DOCSLIB.ORG

Lectures on Supersymmetry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lectures on Supersymmetry 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.
Load more

Recommended publications
  • Compactifying M-Theory on a G2 Manifold to Describe/Explain Our World – Predictions for LHC (Gluinos, Winos, Squarks), and Dark Matter
    Compactifying M-theory on a G2 manifold to describe/explain our world – Predictions for LHC (gluinos, winos, squarks), and dark matter Gordy Kane CMS, Fermilab, April 2016 1 OUTLINE • Testing theories in physics – some generalities - Testing 10/11 dimensional string/M-theories as underlying theories of our world requires compactification to four space-time dimensions! • Compactifying M-theory on “G2 manifolds” to describe/ explain our vacuum – underlying theory - fluxless sector! • Moduli – 4D manifestations of extra dimensions – stabilization - supersymmetry breaking – changes cosmology first 16 slides • Technical stuff – 18-33 - quickly • From the Planck scale to EW scale – 34-39 • LHC predictions – gluino about 1.5 TeV – also winos at LHC – but not squarks - 40-47 • Dark matter – in progress – surprising – 48 • (Little hierarchy problem – 49-51) • Final remarks 1-5 2 String/M theory a powerful, very promising framework for constructing an underlying theory that incorporates the Standard Models of particle physics and cosmology and probably addresses all the questions we hope to understand about the physical universe – we hope for such a theory! – probably also a quantum theory of gravity Compactified M-theory generically has gravity; Yang- Mills forces like the SM; chiral fermions like quarks and leptons; softly broken supersymmetry; solutions to hierarchy problems; EWSB and Higgs physics; unification; small EDMs; no flavor changing problems; partially observable superpartner spectrum; hidden sector DM; etc Simultaneously – generically Argue compactified M-theory is by far the best motivated, and most comprehensive, extension of the SM – gets physics relevant to the LHC and Higgs and superpartners right – no ad hoc inputs or free parameters Take it very seriously 4 So have to spend some time explaining derivations, testability of string/M theory Don’t have to be somewhere to test theory there – E.g.
    [Show full text]
  • Report of the Supersymmetry Theory Subgroup
    Report of the Supersymmetry Theory Subgroup J. Amundson (Wisconsin), G. Anderson (FNAL), H. Baer (FSU), J. Bagger (Johns Hopkins), R.M. Barnett (LBNL), C.H. Chen (UC Davis), G. Cleaver (OSU), B. Dobrescu (BU), M. Drees (Wisconsin), J.F. Gunion (UC Davis), G.L. Kane (Michigan), B. Kayser (NSF), C. Kolda (IAS), J. Lykken (FNAL), S.P. Martin (Michigan), T. Moroi (LBNL), S. Mrenna (Argonne), M. Nojiri (KEK), D. Pierce (SLAC), X. Tata (Hawaii), S. Thomas (SLAC), J.D. Wells (SLAC), B. Wright (North Carolina), Y. Yamada (Wisconsin) ABSTRACT Spacetime supersymmetry appears to be a fundamental in- gredient of superstring theory. We provide a mini-guide to some of the possible manifesta- tions of weak-scale supersymmetry. For each of six scenarios These motivations say nothing about the scale at which nature we provide might be supersymmetric. Indeed, there are additional motiva- tions for weak-scale supersymmetry. a brief description of the theoretical underpinnings, Incorporation of supersymmetry into the SM leads to a so- the adjustable parameters, lution of the gauge hierarchy problem. Namely, quadratic divergences in loop corrections to the Higgs boson mass a qualitative description of the associated phenomenology at future colliders, will cancel between fermionic and bosonic loops. This mechanism works only if the superpartner particle masses comments on how to simulate each scenario with existing are roughly of order or less than the weak scale. event generators. There exists an experimental hint: the three gauge cou- plings can unify at the Grand Uni®cation scale if there ex- I. INTRODUCTION ist weak-scale supersymmetric particles, with a desert be- The Standard Model (SM) is a theory of spin- 1 matter tween the weak scale and the GUT scale.
    [Show full text]
  • String Theory. Volume 1, Introduction to the Bosonic String
    This page intentionally left blank String Theory, An Introduction to the Bosonic String The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal field theory. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactification and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the finiteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersym- metric string theories and goes on to a broad presentation of the important advances of recent years. The first three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal field theory. The final four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries.
    [Show full text]
  • Flux Moduli Stabilisation, Supergravity Algebras and No-Go Theorems Arxiv
    IFT-UAM/CSIC-09-36 Flux moduli stabilisation, Supergravity algebras and no-go theorems Beatriz de Carlosa, Adolfo Guarinob and Jes´usM. Morenob a School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK b Instituto de F´ısicaTe´oricaUAM/CSIC, Facultad de Ciencias C-XVI, Universidad Aut´onomade Madrid, Cantoblanco, 28049 Madrid, Spain Abstract We perform a complete classification of the flux-induced 12d algebras compatible with the set of N = 1 type II orientifold models that are T-duality invariant, and allowed by the symmetries of the 6 ¯ T =(Z2 ×Z2) isotropic orbifold. The classification is performed in a type IIB frame, where only H3 and Q fluxes are present. We then study no-go theorems, formulated in a type IIA frame, on the existence of Minkowski/de Sitter (Mkw/dS) vacua. By deriving a dictionary between the sources of potential energy in types IIB and IIA, we are able to combine algebra results and no-go theorems. The outcome is a systematic procedure for identifying phenomenologically viable models where Mkw/dS vacua may exist. We present a complete table of the allowed algebras and the viability of their resulting scalar potential, and we point at the models which stand any chance of producing a fully stable vacuum. arXiv:0907.5580v3 [hep-th] 20 Jan 2010 e-mail: [email protected] , [email protected] , [email protected] Contents 1 Motivation and outline 1 2 Fluxes and Supergravity algebras 3 2.1 The N = 1 orientifold limits as duality frames . .4 2.2 T-dual algebras in the isotropic Z2 × Z2 orientifolds .
    [Show full text]
  • An Introduction to Supersymmetry
    An Introduction to Supersymmetry Ulrich Theis Institute for Theoretical Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, D–07743 Jena, Germany [email protected] This is a write-up of a series of five introductory lectures on global supersymmetry in four dimensions given at the 13th “Saalburg” Summer School 2007 in Wolfersdorf, Germany. Contents 1 Why supersymmetry? 1 2 Weyl spinors in D=4 4 3 The supersymmetry algebra 6 4 Supersymmetry multiplets 6 5 Superspace and superfields 9 6 Superspace integration 11 7 Chiral superfields 13 8 Supersymmetric gauge theories 17 9 Supersymmetry breaking 22 10 Perturbative non-renormalization theorems 26 A Sigma matrices 29 1 Why supersymmetry? When the Large Hadron Collider at CERN takes up operations soon, its main objective, besides confirming the existence of the Higgs boson, will be to discover new physics beyond the standard model of the strong and electroweak interactions. It is widely believed that what will be found is a (at energies accessible to the LHC softly broken) supersymmetric extension of the standard model. What makes supersymmetry such an attractive feature that the majority of the theoretical physics community is convinced of its existence? 1 First of all, under plausible assumptions on the properties of relativistic quantum field theories, supersymmetry is the unique extension of the algebra of Poincar´eand internal symmtries of the S-matrix. If new physics is based on such an extension, it must be supersymmetric. Furthermore, the quantum properties of supersymmetric theories are much better under control than in non-supersymmetric ones, thanks to powerful non- renormalization theorems.
    [Show full text]
  • Supersymmetric Nonlinear Sigma Models
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server OU-HET 348 TIT/HET-448 hep-th/0006025 June 2000 Supersymmetric Nonlinear Sigma Models a b Kiyoshi Higashijima ∗ and Muneto Nitta † aDepartment of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan bDepartment of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152-8551, Japan Abstract Supersymmetric nonlinear sigma models are formulated as gauge theo- ries. Auxiliary chiral superfields are introduced to impose supersymmetric constraints of F-type. Target manifolds defined by F-type constraints are al- ways non-compact. In order to obtain nonlinear sigma models on compact manifolds, we have to introduce gauge symmetry to eliminate the degrees of freedom in non-compact directions. All supersymmetric nonlinear sigma models defined on the hermitian symmetric spaces are successfully formulated as gauge theories. 1Talk given at the International Symposium on \Quantum Chromodynamics and Color Con- finement" (Confinement 2000) , 7-10 March 2000, RCNP, Osaka, Japan. ∗e-mail: [email protected]. †e-mail: [email protected] 1 Introduction Two dimensional (2D) nonlinear sigma models and four dimensional non-abelian gauge theories have several similarities. Both of them enjoy the property of the asymptotic freedom. They are both massless in the perturbation theory, whereas they acquire the mass gap or the string tension in the non-perturbative treatment. Although it is difficult to solve QCD in analytical way, 2D nonlinear sigma models can be solved by the large N expansion and helps us to understand various non- perturbative phenomena in four dimensional gauge theories.
    [Show full text]
  • Super-Higgs in Superspace
    Article Super-Higgs in Superspace Gianni Tallarita 1,* and Moritz McGarrie 2 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7941169, Chile 2 Deutsches Elektronen-Synchrotron, DESY, Notkestrasse 85, 22607 Hamburg, Germany; [email protected] * Correspondence: [email protected] or [email protected] Received: 1 April 2019; Accepted: 10 June 2019; Published: 14 June 2019 Abstract: We determine the effective gravitational couplings in superspace whose components reproduce the supergravity Higgs effect for the constrained Goldstino multiplet. It reproduces the known Gravitino sector while constraining the off-shell completion. We show that these couplings arise by computing them as quantum corrections. This may be useful for phenomenological studies and model-building. We give an example of its application to multiple Goldstini. Keywords: supersymmetry; Goldstino; superspace 1. Introduction The spontaneous breakdown of global supersymmetry generates a massless Goldstino [1,2], which is well described by the Akulov-Volkov (A-V) effective action [3]. When supersymmetry is made local, the Gravitino “eats” the Goldstino of the A-V action to become massive: The super-Higgs mechanism [4,5]. In terms of superfields, the constrained Goldstino multiplet FNL [6–12] is equivalent to the A-V formulation (see also [13–17]). It is, therefore, natural to extend the description of supergravity with this multiplet, in superspace, to one that can reproduce the super-Higgs mechanism. In this paper we address two issues—first we demonstrate how the Gravitino, Goldstino, and multiple Goldstini obtain a mass. Secondly, by using the Spurion analysis, we write down the most minimal set of new terms in superspace that incorporate both supergravity and the Goldstino multiplet in order to reproduce the super-Higgs mechanism of [5,18] at lowest order in M¯ Pl.
    [Show full text]
  • TASI 2008 Lectures: Introduction to Supersymmetry And
    TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg
    [Show full text]
  • International Theoretical Physics
    INTERNATIONAL THEORETICAL PHYSICS INTERACTION OF MOVING D-BRANES ON ORBIFOLDS INTERNATIONAL ATOMIC ENERGY Faheem Hussain AGENCY Roberto Iengo Carmen Nunez UNITED NATIONS and EDUCATIONAL, Claudio A. Scrucca SCIENTIFIC AND CULTURAL ORGANIZATION CD MIRAMARE-TRIESTE 2| 2i CO i IC/97/56 ABSTRACT SISSAREF-/97/EP/80 We use the boundary state formalism to study the interaction of two moving identical United Nations Educational Scientific and Cultural Organization D-branes in the Type II superstring theory compactified on orbifolds. By computing the and velocity dependence of the amplitude in the limit of large separation we can identify the International Atomic Energy Agency nature of the different forces between the branes. In particular, in the Z orbifokl case we THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 3 find a boundary state which is coupled only to the N = 2 graviton multiplet containing just a graviton and a vector like in the extremal Reissner-Nordstrom configuration. We also discuss other cases including T4/Z2. INTERACTION OF MOVING D-BRANES ON ORBIFOLDS Faheem Hussain The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, Roberto Iengo International School for Advanced Studies, Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy, Carmen Nunez1 Instituto de Astronomi'a y Fisica del Espacio (CONICET), Buenos Aires, Argentina and Claudio A. Scrucca International School for Advanced Studies, Trieste, Italy and INFN, Sezione di Trieste, Trieste, Italy. MIRAMARE - TRIESTE February 1998 1 Regular Associate of the ICTP. The non-relativistic dynamics of Dirichlet branes [1-3], plays an essential role in the coordinate satisfies Neumann boundary conditions, dTX°(T = 0, a) = drX°(r = I, a) = 0.
    [Show full text]
  • Aspects of Supersymmetry and Its Breaking
    1 Aspects of Supersymmetry and its Breaking a b Thomas T. Dumitrescu ∗ and Zohar Komargodski † aDepartment of Physics, Princeton University, Princeton, New Jersey, 08544, USA bSchool of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540, USA We describe some basic aspects of supersymmetric field theories, emphasizing the structure of various supersym- metry multiplets. In particular, we discuss supercurrents – multiplets which contain the supersymmetry current and the energy-momentum tensor – and explain how they can be used to constrain the dynamics of supersym- metric field theories, supersymmetry breaking, and supergravity. These notes are based on lectures delivered at the Carg´ese Summer School 2010 on “String Theory: Formal Developments and Applications,” and the CERN Winter School 2011 on “Supergravity, Strings, and Gauge Theory.” 1. Supersymmetric Theories In this review we will describe some basic aspects of SUSY field theories, emphasizing the structure 1.1. Supermultiplets and Superfields of various supermultiplets – especially those con- A four-dimensional theory possesses = 1 taining the supersymmetry current. In particu- supersymmetry (SUSY) if it contains Na con- 1 lar, we will show how these supercurrents can be served spin- 2 charge Qα which satisfies the anti- used to study the dynamics of supersymmetric commutation relation field theories, SUSY-breaking, and supergravity. ¯ µ Qα, Qα˙ = 2σαα˙ Pµ . (1) We begin by recalling basic facts about super- { } multiplets and superfields. A set of bosonic and Here Q¯ is the Hermitian conjugate of Q . (We α˙ α fermionic operators B(x) and F (x) fur- will use bars throughout to denote Hermitian con- i i nishes a supermultiplet{O if these} operators{O satisfy} jugation.) Unless otherwise stated, we follow the commutation relations of the schematic form conventions of [1].
    [Show full text]
  • Supersymmetric R4 Actions and Quantum Corrections to Superspace Torsion Constraints
    DAMTP-2000-91, NORDITA-2000/87 HE, SPHT-T00/117, hep-th/0010182 SUPERSYMMETRIC R4 ACTIONS AND QUANTUM CORRECTIONS TO SUPERSPACE TORSION CONSTRAINTS K. Peetersa, P. Vanhoveb and A. Westerbergc a CERN, TH-division 1211 Geneva 23, Switzerland b SPHT, Orme des Merisiers, CEA / Saclay, 91191 Gif-sur-Yvette, France c NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark k.peeters, [email protected], [email protected] We present the supersymmetrisation of the anomaly-related R4 term in eleven dimensions and show that it induces no non-trivial modifications to the on-shell supertranslation algebra and the superspace torsion constraints before inclusion of gauge-field terms.1 1 Higher-derivative corrections and supersymmetry The low-energy supergravity limits of superstring theory as well as D-brane effective actions receive infinite sets of correction terms, proportional to in- 2 creasing powers of α0 = ls and induced by superstring theory massless and massive modes. At present, eleven-dimensional supergravity lacks a corre- sponding microscopic underpinning that could similarly justify the presence of higher-derivative corrections to the classical Cremmer-Julia-Scherk action [1]. Nevertheless, some corrections of this kind are calculable from unitarity argu- ments and super-Ward identities in the massless sector of the theory [2] or by anomaly cancellation arguments [3, 4]. Supersymmetry puts severe constraints on higher-derivative corrections. For example, it forbids the appearance of certain corrections (like, e.g, R3 corrections to supergravity effective actions [5]), and groups terms into var- ious invariants [6–9]. The structure of the invariants that contain anomaly- cancelling terms is of great importance due to the quantum nature of the 1Based on talks given by K.P.
    [Show full text]
  • Introduction to Supersymmetry
    Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × .
    [Show full text]