An Introduction to Dynamical Supersymmetry Breaking
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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. -
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 . -
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. -
Clifford Algebras, Spinors and Supersymmetry. Francesco Toppan
IV Escola do CBPF – Rio de Janeiro, 15-26 de julho de 2002 Algebraic Structures and the Search for the Theory Of Everything: Clifford algebras, spinors and supersymmetry. Francesco Toppan CCP - CBPF, Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil abstract These lectures notes are intended to cover a small part of the material discussed in the course “Estruturas algebricas na busca da Teoria do Todo”. The Clifford Algebras, necessary to introduce the Dirac’s equation for free spinors in any arbitrary signature space-time, are fully classified and explicitly constructed with the help of simple, but powerful, algorithms which are here presented. The notion of supersymmetry is introduced and discussed in the context of Clifford algebras. 1 Introduction The basic motivations of the course “Estruturas algebricas na busca da Teoria do Todo”consisted in familiarizing graduate students with some of the algebra- ic structures which are currently investigated by theoretical physicists in the attempt of finding a consistent and unified quantum theory of the four known interactions. Both from aesthetic and practical considerations, the classification of mathematical and algebraic structures is a preliminary and necessary require- ment. Indeed, a very ambitious, but conceivable hope for a unified theory, is that no free parameter (or, less ambitiously, just few) has to be fixed, as an external input, due to phenomenological requirement. Rather, all possible pa- rameters should be predicted by the stringent consistency requirements put on such a theory. An example of this can be immediately given. It concerns the dimensionality of the space-time. -
Supersymmetry Breaking and Inflation from Higher Curvature
Prepared for submission to JHEP Supersymmetry Breaking and Inflation from Higher Curvature Supergravity I. Dalianisa, F. Farakosb, A. Kehagiasa;c, A. Riottoc and R. von Ungeb aPhysics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece bInstitute for Theoretical Physics, Masaryk University, 611 37 Brno, Czech Republic cDepartment of Theoretical Physics and Center for Astroparticle Physics (CAP) 24 quai E. Ansermet, CH-1211 Geneva 4, Switzerland E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: The generic embedding of the R + R2 higher curvature theory into old- minimal supergravity leads to models with rich vacuum structure in addition to its well-known inflationary properties. When the model enjoys an exact R-symmetry, there is an inflationary phase with a single supersymmetric Minkowski vacuum. This appears to be a special case of a more generic set-up, which in principle may include R- symmetry violating terms which are still of pure supergravity origin. By including the latter terms, we find new supersymmetry breaking vacua compatible with single-field inflationary trajectories. We discuss explicitly two such models and we illustrate how arXiv:1409.8299v2 [hep-th] 22 Jan 2015 the inflaton is driven towards the supersymmetry breaking vacuum after the inflationary phase. In these models the gravitino mass is of the same order as the inflaton mass. Therefore, pure higher curvature supergravity may not only accommodate the proper inflaton field, but it may also provide the appropriate hidden sector for supersymmetry breaking after inflation has ended. -
1 the Superalgebra of the Supersymmetric Quantum Me- Chanics
CBPF-NF-019/07 1 Representations of the 1DN-Extended Supersymmetry Algebra∗ Francesco Toppan CBPF, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro (RJ), Brazil. E-mail: [email protected] Abstract I review the present status of the classification of the irreducible representations of the alge- bra of the one-dimensional N− Extended Supersymmetry (the superalgebra of the Supersym- metric Quantum Mechanics) realized by linear derivative operators acting on a finite number of bosonic and fermionic fields. 1 The Superalgebra of the Supersymmetric Quantum Me- chanics The superalgebra of the Supersymmetric Quantum Mechanics (1DN-Extended Supersymme- try Algebra) is given by N odd generators Qi (i =1,...,N) and a single even generator H (the hamiltonian). It is defined by the (anti)-commutation relations {Qi,Qj} =2δijH, [Qi,H]=0. (1) The knowledge of its representation theory is essential for the construction of off-shell invariant actions which can arise as a dimensional reduction of higher dimensional supersymmetric theo- ries and/or can be given by 1D supersymmetric sigma-models associated to some d-dimensional target manifold (see [1] and [2]). Two main classes of (1) representations are considered in the literature: i) the non-linear realizations and ii) the linear representations. Non-linear realizations of (1) are only limited and partially understood (see [3] for recent results and a discussion). Linear representations, on the other hand, have been recently clarified and the program of their classification can be considered largely completed. In this work I will review the main results of the classification of the linear representations and point out which are the open problems. -
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 -
SUPERSYMMETRY 1. Introduction the Purpose
SUPERSYMMETRY JOSH KANTOR 1. Introduction The purpose of these notes is to give a short and (overly?)simple description of supersymmetry for Mathematicians. Our description is far from complete and should be thought of as a first pass at the ideas that arise from supersymmetry. Fundamental to supersymmetry is the mathematics of Clifford algebras and spin groups. We will describe the mathematical results we are using but we refer the reader to the references for proofs. In particular [4], [1], and [5] all cover spinors nicely. 2. Spin and Clifford Algebras We will first review the definition of spin, spinors, and Clifford algebras. Let V be a vector space over R or C with some nondegenerate quadratic form. The clifford algebra of V , l(V ), is the algebra generated by V and 1, subject to the relations v v = v, vC 1, or equivalently v w + w v = 2 v, w . Note that elements of· l(V ) !can"b·e written as polynomials· in V · and this! giv"es a splitting l(V ) = l(VC )0 l(V )1. Here l(V )0 is the set of elements of l(V ) which can bCe writtenC as a linear⊕ C combinationC of products of even numbers ofCvectors from V , and l(V )1 is the set of elements which can be written as a linear combination of productsC of odd numbers of vectors from V . Note that more succinctly l(V ) is just the quotient of the tensor algebra of V by the ideal generated by v vC v, v 1. -
CP Violation Beyond the Standard Model*
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS J. Phys. G: Nucl. Part. Phys. 28 (2002) 345–358 PII: S0954-3899(02)30104-X CP violation beyond the standard model* GLKane1 and D D Doyle2 1 Randall Physics Laboratory, University of Michigan, Ann Arbor, MI 48109, USA 2 CPES, University of Sussex, Falmer, Brighton BN1 9QJ, UK E-mail: [email protected] Received 25 October 2001 Published 21 January 2002 Online at stacks.iop.org/JPhysG/28/345 Abstract In this paper a number of broad issues are raised about the origins of CP violation and how to test the ideas. 1. Introduction The fundamental sources of CP violation in theories of physics beyond the standard model is an important issue which has not been sufficiently studied. In this paper, we begin by discussing the possible origins of CP violation in string theory and the potential influence of its phenomenology on string theory. This naturally develops into a consideration of supersymmetry breaking, specifically the soft-breaking supersymmetric Lagrangian, Lsoft. There exist two extreme scenarios which may realistically accommodate CP violation; the first involves small soft phases and a large CKM phase, δCKM, whereas the second contains large soft phases and small δCKM. We argue that there is reasonable motivation for δCKM to be almost zero and that the latter scenario should be taken seriously. Consequently, it is appropriate to consider what CP violating mechanisms could allow for large soft phases, and hence measurements of these soft phases (which can be deduced from collider results, mixings and decays, experiments exploring the Higgs sector or from electric dipole moment values) provide some interesting phenomenological implications for physics beyond the standard model. -
Atomic Electric Dipole Moments and Cp Violation
261 ATOMIC ELECTRIC DIPOLE MOMENTS AND CP VIOLATION S.M.Barr Bartol Research Institute University of Delaware Newark, DE 19716 USA Abstract The subject of atomic electric dipole moments, the rapid recent progress in searching for them, and their significance for fundamental issues in particle theory is surveyed. particular it is shown how the edms of different kinds of atoms and molecules, as well Inas of the neutron, give vital information on the nature and origin of CP violation. Special stress is laid on supersymmetric theories and their consequences. 262 I. INTRODUCTION In this talk I am going to discuss atomic and molecular electric dipole moments (edms) from a particle theorist's point of view. The first and fundamental point is that permanent electric dipole moments violate both P and T. If we assume, as we are entitled to do, that OPT is conserved then we may speak equivalently of T-violation and OP-violation. I will mostly use the latter designation. That a permanent edm violates T is easily shown. Consider a proton. It has a magnetic dipole moment oriented along its spin axis. Suppose it also has an electric edm oriented, say, parallel to the magnetic dipole. Under T the electric dipole is not changed, as the spatial charge distribution is unaffected. But the magnetic dipole changes sign because current flows are reversed by T. Thus T takes a proton with parallel electric and magnetic dipoles into one with antiparallel moments. Now, if T is assumed to be an exact symmetry these two experimentally distinguishable kinds of proton will have the same mass. -
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]. -
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.