Aspects of Supersymmetry and Its Breaking

Total Page:16

File Type:pdf, Size:1020Kb

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]. In local quantum field theo- B F F ries, the basic objects of interest are well-defined Qα, i (x) j (x) + ∂ k (x) + , local operators. In SUSY field theories all such O ∼ O O · · · Q , F (x) B(x) + ∂ B(x) + , (3) operators must be embedded in multiplets of the " α Oi # ∼ Oj Ok · · · supersymmetry algebra, or supermultiplets. Con- and$ likewise%for the Q¯α˙ commutators, such that served currents furnish an important class of local the SUSY algebra (1) is satisfied. It is straightfor- operators. Of particular interest is the supersym- ward to show that a supermultiplet must contain metry current Sαµ, which satisfies equally many independent bosonic and fermionic operators (see, for instance, [2]). µ 3 0 ∂ Sαµ = 0 , Qα = d x Sα . (2) It is always possible (and very convenient) to ! B,F embed the component fields i (x) of a super- ∗T.D. is supported by a DOE Fellowship in High Energy O ¯ Theory, NSF grant PHY-0756966, and a Centennial Fel- multiplet in a superfield S(x, θ, θ). Here θα is an lowship from Princeton University. anti-commuting superspace coordinate. (For now †Z.K. is supported by DOE grant DE-FG02-90ER40542. we suppress any Lorentz indices carried by S.) Any opinions, findings, and conclusions or recommenda- The component fields are identified with the x- tions expressed in this material are those of the authors ¯ and do not necessarily reflect the views of the National dependent coefficients when S(x, θ, θ) is expanded Science Foundation. as a power series in θ, θ¯. The commutation rela- 2 tions (3) are succinctly encoded in the formula A general superfield does not furnish an irre- ducible representation of supersymmetry. To re- ξQ + ξ¯Q,¯ S = i ξ + ξ¯ ¯ S , (4) duce a supermultiplet, we impose supersymmet- Q Q ric constraints. This is most conveniently done in " # & ' which is the defining property of a superfield.3 terms of the superspace differential operators Here ξα is an arbitrary Grassmann parameter ∂ µ α˙ and α, ¯α˙ are the superspace differential oper- ¯ Dα = α + iσαα˙ θ ∂µ , atorsQ Q ∂θ ∂ D¯ = iσµ ∂ . (7) α˙ α˙ αα˙ µ ∂ µ −∂θ¯ − = iσ θ¯α˙ ∂ , α ∂θα αα˙ µ Q − These anti-commute with the supersymmetry ∂ ¯ = + iσµ ∂ . (5) generators , ¯ , and thus map superfields to α˙ ¯α˙ αα˙ µ α α˙ Q −∂θ superfields.Q Hence,Q any constraint written in ¯ Conversely, the defining property (4) implies that terms of Dα, Dα˙ is automatically supersymmet- the components of any superfield furnish a su- ric. We are now ready to begin exploring various permultiplet. Thus, supermultiplets and super- important supermultiplets. fields are in one-to-one correspondence, and we 1.2. Chiral Multiplets and Lagrangians will treat them synonymously. The most familiar representation of supersym- To see this in a little more detail, consider the metry is the chiral multiplet. It is the basic build- component expansion of a general scalar super- ing block which enables us to write SUSY La- field: grangians describing only scalars and fermions. The bottom component of a chiral multiplet is an- S(x, θ, θ¯) = C(x)+iθαψ (x)+ +θ2θ¯2D(x) . (6) ¯ α · · · nihilated by Qα˙ . For example, the bottom com- ponent φ(x) of a scalar chiral multiplet satisfies As explained above, the defining property (4) de- termines the commutation relations of the super- Q¯α˙ , φ(x) = 0 . (8) charges Qα, Q¯α˙ with the component fields. These commutators show that the supercharges act as The" multiplet# obtained from φ(x) by acting with raising operators for the component fields.4 This the supercharges is organized in a superfield ¯ has two important consequences: which satisfies the constraint Dα˙ Φ = 0. This con- straint can be solved in components: The superfield S(x, θ, θ¯) can be constructed 2 • from its bottom component C(x) by apply- Φ = φ(y) + √2θψ(y) + θ F (y) , (9) ing the supercharges. Thus, any local oper- where yµ = xµ + θσµθ¯. We immediately note two ator can be embedded in the bottom com- key properties of chiral superfields: ponent of a superfield. However, it is not always possible to embed an operator in a Any function which depends on the chiral higher component. This will play a crucial • superfields Φi, but not their complex con- role in our analysis of various supercurrents. jugates, is again a chiral superfield. Such a function is said to be holomorphic in the Φi. The SUSY variation of the top compo- • The SUSY variation of F (x) is a total nent D(x) of any superfield is always a total • derivative. This fact will enable us to write derivative. supersymmetric Lagrangians. From (9) we see that Φ contains a complex scalar φ, a Weyl fermion ψ , and another com- 3 The factor of i in (4) is necessary for Hermiticity in α Minkowski space. plex scalar F which will turn out to be a non- 4 For instance, [Qα, C] = ψα. propagating auxilliary field. Among other things, − 3 F ensures that the chiral multiplet has four (real) taken to vanish.) After solving for the auxiliary bosonic degrees of freedom to match the four fields F i, which are now non-zero, the component fermionic degrees of freedom coming from ψα. We Lagrangian takes the form will abbreviate this by saying that Φ is a 4 + 4 ¯j µ i i µ ¯j L = g ¯ φ, φ¯ ∂ φ¯ ∂ φ + ig ¯ ∂ ψ σ ψ¯ multiplet. As advertised, Φ has exactly the right − ij µ ij µ field content to describe a theory of scalars and i k l µ ¯j 1 i k ¯j ¯l + ig ¯Γ &∂ φ ' ψ σ ψ¯ + R ¯ ¯ψ ψ ψ¯ ψ¯ (14). fermions, and we would like to write a Lagrangian ij kl µ 4 ijkl for such a theory. & ' This theory is known as the supersymmetric non- Up to total derivatives, a SUSY Lagrangian L linear σ-model. The bosonic part is an ordi- must be a real scalar whose variation under su- nary σ-model, whose target space is an N com- persymmetry is a total derivative. We can thus plex dimensional manifold with metric g ¯. Since take ij the Lagrangian (14) does not have a potential L = D + F + F¯ , (10) term, any point φi on the target manifold is a supersymmetric vacuum. (Recall that it follows where D is the top component of a real scalar from the SUSY algebra (1) that supersymmetric 2 superfield K = K¯ and F is the θ -component of vacua have zero energy; vacua with positive en- a chiral superfield W . This ensures that L is ergy spontaneously break SUSY.) In supersym- real and supersymmetric. For reasons that will metric theories, it is customary to refer to such be explained below, K is known as the K¨ahler manifolds of vacua as moduli spaces. As usual, potential, and W is called the superpotential. A the scalars φi should be thought of as coordinates particularly simple choice is to take K = ΦΦ¯ on the target space. This is consistent with the and W = 0. It is standard to pick out different fact that the theory is invariant under holomor- components of a superfield using Grassmann in- phic field redefinitions of the form tegration. For example, we can use d4θ to pick 2 i i j out the top component of a superfield, or d θ Φ" = f (Φ ) . (15) to pick out the θ2 component. We th(us write ( Such field redefinitions can be thought of as co- L = d4θ ΦΦ¯ , (11) ordinate changes on the target manifold under which the metric transforms in the usual tenso- ! up to total derivatives. In components: rial way. In the supersymmetric σ-model, the metric is L = ∂µφ∂¯ φ + i∂ ψσµψ¯ + F¯F . (12) determined by the K¨ahler potential: − µ µ This describes a free complex scalar φ, a free gi¯j = ∂i∂¯jK .
Recommended publications
  • Goldstini Can Give the Higgs a Boost
    Goldstini Can Give the Higgs a Boost The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Thaler, Jesse, and Zachary Thomas. “Goldstini can give the Higgs a boost.” Journal of High Energy Physics 2011.7 (2011). As Published http://dx.doi.org/10.1007/jhep07(2011)060 Publisher Springer Berlin/Heidelberg Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/72016 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike 3.0 Detailed Terms http://creativecommons.org/licenses/by-nc-sa/3.0/ Preprint typeset in JHEP style - HYPER VERSION MIT-CTP 4226 Goldstini Can Give the Higgs a Boost Jesse Thaler and Zachary Thomas Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: [email protected], [email protected] Abstract: Supersymmetric collider phenomenology depends crucially on whether the lightest observable-sector supersymmetric particle (LOSP) decays, and if so, what the LOSP decay products are. For instance, in SUSY models where the gravitino is lighter than the LOSP, the LOSP decays to its superpartner and a longitudinal gravitino via super- current couplings. In this paper, we show that LOSP decays can be substantially modified when there are multiple sectors that break supersymmetry, where in addition to the grav- itino there are light uneaten goldstini. As a particularly striking example, a bino-like LOSP can have a near 100% branching fraction to a higgs boson and an uneaten goldstino, even if the LOSP has negligible higgsino fraction. This occurs because the uneaten goldstino is unconstrained by the supercurrent, allowing additional operators to mediate LOSP decay.
    [Show full text]
  • Introductory Lectures on Quantum Field Theory
    Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties.
    [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]
  • Effective Field Theories, Reductionism and Scientific Explanation Stephan
    To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics.
    [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]
  • A Guiding Vector Field Algorithm for Path Following Control Of
    1 A guiding vector field algorithm for path following control of nonholonomic mobile robots Yuri A. Kapitanyuk, Anton V. Proskurnikov and Ming Cao Abstract—In this paper we propose an algorithm for path- the integral tracking error is uniformly positive independent following control of the nonholonomic mobile robot based on the of the controller design. Furthermore, in practice the robot’s idea of the guiding vector field (GVF). The desired path may be motion along the trajectory can be quite “irregular” due to an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the its oscillations around the reference point. For instance, it is robot’s kinematic model, we design a GVF, whose integral curves impossible to guarantee either path following at a precisely converge to the trajectory. A nonlinear motion controller is then constant speed, or even the motion with a perfectly fixed proposed which steers the robot along such an integral curve, direction. Attempting to keep close to the reference point, the bringing it to the desired path. We establish global convergence robot may “overtake” it due to unpredictable disturbances. conditions for our algorithm and demonstrate its applicability and performance by experiments with real wheeled robots. As has been clearly illustrated in the influential paper [11], these performance limitations of trajectory tracking can be Index Terms—Path following, vector field guidance, mobile removed by carefully designed path following algorithms. robot, motion control, nonlinear systems Unlike the tracking approach, path following control treats the path as a geometric curve rather than a function of I.
    [Show full text]
  • Towards a Glossary of Activities in the Ontology Engineering Field
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid Towards a Glossary of Activities in the Ontology Engineering Field Mari Carmen Suárez-Figueroa, Asunción Gómez-Pérez Ontology Engineering Group, Universidad Politécnica de Madrid Campus de Montegancedo, Avda. Montepríncipe s/n, Boadilla del Monte, Madrid, Spain E-mail: [email protected], [email protected] Abstract The Semantic Web of the future will be characterized by using a very large number of ontologies embedded in ontology networks. It is important to provide strong methodological support for collaborative and context-sensitive development of networks of ontologies. This methodological support includes the identification and definition of which activities should be carried out when ontology networks are collaboratively built. In this paper we present the consensus reaching process followed within the NeOn consortium for the identification and definition of the activities involved in the ontology network development process. The consensus reaching process here presented produces as a result the NeOn Glossary of Activities. This work was conceived due to the lack of standardization in the Ontology Engineering terminology, which clearly contrasts with the Software Engineering field. Our future aim is to standardize the NeOn Glossary of Activities. field of knowledge. In the case of IEEE Standard Glossary 1. Introduction of Software Engineering Terminology (IEEE, 1990), this The Semantic Web of the future will be characterized by glossary identifies terms in use in the field of Software using a very large number of ontologies embedded in Engineering and establishes standard definitions for those ontology networks built by distributed teams (NeOn terms.
    [Show full text]
  • Introduction to Conformal Field Theory and String
    SLAC-PUB-5149 December 1989 m INTRODUCTION TO CONFORMAL FIELD THEORY AND STRING THEORY* Lance J. Dixon Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 ABSTRACT I give an elementary introduction to conformal field theory and its applications to string theory. I. INTRODUCTION: These lectures are meant to provide a brief introduction to conformal field -theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available (or almost available), and most of these go in to much more detail than I will be able to here. Those reviews con- centrating on the CFT side of the subject include refs. 1,2,3,4; those emphasizing string theory include refs. 5,6,7,8,9,10,11,12,13 I will start with a little pre-history of string theory to help motivate the sub- ject. In the 1960’s it was noticed that certain properties of the hadronic spectrum - squared masses for resonances that rose linearly with the angular momentum - resembled the excitations of a massless, relativistic string.14 Such a string is char- *Work supported in by the Department of Energy, contract DE-AC03-76SF00515. Lectures presented at the Theoretical Advanced Study Institute In Elementary Particle Physics, Boulder, Colorado, June 4-30,1989 acterized by just one energy (or length) scale,* namely the square root of the string tension T, which is the energy per unit length of a static, stretched string. For strings to describe the strong interactions fi should be of order 1 GeV. Although strings provided a qualitative understanding of much hadronic physics (and are still useful today for describing hadronic spectra 15 and fragmentation16), some features were hard to reconcile.
    [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]
  • Andrew Thomson Dissertation.Pdf
    PHENOMENOLOGY OF COMPACTIFICATIONS OF M-THEORY ON MANIFOLDS OF G2 HOLONOMY by Andrew James Thomson Supervisor: Kelly Stelle A dissertation submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London 24 September 2010 Contents I. Introduction 3 II. Holonomy and G2 manifolds 7 III. Chirality and singular manifolds 11 IV. Moduli stabilization and the hierarchy problem 14 V. Conclusion 17 References 19 2 I. Introduction The Standard Model is a phenomenally successful theory of particle physics, well tested at quantum mechanical level up to the TeV energy scale, and renormalizable (notwithstanding that the Higgs particle has not been observed yet) [1]. However there are several issues (or at least perceived ones) with the model. First of all, it only describes three of the four fundamental forces (the strong, electromagnetic and weak forces) in a quantum manner, leaving gravity still described only in a classical fashion; however we will not be concerned with this any further during this dissertation. The primary issue with which we will be concerned is the so-called hierarchy problem (also known as the weak-scale instability) – why there are two (electroweak and Planck) mass scales and why they are so far apart, why the electroweak scale is not modified at loop level by the Planck scale and whether there are any other scales between them. [1,2] One wonders why we should be concerned with as yet purely theoretical imperfections which have not yet manifested themselves in any experimental observations. One only has to look to the history of the standard model itself for a precedent for looking for physics beyond it – the earliest theories of the weak interaction, which were based on the interaction of four fermions at a point, broke down when calculated to higher order at the then unimaginably high energies of 300 GeV (Heisenberg 1939) before they had been violated by any observation.
    [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]
  • The Equivalence Theorem and the Production of Gravitinos After Inflation
    CERN-TH/99-373 The equivalence theorem and the production of gravitinos after inflation 1; 2; Antonio L. Maroto ∗ and Jos´eR.Pel´aez † 1CERN Theory Division, CH-1211 Geneva 23, Switzerland 2Departamento de F´ısica Te´orica, Universidad Complutense de Madrid, 28040 Madrid, Spain ABSTRACT We study the application of the high-energy equivalence between helicity 1/2 gravitinos and goldstinos in order to calculate the production of helicity 1/2 gravitinos± in time-dependent scalar and gravitational backgrounds. We derive this equivalence± for equations of motion, paying attention to several subtleties that appear in this context and are not present in the standard derivations of the theorem, mainly because of the presence of external sources. We also propose the Landau gauge as an alternative to the usual gauge choices given in the standard proofs at the Lagrangian level. CERN-TH/99-373 November 1999 ∗E-mail: [email protected] †E-mail: [email protected] 1 Introduction In supergravity theories [1, 2] the superpartner of the graviton field is a spin 3/2 particle called the gravitino. This particle couples only with gravitational strength to the rest of matter fields, and accordingly its lifetime can be very long, with a decay rate of Γ m3 /M 2 .In 3=2 ' 3=2 P particular, gravitinos lighter than m3=2 < 100 MeV will live longer than the age of the Universe. This fact can have important consequences in cosmology and imposes stringent constraints on supergravity models. Owing to their weak couplings, gravitinos freeze out very early when they 3 are still relativistic; therefore their primordial abundance can be estimated as n3=2/s 10− [3].
    [Show full text]