DOCSLIB.ORG

Introduction to Supersymmetry (Phys 661)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Supersymmetry (Phys 661) INTRODUCTION TO SUPERSYMMETRY (PHYS 661) Instructor: Philip Argyres (pronounced “are–JEER–us”) Office: Room 334 Newman Lab., 255-4118, [email protected]. Course Outline I. Qualitative supersymmetry — 4 lectures, few indices a. Coleman-Mandula theorem b. Supersymmetric QM: vacua, superfields, & instantons II. Perturbative supersymmetry — 8 lectures, the basics a. Chiral multiplets b. Nonrenormalization theorems c. Vector multiplets III. Supersymmetric model building — 4 lectures, qualitative issues a. Supersymmetric standard model b. Soft supersymmetry breaking terms c. Messenger sectors IV. Non-perturbative supersymmetry — 9 lectures, mostly SU(n) SQCD a. Higgs vacua (& instantons) b. Coulomb vacua (& monopoles) c. Chiral theories V. Dynamical supersymmetry breaking — 2 lectures, if I get to it There is no text covering the contents of this course. Some useful references for its various parts are: 1. Qualitative supersymmetry: E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B188 (1981) 513; S. Coleman, “The uses of instantons,” in The Whys of Subnuclear Physics (Plenum, 1979), and in Aspects of Symmetry (Cambridge, 1985). 2. Perturbative supersymmetry: J. Wess and J. Bagger, “Supersymmetry and supergrav- ity,” 2nd ed., Ch. I–VIII, XXII, App. A–C. Note: I will try to follow the notation and conventions of this book in the course. 3-5. Phenomenology and non-perturbative methods: Various reviews and papers which I’ll mention as we get closer to these topics. Prerequisites are a basic knowledge of QFT (gauge theories, path integrals, 1-loop RG) on the physics side, and an aquaintance with analysis on the complex plane (holomorphy, analytic continuation) as well as rudimentary group theory (SU(3), Lorentz group, spinors) on the math side. i This is a pass/fail course. There will be no tests or final. The grade will be based on participation and doing or discussing with me problems which will be given during the course of the lectures and will also be posted on the web page http://www.hepth.cornell.edu/~argyres/phys661/index.html which can also be found by folowing the appropriate links from the high energy theory home page. These problems are meant to be simple exercises, not lengthy or difficult research projects—so please keep solutions brief, and be sure to come talk to me if you are having trouble with them. My office hours are the hour after lectures and other times by appointment. ii I. Qualitative Supersymmetry 1. The Coleman-Mandula Theorem 1.1. Introduction This course is an introduction to 4-dimensional global N=1 supersymmetric field theory, so not, in particular, other dimensions, supergravity, or extended supersymmetry (except very briefly). I’ll introduce the basics of perturbative supersymmetry and apply them to a critical survey of models of weak-scale supersymmetry in the 1st half of the semester. The first two weeks will introduce supersymmetric quantum mechanics (QM) to try to separate the features special to supersymmetry from the complications of QFT in 3 + 1 dimensions. The 2nd half will be devoted to exploring the non-perturbative dynamics of supersymmetric FTs. Throughout the course I will introduce and use advanced QFT techniques (effective actions, RG flows, anomalies, instantons, ...) when needed, and I will try to emphasize qualitative explanations and symmetry-based techniques and “tricks.” Many of these techniques, though used frequently in QFT, are not usually taught in the standard QFT graduate courses. The aims of this course are two-fold. The first is to supply you with the wherewithall to evaluate the various claims/hopes for weak-scale supersymmetry. The second, and closer to my interests, is to use supersymmetric models as a window on QFT in general. From this point of view, supersymmetric FTs are just especially symmetric versions of ordinary field theories, and in many cases this extra symmetry has allowed us to solve exactly for some non-perturbative properties of these theories. This gives us another context (besides lattice gauge theory) in which to think concretely about non-perturbative QFT in 3 + 1 dimensions. Indeed, the hard part of this course will be the QFT, not the supersymmetry—someone who is thoroughly familiar with QFT should find the content of this course, though perhaps unfamiliar, relatively easy to understand. Finally, this course owes alot to Nathan Seiberg: not only is his work the main focus of the 2nd half of the course, but also much of this course is closely modeled on two series of lectures he gave at the IAS in the fall of 1994 and at Rutgers in the fall of 1995. 1.2. The Coleman-Mandula theorem, or, Why supersymmetry? Though originally introduced in early 70’s we still don’t know how or if supersymmetry plays a role in nature. Supersymmetry today is like non-Abelian gauge theories before the SM: “a fascinating mathematical structure, and a reasonable extension of current ideas, but plagued with phenomenological difficulties.” 1 Why, then, have a considerable number of people been working on this theory for the last 20 years? The answer lies in the Coleman-Mandula theorem, which singles-out supersymmetry as the “unique” extension of Poicar´einvariance in 3+1 or more dimensional QFT (under some important but reasonable assumptions). In the rest of this lecture, I follow the qualitative description of the Coleman-Mandula theorem given by E. Witten “Introduction to supersymmetry,” in Proc. Intern. School of Subnuclear Physics, Erice 1981, ed. A. Zichichi (Plenum Press, 1983) p. 305. A theory of 2 free bose fields has many conserved currents. x Exercise 1.1. Check that in a theory of 2 free bosons µ µ = ∂µφ1∂ φ1 + ∂µφ2∂ φ2, (1.1) L the following currents are conserved: Jµ = φ1∂µφ2 φ2∂µφ1, − Jµρ = ∂ρφ1∂µφ2 φ2∂µ∂ρφ1, (1.2) − Jµρσ = ∂ρ∂σφ1∂µφ2 φ2∂µ∂ρ∂σφ1. − There are interactions which when added to this theory still keep Jµ conserved. x Exercise 1.2. Check that Jµ is conserved if any interaction of the form 2 2 V = f(φ1 + φ2) is added to the 2 boson theory. However, there are no Lorentz-invariant interactions which can be added so that the others are conserved (nor can they be redefined by adding extra terms so that they will still be conserved). This follows from the Coleman-Mandula theorem Phys. Rev. D159 (1967) 1251: In a theory with non-trivial scattering in more than 1+1 dimensions, the only possible conserved quantities that transform as tensors under the Lorentz group are the usual energy-momentum Pµ, Lorentz transformations Mµν , and scalar quantum numbers Qi. (This has a conformal extension for massless particles.) The basic idea is that conservation of Pµ and Mµν leaves only the scattering angle un- known in (say) a 2-body collision. Additional “exotic” conservation laws would determine the scattering angle, leaving only a discrete set of possible angles. Since the scattering amplitude is an analytic function of angle (assumption # 1) it then vanishes for all angles. Concrete example: Suppose we have a conserved traceless symmetric tensor Qµν . By Lorentz invariance, its matrix element in a 1-particle state of momentum p and spin zero is 1 p Q p p p η p2. (1.3) h | µν | i ∝ µ ν − 4 µν Apply this to an elastic 2-body collision of identical particles with incoming momenta p1, p2, and outgoing momenta q1, q2, and assume that the matrix element of Q in the 2-particle 2 state p p is the sum of the matrix elements in the states p and p . This is true if Q | 1 2i | 1i | 2i is “not too non-local”—say, the integral of a local current (assumption # 2). x Exercise 1.3. Show that conservation of a symmetric, traceless charge Qµν together with energy momentum conservation implies µ ν µ ν µ ν µ ν p1 p1 + p2 p2 = q1 q1 + q2 q2 , (1.4) for an elastic scattering of two identical scalars with incoming momenta p1, p2, and outgoing momenta q1, q2. Show that this implies the scattering angle is zero. For the extension of this argument to non-identical particles, particles with spin, inelastic collision, see Coleman and Mandula’s paper. The Coleman-Mandula theorem does not mention spinor charges, though. So consider a free theory of a complex scalar and a free two component (Weyl) fermion = ∂ φ∂µφ + iψσµ∂ ψ. (1.5) L µ µ Again, an infinite number of conserved currents exist. x Exercise 1.4. In a theory of a free complex boson and a free 2-component fermion, show that the currents ρ Sµα =(∂ρφσ σµψ)α, (1.6) ρ Sµνα =(∂ρφσ σµ∂ν ψ)α, are conserved. (σ and σ obey the Dirac-like algebra σµσν + σν σµ ηµν . We will discuss 2-component spinors in detail in lecture 5.) ∝ Now, there are interactions, e.g. V = g(φψαψ + h.c.) + g2 φ 4, that can be added to this α | | free theory such that Sµα (with correction proportional to g) remains conserved. (We will return to this example in much more detail in later lectures.) However, Sµνα is never conserved in the presence of interactions. We can see this by applying the Coleman-Mandula theorem to the anticommutators of the fermionic conserved charges 3 Qα = d x S0α, Z (1.7) 3 Qνα = d x S0να. Z Indeed, consider the anticommutator Q , Q , which cannot vanish unless Q is iden- { να µβ} να tically zero, since the anticommutator of any operator with its hermitian adjoint is positive definite. Since Qνα has components of spin up to 3/2, the anticommutator has compo- nents of spin up to 3. Since the anticommutator is conserved if Qνα is, and since the Coleman-Mandula theorem does not permit conservation of an operator of spin 3 in an 3 interacting theory, Qνα cannot be conserved in an interacting theory.
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]
  • An Introduction to Quantum Field Theory
    AN INTRODUCTION TO QUANTUM FIELD THEORY By Dr M Dasgupta University of Manchester Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009 - 1 - - 2 - Contents 0 Prologue....................................................................................................... 5 1 Introduction ................................................................................................ 6 1.1 Lagrangian formalism in classical mechanics......................................... 6 1.2 Quantum mechanics................................................................................... 8 1.3 The Schrödinger picture........................................................................... 10 1.4 The Heisenberg picture............................................................................ 11 1.5 The quantum mechanical harmonic oscillator ..................................... 12 Problems .............................................................................................................. 13 2 Classical Field Theory............................................................................. 14 2.1 From N-point mechanics to field theory ............................................... 14 2.2 Relativistic field theory ............................................................................ 15 2.3 Action for a scalar field ............................................................................ 15 2.4 Plane wave solution to the Klein-Gordon equation ...........................
    [Show full text]
  • Localization at Large N †
    NORDITA-2013-97 UUITP-21/13 Localization at Large N † J.G. Russo1;2 and K. Zarembo3;4;5 1 Instituci´oCatalana de Recerca i Estudis Avan¸cats(ICREA), Pg. Lluis Companys, 23, 08010 Barcelona, Spain 2 Department ECM, Institut de Ci`enciesdel Cosmos, Universitat de Barcelona, Mart´ıFranqu`es,1, 08028 Barcelona, Spain 3Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 4Department of Physics and Astronomy, Uppsala University SE-751 08 Uppsala, Sweden 5Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia [email protected], [email protected] Abstract We review how localization is used to probe holographic duality and, more generally, non-perturbative dynamics of four-dimensional N = 2 supersymmetric gauge theories in the planar large-N limit. 1 Introduction 5 String theory on AdS5 × S gives a holographic description of the superconformal N = 4 Yang- arXiv:1312.1214v1 [hep-th] 4 Dec 2013 Mills (SYM) through the AdS/CFT correspondence [1]. This description is exact, it maps 5 correlation functions in SYM, at any coupling, to string amplitudes in AdS5 × S . Gauge- string duality for less supersymmetric and non-conformal theories is at present less systematic, and is mostly restricted to the classical gravity approximation, which in the dual field theory corresponds to the extreme strong-coupling regime. For this reason, any direct comparison of holography with the underlying field theory requires non-perturbative input on the field-theory side. There are no general methods, of course, but in the basic AdS/CFT context a variety of tools have been devised to gain insight into the strong-coupling behavior of N = 4 SYM, †Talk by K.Z.
    [Show full text]
  • Quantum Field Theory*
    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
    [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]
  • Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions
    PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 05 Tuesday, January 14, 2020 Topics: Feynman Diagrams, Momentum Space Feynman Rules, Disconnected Diagrams, Higher Correlation Functions. Feynman Diagrams We can use the Wick’s theorem to express the n-point function h0jT fφ(x1)φ(x2) ··· φ(xn)gj0i as a sum of products of Feynman propagators. We will be interested in developing a diagrammatic interpretation of such expressions. Let us look at the expression containing four fields. We have h0jT fφ1φ2φ3φ4g j0i = DF (x1 − x2)DF (x3 − x4) +DF (x1 − x3)DF (x2 − x4) +DF (x1 − x4)DF (x2 − x3): (1) We can interpret Eq. (1) as the sum of three diagrams, if we represent each of the points, x1 through x4 by a dot, and each factor DF (x − y) by a line joining x to y. These diagrams are called Feynman diagrams. In Fig. 1 we provide the diagrammatic interpretation of Eq. (1). The interpretation of the diagrams is the following: particles are created at two spacetime points, each propagates to one of the other points, and then they are annihilated. This process can happen in three different ways, and they correspond to the three ways to connect the points in pairs, as shown in the three diagrams in Fig. 1. Thus the total amplitude for the process is the sum of these three diagrams. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 ⟨0|T {ϕ1ϕ2ϕ3ϕ4}|0⟩ = DF(x1 − x2)DF(x3 − x4) + DF(x1 − x3)DF(x2 − x4) +DF(x1 − x4)DF(x2 − x3) 1 2 1 2 1 2 + + 3 4 3 4 3 4 Figure 1: Diagrammatic interpretation of Eq.
    [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]
  • 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]
  • SUSY and XD Notes
    Supersymmetry and Extra Dimensions Flip Tanedo LEPP, Cornell University Ithaca, New York A pedagogical set of notes based on lectures by Fernando Quevedo, Adrian Signer, and Csaba Cs´aki,as well as various books and reviews. Last updated: January 9, 2009 ii iii Abstract This is a set of combined lecture notes on supersymmetry and extra dimensions based on various lectures, textbooks, and review articles. The core of these notes come from Professor Fernando Quevedo's 2006- 2007 Lent Part III lecture course of the same name [1]. iv v Acknowledgements Inspiration to write up these notes in LATEX came from Steffen Gielen's excel- lent notes from the Part III Advanced Quantum Field Theory course and the 2008 ICTP Introductory School on the Gauge Gravity Correspondence. Notes from Profes- sor Quevedo's 2005-2006 Part III Supersymmetry and Extra Dimensions course exist in TEXform due to Oliver Schlotterer. The present set of notes were written up indepen- dently, but simliarities are unavoidable. It is my hope that these notes will provide a broader pedagogical introduction supersymmetry and extra dimensions. vi vii Preface These are lecture notes. Version 1 of these notes are based on Fernando Quevedo's lecture notes and structure. I've also incorporated some relevant topics from my research that I think are important to round-out the course. Version 2 of these notes will also incorporate Csaba Cs´aki'sAdvanced Particle Physics notes. Framed text. Throughout these notes framed text will include parenthetical dis- cussions that may be omitted on a first reading. They are meant to provide a broader picture or highlight particular applications that are not central to the main purpose of the chapter.
    [Show full text]
  • Interpreting Supersymmetry
    Interpreting Supersymmetry David John Baker Department of Philosophy, University of Michigan [email protected] October 7, 2018 Abstract Supersymmetry in quantum physics is a mathematically simple phenomenon that raises deep foundational questions. To motivate these questions, I present a toy model, the supersymmetric harmonic oscillator, and its superspace representation, which adds extra anticommuting dimensions to spacetime. I then explain and comment on three foundational questions about this superspace formalism: whether superspace is a sub- stance, whether it should count as spatiotemporal, and whether it is a necessary pos- tulate if one wants to use the theory to unify bosons and fermions. 1 Introduction Supersymmetry{the hypothesis that the laws of physics exhibit a symmetry that transforms bosons into fermions and vice versa{is a long-standing staple of many popular (but uncon- firmed) theories in particle physics. This includes several attempts to extend the standard model as well as many research programs in quantum gravity, such as the failed supergravity program and the still-ascendant string theory program. Its popularity aside, supersymmetry (SUSY for short) is also a foundationally interesting hypothesis on face. The fundamental equivalence it posits between bosons and fermions is prima facie puzzling, given the very different physical behavior of these two types of particle. And supersymmetry is most naturally represented in a formalism (called superspace) that modifies ordinary spacetime by adding Grassmann-valued anticommuting coordinates. It 1 isn't obvious how literally we should interpret these extra \spatial" dimensions.1 So super- symmetry presents us with at least two highly novel interpretive puzzles. Only two philosophers of science have taken up these questions thus far.
    [Show full text]