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Instantons Versus Supersymmetry: Fifteen Years Later Mikhail Shifman and Arkady Vainshtein

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Instantons Versus Supersymmetry: Fifteen Years Later Mikhail Shifman and Arkady Vainshtein View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server TPI-MINN-99/08-T UMN-TH-1743/99 hep-th/9902018 Instantons Versus Supersymmetry: Fifteen Years Later Mikhail Shifman and Arkady Vainshtein Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455 Abstract An introduction to the instanton formalism in supersymmetric gauge theories is given. We explain how the instanton calculations, in conjunction with analyticity in chiral parameters and other general properties following from supersymmetry, allow one to establish exact results in the weak and strong coupling regimes. Some key applications are reviewed, the main emphasis is put on the mechanisms of the dynamical breaking of supersymmetry. Contents 1 Introduction and Outlook 2 1.1 Dynamical SUSY breaking: what does that mean? ........... 2 1.2Hierarchyproblem............................ 3 1.3Instantons................................. 4 1.4 The Higgs mechanism en route ...................... 6 1.5Chiralversusnonchiralmodels...................... 7 1.6Whatwillnotbediscussed........................ 8 1.7Instantonsvs.supersymmetry:literaturetravelguide......... 9 2 Supersymmetric Theories: Examples and Generalities 11 2.1Superspaceandsuperfields........................ 11 2.2ThegeneralizedWess-Zuminomodels.................. 13 2.2.1 Theminimalmodel........................ 13 2.2.2 Thegeneralcase......................... 14 2.3Simplestgaugetheories.......................... 15 2.3.1 Supersymmetricquantumelectrodynamics........... 15 2.3.2 SupersymmetricQCDwithoneflavor.............. 18 2.4Thevacuummanifold:generalities................... 21 2.5Thevacuummanifold:examples..................... 24 2.5.1 SU(5)modelwithonequintetandone(anti)decuplet..... 24 2.5.2 The 3-2 modelofAffleck,Dine,andSeiberg ......... 24 2.5.3 SU(5)modelwithtwoquintetsandtwo(anti)decuplets.... 26 2.6Theimpactofthesuperpotential.................... 28 2.7Theimpactofquantumeffects...................... 29 2.8Anomalousandnon-anomalousU(1)symmetries............ 30 2.9EffectiveLagrangianandtheanomalousU(1)............. 36 2.10 Witten’s index: where to look for the dynamical supersymmetry breaking?................................. 37 3 Fermion-Boson Degeneracy: Magic Background 41 3.1Aprototype–supersymmetricdomainwalls.............. 43 3.2Superpotentialandanomaly....................... 46 3.3 Digression: solitons in two-dimensional theories with the minimal SUSY 48 3.4 Instanton tunneling transitions in gauge theories – continuation to theEuclideanspace............................ 54 3.4.1 Instantonsolutioninthespinornotation............ 56 3.5Instantoncalculusinsupersymmetry.................. 59 3.5.1 Collectivecoordinates...................... 59 3.5.2 Thesymmetrytransformationsofthemoduli......... 63 3.5.3 Themeasureinthemodulispace................ 64 3.5.4 Includingmatter:SQCDwithoneflavor............ 69 1 3.5.5 Orientation collective coordinates as the Lorentz SU(2)R ro- tations............................... 73 3.5.6 Theinstantonmeasureintheone-flavormodel......... 74 3.5.7 Verificationofthe’tHooftterm................. 76 3.5.8 Theinstantonmeasure:generalgaugegroup.......... 77 3.5.9 Theinstantonmeasure:generalmattercontent........ 78 3.6Cancellationofquantumcorrectionstothemeasure ......... 80 4 Sample Applications 82 4.1 Novikov-Shifman-Vainshtein-Zakharov β function........... 83 4.1.1 Exact β functioninsupersymmetricgluodynamics....... 83 4.1.2 Theories with matter: β function via anomalous dimensions . 84 4.1.3 NSVZ β functionandWilsoneanaction............. 86 4.1.4 MakingcontactwiththeperturbativedefinitionofΛ ..... 87 4.1.5 Perturbative versus nonperturbative β functions........ 89 4.2 Gluino condensate in SUSY gluodynamics ............... 90 4.3One-flavormodel............................. 91 4.4Two-flavormodel............................. 94 4.5HoweffectiveistheeffectiveLagrangian?................ 96 4.6Whenoneinstantonisnotenough?................... 99 5 Spontaneous Supersymmetry Breaking in Weak Coupling 102 5.1Traditionaltreemechanisms(brieflookattheoldguidebook)....103 5.2Dynamicalmechanisms:preliminaries..................106 5.3 Dynamical SUSY breaking in the 3-2 model..............108 5.4SU(5)modelwithtwogenerations....................112 5.5 The 4-1 model: SUSY breaking through gluino condensation .....114 5.6Afewwordsonothercalculablemodels.................116 6 Spontaneous Supersymmetry Breaking in Strong Coupling 117 6.1SU(5)modelwithonegeneration....................117 6.2TheIntriligator-Seiberg-Shenkermodel.................120 6.3TheIntriligator-Thomas-Izawa-Yanagidamodel............123 7 Impact of the Chirally Symmetric Vacuum 126 8 Concluding Remarks 129 Acknowledgments 129 References 130 RecommendedLiterature.........................137 2 1 Introduction and Outlook Most theorists firmly believe that the underlying theory of fundamental interactions is supersymmetric. If so, at energies below 100 GeV supersymmetry (SUSY) must be broken, since no traces of the Fermi-Bose degeneracy were detected so far. One can speak of two alternative patterns of the SUSY breaking: explicit or sponta- neous. Although the former pattern is sometimes discussed in the literature, this direction is obviously peripheral. For aesthetical and theoretical reasons the main trend is the spontaneous breaking of supersymmetry. Various mechanisms of the spontaneous SUSY breaking were worked out; a common feature of all of them is the occurrence of the massless Goldstino. When the supergravity is switched on the phenomenon analogous to the Higgs mechanism occurs – the Goldstino mixes with the gravitino and is “eaten up” by the latter – the gravitino becomes massive. Therefore, phenomenology of the spontaneous SUSY breaking cannot be considered in isolation from gravity. To us, the most interesting question is dynamics lying behind the spontaneous SUSY breaking. More exactly, our prime topic is the nonperturbative gauge dynam- ics which, in certain supersymmetric theories, creates non-supersymmetric vacua. Why this happens in some models and does not in others? Which features of the supersymmetric Yang-Mills theories are crucial for this phenomenon and which are secondary? These and other similar questions were first posed in the beginning of the 1980’s. Many breakthrough discoveries were made then, approximately fifteen years ago. In the subsequent decade the issue was in a dormant state. It was revived recently in connection with new breakthrough discoveries in nonperturbative gauge dynamics. The present review is an attempt to give a unified “big picture” of the development of the subject spanning over 15 years. Several reviews and lectures published recently are devoted to the dynamical SUSY breaking. Usually the main emphasis is made on general aspects and phe- nomenological consequences. Not much attention is paid to theoretical tools devel- oped over these years, which allow one to obtain exact results in strongly coupled gauge theories, in certain instances. The formalism of the supersymmetric instan- tons is one of such tools. A significant part of this review is devoted to in-depth studies of the supersymmetric instantons. More exactly, we focus on the aspects of superinstanton calculus that are important for the mechanisms of the dynamical SUSY breaking. The second part of the review presents a survey of such mechanisms. 1.1 Dynamical SUSY breaking: what does that mean? Logically, there are two possibilities for the spontaneous breaking of supersymme- try. It could be broken at the tree (classical) level by virtue of one of two known mechanisms: the Fayet-Iliopoulos mechanism or the O’Raifeartaigh mechanism (see Sec. 5.1). Both were most popular in the 1970’s and early 1980’s. They are out of 3 fashion now. If one believes that the only genuine mass scale in the fundamental theory is that of Grand Unification, 1016 GeV, or the Planck scale, 1019 GeV, then it is natural to expect that the tree-level∼ breaking would produce the∼ splittings between the masses of ordinary particles and their superpartners of the same order of magnitude. Then, next-to-nothing is left of supersymmetry as far as physics of our low-energy world is concerned. In particular, SUSY will have nothing to do with the hierarchy problem – its main raison d’etre in the model-building. If supersymmetry is unbroken at the tree level, it remains unbroken at any finite order of perturbation theory. If the SUSY breaking still occurs in this case, it must C/g2 be exponential in the inverse coupling constant, e− ,whereCis a positive constant. A natural very small parameter appears∼ in the theory. The exponent C/g2 e− suppresses all SUSY breaking effects. Starting from the fundamental scale 1019 GeV, one can expect to get in this case the mass splitting between the superpartners∼ of order of the “human” scale, 102 GeV. Then supersymmetry may be instrumental in the solution of the hierarchy∼ problem and in shaping the major regularities of the electroweak theory. It is just this scenario that is called the dynamical SUSY breaking, that will be in the focus of the present review. The term was put into circulation by Witten [1]. The models where the phenomenon occur are rather sophisticated in their structure and are not so abundant. The searches that eventually led to the discovery of such models present a dramatic chapter in the history of supersymmetry. We will dwell on various aspects of this story and on nuances of the dynamical SUSY breaking in due course. Whether the models developed
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