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Lectures on Supersymmetry

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Lectures on Supersymmetry Matteo Bertolini SISSA June 9, 2021 1 Foreword This is a write-up of a course on Supersymmetry I have been giving for several years to first year PhD students attending the curriculum in Theoretical Particle Physics at SISSA, the International School for Advanced Studies of Trieste. There are several excellent books on supersymmetry and many very good lecture courses are available on the archive. The ambition of this set of notes is not to add anything new in this respect, but to offer a set of hopefully complete and self- consistent lectures, which start from the basics and arrive to some of the more recent and advanced topics. The price to pay is that the material is pretty huge. The advantage is to have all such material in a single, possibly coherent file, and that no prior exposure to supersymmetry is required. There are many topics I do not address and others I only briefly touch. In particular, I discuss only rigid supersymmetry (mostly focusing on four space-time dimensions), while no reference to supergravity is given. Moreover, this is a the- oretical course and phenomenological aspects are only briefly sketched. One only chapter is dedicated to present basic phenomenological ideas, including a bird eyes view on models of gravity and gauge mediation and their properties, but a thorough discussion of phenomenological implications of supersymmetry would require much more. There is no bibliography at the end of the file. However, each chapter contains its own bibliography where the basic references (mainly books and/or reviews available on-line) I used to prepare the material are reported { including explicit reference to corresponding pages and chapters, so to let the reader have access to the original font (and to let me give proper credit to authors). I hope this effort can be of some help to as many students as possible! Disclaimer: I expect the file to contain many typos and errors. Everybody is welcome to let me know them, dialing at [email protected]. Your help will be very much appreciated. 2 Contents 1 Supersymmetry: a bird eyes view 7 1.1 What is supersymmetry? . .8 1.2 What is supersymmetry useful for? . .9 1.3 Some useful references . 18 2 The supersymmetry algebra 22 2.1 Lorentz and Poincar´egroups . 22 2.2 Spinors and representations of the Lorentz group . 25 2.3 The supersymmetry algebra . 29 2.4 Exercises . 37 3 Representations of the supersymmetry algebra 38 3.1 Massless supermultiplets . 40 3.2 Massive supermultiplets . 47 3.3 Representation on fields: a first try . 53 3.4 Exercises . 56 4 Superspace and superfields 57 4.1 Superspace as a coset . 57 4.2 Superfields as fields in superspace . 60 4.3 Supersymmetric invariant actions - general philosophy . 64 4.4 Chiral superfields . 65 4.5 Real (aka vector) superfields . 68 4.6 (Super)Current superfields . 70 4.6.1 Internal symmetry current superfields . 71 4.6.2 Supercurrent superfields . 72 4.7 Exercises . 75 5 Supersymmetric actions: minimal supersymmetry 76 5.1 = 1 Matter actions . 76 N 5.1.1 Non-linear sigma model I . 83 3 5.2 = 1 SuperYang-Mills . 87 N 5.3 = 1 Gauge-matter actions . 91 N 5.3.1 Classical moduli space: examples . 95 5.3.2 The SuperHiggs mechanism . 102 5.3.3 Non-linear sigma model II . 104 5.4 Exercises . 106 6 Theories with extended supersymmetry 108 6.1 = 2 supersymmetric actions . 108 N 6.1.1 Non-linear sigma model III . 111 6.2 = 4 supersymmetric actions . 113 N 6.3 On non-renormalization theorems . 114 7 Supersymmetry breaking 122 7.1 Vacua in supersymmetric theories . 122 7.2 Goldstone theorem and the goldstino . 124 7.3 F-term breaking . 126 7.4 Pseudomoduli space: quantum corrections . 137 7.5 D-term breaking . 141 7.6 Indirect criteria for supersymmetry breaking . 144 7.6.1 Supersymmetry breaking and global symmetries . 144 7.6.2 Topological constraints: the Witten Index . 147 7.6.3 Genericity and metastability . 153 7.7 Exercises . 154 8 Mediation of supersymmetry breaking 156 8.1 Towards dynamical supersymmetry breaking . 156 8.2 The Supertrace mass formula . 158 8.3 Beyond Minimal Supersymmetric Standard Model . 160 8.4 Spurions, soft terms and the messenger paradigm . 161 8.5 Mediating the breaking . 164 8.5.1 Gravity mediation . 165 4 8.5.2 Gauge mediation . 168 8.6 Exercises . 172 9 Non-perturbative effects and holomorphy 174 9.1 Instantons and anomalies in a nutshell . 174 9.2 't Hooft anomaly matching condition . 178 9.3 Holomorphy . 180 9.4 Holomorphy and non-renormalization theorems . 182 9.5 Holomorphic decoupling . 192 9.6 Exercises . 195 10 Supersymmetric gauge dynamics: = 1 196 N 10.1 Confinement and mass gap in QCD, YM and SYM . 196 10.2 Phases of gauge theories: examples . 206 10.2.1 Coulomb phase and free phase . 207 10.2.2 Continuously connected phases . 208 10.3 N=1 SQCD: perturbative analysis . 209 10.4 N=1 SQCD: non-perturbative dynamics . 211 10.4.1 Pure SYM: gaugino condensation . 212 10.4.2 SQCD for F < N: the ADS superpotential . 214 10.4.3 Integrating in and out: the linearity principle . 220 10.4.4 SQCD for F = N and F = N +1................ 224 10.4.5 Conformal window . 231 10.4.6 Electric-magnetic duality (aka Seiberg duality) . 234 10.5 The phase diagram of N=1 SQCD . 242 10.6 Exercises . 243 11 Dynamical supersymmetry breaking 246 11.1 Calculable and non-calculable models: generalities . 246 11.2 The one GUT family SU(5) model . 249 11.3 The 3-2 model: instanton driven SUSY breaking . 251 11.4 The 4-1 model: gaugino condensation driven SUSY breaking . 257 5 11.5 The ITIY model: SUSY breaking with classical flat directions . 259 11.6 DSB into metastable vacua. A case study: massive SQCD . 262 11.6.1 Summary of basic results . 262 11.6.2 Massive SQCD in the free magnetic phase: electric description 264 11.6.3 Massive SQCD in the free magnetic phase: magnetic description265 11.6.4 Summary of the physical picture . 273 12 Supersymmetric gauge dynamics: extended supersymmetry 276 12.1 Low energy effective actions: classical and quantum . 276 12.1.1 = 2 effective actions . 278 N 12.1.2 = 4 effective actions . 284 N 12.2 Monopoles, dyons and electric-magnetic duality . 285 12.3 Seiberg-Witten theory . 294 12.3.1 = 2 SU(2) pure SYM . 296 N 12.3.2 Intermezzo: confinement by monopole condensation . 307 12.3.3 Seiberg-Witten theory: generalizations . 309 12.4 = 4: Montonen-Olive duality . 316 N 6 1 Supersymmetry: a bird eyes view Coming years could represent a new era of unexpected and exciting discoveries in high energy physics, since a long time. For one thing, the CERN Large Hadron Collider (LHC) has been operating for some time, now, and many experimental data have already being collected. So far, the greatest achievement of the LHC has been the discovery of the missing building block of the Standard Model, the Higgs particle (or, at least, a particle which most likely is the Standard Model Higgs particle). On the other hand, no direct evidence of new physics beyond the Standard Model has been found, yet. Still, there are several reasons to believe that new physics should in fact show-up at, or about, the TeV scale. The most compelling scenario for physics beyond the Standard Model (BSM) is supersymmetry. For this reason, knowing what is supersymmetry is rather impor- tant for a high energy physicist, nowadays. Understanding how supersymmetry can be realized (and then spontaneously broken) in Nature, is in fact one of the most important challenges theoretical high energy physics has to confront with. This course provides an introduction to such fascinating subject. Before entering into any detail, in this first lecture we just want to give a brief overview on what is supersymmetry and why is it interesting to study it. The rest of the course will try to provide (much) more detailed answers to these two basic questions. Disclaimer: The theory we are going to focus our attention in the (more than) three-hundred pages which follow, can be soon proved to be the correct mathematical framework where to understand high energy physics at the TeV scale, and become a piece of basic knowledge any particle physicist should have. But it can very well be that BSM physics is more subtle and Nature not so kind to make supersymmetry be realized at low enough energy that we can make experiment of. Or, it can also be that all this will eventually turn out to be just a purely academic exercise about a theory that nothing has to do with Nature. An elegant way mankind has worked out to describe in an unique and self-consistent way elementary particle physics, which however is not the one chosen by Nature (but can we ever safely say so?). As I will briefly outline below, and discuss in more detail in the second part of this course, even in such a scenario... studying supersymmetry and its fascinating properties might still be helpful and instructive in many respects. 7 1.1 What is supersymmetry? Supersymmetry (SUSY) is a space-time symmetry mapping particles and fields of integer spin (bosons) into particles and fields of half integer spin (fermions), and viceversa. The generators Q act as Q F ermion = Boson and viceversa (1.1) j i j i From its very definition, this operator has two obvious but far-reaching properties that can be summarized as follows: • It changes the spin of a particle (meaning that Q transforms as a spin-1/2 particle) and hence its space-time properties.
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