A Quest for Symmetry: Selected Works of Bunji Sakita 9810236433, 9789810236434

Citation preview

World Scientific Series in 20th Century Physics - Vol. 22

HyJLi iy^^J^I

Wwi

ETOUJIIM Selected Works of Bunji Sakita

Editors

Keiji Kikkawa Miguel Virasoro Spenta R. Wadia

World Scientific

_ j QUEST FOR SYMMETRY

World Scientific Series in 20th Century Physics Published Vol. 1 Gauge Theories — Past and Future edited by ft Akhoury, B. de Wit, and P. van Nieuwenhuizen Vol. 2

Scientific Highlights in Memory of Leon van Hove edited by F. Nicodemi

Vol. 3

Selected Papers, with Commentary, of T. H. R. Skyrme edited by G. E. Brown

Vol. 4

Salamfestschrift edited by A. Ali, J. Ellis and S. Randjbar-Daemi Selected Papers of Abdus Salam (with Commentary) edited by A. Ali, C. Isham, T. Kibble and Riazuddin

Vol. 5 Vol. 6

Research on Particle Imaging Detectors edited by G. Charpak

Vol. 7

A Career in Theoretical Physics edited by P. W. Anderson

Vol. 8

Lepton Physics at CERN and Frascati edited by N. Cabibbo

Vol. 9

Quantum Mechanics, High Energy Physics and Accelerators: Selected Papers of J. S. Bell (with Commentary) edited by M. Bell, K. Gottfried and M. Veltman

Vol. 10 How We Learn; How We Remember: Toward an Understanding of Brain and Neural Systems — Selected Papers of Leon N. Cooper edited by L N. Cooper Vol. 11 30 Years of the Landau Institute — Selected Papers edited by I. M. Khalatnikov and V. P. Mineev Vol. 12 Sir Nevill Mott — 65 Years in Physics edited by N. Mott and A. S. Alexandrov Vol. 13 Broken Symmetry — Selected Papers of Y. Nambu edited by T. Eguchi and K. Nishijima Vol. 14 Reflections on Experimental Science edited by M. L Perl Vol. 15 Encounters in Magnetic Resonances — Selected Papers of Nicolaas Bloembergen (with Commentary) edited by N. Bloembergen Vol. 16 Encounters in Nonlinear Optics — Selected Papers of Nicolaas Bloembergen (with Commentary) edited by N. Bloembergen Vol. 17 The Collected Works of Lars Onsager (with Commentary) edited by P. C. Hemmer, H. Holden and S. K. Ratkje Vol. 18 Selected Works of Hans A. Bethe (with Commentary) edited by Hans A. Bethe Vol. 19 Selected Scientific Papers of Sir Rudolf Peierls (with Commentary) edited by ft H. Dalitz and ft Peierls Vol. 21 Spectroscopy with Coherent Radiation — Selected Papers of Norman F. Ramsey (with Commentary) edited by N. F. Ramsey

Forthcoming Selected Papers of Richard Feynman (with Commentary) edited by L M. Brown The Collected Papers of S. Chandrasekhar edited by K. C. Wali

World Scientific Series in 20th Century Physics - Vol. 22

AAQUKI_ QUEST

BtSlWEHY Selected Works of Bunji Sakita

Editors

Keiji Kikkawa Osaka University Japan

Miguel Virasoro ASICTP, Trieste, Italy

Spenta R. Wadia Tata Institute Mumbai, India

Vpfe World Scientific n

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

The editors and publisher would like to thank the following publishers for their assistance and their permission to reproduce the articles found in this volume: American Physical Society (Phys. Rev., Phys. Rev. Lett.), Elsevier Science Publishers (Phys. Lett. B, Nucl. Phys., Phys. Rep.), Institute of Physics Publishing (J. Phys.), Gauthiers-Villars (Ann. Inst. Henri Poincare)

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

A QUEST FOR SYMMETRY Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3643-3

Printed in Singapore by Uto-Print

V

This page is intentionally left blank

vii

CONTENTS Foreword Reminiscences 1.

2.

3.

4.

5.

6.

7.

8.

9.

On the Decay Interaction of Strange Particles B. Sakita and S. Oneda Nucl. Phys. 16, 72 (1960)

xi xiii

1

Application of Dispersion Relations to the Photodisintegration of the Deutron B. Sakita and C. Geoebel Phys. Rev. 137, 1787 (1962)

10

Low-Energy Limit of the Photodisintegration of the Deutron B. Sakita Phys. Rev. 127, 1800 (1962)

23

Discussion of a Suggested Bound on Coupling Constants C. J. Goebel and B. Sakita Phys. Rev. Lett. 11, 293 (1963)

30

Supermultiplets of Elementary Particles B. Sakita Phys. Rev. B136, 1756 (1964)

34

Electromagnetic Properties of Baryons in the Supermultiplet Scheme of Elementary Particles B. Sakita Phys. Rev. Lett. 13, 643 (1964)

39

Group of Invariance of a Relativistic Supermultiplet Theory L. Michel and B. Sakita Ann. Inst. Henri Poincare M. 167 (1965)

43

Relativistic Formulation of the SU(6) Symmetry Scheme B. Sakita and K. C. Wali Phys. Rev. 139, B1335 (1965)

47

Lie Group of the Strong-Coupling Theory T. Cook, C. J. Goebel and B. Sakita Phys. Rev. Lett. 15, 35 (1965)

60

viii

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

Strong Coupling of the Multi-Partial-Wave B. Sakita Phys. Rev. 170, 1453 (1968)

Meson Isotnplet

Extension of the Veneziano Form to N-Particle C. J. Goebel and B. Sakita Phys. Rev. Lett. 22, 257 (1969)

64 Amplitudes 72

Field Theory of Currents B. Sakita Phys. Rev. 178, 2439 (1969)

76

Feynman-like Diagrams Compatible with Duality. I. Planar Diagrams K. Kikkawa, B. Sakita and M. A. Virasoro Phys. Rev. 184, 1701 (1970)

82

Feynman-like Diagrams Compatible with Duality. II. General Discussion Including Nonplanar Diagrams K. Kikkawa, S. A. Klein, B. Sakita and M. A. Virasoro Phys. Rev. 184, 1701 (1970)

95

Dynamical Model of Dual Amplitudes B. Sakita and M. A. Virasoro Phys. Rev. Lett. 1146, 1146 (1970)

104

Formulation of Dual Theory in Terms of Functional Integrations C. S. Hsue, B. Sakita and M. A. Virasoro Phys. Rev. D l , 2857 (1970)

108

Functional-Integral Approach to Dual-Resonance Theory J.-L. Gervais and B. Sakita Phys. Rev. D4, 2291 (1971)

120

Generalization of Dual Models J.-L. Gervais and B. Sakita Nucl. Phys. B34, 477 (1971)

138

Field Theoretical Interpretation of Supergauge in Dual Models J.-L. Gervais and B. Sakita Nucl. Phys. B34, 632 (1971)

154

Ghost-Free String Picture of Veneziano Model J.-L. Gervais and B. Sakita Phys. Rev. Lett. 30, 716 (1973)

162

ix 21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

Functional Approach to Strong-Coupling Theory in Static Models. I. Charge-Scalar Models G. C. Branco, B. Sakita and P. Senjanovic Phys. Rev. D10, 2582 (1974)

166

Quantized Relativistic Strings as a Strong Coupling Limit of the Higgs Model J.-L. Gervais and B. Sakita Nucl. Phys. B91, 301 (1975)

176

Extended Particles in Quantum Field Theories J.-L. Gervais and B. Sakita Phys. Rev. D l l , 2943 (1975)

192

Perturbation Expansion around Extended-Particle States in Quantum Field Theory J.-L. Gervais, A. Jevicki and B. Sakita Phys. Rev. D12, 1038 (1975)

195

Collective Coordinate Method for Quantization of Extended Systems J.-L. Gervais, A. Jevicki and B. Sakita Phys. Reports 230, 281 (1976)

209

The Surface Terms in Gauge Theories J.-L. Gervais, B. Sakita and S. Wadia Phys. Lett. 63B, 55 (1976)

222

WKB Wave Function for Systems with Many Degrees of Freedom: A Unified View of Solitons and Pseudoparticles J.-L. Gervais and B. Sakita Phys. Rev. D16, 3507 (1977)

226

Gauge Degrees of Freedom, External Charges, Quark Confinement Criterion in AQ = 0 Canonical Formalism J.-L. Gervais and B. Sakita Phys. Rev. D18, 453 (1978)

234

Wave Functions and Energy for the Vacuum and Heavy Quarks from WKB Schrodinger Equation for Massive Gauge Theories with Instantons J.-L. Gervais, B. Sakita and H. J. de Vega Phys. Rev. D19, 604 (1979).

244

Field Theory of Strings as a Collective Field Theory of U(N) Gauge Field

X

31.

32.

33.

34.

35.

36.

37.

38.

39.

B. Sakita Phys. Rev. D21, 1067 (1980)

259

The Quantum Collective Field Method and its Application to the Planar Limit A. Jevicki and B. Sakita Nucl. Phys. B165, 511 (1980)

266

SO(2N) Grand Unification in an SU(N) Basis R. N. Mohapatra and B. Sakita Phys. Rev. D 2 1 , 1062 (1980)

283

Derivation of Quenched Momentum Prescription by Means of Stochastic Quantization J. Alfaro and B. Sakita Phys. Lett. 121B, 339 (1983)

288

Large-N Baryonic Soliton and Quarks J.-L. Gervais and B. Sakita Phys. Rev. D30, 1795 (1984)

294

Chiral Symmetry and Chiral Anomaly in Incommensurate Charge-Density- Wave System Zhao-Bin Su and B. Sakita Phys. Rev. Lett. 56, 780 (1986)

304

Local Chiral Symmetry and Charge-Density Waves in Conductors B. Sakita and K. Shizuya Phys. Rev. B42, 5586 (1990)

308

One-Dimensional

Fermions in the Lowest Landau Level: Bosonization, W^ Droplets, Chiral Bosons S. Iso, D. Karabali and B. Sakita Phys. Lett. B296, 143 (1992)

Algebra,

316

WQO Gauge Transformations and the Electromagnetic Interactions of Electrons in the Lowest Landau Level B. Sakita Phys. Lett. B315, 124 (1993)

324

Collective Variables of Fermions and Bosonization B. Sakita Phys. Lett. B387, 118 (1996)

329

Bibliography

336

Vitae

343

xi

Foreword It is a great pleasure for us to edit this volume of the selected papers of Professor Bunji Sakita. This selection was made by him upon our request and it spans a career of more than 40 years. The brief autobiographical notes which precede the selection were written by him with some reluctance born of his characteristic modesty. They are a vivid personal account of some of the most exciting times in high energy physics and they move in a continuous stream from theme to theme. Here, and in the selected papers one encounters among other contributions the beginnings of SU(6) symmetry, the strong coupling group, the string model, supersymmetry and the use of collective variables in quantum field theory. Besides a desire to honour Bunji Sakita, for his contributions, humility and generosity, we wanted to bring to light the origins of some key concepts which have become part of the daily compendium of high energy physics.

Keiji Kikkawa, Miguel Virasoro, Spenta R. Wadia December, 1997

This page is intentionally left blank

XU1

REMINISCENCES

I was an undergraduate student at Kanazawa University, which had been recently established as part of the post-war educational reforms. Many of the professors had moved from the old imperial universities and still followed the old curricula. A few among them formed a research group for theoretical particle physics. Since there were no students senior to us nor graduate students, a few of us were made welcome in their reading club. There, for the first time I was exposed to research in theoretical particle physics. Since Kanazawa university was an undergraduate college at that time, I went to Nagoya university for my graduate studies. Before going to Nagoya I was already engaged in some calculations that I had been asked to carry out by Oneda, my mentor at Kanazawa. These were A — (3 decay calculations, and the results were published in a joint paper by Iwata-Okonogi-Ogawa-Sakita-Oneda, a paper that dealt with the universality of weak iateractions and in a sense, adumbrated the universal V-A interaction. The architects of the paper were Shuzo Ogawa and Sadao Oneda, from whom I learned the phenomenology of strange particles (then called V-particles) and the weak interactions. This was about the time that the strangeness theory was put forward 1by Nakano-Nishijima and by Gell-Mann. In Nagoya at that time, each graduate gr student belonged to a research group. I belonged to Sakata's group. I stayed staye there for two years, and received a Master's degree in 1956. As I look back now, this was one of the most fruitful periods for was proposed in my second year, although I was Sakata's 's group. The Sakata model w; not a part of this activity. By then I had become more and more interested in the collective model of nuclei and the w work in Nagoya by Marumori and others. My master's thesis was on collective motions. mot In 1956 I went to the University of Rochester. When Robert E. Marshak, then at Rochester, visited Japan to attend an the chairman of the physics department departrr international conference in 1953, he expressed his interest in having a number of Japanese graduate students join his group at Rochester. To this end, he requested Yukawa and Tomonaga to select some students. Fourteen students were selected between 1953 and 1959,1 being one of them. I received a research assistantship and a Fulbright travel grant. In Rochester, I had to take regular courses during the first year. Among these courses, I had to take an experimental course; "Modern Physics Laboratory." Before coming to Rochester I had been entertaining the possibility of going over to experimental physics, as Koshiba and Yamanouchi had done. After an unsuccessful X-ray experiment in that course, however, I gave up that dream. In the spring of 1957 there was a Rochester Conference which graduate students were allowed to attend. A highlight of the conference was Lee-Yang's work on parity non-conservation in weak interactions. It was very exciting to see all these noted

XIV

people whom I knew only by name. There were many activities in Rochester going on then too, such as the V-A theory of Marshak and Sudarshan and the MarshakSignel nuclear potential, to name a few. However, I, being timid and merely a new graduate student to boot, could not participate in them. It was very frustrating. While I was at Rochester, I more or less followed the topic that was current - dispersion relations and symmetries (global symmetry, pre-SU(3)). Dispersion relation was the subject that I had never studied in Japan. While I was studying its techniques, I tried to apply it to various problems. The first attempt was on the K^ decay where I mimicked the Goldberger-Treiman calculation of the it meson decay. This calculation led to the conclusion that the strangeness changing current is much weaker than the strangness conserving current, an indication of Cabibbo mixing. Around this time, Marshak's group was working on nuclear forces, computation of nucleon-nucleon phase shifts, the photo-disintegration of deuteron, etc. J. J. de Swart, a student of Marshak, was working on the photo-disintegration of the deuteron and I was attracted to the subject. I discussed with Susumu Okubo, a research associate then, regarding the possibility of using dispersion relations in this problem - and eventually it became a part of my thesis at Rochester. My adviser was Charles J. Goebel, then a young assistant professor, with whom I finished my Ph.D. in 1959. He gave me complete freedom in physics research and provided appropriate advice whenever necessary. During my three years at Rochester, I received much encouragement from my fellow Japanese graduate students, and I also learned a great deal of physics from them, especially particle theory from Susumu Okubo and particle experiments from Taiji Yamanouchi.

I took a postdoctral job at the University of Wisconsin and moved to Madison in the summer of 1959. While I was in Rochester I did not work on weak interaction phenomenology, but I maintained an interest in that subject. In Madison I resumed research on strange particle decays and collaborated with Oneda and Pati of Mary­ land through correspondences. This work is a precursor to the Penguin mechanism of the | A/1 = ^ enhancement in non-leptonic weak interaction. My boss at Wisconsin was Robert G. Sachs, who was keenly interested in high energy experiments and had helped to build up a strong experimental group. With his encouragement, I developed a friendship with the experimental group, in particular with M. L. Good and W. D. Walker. Through the next several years I closely observed the excitement they felt in the discovery of hadronic resonances. At about this time the Fry-Camerini group errorneously claimed the discovery of AS = —AQ events in neutral K meson dacays. Sachs had strongly supported the claim and had created such an atomosphere that for the next several years the theorists in his group could not discuss models of weak interactions without AS = —AQ. Most of the models, based of the constituents of hadrons such as the Sakata model or the quark model, became extremely ugly. I spent fair amount of

XV

time and energy on the model building of weak interactions only to be considerably frustrated. This was the last time I was engaged in weak interaction physics. In Madison the Summer Institute was regularly organized by Sachs. Attending the 1961 Summer Institute, I keenly felt my lack of knowledge in group theory. I only knew angular momentum at the time. In the summer of 1962 we did not have the Summer Institute, but Jan Tarski, then a postdoc at the Institute for Advanced Study (IAS) in Princeton, visited Madison and he and I shared an office. Since I knew that he had given a seminar on group theory in the previous year's Summer Institute, I thought it was a good opportunity to learn group theory from him. Although I wanted to know the representations of SU(3), I asked him about SU(4) instead, because I was shy about revealing my intention and moreover thought that if I understood "4" I would understand "3". He then started explaining it at great speed by drawing a lot of dots on the blackbord. But, when he found me completely foxed, he looked at me pityingly and said that the representations of SU(4) had all been worked out by Winger. This is Wigner's famous paper on the supermultiplets of nuclei. Immediately I went to the library. Although I did not quite understand the group theory part of the paper, I understood very well the intent of the paper, in particular the introduction, where Wigner explains why and how group theory is applied to this problem. My teaching career began in 1963. That year I taught a course called "Special Topices in Theoretical Physics"; the weak interactions in the fall semester and the strong interactions in the spring semester. This was the year of SU(3); the Cabibbo theory of weak interactions and the quark model. I reviewed in detail the status of elementary particle research at that time including some of my own contributions. In the last lecture of the course I discussed the non-relativistic SU(6) theory, a supermultiplet theory of hadrons based on the non-relativistic quark model, which I had developed during the second term (Spring 1964). I was well aware of the limitation of the theory, especially the difficulty associated with the relativistic generalization. I worked on this problem all the time during the summer of 1964. I presented my version of SU(6) in the last seminar of that year's Summer Institute. Louis Michel and Eugene Wigner were there in the audience. After the seminar Wigner came back to the seminar room and informed me of the news he had received from Giirsey and Radicati on their independent work. Next day Wigner invited me to his office, and grilled me with several questions before convincing himself that I had used the same SU(6) group as had Giirsey and Radicati, and then he opened a briefcase to show me their paper. By then I knew the content of the paper because I had called Feza Giirsey the previous day and had learned it from him. There was an important difference, however, between the two. The difference was that I had chosen the anti-symmetric representation "20" of SU(6) for baryons based on the naive quark model, while they correctly chose the symmetric representation "56". During the summer I had been so preoccupied with the relativistic problem of SU(6) that I had neglected to examine the SU(3) contents of some of the possible

XVI

representations of SU(6). When I expressed my concern about the relativization, Wigner politely refused the discussion and suggested that I discuss the matter with Louis Michel. A few days after the seminar, I moved to Argonne National Laboratory, where Sachs became the associate director of the laboratory in charge of high energy physics research and I joined its theory group. The first thing I did in Argonne was a quick calculation of the nucleon's magnetic moment using the representation "56". When I found the ratio —3/2 for the magnetic moment of the proton and the neutron, I became fully convinced of the correctness of the "56", and thought that we might have to abandon the naive quark model. As is well-known, the resolution of this dilemma later led to the introduction of the color degree of freedom, and eventually to QCD. In Argonne, before my family and I found a house in a nearby town, we stayed for one month at the visitors' housing facility. Harry Lipkin, who showed a strong interest in SU(6), was also there with his family. A few days later Michel and his family moved from Madison and stayed in the facility for a month or so before going back to France. I started to discuss with Michel about the feasibility of relativistic SU(6). But we had communication problems, because his group theory was much too sophisticated for me. Nevertheless, we managed to finish a work, which dealt with a discussion of, together with some negative results in the relativistic extension of the SU(6). A joint paper, in actuality written by Michel entirely in his own style, was drafted. To tell the truth, he had agreed to write a more elementary paper with me by using Lie algebra. But, we have never finished it mainly because he was satisfied with his version and I became extremely busy working on the next project with Kameshwar C. Wali. This was the time when Argonne played the role of a center of high energy physics activities in the Midwest. Many physicists from the nearby universities gathered at the weekly seminars. Yoichiro Nambu and Jun J. Sakurai showed up quite frequently. I had known Nambu for some time since he was a frequent visitor to the Summer Institute at Madison. In discussions with him I myself became con­ vinced that an attempt at a phenomenological but relativistic formulation of SU(6) would be worth a try, inspite of the negative results we had had regarding SU(6) being an essential theory. K. C. Wali and I started working on this intensively in the late fall of 1964, and we finished the work just before the 1965 Coral Gables Con­ ference in January, to which I had been invited. The first speaker of the conference was Abdus Salam, who spoke on the relativistic U{12) theory of Salam-DelbourgoStrathdee. I was shocked by the talk since their work was identical to ours, even to the notations. During the talk W. D. McGlinn, who knew our work and was sitting in front of me, turned around several times and kept gesturing to me to speak up. After the talk I spoke up and then handed a hand-written copy of our paper to Salam. At a party that evening Salam returned the paper back to me saying that both works were the same. And then he invited me to visit the International Center

xvii

for Theoretical Physics (ICTP) where he was the director. In the following week at the APS meeting in New York, Pais presented a work with Beg on their version of the relativistic SU(6), which subsequently was reported in the New York Times. A few weeks later, I saw Salam's picture in an English newspaper that had been posted on a bulletin board in Argonne. I was not too happy about the fact that our contribution was not suitably acknowledged. Moreover, the publicity accorded to a work that I regarded as merely phenomenological, made me quite uneasy. In the same Coral Gables conference, Roger Dashen gave a talk on the boot­ strap program, in which he was describing the various vertices in terms of the matrix elements of a few matrices. After I came home I realized that Goebel's strong coupling theory could be formulated algebraically by using matrices, and the next moment I had obtained a Lie algebra of the symmetry group which serves as a spectrum generating algebra for isobar states (hadrons). I immediately informed Goebel, my advisor at Rochester, of this development. In the meanwhile, in 1961, he had joined the faculty at Wisconsin. In Argonne at that time there were several group theorists: Morton Hammermesh, William D. McGlinn, and the mathemati­ cian Robert Hermann. When Michel was still there, he proposed having a series of tutorial lectures by Hermann. Based on these lectures Hermann later wrote his well-known Benjamin book. When I told Hermann about my findings on the strong coupling group (as we named it), he showed a strong interest in it and suggested that I study a few relevant mathematical subjects: group contraction, induced rep­ resentation, and the Peter-Weyl theorem. This was a very valuable suggestion, for Thomas Cook (my first student) and I spent the next one year or so working on these problems. Meanwhile, we published a short paper (Cook-Goebel-Sakita) on the strong coupling group together with a derivation of the representations for a few simple cases by using the method of contractions. Suddenly the SU(6) became fashonable and I started receiving many invita­ tions. In the summer of 1965 I was invited by Delhi University to give a series of lectures in a summer school at Dalhouse, a hill station in the Himalayas. Using this opportunity, I traveled around the globe: India, Japan and England in that order to participate in summer schools and conferences. This visit to Japan was the first since I had left nine years previously. I was well received there and I really felt the difference that the SU(6) had made.

In the spring of 1967 I stayed at the ICTP in Trieste for five months. Towards the end of the stay K. C. Wali and I traveled to Israel, specifically the Weizmann Institute for ten days at the invitation of H. J. Lipkin. When we arrived in Israel we found that the atmosphere was extremely tense and people busy preparing for a war with the neighboring Arabic countries. Although the touristy places were deserted, we could manage to rent a car to visit many places including Jerusalem, Haifa and Acre. Since most of the young Israeli physicists had already been drafted,

xviii

the physicists working at the Institute were mainly foreigners, among whom were H. Rubinstein, G. Veneziano and M. Virasoro. They were working together on superconvergence relations, which was a subject that I was also interested in at that time. In the discussions we had during this visit, the dual resonance program must have come up, since I remember that afterward in Trieste I started discussing with others about the possibility of constructing scattering amplitudes by summing only the s-channel resonance poles. We left Israel as scheduled on June 4 and the very next day in Ankara, Turkey we heard of the outbreak of the Six Day War. In 1968, Keiji Kikkawa and Miguel Virasoro, with whom I had become ac­ quainted in the previous trips to Japan and to Israel, joined our group at Wisconsin as research associates. By then I had returned to the University of Wisconsin to re­ sume teaching, which I missed at Argonne, and I was preparing a course, "Advanced Quantum Mechanics", which was essentially a one-year graduate course on quantum field theory. In that summer Virasoro showed up in Madison with a hand-written paper by Veneziano and he explained to us in detail the activities of the Weizmann Institute. At once Goebel and I got interested in the work and all of us started thinking about generalizations. In that fall after Virasoro succeeded in obtaining the five point Veneziano formula, our activity became intensified and within a few weeks Goebel and I had obtained the N point Veneziano formula. Then Kikkawa, Virasoro and I started to generalize the formula further to include loops. At this point we faced a dilemma. Namely, if one considered the Veneziano for­ mula as a narrow resonance approximation of the true amplitude as was commonly assumed at that time, the construction of loop amplitudes based on this approxi­ mate amplitude did not make sense. After reviewing the logistics of quantum field theory, we arrived at the conclusion that the construction of the loop amplitudes did make sense if we considered the Veneziano amplitude as a Born term of an unknown amplitude for which we had an expansion similar to the standard Feynman-Dyson expansion in perturbative field theory. With this philosophy in mind we decided to construct a new dynamical theory of strong interactions. First we defined the local duality transformation as the crossing transformation at any four point sub-diagram of a Feynman diagram, and invented a Feynman-like diagram compatible with du­ ality as a diagram which contained all the Feynman diagrams related to each other by the local duality transformations. Then we simply wrote down a prescription for the scattering formula corresponding to each of these Feynman-like diagrams. In practice, we used diagrams which were dual to the Feynman diagrams. A three point vertex of a Feynman diagram corresponds to a triangle in the dual diagram. An N point Feynman tree diagram corresponds to a specific triangulation of an N polygon in the dual diagram. A local duality transformation in the dual diagram is the transformation of one triangulation of a quadrangle to another triangulation. In terms of the dual diagram, therefore, an N point Feynman-like tree diagram corresponds to an N polygon. By studying these Feynman-like diagrams, it became clear to us that a dual amplitude corresponded in a one-to-one fashion to a two-

XIX

dimensional surface with boundaries, and equivalently to a Harari-Rosner quark line diagram, which, by the way, we had also invented independently. In the second paper, we discussed the general Feynman-like diagrams by using the classification of two dimensional surfaces, and extended the prescription to non-planar diagrams. This classification is the same as that of open string amplitudes. Kikkawa left Madison in the summer of 1969 for Tokyo, and Virasoro and I left the following summer bound for Berkeley and France respectively. By then the operator formalism of the dual resonance amplitude had been established by S. Fubini, D. Gordon, and G. Veneziano, and independently by Y. Nambu, who further proposed the string interpretation based on this work. When I heard of the string interpretation from Nambu I felt it as natural as if I had known about it beforehand. I remember that I had experienced the same feeling when I had first heard about the Sakata model from Sakata. At Wisconsin Virasoro had used the operator formalism to analyze the possi­ bility of the negative metric ghost states consistently decoupling from the physical states. He obtained a set of operators, which could be used consistently as the operators of subsidiary conditions on the physical states. These operators were later found to be the generators of conformal transformations on a complex plane. This is the origin of the Virasoro algebra. In discussing this problem with him, I realized that these operators were compactly expressed in terms of a scalar field in a fictitious 1+1 space(finite)-time, and the Veneziano formula itself could be expressed in terms of this scalar field operator. At about this time we received a hand-written paper by H. B. Nielsen: "A physical model for the n-point Veneziano model." Inspecting a few mathematical formulae in the paper, I came up with a functional integral representation of the Veneziano formula. There remained several important points to be clarified, such as the Mobius invariant property of the func­ tional integrand, the connection with the operator formalism, and the calculation of non-planar amplitudes. At the end I, together with Virasoro and my student C. S. Hsue, established the functional path-integral formulation of dual resonance amplitudes, and with Virasoro, a physical model of the dual resonance model based on the "fishnet" diagram. I stayed in France for one year before I moved to the City College of New York in 1971. To lessen the financial burden on the Institute, Michel had arranged a joint invitation by his Institute at Bures-sur-Yvette and the Bouchiat-Meyer group at Orsay, a group of physicists that later moved to the Ecole Normale Superieure in Paris. This arrangement turned out to be a very fortunate one for me, as in Orsay I found several young physicists, who were interested in our work. Moreover it was there that I succeeded in starting a long and fruitful collaboration with Jean-Loup Gervais. During this visit, I wrote three papers with Gervais: on the functional integral, conformal field theory, and the super-conformal-symmetry, all in conection with the dual resonance model.

XX

In Wisconsin I had already started working on the factorization of dual reso­ nance amplitudes using the slicing and sewing technique of functional integrals. I had drafted the preliminary results into a paper and had sent it to the Physical Review before I arrived in France. At Orsay, however, I withdrew the paper, as a result of discussions with Gervais, when I convinced myself that a part of the paper was wrong. There were plenty of technical difficulties, on which Gervais and I had to spend another half a year of hard work. In this work we used formally and fully the conformal transformation properties of functional integrals without seriously questioning their validity. Sometime later we suspected the existence of an anomaly, that would explain the critical dimension of the model. I regret that we did not pursue it further. When I received a paper on the new dual pion model of Neveu and Schwarz in the spring of 1971,1 noticed at once that the most important ingredient in the model was the conformal invariance property. One could discuss about the generalization of dual resonance amplitudes in the very general terms of conformally invariant field theories. So, Gervais and I got busy constructing conformally invariant field theories. In this work, we discussed first a general theory of conformal fields by defining the irreducible fields (now known as primary fields) and the conformally invariant Lagrangian, and then we established the functional-integral representation of Neveu-Schwarz model by introducing a fermionic field in the model in addition to the old bosonic field. After the work was completed I wrote a letter to Virasoro (in Berkeley then) informing him of our work, since I heard that he had presented a similar work at a conference in Israel. In the exchange of letters, I learned the Ramond model from Virasoro and that it also could be described by the same Lagrangian simply by changing the boundary condition on the fermionic field. Gervais and I thought that in the functional-integral representation the elimina­ tion of ghost states could be done by factoring out the negative metric components of the fields by using conformal transformations as was done in the standard gauge field theories. The necessary condition for this is, of course, that the Lagrangian is invariant under the conformal transformations. Once we introduced a new field in the new model which generated new ghost states, we had to find out a new set of gauge transformations under which the Lagrangian was invariant. Neveu-SchwarzThorn had just published a paper in which they proposed a set of operators to be used as the subsidiary gauge conditions on the physical states of the dual pion model. We tried to interpret these operators as the Fourier modes of the Noether current associated with the new gauge transformations which involved the new fermionic field, and arrived at the superconformal gauge transformations, under which the Lagrangian we had obtained previously was invariant. I believe that these field transformations are the first instances of supersymmetry transformations in a local field theory. The day after we had drafted this paper, I left France for New York. In this work, we had to use anti-commuting c-numbers (Grassmann numbers) and functional integration of fermionic variables. These, to us were new concepts and

XXI

we were initially reluctant to use them. Apparently, others shared this reluctance and this work and the functional-integral work in general, was not appreciated in our circle. However, I received an impression that when I presented the work later in December at the conference on functional integration at the Lebedev Institute in Moscow, it, as well as the use of anti-commuting c-numbers was well appreciated. When R. E. Marshak became the president of the City College in 1970, I, together with Keiji Kikkawa, accepted a position there. I continued my research on dual resonance theory for a few more years, after I had settled down in the City College. There was a big difference, however, between before and after coming to the City College. Although several faculty members were already there before I came, I was expected to play the role of the leader of the high energy theory group. I felt that it was a great challenge to elevate the group into a quality research group. In a few years, thanks to Marshak's personal connections, we could gather a few talented graduate students into our group. And also we could hire a new faculty member, Michio Kaku and postdocs, such as Yoichiro Iwasaki. Moreover, I could invite J.-L. Gervais for short visits on a few occasions. I intentionally spent more time with students, and shared my insights with them. In the early spring of 1973, I was invited by Ziro Koba to visit the Niels Bohr Institute in Copenhagen for two weeks to deliver a colloquium, and more importantly to discuss the dual resonance string theory with his group, in particular with Holger B. Nielsen and Paul Olesen. By this time at the City College, Gervais and I had already formulated the ghost free Veneziano amplitudes by using the functional-integral representation of the Nambu-Goto string in the light-cone gauge. This work later led to Mandelstam's factorizable functional formulation of lightcone string theory, and eventually to Kaku-Kikkawa's light-cone string field theory. Furthermore in our group at that time, the work of Iwasaki-Kikkawa was near completion. This was an attempt, which I persuaded them to carry out, at a formulation of a light-cone string theory for the Neveu-Schwarz model. I reviewed these activities in Copenhagen. While I was in Copenhagen, David Olive called me up asking me to visit CERN on the way back home. At the CERN seminar, I reviewed the Iwasaki-Kikkawa theory. Later, I was told that this seminar and a conversation after the seminar had led Wess and Zumino to start their seminal work on supersymmetric field theory. I vividly remember the conversation with Zumino at the CERN coffee lounge. When I said, "If you allow me to use anti-commuting c-numbers, Gervais and I have written down a transformation of a fermi field to a bose field in the Nuclear Physics paper", he replied, "It's OK to use anti-commuting c-numbers. Schwinger has frequently used them."

In the June of 1968 there was an international symposium at the ICTP cel­ ebrating its new building at Miramare. At the symposium I was introduced to Faddeev and from him I learned the Faddeev-Popov trick. Being fascinated by the

xxii

method I tried to use it in various problems, and gradually I convinced myself that the method could be useful for a much wider class of problems than simple gauge fixing. My encounter with the strong coupling theory of Wentzel goes back to my student days at Rochester. Since then I had been observing the development of Goebel's S-matrix approach to the strong coupling theory from up close. As I have mentioned before, I even contributed to it by formulating and extending Goebel's theory in the form of an operator algebra including multi-partial waves. Through this work, I became acquainted with Gregor Wentzel and I was even introduced as his grandson at his retirement dinner party in Chicago, since Goebel was his student. But, to tell the truth I had never seriously studied his field theory of the strong coupling model. When I learned the Faddeev-Popov trick, it occurred to me to develop a functional-integral formulation of the strong coupling theory by using this trick. Because of other work that had to be done meanwhile, I could not even get started on this project until I had found two students, Gustavo C. Branco and Pavao Senjanovic, at the City College. At the beginning I thought that the problem was rather easy and one that was appropriate for graduate students. It turned out, however, that we had to overcome many obstacles; of which some were crucial albeit most were technical. I remember that I had to read Tomonaga's strong coupling paper again very carefully. At the end we succeeded in the functional-integral for­ mulation of the strong coupling theory. There were two important general issues involved in this work, namely, (a) the introduction of collective coordinates in field theory, and (b) the semi-classical expansion in field theory. But, I suspect that at that time we did not fully recognize the relationship between the collective coordi­ nates and the zero modes, nor that between the strong coupling limit of the static models and the classical limit. In the summer of 1974, I went to Europe to attend the International Confer­ ence on High Energy Physics in London. Before the conference I stayed in Orsay for a month. During this time, influenced by a seminar given by Neveu on his work with Dashen and Hasslacher, Gervais and I decided to work on the semi-classical quantization of classical solutions, in particular the Nielsen-Olesen vortex solution of the Higgs model. Gervais studied our strong coupling paper very carefully and brought his new insights to bear upon it. In this summer we worked together in As­ pen and at Brookhaven successively for several weeks to finish up this work on the quantized relativistic string as a strong coupling limit of the Higgs model. In this and in the subsequent work on soliton quantization, we used the Faddeev-Popov trick to extract the collective coordinates out of bosonic field theories. With this as the starting point, Gervais and I, together with Antal Jevicki (then a student) had firmly established the collective coordinate method as a method of semi-classical ex­ pansion in field theories, by the time of the following year's workshop on "Extended System in Field Theory" held at the Ecole Normale Superieure.

xxiii

In these works we performed point canonical transformations in the functionalintegral representation of bosonic quantum field theories. Since in functionalintegral representations the operator ordering is not explicit, one often misses a term which is proportional to fi. Sure enough, we missed such a term in our work as was pointed out by E. Tombolis. Although Gervais and Jevicki have shown subsequently that it is possible to incorporate operator ordering into the functional-integral for­ malism, I realized that this was a serious drawback in the actual application of the functional-integral formalism. It was then that I decided to use the Hamiltonian operator formalism whenever a change of variables in quantum mechanics was involved. Meanwhile, I had received a paper from Kikkawa, who had returned to Japan the previous year; the Hosoya-Kikkawa paper on the gauge theory of collective coor­ dinates. The main idea of this paper was to construct for a given theory an artificial gauge theory, which involved the collective coordinates as gauge parameters, such that if one fixed the gauge by setting the collective coordinates zero, the theory re­ verted back to the original theory. A natural question occurred to us: what would happen if one applied this method to a genuine gauge field theory? It turned out that the most of the collective coordinates were absorbed into the vector potentials by gauge transformations except for the collective coordinates at the boundary of the system, which manifested themselves as surface variables. Subsequently we, Gervais-Sakita-Wadia, found that these surface variables were indispensable for a gauge invariant quantum mechanical description of the 't Hooft-Polyakov-Julia-Zee monopole-dyon solution. I encouraged Spenta Wadia (then a student) to investigate this problem further, addressing non-Abelian gauge theories in general, by using the Hamiltonian formalism. At about this time Gervais and I worked together fairly regularly. According to my notes, we worked together in New York, Aspen, and Paris for a total of twelve weeks in two years ('76-'77). We developed the many-variable WKB method, the AQ = 0 canonical formalism for non-Abelian gauge theories, and together with H. J. de Vega, a real time approach to instanton phenomena. In our group at the City College around this time, R. N. Mohapatra, who succeeded Kikkawa, was actively working with his students on his left-right symmetric model of weak interactions. Michio Kaku was productive in conformal supergravity with Townsend and van Nieuwenhuizen of Stony Brook. The main theme of the research surrounding me was the non-perturbative study of non-Abelian gauge field theories. Tamiaki Yoneya (then a postdoc) and Spenta Wadia were very active. I recall one of their works that dealt with the role of surface variables in the vacuum structure of Yang-Mills theory. In this paper, they explicitly transformed the Belavin-Polyakov-SchwarzTyupkin solution to the Coulomb gauge. They obtained a pendulum equation for the transformation function, which led to infinitely many solutions. This infinite multiplicity is now known as the Gribov phenomenon, but their paper predates that of Gribov by almost a year.

xxiv

Large N QCD had been introduced by G. 't Hooft in 1974. One of his mo­ tivations was to remedy the arbitrariness involved in the "fishnet" diagrammatic approach to the string theory of strong interactions. This and the subsequent developements influenced us into thinking about the large N expansion in field theory in general. In the winter of 1978-79, Wadia, then a postoc at the University of Chicago, came back to New York and informed me of a work with Eguchi. This discussion stimulated me to think about a gauge invariant calculation of non-Abelian gauge theories. I tried to rewrite a non-Abelian gauge theory in terms of equal-time Wilson loop variables. For this purpose I used the Hamiltonian canonical formalism and the method of change of variables that I had learnt long ago in Nagoya. To my surprise I obtained a field theory of interacting strings as a large N limit of the SU(N) gauge theory. In order to justify the procedure, I applied the same procedure to known examples: a collection of many identical free harmonic oscillators, and high density bosonic plasma oscillations. It worked correctly only if I made a similarity transfor­ mation such that the resulting Hamiltonian became hermitian. I was mystified by this until I spoke with Jevicki, then a postdoc at the IAS, who pointed out at once that this transformation essentially took care of the contribution from the Jacobian of the change of variables. So, I proposed to him that we write a joint paper on the general theory of the collective field method, so named later, after poUshing up all the calculations. When I was writing a first draft of the paper in the summer of 1979, Jevicki informed me of the work of Brezin-Itzykson-Parisi-Zuber on the large N quantum matrix model, to which we could apply our method. Indeed, it was not difficult to derive their result by the collective field method. As I have mentioned above, in Nagoya I had learned the method of change of variables used in the collective field theory. I was, and still am, curious as to whether I had a prototype of the collective field theory in my master's thesis or not. Sometime later when I went back to Nagoya, I went to the library to look for the thesis. However it was missing from the library. In the fall of 1983 Gervais stayed in the City College for a few months as a visiting Professor to fill in the gap created by the departure of R. N. Mohapatra. I do not remember why, but I was explaining to him the derivation of the strong coupling group and its representation by the method of group contraction, namely the old work of Cook-Goebel-Sakita. To my surprise, he was visibly excited. He said "This could just be the large N QCD." It did not take us long to realize the relation between the large N baryons and the strong coupling theory: y/N « G, once we learnt that Witten had shown that in the large N QCD the masses of baryons are of order N while the meson-baryon Yukawa couplings are of order y/N. We spent several more weeks to complete the work, as we tried to establish its relation to the solitonic Skyrmion physics: y/N « G ss 1/g. I remember, when we

XXV

had finished the work, that I was extremely satisfied with it, since it involved many of my previous works; SU(6), strong coupling algebra, and soliton quantization, which were seemingly unrelated until then.

In the fall of 1980 I spent 4 months at the Yukawa Institute in Kyoto. On the way to Japan I was in Europe for several weeks and I came across the Parisi-Wu stochastic quantization paper in CERN's preprint library. Although I had been in­ terested in statistical mechanics, I had never seriously worked on the subject. When I came back to New York the next spring I decided to spend some time in studying non-equilibrium statistical mechanics and stochastic processes in particular. My source was a Japanese book by R. Kubo and M. Toda entitled Statistical Physics, one of the Iwanami series, which I had bought in Japan. I translated one relevant chapter on stochastic processes and distributed it among my students, Guha, Alfaro and Gozzi, as they were all fascinated by stochastic quantization. Over the next few years several papers on stochastic quantization appeared from our group: on large N reduction by Jorge Alfaro and me, on supersymmetry and stochastic quantization by Ennio Gozzi, on stochastic quantization of supersymmetric theories by Kenzo Ishikawa, and the calculation of the chiral anomaly by Rodanthy Tzani. At about this time I taught on two occasions a special topics course on Field Theory and Statistical Mechanics, which included such topics as the derivation of the Landau-Ginzburg equation in the BCS model and the Lee-Low-Pines theory of polarons in terms of Feynman's variational method. Based on the lecture notes compiled by the students, I wrote a book called Quantum Theory of Many- Variable Systems and Fields, World Scientific Lecture Notes in Physics Vol. 1, published in 1985. In the spring of 19851 met Zhao-bin Su of the Institute of Theoretical Physics in Beijing, who was a visitor to our condensed matter theory group at City College. He introduced me to several topics in condensed matter physics and frankly revealed the problems he was facing. One of them concerned the charge density wave transport phenomenon in a one dimensional system of electrons in a crystal. During the exchange of questions and answers, we gradually realized the importance of chiral symmetry and the chiral anomaly for this phenomenon. This work, together with a later work with Kenichi Shizuya on the same subject, drastically changed the level of my understanding of the chiral anomaly and the physics of anomalies in general. Another topic Su had introduced me to was the fractional quantum Hall effect. I have spent a fair amount of time and energy on this subject over the past ten years, and have written several papers with him, but I must confess that to this date I still do not understand the subject to my satisfaction. In the past ten years I have become more and more interested in many-body problems, which is the subject I was involved in when I was a student at Nagoya. Interestingly, to me, the subject is rich enough to fill the gap between condensed

XXVI

matter physics research and particle physics research. When I learnt about the Woo algebra in a seminar on string theory, I realized that I had obtained the same algebra in my study of the fractional quantum Hall effect. I simply had not thought about the significance of the algebra in the physics of the Hall effect. In this respect I am pleased with a series of works I have done over the past five years with Dimitra KarabaH, Satoshi Iso, and Rashmi Ray, since all of these works illuminate the significance of the Woo (or Woo ) algebra in the physics of low dimensional fermionic systems. J a n u a r y 1997 in New York Bunji Sakita

1 8.B

Nuclear Physics 16 (1960) 72—80; (g) North-Holland

Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

ON THE DECAY INTERACTION OF STRANGE PARTICLES B. SAKITA University of Wisconsin, Madison, Wisconsin and S. O N E D A t University of Maryland, College Park, Maryland tt

Department of Physics,

Department of Physics,

Received 26 October 1959 Abstract: It is proposed t h a t the strength of the coupling constants is different for the strangeness non-conserving and strangeness conserving currents in the scheme of Fermi interactions of an ordinary charged current-current type. First, the consistency with experimental results is analyzed by introducing phenomenologically the direct An interaction. Then, the possibility of the derivation of this interaction as the effective interaction of the primary Fermi interactions is discussed.

1. Introduction There are several theoretical attempts to explain the striking difference between the rates of K+-+n++7i° and K-f-+n+n decay ttt. The simplest explanation would be to impose the so called \Al\ =\ selection rule for the non-leptonic decay processes. Presently available experiments 2) on the hyperon decay and the K-meson decays into pions seem to be consistent with this selection rule. Of course, this rule cannot be strict since the K+ -> n++n° decay, though slow, does take place. It is also usually believed that the electro­ magnetic interactions are too small to be responsible for the violation of \AI\ = \ rule. If we wish to explain all the weak processes in terms of the universal Fermi interaction (U. F. I.), we must construct the U.F.I, in such a fashion that the \A\\ = § part of the matrix elements of K-decay is suppressed about 20 times compared with the |JI| = -| part. This does not seem to be an easy problem, especially if we wish to restrict the form of the U.F.I, to the charge-exchange type in order to forbid many unwanted reactions involving leptons. Another difficulty of the U.F.I, arises from the recently established slow rate of hyperon decays into leptons 3 ). Present experiments on the A0 ->■ p+e~ + v and t On leave of absence from Kanazawa University, Japan. tt Supported in part by the U. S. Atomic Energy Commission, under Contract No. A T ( l l - l ) - 3 0 and in part by the United States Air Force through the Air Force Office of Scientific Research and Development Command. t t t Okubo and Marshak ') have tried to explain this difference of decay rates by taking into account the final state interaction in T = 0 and T = 2 states. It is, however, concluded by Chew and Mandelstam ') that there are no low energy resonances in the S-state i n w - n interaction

2 ON THE DECAY INTERACTION OF STRANGE PARTICLES

73

E± -> n+e±+v decays have shown that the rates of these decays are, at least, about ten times smaller than those expected from the U.F.I. We have, as a matter of fact, conjectured this result from the related experimentally better known processes, viz. the leptonic decay modes of the K-meson. That is, a dispersion calculation of the K -> /i+v decay rate and the comparison of the rate 4) of K^ -*- 7i++n~ to that of K°^-e(ju)+v+n decay also suggest that both the A- and V-part of the strangeness non-conserving baryon-lepton interactions have weaker coupling constants than the usual Fermi coupling constant. Under these circumstances, we feel that we should now try to reconstruct a theory based on the above mentioned facts. In this paper, we would like to propose an improved scheme of Fermi interaction in which the values of the coupling constants depend upon whether the strangeness is conserved or not. Although the situation may certainly not be ideal from the standpoint of universality, it may be compared to the similar situation in strong interactions; that is, the coupling constants for the strong pion and K-meson interactions seem to be different and also the highest possible symmetry of the interactions seems to be violated. 2. Phenomenological Analysis As discussed above, we shall assume that the coupling of strangeness nonconserving current is weaker than that of the strangeness conserving current (/»/')• The Lagrangian density of this interaction is ^ 2 ( ? A + / A + J fxf

) '

(ix+Jx+ J fx))

(A)

where j x , / A and flx a r e leptonic, strangeness conserving baryonic and strangeness non-conserving baryonic currents: h = =ryx(l+Ys){e+e)> 1x *Yx(l+Ys) {(*+*)> Jx = pyA(l+y 6 )n, A

=

VYX^+YMVYX^+YM-

For simplicity, we have in this paper adopted the Sakata modelB), in which the yl-particle is the only fundamental hyperon and all other strange particles are regarded as the appropriate compound states of the nucleons and the Aparticle. Then the choice of the above primary interaction will be unique as long as we insist on limiting the interaction to charged currents. (For another model see the remark in the end of section 3). It should be noted that the nonleptonic part of the interaction (A) contains an appreciable fraction of the \A\\ = f part as well as the \A1\ = \ part. In order to explain the idea in a simpler way, let us, for the moment, introduce

3 74

B . SAKITA A N D S. ONEDA

the following phenomenological interaction (B) in addition to the interaction (A): pmn(l+y5)A+h.c, (B) where p is a parameter to be determined and m denotes the neutron mass. In this interaction scheme (A) and (B), since the interaction (B) behaves as an isospinor, the slowness of the hyperon leptonic decays as well as the \A1\ =\ selection rule for the non-leptonic decay processes could be explained, provided / > /' and the value of parameter p is large enough so that for the non-leptonic processes the contribution from the interaction (B) exceeds that from the nonleptonic part of the interaction (A). We will examine this statement by choosing values of the constants p and /' to be, for example, p = 3 X10- 8 , /' = §/, / = 1X lO- 6 /^ 2 ,

(C)

where / is the usual value of the Fermi coupling constant. 2.1. /l-DECAY

For the non-leptonic decays of A there are two types of decay scheme as shown in fig. 1: one is through (A), the other through (B). The ratio of the

X. (a)

^7i-)IW{A°->n^-^>) = 2 for the diagram (b). For the /J-decay of A, the numerical value of /' in (C) gives

Tf(^°^p+e+v) W{A° -* n+^+W^-^p+jr-)

(4)

which is slightly smaller than, but not inconsistent with, the present experimental value. We have chosen the fig. 1(a) as one of the typical diagrams through (A). However, it contains an appreciable amount of \A1\ = f in addition to |JI| = \ transitions. Therefore, though fig. 1 (b), which satisfies the strict \Al\ = \ rule, is more important than fig. 1 (a), we have to suppress the \A\\ = f part of such diagrams as fig. 1(a) to some extent (possibly by higher order corrections) in order to explain the approximate validity of the |JI| = ^rule for the branching ratio of the ./1-decay. 2.2. K-MESON DECAY INTO PIONS

For the K -> n-\-n decay, the Feynman diagrams through the interaction (A) (diagrams (a) and (b)) and through (B) (diagram (c)) are shown in fig. 2. Let

(«)

(b)

Fig. 2. Feynman diagrams of K - > J I + J I decay,

us compare the matrix element of the diagram (a) with that of the diagram (c). By a perturbation calculation, both of them are logarithmically divergent but their ratio is finite and is given by

rrS^+SL':^,

(6)

where F again represents the black bubble of fig. 2a. Using the values (C), this ratio turns out to be 1:20. (5') The diagram (b) does not contribute to both the K+->jr+-fw0 and the Ki° ->• n-\-n decay. However, if we allow for the radiative corrections between

_5 B . SAKITA A N D S. ONEDA

76

the two bubbles, the diagrams of this type may no longer be unimportant. The contributions of such diagrams however, are not likely to surpass that of the diagrams of the type (a). For the K+ -> 7t++jp decay, the diagram (c) does not contribute and, therefore, if we discard the diagram (b) for the moment, the ratio of the decay rate of K^ -► n++xr to K+ -> n++n° in the present scheme will be approximately derived from the ratio (5'). We obtain 7Z++71?)

agreeing well with the experimental ratio. As mentioned above, even if we include the diagram (b) the qualitative feature will not be changed drastically. The charge branching ratio of the K1°-decay is, of course, compatible with the \A1\ = \ selection rule. In the same way, for the K -> Bn processes the \A1\ = \ part due to the interaction (B) will play a dominant role. Note that in the present model there is no trouble of looking for the component \A\\ = f. 2.3. OTHER PROCESSES

It will now be worthwhile to see in the present scheme what processes are expected to be different from U.F.I, (by this we mean the scheme with / = /' and p = 0) and what processes remain unchanged. i) It is obvious that strangeness conserving processes such as //-decay, jr-decay, nuclear /?-decay, etc., are the same as in U.F.I. ii) As to the leptonic decay of the K-meson, dispersion relation calculation of the K -*■ p+v decay is now consistent with experiment since /' tv \f. The ratio W(K+-^e((Ji)+v+7t0)IW(K+^>-a++*P) is the same as U.F.I. If we compare the processes K+ -*■ /*(e)+v+n° with the K + ->ji++n° decay due to fig. (2a) we get the following estimate. We write the matrix-elements of K+ ->tt°+jK(e)+v decay in the form4) J- K(k)„*(k-p) (kaM+paN)fiya(l+ys)v, where ka and (k—P)a denote the four momenta of the K-meson and the Ja­ meson respectively and M and N are the form factors which will probably be approximately constant. We get W(K+ -» 7i?+n++v) W(K+ -► n»+e++v)

0.80+0.33 ( £ ) + 0.077

Q*

and W{K+ -* 7i°+e++v) W(K+ -> ^++7r°)

M2 * (6.3M+iV) 2 '

The experimental ratio W{K+ ->nP+p++v) *» W(K+ -»• 7iP+e++v) is obtained

6 ON THE DECAY INTERACTION OF STRANGE PARTICLES

77

for the value \NjM\ *w 1. In fact, the lowest order calculation with cut off momentum on m (nucleon mass) gives N «s — M, which implies W(K+ -> n°+e++v)

1 2\r 2m m 1 2H8 + ff 4 ff8+ 5wVL E+m £+*»

m2 ~\ ff u , £(£+»») JA_O

(2.18a)

(2.18b)

(2.18c)

(2.18d)

14 APPLICATION

OF D I S P E R S I O N

X'*>

1 2 3 4 5 6 7 8 9 10 11 12

(«-I0(«F-#)

(e-l/)M) (e-#(P-«r) (e-£)(l7-£)(«r.# (e-£)(tf£)M) («-#)(^)(«F-^)

(e-£)(U-£)M) (e-a)(tf-£) (e-oXU-k) i(.e-t)(U-fixk) i(e- UXi>) i(e- Uxk)

a=-{p-k)/m,

(8=-y.*)/m;

»=(a+/J)/2,

(3.2)

(2.19)

Im{£i( 3 Ps)cos€+£i( 3 F 2 )sine} = (ten« 2 )Re{£i( 8 P 2 )cose+£i( 3 F 2 )sine},

(2.20a)

Im{-E l (lP 2 )sine+.Ei( 3 F 2 )cos€} = (tan5 / )Re{-£i( 3 P 2 )sine+£i( 3 F 2 )cose},

(2.20b)

«-J"

~ 6

1 «r(P)» [dx{p'/\T(J(x)U0))\d)

L(27r) 2a>£J

J Xtr*'-+P,

3

3. DISPERSION RELATIONS We shall assume that the scalar functions H,■ with fixed A are analytic on the entire complex v plane except for possible singularities on the real axis. We will look for these singularities by graphs. First, the class of diagrams shown in Fig. 1 (a) have singularities for positive v. The first diagram, which has a deuteron in its intermediate state, has an isolated pole at v=0, while the others have continuous singular­ ities (branch cut) for v larger than the physical threshold of this reduction, namely v=B, the binding energy of the deuteron. Second, the class of diagrams shown in Fig. 1(c) have singularities for negative v— |A|. The first two diagrams, which have a one-nucleon intermediate state, give the isolated poles at i can be obtained from B+«) by using the symmetry of A. 7i/1 arcO )=3K -r~ / The charge triplet terms can be obtained by replacing q+ by *~ and putting w J B v'—v—ie all BoB is the phase shift of n-p scattering in the 'So

(3.12)

J,.^,.

The meaning of the notation in these expressions is explained in the Appendix.

' I t might be noted that the amplitude MipSi) would vanish completely in the absence of the n-p tensor interaction; in the usual calculation the reason would be the orthogonality of the initial and final'S wavefunctions, and in the present calculation the equality of the magnetic moments of the initial and the final states.

16 APPLICATION

OF

DISPERSION

we find F(p) to be

TABLE IV. Dipole transition amplitudes. Dipole amplitude

Final np state

JfiPSo) E.C-Po)

'So

m

'Pi l D% 'St 'Pt 'Pi 'P, 'Dt >D, *F,

MADt) Mil'S,) Ed*/,) EiPPi) E^P,) JtfiM>i)

M^D,) £.(*Pi)

1793

RELATIONS

=

smS(p)\ smS(p)r

1—ya—ip*ra

op L

(f+y*)y*ar (4.8)

2

7 [l+(l-2r/o)*+7r]J In the zero-range approximation (r=0), F(p) becomes simply F(^)|r_0=(l-7 »+/> structure due to the long-range part of the n-p potential, (b) that the deuteron can be treated as an ordinary structure of the n-pfinalstate due to the long-range part of the n-p potential, and (c) the long-range part of the meson current. particle.10 But, in fact, the deuteron exhibits itself as The heavier lines are off the mass shell. a not quite "ordinary particle" by the existence of anomalous singularities, corresponding to an inner the ordinary calculation. This can result if, for instance, structure which arises from the long range nature of the there is cancellation between the contributions of n-p potential. Although it is clear in principle how these Figs. 2(a), (b), and 2(c). An experimental check on our singularities can be included," in the present work we results is in principle possible through the observation have neglected them, keeping only the elastic rescatterof the energy dependence of the Ml matrix element. ing cut. Thus, our result for the Ml matrix element The argument goes as follows. Eq. (4.8) depends on the *S scattering parameters, For our process, y+d—* n+p, we have, as usual, 8 but not on the 5 parameters except through the binding "threshold theorems," or better termed, "zero-energy 8 energy and the »^x (3*32-x!!)/)(*)U >

The operator inside is just the quadrupole moment operator so that the above quantity is equal to -(2/5)»eQ,

APPENDIX A In this appendix we derive the y—d and np-d vertex functions. f-d Vertex Lorentz convariance and gauge invariance determine the following unique form for the matrix element of the current operator between one deuteron states:

(A5)

(A6)

when initial and final deuteron polarization are in z direction in its rest system. In above expression Q denote the quadrupole moment of the deuteron. Apply the same operation to (Al), and we find y(0)=eQ/2(10)i,

(A7)

by comparing with (A6). np-d Vertex

U„*(d')U,(d) D ](«"'- l)/2ip,

where y=(mB)i (B: binding energy of deuteron) and p is the triplet n-p effective range. If we put (A19) into (A18), then we find that this has a pole at p=iy. This is nothing but the pole of the bound state in S-matrix theory. Finally, we obtain (A20)

B(-m*)~3(nfi/y>)aT, where &iry/m

e

tvkXe ftp , 2m (2*)»

Mi yd^d:

e M *(-*S/Xe,)-

2m

'

rtkXe-I —— , L(2*)*J

(B4)

where £,./. is the initial (final) polarization unit pseudovector of the deuteron. d-*np:

Gxp*(1)

**(t?-pf)i *t*M-++d, (B8) where ^p0(r) = sin(^r+8)/sin5 and p(r) is the exact radial wavefunction, normalized so as to equal p0 outside the range of the potential. When p=iy then S=— too and so ^ T °=« - 1 "' (naturally), while iy=u/f!l where u is the normalized bound state radial wave function, 31 being its asymptotic amplitude, i.e., w=3ltf-Tr outside the potential (for a zero-range potential, 31= (27)*. Thus, (B8) gives us {dpp cot5} 1 _. T =2t7C(l/2y)- (1/312)],

(B9)

22 APPLICATION

OF D I S P E R S I O N

RELATIONS

1799

due to the diagrams of Fig. 2. Actually, in this approx­ imation it is unnecessary to write down the dispersion /w=(i3P/27)[l/(?-i7)]. (BIO) representation, for we can work directly with the Comparing the results (B6) and (BIO), we conclude that analytic properties of the matrix element. For definiteness, let us work with the M1 matrix G=-(4T)»31/», (Bll) element, which we shall call M" in our rescattering approximation. It has the following properties: where the sign is fixed by considering the calculation of (a) M"=MB at k=0 (threshold theorem), because the amplitude of the tip component of the deuteron by all Ml matrix elements contain a factor k, and only the perturbation theory, which also immediately makes Born term has an energy denominator which vanishes clear that the coupling constant is proportional to the at * = 0 . asymptotic bound-state amplitude. (b) M"(k) is analytic in the complex k plane In the effective-range approximation, i.e., [equivalently the p2 plane, since mk=j?-\-~fr\ except for the "elastic cut" k>B=y2/m, where its phase is F"~ L-y~ip+ (7 2 +? 2 )7/2]- 1 the scattering phase shift of the x 5 n-p state. It follows and so that ^•«-.««p/(l-7f)][l/(#-jy)l (B12) M n '(£) = [D(O)/.D(/0]MB0fe), (B18) we have immediately, by comparison with (B6), where D(p) is the so-called denominator function of G««(l/wi)[8iry/(l-7r)]*. (B13) the n-p scattering amplitude; it is analytic in the energy plane except that on the elastic cut it has the We can now proceed to calculate the Born approxima­ phase — 5. Omnes has given the formula for D(k): tion photodisintegration matrix elements. For £ 1 , the dp'* term in which the photon is absorbed by the deuteron D(p) e X P contributes nothing since k - e = 0 ; so the only term is ~S(P') , (B19) n l W o p'p2-f-u that in which the proton absorbes the photon: and so, according to (B7),

DE

M„ B = e

1

=0, where | d,*) is a deuteron state with momentum d and spin direction i while |0) is the vacuum state. We shall assume that this adjustment is possible and the first equation of (2.7) is finite. We can easily demon­ strate the usual translation relation tr"-''Bli(x,k)ei"=Bll(x+a,

{).

(2.8)

In the limit $ —» 0, Bt behaves like a local pseudovector field operator under a Lorentz transformation. Due to the Lorentz covariance, the first equation of (2.8) can be written as

™*™M-&j^'iC{*A-

(2 9)

'

Using the relation (2.1) and the second equation of (2.7), we can see that

/„i{«l+T0^«^(*-r)4^^(l+T^w4(y-r)-«1^w^(«-r)}.

Taking the Fourier transform of the above equation and applying the generalized Ward's identity for the nucleon " From the charge-conjugation invariance we obtain F,(p',p)= —Ty.T(—p, —p'), which implies that p and /u(0) are symmetric with respect to p* and p* while T where u is the energy of the y ray. The Born term starts from the term ~l/&>, while ant^,' starts from the term of zeroth power in u. Due to (3.11), the term of zeroth order in w in 31Z^' does not depend on & at all so that it is independent of the direction of the incident y ray h, while the term of zeroth power in the Born term does depend on k. Therefore, we can say that the l/o> term of the Bom approximation is exact and also that the part of the Bom term which approaches a constant as o> —* 0 and which is odd under the operation k —» — k is exact in the limit w —> 0. The former corresponds to the El transition while the latter corre­ sponds to the Ml. The last term added to Eq. (3.12) is an even term, so that if we keep only the odd part for the zeroth power of w in Eq. (3.11) it does not contribute at all. Along this line, if we make in (3.12) a nonrelativistic expansion and neglect y*/m2 and pP/m*, then we obtain ff--^-i^|-(l^)|>,+M.-i(tf/^ loirm1 I

L\ u J + (Mp-M»){(l-2a)i[e-UX^]-3at(e-/.)[U-px£]}l) )

where p* is the direction of the relative momentum of two final nucleons in the center-of-mass system while k is the direction of the incident y ray. 5 is the transition amplitude, such that the differential cross section of the photodisintegration of the deuteron in the center-ofmass system is = V | - ] \(s) 12 (314) dtl \w/ '

(3.13)

where the matrix element indicated is taken between the twofinalPauli spinors. £ is an abbreviation for the s u m 0f the ordinary spin and isospin for the final state and the average of the polarization for the initial y ray and deuteron. The amplitude ff in Eq. (3.13) is the exact amplitude ™ t n e long-wavelength limit of the incident y ray except the small corrections of order of B/m.

29 BUNJI IV. APPLICATION The low-energy limit derived in the previous section is the result of the theory of composite particles and of gauge invariance. It is interesting, therefore, to apply this result to the n-p capture (Ml transition) in order to compare the theory with experiment. As we have shown,1 the effect of rescattering in the n-p state can be obtained as a function of the n-p phase shift by using unitarity. The low-energy limit obtained in the previous section and the enhancement factor

SAKITA

1806

due to the rescattering effect will give a good approxima­ tion for the Ml amplitude and allows comparison with experiment. This subject has been discussed in detail in reference 1. ACKNOWLEDGMENTS The author wishes to thank Professor R. G. Sachs and Professor R. F. Sawyer for their interest and valuable discussions. He also wishes to thank Professor K. Nishijima and Professor C. Goebel for their helpful correspondence.

30 PHYSICAL REVIEW

VOLUME 11, NUMBER 6

LETTERS

15 SEPTEMBER 1963

DISCUSSION OF A SUGGESTED BOUND ON COUPLING CONSTANTS C. J. Goebel* and B. Sakitat University of Wisconsin, Madison, Wisconsin (Received 12 August 1963) In a recent Letter, Geshkenbein and Ioffe1 have derived an upper bound on coupling constants, i.e., the mass-shell value of three-leg vertices. Their form of bound is remarkable in comparison to previous results 2 because it depends only on the masses of the three particles involved, and not on the nature and range of the forces between the particles or on the nonexistence of stable states in other (crossed) channels. In the present com­ munication we conclude that an assumption on which the G-I bound is based (namely, that the proper vertex function has no pole) has no direct (i. e . , phenomenological) physical significance, and hence that their bound on coupling constants likewise has no direct physical significance. We recapitulate briefly their argument: The propagator of a spinless boson a has the represen­ tation

tion constant, should be nonnegative; this imposes the condition /pds«l-Sc.«l.

The spectral weight p(s) is a linear combination of positive definite t e r m s , each contributed by a state into which particle a can transform (con­ serving everything except energy). The contri­ bution to p of a two-body state consisting of par­ ticles 6 and c is £(s^Mrfe) I2, where £(s) is a phase-space factor, g is the abc coupling constant, and r is the proper vertex part (as defined in r e normalized field theory) of the abc vertex, nor­ malized to unity on the mass shell, i . e . , T(jna2) = 1. This implies the inequality p > £ (s)g21T I2, which, used in Eq. (2), leads to

«*fW-m

i]|-

(1)

with c,- 5*0. In the limit of infinite energy, G"1 s " ° > Zs, where Z, the propagator renormalizaQ534

(2)

dst.(s) in 2 J 0 ■m a 2 )

(3)

If a lower limit can be put on $, we have an upper limit on g2. G-I show that * does have a minimum value, if it is assumed that r has no singularities on the 1-4

293

31 VOLUME 11,NUMBER6

PHYSICAL

REV

physical sheet besides its right-hand cut, s » (w ■ +mc)2. G-I did consider that r might have a pole, noting that if G has a zero at the same point GT will be nonsingular (for the significance of this, see below); but they rejected this possibility, on the grounds that such a pole would correspond to a bound state of particles b and c. They also remarked that this pole will appear in the scatter­ ing amplitude of particles b and c through the ir­ reducible contribution of particle a as intermediate state: g2TGV. Here we shall argue that such a pole of r has, in fact, no direct physical signifi­ cance and does not appear in the scattering am­ plitude. [The pole of g'TGT due to the pole of r is actually a "ghost," i . e . , its residue has the wrong sign for it to represent a bound state.] We shall first show that it is natural, when the cou­ pling is strong, for G to have a zero, hence r to have a pole there, thus allowing the G-I bound to be violated. We shall then explain from the point of view of "dynamics" how the proper ver­ tex part can have a pole, and how the consequent pole of g*rGT is canceled from the b-c scattering amplitude. We start from the original KS116n- Lehmann representation of the propagator of particle a:

G—^-rds'^, s-m 2 J a

s'-s'

(4) (4)

EW L E T T E R S

b-c threshold is defined by saying that in the b-c scattering amplitude/, the pole term due to particle a is fB = -0/(*2+y2) • The vertex function r e p ­ resentation, the mass-splitting Hamiltonian should be a product of two 35 tensors, which is reduced as 358) 35 = 1 + 35 + 35 +189 + 280 + 280* + 405. The mass term of the baryons formed from * * is reduced as 56»8> 56 = 1 +35 + 405 + 2695. The number of representations in common in these two products is the number of parame­ t e r s (number of independent couplings) in the effective electromagnetic mass-splitting term of baryons. One of the 3J>'s in the first product is antisymmetric with respect to the inter­ change of two tensors in the product, so that it does not contribute to the electromagnetic mass term because these two tensors in the product in the electromagnetic mass-splitting Hamiltonian are symmetric. Therefore, we have three independent couplings: 1_, 3j>, and 405. Since the terms from 1_ do not contribute to the mass difference, however, we have e s ­ sentially two parameters in an effective mass t e r m . Using tensor notations, we may write the mass term of baryons as follows: tl,A,B

(4)

il,jl,A

The first term is the contribution from 35 and 405, while the second term is from 405. If we insert the expression for * [Eq. (1)], we obtain

= Yl

=S+-S°,

* - _ y 1 *o

= H--H° = 2--L°> ++

23NOVEMBER 1964

where we have denoted the mass of the parti­ cle by its symbol. The relations among the decuplet states are the consequences of SU(3) symmetry alone. 12 Experimental values of the mass differences are 1 3 p-n =-1.3 MeV, S + - 2 ° =-2.85 ±0.30 MeV; E - - H ° = 6.1±1.6MeV, S - - S ° = 4.75±1.0 MeV. Agreements with the theoretical predictions a r e not so impressive as for the magnetic moment. We note that our results, of course, satisfy the Coleman-Glashow mass relation for bary­ ons, 11 but are not compatible with some of the predictions of the tadpole model. 14 We would like to thank Professor L. Michel and Dr. H. J . Lipkin for their illuminating d i s ­ cussions, and Professor R. G. Sachs for his careful reading of this note. Note added in proof. -After we had written this note, Professor L. Michel showed us a preprint by M. A. B. Be"g, B. W. Lee, and A. Pais, in which they also obtained the ratio of magnetic moments. "Work performed under the auspices of the U. S. Atomic Energy Commission. *B. Sakita, Phys. Rev., to be published. 2 F. GUrsey and L. A. Radicati, Phys. Rev. Let­ ters 13, 173 (1964), S A. Pais, Phys. Rev. Letters 13, 175 (1964). 4 F. GUrsey, L. A. Radicati, and A. Pais, Phys. Rev. Letters 13, 299(1964). S E. Wigner, Phys. Rev. 51, 106 (1937). 6 M. Gell-Mann. Phys. Letters J3, 214(1964); G. Zweig, to be published. 7 The separation of "electric" and "magnetic" cur­ rents is based on the behavior of the current in the static limit. In the static limit, the electric current of a spin-{ particle has the form **V *-(V**)* so that it transforms as a scalar under the SU(2) spin transformation, while the magnetic current has the form **S* xk so that it transforms as a vector. Combining it with the SU(3) tensor property of the electromagnetic current, i.e., Ttl, therefore, the electric current should be a tensor of the (1,8) rep­ resentation of SU(2)®SU(3), while the magnetic cur­ rent is a tensor of the (3,8) representation. The lowest representation which has (1,8) and (3,8) as its components is the regular representation 35. *Daoy has 10 components, identified as follows: Dm = Q~, = r

Z>,„ = E*%/3, +

,0,SJ=E*-/V3",

0iu i i* />/3,

o I2S = y,*°/^6,

j>M, = y 1 *-/V3,

Dut=N*++,

Dm=N*+/J3,

/>„,=iV*%/3,

Dm=N

+

Ar* = 3(Ar* -Ar*°)+Ar*-,

'The type of coupling obtained here is the same as

U511 3-4

645

42 VOLUME 13,NUMBER21

PHYSICAL

REVIEW

the type of coupling obtained in reference 4 for the pion-nucleon interactions. In our point of view, however, the Yukawa-type pion-nucleon interactions are symmetry-violating interactions of SU(6). If we assume that the Yukawa interaction transforms as the 35 representation [simplest SU(6)-violating interaction], we have two independent couplings so that we do not necessarily have a definite type of coupling for the pion-nucleon interactions. ,0 Since we are not convinced of the relativistic extension of the group SU(6), contrary to the statement in reference 2, we apply this group in the static limit by assuming that it is an approximate symme-

646

LETTERS

23NOVEMBER 1964

try valid only in this limit. We thank Professor L. Michel for his informative discussion on the extension of the Poincart group. n S . Coleman and S. L. Glashow, Phys. Rev. Letters j5, 423 (1961). "S.~P. Rosen, Phys. Rev. Letters 11, 100(1963). ,S D. D. Carmony et al., Phys. Rev. Letters W, 482 (1964); R. A. Burnstein, T. B. Day, B. Kehoe, B. Sechi-Zorn, and G. A. Snow, Phys. Rev. Letters 13, 66 (1964). il S. Coleman and S. L. Glashow, Phys. Rev. 134, B671 (1964).

U511 4 - 4

43

Ann. Inst. Henri-Poincari, Vol. H, n° 2, 1965, p. 167-170.

Section A : Physique thiorique.

Group of invariance of a relativistic supermultiplet theory (*) par Louis MICHEL Institut des HautesfitudesScientifiques, Bures-sur-Yvette (S.-et-O.) and Bunji SAKITA Argonne National Laboratory Argonne (Illinois).

Recently Sakita [7], Giirsey and Radicati [2] [4] and Pais [3] [4] have proposed a generalization of Wigner supennultiplet theory [5] for the nucleus to baryons and mesons [6]. This raises the question: what is a relativistic supermultiplet theory ? In this paper we shall consider only the problem of defining the invariance group G for such a theory [7]. We denote by P the connected Poincar6 group. It is the semi-direct product P = T x L where T is the translation group and L is the homo­ geneous Lorentz group. CONDITION 1. — The invariance group G of a relativistic theory contains P. We shall not discuss here the discrete invariance P, C, T, so we shall add. CONDITION 2. — G is a connected topological group (with P as topological subgroup) [8]. Invariance under G is considered as the largest symmetry for strong coupling physics [7] [2] [3] [4], The particles of a supermultiplet have

(*) Work performed under the auspices of the U. S. Atomic Energy Commission.

44

168

LOUIS MICHEL

the same mass and for a given energy momentum p, all possible states (spin, charges, etc.) of these particles form a finite dimensional Hilbert space which is the space of an irreducible unitary representation of a compact group Sp, the « little group » of p. From the classical Wigner analysis it is easy to translate this as conditions on G. 3. — The translation group T is invariant subgroup of G. The action of G on T (by its inner automorphisms) preserves the Minkowski metric and the little group (mathematicians say stabiliser or isotopy group) in G/T of a time-like translation a e T is a compact group S. CONDITION

— If a group G satisfies condition 1, 2 and 3, then G/T is a direct product of H 0 L. THEOREM.

Proof : The condition 3 implies that for every g e G , a ( e T ) and its transformed g(a) = gag~1 have same Minkowski length : a.a = g(a).g(a). In the dual of T [i. e., the four dimensional vector space of energy momentum] the orbits of G are the connected sheets of mass hyperboloid. Denote / : G -£ Aut T, the homomorphism of G which describes its action, by inner automorphisms, on its invariant subgroup T. It is easy to prove [9] that the connected group of continuous automorphisms of G which preserves the Minkowski metric is L. So the image o f / i s L: I m / = L. Since T is abelian T P, the covering of the Poincard group. We proceed now to build such a group G. Among all subgroups of the linear group with enumerable dimension let us look for the smallest group H such that: SU(6) c H, SL(2, C) 0 and the unitary representations of SU(6), is the group G we just defined. It is a 106 para­ meter Lie group [//]. As we shall explain elsewhere the use of such G as invariance group for a relativistic supermultiplet theory of elementary particle is possible, but we do not like it. One of us (L. M.) has benefitted very much from a two-month visit to the Argonne National Laboratory, where this work was done. We would like to thank Dr. W. D. McGlinn for his reading of the manuscript.

46 LOUIS MICHEL

170

REFERENCES [i] B. SAKITA, Phys. Rev., t. 136, B, 1964, p. 1756.

[2\ F. GURSEY and L. RADICATI, Phys. Rev. Letters, t. 13, 1964, p. 173. [3\ A. PAIS, Phys. Rev. Letters, t. 13, 1964, p. 175. [-fl F. GURSEY, A. PAIS and L. RADICATI, Phys. Rev. Letters, t. 13, 1964, p. 299. [5] E. P. WIGNER, Phys. Rev., t. 51, 1937, p. 106.

[6] More than twenty papers on this subject have been published or mimeo­ graphed. [7] For physicists who may find our simple rigorous proof too abstract we are writing a more detailed paper on the subject in terms of Lie algebra. We will alto show in this paper that the invariance properties of such a theory cannot be reduced to the study of a group. [8] This is a purely technical condition; with the work of E. C. ZEEMAN, /. Math. Phys., t 5, 1964, p. 491, we can obtain that G/T is the semi-direct product H x L without condition 2. [9] Indeed ZEEMAN [8], has proven it without the assumptions of continuity and automorphisms. [10\ K. IWASAWA, Ann. Math., t 50,1949, p. 507. [11] This group has been mentioned at the end of reference [1]. (Manuscrit recu le 28 Janvier 1965).

Directeur de la publication : p. GAUTHEER-VILLARS. IMPRIME EN FRANCE DBPdT LEOAL £ D . N° 1304a.

IMPRMERIE BARNEOUD S. A. LAVAL, N° 5014. 3-1965.

65-01

47 Reprinted from THE PHYSICAL REVIEW, Vol. 139, No. SB, B1355-B1367, 6 September 1965 Printed in U. S. A.

Relativistic Formulation of the S 17(6) Symmetry Scheme* Btmji SAKITA AND KAMESHWAK C. WALI

Argonne National Laboratory, Argonne, Illinois (Received 22 March 1965) A relativistic formulation of the SU(6) symmetry scheme is presented, starting with the basic assumption that the fields corresponding to elementary particles are tensors of M(12) for U(12) or SU(12)£}. In particular a mixed second-rank tensor and a totally symmetric third-rank tensor are associated with the meson and baryon fields, respectively. It is shown that if these fields are required to satisfy prescribed free-field equations of motion, then one is led to a particle supermultiplet structure which corresponds to the 3501 and 56-dimensional representations of SU(6) for the mesons and baryons. It is also shown that the spin-dependent and S£/(3)-spin-dependent mass splittings can be included in the theory and that solutions in terms of physical particle fields can be obtained. Effective trilinear meson-meson and meson-baryon vertex functions, using these solutions and an interaction Lagrangian which is invariant under if (12), are calculated in the lowest order perturbation. We would like to note especially the following results: (a) From the known pion-nucleon coupling constant, the width of the pion-nucleon (3,3) resonance is calculated to be 94 MeV. (b) The ratio of the magnetic form factors for the neutron and proton is — J for all momentum transfers and np=-{l+2Mp/m,) nuclear magnetons, (c) The charge form factor of the neutron is zero for all momentum transfers.

The problems connected with a relativistic formula­ tion of the SU(6) theory may be discussed from a purely HERE has been considerable interest recently mathematical point of view of finding an appropriate in the SU(6) symmetry scheme for elementary group of invariance. For this purpose, we recall that the particles.1 It is conceived as an extension of Wigner's irreducible representations of SU(6) can be decomposed nuclear-supermultiplet theory2 to elementary-particle into irreducible representations of SU(2) ® SU(3), where phenomena. Unlike other higher symmetry schemes,* the SU(2) can be identified as the ordinary spin group the SU(6) theory proposes to treat the ordinary spin and the SU(3) as the familiar internal symmetry group on the same footing as the isotopic spin and hyper- SU(3>). Ii the theory has to contain orbital angular charge. Clearly such a formulation is possible only if the momentum and spin mixed in a Lorentz-invariant space-time variables and the spin variables are com­ manner, the spin groups SU(2) has to be extended to pletely decoupled. This is possible only in a nonrela- SL(2,C) which is the covering group of the restricted tivistic theory as in the case of Wigner's supermultiplet Lorentz group. A fully relativistic SU(6) theory must theory. Since Lorentz transformations mix the intrinsic include in addition to the homogeneous Lorentz trans­ spin and the orbital angular momentum in an intricate formations, space and time translations. The required manner, it is not obvious whether the SU(6) theory can group G therefore must contain SU(6) and the Poincare be extended to the relativistic domain. It is therefore group as subgroups in such a manner that the intersec­ not surprising that several attempts 4 have been made tion of SL(2,C)®SU(3) and SV(6) is SU(2)®SU(3). towards an understanding of this problem. It has been shown5 that G must then contain the group SL(6,C). Now depending on how the translations are * Work performed under the auspices of the U. S. Atomic Energy imbedded in the group, one obtains two types of struc­ Commission. 1 F. GUrsey and L. A. Radicati, Phys. Rev. Letters 13, 173 tures for G: (i) G is given by P'®Q, where P' is a (1964); A. Pais, ibid. 13, 175 (1964); B. Sakita, Phys. Rev. 136, group isomorphic to the physical Poincare" group and B1756 (1964); F. Gursey, A. Pais, and L. A. Radicati, Phys. QZ)SL(6,C)a; (ii) G is a semidirect product of T» by Rev. Letters 13, 299 (1964). SL(6,C) where T» is the group of translations in a • E. Wigner, Phys. Rev. 51, 106 (1937). »J. Schwinger, Phys. Rev. Letters 12, 237 (1964); Phys. Rev. 36-dimensional space.7 135, B816 (1964); 136, B1821 (1964); F. Gursey, T. D. Lee, and Once a group G is given, its unitary representations on M. Nauenberg, ibid. 135, B468 (1964); P. G. 0. Freund and Y. Nambu, Phys. Rev. Letters 12, 714 (1964); 13, 221 (1964); M. Hilbert space provide a set of symmetry transformations Gell-Mann, Physics 1, 63 (1964); Z. Maid, Progr. Theoret. Phys. on the physical states which are characterized by the (Kyoto) 31,331 (1964); P. Tarjanne and V. L. Teplitz, Phys. Rev. bases of the representations. The basis of an irreducible Letters 11, 447 (1963); Y. Hara, Phys. Rev. 134, B701 (1964). *R. P. Feynman, M. Gell-Mann and G. Zweig, Phys. Rev. representation gives a set of physical states which are Letters 13, 678 (1964); K. Bardaci, J. M. Cornwall, P. G. O. commonly identified as the particles belonging to a Freund, and B. W. Lee, ibid. 13,698 (1964) and 14,48 (1965); S. I. INTRODUCTION

T

Okubo and R. E. Marshak, ibid. 13, 818 (1964) and 14, 156 (1965); W. RUM, Phys. Letters 13, 349 (1964); 14, 334 (1965); •L. Michel and B. Sakita, Ann. Inst. Henri-Poincare' 11, 167 A. Salam, ibid. 13, 354 (1964); T. Fulton and J. Wess, ibid. 14, (1965). 57 (1965); P. Roman and J. J. Aghassi, ibid. 14, 68 (1965); M. A. 'L. Michel, Second Coral Gables Conference on Symmetry B. Beg and A. Pais, Phys. Rev. Letters 14,267 (1965); Y. Ne" eman, Phys. Letters 14, 327 (1965); F. Gursey, ibid. 14, 330 (1965); Principles at High Energy, January 1965 (to be published). ' L. Michel and B. Sakita (Ref. 5); W. RUhl (Ref. 4), T. Fulton K. T. Mahanthappa and E. C. G. Sudarshan, Phys. Rev. Letters 14, 458 (1965); Riazuddin and L. K. Pandit, ibid,, }4,462 (1965). and J. Wess (Ref. 4).

B 1355

48 B1356

B. B . SSAAKKIITTAA AANNDD K. K. C.

WALI WALI

supermultiplet of the system. One can construct a an irreducible representation of SU(6) for a fixed mo­ unitary representation of G in case (i) by using Wigner's mentum q. This suggests the possibility that the basic method for i". Since G is a direct product of P' by Q, fields are tensors of SL(6,C) whereas the solutions to the basis of such a representation must be a tensor appropriate equations of motion for these fields give the product of the basis of P' and the basis of the unitary desired particle-multiplet structure, even though the representation of Q. But QZ)SL(6,C) which is noncom- equations themselves are not covariant under SL(6,C). pact. An irreducible unitary representation of Q is of The purpose of the present paper is to examine the infinite dimensions. This corresponds, therefore, physi­ possibility and consequences of such an approach.10,11 cally to an infinite number of particles belonging to a If one assumes that the elementary particles are the supermultiplet. bound states of one or several quarks and antiquarks, In case (ii), G contains additional translations other the bound-state wave function (or field in a phenomenothan the usual four space-time ones. Clearly the physical logical Lagrangian theory) can be described as a product interpretation of the extra translations is not easy. of the fundamental fields ^'s and #'s. In the following Further if one identifies the usual space-time transla­ discussion, however, it is not necessary to assume ex­ tions with four of the translations in Tn, the physical plicitly such a quark model. We shall only assume that mass is no longer invariant under SL(6,C). It will change the fields associated with the elementary particles trans­ continuously under the transformations of SL(6,C). form like the products of ^'s and $ 's. In particular, the Physically this corresponds to a continuous mass dis­ meson field is represented by a second rank mixed tribution for a given particle state. Since the physical tensor fu B (144 components). A totally symmetric third world does not appear to admit either an infinite number rank tensor ^ABC (364 components) is associated with of one-particle states for fixed four-momentum or a con­ the baryon field. These tensor representations of tinuous mass distribution for a given particle state, we 5L(6,C) together with their properties under space re­ are forced to conclude that there is no physically in­ flections are discussed in Sec. II. It also contains the teresting group of invariance which contains the Poin- interaction Lagrangian which is assumed to be invariant car6 group and the SU(6) group in a nontrivial way.8 under SL(6,C) and space reflections. The interaction A more physical description of the SU(6) theory is Lagrangian assumed in the present discussion is in­ provided by the quark model. A relativistic quark model variant under a larger group of transformations M (12) 12 can be constructed along the lines of the three-field [or £7(12) or S*7(12)£]. Section III is devoted to the decomposition of $ and V into appropriate auxiliary Thirring model. The fundamental field in this model can be described by a 12-component spinor ^M. A pair of fields and to the discussion of the symmetry properties indices ia can be assigned to A, where i runs from 1 to 4 with respect to the interchange of Dirac and 51/(3) spin and can be identified as the Dirac spinor index, a takes indices. In Sec. IV, the free field equations of motion for the values 1 to 3 and corresponds to the S£/(3)-spin the meson and baryon fields are given. It is shown that index. If one decomposes a Dirac spinor field into two the meson field equations admit solutions which corre­ two-component spinors (Weyl decomposition), f decom­ spond to a nonet of 0 and a nonet of 1~ mesons. The poses into two six-component fields and x- The fields baryon field equations lead to solutions which corre­ + + andx then provide vector and conjugate-vector repre­ spond to a decuplet of $ and an octet of \ baryons. sentations of SL(6,C) (Sec. II). As pointed out by The desired mass splittings are introduced and the several authors,9 one can construct an interaction solutions for * and * are obtained in terms of physical Lagrangian which is invariant under SL(6,C). However, particle fields. These solutions are used to calculate it is impossible to construct a free Lagrangian which is effective vertex parts in the lowest order perturbation also invariant under 5L(6,C) without encountering the calculation in Sec. V. The relations between different difficulties mentioned earlier in connection with the coupling constants and some of their consequences are group of invariance. Without a free Lagrangian, the also discussed. Finally, the concluding section is devoted standard quantization procedures and the particle inter­ to a summary and the discussion of some of the diffi­ culties of the theory. pretation of the fields cannot be carried out. In spite of this apparent difficulty, the quark model suggests an alternative approach. If we consider a model » B. Sakita and K. C. Wali, Phys. Rev. Letters 14,404 (1965). of noninteracting quarks and construct a free La­ present paper is an extended version of this letter. 11 grangian in terms of ^ which is invariant under The A. Salam, Proceedings of the Second Coral Gables Conference P®5i7(3) (PsPoincare' group), we can obtain free on Symmetry Principles at High Energy, 1965 (W. H. Freeman field equations of motion (Dirac equation) and com­ and Company, San Francisco, 1965); A. Salam, R. Delbourgo and J. Strathdee, Proc. Roy. Soc. (London) 284, 146 (1965); mutation relations. The solutions to these equations can A. Salam, R. Delbourgo, J. Strathdee, and M. A. Rashid, Proc. be interpreted as particle states which form a basis of Roy. Soc. (London) 285, 312 (1965); M. A. B. Beg and A. Pais, Phys. Rev. Letters 14, 267 (1965); W. Riihl, Phys. Letters 14,334 (1965); 15, 99 (1965); 15, 101 (1965). « S. Coleman, Phys. Rev. 138, B1262 (1965), » K. Bardakci J. M. Cornwall, P. G. 0. Freund, and B. W. Lee, • K. Bardaka, J. M. Cornwall, P. G. O. Freund, and B. W, Lea Phys. Rev. Letters 14, 48 (1965); R. Delbourgo, A. Salam, and J. Strathdee (to be published). (Ref. 4) j S. Okubo and R. E. Marshak (Ref, 4).

49 RELATIVISTIC

F O R M U L A T I O N OF SU(6)

n . ELEMENTARY FIELDS AND INTERACTIONS Consider the group 6X6 complex matrices of determi­ nant one, which we shall denote SL(6,C). To every matrix A (det4 = l) belonging to this group, there corresponds a linear unimodular transformation T(A) over a six-dimensional complex vector space. A vector 4> in this space is a fundamental representation of SL(6,C) and undergoes the transformation ^'=A?/=7(.75$'>

^=-7*?.

(3.19)

IV. FREE-FIELD EQUATIONS OF MOTION FOR MESONS AND BARYONS The spinor fields described so far can lead to no physi­ cal consequences unless they are made to represent physical particle states. As stated in the Introduction, we may accomplish this by requiring that * and ¥ satisfy prescribed free field equations of motion. The guiding principle in the choice of such equations is provided by one of the most remarkable successes of the SU(6) symmetry, namely the assignment of representa­ tions for the low lying baryonic and mesonic states. The octet of | + and the decuplet of | + baryons can be identified as belonging to the 56-dimensional repre­ sentation. The octet 0~ and the nonet 1~ mesons can be fitted into the 35-dimensional representation whereas the X" meson17 can belong to the singlet representation. We shall choose the wave equations which lead to solu­ tions that correspond to the SU(6) supermultiplet states.

O F 5 V (6)

SYMMETRY

SCHEME

B 1359

where 8 is given by

ro 0 °] 5= 0 0 0 lo 0 1J We shall continue to use the direct-product notation with the convention regarding matrix multiplication as defined in (2.7) and (2.9). (Also see footnote 15.) In Eq. (4.2), the first term gives a common mass m0 to all mesons. The second term is a spin-dependent term and is introduced to split the vector mesons from the pseudoscalar ones. The third term is responsible for Gell-Mann-Okubo (GMO) 1 ' splittings as well as singlet and octet mixing. The last term splits the singlet pseudoscalar meson from others, since it is proportional to the projection operator of the singlet, pseudoscalar term. If we assume a sufficiently large value for ms, there would be little mixing between the singlet Q~(X°) and the octet 0~(ri°) states. Since the GMO mass sum rule is satisfied very well for the octet of 0~ mesons, we assume this to be the case. If we insert the expansion (3.2) for * in (4.1), multiply by the Dirac matrices 1, y,,, o-M,, yf/s, and 75, respec­ tively, and take trace, we obtain the following set of equations in SU(S) space: [ ( w o + 4 « i ) l ® l ' - r - w ' ( 8 ® l ' + l ® 5 ' ) ] S = 0 , (4.3) C«ol®r]V M -r-l®l'dxTx„=0, (4.4) | > o l ® l ' + » ' ( 5 ® l'-f 1® 5')]T„X +l®l'(d M Vx-axV M )=0, (4.5)

The meson field * is required to satisfy f [ 7 - a ® l ' ® l ® r - l ® 7 d ® l ® l ' ] * - r - f » * = 0 , (4.1)

[ w o l ® l ' ] A M + l ® l ' d „ P = 0 , (4.6)

which is the Duffin-Kemmer equation18 rewritten in a form more convenient for our purpose. To introduce the desired mass splittings, we regard m as a matrix in both Dirac-spin and 5'7(3)-spin space. It can be chosen in a number of different ways. One of the forms which leads to the well-known empirical relations for the masses of the pseudoscalar and vector mesons is as follows:

[(wo-4mi)l®l'+m'(8®l'+l®6')]P + l ® l ' d A = 0 . (4.7) From these equations it follows that

m=«0(i®i'®i®r)

[(i®r)n-«o{»oi®r

£

(4.9)

+»»'(8®1'+1®8')}]P=0.

8

7 A ® T V W 5 ® E a.

d„V„=0,

(4.10)

[(i®r)n-»io{(>»o-4mi)i®i'

-ri»'(l®r+75®75)®(5®r+l®&') +ms

(4.8)

+»»'(8®1'-1-1®5')}]V|1=0>

+ J m i ( 7 x ® 7 x + 7 » 7 s ® 7 x 7 « ) ® (1® 1')

It

S=0,

(4.2)

» G. R. Kalbfleisch tl al., Phys. Rev. Letters 12, 527 (1964); M. Gundzik el al., ibid. 12, 546 (1964). »R. J. Duffin, Phys. Rev. 54, 1114 (1938); N. Kemmer, Proc. Roy. Soc. (London) A173, 91 (1939). Please note that (for typographical reasons) primes are used with identity and 5 matrices instead of tildes to indicate the multiplication from the right [Eqs. (2.7)- (2.10)].

(4.11)

Equations (4.9), (4.10), and (4.11) represent the free wave equations of motion for a nonet of 1~ mesons and an octet of 0~ mesons. The physical states can be ob­ tained by diagonalizing the mass matrix in (4.10) and »M. Gell-Mann, Phys. Rev. 125, 1067 (1962); S. Okubp, Progr. Theoret. Phys. (Kyoto) 27, 949 (1962).

52 B.

B1360

SAKITA

A N D K.

fp0+w0

lyd+M2*=0,

K*+

w°-p° V=

V2

K*°

M=[iWol®l®l+(M'-itfi-3Jlfj)PD +(Pj>+P*)(iTiGi+J/iGO:i,

K* K+ V&

+—

P= V2

K-

\/6

X°

(4.17)

where Pz> and FB are the projection operators defined in Eqs. (3.10) and (3.11). The first term gives a common mass to all the baryons. The second term splits the decuplet masses from the octet masses. Gi and G2 in the third term are introduced so that they not only give the GMO splittings, but also give the observed relations between the decuplet spacings and the octet spacings. They are given by

i?

K-

(4.16)

where M is again a matrix. The desired mass splittings can be introduced by choosing M to be a matrix in SU(S) space21 alone. The required form is

v2

V2~

WALI

generalized Bargmann-Wigner equations20:

(4.11), and are described by

fir 0

C.

-V/6J

Gi=5®l®l-t-l®8®H-l®l®8,

The masses corresponding to these particles are as follows:

G2=LC^®(83t,®l+l®85.,)

(4.18)

i-0

mp2=tnJ=mo2;

WK»2=m0(»»o+W); mf= »»o(»*o+ 2mo),

w, 2 =w 0 (»»o—4«x);

*»jt2=*»o(»io—4WI+W); w,2=«ii(w«- 4 w i + f W ) •

+symmetrizing terms]. (4.12)

It is clear that the squares of the 0~ meson masses satisfy the GMO sum rule for an octet and that 2

2

2

2

2

m^-mK' -=mK' -m =mK -mr .

(4.13)

From Eqs. (4.5), (4.6), and (4.12), r„x.-'= [ l / ( « r ) / ] ( 3 x » V / - «, VX.J), .4,./=(l/mP)6VP/,

OTT=

mp mp mK'2/tnr

2

mK' /m, ntK^/mp m^/mp.

(4.14)

The solution of the wave equation (4.1) can therefore be explicitly written as

L

(GI*)«»T= D ^ A ' + t f a a ^ + ^ s V ] , (G»¥)«|JT=[(*,S8T+^3)S«3+(¥«.y3+¥8«7)V

+ (*3/>«+^8u)«T8],

2(mT)J

(4.19)

(4.20)

where for convenience we have Suppressed the Dirac indices. As in the meson case, if we insert the expansion (3.7) in (4.16), it follows that (4.21)

e„PyGijic=0,

where (mr)J are given by the elements of the matrix tfla m, ,mK'2/mP

To be more explicit,

[7- d+Wt+M'-Mi-3M 2) +(Mi+2MM.l+*f+S7*)YDeit,mto-0,

(4.22)

lyd+iMo-MdYxi-ikJ + p f 2 [ ( « a 8 + StfxwJ-lSJxijk.fl=0.

(4.23)

With the identification, Dui=N*++,

D112=N*+/yJ3,

Dm= N*"/y/S,

Z)m=A'*-, Dn3=Y*-/rt,

Z?„,= F*+/VJ, £>138=E*°/v3~,

Dm=Y*»/V6, Z? 23 3=E*-/v3,

Di33=Q~,

-(7,7.V(—+ - ) , (4-15) \ m,, ms V3 / J where ms is the mass of X° and Fll,= d^V,—d,Vll. For the baryons, we assume that V satisfies the

*> V. Bargmann and E. P. Wigner, Proc. Natl. Acad. Sci. U. S. 34, 211 (1948). » We split the decuplet mass from the octet by using the SU(S) spin-dependent term. We may, of course, use the ordinary spindependent splitting. Because of the over-all symmetry of * , however, it can be proved that both methods are equivalent when the mass matrix is operated on ¥ .

53 RELATIVISTIC

2°A - + — v2 y/6

FORMULATION

S U (6)

SYMMETRY

SCHEME

B1361

Thus Dijk.afiy is given by

2+ 2°

OF

Dijt,affr=\ i(.y^C)Jk4'ltitaffy (ff„/r),tAM„-,„?T . L 4Ma0y J

A

+—

N

where (4.31)

2A

Similar considerations lead to

V6 the masses of the decuplet and the octet states are given by the mass formulas

XHkJ=WyiJ L

MD= (M0+Mt)+M'-

where ^ satisfies the Dirac equation

M o = (Mo+Mi)-(M

(Mi+Wi)y

- J / j [ 7 ( / + l ) - i 7 2 ] , (4.24) i+WJY -M£I(I+1)-IY*1.

(4.25)

If we rewrite (4.22) as [no summation is intended for re­ peated SU(3) indices]. lyd+MafiyVDi.jk,a$y=0,

(4.26)

where Main-

(Mo+M'-MtSMi)

and substitute (3. IS) in (4.26), we obtain (4.27)

[7-a+ikf^ 7 >„, < ,o r =0; [y-d+ MaSyil'^.afy = 0 .

Since Dijt.afiy it totally symmetric, Eq. (4.26) can also be written as Zyd+Ma$y]ii'Dii.k,a0y=O. If we now substitute the expansion Djfk,a»y= [4&../7«C).-'*+llA/«,>(+MN)/mfy 9

50/ gpp..*=-{l 81\ 2r