Supersymmetry: Lectures and Reprints, 2 Volume set 9789814415095, 981441509X

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SUPERSYMMETRY

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SUPERSYMMETRY Sergio Ferrara

Volume 1

~ North-Holland V?$ Amsterdam • Oxford • New York • Tokyo

UfeWorld Scientific JKm

Singapore • New Jersey • Hong Kong

© ELSEVIER SCIENCE PUBLISHERS, B.V., AND WORLD SCIENTIFIC PUBLISHING CO. PTE. LTD., 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval systems, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, Elsevier Science Publishers B. V. (NorthHolland Physics Publishing Division), P O Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about con­ ditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. ISBN North-Holland ISBN World Scientific

0-444-87079-2 0-444-87085-7 pbk 9971-966-21-2 9971-966-22-0 pbk

Published by: North-Holland Physics Publishing, a division of Elsevier Science Publishers B.V., P O Box 103,1000 AC Amsterdam, The Netherlands and World Scientific Publishing Company Pte Ltd, Singapore P O Box 128, Farrer Road, Singapore 9128 Sole distributors for Europe: Elsevier Science Publishers, B.V., P O Box 211, Amsterdam, The Netherlands Sole distributors for the USA and Canada: Elsevier Science Publishing Company, Inc., 52 Vanderbilt Avenue, New York, NY 10017, USA Sole distributors R.O. W.: World Scientific Publishing Company Pte Ltd, Singapore

Library of Congress Cataloging-in-Publication Data Supersymmetry 1. Supersymmetry. 2. Supergravity. I. Ferrara, S. QC174.17.S9S972 1987 530.1'42 87-21968 ISBN 0-444-87079-2 (Elsevier) ISBN 0-444-87085-7 pbk (Elsevier)

Printed by Singapore National Printers Ltd.

V

PREFACE

Supersymmetry, i.e., Fermi-Bose symmetry is one of the most peculiar discoveries in the history of physics. The peculiarity lies in the fact that although a tremendous theoretical effort has been made in this field, no experimental evidence of this symmetry has shown up. To see whether supersymmetry will remain a subject of mathematical physics or rather will change the future of particle physics, one has probably to wait for the next generation of accelerators exploring physics in the TeV energy region. In spite of the lack of experimental evidence, supersymmetry has attracted the interest of the high energy physics community for a number of reasons. Among them let me quote the symmetry between fermions and bosons, the building blocks of matter, and their interactions; the softening of quantum divergences of relativistic quantum field theories. The local (or gauged) extension of supersymmetric theories, i.e., supergravity theories, provides a natural framework for the unification of all fundamental interactions of elementary particles in a single superunified theory. The latter program experienced a major advance in the last two years with the discovery of realistic superstring theories with gauge quanta and matter fields encompassing the four fundamental forces of nature: electroweak, strong and gravitational interactions. Surprisingly enough, the general field of supersymmetry gave rise to some feedback in other branches of physics and other aspects of quantum field theory. Let us mention the possible evidence of a dynamical supersymmetry in a unified description of the spectra of odd and even nuclei, the relation between supersymmetry and stochastic phenomena and the connection of supersymmetry with topology. Last but not least we mention the Becchi-Rouet-Stora symmetry which is an essential ingredient for the formulation of quantized Yang-Mills theories. The present volumes are a collected series of contributions to the field of supersymmetry and supergravity, regarded as four-dimensional space-time symme­ tries of Lagrangian field theory. Special emphasis is given to papers dealing with broken supersymmetry and supergravity and their application to elementary particle physics. Due to the extension of the subject, it is the opinion of the editor that it is impossible to give a comprehensive set of collected papers encompassing the whole

vi field of supersymmetry. As a consequence, because of the lack of space and time, any review can only concentrate on particular aspects of the field. In the present case, there is an imbalance, deliberately intended, between papers connected to technical developments with respect to papers related to model building or spontaneously broken supersymmetry. In particular, the papers mainly focus on four-dimensional space-time N = 1 supersymmetry and supergravity. Also important topics such as the superspace formulation of supersymmetric theories as well as extended supersymmetric theories and higher dimensional theories are only marginally considered. Supersymmetry has become a natural framework for the study of unified theories only after the discovery of a supersymmetric extension of a theory of gravity. The minimal theory incorporating gravity is the N = 1 supergravity theory. This theory was proposed in March 1976 by Freedman, van Nieuwenhuizen, and the editor (Chapter VIII). The theory was constructed by the requirement that a new gauge field, carrying spin f (gravitino), could be coupled to Einstein gravity with a locally supersymmetric interaction. Soon afterwards the same theory was formulated in simplified form (Palatini formalism or first order formulation) by Deser and Zumino (Chapter VIII). The merging framework of superstring theories is not treated in these volumes except in the last chapter of the second volume. Superstring theories and their interplay with supersymmetry and supergravity do have a bizarre history. World sheet two-dimensional supersymmetry in string theories was discovered in 1971 simultaneously with the first paper on the four-dimensional super-Poincare algebra by Gol'fand and Likhtman (Chapter I). However, the Russian paper passed almost unnoticed by the physics community and did not motivate additional investigation on the subject. Instead the world-sheet supersymmetry of the Neveu-Ramond-Schwarz fermionic strings inspired the work of Wess and Zumino (Chapter I) in 1973 which then boosted the interest in supersymmetry in Lagrangian field theory and in particle physics. In the same year, 1973, another independent formulation of the fourdimensional super-Poincare algebra was provided by Volkov and Akulov (Chapter I) who formulated a nonlinear realization of this algebra. Based on this realization, they wrote a geometrical Lagrangian for the self-interaction of the Goldstino, the Goldstone fermion of supersymmetry. The main motivation of this second Russian work was to try to explain the masslessness of the neutrino in exact analogy to the masslessness of the pion in the limit of spontaneously broken chiral symmetry in QCD. However, this idea was soon abandoned because of the wrong prediction for the energy spectrum of the neutrino in /?-decay if it were considered as a Goldstone fermion. Superstring theories, i.e., fermionic strings with space-time supersymmetry were discovered instead after the emergence of supergravity theory in 1976. The fundamental paper of Gliozzi, Olive and Scherk (Chapter XIV) was clearly motivated by the discovery of supergravity and was also the first paper in which the existence of the massless spin-f excitation predicted by supergravity, the gravitino, was seen to occur in the spectrum of closed strings. In retrospect, we can say that while global space-time supersymmetry was motivated by (two-dimensional) world-sheet super-

vii symmetry, the supersymmetric strings or superstrings of Green and Schwarz originated from supergravity theories. What is most important is that the consistency and finiteness of superstrings and their heterotic version seem due to an interplay between world-sheet and space-time supersymmetry. Ironically, one could say that superstrings are supergravity theories twice, because of the existence of a world-sheet and of a target space supersymmetry in these theories. An excellent reprint volume on superstrings and their relation to supersymmetric point-field theories, with a slightly different view of the history of these theories, was published two years ago by World Scientific Publishing under the editorship of Professor John Schwarz of Caltech. The very existence of these volumes was the main reason for excluding almost all of the string papers in this new collection. As a final remark, I would like to apologize to those authors whose contribution, have not been included or not emphasized in these volumes. The present reprint collection is in two volumes and is divided into thirteen chapters, describing global and local supersymmetry. Each chapter opens with some commentary notes by the editor.

References 1. L. Corwin, Y. Ne'eman and S. Sternberg, "Graded Lie Algebras in Mathematics and Physics (Bose-Fermi Symmetry)," Rev. Mod. Phys. 47, 573 (1975). 2. L. O'Raifeartaigh, Commun. Dublin Institute for Advanced Studies, Series A (Theor. Phys.) No. 22 (1975). 3. S. Ferrara, "Supersymmetry (Fermi-Bose Symmetry): A New Invariance of Quantum Field Theory," Rivista del Nuovo Cimento 6, 106 (1975). 4. P. Fayet and S. Ferrara, "Supersymmetry," Phys. Reports 32, 249 (1977). 5. A. Salam and J. Strathdee, "Supersymmetry and Superfluids," Fortschz. Phys. 26, 56 (1978). 6. D. Z. Freedman and P. van Nieuwenhuizen (eds.), Supergravity (Stony Brook, 1979), North-Holland, 1979. 7. S. Ferrara, J. Ellis and P. van Nieuwenhuizen (eds.), Unification of the Fundamental Particle Interactions (Erice, 1980), Plenum, 1983. 8. S. Ferrara and J. G. Taylor (eds.), Supersymmetry and Supergravity '81 (Trieste), Cambridge University Press, 1982. 9. P. van Nieuwenhuizen, "Supergravity" Phys. Reports C68, 189 (1981). 10. J. Ellis and S. Ferrara (eds.), Unification of the Fundamental Particle Interactions (Erice, 1981), Plenum, 1984. 11. S. W. Hawking and M. Rocek (eds.), Superspace and Supergravity, Cambridge University Press, 1981. 12. S. Ferrara, J. G. Taylor and P. van Nieuwenhuizen (eds.), Supersymmetry and Supergravity '82 (Trieste), World Scientific, 1983. 13. J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, 1983. 14. S. J. Gates, M. T. Grisaru, M. Rocek, and W. Siegel, Superspace or One Thousand and One Lessons in Supersymmetry, Frontiers in Physics, Benjamin, 1983. 15. D. V. Nanopoulos and A. Savoy-Navarro, "Supersymmetry Confronting Experiment," Phys. Reports C105, 1 (1984).

viii 16. H. P. Nilles, "Supersymmetry, Supergravity and Particle Physics," Phys. Reports CllO, 1 (1984). 17. B. Zumino, "Supersymmetry and Supergravity," Phys. Reports C104, 87 (1984), (XVIII Solvay Conference, Austin, Texas, 1982), p. 113, ed. L. Van Hove. 18. B. de Wit, P. Fayet and P. van Nieuwenhuizen (eds.), Supersymmetry and Supergravity '84 (Trieste), World Scientific, 1984. 19. H. E. Haber and G. L. Kane, "The Search for Supersymmetry: Probing Physics Beyond the Standard Model," Phys. Reports C117, 75 (1985). 20. M. Sohnius, "Introducing Supersymmetry," Phys. Reports C128, 39 (1985). 21. V. A. Kostelecky and D. K. Campbell (eds.), Supersymmetry in Physics, Physica 15D, Nos. 1 and 2, p. 3 (1985). 22. M. Jacob (ed.), Supersymmetry and Supergravity, North-Holland and World Scientific, 1986. 23. P. C. West, Introduction to Supersymmetry and Supergravity, World Scientific, 1986. 24. A. B. Lahanas and D. V. Nanopoulos, "No-Scale Supergravity," Phys. Reports 145, 1 (1987). 25. J. H. Schwarz (ed.), Superstrings, the First 15 Years of Superstring Theory, Vols. I and II, World Scientific, 1985. 26. M. Green, J. H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, 1986. 27. F. Iachello, "Dynamical Supersymmetries in Nuclei," Phys. Rev. Lett. 44, 772 (1980). 28. G. Parisi and N. Sourlas, "Random Magnetic Fields, Supersymmetry, and Negative Dimensions," Phys. Rev. Lett. 43, 744 (1979). 29. C. Becchi, A. Rouet and R. Stora, "The Abelian Higgs Kibble Model, Unitarity of the Soperator," Phys. Lett. 52B, 344 (1974).

IX

CONTENTS

Volume 1 Preface

v

Chapter I. DISCOVERY OF SPACE-TIME GLOBAL SYPERSYMMETRY 1. Yu. A. Gorfand and E. P. Likhtman, "Extension of the Algebra of Poincare Group Generators and Violation of P Invariance," JETP Lett. 13 (1971) 323-326 2. D. V. Volkov and V. P. Akulov, "Is the Neutrino a Goldstone Particle?" Phys. Lett. 46B (1973) 109-110 3. J. Wess and B. Zumino, "Supergauge Transformations in Four Dimensions," Nucl. Phys. B70 (1974) 39-50 4. B. de Wit and D. Z. Freedman, "Phenomenology of Goldstone Neutrinos,,, Phys. Rev. Lett. 35 (1975) 827-830

3 7

Chapter II. SUPERMULTIPLETS AND SUPERFIELDS 1. A. Salam and J. Strathdee, "Supergauge Transformations," Nucl. Phys. B76 (1974) 477-482 2. S. Ferrara, B. Zumino and J. Wess, "Supergauge Multiplets and Superfields," Phys. Lett. 51B (1974) 239-241 3. A. Salam and J. Strathdee, "Unitary Representations of Supergauge Symmetries," Nucl. Phys. B80 (1974) 499-505 4. R. Haag, J. Lopuszanski and M. Sohnius, "All Possible Generators of Supersymmetries of the 5-Matrix," Nucl. Phys. B88 (1975) 257-274 Chapter ffl. SUPERSYMMETRIC LAGRANGIANS AND CURRENTS 1. J. Wess and B. Zumino, "A Lagrangian Model Invariant Under Supergauge Transformations," Phys. Lett. 49B (1974) 52-54 2. J. Wess and B. Zumino, "Supergauge Invariant Extension of Quantum Electrodynamics," Nucl. Phys. B78 (1974) 1-13 3. S. Ferrara and B. Zumino, "Supergauge Invariant Yang-Mills Theories," Nucl. Phys. B79 (1974) 413-421

11 13 25 31 35 41 44 51 71 77 80 93

4.

A. Salam and J. Strathdee, "Supersymmetry and Non-Abelian Gauges," Phys. Lett. 51B (1974) 353-355 5. B. de Wit and D. Z. Freedman, "Combined Supersymmetric and Gauge-Invariant Field Theories," Phys. Rev. D12 (1975) 2286-2297 6. S. Ferrara and B. Zumino, "Transformation Properties of the Supercurrent," Nucl. Phys. B87 (1975) 207-220 7. P. Fayet, "Supergauge Invariant Extension of the Higgs Mechanism and a Model for the Electron and Its Neutrino," Nucl. Phys. B90 (1975) 104-124 8. P. Fayet, "Fermi-Bose Hypersymmetry," Nucl. Phys. B113 (1976) 135-155 9. P. Fayet, "Spontaneous Generation of Massive Multiplets and Central Charges in Extended Supersymmetric Theories," Nucl. Phys. B149 (1979) 137-169 10. B. Zumino, "Supersymmetry and Kahler Manifolds," Phys. Lett. 87B (1979) 203-206 11. J. A. Bagger and E. Witten, "The Gauge Invariant Supersymmetric Nonlinear Sigma Model," Phys. Lett. 118B (1982) 103-106

102 105 117 131

152 173

206 210

Chapter IV. SUPERSYMMETRY AND RENORMALIZATION 1. J. Iliopoulos and B. Zumino, "Broken Supergauge Symmetry and Renormalization," Nucl. Phys. B76 (1974) 310-332 2. S. Ferrara, J. Iliopoulos and B. Zumino, "Supergauge Invariance and the Gell-Mann-Low Eigenvalue," Nucl. Phys. B77 (1974) 413-419 3. B. Zumino, "Supersymmetry and the Vacuum," Nucl. Phys. B89 (1975) 535-546 4. P. C. West, "The Supersymmetric Effective Potential," Nucl. Phys. B106 (1976) 219-227 5. M. T. Grisaru, W. Siegel and M. Rocek, "Improved Methods for Supergraphs," Nucl. Phys. B159 (1979) 429-450 6. D. R. T. Jones, "Charge Renormalization in a Supersymmetric Yang-Mills Theory," Phys. Lett. 72B (1977) 199 7. O. Piguet and K. Sibold, "The Anomaly in the Slavnov Identity for N=l Supersymmetric Yang-Mills Theories," Nucl. Phys. B247 (1984) 484-510

217 222

8.

323

D. R. T. Jones, "Asymptotic Behavior of Supersymmetric Yang-Mills Theories in the Two-Loop Approximation," Nucl. Phys. B87 (1975) 127-132 9. E. Poggio and H. Pendleton, "Vanishing of Charge Renormalization and Anomalies in a Supersymmetric Gauge Theory," Phys. Lett. 72B (1977) 200-202 10. L. V. Avdeev, O. V. Tarasov and A. A. Vladimirov, "Vanishing of the Three-loop Charge Renormalization Function in a Supersymmetric Gauge Theory," Phys. Lett. 96B (1980) 94-96

245 252 264 273 295 296

329

332

xi 11. M. T. Grisaru, M. Rocek and W. Siegel, "Zero Value for the Three-Loop /3 Function in N=4 Supersymmetric Yang-Mills Theory," Phys. Rev. Lett. 45 (1980) 1063-1066 12. W. E. Caswell and D. Zanon, "Zero Three-Loop Beta Function in the N=4 Supersymmetric Yang-Mills Theory," Nucl. Phys. B182 (1981) 125-143 13. S. Mandelstam, "Light-Cone Superspace and the Ultraviolet Finiteness of the N=4 Model," Nucl. Phys. B213 (1983) 149-168 14. L. Girardello and M. T. Grisaru, "Soft Breaking of Supersymmetry," Nucl. Phys. B194 (1982) 65-76 15. A. J. Parkes and P. C. West, ' W = l Supersymmetric Mass Terms in the N=4 Supersymmetric Yang-Mills Theory," Phys. Lett. 122B (1983) 365-367 16. A. Parkes and P. West, "Explicit Supersymmetry Breaking can Preserve Finiteness in Rigid N=2 Supersymmetric Theories," Phys. Lett. 127B (1983) 353-359

335

Chapter V. SPONTANEOUS SUPERSYMMETRY BREAKING 1. P. Fayet and J. Iliopoulos, "Spontaneously Broken Supergauge Symmetries and Goldstone Spinors," Phys. Lett. 51B (1974) 461-464 2. L. O'Raifeartaigh, "Spontaneous Symmetry Breaking for Chiral Scalar Superfields," Nucl. Phys. B96 (1975) 331-352 3. P. Fayet, "Spontaneous Supersymmetry Breaking Without Gauge Invariance," Phys. Lett. 58B (1975) 67-70 4. S. Ferrara, L. Girardello and F. Palumbo, "General Mass Formula in Broken Supersymmetry," Phys. Rev. D20 (1979) 403-408 5. E. Witten, "Dynamical Breaking of Supersymmetry," Nucl. Phys. B185 (1981) 513-554

403 407

358 378 390

393

411 433 437 443

Chapter VI.

487

1.

490

2.

3. 4.

5.

6.

NON-PERTURBATWE ASPECTS OF GLOBAL SUPERSYMMETRY E. Witten, "Constraints on Supersymmetry Breaking," Nucl. Phys. B202 (1982) 253-316 S. Cecotti and L. Girardello, "Functional Measure, Topology and Dynamical Supersymmetry Breaking," Phys. Lett. B110 (1982) 39-43 L. Alvarez-Gaume, "Supersymmetry and the Atiyah-Singer Index Theorem," Comm. Math. Phys. 90 (1983) 161-173 D. Friedan and P. Windey, "Supersymmetric Derivation of the Atiyah-Singer Index and the Chiral Anomaly," Nucl. Phys. B235 (1984) 395-416 G. Veneziano and S. Yankielowicz, "An Effective Lagrangian for the P u r e N = l Supersymmetric Yang-Mills Theory," Phys. Lett. 113B (1982) 231-236 I. Affleck, M. Dine and N. Seiberg, "Dynamical Supersymmetry Breaking in Chiral Theories," Phys. Lett. 137B (1984) 187-192

339

554

559 572

594

600

xii 7.

8.

9.

10.

11.

12. 13.

V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, "Instanton Effects in Supersymmetric Theories, ,, Nucl. Phys. B229 (1983) 407-420 G. C. Rossi and G. Veneziano, "Non-Perturbative Breakdown of the Non-Renormalization Theorem in Supersymmetric QCD," Phys. Lett. 138B (1984) 195-199 S. Ferrara, L. Girardello and H. P. Nilles, "Breakdown of Local Supersymmetry through Gauge Fermion Condensates," Phys. Lett. 125B (1983) 457-460 W. Buchmuller, R. D. Peccei and T. Yanagida, "Quarks and Leptons as Quasi Nambu-Goldstone Fermions, ,, Phys. Lett. 124B (1983) 67-73 R. Barbieri, A. Masiero and G. Veneziano, "Hierarchy of Fermion Masses in Supersymmetric Composite Models," Phys. Lett. 128B (1983) 179-184 H. Nicolai, "On a New Characterization of Scalar Supersymmetric Theories," Phys. Lett. 89B (1980) 341-346 H. Nicolai, "Supersymmetry and Functional Integration Measures," Nucl. Phys. 176B (1980) 419-428

Chapter V n . 1.

2.

3. 4. 5.

6. 7.

8. 9.

606

620

625

629

636

642 648

SUPERSYMMETRIC MODELS FOR PARTICLE PHYSICS

661

P. Fayet, "Spontaneously Broken Supersymmetric Theories of Weak, Electromagnetic and Strong Interactions," Phys. Lett. 69B (1977) 489-494 P. Fayet, "Relations between the Masses of the Superpartners of Leptons and Quarks, the Goldstone Couplings and the Neutral Currents," Phys. Lett. 84B (1979) 416-420 S. Dimopoulos and H. Georgi, "Softly Broken Supersymmetry and SU(5)," Nucl. Phys. B193 (1981) 150-162 E. Witten, "Mass Hierarchies in Supersymmetric Theories," Phys. Lett. 105B (1981) 267-271 L. E. Ibanez and G. G. Ross, "Low-Energy Predictions in Supersymmetric Grand Unified Theories," Phys. Lett. 105B (1981) 439-442 S. Weinberg, "Supersymmetry at Ordinary Energies: Masses and Conservation Laws," Phys. Rev. D26 (1982) 287-302 G. R. Farrar and S. Weinberg, "Supersymmetry at Ordinary Energies. II: R Invariance, Goldstone Bosons, and Gauge-Fermion Masses," Phys. Rev. D27 (1983) 2732-2746 S. Dimopoulos, S. Raby and F. Wilczek, "Supersymmetry and the Scale of Unification," Phys. Rev. D24 (1981) 1681-1683 L. Ibihez and G. G. Ross, "SU(2)L X U(l) Symmetry Breaking as a Radiative Effect of Supersymmetry Breaking in GUTs," Phys. Lett. HOB (1982) 215-220

666

672

677 690 695

699 715

730 733

xiii 10. R. Barbieri, S. Ferrara and D. V. Nanopoulos, "Symmetry Breaking in a Supersymmetric Model of Strong and Electroweak Interactions," Z. Phys. C13 (1982) 267-271 11. S. Dimopoulos, S. Raby and F. Wilczek, "Proton Decay* in Supersymmetric Models," Phys. Lett. 112B (1982) 133-136 12. N. Sakai and T. Yanagida, "Proton Decay in a Class of Supersymmetric Grand Unified Models," Nucl. Phys. B197 (1982) 533-542 13. J. Ellis, D. V. Nanopoulos and S. Rudaz, "A Phenomenological Comparison of Conventional and Supersymmetric GUTs," Nucl. Phys. B202 (1982) 43-62 14. J. Ellis, L. Ibaiiez and G. G. Ross, "Grand Unification with Large Supersymmetry Breaking," Phys. Lett. 113B (1982) 283-287 15. D. Polchinski and L. Susskind, "Breaking of Supersymmetry at Intermediate Energy," Phys. Rev. D26 (1982) 3661-3673 16. B. Grinstein, "A Supersymmetric SU(5) Gauge Theory with No Gauge Hierarchy Problem," Nucl. Phys. B206 (1982) 387-396 17. A. Masiero, D. V. Nanopoulos, K. Tamvakis, and T. Yanagida, "Ordinary SU(5) Predictions from a Supersymmetric SU(5) Model," Phys. Lett. 115B (1982) 298-300 18. L. Alvarez-Gaume, M. Claudson and M. B. Wise, "Low-Energy Supersymmetry," Nucl. Phys. B207 (1982) 96-110 19. S. Dimopolous and S. Raby, "Geometric Hierarchy," Nucl. Phys. B219 (1983) 479-512 20. N. S&kai, "Proton Decay in Models with Intermediate Scale Supersymmetry Breaking," Phys. Lett. 121B (1982) 130-134

739

744 748

758

778 783 796 806

809 824 858

Volume 2 Chapter VIII. THE DISCOVERY OF SUPERGRAVTTY 1. D. Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, "Progress Toward a Theory of Supergravity," Phys. Rev. B13 (1976) 3214-3218 2. S. Deser and B. Zumino, "Consistent Supergravity," Phys. Lett. 62B (1976) 335-337 3. S. Ferrara, J. Scherk and P. van Nieuwenhuizen, "Locally Supersymmetric Maxwell-Einstein Theory," Phys. Rev. Lett. 37 (1976) 1035-1037 4. S. Ferrara, F. Gliozzi, J. Scherk, and P. van Nieuwenhuizen, "Matter Couplings in Supergravity Theory," Nucl. Phys. B117 (1976) 333-355 Chapter IX. OFF-SHELL SUPERGRAVITY 1. S. Ferrara and P. van Nieuwenhuizen, "The Auxiliary Fields of Supergravity," Phys. Lett. 74B (1978) 333-335 2. K. S. Stelle and P. C. West, "Minimal Auxiliary Fields for Supergravity," Phys. Lett. 74B (1978) 330-332

865 868 873 876

879

905 909 912

XIV

3. 4.

S. Ferrara and P. van Nieuwenhuizen, "Tensor Calculus for Supergravity," Phys. Lett 76B (1978) 404-408 K. S. Stelle and P. West, "Tensor Calculus for the Vector Multiplet Coupled to Supergravity," Phys. Lett. 77B (1978) 376-378

920

Chapter X.

925

1.

930

2. 3.

4. 5.

6.

SPONTANEOUSLY BROKEN SUPERGRAVITY: THE SUPERfflGGS EFFECT D. V. Volkov and V. A. Soroka, "Higgs Effect for Goldstone Particles With SpinVi^ JETP Lett. 18 (1973) 312-314 S. Deser and B. Zumino, "Broken Supersymmetry and Supergravity," Phys. Rev. Lett 38 (1977) 1433-1436

915

E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello, and P. van Nieuwenhuizen, "Spontaneous Symmetry Breaking and Higgs Effect in Supergravity without Cosmological Constant," Nucl. Phys. B147 (1979) 105-131 E. Witten and J. Bagger, "Quantization of Newton's Constant in Certain Supergravity Theories," Phys. Lett. 115B (1982) 202-206 E. Cremmer, S. Ferrara, L. Girardello, and A. van Proeyen, "Yang-Mills Theories with Local Supersymmetry: Lagrangian, Transformation Laws and Super-Higgs Effect," Nucl. Phys. B212 (1983) 413-442 J. A. Bagger, "Coupling the Gauge-Invariant Supersymmetric Non-Linear Sigma Model to Supergravity," Nucl. Phys. B211 (1983) 302-316

Chapter XL SUPERGRAVITY MODELS FOR PARTICLE PHYSICS 1. P. Fayet, "Mixing between Gravitational and Weak Interactions through the Massive Gravitino," Phys. Lett 70B (1977) 461-464 2. S. Weinberg, "Cosmological Constraints on the Scale of Supersymmetry Breaking," Phys. Rev. Lett. 48 (1982) 1303-1306 3. H. P. Nilles, "Dynamically Broken Supergravity and the Hierarchy Problem," Phys. Rev. Lett. 115 (1982) 193-196 4. A. H. Chamseddine, R. Arnowitt and P. Nath, "Locally Supersymmetric Grand Unification," Phys. Rev. Lett. 49 (1982) 970-974 5. R. Barbieri, S. Ferrara and C. A. Savoy, "Gauge Models with Spontaneously Broken Local Supersymmetry," Phys. Lett. 119B (1982) 343-347 6. H. P. Nilles, M. Srednicki and D. Wyler, "Weak Interaction Breakdown Induced by Supergravity," Phys. Lett. 120B (1983) 346-348 7. E. Cremmer, P. Fayet and L. Girardello, "Gravity-Induced Supersymmetry Breaking and Low Energy Mass Spectrum," Phys. Lett. 122B (1983) 41-48 8. L. Ibanez, "Grand Unification with Local Supersymmetry," Nucl. Phys. B218 (1982) 514-544

933 937

964 969

999

1017 1021 1025 1029 1033

1038

1043

1046

1054

XV

9.

10. 11. 12.

13.

H. P. Nilles, M. Srednicki and D. Wyler, "Constraints on the Stability of Mass Hierarchies in Supergravity," Phys. Lett. 124B (1983) 337-340 L. Hall, J. Lykken and S. Weinberg, "Supergravity as the Messenger of Supersymmetry Breaking," Phys. Rev. D27 (1983) 2359-2378 L. Alvarez-Gaume and J. Polchinski, "Minimal Low-Energy Supergravity," Nucl. Phys. B221 (1983) 495-523 J. Ellis, J. S. Hagelin, D. V. Nanopoulos, and K. Tamvakis, "Weak Symmetry Breaking by Radiative Corrections in Broken Supergravity," Phys. Lett. 125B (1983) 275-281 C. Kounnas, A. B. Lahanas, D. V. Nanopoulos, and M. Quiros, "Supergravity Induced Radiative SU(2) X U(l) Breaking with Light Top Quark and Stable Minimum," Phys. Lett. 132B (1983) 95-102

Chapter XII. NO-SCALE SUPERGRAVITY 1. E. Cremmer, S. Ferrara, C. Kounnas, and D. V. Nanopoulos, "Naturally Vanishing Cosmological Constant in N=l Supergravity," Phys. Lett. 133B (1983) 61-66 2. J. Ellis, A. B. Lahanas, D. V. Nanopoulos, and K. Tamvakis, "No-Scale Supersymmetric Standard Model," Phys. Lett. 134B (1984) 429-435 3. J. Ellis, C. Kounnas and D. V. Nanopoulos, "No-Scale Supersymmetric GUTs," Nucl. Phys. B247 (1984) 373-395 4. R. Barbieri, E. Cremmer and S. Ferrara, "Flat and Positive Potentials in N=l Supergravity," Phys. Lett. 163B (1985) 143-147 Chapter Xffl. BEYOND SUPERGRAVITY: SUPERSTRINGS 1. F. Gliozzi, J. Scherk and D. Olive, "Supersymmetry, Supergravity Theories and the Dual Spinor Model," Nucl. Phys. B122 (1977) 253-290 2. A. H. Chamseddine, "Interacting Supergravity in Ten Dimensions: The Role of the Six-Index Gauge Field," Phys. Rev. D24 (1981) 3065-3072 3. E. Bergshoeff, M. de Roo, B. de Wit, and P. van Nieuwenhuizen, "Ten-Dimensional Maxwell-Einstein Supergravity, Its Currents, and the Issue of Its Auxiliary Fields," Nucl. Phys. B195 (1982) 97-136 4. G. F. Chapline and N. S. Manton, "Unification of Yang-Mills Theory and Supergravity in Ten Dimensions," Phys. Lett. 120B (1983) 105-109 5. M. B. Green and J. H. Schwarz, "Anomaly Cancellations in Supersymmetric D = 10 Gauge Theory and Superstring Theory," Phys. Lett. 149B (1984) 117-122 6. D. J. Gross, J. A. Harvey, E. Martinec, and R. Rohm, "Heterotic String," Phys. Rev. Lett. 54 (1985) 502-505 7. P. Candelas, G. Horowitz, A. Strominger, and E. Witten, "Vacuum Configurations for Superstrings," Nucl. Phys. B258 (1985) 46-74 8. E. Witten, "Dimensional Reduction of Superstring Models," Phys. Lett. 155B (1985) 151-155

1085

1089 1109 1138

1145

1155 1159

1165

1172 1195 1203 1209

1247

1255

1295

1300

1306 1310 1339

3

Chapter I DISCOVERY OF SPACE-TIME GLOBAL SUPERSYMMETRY

This chapter deals with the three basic papers on four-dimensional space-time supersymmetry (1971, 1973 and 1974). The fourth shows that the idea of having the neutrino as the Goldstone fermion of spontaneously broken supersymmetry conflicts with experment. Let us recall that the basic algebraic structure underlying the notion of supersym­ metry is a graded Lie algebra. Graded Lie algebras are extensions of ordinary Lie algebras in which a grading is introduced, namely, a distinction between even and odd elements: even elements belong to an ordinary Lie algebra and obey commutation relations; odd elements, responsible for the grading, obey anticommutation relations among themselves and commutation relations with the even elements. These latter relations indeed specify that the odd elements are a representation of the Lie algebra, the grading representation. By denoting the set of elements of a D-dimensional Lie algebra by Am(m = 1, . . . , D) and the set elements of arf-dimensionalgrading representation by Qa( 2, the solution of this equation has only a finite number of parameters ( ^ quadratic in xM) f/X*) = cM + co^x" + exM + aMx2 - Ix^a-x,

coMV = -a>v/i9

(1.8)

while for D = 2, Eq. (1.7) reduces to the Klein-Gordon equation and gives an infinite

5

parameter Lie algebra. World-sheet supersymmetry of string theories corresponded to a grading of the two-dimensional conformal algebra. Instead in D = 4, one has a finite-dimensional grading with generators g a , Sa obeying the following commutation relations: [0«, D] = £&,, [0«, ^ J = / ( ^ 0 „

[2a, /y = o,

[S„ D] = - £ $ . , [Sa, M,v] = 1(^5).,

[s„ * j = o,

[&, *J = -'fys).,

[s„ /y = /(y„g).,

[ga, n] = -Jifog).,

P., n] = H(y5S)a9

{&> e^} = 2(yvw^,

(1.9)

{$., ^} = - 2 ( ? V M ; ,

{ga, S„} = -2(^>°M, V - y°Z) + 2y 5 ^n) a/? , where D and A^ are the generators of dilatations and conformal boosts, respectively, and II is a generator of a U(l) chiral transformation. If one introduces now anticommuting spinorial parameters £«(£«£/? = —fy£a),the previous anticommutators between odd elements can be written as commutators for the infinitesimal transformations sQ, TjS.

For example, for two Q transformations, one gets [Si, S2] = fag, e2Q] = -2eiy»e2PM.

(1.10)

The two constants spinors e, rj can be imbedded in a (linearly) x-dependent spinor which is the general solution of the differential equation (in D = 4)

(rA + YA -\g,vrd)e{x) = o.

(l.ii)

This ensures that the bilinear object

-1/2V *!?*,*•+ *2«*T«>-

(7)

2 ^A separate paper will be devoted to a justification of this postulate, and also to its comparison with the ordinary formulation of the requirement that the theory be invariant against a transformation group.

325

10 Thus, we have obtained a model for the interaction of quantized fields with parity nonconservation, invariant against the algebra (1) [1] S. Schweber, Introduction to Relativlstic Quantum Field Theory, Harper, 1961.

326

11 Volume 46B, number 1

PHYSICS LETTERS

3 September 1973

IS THE NEUTRINO A GOLDSTONE PARTICLE? D.V. VOLKOV and V.P. AKULOV Physico-Technical Institute, Academy of Sciences of the Ukrainian SSR, Kharkov 108, USSR Received 5 March 1973 Using the hypotheses, that the neutrino is a goldstone particle, a phenomenological Lagrangian is constructed, which describes an interaction of the neutrino with itself and with other particles.

Recently much attention has been paid in the ele­ mentary particle physics to the problem of spontane­ ously broken symmetries and the related degeneracy of the vacuum state. An immediate consequence of the vacuum degeneracy is that it gives rise to a possible existence of zero mass particles, the so-called Goldstone particles [1]. Among known elementary particles only the neu­ trino, the photon and the graviton have zero masses. However, the last two correspond to the gauge fields and do not require the vacuum degeneracy for their existence. Therefore the neutrino is the only elemen­ tary particle the existence of which may be immedi­ ately related to the vacuum degeneracy. We will restrict our attention to the following. If the neutrino is regarded as a Goldstone particle then this leads to a certain type of interaction of the neu­ trino with itself as well as with other particles. The interaction is completely defined by phenomenologi­ cal constants and in this sense is universal. For the determination of the type of spontaneously broken symmetry that causes the degeneracy of the vacuum and the corresponding properties of the neu­ trino as a Goldstone particle, let us consider the equa­ tion for a free neutrino ia M di///3x M =0

(1)

Eq. (1) is invariant under transformations of the Poincare group and the chiral transformations as well as under translations in the spinor space, i.e. under the transformations of the type i//->i/>' = i//+r

x^x'^x^

(2)

where f is a constant spinor, anticommuting with i>. Leaving the transformation properties of x^ and \}J under the Poincare group unchanged, let us replace

the transformations (2) by the transformations:

*M -* *M = *M " 5J (f+C7M ^ - * + a M 0 '

^

The resulting structure is a group with ten commuting and four anticommuting parameters*. It is the only possible generalization u>f *(2) and the Poincare group if the dimension of the group space is not enlarged. In the transformations (3) a is an arbitrary constant. Its dimension is the fourth power of length. Let us assume that in the presence of interaction the equations for the neutrino are invariant under the transformations (3). In the following we also assume that the number of the derivatives of the neutrino ifieM is a minimal one that is compatible with the ixraariance requirement. To construct the phenomenological action integral that satisfies the above assumptions it is sufficient to use the following differential forms coM = dxM + | (nvxV + exn

cM = 2/aP>7 M c4 0 ) ,

+ a

n*2 ~ 2xnax >

(6)

16 42

/. Wess, B. Zumino, Supergauge transformations

aM = 2ia[l\aP

.

For later use we calculate the expression T? = /3 M a 1 7 5 7^a 2 - /dMa2757"c*7 .

(7)

Using the explicit form for c^ and a 2 one finds T? = 4 / ( a P 7 5 4 0 ) - a i 1 ) 7 5 a f ) . Therefore 17 is x-independent.

3. Supergauge transformations Consider a multiplet consisting of a Majorana spinor i// and four scalar fields^, B, Fand G. Let us define an infinitesimal supergauge transformation by dA = ioc\jj , dB = iay5\lj , 8\p = b^(A - y5B)y»a + n(A - y5B)y^b^a + Fa + Gysa , 6 F = iay^d^ 6C = iay^d^

+ i(w - i)3 M ary^ , + Kn - x)d M a7 5 7 M ^ >

(8)

where the parameter a(x) is an infinitesimal spinor which anticommutes with itself and with 1// and commutes with the other fields. Furthermore a satisfies the diffe­ rential equation discussed in the previous section. The number n is arbitrary (it need not be an integer) and gives the weight of the multiplet. We say of a multiplet transforming as in (8) that it belongs to a scalar representation of weight n. The supergauge transformations generate a closed algebraic structure, similar to a Lie algebra. To see this, let us evaluate the commutator of two infinitesimal supergauge transformations 6 j and 6 2 , of parameters o^ and a 2 . Starting with the scalar field A we have 8 2 M = /a 1 (« 2 *) = ^ I C V ^ " ysBW where we imply summation over all sixteen matrices yA. Similarly, for the other terms, - '(a 2 7 5 3 M *)757 M ai = i*( /(a 2 7 M 3 M ^)ai = ~ i*X 2 ,

(9)

belongs to a scalar representation of weight n± + n2, which means that it transforms like (8) with n = nl +n2. This can be verified directly. For instance SA =Alia\l/2 +A2ia\pl

- Blioty5\l/2 - B2iocy5\IJl

= /a(^ 1 i// 2 + A2i*i - B1y5\p2 - B2y5\px) =" ia\p . Similarly for the other fields. For the spinor field one must use the rearrengement formula described in the appendix. The combination of two scalar representations into a third scalar representation just described gives only the simplest example of a generalized tensor calculus for supergauge transformations, the theory of which we have not yet fully developed. It is clearly the main tool for the construction of in­ variant interactions. Before closing this section we give another example of a representation. Let us consider a multiplet consisting of four scalar fields Z), C, M, N, of the vector field uM and of the two spinor fields x and X. We call it a vector multiplet, because of the presence in it of the vector field v . We shall say that this vector multiplet trans­ forms according to a vector representation of weight n if, under a supergauge trans­ formation, hD - iay^d^X

+ ?i(n - l)3 Ju a7 5 7' i A + i(n - l)3^a7 5 3 M x ,

6C=iay5x, 8M=IOL\

+ iayvd^x + kKn - 3)9Ma7Mx ,

SN= iay5X + iay5y^d^x + kKn - 3)3 a7 5 7^x » dv

n

=

tny^

+

'^M*

+ ind

,fiX ,

M

6x = 7 y M a - 3 M C 7 5 7 ^ a - K « - l)C7 5 7^3 M a + (M + y5N)a , SX = -$(dtlvv-dvvJy»y*>a+Dy5a

(10)

+ (n--l)v»dtiy,«(/)]/ l / -% + 7) p .

(7)

The first term gives the connection between the commutator of supersymmetry currents and the ener­ gy-momentum tensor 0PP. The second term, which has no analog for chiral pions, may be described as a covariant Schwinger term which contributes to the low-energy theorem because it brings in a pho­ ton propagator which is singular. It is closely related to the second term in the current (6). Other Schwinger and seagull terms 13 which do not contribute to the low-energy theorem have been omitted in (7). To relate the amplitude Mm(q,l) to the right-hand side of (7), we use the same combinatoric tech­ nique applied by Weinberg16 to multiple-pion emission. We define amplitudes T^iq.l), T^{q,l)u{l), and u{qffp{q, I) as the Fourier transforms of the matrix elements (B\TS^{xySp{y)\A), , fttlOiX • • •> G/t • • • fiillOi), • •., Qt • • • e,+J^>There are I

1 states with helicity X - k/2 while the last state has helicity X — N/2.

The P C r conjugate states are obtained by doubling the representation starting with a PCT conjugate Clifford vacuum with helicity N/2 - X. In this way one gets a set of 2N+i particle and antiparticle states. The representation with lowest helicity states contain particles with helicities from X = 0 up to X = N/4(N + 1/4) for N odd). Moreover, for a Clifford vacuum of helicity X = N/4 (and N multiples of four), the PCT doubling is not needed and the representation is PCT self-conjugate. This phenomenon occurs in N = 4 and N = 8 theories, for Clifford vacua with helicity X = 1 and 2, respectively. The very fact that the maximal helicity state in a given supermultiplet cannot be lower than N/4 gives a bound for the possible supersymmetric theories describing quanta up to spin two. The maximal algebra containing at most a spin-2 massless state, to be identified with the graviton, is the N = 8 extended algebra. The maximal algebra with representations containing helicities up to one is the N = 4 algebra. Finally the maximal algebra containing multiplets with spin 0 and \ is the N = 2 algebra. When gravitational interactions are absent, interacting supersymmetric theories exist up to N = 4. In these theories gauge Yang-Mills symmetries can be introduced provided they commute with the spinorial charges Qla. This fact implies that particles

(2.9)

34 within the same supermultiplets transform in the same representation of the gauge internal symmetry. The major consequence of this assignment of internal quantum numbers is that chiral fermions can occur only in supermultiplets in the case of the N = 1 supersymmetry algebra. The chiral representation is the Wess-Zumino multiplet (with spin content 0 + , 0", J) with the particles in a complex representation of the internal symmetry group. In the N = 2 case, although the Clifford vacuum can be in a complex representation of the internal symmetry group, the application of two supersymmetry charges to this state transforms it into a mirror particle (rather than the antiparticle) with opposite helicity but in the same complex representation (rather than the complex conjugate one). In the case of massive representations, the situation is quite different. In the absence of central charges, the smallest representation of the algebra given by Eq. (2.7) has 22N particles with spin range s = 0, s = £, . . . , s = N/2. However, for N > 1, a multiplet shortening can occur in the presence of central charges. Massive multiplets with non-vanishing central charges can have particles with spins ranging from s = 0 up to s = N/4 in N = 2 and N = 4 supersymmetric theories. These multiplets naturally occur in extended supersymmetric Yang-Mills theories where a sponta­ neous breaking of the gauge symmetry occurs (see Refs. 9-11 in Chapter III).

References 1.* A Salam and J. Strathdee, "Super-gauge Transformations," Nucl. Phys, B76, 477 (1974). 2.* S. Ferrara, J. Wess and B. Zumino, "Supergauge Multiplets and Superfields," Phys. Lett. 51B, 239 (1974). 3. References 5, 11, 13, 14 in the Preface. 4.* A. Salam and J. Strathdee, "Unitary Representations of Super-gauge Symmetries," Nucl. Phys. B80, 499 (1974). 5. M. Gell-Mann, A.P.S. Washington Meeting (1974), unpublished. 6.* R. Haag, J. Lopuszanski and M. Sohnius, "All Possible Generators of Supersymmetries of the S-Matrix," Nucl. Phys. B88, 257 (1975). 7. W. Nahm, "Supersymmetries and their Representations," Nucl. Phys. B135, 149 (1978). 8. M. Gell-Mann, P. Ramond and R. Slansky, "Color Embeddings, Charge Assignments, and Proton Stability in Unified Gauge Theories," Rev. Mod. Phys. 50, 721 (1978); M. Gell-Mann in Ref. 6 of Preface. 9. S. Ferrara, C. A. Savoy and B. Zumino "General Massive Multiplets in Extended Supersymmetry," Phys. Lett. 100B, 393 (1981). 10. S. Ferrara and C. Savoy, "Representations of Extended Supersymmetry on One- and Two- Particle States," in Ref. 8 of the Preface, p. 47; D. Z. Freedman, in Recent Developments in Gravitation (Cargese, 1978), eds. M. Levy and S. Deser, Gordon and Breach, 1979.

35 Nuclear Physics B76 (1974) 4 7 7 - 4 8 2 . North-Holland Publishing Company

SUPER-GAUGE TRANSFORMATIONS Abdus SALAM International Centre for Theoretical Physics, Trieste, Italy and Imperial College, London, England

J. STRATHDEE International Centre for Theoretical Physics, Trieste, Italy Received 26 February 1974

Abstract: A systematic method for constructing Wess-Zumino supergauge transformations is exhibited.

In a recent article Wess and Zumino [1] have invented an interesting new symme­ try. Generalizing from the dual model super-gauge symmetry [2] these authors suc­ ceeded in defining an analogous transformation group in four-dimensional spacetime. This invention is quite remarkable in at least two respects: (i) the irreducible representations of this symmetry combine fermions with bosons and (ii) the stric­ tures of O'Raifeartaigh's theorem are circumvented — we seem to have here a relativistic spin-containing symmetry which is consistent with unitarity*. Moreover, in a simple Lagrangian model involving two scalars and a Majorana spinor, Wess and Zumino found that, in the one-loop approximation, there is only one (logarithmic) divergence [3]. The purpose of this paper is to present a rather simple method which can be used for the construction of at least some of the representations of this symmetry. We shall confine our considerations to the 14-parameter subalgebra of the Wess— * The group of Wess and Zumino can be looked upon as a sort of quasi U(2,3): the set of unitary 5 X 5 matrices,£, whose elements^, a,0 = 1, 2, 3 , 4 , and£5 are ordinary complex numbers while gj and gf are anti-commuting c-numbers. The subgroup SU(2,2) X U(l) of the ordinary sort is identified with the product of the 15-parameter conformal group of space-time and a 1-parameter group of ys transformations. The anticommuting parts are identified with supergauge transformations. Looked at in this way an immediate generalization to U(2,4) (or U(2,S)) is suggested. The ordinary subgroup SU(2,2) X U(2) (or SU(2,2) X U(3)) might then be said to include a strictly internal SU(2) (or SU(3)) symmetry. Contrasted with this marriage of in­ ternal symmetries with space-time symmetries, one may also consider a rather trivial generali­ zation of Wess and Zumino's work where each one of their fields is considered as (for example) the adjoint representation of an internal symmetry U(w).

36 478

A. Salam, J. Strathdee, Super-gauge transformations

Zumino system which is generated by the Poincare operators J^P^ and the Majorana spinor Sa. In addition to the usual commutation rules involving/^ and Pp only, we have

[^■y-JMiv

o)

where C denotes the charge conjugation matrix*. The last of these rules can be ex­ pressed in the alternative version [f 1 5 > 5€ 2 ]=-e 1 7 M 6 2 /> M ,

(2)

where €j and e 2 are two arbitrary Majorana spinors which anticommute with one another and with 5. (Notice that IS = Se is a hermitian operator and that e j7^e2 = — ^2%i6l *s a n *maS*nary 4-vector*.) Our basic approach is to work out the group action on the space of left cosets with respect to the subgroup of homogeneous Lorentz transformations. This "space" is essentially eight-dimensional, being parametrized by the 4-vector *M and the (anticommuting c-number) Majorana spinor 0a. A simple way to obtain the group action on this homogeneous space is to define the unitary operators /.(*. 0) = exp [ixMPM] exp [HPSJ ,

(3)

and consider what happens to them when any one of the operators representing, respectively, a translation, a homogeneous Lorentz transformation or a super-gauge transformation is applied on the left. One finds,

^v[ic/^L{xfe)

= L(x + c, 0),

exp [Jic^y^] L(xt 8) = L(Ax, a(A)d) exp [J/co^/J , exp [ieS] L(x, 6)

(4)

=L [x^ - \feyj, 6 + e} ,

where a (A) = exp^Ocj^a^ denotes the usual spinor representation of the homo­ geneous Lorentz group**. (Notice that, because of the Majorana constraints on€ * Our notational conventions are as follows. The Dirac matrices satisfy (j) {7 , 7„} = q» y = = diag (+ ) and adjoint spinors are defined by $ s $' 7 0 . The matrices 70,7^7^i» 7oauv* y y oi?H s> ^o^s a r c hwmitian. The charge conjugate oftf/is defined bytf/c= C^* where C* = -Cand C~l y^C- - 7 J . By a Majorana spinor we mean tf/c = ^. It is useful to remember that the matrices y'^Cand aM„C = (\i) [7„, yv)Care symmetric while C, ysCand iy„ysC are antisymmetric. In particular, it follows that ^fx^7 ~ ^2^i» ^1*12^2 = ~?2^11^ i»^iaiiy^2 ' = - ^2a/iy^i» * i ' V ^ * = ^ ' V ^ s ^ i * ^^5^2 = ^2">s^i " * i ^ d V 2 are anticommuting Majorana spinors. ** Space reflections are incorporated by requiring that $ transform according to the rule 6 -» -+ iy06. Likewise for dilations, JT -♦ \xt $ — \ie and 75 transformations, B -+ (cos a + 75sina)0 with real or.

37 A. Salam, J. Strathdee, Super-gauge transformations

479

and 0, the displacement in *M caused by a super-gauge transformation is real.) Eqs. (4) serve to define the action of the group on the space of the parameters x and 0 and indicate how any field defined over this space should transform. Thus, for ex­ ample, the scalar super-field 4>(x, 0) should satisfy exp[/eS] 4>(x, 0)exp[-/?S] = 4>[x~ ^fey^d, 0 + e] .

(5)

By appending a Lorentz index* one could define a vector super-field *M(x, 0), a spinor super-field ^ a (x, 0), etc. If we were dealing with an arbitrary group then we should not be very pleased with fields defined on an eight-dimensional space-time. It was this aspect of the old attempts at combining internal symmetries in a non-trivial way with the Poincare group which hindered their development. The truly remarkable and exciting feature of the Wess-Zumino group is that the superfield (x, 0) in eight dimensions is exactly equivalent to a 16-component set of ordinary fields in four dimensions. One simply has to expend 4> in powers of 0 a and observe that the series must term­ inate in the fourth order. This is due, of course, to the fact that the monomials

must be completely antisymmetric and therefore vanish for n > 4. Therefore we can write ^ , 0 ) = 0(x) + 0 a (x)0 a + ^ ^ ) ( x ) 0 ^ 0 a

+ i* | f l W W» 7 V. + n # | f t W l W « t

(6)

The number of independent real components involved here can be halved by im­ posing the reality condition

* ( * . * ) • «*(x,0), which reads, in terms of the component fields,

4>(x)

=*00,

The super-gauge transformations induce no Lorentz transformation.

(7)

38 480

A. Salam, J. Strathdee, Super-gauge transformations

where the barred quantities are defined in the usual way, 0(x) = 0(x)*, 0 a (x) =

laf>l

=♦JWCTO);. *

w - v

(10)

If the reality conditions (8) are imposed then all boson components are real and the fermion components \j/ and A are Majorana spinors. The axial-vector field aM is constrained to be transverse, d^a^ = 0. The transformation rules (9) now take the form* 6A

=etp,

SB =e-ys\l/- —

eysi\,

* Some of the details in these rules are affected by conventions in the definition of C. Our C is defined such that (\Uaf)y6CyS = - (C~l)a^ (4)e 0 ^*(7 s CL 8 = + U T S S ) 0 * * and ( i ) e ^ * ( « 7 M 7 s 0 7 6 = + = X = * = 0. The combining of representations into products is, at least in some cases, quite easy. One simply multiplies the super-fields. The detailed combinations of compo* We have adapted our notation here to that of Wess and Zumino, ref. [ 1 ]. Thus, our eq. (11) with X, a and D set equal to zero corresponds to their eq. (8) with d^or = 0. Similarly, our (12) corresponds to their formulae on p. 48.

40 482

A. Salam, J. Strathdee, Super-gauge transformations

nents will be revealed by expanding the result in powers of 0. For example if

then, using the components defined by (6),

V

=*1*2,

^f

= iff

+ *« ^ - ^ ««+0f *2,

*«** -♦,♦«'* + S [♦?*? + *f*Jl + *f 7*2 » eye.

?f7* = ♦,♦$** + £ * ? ♦f1* + £ * f ♦£ + S * ^ 0$+0f 76 «2 • (16> In particular, with * j = * 2 satisfying #„„ = 0, it is a simple matter to show that ^«07« = ei transforming as in (14), one can construct another one by taking its derivative with respect to 0, since 0 does not enter in the differential operator (14). Similarly, the differentiation with re­ spect to 0 is an invariant operation on a superfield $ 2 transforming as in (15). Since j and 4>2 are related by (8) one sees that there are two invariant differen­ tiation operators on each type of field. They are

(8) Operating from the left with the group element

5 August 1974

*r5 + 2 i o M*£ ( 1 , n d - £ and

*ri

°7)

-&- 2 i f l 0 M37;( 1 8 )

In particular, on a field of the type i one can impose the invariant constraint that it be independent of 0, or linear in 0 or a general function of 0 (which means qua­ dratic in 0, since the square of each component of 0 vanishes). These three cases correspond to three types of multiplets, of which the first (together with its com­ plex conjugate) gives the scalar multiplet of ref. [1 ] and the third (which by a shift can be changed to type 4>) is the vector multiplet of ref. [1J. The second case corresponds to a new type of multiplet, which we may call the "linear'' multiplet. It contains two scalars, three spinors and one vector, all complex. This new kind of multiplet, like the scalar multiplet, is inequivalent to its complex conjugate, while the vector multi­ plet is equivalent to its complex conjugate. Superfields do not need to be scalar functions of JC, 0 and 0 but can have dotted and undotted spinor indices. The in­ variant operations of differentiation given by (16)(18) endow the superfield with one additional spinor index. For the sake of clarity we give now a few examples. Let ! be independent of B *&.$)

=^W+0^aW+0aea/F(jc),

(19)

then (14) implies M»W,

5* = 2ia M F^->I+2r/s

6F--igoJ,

(20)

43 Volume 51B, number 3

PHYSICS LETTERS

the transformation law for a scalar multiplet 3 . From (19) one can obtain another superfield of the same type by complex conjugation, shift and double differentiation

£

*i - - ^ h *H*,-™;'&

(20

It is easy to check that its components are A'=F*t

*' = _ i a M ^ -

ft

F ' = DA*.

(22)

The equations of motion for a self-interacting scalar multiplet investigated in the papers of ref. [2] can be written in a manifestly supergauge invariant way as

+ m* 1 (jc | | ,a) + 2 ^ J ( j c | < f a ) = 0.

(23)

The components of this superfield equation (plus those of the complex conjugate) give the equations of mo­ tion of the individual fields of the scalar multiplet. According to (8), ^ ( x ^ + i0a M 0,0) is a vector multi­ plet constructed from the scalar multiplet * J ( J C , 0 ) . It has been called previously the gradient of the scalar multiplet because its vector component is the gradient of the scalar A occurring in (19). Another example is the construction of a vector multiplet from two scalar multiplets $ t ( x , 0 ) and $ 2 ( x >^) o{ tyP^ o n e a n d t w o

As a special case one can take 2 = &l and then (24) is real. This construction, together with the gradient operation described above, has been used extensively in ref. [ 3 ] . It is quite striking how the relations (7) and (8) of that paper can be proved almost trivially in the superfield formalism, while their proof using ex­ plicitly the fields of the multiplets is rather lengthy. As further interesting examples we mention the sca­ lar, the linear and the vector multiplet each with an un­ do t ted spinor index. Together with their complex con­ jugates, these superfields correspond to the following real field multiplets. The first, * a ( x , 0 ) , contains two 3

In the notation of ref. [1 ], the fields A and F would be called (A - iB)/2 and (F+iG)/2, respectively. The two-com­ ponent spinor 4> consists of the upper components of the spinor \(i of ref. [1 ] , in a representation in which ys is diag­ onal

5 August 1974

Majorana spinors, a scalar, a pseudoscalar and an anti­ symmetric tensor. One can impose on it the supergauge invariant constraint

-^VI*A+2i^M) ='("M3*. (25) which implies that the scalar is constant, the two spi­ nors are related by the Dirac operator and the tensor is the curl of a vector. The product $ a ( x t 0 ) * ° ( x , 0 ) of two such representations contains the free Lagrangian for a vector and for a Majorana spinor (see eq. (14) of ref. [1J). The second, 4> a (x,0,0) (linear in 0) contains two Majorana spinors, a scalar, a pseudoscalar, two vectors, two pseudovectors, an antisymmet­ ric tensor and a Majorana vector-spinor. This multiplet is contained in the decomposition of the product of two scalar multiplets. Finally, the third, 3> a (x,0,0) contains four Majorana spinors, two scalars, two pseudoscalars, two vectors, two pseudovectors, two antisymmetric tensors and a Majorana vector-spinor. It is contained in the product of two vector multiplets. The compactness of the superfield notation simpli­ fies considerably calculations and proofs. We leave it to the reader to show that, given any two multiplets constructed with the operations described above, the product of any field of one by any field of the other can be expanded as a linear combination of fields be­ longing to some multiplets. The proof would be ex­ tremely tedious and almost unmanageable if one used the field multiplets explicitly.

References (1) J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. (2) J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52; J. Iliopoulos and B. Zumino, CERN preprint TH. 1834 (1974), to be published in Nuclear Physics B; S. Ferrara, J. Iliopoulos and B. Zumino, CERN preprint TH. 1839 (1974), to be published in Nuclear Physics B; A. Salam and J. Strathdee, Trieste preprint IC/74/17 (1974), submitted to Physics Letters B. {31 J. Wess and B. Zumino, CERN preprint TH. 1857 (1974), submitted to Nuclear Physics B. (41 S. Ferrara, CERN preprint TH. 1824 (1974), to be pub­ lished in Nuclear Physics B.; A. Salam and J. Strathdee, Trieste preprint IC/74/16 (1974), submitted to Physical Review Letters. (51 A. Salam and J. Strathdee, Trieste preprint IC/74/11 (1974), to be published in Nuclear Physics B.

241

44 Nuclear Physics B80 (1974) 4 9 9 - 5 0 5 . North-Holland Publishing Company

UNITARY REPRESENTATIONS OF SUPER-GAUGE SYMMETRIES Abdus SALAM International Centre for Theoretical Physics, Trieste, Italy, and Imperial College, London, England

J.STRATHDEE International Centre for Theoretical Physics, Trieste, Italy Received 14 June 1974

Abstract: A method is given for constructing some of the unitary irreducible representations of the Wess-Zumino super-gauge symmetry. Application of this symmetry to the analysis of ^-matrix elements is considered. A new super-gauge symmetry which includes isospin is in­ troduced and some of its representations are constructed.

The super-gauge operations invented by Wess and Zumino [ 1 ] have the remark­ able property of transforming bosons into fermions and vice versa. These authors define such transformations on a multiplet of fields which is then*built into an in­ variant Lagrangian. The absence of ghosts (in the perturbative development at least) shows, among other things, that the symmetry is consistent with unitarity. Indeed, it is possible to proceed directly to the construction of unitary representations. This we shall do in the following. The super-gauge symmetry of Wess and Zumino may be looked upon as the first example of a unitary and relativistic spin-containing theory. Particles with distinct intrinsic spins are here combined into irreducible multiplets. In this letter we present a generalized super-gauge symmetry which contains isospin as well. It is not a rela­ tivistic version of Wigner's SU(4), though it resembles it. It appears to be a potent new symmetry, though we do not speculate at present on its usefulness in particle physics. The full super-gauge symmetry involves dilatations, conformal and 7 5 transforma­ tions. In a recent note [2] we considered the more easily manageable subalgebra which is generated by the Poincare operators J' , P and the Majorana spinorS^ for which we adopted the following algebra:

[sa,ptt]=o, [sa,.V] = Kv)aV tf«,V

=

-(VVi.-

0)

45 500

A. Salam, J. Strathdee, Super-gauge symmetries

Here C denotes the charge conjugation matrix* and the spinor Sa is constrained to be real in the sense,

stt = caftse.

(2)

Under space reflections it transforms according to**

Sa-/(70)£V

(3)

The above-mentioned generalized super-gauge symmetry is obtained by replacing Sa with the isospinor Sai (/ = 1,2) for which a modified anticommutator is postu­ lated.

{Sai9Sfi,} = eyOT^CV^ .

^

The space reflection rule (3) is unchanged but the Majorana constraint (2) is re­ placed by the SU(2) covariant form Sai = ieij(l5C)aeS^.

(5)

The construction of unitary representations begins with the observation that super-gauge transformations must leave invariant the manifold of states with fixed 4-momentum since Sa commutes withPM. On this manifold the anticommutator {Sa, Sp} becomes a fixed set of numbers and we see that the operators generate a Clifford algebra. Since this algebra has just sixteen independent members, its one and only finite-dimensional irreducible representation is in terms of 4 X 4 matrices [3]. As we shall see, these matrices are hermitian (in the sense of (2)) when pM is timelike. In this case the super-gauge transformations serve to resolve the manifold of states with fixed pM into four-dimensional invariant subspaces. If pM is light-like, these four-dimensional spaces involve two states of positive norm and two orthogonal states of zero norm. One can contemplate a complete classification of the unitary irreducible repre­ sentations according to whether the four-momentum is timelike, lightlike, space­ like or null. In all but the first case one would in general meet infinite-dimensional representations of the Clifford algebra***. Here we shall be dealing exclusively with timelike representations. In addition to the operators Sa there are, of course, the well-known rotations of Wigner's little group which leave invariant the manifold of states with fixed p^. The * Our notational conventions are as follows. The_Dirac matrices satisfy \ {7^, yv} = r\^v = diag (+ ). Adjoint spinors are defined by $ = ^ + 7o- The matrices 70, 7o7 M , 7oaMi>> 7o'7/i7s> 7075 a r e Hermitian. The antisymmetric matrix C defined by C^y^C = - 7 T is real. ** For consistency with the Majorana constraint the factor i is necessary. *** Such representations are relevant for group theoretical analysis in the crossed channels. See for example ref. [4].

4A

A. Salam, J. Strathdee, Super-gauge symmetries

501

generators of these rotations, taken together with the Clifford elements, Sa, S^Sp], etc., span an algebra which might be called the little algebra. The basic problem is to find the unitary irreducible representations of this algebra. These representations are in one-one correspondence with the unitary irreducible representations of the full group. Similar considerations apply to the generalization (4), (5) where the irre­ ducible representations of the timelike Clifford algebra are sixteen-dimensional. In the rest frame, p 0 = M, p = 0, the little algebra is generated by the angular momentum operators J and the Sa which here obey the rule {Sa9Sfi} = -M{y0C)afi.

(6)

In a basis where 7 0 js diagonal and C is real, the Majorana constraint (2) implies 5 4 = -S\, 5*3 = $2 and the anticommutators (6) can be expressed in the suggestive form: {Sa,Sb} = 0,

{5a+,^} = 0,

{Sa,St}=M&ba,

(7)

where Sa is a two-component spinor under space rotations and transforms accord­ ing to Sa-+iSa

(8)

under reflections. Viewed as creation and annihilation operators, the spinors Sa and S* can be used in the familiar way to set up a four-dimensional "Fock space" with positive metric. The procedure is as follows. Choose a set of 2 9 + 1 vectors 19, 9 3^ ~~9 ^ 9l ^ 9 > t 0 represent the states of a particle at rest with mass M and spin g. Let these states constitute the "Clif­ ford vacuum", i.e.

s«l2.9 3 > = 0-

(9)

Define the orthonormal vectors

l29 3 n,ii 2 >=tf-* ( " ,+B2) sWlSW.

0°)

with the pair {n^, n2) taking the values (0,0), (0,1), (1,0) and (1,1). These states span a 4(2 9 + l)-dimensional irreducible representation of the little algebra*. The (spin)Paritv content of the multiplet is ( 9 ~ \y*, 9 / T », 9 _/r >, ( 9+ £)*», where T? takes one of the values ± / (for integer 9 ) or ± 1 (for half-integer 9)Basis vectors for the representation of the full group are obtained in the usual way by applying Lorentz boosts to the rest frame states (10), * The three-vector operator 9 = / - (2M)~l S+aS commutes with the Clifford elements and satisfies the commutation rules of SU(2). It coincides in the rest frame with the transverse part of the generalized Pauli-Lubanski vector, K^ introduced in ref. [2]. The singlet (O)2 is a Casimir operator. Its eigenvalues 9^ 9 + ^ s e r v e t 0 ^ ^ t n e ifreducible representations.

47 502

A. Salam, J. Strathdee, Super-gauge symmetries

ipgg^n^-uiLp)^^^),

(ii)

where L denotes a three-parameter boost which carries the 4-vector (Af, 0) into p . The behaviour of these states under super-gauge transformations is easily obtained. Apply S& o n b ° t n sides of (11) and take it through the operator U(Lp) using the knowledge that it transforms as a Dirac spinor under the Loreritz group. One finds Sa\p$)=7}\pl;'HS'\Sa(p)\$),

(12)

where £ denotes the set of labels in (10) and ^\S0i(pW = a^Lpn,\S^)i

(13)

where a^(Lp) denotes the spinor representation of the Lorentz transformation Lp. The p-dependence of the 4(2 9 + l)-dimensional matrix Sa(p) is thereby given ex­ plicitly. In defining the action of Sa on two-particle states there is one subtle point which must not be overlooked. Suppose we take5 Q =5^ 1 ) +5^ 2 ) where 3 it is necessary to discard the Majorana constraint and treat the generators S^P as independent. The fundamental anticommutators would take the form:

In this case the smallest representation has 2 4 " components since there are now An independent anticommuting creation operators S . For the case n = 3, the rest frame algebra contains SU(12) which can therefore be used to classify the 2 1 2 =4096 states. One finds all the antisymmetric tensors of SU(12). In terms of dimensions the SU(12) decomposition reads: 212= 1 + 12 + 66 + 220 + 495 + 792 + 924 + 792 + 495 + 220 + 66 + f2+ 1 . In this multiplet one finds spins up to the value / = 3. Although the examples sketched here may not be realistic, they do, at least, show that algebraic generalizations of the Fermi-Bose symmetry are possible. It now be­ comes important to search out the economical ones.

50 A. Salam, J. Strathdee, Super-gauge symmetries References [1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39; Phys. Letters B49 (1974) 52. [2] Abdus Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477. [3] J.M. Jauch and F. Rohrlich, Theory of photons and electrons (Addison-Wesley, 1955), appendix A2, p. 425. 1955), [4] J.F. Boyce, J. Math. Phys. 8 (1967) 675; A. Sciarrino and M. Toller, J. Math. Phys. 8 (1967) 1252.

505

51 Nuclear Physics B88 (1975) 257-274 © North-Holland Publishing Company

ALL POSSIBLE GENERATORS OF SUPERSYMMETRIES OFTHES-MATRIX Rudolf HAAG* CERN, Geneva

Jan T. LOPUSZA1SISKI** Institute for Theoretical Physics, University of Wroclaw

Martin SOHNIUS* Institut fiir Theoretische Physik, Universitat Karlsruhe Received 15 November 1974

A generator of a symmetry or supersymmetry of the S-matrix has to have three simple properties (see sect. 2). Starting from these properties one can give a complete analysis of the possible structure of the pseudo Lie algebra of these generators. In a theory with non-vanishing masses one finds that the only extension of previously known relations is the possible appearance of "central charges" as anticommutators of Fermi charges. In the massless case (disregarding infrared problems and symmetry breaking) the Fermi charges may generate the conformal group together with a unitary internal symmetry group.

1. Introduction We have chosen the title to indicate the close relationship of this study to ref. [1]. The results of the latter paper were generally accepted as the last and most powerful in a series of "no-go" theorems, destroying the hope for a fusion between internal symmetries and the Poincare group by a relativistic generalization of SU(6)< Recently Wess and Zumino discovered field theoretical models with an unusual type of symmetry (originally called "supergauge symmetry" and now "supersym­ metry") which connects Bose and Fermi fields and is generated by charges trans­ forming like spinors under the Lorentz group [2]. These spinorial charges give rise to a closed system of commutation-anticommutation relations, which may be called * On leave of absence from II. Institut fiir Theoretische Physik, Universitat Hamburg. ** Supported in part by the German Bundesministerium fur Forschung und Technologie and by CERN-Geneva. * Supported by Studienstiftung des Deutschen Volkes.

52 258

R. Haag et al. / Supersymmetries

a "pseudo Lie algebra"*. It turns out that the energy-momentum operators appear among the elements of this pseudo Lie algebra, so that in some sense a fusion be­ tween internal and geometric symmetries occurs [2,4]. The possibility of supersymmetries was not envisaged in [1]** but most of the ideas in [1] apply also to this case. We shall use them to determine all supersym­ metry structures which are allowed in a theory without zero-mass particles and longrange forces. An equally satisfactory and complete discussion of the zero-mass situation is not attempted here. However, the simplest case in which there is no infrared problem and no degeneracy of the vacuum will be treated in sect. 5. The allowed supersym­ metry structure is then much more interesting than in the massive case, because it gives a complete fusion between internal and geometric symmetries which is fur­ thermore essentially unique. In assessing the results, one should bear in mind that the scope of this investiga­ tion is limited in three directions: First, we deal only with visible symmetries, i.e., with symmetries of the S-matrix; the fundamental equations may have a higher symmetry. Second, all impossibility statements below have to be reexamined when infrared problems or vacuum degeneracy occur. Third, we assume that each mass multiplet contains only a finite number of different types of particles. This again is eminently reasonable in the massive case, but there is one interesting alternative, namely to use the supersymmetry structure in the context of an idealization which assigns zero mass to all particles. If the total number of particle types is infinite, then this idealization is not covered here. With these limitations in scope being un­ derstood, the conclusions of our analysis may be summarized as follows: the most general pseudo Lie algebra of generators of supersymmetries and ordinary sym­ metries of the S-matrix in a massive theory involves the following Bose type oper­ ators: the energy-momentum operators PM; the generators of the homogeneous Lorentz group A/M„; and a finite number of scalar charges B\. It will involve Fermitype operators, all of which commute with the translations and transform like spinors of rank 1 under the homogeneous Lorentz group. Using the spinor notation of var^der Waerden*, we may divide them into a set Q% (L = 1,..., v\ a = 1, 2) and a set 2^, indicating the different transformation character by dotted or undotted indices. Since the Hermitian conjugate of a supersymmetry generator is again such a generator, we can choose the basis of the pseudo Lie algebra so that QL = (Q%)t,

* The spinorial charges may be considered as generators of a continuous group whose parameters are elements of a Grassmann algebra [3]. '* Spinorial charges were considered but prematurely discarded in [5]. * €ap = -*(&> ^ = -****' * 1 2 = *2i = 1 (same for dotted indices). (o»)afc = (l.of); &*$# = (1, - a , ) . a^v = \i(o*iop - ovo*) and a^v = \i(a^ o"-ov o»). A quick orientation about this formalism may be obtained from [6]. The conventions adopted there are, however, slightly different from ours.

53 R. Haag et al / Supersymmetries

259

which is equivalent to a Majorana condition in a four-spinor notation. The algebra of these quantities can be reduced to the following form*: {QLa,Qf}-ea^{al)LMB^e^ZLM,

(1.1)

[ZLM,G]=0

(1.2)

where

for all G in the pseudo Lie algebra, G2f. Q1$} = i>LMo\i)Ptl,

(1.3)

[), except for v = 4 where it may be either U(4) or SU(4).

2. Assumptions and basic facts A generator of a symmetry or supersymmetry of S is any operator G in the Hilbert space of physical states which has two properties: (i) it commutes with the S-matrix; (ii) it acts additively on the states of several incoming particles. The second of these requirements can be most conveniently expressed in the following way: Let affin\p) denote the creation operator of an incoming particle of type / with momentum p and spin orientation r and a\*rn\p) the corresponding destruction operator*. Then G = TJ /d 3 pd 3 p'afi(p')K i t ; l r (p',

p)a£(p),

(2.2)

r,s

where AT is a onumber kernel. If in the sum over particle types only pairs (/,/) occur which refer to particles of equal statistics (both Bose or Fermi), G is of Bose type and generates an ordinary symmetry; if only pairs (/,/) with opposite statistics occur, G is of Fermi type and generates a supersymmetry. Since the ^-matrix conserves statistics, both the Bose part and the Fermi part separately have the property (i), so that we can always take G to be either of pure Bose or of pure Fermi type. The justification of the requirements (i) and (ii) can be given in a variety of ways. In the frame of a local field theory, the more fundamental requirement would be that G induces an infinitesimal transformation of the basic field quantities 4f -* \j/ + edG\lj such that** * We use the canonical normalization [*jf W « $ i n ) < W ] -^ij^s^iP-Q)

•

(2.1)

** In (2.3) we denote the anticommutator by [...]+, while elsewhere curly brackets {...}are used.

_55 R. Haag et al. / Supersymmetries

SG(Hx)) = i[G,Hx)]±,

261

(2.3)

is again local. On the right-hand side of (2.3) the anticommutator occurs when both G and \J/ are of Fermi type, the commutator in all other cases. The usual way to con­ struct such a G is to start from a conserved local current, dM/*(x) = 0 ,

(2.4)

and define

G=J

d3x/°(x).

(2.5)

x0 = t

Due to (2.4) this yields the same operator for an arbitrary choice of the time f, and hence &Q(\IJ (X)) is local for arbitrary x. Note that no assumptions about the covariance of/M(JC) under the Poincare group have to be made at this point, i.e.,/M(x) need not be a four-vector field, and it may depend not only on the basic fields at the point x but also explicitly on the position coordinates*. In a massive theory where the assymptotic relations as described in [9] hold, the locality of &Q\I/(X) implies that G has the properties (i) and (ii) (compare [10-12, 5]). In fact it also implies a third property: (iii) G connects only particle types (/,/) which have the same mass. The kernel K in (2.2) is of the form K{p\p) = Tj K(n\p)iT

8(p-p'),

(2.6)

n

where dn stands for a monomial in the derivatives d/bpj and the sum has a finite number of terms. It is interesting to note that property (iii) follows on the one hand from the locality of 6Gi// as indicated in [10-12, 5]**, and that it can also be derived from the properties (i) and (ii) and the assumption that the S-matrix is not trivial. This was done in [1]. We now look first at those generators which commute with the translations. Let us call the set of these6 \. Thus the mentioned anticommutator must vanish unless the original Q was a spinor (j,0) or (0, \). However, QQ* + Q' Q = 0 implies 2 = 0. Hence no spinors of rank higher than 1 are allowed and for a spinor Qa of rank 1 [representation (5, 0)] we must get: * Actually pt p should be so chosen that the elastic scattering does not vanish for a pair of particles with these momenta. ** Note that the property (ii) is conserved for the anticommutator (not for the commutator) of two Fermi-type generators. This is the reason why the last part of the analysis in [ 1 ] is not applicable to the Fermi-type generators.

57 R. Haag et al ISupersymmetries

263

since (5, | ) is the spinorial description of a four-vector and the only four-vectors in c$(°) are multiples of I*. If Qa =£ 0 then c =£ 0. If we have several spinorial charges, say £?f > we must have [remember ()f = (QL)^]: {G^2f} =cIAfa"a/M,

(3.1)

where c ^ is a positive definite Hermitian numerical matrix, which can be brought into the form cLM = 8LM,

(3.2) L

LM

and normalization). Consider by suitable choice of the Q (diagonalization of c now {Q£, Op*}, it also belongs tocS (°). The antisymmetric part in the indices a, j3 is a scalar, which could be a linear combination of the internal symmetry generators £/. The symmetrical part belongs to the representation (1,0) and corresponds to a self-dual skew-symmetric tensor. Since there is no such thing in 6^\ the symmetric part must vanish and we can only have

«& Of)-«.,S («')"%

(3.3)

with ^LM

^ _^ML

( 3 4 )

Finally, the commutators [Bh Q%] belong to the (5,0) part of c5 (0) and therefore:

[(#,*,] = E^ef.

(3.5)

M

Relations (3.1)—(3.5), together with the structure relations of the ordinary internal symmetries

tMml-'?«£,**•

(36)

and the fact that PM commutes with all Bt and Qa, give the structure of the part of our pseudoLie algebra which lies inc5 W. The matrices a1, S/ are still restricted by the Jacobi identities*. The one involving

* If B denotes a Bose and F a Fermi operator, the relevant identities are: [B,{FhF2}]

+ {Flt lF2,B]}-{F»[B.Fi\}*0,

[Fu fa. FJ] + [F2. fa, Fx}] + [F 3 , {Fh F2}) = 0 .

(3.7) (3.8)

58 R. Haag et al. /Supersymmetries r'

264

BbQZAQ?)*

tells us that

SLM^-ML

(39)

i.e., that 5/ is Hermitian. The Jacobi identity involving 2?/, Bm, Q& tells us that the matricesfyform a representation of the Lie algebra of the /?/. The remaining three identities are (Q.Q.Q):

B(alfLsfM=ip(al)^sfM,

(3.10)

(Q,Q,Qi>

(al)KLsfIN = 0>

(3.11)

(0, Q, B).

(smal)KL

- (smal)LK

= / Ticlnm(an)KL

■

(3.12)

n

Eq. (3.10) is also a consequence of (3.11) with (3.9). For the analysis of the remain­ ing relations , we remember, that £ = £± © £2 where £± is semisimple, i? 2 Abelian. Consider the elements ZKL=H(al)KLBr

(3.13)

Eqs. (3.6) and (3.12) tell us that the linear span of the ZKL is an invariant subalgebra, say £ 3 C £. Using in addition (3.11), we see that [ZKLfZMN]=0.

(3.14)

Thus the intersection of £1 and £ 3 would have to be an Abelian, invariant sub­ algebra of £1 which does not exist because £± is semisimple. Thus £ 3 C i ? 2 , i.e., [ZKL,Bm]=0.

(3.15)

Finally, eq. (3.11) tells us that ZKL lies in the kernel of the representation s or, in other words, by (3.5), that [ZKL9QM]

= 0.

(3.16)

Taking (3.15) and (3.16) together, we see that £3 must be a part of the center X of the whole pseudo Lie algebra. This result has been stated above as eq. (1.2). We may write

£=£'oZ, where £' does not contain any central elements. We choose a basis Zp ini£, and we can replace the right-hand side of (1.1) or (3.3) by 2p€ap(af)Y'MZp. The matrices a** are still restricted by (3.12): putting there / = p and m arbitrary,

59 R. Haag et al. I Supersymmetries

265

the right-hand side of (3.12) is zero, because the structure constant then vanishes. Using (3.4), we can now rewrite (3.12) in the form sma" = a"tm,

(3.17)

where t

=-s (3.18) m m w- l v v is the complex conjugate representation of the Lie algebra. This means, that every matrix a** must be an intertwiner of the representations s and t of £'. This limits the number of central charges which can appear in (1.1) and (3.3). To give an ex­ ample: Suppose that £' is the Lie algebra of SU(2) and s its basic two-dimensional representation, so that we have two spinors Q% (L = 1, 2). Then the complex con­ jugate representation is unitarily equivalent to s and there is only one linearly in­ dependent intertwiner between s and f, namely a complex multiple of the matrix eLM Thug t h e r e c a n b e a t m o s t t w o r e a i c e n t r al charges Z j , Z 2 in the pseudo Lie algebra which are not completely trivial, and (1.1) becomes for this example

4. Symmetry generators which do not commute with P^ Let us call c J ^ the set of those symmetry generators for which the kernel (2.6) contains only derivatives up to (and including) the order N. An operator which be­ longs to cJ W but not to c5 OW-D w yi be called a symmetry of degree TV*. The analysis proceeds now from the following observations: (a) All symmetries commute withP 2 (sect. 2, property (iii))**. G]^6^N~^. OOIfGGcJWthen [PM, This is, because the commutation with PM conserves the three properties (i), (ii), (iii) and the order of the derivatives in (2.6) (or, alternatively, the degree of the polynomial of x in the current density) is lowered. In fact, the degree of a symmetry is lowered in all cases precisely by one. * Such operators arise typically if in (2.5)

/o^ZIKx,)"^^^), where fw are Poincare covariant fields and TV = sup L nv. ** This holds for the restriction of the symmetry operators to the single particle subspace [i.e. for the kernel in (2.2)], which we need to consider only and which we also denote by G. Furthermore, in the zero-mass case, the discussion refers to a multiplet of particles all of which are massless.

60 266

R. Haag et al. / Supersymmetries

(c)cJ W contains a finite number of linearly independent elements because G is fixed up to addition of elements from cJ W> by [P^, G]. cJ W is also closed under Lorentz transformations. We can then classify the ele­ ments of cJ (M according to their Lorentz transformation character (/, /'). 4.1. Bose symmetries of degree 1 Here [P^ G] €d. Since all Bose symmetries in d belong either to ($, j ) or (0,0), G can belong only to (£, f ), (0,0), (1,0), (0,1), (1,1). The last four can be combined into a general second rank tensor T^. The first is a vector KM. Then we must have [Pll,Vv)=gllvTiClBl, [Pp. T„v] = icPpg^ + ib+(gpiiPv -gp¥P„ +

ie^P")

~Vv ~ *,A - iepwP°) + "(Zp^v+gpA - ^pp)

+ ib

• (4-0

where the four terms on the right-hand side correspond respectively to the parts (0,0), (1,0), (0, 1), (1,1) of rM„ and cb c, Z>+, b~, a are numerical constants. The commutativity of G withi* 2 gives 0 = PllTiclBr

0 = /a(2/> M />,-j* MI ,/> 2 ),

O^icg^P2 .

Thus ct = 0 ,

a = 0 , and for

m2 ^ 0

also

c=0.

The second and third terms in (4.1) agree up to the numerical factors with the commutators between Pp and the (1,0) respectively (0, Imparts of the angular momenta M^v. Since &(& does not contain any skew tensor, the irreducible parts of rM„ - TVIJL are multiples of those of M^v. Thus we have the result (A) In the massive case all Bose symmetry generators of degree 1 are linear com­ binations of AfM„. (B) For zero-mass one may have in addition one scalar element DE6^ with commutation relations [/>„, £ > ] = / V (4.2) By this/) is fixed up to an additive scalar from c5 (0) . 4.2. Bose symmetries of degree 2 Since the Bose part of 6^ contains at most the covariants (0,0), (1, 0), (0, 1), the Bose part ofc$ can contain at most (5, 5), (f, \ ) , (j, §).

61 R. Haag et al. /Supersymmetries

267

The case (f, \) may be ruled out as follows (see appendix for definitions of PQL&*

Ma

a2

and their commutation relations):

(where the sum runs over the permutations of 7 l f 7 2 , 73), is the only possible covariant ansatz. This yields

Now this quantity should be symmetric under the simultaneous interchange (*! «* a 2 ,fr\** $i (Jacobi identity). Specializing to the choice Ji = 72 = 73 = 1, a! = a 2 = 2,7i = P\ = 1 and 0 2 = 2, this symmetry requirements becomes -12/*//>! r 0 ,

i.e.,

/=0.

A corresponding argument rules out the case (5, §). For a (5, | ) covariant K^ we have the ansatz [P,K]=ag

D + bM

+ce

,Af" x .

We look at the Jacobi identity involving ?M, Pv, Kp and find with the help of (4.2) that c = 0, b = -a. The existence of D =£ 0 is therefore a necessary condition for the non-vanishing of Kv. Result: (C) In the massive case there is no Bose symmetry of degree 2. (D) For zero-mass, if there is a D G 6^ then there may be a Kv G 6 (2\ whose commutator with PM may be normalized to [P^K^^liig^D-M^).

(4.3)

By (4.3), Kv is only fixed up to an additive multiple of />M. We shall make use of this freedom below. 4.3. Bose symmetries of degree N>2 By the same technique one finds (E) No Bose symmetry of degree 3 exists and hence no Bose symmetry of any higher degree. We only indicate the steps of the argument. A G of degree 3 can only be a com­ ponent of a general tensor A^v, whose commutator with Pp will be of the form (4.1) with K replacing P. Commuting again withP a , the result sould be symmetric in p, a (Jacobi identity). Evaluating the expressions, using (4.3), one finds that this requires the vanishing of all coefficients, i.e., G = 0.

62 268

R. Haag et al ISupersymmetries

4.4. Fermi charges of degree 1 Possible covariants are (\, 0), (1, \) and their conjugates. (1, \) is excluded by (1) t.IfO .... iiss of degree 1 then If G the following argument.

lP^Q{y%;i)-ae^(eayiQy2

+6^,) ,

where Q on the rkht-hand side is some (non-vanishing) Fermi charge in 6^°\ The condition [P2, Qy\\2^] = ° 8 ives 0 = a(P . 0 V

7i7^72

+P

.0 ),

727^7l''

and if we anticommute that with £? .+/> .P , ) .

7 i 7 72 w^~ "

(33)

'

once the transformation laws of the fields, ly 9^2 ~ $2* 3 ^ . There exists another way of combining two scalar multiplets to a vector multiplet, this one antisymmetric. We denote it by S± A 5 2 = - 5 2 A 5 j . It& components are

* In ref. [ 1 ] a larger supergauge (extended) group was considered which contains the conformal group as a subgroup. In the papers of ref. [3] and in the present paper the invariance group consists of restricted supergauge transformations having constant parameters, whose commu­ tator is a four-dimensional translation, and of Lorentz transformations. It is interesting to ob­ serve that the generators of the larger supergauge group appear to be in some sense the quan­ tum analogues of the classical entities which were called twistors by Penrose [4].

83 4

J. Wess, B. Zumino, Supergauge invariant extension

^'AxB2-A2Blt X =(A1+y5Bl)iP2-(A2+y5B2W1

,

M'=AlF2-A2Fl-BlG2+B2Gl N'=AlG2

,

- A2Gy +BXF2 - B2Fy ,

% = Al V * 2 - ^2 V 4 ! X' =(F2-ysG2Wl

+ B B

l\ 2

(4)

~ BlhBl

-(Fl-y5GlW2

+

" ^1^*2>

dti(A2-y5B2)y»+1-dii(Al-y5Bl)y»1J2

D' = 2F2Gl - 2FXG2 + 2bA2 ■ dflj - 2bAx ■ bB2-vi^xy5y

,

■ di//2 - i$2y5y ■ d\px .

We shall also need the symmetric composition law of two vector multiplets Vx and V2, of components Cj, Xi, v^i, Xj, Dx and C2, x2> etc., to another vector multiplet Vx- V2=V2- Kx. It is C = CjC2 , X^CjXj + C ^ , C

V

lV

+ C U

2 Ml - EXIT'S V 2 >

M' = CjJMj + C2MX - JJX175X2 .

x' = q x 2 + c 2 Xj - £7 • 9Cjx 2 - \y ■ 9C2Xi + |Mj7 5 x 2 + j^ 2 7 5 Xi + \NXX2 + 5^2*1 - K l ^ S ? " ^ - 5»M275r''X1 , £>' = CjD 2 + C2DX - 3Cj • 9C2 - Uj • u2 +MXM2 +NXN2 - ixfa - /'x2Xj +

21'9/iX17MX2 + ^3 M X 2 7 M X 1 .

(5)

This combination law is associative (Kj • V2) • V^ = V± • (K2 • F3). Finally we observe that, if 5 is a scalar multiplet of components A, 2?, i/>, F, G, we can construct from it a vector multiplet of components

D =0

' -

(6)

84 /. Wess, B. Zumino, Supergauge invariant extension

5

We denote this vector multiplet by bS. With the notations introduced above, it is easy to verify the validity of the two relations (SSj) X S2 - (SS2) XSX= 2(SX A S2) • 3 5 ,

(7)

(55)A51=-(51X51)-35,

(8)

where S, 5j and S2 are any three scalar multiplets. We shall use these relations in sect. 3.

3. Construction of the Lagrangian In this section we construct a Lagrangian describing the interaction of a vector and two scalar multiplets. We shall require that it be invariant under supergauge as well as ordinary gauge transformations. The two scalar multiplets can be thought of as the real and the imaginary part of a complex scalar multiplet, however, we prefer to keep the real description in this paper. An ordinary (infinitesimal) gauge transformation is given by 5yM = 3M A ,

S ^ = gA$2 ,

54,

5X = 5Z) = 0 ,

and for the scalar multiplets 8AX

=g(AA2-BB2),

8B1^g(AB2+BA2),

(11)

85 6

/. Wess, B. Zumino, Supergauge invariant extension

8*1 =g[(A-y5B)+2 Sfj

+ (A2-y5B2W]

,

=g(AF2+FA2+BG2+GB2-$\l>2),

5Gj

=g(AG2+GA2-BF2-FB2+$y5\p2),

5A2=-g(AAl-BBl), 8B2=-g(ABl+BAl), W2 = -g[(A-y5BWl+(Al-y5B1)M 8F2 =

,

-giAF^FA^BG^GB^i^J^^,

8G2=-g(AGl+GAl-BFl-FBl+$ys\IJ1).

(12)

If the Lagrangian is invariant under supergauge transformations and gauge trans­ formations, it is automatically invariant under the generalized gauge transformations given by (10) or (11) and (12). We now proceed to construct a Lagrangian having these properties. It will be a function of the coupling constant g which, foig = 0, must reduce to the sum of the free Lagrangians for the three multiplets V, S^ and 52- Our method will consist in constructing, with V, Sy and 52, a vector multiplet invariant under generalized gauge transformations. Its D component will then give a Lagrangian which is also invariant under supergauge transformations. First, with the scalar multiplets Sj and S2 w e construct the two vector multiplets Vl =

1

i(SlXSl

+S2XS2),

rn=5lA52-

(13) (14)

Under the generalized gauge transformation (10) they transform into each other 6 F, =g(SS2) X S{ - giSSJ X S2 = -2?(5 1 A S2) -dS = -7g Vu ■ dS ,

(15)

6 Vu=g(SS2) A S2-gS1A(SS1) = -g(s2 x s2 + Sj x Sj) • as = -ig vx ■ bs. Here we have made use of the identities (7) and (8). The combinations

(16)

86 7

J. Wess, B. Zumino, Supergauge invariant extension

^a = n + ^n.

0?)

transform simply as SV^-lgV^bS,

8Vb = 2gVb-bS.

(19)

Now, using the multiplication law (5) for vector multiplets, one can define the ex­ ponential of a vector multiplet. One sees from (10) that 8e*v = 2ge*v-dS,

5 e ~ * F = -2ge~*v - bS.

(20)

Therefore, we find two invariant expressions, since 6 ( F a - e ^ F ) = - ^ ( F a - 3 5 ) - e ^ F + ^ F a - ( e ^ K . a 1 S ) = 0,

(21)

and similarly 8(Vb-e-*v)

= 0.

(22)

We choose the particular combination l(Fa.e*»r+Fb.e-*»r),

(23)

because, for g = 0, it gives k(V!ltVh) = kVl,

(24)

whose D component is just the sum of the free Lagrangians for the scalar multiplets Si and ^2- Our invariant Lagrangian consists of the D component of (23) to which one must still add a mass term and the free Lagrangian for the vector multiplet V, which are by themselves gauge and supergauge invariant. It is easy to see that, with the restrictions imposed, our Lagrangian is unique. We do not write out the obtained invariant Lagrangian in detail. It is an infinite power series in the coupling constant g and contains all kinds of apparently non-renormalizable couplings. However, it is possible, by choosing a special gauge, to trans­ form it into a much simpler and tractable form. This will be done in sect. 4. 4. Special gauge Since the Lagrangian obtained in sect. 3 is invariant under generalized gauge trans-

87 8

/. Wess, B. Zumino, Supergauge invariant extension

formations, we can choose a special gauge. Now it is obvious that, by means of a (finite) transformation corresponding to (11), one can bring to zero the fields C, x, Mand TV of the vector multiplet V, so that only v^, X and D survive. Using (5) one sees then that, for the vector multiplet V2 only the D component survives, and equals —v2, while the higher powers Vn (n > 2) vanish identically in the special gauge. The expression (23) simplifies now to iC^+F^ +i ^ - F y - F + l ^ + F ^ - F

2

-kVl^gVll'V^g2Vl'V2.

(25)

The Lagrangian in the special gauge consists of the D component of (25), plus a mass term for the scalar multiplets, plus the free Lagrangian of the vector multiplet. Written out in full, it is L = -\ [ ( a ^ ) 2 + (^4 2 ) 2 + ( 3 i y 2 + (3* 2 ) 2 - F\ - F\ - G\ - G\ + /^17-3^1+/^27-3I//2]

+

m(F1Al+F2A2+GlBl+G2B2-\rt1\IJl-irt2\l>2)

- i X {(Al+y5Bl)+2

- 042+75*2)^}] - k V ( ^ 2 + 4 + £ 2 + ^ ) •

( 26 >

Since we have not fixed completely the gauge, it is still invariant under the ordinary gauge transformations (9). However, the supergauge invariance is no longer manifest, a consequence of the fact that the special gauge chosen has no invariant meaning. Nevertheless the Lagrangian (26) is invariant under transformations which combine a supergauge transformation (which violates the gauge condition) with a generalized gauge transformation (which re-establishes the gauge condition). In infinitesimal form these transformations can be written as 5=5S +5G,

(27)

where 6 S is the usual supergauge transformation on the fields given by (i =1,2)

8sArmt, dsBriaj5^i9

88 /. Wess, B. Zumino, Supergauge invariant extension

isFf =

9

107-9^,

and

8sD = iay5yd\,

(29)

while 5 G is the gauge transformation which re-establishes the special gauge, given by 0=1,2)

5 ^ = -^(aX^ 2 +o7 5 X5 2 +i; M a7 M i// 2 ) ,

5Gi//2=^1-75^)7-^, 5 G F 2 = ^(aX^ 1 +a7 5 X^ 1 +y M a7 M i// 1 ), 6GG2 = Wffy5Xi41-aX51+u'lflfy57M*1) ,

(30)

and 5

GyM=5GZ)

= 5

G

X = 0

-

(31>

These formulae are easily deduced from the supergauge transformation formulae and from the formulae (11)—(12) giving the generalized gauge transformation. Observe, for instance, that in the first of (29) the term *a3Mx is missing because one starts from the special gauge where x = 0. The commutator of two transformations of the type (27) is the combination of a four-dimensional translation and of an ordinary gauge transformation, but with a field dependent parameter. For instance

89 10

/. Wess, B. Zumino, Supergauge invariant extension

[ 5 2 , 5 1 ] ^ 1 = 2 i a 1 7 p a 2 V * l - 2igaiyf>

(35)

where 5

G*V

=

'3 M (* p *r P >0,

$GX

=

&GD

=

° '

and 6 ^ i = igap ayp\ m2 the Goldstone spinor is a linear combination of the fields X, y$ i//j and ^ 2 - *n this case gauge invariance is also spontaneously broken, the Higgs mechanism is operative and the vector field acquires a mass.

References [1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. [2] S. Ferrara, Nucl. Phys. B77 (1974) 73; A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477; Trieste preprint IC/74/16 (1974), Phys. Rev. Letters, submitted. [3] J. Wess and B. Zumino, Phys. Letters 49B (1974) 52; J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310; J. Iliopoulos, S. Ferrara and B. Zumino, Nucl. Phys. B77 (1974) 413; A. Salam and J. Strathdee, Trieste preprint IC/74/17 (1974), Phys. Letters B, submitted. [4] R. Penrose and M.A.H. MacCallum, Phys. Reports 6 (1973) 241. [5] B. Zumino, J. Math. Phys. 1 (1960) 1. [6] P. Fayet and J. Iliopoulos, Orsay preprint (June, 1974).

13

Q3

Nuclear Physics B79 (1974) 4 1 3 - 4 2 1 . North-Holland Publishing Company

SUPERGAUGE INVARIANT YANG-MILLS THEORIES S. FERRARA and B. ZUMINO CERN, Geneva Received 27 May 1973

Abstract: We construct Lagrangian theories which are simultaneously invariant under supergauge transformations and under Yang-Mills transformations. The simplest of these turns out to be just the usual theory describing the interaction of a Yang-Mills field with a Majorana spinor belonging to the regular representation of the internal symmetry group. This theory is asymptotically free. Other examples, involving in addition to spinors also scalar and pseudoscalar fields are described. They also are asymptotically free, provided the number of scalar supermultiplets is not too high and supergauge invariance, as expected, is preserved by renormalization.

1. Introduction In a recent paper [1], a field theory model was constructed which is invariant under supergauge transformations [2] as well as under ordinary (Abelian) gauge transformations. It can be considered as a supergauge invariant extension of quan­ tum electrodynamics. The model was shown to be renormalizable in the one loop approximation in a manner consistent with gauge and supergauge invariance. Prelim­ inary calculations in higher orders appear to give the same result. In the present paper we show how to construct field theories invariant under both supergauge transformations and non-Abelian gauge transformations of the Yang-Mills type. This is done by using the technique of superflelds, introduced by Salam and Strathdee [3] and extended by the present authors in collaboration with Wess [4]. As in the Abelian case, supergauge transformations enlarge the nonAbelian gauge group to a generalized gauge group, which has a simple description in terms of superflelds. A Lagrangian invariant under gauge and supergauge transforma­ tions is automatically invariant under these generalized gauge transformations and is an infinite power series in the coupling constant. In this manifestly gauge and supergauge invariant form, the theory is not obviously renormalizable. However, by means of the generalized gauge transformations, one can go to a special gauge [1] where some of the components of the vector field multiplet vanish. In this gauge the Lagrangian has a much simpler form, renormalizable by power counting, while the supergauge invariance is expressed in a more complicated way. For the case of the vector multiplet alone, which - in the special gauge - reduced essentially to a

94 & Ferrara, B. Zumino, Yang-Mills fields

414

Yang-Millsfieldplus a Majorana spinor, both belonging to the regular representation of the group, the Lagrangian reduces to the ordinary Yang-Mills Lagrangian for thosefields.One obtains therefore the interesting result that the Yang-Mills theory with a Majorana spinor in the regular representation is automatically supergauge in­ variant*. Assuming that the Lagrangians constructed in the present paper are indeed renormalizable to all orders in a manner consistent with supergauge invariance, they provide simple examples of asymptotically freefieldtheories [6]. This is true not only of the model of sect. 3, containing only the vector multiplet, but also of the model with scalar multiplets described in sect. 4, provided the number of these multiplets is not too high. Contrary to the usual situation, in our case the presence of scalarfieldsshould not pose any problems, since supergauge invariance implies that all couplings are expressible in terms of only one coupling constant. In sect. 5 we give a Lagrangian which could almost be taken as a realistic model of strong interactions. The main outstanding problem is that of generating masses for the var­ iousfields,and especially for the vectorfields.We are faced here with the difficulty that, in spite of the occurrence of scalarfields,spontaneous symmetry breaking does not seem possible in a supergauge invariant theory, at least in the tree approxi­ mation. It seems therefore desirable to study the properties of the effective potential in the one-loop approximation. Finally, it would be very interesting to see whether supergauge invariant Yang-Mills theories exhibit to all orders the same kind of com­ pensation of divergences found earlier in models with only scalar multiplets [7]. 2. Vector and scalar multiplets In this section we recall some formulae for calculating with superfields. They shall be used in the rest of this paper for the construction of invariant Lagrangians. In the notation of ref. [4], the superfield for a real vector multiplet is given by r(*,M)«C + /9%-/*dx* + OaeJi(M + iN) - 0 d 0* \i(M - W) - Oo/v*

+ 0*9Jf*§D

+ \oQ9

(1)

where thefieldsCfM,N and D are real. The two-component spinors \a and X * can be arranged into a four-component Majorana spinor * This result was obtained independently by Salam and Strathdee [5]. For the construction of invariant Lagrangians, these authors, like us, make essential use of the techniques of ref. (4). We thank them for sending us an advance copy of their preprint

95 5. Ferrara, B. Zumino, Yang-Mills fields

415

(2)

- & ) •

Similarly, for x and x- The particular way of writing the various terms of (1) (factors /, etc.) is chosen so as to bring agreement with conventions used in earlier papers. Under a supergauge transformation of parameters f and J*, V transforms according to ^ = [ ? ^ + f ^ + m*S-So*0)h^V.

(3)

The left- and right-handed covariant derivatives defined in ref. [4] shall be de­ noted here by the Da and D&. So, on a superfield transforming like V,

D

*" ^ + ' ( a M f f ) « d " '

B =

(4)

* ~ iJs" ^^'

They satisfy

{Da,Dh}=-W\b\,

(5)

{Da,D§}^{D&,D^}^0.

(6)

In particular, it follows that A left-handed superfield S(x, 0, 6 ) satisfies D d 5 = 0.

(8)

According to ref. [4], it can be shifted to a scalar superfield of type one Sl(X^d) = S(x^-ieo7td,6),

(9)

which is independent of 6 in virtue of (8) because, after the shift, the covariant de­ rivatives take the form

Da = A

+ 2 /(a^) a 3

£d = - 4 r .

(10)

They still satisfy (5) and (6), of course. The components of the scalar multiplet are defined by the expansion in 6 Sx(x9 6) = \{A - iB) + d«+a + 0 a 0 j ( F + /C) ,

(11)

where thefieldsA,B9F and G are real. With the two-component spinorfieldi//a and its conjugate, one can construct a Majorana spinor

♦-(£)• Corresponding formulae are valid for a right-handed superfield.

(n)

%

416

& Ferrara, B. Zumino, Yang-Mills fields

3. Lagrangian for the vector multiplet In this section we construct the invariant Lagrangian for the self-interacting vector multiplet. To be definite, we consider the case of SU(N), although our treat­ ment could be easily extended to other internal symmetry groups. We take a set of vector superfields belonging to the regularjepresentation of the group. This can be done most simply by considering V(x, 0,0) to be a N X N Hermitian matrix. The generalized Yang-Mills transformations mentioned in sect. 1 are then defined by c^e-'^Ve^,

e-F-e-,Ae-'7At,

(13)

where A is a matrix left-handed superfield and At its right-handed Hermitian con­ jugate DdA = DaAi=0.

(14)

Expanding (13), we find F - K + i ( A - A t ) - i ( [ A , A t ] + / [ A + At,F]) + ....

(15)

The term i(A — At) is the "gradient" of the superfield A. It was written out ex­ plicitly in ref. [1]. Using (14) one verifies easily that (13) implies e"yDaev ->e-iA(e~vDaev)eiA

+ e - ,A Z> a e ,A ,

(16)

- e - ' A [ ^ ( e - v D < / ) ] t ^ - 2/(o")^ e"' A 3 M e' A .

(17)

and therefore, using also (5), DffTvDy)

Now, applying D, to this relation, the last term is annihilated and we find 5f^(e-KZ)aeK)-e-/A[^^(e-KPaeK)]e/A.

(18)

In particular, the superfield

Jfa=^#W)

(19)

transforms as Wa^t'iAWaeiA

(20)

under generalized Yang-Mills transformation. Using (7), we see that it satisfies D.Wa = 0.

(21)

It is therefore a left-handed multiplet with an undotted spinor index. As explained in sect. 2, it can be shifted to a superfield of type one, defined by

^ ( V ») s K&* ~ ' «*/• 6>W) »

which is independent of 0" as a consequence of (21). Among thefieldsof the super-

97 S. Ferrara, B. Zumino, Yang-Mills fields

417

gauge multiplet corresponding to Wla one finds an antisymmetric tensor which is a sum of terms, one of which is just the Yang-Mills field strength associated with the vector t>M. So W1(x appears as the natural generalization of the Yang-Mills field strength in our case. The superfield Tt(WlaW«)

(23)

(trace over the unwritten matrix indices) corresponds to an ordinary scalar multiplet. Its F component can be taken as Lagrangian for a gauge and supergauge invariant theory. This way of constructing the Lagrangian for a vector multiplet was already described in ref. [4]. The superfield (23) is an infinite power series in the coupling constant £. Just as in ref. [1], we now make use of the invariance under the (generalized) gauge trans­ formations (13). From (15) it is clear that one can go to a special gauge* where_the components C9M,N, x and x of the vector multiplet vanish, so that only uM, X, X and D survive. For the multiplet V2 only the D component survives, and equals —v2, while the higher powers Vn(n>2) vanish identically in this gauge. Now the ex­ pression (23) is easily calculated. Its F component can be seen to become simply Tr(- * £ - \i\y»

q>MX + \D2) ,

(24)

up to an over-all normalization constant. Here

q>MX«3MX +

fr[uM,X],

(26)

and the Majorana field X is given by (2). We have introduced the coupling constant g by the replacement uM -» 2guM, X -+ 2g\, D -* 2gD. Except for the trivial extra term containing the auxiliary field D (which will, however, become important when interaction with other fields is introduced), this * The gauge transformation which leads from a general gauge to the special gauge can be con­ structed by iteration. If, for instance, one writes A in the base one as

Al(?9e) = \W A - D RaV~ iV

Also note that

(15)

(16)

0 D R O - D R and , * ^ (16) - it is easy to see that a suitable LaUsing(lS)

grangian for the gauge field is given by TtDD(C- 1 f ( , ((D R a ^ X D R , V^yHPu, KjXDL() I*))

■tTrOMJft^V'W'

(17)

when a surface term is disgarded. In a general gauge the Lagrangian (17) with V^ de­ fined by (13) is not a polynomial in thefieldvariables and therefore the S matrix is not manifestly renormalizable. However, there does exist a remarkable gauge (due to Wess and Zumino [4]) in which the La­ grangian assumes a manifestly renormalizable form. To lowest order in A and ^ the transformation rule (11) reads * -► * - iA x + iA

(18)

+ ...

This indicates that from among the sixteen compo­ nents in ^ we can gauge away eight and leave ^f in the special form: * = I0ry„ 7 5 BAv + Wjh5

X + A0*)2D,

(19)

where Av is a transverse vector (dvAv = 0). In this gauge, Vp reduces to a polynomial in ^ ,

_

19 August 1974

PHYSICS LETTERS

l+i7 5

- iZ)^-yM D* - i [*, Z>yM - y ^ 0 * ]

(20)

and the Lagrangian (17) reduces to

Trt-iO^.-a^+i^./iJ) 2 +

fiTM0M*+iM„.M) + l / ) 2 ] .

CM)

Apart from the J/) 2 term (which on variation gives thefieldequation D s 0 in the absence of matter), this is the Yang-Mills Lagrangian for the gauge fields A^ and the Majorana spinors X. The super-symmetry implies that each of the fields >4M and X acts as a gauge particle for the other. It would be interesting to investigate if this system (with a Majorana X) shows the same diminution of infinities as has been noticed by Illiopoulos and Zumino [6]) for other super-sym­ metric Lagrangians. For example, it is amusing that charge renormalization in this theory is finite in the lowest order provided there are in the theory three distinct matter super-multiplets (each containing two spin-zero bosons and a Majorana fermion) belonging to the adjoint representation of the internal symmetry group. Whether on account of super-invariance of the theory such a result persists in higher orders is an open question.

References [1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. [2] J-L. Gervais and B. Sakita, Nucl. Phys. B34 (1971) 632; A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86; P. Ramond, Phys. Rev. D3 (1971) 2415; Y. Aharonov, A. Casher and L. Susskind, Phys. Letters 35B (1971) 512; Y. Iwasaki and K. Kikkawa, City College (NY) preprint (1973). [3] J. Wess and B. Zumino, Phys. Letters 49B (1974) 52. [41 J. Wess and B. Zumino, CERN preprint TH.1857 (1974), submitted to Nucl. Phys. B. [5] A. Salam and J. Strathdee, ICTP, Trieste, preprint IC/74/11, to be published in Nucl. Phys. B; C. Fronsdal, ICTP, Trieste, preprint IC/74/21; S. Ferrara, J. Wess and B. Zumino, CERN preprint TH.1863 (1974). [6] J. Illiopoulos and B. Zumino, CERN preprint TH.1834 (1974), to be published in Nucl. Phys. B.

355

105 PHYSICAL REVIEW D

VOLUME 1 2 , NUMBER 8

15 OCTOBER 1975

Combined supersymmetric and gauge-invariant field theories* Bernard de Wit+ and Daniel Z. Freedman Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794 (Received 9 July 1975) The combined supersymmetric and gauge-invariant field theories are formulated using ordinary fields, not superfields, from the start, thereby eliminating reference to superfluous components. The construction is a generalization to supersymmetry of the minimal-coupling procedure of ordinary gauge theories. The Lagrangians obtained coincide with those of the superfield approach in the Wess-Zumino gauge. The WardTakahashi identities of Green's functions for fields and supersymmetry currents are obtained by functional methods. The Ward-Takahashi identities are presented in a form from which low-energy theorems for Goldstone fermions can be directly derived, and the proof of the Goldstone theorem in these combined theories is clarified. Discussion of the closure of the supersymmetry algebra leads to the concept of gaugeinvariant translations for which the conserved current is a gravitational energy-momentum tensor. Some of the results are valid in a broader context than supersymmetry and are applicable to gauge theories in general.

I. INTRODUCTION

An important step in the development of s u p e r symmetric field theories was the demonstration that Lagrangian models with combined s u p e r s y m ­ metry and Abelian 1 and non-Abelian 2 , 3 gauge invariance can be constructed. In this paper we d i s ­ cuss several aspects of these theories. First we give an alternate formulation of the combined theories, for a general gauge symmetry group, in which ordinary fields, not superfields, 4 are used from the s t a r t . This formulation is a generalization to supersymmetry of the minimalcoupling procedure of ordinary gauge fields. The resulting Lagrangians coincide with what would be obtained by the superfield technique used in the original construction 1 " 3 after eliminating s u p e r ­ fluous components by passing to the Wess-Zumino gauge. 1 Second we study the nature of the supersymmetry transformation of the fields under which our L a ­ grangians a r e invariant, and obtain the corre sponding conserved currents 5 and Ward-Takahashi (WT) identities. Some of these results are valid for a more general class of symmetry transforma­ tions in gauge field theories and are not restricted to supersymmetry. It is known1 that the supersymmetry t r a n s f o r m a ­ tions in the Wess-Zumino gauge differ from o r ­ dinary supersymmetry transformations, and that their commutator is not exactly a translation. As the third topic of the paper we show that the com mutator differs from a translation in that ordinary derivatives of fields a r e replaced by gauge-covariant derivatives. Surprisingly, the conserved c u r ­ rent of such a gauge-covariant translation is the gravitational energy-momentum tensor of the t h e ­ ory which occurs when the fields a r e coupled to 12

gravity in a variant of the general theory of r e l a ­ tivity. Tt differs from the canonical e n e r g y - m o ­ mentum tensor. Other aspects of the closure of the algebra generated by supersymmetry t r a n s ­ formations a r e also discussed. It is not at all clear that supersymmetric field theories have a physical application; however, possible applications to weak interactions have been discussed in which the neutrino is interpret­ ed as the Goldstone particle due to spontaneous breakdown of the supersymmetry. 6 In the com­ bined theories a mechanism for spontaneous breakdown has been devised by Fayet and Iliopoulos, 7 and Fayet 8 has given an admirable s e m i realistic model of weak and electromagnetic in­ teractions. The low-energy theorems for Goldstone neu­ trinos provide important experimental constraints on this approach to the weak interactions, and careful elucidation of the low-energy theorems was the motivation for our study of the combined mod­ els. The more general and technical results of our study a r e given in this paper, while specific results on the low-energy theorems will be pub­ lished separately. 9 Although superfield techniques a r e not used here, it must be said that we do not dispute the elegance of the superfield concept and its applications. They a r e elegant, and we lean heavily on r e s u l t s which were first obtained in the superfield f r a m e ­ work. However, the superfield approach has a very formal character and involves complicated manipulations of a type unfamiliar to most physi­ cists, so we feel that the present approach may make the physical aspects of the combined theo­ r i e s and the strategy of phenomenological model building c l e a r e r . Certainly the phenomenon of spontaneous breakdown of gauge invariance and 2286

106 12

COMBINED SUPERSYMMETRIC

supersymmetry is best discussed when L a g r a n ­ gians a r e written in the form given h e r e . On the other hand, the superfield approach and superfield perturbation theory 1 0 have advantages in the t r e a t ­ ment of renormalization and regularization. 1 1 In Sec. II we describe the construction of the combined supersymmetric and gauge-invariant L a ­ grangians. Section i n contains the derivation of the Ward-Takahashi identities for supersymmetry currents. The closure of the supersymmetry a l ­ gebra and the gauge -covariant translations a r e discussed in Sec. IV. Some technical material r e l ­ evant to Sees. II and III is presented in the Appen­ dix. II. CONSTRUCTION OF GAUGE-INVARIANT SUPERSYMMETRIC LAGRANGIANS The main result of this section is a procedure for the construction of field theories with com bined supersymmetry and gauge invariance. Our procedure is analogous to the minimal-coupling method of ordinary gauge field theories: We start from a r a t h e r general class of Lagrangians which are supersymmetric and invariant under a global internal symmetry. In order to obtain Lagrangians which a r e invariant under the corresponding local symmetry, we introduce gauge fields B£(x) t r a n s ­ forming in the adjoint representation of the i n t e r ­ nal-symmetry group and accompanying Majorana spinor fields xa(x) and auxiliary (canonical dimen­ sion 2) s c a l a r fields If(x) necessary to maintain supersymmetry. The Lagrangian for the i n t e r a c ­ tion of the "gauge multiplet" 5 M , x , # with the o r i g ­ inal " m a t t e r fields" is then written. It involves the expected covariant derivatives together with other t e r m s required by supersymmetry. The L a ­ grangian for the gauge multiplet is also obtained. These Lagrangians can be regarded as ordinary gauge field theories which happen to have a f e r mionic transformation (i.e., the supersymmetry) as an additional invariance. It is then clear that these theories can be quantized using the p r o c e ­ dures 1 2 of ordinary gauge t h e o r i e s . A suitable regularization procedure is necessary in o r d e r to maintain simultaneously the WT identities of s u ­ p e r s y m m e t r y and gauge invariance. The r e g u l a r ­ ization question is not treated h e r e . The Lagrangians we start from a r e simple gen­ eralizations (to include global internal symmetry) of the earliest Wess-Zumino-Uiopoulos model. 13 Readers familiar with that model should be able to follow our procedures without difficulty. The final gauge-invariant Lagrangians also coincide with those obtained via the superfield procedure. This fact was noted in the Introduction, and a brief discussion of the relative merits of the two p r o c e ­

A N D GA U G E - I N V A R I A N T . . .

2287

dures was given t h e r e . Therefore, we begin with multiplets of complex spinless fields At and F< (i = 1, 2 , . . . , « ) with c a ­ nonical dimension 1 and 2, respectively, and Ma­ jorana spinors ip{. These a r e joined in column vectors A, F , and ip, and internal-symmetry transformation properties a r e described by 6A=iL-&4,

6^j = -iLT*~$ipR, 6F=*'L-3F,

where we have introduced chiral components ipR>L = | ( 1 +y5)''P of the Majorana spinor ip. The Majorana constraint ip=Clpr implies that ipR =C4>[. Here L'^ = La0tf where the 9* a r e group p a r a m e t e r s and the La are each n x n matrices of a representation, not necessarily irreducible, of the Lie algebra of the internal-symmetry group. Group structure constants a r e specified by commutators [La,Lb] = ifcabLc.

(2) r

Note that the negative transpose matrices -L a a r e also a representation of the Lie algebra. Under supersymmetry transformation, A, , and F constitute a s c a l a r or chiral supermultiplet 4 1 4 and transform as «A=2Tfe, 6^=l(l-y5)(F-#A)e,

(3)

6F = -2*T?tyL, where € is an anticommuting space-time-indepen­ dent Majorana spinor. Our notation, we decided after much internal debate, is to use the conven­ tions of the textbook of Bjorken and Drell 1 5 for Dirac m a t r i c e s . The normalization of s u p e r s y m ­ metry transformations 1 4 is such that (636, -6162)A(x)

= 2i(eyel)dllA(x);

(4)

the relations for ip(x) and F(x) a r e s i m i l a r . The Lagrangians for the " m a t t e r fields" A,ip,F a r e of the form £> = £ ( V * V A +i4>?il> + FfF)+£'(A,il>,F).

(5) 14

The kinetic t e r m is known to be supersymmetric [it changes by a total derivative under the t r a n s ­ formation (3)], and it is obviously group invariant. We shall assume that £'(A, ip, F), which includes possible m a s s and interaction t e r m s of the matter fields, is (a) independent of derivatives of the fields; (b) group invariant, 6 £ ' =0 under (1); and (c) supersymmetric, 6 £ ' =ra;iK''J(A, ip, F) under (3). From these requirements one can derive general properties of £ ' and K? which a r e needed later in

107 2288

BERNARD DE WIT AND DANIEL Z. FREEDMAN

this section and Sec. III. These a r e given in the Appendix. It is worth remarking that in model building one would normally consider only an £ ' for which the global theory (5) is renormalizable. In practice this means that only group-invariant mass and trilinear interaction t e r m s of the Wess-ZuminoIliopoulos type occur. In some cases, such as the example of an SU(2)-doublet matter multiplet, the combined constraints of renormalizability, i n t e r ­ nal symmetry, and supersymmetry may not allow any mass or interaction t e r m s . We have used Majorana fields and their chiral components because they occuf naturally in s u p e r symmetry. The framework defined by Eqs. (1), (3), and (5) is actually sufficiently flexible to in­ clude cases where the Majorana fields combine into ordinary Dirac fields. A chiral internal s y m ­ metry can also be treated. This is related to the question of fermion-number conservation, which has been discussed in the literature. 1 6 Before generalizing to a local internal s y m m e ­ try we note that under a supersymmetry transfor­ mation (3) our Lagrangian changes by the total derivative:

12

(s> M x) a = 9 ,x a +£r 6c B;x c ,

(8) (9)

and note that the standard gauge transformation properties can be written as

(10) 6Da

=f*bcDbec,

6F;„

=/a6cF>c,

where the field strengths a r e defined by F% =9„BJ -8 M B, a +gf^cB^Bcv .

(ID

We now turn to supersymmetry transformation properites of B M , x> and D and a necessary r e ­ vision of the properties (3) of the matter fields. The transformation properties are presented first and discussed later: 6B M

= -ey^sX,

6x = 4 ; F M , , r V y 5 € + D € , 6D= -tey M » M x, (12)

6A=2e^, ^L=l2(l-y5)(F-iy^llA)e,

+ i [ - ^ M F r + (Mr>yVtoii+K'1}.

(6)

For invariance under local internal symmetry transformations (2) with a r b i t r a r y space-time functions 0"(x), we expect to introduce gauge fields Bj(x) and to replace derivatives in £ by covariant derivatives 1 7

(7) S>»R=(*v+igtT 'BjtyR. If this were the only change in £ , with BM inert under supersymmetry transformations, the s u p e r symmetry would be lost, since with an inert field it is not possible to p r e s e r v e the connection (4) b e ­ tween supersymmetry transformations and t r a n s ­ lations. It is therefore necessary to include new fields in the adjoint representation of the internal symmetry so that the resulting set constitutes a super multiplet closed under supersymmetry t r a n s ­ formations. From the general properties of m a s s less representations 1 8 of supersymmetry we would expect Majorana spinor fields xa(*) and the a s s o ­ ciated particles, and it is also necessary to i n t r o ­ duce auxiliary spinless fields Da{x) of canonical dimension 2. We introduce covariant derivatives of quantities in the adjoint representation,

6F= -Zity^to^L

+ge(l +y 5 )L-xA

(we refer to them as gauge-covariant s u p e r s y m ­ metry transformations). Once one accepts that the fields BM, J , and D a r e necessary to maintain supersymmetry, one can obtain a virtually com­ plete motivation (which we give below) for (12) in­ dependent of the superfield method. However, it must be confessed that we found (12) by using the method of Refs. 1 and 3 for determining the r e s i d ­ ual fermionic symmetry in the Wess-Zumino gauge. The independent motivation will now be given. Consider first the transformation equations of B^, X, and D. The individual t e r m s in those equations are completely dictated by the following r e q u i r e ­ ments. (a) Since supersymmetry is a fermionic s y m m e ­ try it must change Bose fields into F e r m i fields, and vice v e r s a . (b) The transformation must be gauge covariant, so that, as we will show later, the resulting c u r ­ rent is gauge invariant and therefore leads to a meaningful symmetry of the S matrix. (c) The transformation must be compatible with with the parity properties e — iy°e and x — -iy\, D a pseudoscalar and BM a vector. (d) The transformation must be compatible with the canonical dimensions of e, £?„, x> a n ( * #> namely - | , 1, | , and 2.

108 12

COMBINED SUPERSYMMETRIC AND G A U G E - I N V A R I A N T . . .

The coefficients of the various terms are then fixed by the powerful requirement19 that the com­ mutator of two covariant super symmetry transfor­ mations be a gauge-covariant translation, to be discussed in Sec. IV. It is also clear that because of the gauge -covariance requirement, the previous transformation properties (3) of the matter field must be modified at least to the extent of replacing all derivatives by covariant de rivatives. The additional term in the transformation of F is then determined by gauge invariance, dimen sional considerations, and the commutator require­ ment. We must now write down Lagrangians which are gauge invariant and invariant under covariant supersymmetry transformations. For the gauge supermultiplet the Lagrangian is 20 The additional term £B = ?lf

(14)

is permitted when the constants £* satisfy the con­ dition f\e£ = 0 for all b and c. This condition auto­ matically selects components Da in invariant Abelian subgroups of the internal-symmetry group and ensures that (14) is supersymmetric and gauge invariant. It is parity noninvariant and plays an important role in the spontaneous breakdown of supersymmetry. 7 The matter-field Lagrangian, which is the gauge -invariant generalization of (5), is given by £ v = i(j)MA)t(»'iA)+J>Ly'1a)M0L + | F f F -gf-A*LipL - I T ^ L A - X

+ kgAfZA-D + £'nFvp)a + (j^Fp, )a + (©pF„ „ )a = 0, S^Dy -3>y©M = - t e L - ¥„„,

(18)

/WxVx'H^x')^. The first of these is the Bianchi identity of gauge fields. The second is the Ricci identity which holds for covariant derivatives in any representa­ tion of the group. The third relation is proved using a Fierz transformation, the Majorana con­ straint on x, and the total antisymmetry (in uncontracted indices) of fbca. It is not obvious that £'(A, , F), which was in­ variant under the transformations (3), is also in­ variant under the covariant supersymmetry trans formations (12), and that its contribution to (17) is the same If*1 (A, , A, */>, F and we use J{x) to de note their s o u r c e s . The effective Lagrangian has the form -iC«['[4>(x)Mx)] =

4>b)+€(x)f[4>(x)]+0(€*)"> (23)

depending on a local p a r a m e t e r e(x), which may be a commuting o r an anticommuting parameter, depending on the type of transformation. If we replace the integration variables

W*>)] a °. (25)

where the t r a c e extends over the various fields and over space-time points. We have assumed that the Lagrangian depends on fields and their first derivatives only. In the transformation considered in this section the t r a c e t e r m will vanish, and it will be ignored henceforth. Let us now consider transformations which, for constant p a r a m e t e r s €, a r e s y m m e t r i e s of the invariant Lagrangian, £ i n v . This means that £ i n v changes by an explicit local total derivative. In this case the Noether current SM(x) associated with the symmetry is given by fd*x€(x)9rS»(x)=

jd*xe(x)(^

-*»%&)'

For such a symmetry transformation (24) gives the identity f [d)[dc][dc]exp(i J d V ( £ e f f + = 0,

(32)

i.e., the matrix elements of the supersymmetry current between physical states are conserved. The WT identity and therefore (32) are valid12 in both the normal and the spontaneously broken realization of su­ persymmetry. In the latter case (32) leads to low-energy theorems for Goldstone fermion amplitudes.9 Before going further it may be useful to give one concrete example of some of the rather formal quan tities which have entered the derivation of (31). For a supersymmetric model of the type described in Sec. II, we choose the common gauge function Ca[] = ad^B'(x). Using (7), (10), and (12) we then have the explicit formulas 6C'(*) = -aaJr(*)y^ 5 x a (*)], 6GCa(x) =g~latd»[dtl6«> +gfacbBcil(x)]=Mab(x), 6bGBZ(z)cb(z)=g-1[s>ilc(z)¥,

(33)-

t>bGXa{z)cb{z) =facbxc(*)cb(z), etc. Further physical information can be obtained from WT identities for Green's functions with two or more supersymmetry currents. To derive them we start from a generating functional with an additional source term for the current operator, W[JfHli]= j[d(f>]ldc][dc] exp/i fd'x'lSMx^

+ tixVW+S^x'Wix')]),

(34)

Ill 2292

B E R N A R D DE W I T AND D A N I E L Z .

FREEDMAN

tt

where H* is a fermionic vector source that satisfies the Majorana constraint. It should be noted that matrix elements of current operators between physical states a r e gauge indepen­ dent because the current is gauge invariant. This follows straightforwardly from a simple adaptation of the standard proof of the gauge independence of the S matrix 2 4 to the generating functional (34). The derivation of the multiple-current WT identity follows the same manipulations that led to (31), and we find the result f [d$\dc\[dc}expU

J d V ( £ e f f + 4>J

+S^V)

x (e(x)duSll(x)+6Sll(x)H'1(x)

- j d4z\6Ca(x)cLy»y &-H)

pa)+£„(*

-y5)(F -t$(A) + (l+y5)[F+ -i($$A)+].*Vi

- i « " ( i +y 5 >y f ,&y M y l, £* J? )

-UeyM2^v^yn^}+hey,y5m2g^+y"yv)y5^} - | f € ( x > ^ ) - JtTy5(xy"y5£ir) - £ i € [ y „ yp](x~y V / £ * ) .

In this result we have suppressed the summation over internal-symmetry indices. We have distin­ guished four different contributions, which we will now discuss separately. The first t e r m (S,) contains the energy-momen­ tum tensor 0M„, which will be defined and exten­ sively discussed in the next section. This t e r m reflects the well-known relation of supersymmetry transformations and translations, but now in a l o ­ cal form. The t e r m s Su and Sm a r e roughly anal­ ogous to Schwinger and "seagull" t e r m s . I n S „ the field operator which occurs is a total divergence (for the last t e r m in Su this is shown in the Ap­

pendix). The t e r m i€ l;Mpo ey, i Fj 0 £ a is physically quite relevant. It may contribute to low-energy theorems 9 because the field tensor Fapa can bring in a m a s s l e s s propagator whose singularity can­ cels the effect of the derivative. Other t e r m s in S u and Sm do not contribute to low-energy theo­ r e m s . Results similar to S,-S„, have been found before, either by standard canonical manipula­ tions 2 5 or by considering the transformation p r o p ­ e r t i e s of the super cur rent. 2 3 In the latter case it was found that the t e r m s Su and S m can be s i m p l i ­ fied by adding so-called improvement t e r m s to the current.

112 COMBINED S U P E R S Y M M E T R I C AND G A U G E - I N V A R I A N T . . .

12

The terms S, v are proportional to the variational derivatives of the invariant Lagrangian. We have used the following notation: and similarly for £r and £ F t ,

£D=-

(38) £v=-

«X

-a„M- ^ S - , and similarly for JC? M„X

J [d][dc][dc]exp(i J > * ' ( £ e f f + &)) = 0, (39) j[dc(>}[dc][dc}exp(i

J*rfV(£ eff + 4>J)j

*{D(x)[£,D(x)+JD(x)]

-iN6*(0)}

definiteness, although we could easily work with F, also. Then using (13) we see that de * 0 . We differentiate the WT identity (31) with respect to the source of x*()0, and obtain, at zero value of all sources, and for constant € ,

-5-vD)a, [6 2 6, -6t62]A [626l-6lS2}tL,R

=2i(e2yvcl)£>vA, =

(46)

v

2i(€2Y el)vvhtR,

f6 2 6 1 -6 1 6 2 lF = 2f(e 2 ^e 1 )D I / F. This commutator generates a new transformation that can be described as a translation plus gauge transformation with the field-dependent p a r a m e ­ ter 1 -2(e 2 y"e JgBlix), or as a translation in which covariant derivatives replace ordinary derivatives for all fields except the gauge field Bj where the field strength Favtl appears. The latter description suggests that we call the transformation (46) a gauge-covariant translation. Covariant translations a r e a symmetry of the in­ variant Lagrangian (16) for a r b i t r a r y ( s p a c e - t i m e independent) translation p a r a m e t e r s av replacing 2(e2yvel). This is guaranteed either by the d e ­ scription as ordinary translation plus gauge t r a n s ­ formation or by the Poisson-bracket structure of s y m m e t r i e s of dynamical systems. It is therefore of interest to ask what is the conserved tensor cur rent associated with this symmetry, and the Noether procedure gives the answer, which is the gauge -invariant expression

6 r («i) 5 r(« 2 ) - M a 2 ) 6 r ( « i ) = Mfe)

(49) b

with field-dependent gauge p a r a m e t e r h = -aJVz^F* v and (50)

Ma)6 s (€)-6 s (e)6 T (fl) = Mft')

+ !(5V4) t aVA +i(»„A)+3DtfA

with field-dependent gauge p a r a m e t e r h' =a eyvyM^x +r„.

(6)

Using the equations of motions (2), /M can al so be written as /M = [y*dx(A - y5B)]y^

- (F + 7 5 G)7 M * .

(7)

The above formulae were already given in ref. [5]. The current 7M is not expressible in terms of/M alone. This can be remedied by defining an improved spinor current

■Cpr=/M + kb^h

- yxhy^ kA+ysBM.

(8)

Clearly/{Jnpj is also conserved and the three-dimensional integral of its time compo­ nent has the same value as for / ° . Furthermore, using the equations of motion (2) one can easily verify that

VW

=2

"^-?5*)*-

We can now define as improved second spinor current simply

(9)

120 210

S. Ferrara, B. ZuminoI Supercurrent

Cpr = - ^ v C p r -

(10)

Clearly it satisfies the same partial conservation equation (6) an/ M . Furthermore, the three-dimensional integral of /j}npr is the same as that of 7°. We see, therefore, that it is sufficient to introduce one local operator current, namely/fmpr, and that both the restricted and the special supersymmetry generators can be obtained from it as moments of order zero and one. In the presence of interaction, it is preferable to introduce the "modified" cur­ rent, defined by Cod=^npr+5(V>X3x-7^7^.

0D

Clearly, it is conserved

but the extra term in (11) changes eq. (9) into (13)

Viod—T*'

as seen using once more the equations of motion (2). Therefore, the modified sec­ ond current

satisfies

The three-dimensional integral of 7jj o d is again the same as that of 7°, the generator of restricted transformations

The integral of / ^ o d differs from that of 7°, but only if m f 0, since7^ od and/*fmpr are equal for m = 0. Since the larger algebra is uniquely fixed only for m = 0, this is not a difficulty. Using the equations of motion (2), it is not difficult to verify that

3Cpr= frW ' V*)l r+ + (F+JsW* -2(A+ysB)^it^ where

+ 2m(A+ysB)yit^,

(17)

121 S. Ferrara, B. Zumino I Supercurrent

211

and that K o d = bx*\(A - is*)]

y^+V+75^)7^

- %A + 7 5 5)!"i// - — 3M^/ _ ^ _ 7 M ^ .

(19)

As we shall see in sect. 3, this last form has a very simple meaning in terms of superfields. Since we shall need them later, we now recall [5] the expressions for the energymomentum tensor and for the axial-vector current associated with the Lagrangian (1). The canonical energy-momentum tensor is

rM„ = a ^ a ^ + d^Bdji + i*(A3U.

(31)

We now define the real superfield V . = /0O*. 3U0* + \DJ>D. 0* - -£ IO" . du(0 - 0*) ,

(32)

which is endowed with one dotted and one undotted index (or equivalently one four-vector index). As we shall see below, the componentfieldsof (32) are related in a simple way to the modified currents and energy-momentum tensor of sect. 2. Therefore, we shall call Vait the supercurrent. Observe that the last term of V^ can be generated* by adding a constant to the superfield 0. Using the relations (29), * The first two terms of (32) give a supercurrent which corresponds to the improved, rather than the modified, quantities of sect. 2. If only thefirsttwo terms are kept, (33) must be re­ placed by

but (34) and (35) are still valid. Clearly this is the procedure to be followed, in particular, when g = 0.

123 S. Ferrara, B. ZuminoI Supercurrent

213

(30) and (31), it is easy to verify that the equations of motion (28) imply D«V . = - ^ - 0 . 0 * ,

(33)

4g a

act

to gether with its complex conjugate. From (33) we abstract the equations D

°'Vac,=DclS*'

D S

«*

= 0

W>

Mmod = ~ ! x M »

(61)

^XMmod = ~3r\M

(62)

'

where f^ is, as before, the symmetric part of u^, given by (45) (clearly, for m = 0, the improved and the modified auantities are identical). The fields^ and B vanish in the case m = 0. Therefore, /£ 5 m o d , ^M mo d and 0 ^ m o d just transform into each other by a restricted supersymmetry transformation, as given by (46) with the ap­ propriate simplifications. We now wish to find out how the supercurrent multiplet *^!fmo Jp mod ^ ^ 0^ m o d transforms under the larger supersymmetry algebra. We observe that the conditions (51) or (52) to (59) are invariant under the larger algebra. The desired transformation law can be obtained from the transformation of the superfield Va^ under the larger algebra, which is known. Alternatively, they can be obtained by combining the transformation law of the supercurrent multiplet under restricted supertransformations with its transformation law under conformal transformations*, using the algebra given explicitly in ref. [2]. The result of these calculations is

8*Aji = §fi(3xX„

+

*»XX) ~ J/5(7x7 • dXM + 7M7 ' 3Xx)

+ 5/(3xaxM + 3 M 5 Xx ) ,

(63)

where the coordinate dependent parameters a are defined in ref. [1]. As a check one can calculate the commutator of two supertransformations of parameters (Kj and a2With some algebra one verifies that (63) implies

[ M i K M - !xdxCM + §ax**cM + a„**c x , [«2.*i]xM = ^3xxM + la^x M +iO x «p - W z x % + |0 M * X - \&xx - |w 5 x M , * It is essential here that the modified (improved) currents and energy-momentum tensor trans­ form irreducibly under corformal transformations when m = 0. This is not true of the original J

H

and T

iur.

127 S. Ferrara, B. Zumino/Supercurrent

[ M i l 'M„ = * X V M , + I M V

+

V

X

^

+ 9

^XM '

111

(64>

which are the correct transformation formulae under conformal transformations. Here, as in ref. [1] ^ = 2/^7^2 , T? = /9 M a 1 7 5 7 M a 2 - ib^y^^

.

(65)

Using (16) and G (l)

=_/7X^70modd3X)

(66)

it is also easy to see that (63), for /x = 0, gives rise, by three-dimensional integration, to the algebra written out explicitly in ref. [2]. Therefore, (60) to (63) provide the local equations corresponding to the larger supersymmetry algebra. Appendix A In this appendix we give two more examples of supercurrents, and show that they satisfy, respectively, (34) and (51). The first example* is that of the free mas­ sive "vector" multiplet. This multiplet is described by a real superfield V satisfying the equation of motion ±(D«DDDa + D^DDD*) V+m2V=0.

(A. 1)

In real notation the components of the superfield V are defined by an expansion perfectly analogous to the right-hand side of (36), with the omission of the index jtx. Using (30) and (31) (A. 1) can be transformed into {&DDDD + DDDD) - D + m2} V = 0 .

(A.2)

On the other hand, from (31) we see that [DD, DD] = 16D+ WdWdp

= - 1 6 D - SiDo^Db^ .

(A.3)

Therefore, from (A.2) it follows that DDV = DDV=0,

(A.4)

which in turn implies that one has separately \D«DDDotV + m2V=0y

(A.5)

\D.DDD«V+m2V= 0.

(A.6)

* The results on this example are based on work done in collaboration with J. Wess.

128 218

S. Ferrara, B. ZurrtinoI Supercurrent

If we now define the (real) supercurrent to be (up to an over-all proportionality fac­ tor) F a& -(/Hty,F0(Z>Z^^

(A.7)

we can easily verify that it satisfies eqs. (34) and their conjugate, with S* = \m2DDV2 .

(A.8)

The second example is that of the massless self-interacting Yang-Mills multiplet discussed in ref. [9]. We recall that the Yang-Mills multiplet is described by a superfield V which is also a hermitian NXN matrix. The generalized Yang-Mills trans­ formations are defined by + eK_e-iA eKeiA>

e-V-+e-iAe-VeiA\

(A.9)

where A is a (matrix) left-handed superfield and A+ its right-handed hermitian con­ jugate DaA+=/5JV«=0.

(A.10)

The superfield Wa=DD{t-VDy)

(A.11)

is left-handed D&Wa-0,

DaW& = 0,

(A.12)

and corresponding to (A.9), transforms as Wa -+ e- /A Wae'A,

W^ -»e~iA+W^

.

(A.13)

The equations of motion D*(evWae-v) = 0, 5 * ( e - ^ e F ) = 0,

(A.14)

are supersymmetric and also invariant under (A.9). If we now define the supercur­ rent, up to a proportionality constant, to be V^ = K e ^ e - r S y = T r ^ e - ^ e O ,

(A.15)

we see immediately that it satisfies (51) and the complex conjugate, as a conse­ quence of (A.12) and (A.14). In the special gauge described in ref. [9], the spinor current contained in (A.15) agrees with that given by eq. (41) of ref. [6]. Clearly,

129 S. Ferrara, B. Zumino/ Super current

219

the present example contains as a special case that of the free massless "vector" multiplet, in which case there is no matrix multiplication and eqs. (A.9) to (A.l5) take a slightly simpler form.

Appendix B As mentioned in sect. 1, the formulae of sect. 4 can be used to describe the supermultiplet containing free fields of mass m and spin 2, §, § and 1. One can use the superfield equations 0 * ^ = 0,

(B.l)

(D-ml)Va^0.

(B.2)

Indeed, as we already know, (B.l) gives, for (B.3)

'XM = 'MX

the equations 'ppas°'

3 ^ = 0,

(B.4)

and for CM = mC^ the equation d"CM = 0 .

(B.5)

Together with the Klein-Gordon equation for each component, (B.3) to (B.5) de­ scribe free fields of spin 2 and 1. Furthermore, (B.l) gives V 7 - d X M = 0.

(B.6)

Applying 7 * 3 to this equation and using the Klein-Gordon equation for xM, one finds 7-3AM + m2XM = 0 .

(B.7)

Therefore, \ = \ + ^XM and xM = y$(\ - "*XM) both satisfy the Dirac equation for mass m. At the same time (56), (57) and (59) insure that the Rarita-Schwinger conditions d-X = 3 - x = 7-X = 7-X = 0

(B.8)

are satisfied. Therefore, XM and xM are each a free field of spin | . The otherfieldsof the multiplet are not independent, for intance D^-mC„.

(B.9)

130 220

S. Ferrara, B. Zumino/Supercurrent

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

J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. S. Ferrara, Nucl. Phys. B77 (1974) 73. A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477. J. Wess and B. Zumino, Phys. Letters 49B (1974) 52. J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310. B. Zumino, to be published in Proc. 17th Int. Conf. on high-energy physics, London, 1974; also available as CERN preprint TH. 1901. [7] (a) C. Callan, S. Coleman and R. Jackiw, Ann. of Phys. 59 (1970) 42. (b) B. Zumino, Lectures at the 1970 Brandeis University Summer Institute, (M.I.T. Press, Cambridge, Mass). [8] S. Ferrara, J. Wess and B. Zumino, Phys. Letters 51B (1974) 239. [9] S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413.

131 Nuclear Physics B90 (1975) 104-124 © North-Holland Publishing Company

SUPERGAUGE INVARIANT EXTENSION OF THE HIGGS MECHANISM AND A MODEL FOR THE ELECTRON AND ITS NEUTRINO Pierre FAYET Laboratoire de Physique Theorique, Ecole Normale Superieure, Paris, France* Received 4 December 1974

We extend the Higgs mechanism in a supergauge invariant way. Spontaneous sym­ metry breaking occurs for all allowed values of the parameters of the scalar potential: With SU(2) X U(l) as gauge group and after spontaneous breaking of supergauge invari­ ance, we obtain a model of weak and electromagnetic interactions for the electron and its neutrino, which includes also heavy leptons and scalar particles. The neutrino is the Goldstone fermion, associated with the photon in a massless vector multiplet.

1. Introduction Recently, we presented a renormalizable model of weak and electromagnetic interactions in which parity is spontaneously broken, together with SU(2) X U(l) gauge invariance, by the Higgs mechanism [1]. An extra invariance called Q is im­ posed in order to obtain a massless neutrino. The Higgs field sector is formed by two complex doublets \p and M/1£ V ) - /i V V + / t / )

- (D^D^'

-b [(ip'^tp')2 + (^""IV')2] _ cy^'ip'^tp" - d

(3)

where we have denoted the £(1 -175) and ^(1 + J*75) parts of the Dirac fields by the indices L and R respectively. Q-invariance restricts the possible terms in the Lagrangian considerably. Spontaneous symmetry breaking occurs (see appendix of ref. [l])if M 2 0,

c-2b>Sup(-di0).

(4)

Then one, and only one, of the fields y and " = - V J / ( 4 - iB) ,

H" = WlHF- iG) ,

and the expression (1) of the covariant derivative D^, we find: •£= - i W W>» - \ilB\

+ \D2

(27)

141 P. Fayet/Higgs mechanism

114

-{WjrW'i" - \iX'?X' + %D'2 + $D' + (H'iH' +H"iH" - D ^ t f l V - Dy^D**"

-

typ})

+ m ( t f ' V + $'1H' + H"i V I •

(28)

Supergauge invariance is spontaneously broken for % # 0, together with gauge invariance if \\%g'\ > m2. Then ki and 6 by eia and e~2l0£ respectively. All terms in the Lagrangian density are {/-invariant with the exception of s ( / 0 cos u + g$ sin u}. Thus this term can be cast into the simpler form sf0 if convenient phase transformations for H)^ and 6 are performed. The whole Lagrangian density is now: £ = i T r [ Q ) X ^ t c h ( ^ + ^ V ' ^ A ^ t s h U c V + ^ c V , ) ] Z ) + i? 0 + |Z) , + i[dXcJ] Z ) +s/ 0 +i/z[q)tc7)cJ] F ,

(33)

It reads in the special gauge*:

+ [H'tH'+H'^H" - D^Dty'

- D^D^"

- ijpip]

+ * L C* t WxV + ^(s*Wx')* L ] + i [ip^igxD +g'D'W - ip'HgxD+g'D')*'] +

\ Ul +So - Wa h

+

+ i i(fo *#o W ' + {a + ib) ( # V

- bJWb - ipfp] + s/ 0

+ (/o - teo W

+ gfj of motion. This leads to the potential:

are

V(. Using expression (34) of the Lagrangian density we can show that time reversal is a symmetry of the broken theory, con­ trary to models such as T.D. Lee's, where T invariance is spontaneously broken [18].

144 P. Fayet/Higgs mechanism

117

\2 = X3 cos 0 + X' sin 0 , X_ =%/!(*!+/A 2 ), (40) XT = —X3 sin 0 + X' cos 0, and similarly for Dz, Dy and Z>_. Among the auxiliary fields D, £>', H\ H", / 0 and g0, only Z>7 takes a nonvanishing vacuum expectation value: (41) = - $ c o s 0 . If £,# andg' are non zero, supergauge invariance is spontaneously broken. Under a gauge transformation of infinitesimal parameter a, the only term proportional to the vacuum expectation values in the variations of thefieldsis: 6X7 = -£cos075 s i n ° +

e_L+£,-R

X' = -(E*QL + £-0R) sin 6 + {vh + i $ ) cos 0 , e

P=

0L-e0R

(49)

I

We recall here that X' and p are Majorana spinors, singlets of SU(2) X U(l); X is a triplet of Majorana spinors and i// a doublet of Dirac spinors with U(l) quantum numbers F = 0 and F = 1 respectively. In these expressions we clearly read the con­ servation of lepton number in vector boson exchanges. Formula (49) depends on two mixing angles between neutrino fields, 0 and 5. It is remarkable that in this supergauge theory 0 is at the same time the SU(2) X U(l) mixing angle of Weinberg-type models [9], and the neutrino/heavy neutrino mixing angle of Georgi-Glashow type models [10]. Both 0 and 6 are determined by simple mass ratios: m„ cos0 =-

(50) mT In the heavy W limit, a W-exchange is equivalent to a local four-fermion interac­ tion, for which we define the effective coupling constant: tg« =

—fi= rsin 2 0 = V2 4 m 2

z2 4m

(51)

G, and hence w w , can in principle be determined from *>ee scattering (see next sub­ section). 5.3.3, Scalar bosons. The combinations: L + iy/2 cos5 w_e_RvL)

+ h.c.,

(55)

and we find:

(56)

V~V—2^~T'

where G has been defined in (51). These cross sections determine m w , indepen­ dently of the angle 0. Present data on y"ee scattering [14] give a lower bound ~ 60 GeV/c 2 form w . 5.5.2. Electron anomalous magnetic moment The problem of fermion anomalous magnetic moment is important in supergauge theories. Ferrara and Remiddi [11] have shown that it vanishes identically in the model of ref. [4], as a consequence of supergauge invariance. The difficulty can be solved if supergauge invariance is spontaneously broken, as in ref. [5], but with the U(l) gauge group of quantum electrodynamics remaining unbroken. Both conditions are realized in the present model. The diagrams which contribute at the one loop level are shown in fig. 1.

a)

b)

d)

c)

e)

Fig. 1. One-loop diagrams associated with the electron anomalous magnetic moment. The arrows show lepton number conservation.

149 122

P. Fayet/Higgs mechanism

V (W_.,E^A

a)

V {VL/E&AK

i' (W.,E.^W^X

b)

c)

Fig. 2. One-loop "super-diagrams" associated with the electron anomalous magnetic moment.

Diagram la gives the well known contribution ajl-n to the electron anomaly \{g - 2). Diagrams lb give contributions proportional to the squared electron mass (with a possible logarithmic factor). For the other ones, whenever there is no extra damping factor sin 5 = w e _/m w _ >/2, the initial and final electrons have the same chirality and we can retain only the chirality-conserving parts in the internal fermion propagators; thus these contributions are also proportional to the squared electron mass. So the weak contributions to the electron anomaly are negligible, provided h is not too small*. The diagrams of fig. 1 can be associated in three diagrams shown in fig. 2, where particles of a given supermultiplet are exchanged: the photon is coupled to the charged multiplet (W_; E_, e_; w_), and one of the three neutral multiplets (7; vL), (Z; E 0 ; z) and (e 0 ; 00, y) is exchanged. All diagrams of fig. 1 follow immediately from superdiagrams of fig. 2 using Poincar^ invariance and leptonic number conser­ vation.

6. Conclusion We began with building an SU(2) X U(l) gauge theory of weak and electromag­ netic interactions; as in ref. [1] our leading principle was the idea that the initial theory should be as symmetric as possible, i.e. gauge- and parity-invariant; the exis­ tence of a massless neutrino was used as a phenomenological input and led to Q-invariance. Spontaneous symmetry breaking provided both the vector electromagnetic current and the usual V—A charged currents. A connection has been established between this model, the supergauge invariant extension of quantum electrodynamics of ref. [4] and an SU(2) X U(l) supergauge theory, using the interpretation of 2-invariance in terms of superfields. But the last theory, after spontaneous symmetry breaking leads only to a massless "electron". In order to improve it, we used a new method of spontaneous symmetry breaking which allows us to break gauge-invariance in a supergauge invariant way. We ob­ tained the supergauge invariant extension of the Higgs-Kibble model; for every vector multiplet acquiring a mass, a real scalar multiplet is eliminated by a "super Higgs mechanism". After reduction of SU(2) X U(l) gauge invariance to U(l), the super* For small h, / exchange can be large.

150 P. Fayet/Higgs mechanism

123

gauge invariant theory describes the interaction of the four vector multiplets associ­ ated with 7, Z, and W_ (charged) and of a physical "super Higgs multiplet" gener­ ated by a massive Dirac spinor eQ. Supergauge invariance was broken spontaneously as in ref. [5] but in a very soft way. The Goldstone spinor X7 forms with the photon y a massless vector multiplet; it is the Majorana spinor associated with the two-component neutrino field vL. The masses in the multiplets remain equal, at least at lowest order, except those of the Dirac spinors e_ and E_ associated with W_: they are split on each side of the W_ mass. e_ and E_ are interpreted as the electron and heavy electron respectively. If G-invariance were also demanded, the electron mass would vanish; the smallness of electron mass is then much more understandable than in other models with heavy leptons. R-invariance, which is still a symmetry for the broken theory, is associated with lepton number conservation. There is no trouble with the electron anomalous magnetic moment: the weak corrections are very small. Now we know two methods for spontaneous symmetry breaking: with the method of ref. [5] supergauge invariance is broken, and gauge invariance may either be conserved, or not. We found here a supergauge-invariant way to break gauge in­ variance, which generalizes the Higgs mechanism without any restriction on the parameters. Applied together for an SU(2) X U(l) gauge group, the two methods give a possible model of weak and electromagnetic interactions for the electron sector. The introduction of supergauge invariance in gauge theories has many interesting features. Spontaneous symmetry breaking is necessary. Instead of three sorts of objects in "old type" gauge theories (gauge vector fields, spinor multiplets and Higgs multiplets), we have two only: vector multiplets of generalized gauge fields ("radiation") and scalar multiplets ("matter"). Spontaneous symmetry breaking mixes the fermions of radiation and matter multiplets to give the leptons. It pro­ vides a fundamental reason for the vanishing of neutrino mass, and yields rather attractive associations of particles in multiplets: the photon with the neutrino; the charged vector boson and a scalar of the same mass with the electron and heavy electron, etc. The model has all features desirable for a theory of weak and electromagnetic interactions, if the muon sector is ignored. Nevertheless it suggests that supergauge invariance may be of great physical interest in weak interaction theory. I am very grateful to Professor J. Iliopoulos for many helpful discussions.

References [1] P. Fayet, Nucl. Phys. B78 (1974) 14. [2] J. Goldstone, Nuovo Cimento 19 (1961) 154; J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965.

151 124

P. Fayet/Higgs mechanism

[3] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39. [4] J. Wess and B. Zumino, Nucl. Phys. B78 (1974) 1. [5] P. Fayet and J. Iliopoulos, Phys. Letters 5IB (1974) 461. [61 A. Salam and J. Strathdee, Nucl. Phys. B76 (1974) 477. [7] S. Ferrara, J. Wess and B. Zumino, Phys. Letters 5 IB (1974) 239. [8] B. Zumino, Lectures given at the Cargese Summer Institute, July 1972, and references therein. [9] S. Weinberg, Phys. Rev. Letters 19 (1967) 1264; A. Salam, Proc. of the 8th Nobel Symposium (Wiley New York, 1968). [10] H. Georgi and S.L. Glashow, Phys. Rev. Letters 28 (1972) 1494. [11] S. Ferrara and E. Remiddi, Preprint CERN TH 1935 (1974). [12] P. Fayet, Proc. of the 9th Rencontre de Moriond, ed. Tran Thanh Van (1974). [13] F.J. Hasert et al., Phys. Letters 46B (1973) 121; D.C. Cundy, Report to the 17th Int. Conf. on high-energy physics, London (1974). [14] H.S. Gurr, F. Reines and H.W. Sobel, Phys. Rev. Letters 28 (1972) 1406. [15] J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310; A. Salam and J. Strathdee, Phys. Letters 49B (1974) 465. [16] A. Salam and J. Strathdee, Phys. Letters 5IB (1974) 353. [17] S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413. [18] T.D. Lee, Phys. Rev. D8 (1973) 1226.

152 Nuclear Physics Bl 13 (1976) 135-155 © North-Holland Publishing Company

FERMI-BOSE HYPERSYMMETRY P. FAYET Laboratoire de Physique Theorique de VEcole Normale Superieure Received 22 September 1975 (Revised 9 February 1976) A new algebra, combining supersymmetry and internal symmetry, is presented. A massless vector hypermultiplet contains a vector, an isodoublet of left-handed Dirac spinors, and a complex scalar. These can be used as generalized gauge fields. Abelian as well as non-Abelian gauge theories are studied, and the Higgs mechanism is extended in a hy­ per symmetric way. We present, also, a (non-realistic> SU(2) X U(l) model; gauge invariance and hypersymmetry are spontaneously broken; two Goldstone spinors appear. Hypersymmetry allows one to define "electronic" and "muonic" numbers, and suggests that a weakly interacting scalar particle tjy is associated with the photon and the two neutrinos.

1. Introduction Supersymmetry generators [1] are charges transforming under the Lorentz group as the components of a Majorana spinor. Any linear representation contains both fermions and bosons, which have equal masses if the symmetry is unbroken. Gauge invariance has been generalized to such theories [2]. Supersymmetry can be spontaneously broken, thus providing a Goldstone spinor [3]. It is tempting to consider the (electron) neutrino as the massless Goldstone particle. In ref. [4], we constructed a model of weak and electromagnetic interactions for the electron sec­ tor. The "gauge fields" associated with the unbroken electromagnetic gauge invari­ ance are the vector FM and the Majorana spinor X * the former describes the photon, the latter, which is the Goldstone spinor, the (electron) neutrino. We defined, recently, a self-interaction for a vector multiplet; this supersymmetric extension of the Higgs model depended only on two parameters [5]. If they vanish, we are left with a vector V*1, two Majorana spinors X and p, and a complex scalar co, all free and massless; we suggested that they could be used to represent the photon, the two neutrinos, and a weak-interacting scalar particle, with electronic number 1, muonic number 1. Laboratoire propre du CNRS associe a FEcole Normale Superieure et a l'Universite de Paris-Sud. Postal address: Ecole Normale Superieure, 24, rue Lhomond, 75231 Paris Cedex 05, France. 135

153 136

P. Fayet / Hypersymmetry

Now, we have to define an interaction. In order that the two neutrinos play the same role, we introduce an internal symmetry group SU(2); (^ L ) is an isodoublet of left-handed Dirac spinors. The mixing of supersymmetry and internal symmetry has already been realized [6], but massive representations contain a high number of particles, and it is not clear whether one can construct a renormalizable theory [7]. To avoid these prob­ lems, we introduce a new structure. In sect. 2 of this paper, we give a general definition of "hypersymmetry"; it com­ bines, in a new way, supersymmetry with an internal invariance group 9 . In our examples, we shall choose SU(2) or one of its U(l) subgroups; in this case, the hyper­ symmetry algebra contains two ordinary isomorphic supersymmetry algebras. A vec­ tor hypermultiplet is composed of a vector and a scalar supermultiplet, both massless; it contains the isoscalar antisymmetric tensor V^u, the isodoublet of left-handed Dirac spinors (\^), and the complex isoscalar scalai co, as physical fields. We also de­ fine scalar hypermultiplets. In sect. 3, we construct a hypersymmetric extension of quantum electrodynamics, where {V^\ X, p\ co) appears as a vector hypermultiplet of U(l) "gauge fields" inter­ acting with "matter" hypermultiplets. In sect. 4, U(l) gauge invariance is spontaneously broken but hypersymmetry is not: a massless vector hypermultiplet and a massless scalar one join together into a single massive vector hypermultiplet (this was studied in detail, for supermultiplets, in ref. [5]); the latter describes one vector, two Dirac spinors, and five real scalar fields, with equal masses. Two conserved quantum numbers can be defined. We show in sect. 5 that the interaction [8] between a Yang-Mills vector super­ multiplet ^,-, and a (massless) scalar supermultiplet 9£,-, belonging to the regular representation of the gauge group SU(/V), is indeed hypersymmetric: the model des­ cribes the self-interaction of a Yang-Mills vector hypermultiplet. We present briefly, in sect. 6, an example of spontaneous hypersymmetry break­ ing: the SU(2) X U(l) gauge-invariant model of ref. [9] is made hypersymmetric. SU(2) X U(l) is spontaneously broken as usual, together with both supersymmetry invariances: the Goldstone spinors are X7 and pyi associated with the photon y and the scalar particle coy in a vector hypermultiplet. This unifies the results obtained in refs. [4,9]: in the former, X7 was the Goldstone spinor, and (7, X7) a vector super­ multiplet; in the latter py was the Goldstone spinor, and (py9 coy) a scalar supermul­ tiplet. The main purpose of the paper is to show that hypersymmetry allows the con­ struction of theories invariant under supersymmetry and (global) internal symmetry, combined in a non-trivial way; two conserved quantum numbers can be defined. Finally, we state some remarks about the possible application of hypersymmetry to weak and electromagnetic interactions theory ..In a realistic model, neutrinos can­ not be Goldstone particles (but those could exist as still-unobserved massless particles). The electron and muon neutrinos might be associated with the photon, and a weakinteracting scalar particle co , in a vector hypermultiplet.

154 P. Fayet / Hypersymmetry

137

2. The hypersymmetry algebra 2.1. The supersymmetry algebrasrt.{6) Let M^v and P^ be the Poincare' group generators. Qa are four charges transform­ ing under the Lorentz group as the components of a Majorana spinor, and satisfying:

{Qa,Q,} = - 2 y ^ ,

where Q = Qy°. The supersymmetry algebra 5]=i7 5 *X. We demand that (p, X) transform as (G t , Q^) under internal symmetry:

* - © • *-(-£)■■

(12)

. -+ e*/«+«>

co.,

\x^e/aX.L.

(55)

The two conserved quantum numbers Le and L are given in table 2. Table 2 Field

n

"iL

\X

"/

0

1

0

1

0

0

1

1

The difference: L e - I M = 2/3

(56)

The one already exhibited in ref. [8] can now be interpreted as a result of invariance under "isospin" rotations generated by 7 2 . Here we prefer to use 7 3 .

167 150

P. Fayet I Hypersymmetry

allows the definition of a conserved "fermionic" number, which is 0 for bosons, 1 for Dirac fermions. In this section, we have been concerned, only, with the self-interaction of a YangMills vector hypermultiplet of "gauge fields". In the next section, we describe briefly. interaction with scalar hypermultiplets, for an example in which spontaneous hyper­ symmetry breaking occurs.

6. Spontaneous hypersymmetry breaking In refs.[4,5] supersymmetry was spontaneously broken and the Goldstone spinor was the Majorana spinor Xy associated with the photon y. In ref. [9] it was the Majorana spinor py associated with the coy particle. In the previous sections we learned how to associate X and p spinors by means of SU(2) internal symmetry. Our purpose is to unify our previous examples of spontaneous supersymmetry breaking: we shall construct a hypersymmetric model which will provide two Goldstone spi­ nors \ y and pyi belonging to a vector hypermultiplet (7; X ,py\ co7). In this paper, we expose the model only very briefly. 6.1. The model It is the one we presented in ref. [9] with gauge invariance realized locally. It described the supersymmetric, SU(2)G X U(l),/?- and parity-invariant interaction of a triplet^ and a singlet^' of vector multiplets; a triplet 9C and a singlet 9£' of real scalar multiplets; a doublet H) of complex scalar multiplets. Four coupling con­ stants were involved: the gauge coupling constants g and g\ and the coupling con­ stants of the scalar multiplets9C and 9£' with