads

which, for

‡

Recall that

yields for our original integral

is single valued and analytic over the entire complex plane, save for the points where it possesses simple poles with residue

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Since is an analytic function in the complex plane except for poles of first order in 0, –1, –2, …, and is an analytic function in C, the right-hand side of Eq. (A.36) can be thought of as a function of a complex variable which is analytic in C except for poles of first order for Making use of

we define (for complex n)

Of course, for the Gamma function has a pole and we want to investigate how this pole is approached. The property allows one to rewrite

where we defined Making use use of we expand the integral for small

where

where

is Euler’s constant. We finally obtain

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Using the same techniques one can easily derive a very useful expression for the more general integral

We first assume and The last condition is used in the Wick rotation to guarantee that the quarter circles at infinity do not contribute to the integral. The transition to the Euclidian metric produces the factor The angular integral in dimensions is then performed as in Eq. (A.31). The remaining radial integration is done using Eq. (A.33) with the substitution and The analytic continuation of the right-hand side of Eq. (A.39) is used to also define expressions with (integer) in dimensional regularization. In the context of combining propagators by using Feynman’s trick one encounters integrals of the type of Eq. (A.39) with replaced by where A is a real number. In this context it is important to consistently deal with the boundary condition [311]. For example, let us consider a term of the type To that end one expresses a complex number in its polar form where the argument of is uniquely determined if, in addition, we demand For A > 0 one simply has For A < 0 the infinitesimal imaginary part indicates that — |A| is reached in the third quadrant from below the real axis so that we have to use the We then make use of and obtain

Both cases can be summarized in a single expression

The preceding discussion is of importance for consistently determining imaginary parts of loop integrals. Let us conclude with the general observation that (ultraviolet) divergences of one-loop integrals in dimensional regularization always show up as single poles in

A.3. Loop Integrals In Appendix A.2 we discussed the basic ideas of the method of dimensional regularization. Here we outline the calculation of more complicated one-loop

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integrals of mesonic as well as heavy-baryon chiral perturbation theory. We restrict ourselves to the cases needed to reproduce the examples discussed in the main text and refer the interested reader to Refs. [197, 229, 198, 91, 94, 311] for more details.

A.3.1.

One-Loop Integrals of the Mesonic Sector

In the mesonic sector we will use the following definition and nomenclature for the scalar loop integrals (i.e., no Lorentz indices) extended to dimensions:

where we omit an explicit reference to the scale µ and the “number of dimensions” § In the SU(3)×SU(3) case one also needs loop integrals with different masses such as, e.g.

A.3.1.1. We define

Using a shift

(in the regularized integral) we obtain

However, this is just the basic integral I we discussed in detail in Appendix A.2:

where

Later on we will also use the common notation §

If

the integral converges for

for the integral

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A.3.1.2. In the calculation of the one-loop contribution of Fig. 4.12 to the electromagnetic form factor of the pion, Eq. (4.163), we encounter an integral of the type

Using a shift

we obtain

It is thus sufficient to consider denominators using Feynman’s trick:

with

and perform the shift

To that end, we first combine the

and

to obtain

resulting in

Performing the Wick rotation, applying the results of Eqs. (A.31) and (A.35), expanding the result in and finally performing the integration, we obtain

where [308]

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with

Note that Eq. (A.47) represents a case where the boundary condition has to be treated consistently, as discussed at the end of Appendix A.2. For later use we introduce the notation

where the subscript 0 refers to the scalar character of the integral. Next we want to determine the tensor integrals appearing in Eq. (4.163) by reducing them to already known integrals. The general idea consists of parameterizing the tensor structure in terms of the metric tensor and products of external four-vectors and multiplying the results by invariant functions of Lorentz scalars. We first consider

which must have the form where the subscript 1 refers to one four-vector in the numerator of the integral. We contract Eq. (A.49) with and make use of to obtain

where we used the argument in Appendix A.3.1.1 to show that the first two integrals cancel. We have thus reduced the determination of Eq. (A.48) to an already known integral:

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Finally, we also need

where the first subscript 2 refers to two four-vectors in the numerator of the integral and the second subscripts 0 and 1 refer to the number of metric tensors in the parameterization, respectively. Contracting with and making use of in dimensions we obtain Similarly, contracting with

we obtain

By subtracting Eq. (A.53) from (A.52) we find

where we made use of

and Eqs. (A.43) and (A.47). From Eq. (A.53) we obtain

In working out Eqs. (A.54) or (A.55) it must be remembered that contains a term which, when multiplied by gives a term of order This must be done carefully to obtain, e.g., the term in Eq. (A.54) and the 5/6 term in Eq. (A.55).

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One-Loop Integrals of the Heavy-Baryon Sector Basic Loop Integral

The structure of the one-loop integrals in the heavy-baryon approach is slightly different from the integrals of the mesonic sector discussed in Sect. 6. Here we will outline the calculation of a basic loop integral which serves as a starting point for more complicated calculations. Consider an integral of the type

where and A are arbitrary real numbers and is the four-velocity of the heavy-baryon approach. Counting powers of the momenta, the integral is linearly divergent. An integral of this type appears in the calculation of the nucleon self energy in Sect. 5.5.9. We combine the denominators using the following Feynman trick:

Below, this choice will allow us most easily to combine the integration with the “radial” integration of the loop momentum after the Wick rotation. Inserting and we obtain

We now generalize to

dimensions:

and perform a shift of integration variables no terms linear in in the denominator:

so that there remain

where we made use of Finally, we shift the integration variable in order to eliminate terms linear in in the denominator:

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The integration is split into and Making use of a Wick rotation and Eqs. (A.31), (A.35), and (A.40) we obtain for the first integral

where

Performing a Wick rotation and using Eq. (A.31), the second integral reads

Using polar coordinates

and

together with

we rewrite the second integral as

Finally, applying again Eq. (A.35) for the radial integral we obtain for the second integral (in four dimensions)

where we made use of and The remaining integration of the first integral is elementary and we obtain as the final expression

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with

A.3.2.2. In analogy to the mesonic case we define

Using a shift

we obtain

It is thus sufficient to consider is given by¶

¶ Our

corresponds to

which, using the result of Eq. (A.65),

of Ref. [44].

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In the calculation of the nucleon self energy we also need tensor integrals which, as in the mesonic case, one may reduce to already known integrals. Let us introduce the notation

where the subscript 0 refers to the scalar character of the integral. Once again, the general idea in the determination of tensor integrals consists of parameterizing the tensor structure in terms of the metric tensor and products of the four-velocity We first consider

which must have the form where the subscript 1 refers to one four-vector in the numerator of the integral. We contract Eq. (A.69) with make use of and add and subtract in the numerator of the integral, obtaining

where is defined in Eq. (A.43). We have thus reduced the determination of Eq. (A.68) to the already known integrals and As our final example, let us discuss

which must be of the form

where the first subscript 2 refers to two four-vectors in the numerator of the integral and the second subscripts 0 and 1 refer to the number of metric tensors in the parameterization, respectively. Contracting with and adding and subtracting in the numerator, we obtain

Introduction to Chiral Perturbation Theory

521

where we made use of Eq. (A.70) and

Finally, contracting with we obtain

and making use of

in

dimensions

where we made use of

in dimensional regularization. In order to verify Eq. (A.74), one writes

The first term vanishes, because the integrand is odd in For the evaluation of the second term we choose Applying the residue theorem, we obtain for the integral

so that we may define for arbitrary

However, the last term vanishes in dimensional regularization. From Eqs. (A.72) and (A.73) we obtain

522

Stefan Scherer

and

where, as in the mesonic case, it is important to identify the finite terms resulting from the product of terms and . Finally, when discussing the relation between the infrared regularization method of Ref. [71] and the heavy-baryon approach in Sect. 5.6., we made use of the fact that an integral of the type

vanishes in dimensional regularization. This can be seen by substituting and relabeling

Since is arbitrary and, for fixed and the result is to hold for arbitrary Eq. (A.77) is set to zero in dimensional regularization. We emphasize that the vanishing of Eq. (A.77) has the character of a prescription [201]. The integral does not depend on any scale and its analytic continuation is ill defined in the sense that there is no dimension where it is meaningful. It is ultraviolet divergent for and infrared divergent for

A.4.

Different Forms of

in SU(2) × SU(2)

A.4.1. GL Versus GSS The purpose of this appendix is to explicitly relate the two commonly used Lagrangians of Refs. [163] (GL) and [144] (GSS). In their pioneering work on mesonic SU(2) × SU(2) chiral perturbation theory [163], Gasser and Leutwyler used a notation adopted from the O(4) nonlinear model, because the two Lie groups SU(2) × SU(2) and O(4) are locally isomorphic, i.e., their Lie algebras are isomorphic. The effective Lagrangian was written in terms of invariant scalar products of real four-vectors in contrast to the nowadays standard trace form. The dynamical pion degrees of freedom were expressed in terms of a four-component real vector field of unit

Introduction to Chiral Perturbation Theory

523

length with components A = 0, 1, 2, 3. The connection to the SU(2) matrix of Sect. 4.2.2., can be expressed as

with

so that U is unitary. The lowest-order Lagrangian in the trace notation is given by the SU(2) × SU(2) version of Eq. (4.70), and the transcription of the SU(2) × SU(2) Lagrangian [see Eq. (5.5) of Ref. [163]] reads||

When comparing with the SU(3)×SU(3) version of Eq. (4.104) we first note that Eq. (A.79) contains fewer independent terms which results from the application of the trace relations, as discussed in Sect. 4.10.1. The bare and the renormalized low-energy constants and are related by

||

Note that Gasser and Leutwyler called the tively.

and

Lagrangians

and

respec-

524

where

Stefan Scherer

and

In the SU(2)×SU(2) sector one often uses the scale-independent parameters which are defined by

where Since ln(1) = 0, the are proportional to the renormalized coupling constant at the scale µ = M. Table A.1 contains numerical values for the scale-independent low-energy coupling constants as obtained in Ref. [163] together with more recent determinations.

Secondly, the expression proportional to can be rewritten so that the U’s completely drop out, i.e., it contains only external fields. The trick is to use

Introduction to Chiral Perturbation Theory

525

The terms proportional to the agree with Eq. (4.2) of Ref. [32] but the term is not yet in the form of the SU(3)×SU(3) version of Eq. (4.104). By means of a total-derivative argument in combination with a field transformation as discussed in Sect. 4.7., we will transform of Eq. (A.79) into another form which is often used in the literature. To that end let us make use of

rewrite the

and

terms by using

to obtain

where “tot. der.” refers to a total-derivative term which has no dynamical significance. We make use of a trace relation for arbitrary 2 × 2 matrices [see Eqs. (4.192) and (4.194)],

and [see Eq. (4.67)] to rewrite the first term of (A.83) as the product of two trace terms,

By adding and subtracting appropriate terms to generate an expression proportional to the lowest-order equation of motion which, for SU(2)×SU(2),

526

Stefan Scherer

reads [see Eq. (4.77)]

we re-express the last term of Eq. (A.83) as

The

term can thus be written as

which, except for the total derivative and the equation-of-motion term, is the same as Eq. (5.9) of Gasser, Sainio, and [144]. The difference between the Lagrangians of [163] and [144] then reads

which agrees with Eq. (26) of Ecker and [121] once their expressions are rewritten in the above notation. Let us also specify the field transformation required to connect the two Lagrangians. For that purpose we rewrite Eq. (A.88) in accord with Eq. (2.11) of [298],

According to Eqs. (4.118) and (4.119) we need to insert and in Eq. (4.116) in order to relate the two Lagrangians. Finally, making use of Eqs. (A.87) and (A.82) and dropping the total derivative as well as the equation-of-motion term let us explicitly write out the GSS

Introduction to Chiral Perturbation Theory

527

Lagrangian:

A.4.2.

Different Parameterizations

In Appendix A.4.1 we saw that two versions of the mesonic Lagrangian, Eqs. (A.79) and (A.90), are used in the literature. Since they are related by a field transformation, they must yield the same results for physical observables [89, 210]. Furthermore, in SU(2)×SU(2) two different parameterizations of the SU(2) matrix [see Eqs. (4.87) and (4.88)] are popular,

where the pion fields of the two parameterizations are non-linearly related [see Eq. (4.89)]. In this appendix we collect the pion wave function renormalization constants entering a calculation at depending on which Lagrangian and parameterization is used. The actual calculation parallels that of Sect. 4.9.1., and will not be repeated here. The self energies up to can be written as

528

Stefan Scherer

The renormalized mass and the wave function renormalization constant are, respectively to and given by

where is the prediction at The different values for A, B, and Z are given in Table A.2. Note that the result for the pion mass is, as expected, independent of the Lagrangian and parameterization used:

where

and

[see Eqs. (A.80) and (A.81)]. On the other hand, the constants A, B, and Z are auxiliary mathematical quantities and thus depend on both Lagrangian and parameterization.

Introduction to Chiral Perturbation Theory

529

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INDEX Anomalies, 282, 304, 307 axial, 307-308, 322 Asymptotic freedom, 278 Axial vector current, 314-315, 321, 345, 354-356, 372, 375, 418, 427, 431

Cayley-Hamilton theorem, 413 C14 beta decay. 68-72 and tensor forces, 68-69 Charge conjugation, 317-318, 387 Charge densities, 308 equal-time commutation relations, 311 Charge independence, 10, 159-160 Charge operators, 301-306 axial, 345 Lie algebra, 302 Chiral algebra, 304-306 Chiral anomaly, 284 Chiral Lagrangian, 276-277, 380384 fourth order, 380-389 sixth-order, 406-414 Chiral limit, 276, 280, 286, 299, 304, 314-316, 340, 344, 358, 425, 438-440 Chiral perturbation theory (ChPT), 275-537 convergence. 284-285, 364, 473, 497-499 for baryons, 419-499 for mesons, 339-417 heavy-baryon formulation (HBChPT), 286-288, 442, 449-483 Lagrangian, 457, 471 light and heavy nucleon components, 454-458 power counting scheme, 476479

Bardeen-Cooper-Schrieffer (BCS) theory, 180-193, 216, variational wave function, 182 ground state, 183, 220 Bartlett force, 30, 59 Baryon number conservation, 280, 340 Baryon octet, 420-421 Beta decay, 44-58, 68-72, 95, 106, 118, 177 favoured, 219 first forbidden, 149 retarded, 158 unfavoured, 114 Bethe Goldstone equation, 170 Bianchi identity, 410-412 Bjorken scaling, 278 Blomqvist interaction, 229 Bogolyubov-Valatin transformation, 183, 185, 191 Broken pair approximation, 221 Brueckner theory, 72, 89 Cabbibo-Kobayashi-Maskawa matrix, 320, 371 Canonical quantization, 300 539

540

reparameterization invariance, 473 fourth-order applications, 388405 free parameters, 286 mesonic, 277, 359 orders, 277, 285, 359 relativistic baryon, 439 renormalization scale, 364 sixth-order, 405-417 two-flavour version, 320 Chiral symmetry, 277-322 breaking, 496 due to quark masses, 306308, 322, 359 breaking scale, 284 of QCD Lagrangian, 322 Chung-Wildenthal interactions, 237 Cloudy bag model, 277 Coleman’s theorem, 280-281 Collective model, 3-7, 46-49, 74, 100, 185 collective Hamiltonian, 186 collective parameter, Q, 186 relation to Shell model, 6, 78, 205 SU(3) scheme, 202-214 Configuration mixing, 36, 46, 48, 62-68, 74, 89, 113, 167168, 173, 259 Continuous symmetry, 323 Cooper pairs, 180 Core excitation, 108, 129-130, 172, 216 Core polarization, 99, 110 Coulomb energy of nuclei, 10-11, 62, 91 Cross conjugate nuclei, 116-118 symmetry, 117 Cross-over of levels, 25 Current algebras, 281, 283, 287, 375,

Index

415, 496 Current conservation, 299, 302, 308 vector, 308 Current density four-current, 300 Cutkowsy rules, 486 Darwin term, 453 Deep-inelastic lepton-hadron scattering, 272 Dimensional regularization, 381, 394, 446, 449, 460, 507-513 Dirac equation, 297 Discrete internal symmetry, 323

Drell-Yan processes, 278 Dynamical symmetry, 209 Effective charge, 50-52, 86, 152, 192, 218, 228 Effective field theory, 281-288, 339, 359, 497-499 Lagrangian, 282 nonrenormalizability, 283 Effective g-factors, 38, 96, 174 Effective interaction, 2, 4, 75, 89180, 230, 253-265 figure of merit, 157 from experimental nuclear energies, 89-180, 97 in the p-shell, 101-109, 175176 matrix elements, 102-104 in the d-f-shell, 108-109, 115120, 133-140 in the g-shell, 111 matrix elements, 146 in mixed configurations, 121132 matrix elements, 139 isopspin components, 113-114, 255-256

Index

Kuo-Brown interaction, 130132, 135 monopole part, 242 pairing plus quadrupole, 180192 Effective Lagrangian, 359-365, 418 construction, 365-369 Goldstone boson fields, 373 lowest order, 352-359 baryonic, 422-427 sixth-order, 406 Eightfold way, 288 Electromagnetic form factors, 497 neutral kaons, 404 mean-square-radius, 405 neutral pions, 400-405 Electromagnetic multipole transitions, 34, 44-52, 78, 115, 139, 158, 172, 217-218, 228, 254 Energy gap, 98, 181-188 Equal-time commutation relations, 309-313 Euler-Lagrange equations, 369 Explicit symmetry breaking, 328, 496 Fermi fields, 305 Fermion creation operators, 193 Fermion Dynamical Symmetry Model (FDSM), 214 Ferromagnets, 358 Flavour conservation, 279 Folded diagram expansion method, 228 Foldy-Wouthuysen method, 287, 450, 453, 469 Form factors, 277, 288 Fractional parentage coefficients, 42, 75, 91-93

541

Gap equation, 182-183 Gasser-Leutwyler convention, 366 Gasser-Leutwyler Lagrangian, 286, 315, 339, 367, 379, 384, 397-400, 416, 497, 522 Gasser, Sainio and Svarc Lagrangian, 521 Gauge fields external, 319 Gauge invariance, 292 Gauge principle, 291 and QCD Lagrangian, 293 Gauge transformations, 292, 294 Gell-Mann-Levy method, 501 Gell-Mann-Low formula, 461-462, 479 Gell-Mann matrices, 289, 299, 350, 368 action in colour space, 294 commutation relations, 290 Gell-Mann, Oakes and Renner relations, 358, 417 Gell-Mann-Okubo relation, 358 Gerimasov-Drell-Hearn sum rule, 484 Giant Dipole Resonance, 82-83 Ginocchio models, 210-214 Global symmetry currents light quark sector, 303-304 Gluon fields, 278 Goldberger-Treiman relation, 425432, 435-437, 498 Goldstone boson, 276, 281-286, 324, 331-342, 352-374, 418496 masses, 357, 388-398, 497, 507 number of, 334 transformation properties, 346351 and WSW action, 384-388 Goldstone theorem, 276, 281 334-

542

338, 342 and spontaneous symmetry breaking, 322-339 Green functions, 276-277, 281-282, 287, 306, 308-322, 364, 501-506 and generating functional, 315316 chiral, 308-311 four-point, 378-379 n-point, 314 QCD, 375-377,497-499 single-baryon, 422 two-point, 389 Gross-Frenkel interaction, 164-167 Group properties cosets, 347-351 nonlinear realization, 346

Index

228 mapping nucleon states onto boson states, 214 proton and neutron bosons, 207208 F-spin, 207 Intermediate coupling, 31, 33, 63, 77 Intruder states, 126, 173, 238, 243, 251, 259 Island of inversion, 239-240, 251 Isomeric states, 27 Jacobi identity, 290, 412 Kingston International Conference on Nuclear Structure (1960), 189 Kuo-Brown interaction, 130-132, 136, 241-250

Hadron production in electron positron annihilation, 278 Hadron spectrum, 277-279, 309, 340342 and spontaneous symmetry breaking, 341 global SU(3) symmetry, 288 Hard-core interactions, 73-85, 87 Hartree-Fock calculations, 6-8, 12, 201 Cranking model, 7 Heavy baryon formulation, 449-483, 521 Heat-kernel technique, 381 Heavy quark physics, 286, 296 masses, 380 Heisenberg force, 30

Large Scale Shell Model Calculations, 232-252, 260 size of matrices. 249, 260-263 Lehmann-Symanzik-Zimmermann reduction formalism, 309310, 376, 464 Lepton pair production in Drell-Yan processes, 278 Levi-Civita tensor, 295, 359 Lie group compact, 331-333, 346 Linear sigma model (O(N)), 346 Liquid drop model, 13 Loop integrals, 513-522 baryon sector, 517 mesonic sector, 513

Infrared Regularization method, 483496, 499, 521 Interacting boson model, 206-214,

Magic numbers, 2-5, 13-19, 36, 216, 249 Majorana interaction, 18-26, 30, 59

Index

543

Margenau-Wigner lines, 24 Neutron electric dipole moment, 295 Mexican Hat potential, 331-332 Migdal force, 201 0(3) group, 208, 210, 330 Millener-Kurath interaction (MK), 177, 239 Casimir operator, 209, 213 Minkowski space, 347, 385 O(5) group, 208, 210 Modified-surface-delta-interaction(MSDI),O(6) group, 210-211 165-167 Casimir operators, 212 Moment of Inertia, 8, 128, 171 quadrupole operator, 210 Monopole force, 200 Odd-group model, 35, 39 Monte Carlo Shell Model (MCSM), 247, 258 Pair correlations, 141 Moszkowski-Scott separation method, Pairing interaction, 82, 113, 142, 134 167, 181-194, 255 Multidetector arrays, 163 general wave functions, 220 Hamiltonian, 182 Nambu-Goldstone mode, 281, 325, schematic, 134 389, 397 Pandya transformation, 157 Nambu Goldstone realization of symparticle-hole, 159 metry, 323 Partially conserved axial-vector curNilsson model, 5, 79, 169 rent (PCAC), 281, 286, Noether’s theorem, 299-303, 311, 308, 321, 375, 496 503 Pauli-Lubanski vector, 459 Nordheim rules, 59=62 Permutation group, 21 Nuclear magnetic moments, 37-52. Phase transition, 155 76, 96, 116-121, 153, 175, Pion crossing symmetry, 433-435 180, 244 Pion Decay, 370-372 effective moments, 123-124 Pion electromagnetic form factor, Schmidt-Lande model, 24, 37398-407 52, 153, 180 Pion-nucleon form factor, 460 deviations from, 38-40, 121 Pion-nucleon scattering, 432-440 meson-exchange corrections, 41 correction due to pion loops, Nucleon electromagnetic form fac445-449 tor lowest order, 468-470 isovector spectral function, 287 Pion-pion scattering, 372-380 Nucleon magnetic moments at sixth-order, 413-416 anomalous, 277, 477 s-wave scattering length, 415 quark model analysis, 278 Roy equations, 416 Nucleon mass Weinberg-Tomozawa predicdue to pion loops, 441-445, 478tions, 432, 438-440, 498 483 Pion self energy, 393

544

Poincare group, 459 Power counting, 282-287, 497 Pseudonium nuclei, 179-180, 225 Quadrupole-quadrupole interaction, 185-192, 204, 206, 258 and seniority, 188 Quantum chromodynamics (QCD) and chiral symmetry, 288-322 global symmetry, 276, 279, 281 accidental, 296-308 flavour, 339 spontaneous breakdown, 276277 Lagrangian, 279, 281, 289-303, 315-316 gluon part, 295 in chiral limit, 299 Levi-Civita tensor, 295 light flavour version, 296 phenomenological, 281 right and left-handed fields, 298 local colour symmetry, 279 open problems gap problem, 278 quark confinement, 278 spontaneous chiral symmetry breaking, 278, 285, 340346 vs. Quantum Electrodynamics, 279, 291 Quantum Monte Carlo Diagonalization (QMCD), 247, 250 Quark condensate chiral, 357 scalar, 343-346 Quark confinement, 278 colour confinement, 279 Quark fields, 294, 317 left and right-handed fields, 297,

Index

318 Quark mass terms, 293, 306 quark mass matrix, 306 Quark properties, 293, 296 Quasi particles, 183 random phase approximation, 191 single particle energy, 184, 186 vacuum, 183-185 Quasi-spin scheme, 192-201, 212, 220, 226-227 Relativistic framework infrared regularization, 276-277, 287 Relativistic mean field theory, 8-10, 206 Riemann-Christoffel curvature tensor, 412 Rosenfeld mixture, 30, 32-34, 49, 65, 80-83, 87, 110, 134, 155 Rotation Group, 20 Running coupling constant, 288 Saddle-point method, 381 Schiffer method, 200 Schiffer-True interaction, 218 Schwartz theorem, 412 Schwinger term, 312-313 Seagull terms, 313 Seniority Scheme, 28-30, 39-40, 5358, 93-99, 105, 109-184, 192-200, 214-232, 242, 257 and the delta potential, 53-58 generalized seniority, 36, 133, 140, 180, 220-232, 261 double commutator condition, 223 and odd-even nuclei, 225 powerful truncation scheme,

Index

228, 230 and semi-magic nuclei, 220 variational wave function, 221 in h-orbits, 216-219 and jj-coupling, 54 and spacing between levels, 215, 228 and surface delta interaction, 194 and SU(2) symmetry, 54 and U(21+l) & O(3) symmetry, 55 Racah’s scheme, 181, 193 Serber mixture, 34, 123 Shape coexistence, 246 Shell Model centre-of-mass motion, 80 Mayer-Jensen shell model, 40 and neutron drip line, 257 origins, 2, 11-19 LS-coupling vs jj-coupling, 15-18 rules of jj-coupling, 16-18 validity, 13, 19 Single nucleon tensor operators, 51 irreducible, 51 Skyrme interaction, 201 SO(3) group, 330 Spectroscopic factors single nucleon, 137-138, 173 Spin gap, 143-144 Spontaneous symmetry breaking and Goldstone theorem, 322339 and hadron spectrum, 341 comparisons, 346 global, continuous, non-Abelian, 330-334 Spurious states, 103, 108, 177-178, 239, 249 Stabilization effect, 35

545

Stretched states, 145 SU(2) group and QCD, 399, 413-414, 420422 SU(2) Lie algebra, 193, 214, 221224, 226, 284, 309, 373 Casimir operator, 222 SU(3) group, 79, 107, 279, 382, 408, 496 Lie algebra, 288-290 Clebsch-Gordan series, 289 SU(3) scheme, 202-214, 254 broken by the spin orbit interaction, 206 Casimir operator, 204, 209, 213 colour transformations, 279-322 as underlying gauge group of QCD, 292, 365 diagonal interaction, 204 energy eigenvalues, 209 flavour symmetry, 280 gauge theory, 276-278 generators, 203 global transformations, 352, 419421 Structure constants, 290, 293 SU(4) group, 202 SU(4) symmetry, 19-26, 40, 47, 55 SU(N) group, 353, 505 Super deformed bands, 260 Supermultiplet symmetry, 4, 19-26 abandonment, 26 and favoured beta decay, 2426, 47, 115 Surface Delta interaction (SDI), 192201 and generalized seniority, 194 modified (MSDI), 197, 201 Symmetry energy term, 94, 113, 255 Symplectic group, 57, 120

546

Index

Tensor operators, 53 Three-body interactions, 3, 255 Toy model, 330-334, 340

Yang-Mills theories, 278 Yrast states, 143, 158, 178, 199, 229, 238, 245, 248, 259, 265

U(l) group, 279, 282, 288, 291, 309, 340, 496 U(3) group, 203, 280, 299 U(4) group, 21 U(5) group, 208, 209 U(6) group, 207, 209 and SU(3) symmetry, 209 Unified model, 3-7

Zero range potential, 53-62, 85, 87, 113, 121-122, 168, 194

Vector currents, 315, 354 Vibrational nuclei, 133, 141 Vielbein, 424, 428, 460 Ward identities, 276-277, 282, 306, 308-322, 364, 422, 427506 anomalous, 385 chiral, 309-322, 493 Ward-Fradkin-Takahashi identity, 309 Weinberg’s power counting scheme, 282, 339, 359-365, 380, 497 Weinberg theorem, 282 Weisskopf units, 45, 161 Weizmann Institute conference, (1957), 100 Wess-Zumino-Witten construction, 282, 285, 340, 365 effective action, 384-388, 406 Wick rotation, 447, 509-515 Wick’s theorem, 461 Wigner-Eckart theorem, 51, 58, 497 Wigner force, 52, 142 Wigner interaction 19-26, 30 Wigner term, 23 Wigner-Weyl mode, 280, 323