DAMTP-2010-90IC/2010/xxMPP-2010-143CambridgeLecturesonSupersymmetryandExtraDimensionsLecturesby: FernandoQuevedo1,2,Notesby: SvenKrippendorf1,OliverSchlotterer31DAMTP,UniversityofCambridge,WilberforceRoad,Cambridge,CB30WA,UK.2ICTP,StradaCostiera11,Trieste34151,Italy.3Max-Planck-Institutf urPhysik,FohringerRing6,80805M unchen,Germany.November8,2010arXiv:1011.1491v1 [hep-th] 5 Nov 20102Contents1 PhysicalMotivationforsupersymmetryandextradimensions 91.1 BasicTheory: QFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 BasicPrinciple: Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 Classesofsymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Importanceofsymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Basicexample: TheStandardModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 ProblemsoftheStandardModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Supersymmetryalgebraandrepresentations 172.1 Poincaresymmetryandspinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 PropertiesoftheLorentzgroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 RepresentationsandinvarianttensorsofSL(2, C) . . . . . . . . . . . . . . . . . . . . . 182.1.3 GeneratorsofSL(2, C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.4 ProductsofWeylspinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Supersymmetryalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Historyofsupersymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Gradedalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.3 RepresentationsofthePoincaregroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.4 A= 1supersymmetryrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.5 Masslesssupermultiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.6 Massivesupermultiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Extendedsupersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Algebraofextendedsupersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.2 Masslessrepresentationsof A> 1supersymmetry . . . . . . . . . . . . . . . . . . . . 302.3.3 Massiverepresentationsof A> 1supersymmetryandBPSstates . . . . . . . . . . . . 323 Supereldsandsuperspace 353.1 Basicsaboutsuperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.1 Groupsandcosets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 PropertiesofGrassmannvariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3734 CONTENTS3.1.3 Denitionandtransformationofthegeneralscalarsupereld . . . . . . . . . . . . . . 383.1.4 Remarksonsuperelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Chiralsuperelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 Vectorsuperelds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Denitionandtransformationofthevectorsupereld . . . . . . . . . . . . . . . . . . 413.3.2 WessZuminogauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.3 Abelianeldstrengthsupereld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.4 Non-abelianeldstrength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 FourdimensionalsupersymmetricLagrangians 454.1 A= 1globalsupersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.1 ChiralsupereldLagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1.2 Miraculouscancellationsindetail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.3 AbelianvectorsupereldLagrangian. . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.4 Actionasasuperspaceintegral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Non-renormalizationtheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.2 Proofofthenon-renormalizationtheorem. . . . . . . . . . . . . . . . . . . . . . . . . 534.3 A= 2, 4globalsupersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.1 A= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.2 A= 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.3 Asideoncouplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Supergravity: anOverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.1 Supergravityasagaugetheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.2 Thelinearsupergravitymultipletwithglobalsupersymmetry . . . . . . . . . . . . . . 594.4.3 Thesupergravitymultipletwithlocalsupersymmetry . . . . . . . . . . . . . . . . . . 604.4.4 A= 1SupergravityinSuperspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.5 A= 1supergravitycoupledtomatter . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 Supersymmetrybreaking 655.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 F-andDbreaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.1 Ftermbreaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.2 ORaifertaighmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2.3 Dtermbreaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.3 Supersymmetrybreakingin A= 1supergravity. . . . . . . . . . . . . . . . . . . . . . . . . . 706 TheMSSM-basicingredients 716.1 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71CONTENTS 56.3 SupersymmetrybreakingintheMSSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.4 Thehierarchyproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.5 Thecosmologicalconstantproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 Extradimensions 777.1 BasicsofKaluzaKleintheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.1.2 Scalareldin5dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.1.3 Vectoreldsin5dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.1.4 Dualityandantisymmetrictensorelds . . . . . . . . . . . . . . . . . . . . . . . . . . 817.1.5 GravitationinKaluzaKleintheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 Thebraneworldscenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2.1 Largeextradimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2.2 Warpedcompactications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2.3 Braneworldscenariosandthehierarchyproblem. . . . . . . . . . . . . . . . . . . . . 898 Supersymmetryinhigherdimensions 918.1 Spinorsinhigherdimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.1.1 SpinorrepresentationsinevendimensionsD = 2n . . . . . . . . . . . . . . . . . . . . 918.1.2 SpinorepresentationsinodddimensionsD = 2n + 1 . . . . . . . . . . . . . . . . . . . 928.1.3 Majoranaspinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2 Supersymmetryalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 938.2.1 Representationsofsupersymmetryalgebrainhigherdimensions . . . . . . . . . . . . . 948.3 DimensionalReduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97AUsefulspinoridentities 99A.1 Bispinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.2 Sigmamatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.3 Bispinorsinvolvingsigmamatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100BDiracspinorsversusWeylspinors 103B.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103B.2 Gammamatrixtechnology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104B.3 TheSupersymmetryalgebrainDiracnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105CSolutionstotheexercises 107C.1 Chapter2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107C.2 Chapter3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110C.3 Chapter4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114C.4 Chapter5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 CONTENTSC.5 Chapter7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123C.6 AppendixA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127CONTENTS 7GuidethroughthenotesTheselecturesonsupersymmetryandextradimensionsareaimedatnishingundergraduateandbeginningpostgraduatestudentswithabackgroundinquantumeldtheoryandgrouptheory. Basicknowledgeingeneralrelativitymightbeadvantageousforthediscussionofextradimensions.Thiscoursewastaughtasa24+1lecturecourseinPartIIIoftheMathematicalTriposinrecentyears. Therstsixchaptersgiveanintroductiontosupersymmetryinfourspacetimedimensions, theyll abouttwothirds of the lecture notes and are in principle self-contained. The remaining two chapters are devoted to extraspacetime dimensions which are in the end combined with the concept of supersymmetry. Understanding theinterplaybetweensupersymmetryandextradimensionsisessentialformodernresearchareasintheoreticalandmathematicalphysicssuchassuperstringtheory.Videosfromthecourselecturedin2006canbefoundonlineat:http://www.sms.cam.ac.uk/collection/659537Therearealotofotherbooks,lecturenotesandreviewsonsupersymmetry,supergravityandextradimen-sions,someofwhicharelistedinthebibliography[1].AcknowledgmentsWeareverygrateful toBenAllanachforenumeroussuggestionstothesenotesfromteachingthiscourseduringthelastyear, JoeConlonforelaboratingmostsolutionsduringearlyyears, ShehuAbdusSalamforlming and Bjorn Hasler for editing and publishing the videos on the web. We wish to thank lots of studentsfromPart III 2005to2010andfromvarious other places for pointingout various typos andfor makingvaluablesuggestionsforimprovement.8 CONTENTSChapter1PhysicalMotivationforsupersymmetryandextradimensionsLetusstartwithasimplequestioninhighenergyphysics: Whatdoweknowsofarabouttheuniversewelivein?1.1 BasicTheory: QFTMicroscopicallywehavequantummechanicsandspecial relativityasourtwobasictheories.Theframeworktomakethesetwotheoriesconsistentwitheachotherisquantumeldtheory(QFT).Inthistheory the fundamental entities are quantum elds. Their excitations correspond to the physically observableelementaryparticleswhicharethebasicconstituentsof matteraswell asthemediatorsof all theknowninteractions. Therefore, eldshaveparticle-likecharacter. Particlescanbeclassiedintwogeneralclasses:bosons (spin s = nZ) and fermions (s = n+12 Z+12). Bosons and fermions have very dierent physicalbehaviour. ThemaindierenceisthatfermionscanbeshowntosatisfythePauli exclusionprinciple,whichstatesthattwoidentical fermionscannotoccupythesamequantumstate, andthereforeexplainingthevastdiversityofatoms.All elementarymatter particles: the leptons (includingelectrons andneutrinos) andquarks (that makeprotons, neutrons and all other hadrons) are fermions. Bosons on the other hand include the photon (particleoflightandmediatorofelectromagneticinteraction),andthemediatorsofalltheotherinteractions. TheyarenotconstrainedbythePauli principleandthereforehaveverydierentphysical propertiesascanbeappreciatedinalaserforinstance. Aswewill see, supersymmetryisasymmetrythatuniesbosonsandfermionsdespitealltheirdierences.910 CHAPTER1. PHYSICALMOTIVATIONFORSUPERSYMMETRYANDEXTRADIMENSIONS1.2 BasicPrinciple: SymmetryIfQFTisthebasicframeworktostudyelementaryprocesses, thebasictooltolearnabouttheseprocessesistheconceptofsymmetry.Asymmetryisatransformationthatcanbemadetoaphysicalsystemleavingthephysicalobservablesun-changed. Throughout the history of science symmetry has played a very important role to better understandnature. Letustrytoclassifythedierentclassesofsymmetriesandtheirphysicalimplications.1.2.1 ClassesofsymmetriesThereareseveralwaystoclassifysymmetries. Symmetriescanbediscreteorcontinuous. Theycanalsobeglobalorlocal. Forelementaryparticles,wecandenetwogeneralclassesofsymmetries: Spacetimesymmetries: Thesesymmetriescorrespondtotransformationsonaeldtheoryactingex-plicitlyonthespacetimecoordinates,x xt(x) , , = 0, 1, 2, 3.Examples are rotations, translations and, more generally, Lorentz-andPoincaretransformations den-ingtheglobal symmetriesof special relativityaswell asgeneral coordinatetransformationsthatarethelocalsymmetriesthatdenegeneral relativity. Internal symmetries: Thesearesymmetriesthatcorrespondtotransformationsof thedierenteldsinaeldtheory,a(x) Mab b(x).Romanindicesa, blabel thecorrespondingelds. If Mabisconstantthenthesymmetryisaglobalsymmetry;incaseofspacetimedependentMab(x)thesymmetryiscalledalocal symmetry.1.2.2 ImportanceofsymmetriesSymmetriesareimportantforvariousreasons: Labellingandclassifyingparticles: Symmetrieslabel andclassifyparticlesaccordingtothedierentconservedquantumnumbersidentiedbythespacetimeandinternalsymmetries(mass, spin, charge,colour, etc.). ThisisaconsequenceofNoetherstheoremthatstatesthateachcontinuoussymmetryimpliesaconservedquantity. Inthisregardsymmetriesactuallydeneanelementaryparticleac-cordingtothebehaviourofthecorrespondingeldwithrespecttothecorrespondingsymmetry. Thispropertywasusedtoclassifyparticlesnotonlyasfermionsandbosonsbutalsotogrouptheminmul-tipletswithrespecttoapproximateinternalsymmetriesasintheeightfoldwaythatwasattheoriginofthequarkmodelofstronginteractions. Symmetries determine the interactions among particles by means of the gaugeprinciple. By promotingaglobal symmetrytoalocal symmetrygaugeelds(bosons)andinteractionshavetobeintroduced1.2. BASICPRINCIPLE:SYMMETRY 11accordingly dening the interactions among particles with gauge elds as mediators of interactions. Asanillustration,considertheLagrangianL = V (, )whichisinvariantunderrotationinthecomplexplane exp(i) ,aslongasisaconstant(globalsymmetry). If = (x),thekinetictermisnolongerinvariant: exp(i)_+i()_.However,thecovariantderivativeD,denedasD := +iA,transformslikeitself,ifthegauge-potentialAtransformstoA:D exp(i)_+i() +i(A) _= exp(i) D,sorewritetheLagrangiantoensuregauge-invariance:L = DDV (, ) .Thescalareldcouplestothegauge-eldAviaAA,similarly,theDiracLagrangianL = DhasaninteractiontermA. Thisinteractionprovidesthethreepointvertexthatdescribesinter-actionsofelectronsandphotonsandillustratehowphotonsmediatetheelectromagneticinteractions. Symmetriescanhideorbespontaneouslybroken: Considerthepotential V (, )inthescalareldLagrangianabove.IfV (, ) = V ([[2),thenitissymmetricfor exp(i). IfthepotentialisofthetypeV = a [[2+b [[4, a, b 0,thentheminimumisat = 0(here 0[[0denotesthevacuumexpectationvalue(vev)oftheeld). Thevacuumstateisthenalsosymmetricunderthesymmetrysincetheoriginisinvariant.HoweverifthepotentialisoftheformV =_ab [[2_2, a, b 0,the symmetry of Vis lost in the ground state , = 0. The existence of hidden symmetries is importantforatleasttworeasons:12 CHAPTER1. PHYSICALMOTIVATIONFORSUPERSYMMETRYANDEXTRADIMENSIONSFigure1.1: TheMexicanhatpotentialforV=

ab ||2

2witha, b 0.(i) Thisisanaturalwaytointroduceanenergyscaleinthesystem,determinedbythenonvanishingvev. Inparticular, wewill seethatforthestandardmodel Mew 103GeV, denesthebasicscale of mass for the particles of the standard model, the electroweak gauge bosons and the matterelds,throughtheirYukawacouplings,obtaintheirmassfromthiseect.(ii) Theexistenceof hiddensymmetriesimpliesthatthefundamental symmetriesof naturemaybehugedespitethefactthatweobservealimitedamountof symmetry. Thisisbecausetheonlymanifestsymmetrieswecanobservearethesymmetriesofthevacuumweliveinandnotthoseof the fullunderlying theory. This opens-up an essentiallyunlimitedresource to considerphysicaltheorieswithanindenitenumberofsymmetrieseventhoughtheyarenotexplicitlyrealisedinnature. Thestandardmodel is thetypical exampleandsupersymmetryandtheories of extradimensionsarefurtherexamples.1.3 Basicexample: TheStandardModelThe concrete example is the particular QFT known as The Standard Model which describes all known particlesandinteractionsin4dimensionalspacetime. Matter particles: Quarks and leptons. They come in three identical families diering only by their mass.Onlytherstfamilyparticipateinmakingtheatomsandall compositematterweobserve. Quarksandleptonsarefermionsof spin

2andthereforesatisfyPaulisexclusionprinciple. Leptonsincludetheelectrone, muonand aswell asthethreeneutrinos. Quarkscomeinthreecoloursandarethebuildingblocksofstronglyinteractingparticlessuchastheprotonandneutronintheatoms. Interactionparticles: Thethreenon-gravitationalinteractions(strong, weakandelectromagnetic)aredescribedbyagaugetheorybasedonaninternalsymmetry:GSM= SU(3)C. .strongSU(2)LU(1). .electroweak1.4. PROBLEMSOFTHESTANDARDMODEL 13HereSU(3)Crefers toquantumchromodynamics part of thestandardmodel describingthestronginteractions,thesubindexCreferstocolour. AlsoSU(2)LU(1)referstotheelectroweakpartofthestandard model, describing the electromagnetic and weak interactions. The subindex L in SU(2)L referstothefactthattheStandardModel doesnotpreserveparityanddierentiatesbetweenleft-handedandright-handedparticles. IntheStandardModel onlyleft-handedparticlestransformnon-triviallyunderSU(2)L. Thegaugeparticleshaveallspins = 1andmediateeachofthethreeforces: photons()forU(1)electromagnetism, gluonsforSU(3)Cofstronginteractions, andthemassiveWandZfortheweakinteractions. TheHiggsparticle: This is the spin s = 0 particle that has a potential of the Mexican hat shape (seegure)andisresponsibleforthebreakingoftheStandardModelgaugesymmetry:SU(2)LU(1))103GeV UEM(1)ForthegaugeparticlesthisistheHiggseect, thatexplainshowtheWandZparticlesgetamassand therefore the weak interactions are short range. This is also the source of masses for all quarks andleptons. Gravity: Gravitycanalsobeunderstoodas agaugetheoryinthesensethat theglobal spacetimesymmetriesofspecial relativity, denedbythePoincaregroup, whenmadelocal giverisetothegen-eral coordinatetransformationsof general relativity. Howeverthecorrespondinggaugeparticle, thegraviton, corresponds toamassless particleof spins=2andthereis not aQFTthat describesthese particles to arbitrarily small distances. Therefore, contrary to gauge theories which are consistentquantummechanicaltheories,theStandardModelonlydescribesgravityattheclassicallevel.1.4 ProblemsoftheStandardModelThe Standard Model is one of the cornerstones of all science and one of the great triumphs of the XX century.It has been carefully experimentally veried in many ways,especially during the past 20 years,but there aremanyquestionsitcannotanswer: QuantumGravity: TheStandardModel describesthreeof thefourfundamental interactionsatthequantumlevel andtherefore microscopically. However, gravityis onlytreatedclassicallyandanyquantumdiscussionofgravityhastobeconsideredasaneectiveeldtheoryvalidatscalessmallerthanthePlanckscale(Mpl=_Ghc31019GeV). Atthisscalequantumeectsofgravityhavetobeincluded and then Einstein theory has the problem of being non-renormalizable and therefore it cannotprovideproperanswerstoobservablesbeyondthisscale. WhyGSM=SU(3) SU(2) U(1)? Whytherearefourinteractionsandthreefamiliesoffermions?Why 3 + 1 spacetime - dimensions?Why there are some 20 parameters (masses and couplings betweenparticles)intheStandardModelforwhichtheirvaluesareonlydeterminedtotexperimentwithoutanytheoreticalunderstandingofthesevalues?14 CHAPTER1. PHYSICALMOTIVATIONFORSUPERSYMMETRYANDEXTRADIMENSIONS Connement: Whyquarkscanonlyexistconnedinhadronssuchasprotonsandneutrons?Thefactthatthestronginteractionsareasymptoticallyfree(meaningthatthevalueofthecouplingincreaseswithdecreasingenergy)indicatesthatthisisduetothefactthatattherelativelylowenergieswecanexplore the strong interactions are so strong that do not allow quarks to separate. This is an issue aboutourignorancetotreatstrongcouplingeldtheorieswhicharenotwell understoodbecausestandard(Feynmandiagrams)perturbationtheorycannotbeused. Thehierarchyproblem: WhytherearetotallydierentenergyscalesMew 102GeV, Mpl=_Ghc3 1019GeV =MewMpl 1015Thisproblemhastwoparts. Firstwhythesefundamental scalesaresodierentwhichmaynotlookthatserious. Thesecondpartreferstoanaturalnessissue. AnetuningofmanyordersofmagnitudehastobeperformedorderbyorderinperturbationtheoryinordertoavoidtheelectroweakscaleMewtotakethevalueofthecutoscalewhichcanbetakentobeMpl. ThestrongCPproblem: ThereisacouplingintheStandardModeloftheformFFwhereisaparameter,Freferstotheeldstrengthofquantumchromodynamics(QCD)andF= F.ThistermbreaksthesymmetryCP(chargeconjugationfollowedbyparity). Theproblemreferstothefact that theparameter is unnaturallysmall 1ofanextendedSUSY.Inthischapter,wewillonlydiscuss A= 1.We know the commutation relations [P, P], [P, M] and [M, M] from Poincare - algebra, so we needtond(a)_Q, M_, (b)_Q, P_,(c)_Q, Q_, (d)_Q,Q_,also(forinternalsymmetrygeneratorsTi)(e)_Q, Ti_.2.2. SUPERSYMMETRYALGEBRA 23 (a)_Q, M_SinceQisaspinor,ittransformsundertheexponentialoftheSL(2, C)generators:Qt= exp_i2_Q_1i2 _Q,butQisalsoanoperatortransformingunderLorentztransformationsU= exp_i2M_toQt= UQU _1+i2 M_Q_1i2 M_.ComparethesetwoexpressionsforQtuptorstorderin,Qi2()Q= Qi2_QMMQ_+O(2)=_Q, M_= ()Q (b)_Q, P_c () Q is the onlywayof writing a sensible termwithfree indices , whichis linear inQ. Toxtheconstant c, consider [ Q , P] =c( ) Q(takeadjoints using(Q)=Q and( Q)= (Q) ). TheJacobiidentityforP,PandQ0 =_P,_P, Q__+_P,_Q, P__+_Q,_P, P_. .0_= c () _P,Q _+c () _P,Q _= [c[2() ( ) Q[c[2() ( ) Q= [c[2( ). .,=0QcanonlyholdforgeneralQ,ifc = 0,so_Q, P_=_Q , P_= 0 (c)_Q, Q_Duetoindexstructure,thatcommutatorshouldlooklike_Q, Q_= k ()M.SincethelefthandsidecommuteswithPandtherighthandsidedoesnt,theonlyconsistentchoiceisk = 0,i.e._Q, Q_= 024 CHAPTER2. SUPERSYMMETRYALGEBRAANDREPRESENTATIONS (d)_Q,Q_Thistime,indexstructureimpliesanansatz_Q,Q_= t () P.Thereisnowayofxingt,so,byconvention,sett = 2:_Q,Q_= 2 () PNotice that two symmetry transformations Q Qhave the eect of a translation. Let [B be a bosonic stateand [Fafermionicone,thenQ[F = [B,Q[B = [F = Q Q : [B [B(translated). (e)_Q, Ti_Usually,thiscommutatorvanishes,exceptionsareU(1)automorphismsofthesupersymmetryalgebraknownasRsymmetry.Q exp(i) Q,Q exp(i)Q .LetRbeaU(1)generator,then_Q, R_= Q,_Q , R_= Q .2.2.3 RepresentationsofthePoincaregroupRecalltherotationgroup Ji: i = 1, 2, 3satisfying_Ji, Jj_= iijk Jk.TheCasimiroperatorJ2=3

i=1J2icommuteswithalltheJiandlabelsirreduciblerepresentationsbyeigenvaluesj(j + 1)ofJ2. Withintheserepresentations,diagonalizeJ3toeigenvaluesj3= j, j + 1, ..., j 1, j. Statesarelabelledlike [j, j3.AlsorecallthetwoCasimirsinPoincaregroup,oneofwhichinvolvesthePauliLjubanskivectorW,W=12

PM(where0123= 0123= +1).Exercise2.3: ProvethatthePauliLjubanskivectorsatisesthefollowingcommutationrelations:_W, P_= 0_W, M_= iWi W_W, W_= i WP_W, Q_= i P ()Q2.2. SUPERSYMMETRYALGEBRA 25Inintermediatestepsonemightneed

= 6 [ ],

= 4 [],anditisuseful tonotethatW= 3M[P].ThePoincareCasimirsarethengivenbyC1= PP, C2= WW.theCicommutewithall generators.Exercise2.4: ShowthatC2indeedcommuteswiththePoincaregeneratorsbutnotwiththeextensionQtosuperPoincare.Poincaremultiplets arelabelled [m, , eigenvalues m2of C1andeigenvalues of C2. States withinthoseirreducible representations carrythe eigenvalue pof the generator Pas alabel. Notice that at thislevel thePauli Ljubanski vector onlyprovides ashort waytoexpress thesecondCasimir. EventhoughWhasstandardcommutationrelationswiththegeneratorsofthePoincaregroupM, PstatingthatittransformasavectorunderLorentztransformationsandcommuteswithP(invariantundertranslations),thecommutator [W, W] WPimplies that theWs bythemselves arenot generators of anyalgebra.Tondmorelabels,takePasgivenandlookforallelementsoftheLorentzgroupthatcommutewithP.Thisdeneslittlegroups: Massiveparticles,p= (m, 0, 0, 0. .invariantunderrot.),haverotationsastheirlittlegroup. Duetotheantisym-metricintheW,itfollowsW0= 0, Wi= mJi.Every particle with nonzero mass is an irreducible representation of Poincare group with labels [m, j; p, j3. Masslessparticlesmomentumhastheformp= (E, 0, 0, E)whichimplies(W0, W1, W2, W3) = E_J3, J1+K2, J2 K1, J3_=_W1, W2_= 0,_W3, W1_= iE W2,_W3, W2_= iE W1.CommutationrelationsarethoseforEuclideangroupintwodimensions. Fornitedimensional rep-resentations, SO(2)isasubgroupandW1, W2havetobezero. Inthatcase, W=Pandstatesarelabelled [0, 0; p, =: [p, , whereiscalledhelicity. UnderCPT, thosestatestransformto[p, . Therelationexp(2i) [p, = [p, requires to be integer or half integer = 0,12, 1, ..., e.g. = 0 (Higgs), =12(quarks, leptons), = 1(,W,Z0,g)and = 2(graviton).26 CHAPTER2. SUPERSYMMETRYALGEBRAANDREPRESENTATIONS2.2.4 A= 1supersymmetryrepresentationsFor A=1supersymmetry, C1=PPisstill agoodCasimir, C2=WW, however, isnot. Soonecanhaveparticlesofdierentspinwithinonemultiplet. TogetanewCasimirC2(correspondingtosuperspin),deneB:= W14Q ( ) Q, C:= BPB PC2:= C C.PropositionInanysupersymmetricmultiplet,thenumbernBofbosonsequalsthenumbernFoffermions,nB= nF.ProofConsiderthefermionnumberoperator(1)F= ()F,denedvia()F[B = [B, ()F[F = [F.Thisnewoperator()FanticommuteswithQsince()FQ[F = ()F[B = [B = Q[F = Q ()F[F =_()F, Q_= 0.Next,considerthetraceTr_()F_Q,Q__= Tr_()FQ. .anticommuteQ+()FQ Q. .cyclicperm._= Tr_Q ()FQ+Q ()FQ_= 0.Ontheotherhand,itcanbeevaluatedusing Q,Q = 2() P,Tr_()F_Q,Q__= Tr_()F2 () P_= 2 () p Tr_()F_,wherePisreplacedbyitseigenvaluespforthespecicstate. Theconclusionis0 = Tr_()F_=

bosonsB[ ()F[B +

fermionsF[ ()F[F=

bosonsB[B

fermionsF[F = nBnF.2.2.5 MasslesssupermultipletStates of massless particles have P- eigenvalues p=(E, 0, 0, E). The Casimirs C1=PPandC2= CCarezero. Considerthealgebra_Q,Q_= 2 () P= 2 E_0+3_ = 4 E__1 00 0__ ,2.2. SUPERSYMMETRYALGEBRA 27whichimpliesthatQ2iszerointherepresentation:_Q2,Q2_= 0 = p, [Q2Q2[ p, = 0 = Q2= 0TheQ1satisfy Q1,Q1 = 4E,sodeningcreation-andannihilationoperatorsaandaviaa :=Q12E, a:=Q12E,gettheanticommutationrelations_a, a_= 1,_a, a_=_a, a_= 0.Also,since[a, J3] =12(3)11a =12a,J3_a [p, _=_a J3_a, J3__[p, =_a J3a2_ [p, =_ 12_a [p, .a[p, hashelicity 12, andbysimilarreasoning, ndthatthehelicityof a[p, is +12. Tobuildtherepresentation, startwithavacuumstateof minimumhelicity, letscall it [. Obviouslya[=0(otherwise [wouldnothavelowesthelicity)andaa[ = 0[ = 0,sothewholemultipletconsistsof[ = [p, , a[ = [p, +12.AddtheCPTconjugatetoget[p, , [p, _ +12_.Thereare, forexample, chiral multipletswith=0,12, vector- orgaugemultiplets(=12, 1gaugeandgaugino) = 0scalar =12fermionsquark quarkslepton leptonHiggs Higgsino =12fermion = 1bosonphotino photongluino gluonWino, Zino W, Z,aswellasthegravitonwithitspartner: =32fermion = 2bosongravitino graviton2.2.6 MassivesupermultipletIncaseofm ,= 0,thereareP-eigenvaluesp= (m, 0, 0, 0)andCasimirsC1= PP= m2,C2= C C= 2 m4YiYi,whereYidenotessuperspinYi= Ji14mQ iQ =Bim,_Yi, Yj_= iijk Yk.28 CHAPTER2. SUPERSYMMETRYALGEBRAANDREPRESENTATIONSEigenvalues to Y2= YiYiare y(y +1), so label irreducible representations by [m, y. Again, the anticommu-tation-relationforQandQisthekeytogetthestates:_Q,Q_= 2 () P= 2 m(0) = 2 m__1 00 1__ SincebothQshavenonzeroanticommutatorswiththeirQ-partner,denetwosetsofladderoperatorsa1,2:=Q1,22m, a1,2:=Q1,22m,withanticommutationrelations_ap, aq_= pq,_ap, aq_=_ap, aq_= 0.Let [bethevacuumstate,annihilatedbya1,2. Consequently,Yi[ = Ji[14mQ i2ma[..0= Ji[,i.e. for [thespinnumberjandsuperspin-numberyarethesame. Soforgivenm, y:[ = [m, j= y; p, j3Obtaintherestofthemultipletusinga1[j3 = [j312, a1[j3 = [j3 +12a2[j3 = [j3 +12, a2[j3 = [j312,where apacting on [ behave like coupling of two spins jand12. This will yield a linear combination of twopossibletotalspinsj +12andj 12withClebschGordancoecientski(recallj 12= (j 12) (j +12)):a1[ = k1[m, j= y +12; p, j3 +12 +k2[m, j= y 12; p, j3 +12a2[ = k3[m, j= y +12; p, j312 +k4[m, j= y 12; p, j312.Theremainingstatesa2a1[ = a1a2[ [representspinj-objects. Intotal,wehave2[m, j= y; p, j3. .(4y+2) states, 1[m, j= y +12; p, j3. .(2y+2) states, 1[m, j= y 12; p, j3. .(2y) states,ina [m, ymultiplet, whichisof courseanequal numberof bosonicandfermionicstates. Noticethatinlabellingthestateswehavethevalueofmandyxedthroughoutthemultipletandthevaluesofjchangestatebystate,asitshouldsinceinasupersymmetricmultiplettherearestatesofdierentspin.Thecasey= 0needstobetreatedseparately:[ = [m, j= 0; p, j3= 0a1,2[ = [m, j=12; p, j3= 12a1a2[ = [m, j= 0; p, j3= 0 =: [t2.3. EXTENDEDSUPERSYMMETRY 29Parity interchanges (A, B) (B, A), i.e. (12, 0) (0,12). Since Q,Q = 2() P, need the followingtransformation-rulesforQandQ underparityP(withphasefactorPsuchthat [P[ = 1):P QP1= P(0) QPQ P1= P( 0) QThatensuresP(P0,

P)andhastheinterestingeectP2QP2= Q. Moreover, considerthetwoj =0states [ and [t: Therst is annihilatedbyai, thesecondonebyai. DuetoQ Q, parityinterchangesaiandaiandtherefore [ [t. Togetvacuumstateswithadenedparity,weneedlinearcombinations[ := [[t, P [ = [.Thosestatesarecalledscalar([+)andpseudoscalar([).2.3 ExtendedsupersymmetryHaving discussed the algebra and representations of simple (A= 1) supersymmetry,we will turn now to themoregeneralcaseofextendedsupersymmetry A> 1.2.3.1 AlgebraofextendedsupersymmetryNow,thespinorgeneratorsgetanadditionallabelA, B= 1, 2, ..., A. Thealgebraisthesameasfor A= 1exceptfor_QA,QB_= 2 () PAB_QA, QB_= ZABwithantisymmetriccentral chargesZAB= ZBAcommutingwithallthegenerators_ZAB, P_=_ZAB, M_=_ZAB, QA_=_ZAB, ZCD_=_ZAB, Ta_= 0.Theyformanabelianinvariantsubalgebraofinternalsymmetries. Recallthat[Ta, Tb] = iCabcTc. LetGbeaninternal symmetrygroup, thendenetheRsymmetryH Gtobethesetof Gelementsthatdonotcommutewiththesupersymmetrygenerators,e.g. Ta Gsatisfying_QA, Ta_= SaAB QB,= 0is anelement of H. If ZAB=0, thentheRsymmetryis H=U(N), but withZAB,=0, Hwill beasubgroup. Theexistenceof central chargesisthemainnewingredientof extendedsupersymmetries. Thederivationofthepreviousalgebraisastraightforwardgeneralizationoftheonefor A= 1supersymmetry.30 CHAPTER2. SUPERSYMMETRYALGEBRAANDREPRESENTATIONS2.3.2 Masslessrepresentationsof A> 1supersymmetryAswedidfor A=1, wewill proceednowtodiscussmasslessandmassiverepresentations. Wewill startwiththemasslesscasewhichissimplerandhasveryimportantimplications.Letp= (E, 0, 0, E),then(similarto A= 1)._QA,QB_= 4 E__1 00 0__ AB= QA2= 0WecanimmediatelyseefromthisthatthecentralchargesZABvanishsinceQA2= 0impliesZAB= 0fromtheanticommutators_QA, QB_= ZAB.Inordertoobtainthefullrepresentation,deneNcreation-andannihilation-operatorsaA:=QA12E, aA:=QA12E=_aA, aB_= AB,togetthefollowingstates(startingfromvacuum [,whichisannihilatedbyalltheaA):states helicity numberofstates[ 01 = (A0)aA[ 0 +12A= (A1)aAaB[ 0 + 112!A(A 1) = (A2)aAaBaC[ 0 +3213!A(A 1)(A 2) = (A3).........aAa(A1)...a1[ 0 + A21 = (AA )NotethatthetotalnumberofstatesisgivenbyA

k=0__ Ak__=A

k=0__ Ak__1k1Ak= 2A.Considerthefollowingexamples: A= 2vector-multiplet(0= 0) = 0 =12 =12 = 1Wecanseethatthis A=2multipletcanbedecomposedintermsof A=1multiplets: one A=1vectorandone A= 1chiralmultiplet. A= 2hyper-multiplet(0= 12) = 12 = 0 = 0 =12Againthiscanbedecomposedintermsoftwo A= 1chiralmultiplets.2.3. EXTENDEDSUPERSYMMETRY 31 A= 4vector-multiplet(0= 1)1 = 14 = 126 = 04 = +121 = +1This is thesingle A=4multiplet withstates of helicity 2 exist (so only have A 8). One argument is the fact that massless particle of [[ >12and low momentum couple to some conserved currents (j= 0 in = 1 - electromagnetism, Tin = 2-gravity). Buttherearenofurtherconservedcurrentsfor [[ > 2(somethingthatcanalsobeseenfromtheColemanMandulatheorem). Also, A>8wouldimplythatthereismorethanonegraviton. Seechapter13in[3]onsoftphotonsforadetaileddiscussionofthisandtheextensionofhisargumenttosupersymmetryinanarticle[4]byGrisaruandPendleton(1977). Noticethisisnotafullno-gotheorem,inparticulartheconstraintoflowmomentumhastobeused. A>1supersymmetriesarenon-chiral. WeknowthattheStandardModel particlesliveoncomplexfundamental representations. Theyarechiral sincerighthandedquarksandleptonsdonotfeel theweakinteractionswhereasleft-handedonesdofeel it(theyaredoubletsunderSU(2)L). All A>1multiplets, exceptforthe A=2hypermultiplet, have= 1particlestransformingintheadjointrepresentationwhichisreal(recallthatinSU(N)theoriestheadjointrepresentationisobtainedfromtheproductoffundamentalandcomplexconjugaterepresentationsandsoisreal)andthereforenon-chiral. Thenthe= 12particlewithinthemultipletwouldtransforminthesamerepresentationandthereforebenon-chiral. Theonlyexceptionisthe A=2hypermultiplets-forthisthepreviousargument doesnt work because they do not include = 1 states, but since =12- and = 12states32 CHAPTER2. SUPERSYMMETRYALGEBRAANDREPRESENTATIONSareinthesamemultiplet,therecantbechiralityeitherinthismultiplet. Thereforeonly A= 1, 0canbechiral,forinstance A= 1with_120_predictingatleastoneextraparticleforeachStandardModelparticle. Buttheyhavenotbeenobserved. Thereforetheonlyhopeforarealisticsupersymmetrictheoryis: broken A= 1supersymmetryatlowenergiesE 102GeV.2.3.3 Massiverepresentationsof A> 1supersymmetryandBPSstatesNowconsiderp= (m, 0, 0, 0),so_QA,QB_= 2 m__1 00 1__AB.Contrary to the massless case, here the central charges can be non-vanishing. Therefore we have to distinguishtwocases: ZAB=0Thereare2Acreation-andannihilationoperatorsaA:=QA2m, aA :=QA 2mleadingto22Nstates,eachofthemwithdimension(2y + 1). Inthe A= 2case,wend:[ 1 spin0aA [ 4 spin12aA aB[ 3 spin0, 3 spin1aA aBaC [ 4 spin12aA aBaC aD[ 1 spin0,i.e. as predicted 16 = 24states in total. Notice that these multiplets are much larger than the masslessones with only 2Astates, due to the fact that in that case, half of the supersymmetry generators vanish(QA2= 0). ZAB,= 0Denethescalarquantity 1tobe1 := ( 0)_QAA,QAA_ 0.AsasumofproductsAA, 1issemi-positive,andtheAaredenedasA:= UABQ ( 0) forsomeunitarymatrixU(satisfyingUU= 1). Anticommutators QA,QBimply1 = 8 mA2 Tr_Z U+U Z_ 0.2.3. EXTENDEDSUPERSYMMETRY 33Duetothepolardecompositiontheorem, eachmatrixZcanbewrittenasaproductZ=HV of apositivehermitianH= HandaunitaryphasematrixV= (V)1. ChooseU= V ,then1 = 8 mA4 Tr_H_= 8 mA4 Tr_ZZ_ 0.ThisistheBPS-boundforthemassm:m 12ATr_ZZ_States of minimal m=12A Tr_ZZ_are calledBPSstates (due toBogomolnyi, PrasadandSommerfeld). Theyare characterizedbyavanishingcombinationQA A, sothe multiplet isshorter(similartothemasslesscaseinwhichQa2= 0)havingonly2Ainsteadof22Astates.In A= 2,denethecomponentsoftheantisymmetricZABtobeZAB=__0 q1q10__= m q12.Moregenerally,if A> 2(but Aeven)ZAB=___________________0 q10 0 0 q10 0 0 0 0 0 0 q20 0 0 q20 0 0 0 0 0..................0 qN2qN20___________________,theBPSconditionsholdsblockbyblock: 2m qi. Toseethat,denean 1foreachblock. Ifkoftheqiareequalto2m,thereare2A 2kcreationoperatorsand22(Ak)states.k = 0 = 22Astates,longmultiplet0 < k Texp(io)54 CHAPTER4. FOURDIMENSIONALSUPERSYMMETRICLAGRANGIANSwherethepathintegral isunderstoodtogoforall theeldsinthesystemandtheintegrationisonlyoverallmomentagreaterthaninthestandardWilsonianformalism(dierentfromthe1PIactioninwhichtheintegralisoverallmomenta). Ifsupersymmetryispreservedbythequantizationprocess, wecanwritetheeectiveactionas:o=_d4x_d4_J_, , eV, X, Y, T..._+(X, X, Y, Y) VU(1)_+_d4x_d2_H(, X, Y, W). .holomorphic+h.c._.DuetoU(1)Rtransformationinvariance,HmusthavetheformH = Yh(X, ) +g(X, ) WW.Invariance under shifts in Ximply that h = h() (independent of X). But a linear Xdependence is allowedinfrontofWW(duetoFFasatotalderivative). SotheXdependenceinhandgisrestrictedtoH = Yh() +_X+g()_WW.LimitsInthelimitY 0,thereisanequalityh() = W()attreelevel,soW()isnotrenormalized! Thegaugekineticfunctionf(),however,getsa1loopcorrectionf() = X..treelevel+g()..1loop.Notethat gaugeeldpropagators areproportional to1x(sincegaugecouplings behaveas xFF X[A][A],gaugeselfcouplingstoX3correspondingtoavertexof3Xlines).CountthenumberNxofx-powersinanydiagram;itisgivenbyNx= VWIWandisthereforerelatedtothenumbersofloopsL:L = IWVW+1 = Nx+1 = Nx= 1LL = 0(treelevel) : Nx= 1, = 1L = 1(oneloop) : Nx= 0ThereforethegaugekinetictermX +g()iscorrectedonlyat1loop! (Allother(innite)loopcorrectionsjustcancel.)4.3. A= 2, 4GLOBALSUPERSYMMETRY 55XFigure4.3: ThreevertexofXeldsmultiplyingthegaugeeldselfcouplingsAFigure4.4: LoopcorrectiontotheKahlerpotentialduetoaFIterm.On the other hand, the Kahler potential, being non-holomorphic, is corrected to all orders J(Y, Y, X+X, ...).For the FI term (X, X, Y, Y)VU(1)D, gauge invariance under V V +i() implies that is a constant.Theonlycontributionsareproportionalto

iqi= Tr_QU(1)_.ButifTrQ , = 0,thetheoryisinconsistentduetogravitationalanomalies:Therefore,iftherearenogravitationalanomalies,therearenocorrectionstotheFIterm.4.3 A= 2, 4globalsupersymmetryFor A=1SUSY,wehadanaction odependingonK, W, fand. Whatwillthe A 2actionsdependon?56 CHAPTER4. FOURDIMENSIONALSUPERSYMMETRICLAGRANGIANSAg~ Tr{Q}Figure4.5: GravitationalanomaliesduetoU(1)chargedeldsrunningintheloop.We knowthat inglobal supersymmetry, the A=1actions are particular cases of non-supersymmetricactions(inwhichsomeofthecouplingsarerelated,thepotentialispositive,etc.). Inthesameway,actionsforextendedsupersymmetriesareparticularcasesof A=1supersymmetricactionsandwill thereforebedetermined by K, W, fand . The extra supersymmetry will put constraints to these functions and thereforethe corresponding actions will be more rigid. The larger the number of supersymmetries the more constraintsonactionsarise.4.3.1 A= 2Considerthe A= 2vectormultipletA wheretheAandaredescribedbyavectorsupereldV andthe,byachiralsupereld.WeneedW=0inthe A=2action. K, fcanbewrittenintermsofasingleholomorphicfunction T()calledprepotential:f() =2T2, K(, ) =12i_exp(2V )Th.c._Thefullperturbativeactiondoesnotcontainanycorrectionsformorethan1loop,T =___2: (treelevel)2ln_22_: (1loop)wheredenotessomecuto. ThesestatementsapplytotheWilsonianeectiveaction. Notethat: Perturbativeprocessesusuallyinvolveseries

nangnwithsmallcouplingg 1. exp_cg2_isanon-perturbativeexample(noexpansioninpowersofgpossible).4.3. A= 2, 4GLOBALSUPERSYMMETRY 57ThereareobviouslymorethingsinQFTthanFeynmandiagramscantell,e.g. instantonsandmonopoles.Decomposethe A= 2prepotential TasT() = T1loop+Tnon-pertwhere Tnon-pertfor instancecouldbetheinstantonexpansion

k ak exp_cg2k_. In1994, SeibergandWittenachievedsuchanexpansionin A= 2SUSY[12].Ofcourse, therearestill vector-andhypermultipletsin A=2, butthosearemuchmorecomplicated. Wewillnowconsideraparticularlysimplecombinationofthesemultiplets.4.3.2 A= 4AsanN= 4example,considerthevectormultiplet,____A 11____. .A=2vector+____2323____. .A=2hyper.Wearemoreconstrainedthaninabovetheories,therearenofreefunctionsatall,only1freeparameter:f = =2..FF+4ig2..FFN= 4isanitetheory,moreoveritsfunctionvanishes. Couplingsremainconstantatanyscale,wehaveconformal invariance. TherearenicetransformationpropertiesundermodularSduality, a +bc +d,wherea,b,c,dformaSL(2, Z)matrix.Finally,asanaside,majordevelopmentsinstringandeldtheorieshaveledtotherealizationthatcertaintheories of gravity in Anti de Sitter space are dual to eld theories (without gravity) in one less dimension,that happen to be invariant under conformal transformations. This is the AdS/CFTcorrespondence allowingto describe gravity (and string) theories to domains where they are not well understood (and the same benetappliestoeldtheoriesaswell). TheprimeexampleofthiscorrespondenceisAdSin5dimensionsdualtoaconformaleldtheoryin4dimensionsthathappenstobe A= 4supersymmetry.4.3.3 AsideoncouplingsForallkindsofrenormalization,couplingsgdependonascale. ThecouplingchangesunderRGtransfor-mationsscalebyscale. Denethefunctiontobedgd= (g) = b g3. .1loop+....Thetheoryscutodependsontheparticlecontent.58 CHAPTER4. FOURDIMENSIONALSUPERSYMMETRICLAGRANGIANSSolveforg()upto1looporder:M_mdgg3= b+_d= 12_1g2M1g2m_= bln_Mm_= g2m=11g2M+bln_m2M2_Thesolutionhasapoleatm0=: = Mexp_b2g2_whichis the natural scale of the theory. For m , get asymptotic freedomas longas b >0, i.e.mg2b > 0mg2b < 0Figure4.6: Landaupoleintherunningof thegaugecouplingdependingonthesignof the bg3contributiontothefunction: If b >0, thestrengthof theinteractionmonotonicallydecreases towards higher mass scales m. Negativevaluesb 0,soinbrokenSUSY,theenergyisstrictlypositive,E> 06566 CHAPTER5. SUPERSYMMETRYBREAKING5.2 F-andDbreaking5.2.1 FtermbreakingConsiderthetransformation-lawsunderSUSYforcomponentsofachiralsupereld, =2 =2F +i2 F = i2 .Ifoneof,,F ,= 0,thenSUSYisbroken. ButtopreserveLorentzinvariance,need = = 0as theywouldbothtransform undersome representationofLorentzgroup. So ourSUSY breaking conditionsimpliesto, SUSY F ,= 0.Onlythefermionicpartofwillchange, = F = 0, =2F ,= 0,so call a Goldstone fermion or the goldstino (although it is not the SUSY partner of some Goldstone boson).RememberthattheFtermofthescalarpotentialisgivenbyV(F)= K1ijWiWj,soSUSYbreakingisequivalenttoapositivevacuumexpectationvalue, SUSY V(F) > 0.5.2.2 ORaifertaighmodelThe ORaifertaigh model involves a triplet of chiral superelds 1, 2, 3 for which Kahler- and superpotentialaregivenbyK = i i, W = g 1 (23m2) +M 2 3, M m.FromtheFequationsofmotion,iffollowsthatF1=W1= g (23m2)F2=W2= M 3F3=W3= 2 g 13+M 2.5.2. F-ANDDBREAKING 67SUSY + gauge s ymmetry pres erved SUSY pres erved + gauge s ymmetry brokenSUSY + gauge s ymmetry broken SUSY broken + gauge s ymmetry pres ervedVVVVFigure5.1: Varioussymmetrybreakingscenarions: SUSYisbroken, whenevertheminimumpotential energyV (min)isnonzero. Gaugesymmetryisbrokenwheneverthepotentialsminimumisattainedatanonzeroeldcongurationmin = 0ofagaugenon-singlet.We cannot have Fi= 0 for i = 1, 2, 3 simultaneously, so the form of Windeed breaks SUSY. Now, determinethespectrum:V =_Wi_ _Wj_= g223m22+M2[3[2+2 g 13+M 22Ifm2 0.Thisarbitrarinessof1implieszeromass,m1= 0. Forsimplicity,set 1 = 0andcomputethespectrumoffermionsandscalars. Considerthefermionmassterm_2Wij_ij=____0 0 00 0 M0 M 0____ijintheLagrangian,whichgivesimassesm1= 0, m2= m3= M.68 CHAPTER5. SUPERSYMMETRYBREAKINGFigure5.2: Exampleof aatdirection: If thepotential takesitsminimumforacontinuousrangeof eldcongurations(here: forany1 C),thenitissaidtohaveaatdirection. Asaresult,thescalareld1willbemassless.1turnsouttobethegoldstino(dueto1 F1 , =0andzeromass). Todeterminescalarmasses, lookatthequadratictermsinV :Vquad= m2g2(23+23) +M2[3[2+M2[2[2= m1= 0, m2= MRegard3asacomplexeld3= a +ibwherereal-andimaginaryparthavedierentmasses,m2a= M22 g2m2, m2b= M2+2 g2m2.Thisgivesthefollowingspectrum:M22g2m2M2+ 2g2m2mM0(1, 1)(2, 2), 3Im(3)Re (3)Figure5.3: Masssplittingofthereal-andimaginarypartofthethirdscalar3intheORaifertaighmodel.5.2. F-ANDDBREAKING 69We generally get heavier and lighter superpartners, the supertrace of M(treating bosonic and fermionic partsdierently) vanishes. This is generic for tree level of broken SUSY. Since Wis not renormalized to all ordersin perturbation theory, we have an important result: If SUSY is unbroken at tree level, then it also unbrokentoallordersinperturbationtheory. Thismeansthatinordertobreaksupersymmetryweneedtoconsidernon-perturbativeeects:= , SUSYnon-perturbativelyExercise5.1: Byanalysingthemassmatrixforscalarsandfermions,verifyfortheORaifertaighmodelSTr_M2_:=

j(1)2j+1(2j + 1) m2j= 0,wherejrepresentsthespinoftheparticles.5.2.3 DtermbreakingConsideravectorsupereldV= (, A, D), D = D ,= 0 = , SUSY.isagoldstino(which, again, isnotthefermionicpartnerof anyGoldstoneboson). Moreonthatintheexamples.Exercise5.2: Considerachiral supereldof chargeqcoupledtoanAbelianvectorsupereldV. WritedowntheD-termpart of thescalarpotential. Showthat anon-vanishingvacuumexpectationvalueof D,theauxiliaryeldofV, canbreaksupersymmetry. FindtheconditionthattheFayet-Iliopoulostermandthecharge qhave to satisfy for supersymmetry to be broken. Find the spectrum of this model after supersymmetryisbrokenanddiscussthemasssplittingofthemultiplet.Exercise 5.3 Consider a renormalisable A= 1 supersymmetric theory with chiral superelds i= (i, i, Fi)andvector superelds Va=(a, Aa, Da) withbothDandFtermsupersymmetry breaking (Fi,=0andDa ,= 0). ShowthatinthevacuumVi= Fj2Wij+gaDaj(Ta)ji= 0 .HeregaandTarefertothegaugecouplingandgeneratorsof thegaugegrouprespectively. Also, sincethesuperpotential Wisgaugeinvariant,thegaugevariationofWis(a)gaugeW=Wi(a)gaugei= Fi (Ta)ijj.Writethesetwoconditionsintheformof amatrixMactingonatwo-vector withcomponents FjandDa. Identifythismatrixandshowthat it isthesameasthefermionmassmatrix. Arguethatit hasonezeroeigenvaluewhichcanbeidentiedwiththeGoldstonefermion.70 CHAPTER5. SUPERSYMMETRYBREAKING5.3 Supersymmetrybreakingin A= 1supergravity Supergravitymultipletaddsnewauxiliary-eldsFgwithnonzero FgforbrokenSUSY. TheF-termisproportionaltoF DW =W+1M2plKW. ThescalarpotentialV(F)hasanegativegravitationalterm,V(F)= exp_KM2pl_ _(K1)ijDiW DjW3 [W[2M2pl_.That is why both V = 0 and V , = 0 are possible after SUSY breaking in supergravity, whereas brokenSUSY in the global case required V > 0. This is very important for the cosmological constant problem(which is the lack of understanding of why the vacuum energy today is almost zero). The vacuum energyessentiallycorrespondstothevalueofthescalarpotentialattheminimum. Inglobalsupersymmetry,weknowthatthebreakingofsupersymmetryimpliesthisvacuumenergytobelarge. Insupergravityit is possible to break supersymmetry at a physically allowed scale and still to keep the vacuum energyzero. This does not solve the cosmological constant problem, but it makes supersymmetric theories stillviable. ThesuperHiggseect: Spontaneously broken gauge theories realize the Higgs mechanism in which thecorrespondingGoldstonebosoniseatenbythecorrespondinggaugeeldtogetamass. Asimilarphenomenon happens in supersymmetry. The goldstino eld joins the originally massless gravitino eld(which is the gauge eld of A= 1 supergravity) and gives it a mass, in this sense the gravitino receivesitsmassbyeatingthegoldstino. Amassivegravitino(keepingamasslessgraviton)illustratesthebreakingof supersymmetry. ThesuperHiggseectshouldnotbeconfusedwiththesupersymmetricextensionofthestandardHiggseectinwhichamasslessvectorsupereldeatsachiralsupereldtoreceiveamassturningitintoamassivevectormultiplet.Exercise5.4: Consider A= 1supergravitywiththreechiral supereldsS,T, andC. InPlanckunits,theKahlerpotential andsuperpotential aregivenbyK = log (S +S) 3log (T+TCC)W = C3+a eS+b,wherea,barearbitrarycomplexnumbersand>0. Computethescalarpotential. FindtheauxiliaryeldforS, T, Candverifythatsupersymmetryisbroken. AssumingthatCdenotesamattereldwithvanishingvev,ndaminimumofthepotential. Arethereatdirections?Chapter6TheMSSM-basicingredientsIn this chapter, we will shed light on some aspects of the minimally supersymmetric extension of the StandardModelwhichisobviouslycalledMSSMinshorthand.6.1 ParticlesFirstofall, wehavevectoreldstransformingunderSU(3)CSU(2)L U(1)Y , secondlytherearechiralsupereldsrepresenting quarksQi=_3, 2, 16_. .left-handed, uci=_3, 1,23_,dci=_3, 1, 13_. .right-handed leptonsLi=_1, 2,12_. .left-handed, eci= (1, 1, 1) , ci= (1, 1, 0). .right-handed higgsesH1=_1, 2,12_, H2=_1, 2, 12_the second of which is a new particle, not present in the Standard Model. It is needed in order to avoidanomalies,liketheoneshownbelow.ThesumofY3overalltheMSSMparticlesmustvanish(i.e. multiplythethirdquantumnumberwiththeproductofthersttwotocoverallthedistinctparticles).6.2 Interactions K= exp(2qV )isrenormalizable. fa= awhere Rea =4g2adetermines the gauge coupling constants. These coupling constants changewithenergyasmentionedbefore. Theprecisewaytheyrunisdeterminedbythelowenergyspectrum7172 CHAPTER6. THEMSSM-BASICINGREDIENTSFigure6.1: AnomalousgraphproportionaltoTr{Y3}whichmustvanish.ofthemattereldsinthetheory. WeknowfromprecisiontestsoftheStandardModel thatwithitsspectrum,therunningofthethreegaugecouplingsissuchthattheydonotmeetatasinglepointathigherenergies(whichwouldsignalagaugecouplingunication).However, withthemattereldspectrumoftheMSSM, thethreedierentcouplingsevolveinsuchawaythattheymeetatalargeenergyE. Thisisconsideredtobethemainphenomenologicalsuccessofsupersymmetrictheoriesandithintstoasupersymmetricgranduniedtheoryatlargeenergies.EgaEgaEGUTFigure6.2: RunningofthegaugecouplingsassociatedwiththeSU(3)C,SU(2)L,U(1)Yfactors: Theleftplotreferstothenon-supersymmetricStandardModelwhereallthreecouplingsnevercoincideforanyenergy. Ontheright,therunningoftheMSSMgaugecouplingsisplotted,theyareuniedatsomelargeenergyE= EGUT 1016GeV. FIterm: need= 0,otherwisebreakchargeandcolour. ThesuperpotentialWisgivenbyW = y1QH2 uc+y2QH1dc+y3LH1 ec+H1H2+W, BL,W, BL= 1LL ec+2LQdc+3 ucdcdc+tLH2The rst three terms in Wcorrespond to standard Yukawa couplings giving masses to up quarks, downquarksandleptons. ThefourthtermisamasstermforthetwoHiggselds. Buteachof the ,BLtermsbreaksbaryon-orleptonnumber. ThesecouplingsarenotpresentintheStandardModel that6.3. SUPERSYMMETRYBREAKINGINTHEMSSM 73automaticallypreservesbaryonandleptonnumber(asaccidentalsymmetries). Theshowninteractionwouldallowprotondecayp e++ 0withinseconds. Inordertoforbidthosecouplings, anextraudde+ uu u0pFigure6.3: Protondecayduetobaryon-andleptonnumberviolatinginteractions.symmetryshouldbeimposed. ThesimplestonethatworksisRparitydenedasR := (1)3(BL)+2S=___+1 : allobservedparticles1 : superpartners.ItforbidsallthetermsinW, BL.ThepossibleexistenceofRparitywouldhaveimportantphysicalimplications: Thelightestsuperpartner(LSP)isstable. Usually,LSPisneutral(higgsino,photino),theneutralinoisbestcandidatefordarkmatter(WIMP). In colliders, superparticles are produced in pairs which then decay to LSP and give a signal of missingenergy.6.3 SupersymmetrybreakingintheMSSMRecallthetwosectorsoftheStandardModel:__observablesector(quarks)__Yukawa__symmetry-breaking(Higgs)__Supersymmetryhasanadditionalmessengersector__observablesector____messenger-sector____SUSY-breaking__involvingthreetypesofmediation74 CHAPTER6. THEMSSM-BASICINGREDIENTS gravitymediationIfthemediatingeldcoupleswithgravitationalstrengthtothestandardmodel,thecouplingswillbesuppressed by the inverse Planck mass Mplwhich is the natural scale of gravity. We must include somemass square to get the right dimension for the mass splitting in the observable sector. That will be thesquareofSUSYbreakingmassM,SUSY:m =M2,SUSYMpl.Wewantm 1TeVandknowMpl 1018GeV,soM,SUSY=_m Mpl 1011GeV.Thegravitinogetsamassm32ofmorderTeVfromthesuperHiggsmechanism. gaugemediationG =_SU(3)SU(2)U(1)_G,SUSY=: G0G,SUSYMattereldsarechargedunderbothG0andG,SUSYwhichgivesaM,SUSYoforderm, i.e. TeV. Inthatcase,thegravitinomassm32isgivenbyM2SUSYMpl 103eV. anomalymediationInthiscase, auxiliaryeldsof supergravity(orWeyl compensator)getavacuumexpectationvalue.Theeectsarealwayspresentbutsuppressedbyloopeects.Eachif thesescenarioshasphenomenological advantagesanddisadvantagesandsolvingtheirproblemsisanactingeldsof researchatthemoment. Inall scenarios, theLagrangianfortheobservablesectorhascontributionsL = LSUSY+L,SUSYWhere:L,SUSY= m20. .scalarmasses+__M. .gauginomasses+h.c.__+(A3+h.c.)M, m20, Aarecalledsoftbreakingterms. Theydeterminetheamountbywhichsupersymmetryisexpectedtobebrokenintheobservablesector andarethemainparameters tofollowintheattempts toidentifysupersymmetrictheorieswithpotentialexperimentalobservations.6.4 ThehierarchyproblemInhighenergyphysicsthereareatleasttwofundamentalscales-thePlanckmassMpl 1019GeVdeningthescaleof quantumgravityandtheelectroweakscaleMew 102GeV, deningthesymmetrybreakingscaleoftheStandardModel. Understandingwhythesetwoscalesaresodierentisthehierarchyproblem.Actuallytheproblemcanbeformulatedintwoparts:6.4. THEHIERARCHYPROBLEM 75(i) WhyMew Mpl?Answeringthisquestionistheproperhierarchyproblem.(ii) Isthishierarchystableunderquantumcorrections? Thisisthenaturalnesspartof thehierarchyproblemwhichistheonethatpresentsabiggerchallenge.Letustrytounderstandthenaturalnesspartofthehierarchyproblem.IntheStandardModelweknowthat: Gaugeparticlesaremasslessduetogaugeinvariance, thatmeans, adirectmasstermforthegaugeparticlesMAAisnotallowedbygaugeinvariance(A A +foraU(1)eld). Fermionsmassesmarealsoforbiddenforall quarksandleptonsbygaugeinvariance. Recall thatthese particles receive a mass only through the Yukawa couplings to the Higgs (e.g. Hgiving a masstoafterHgetsanon-zerovalue). TheHiggsistheonlyscalarparticleintheStandardModel. TheyaretheonlyonesthatcanhaveamasstermintheLagrangianm2HH. Sothereisnotasymmetrythatprotectsthescalarsfrombecoming very heavy. Actually,if the Standard Model is valid up to a xed cuto scale (for instance Mplasanextremecase),itisknownthatloopcorrectionstothescalarmassm2inducevaluesoforder2tothescalarmass. Thesecorrectionscomefrombothbosonsandfermionsrunninginloops.ThesewouldmaketheHiggstobeasheavyas. Thisisunnaturalsincecanbemuchlargerthanthe electroweak scale 102GeV. Therefore even if we start with a Higgs mass of order the electroweakscale, loopcorrectionswouldbringituptothehighestscaleinthetheory, . Thiswouldruinthehierarchybetweenlargeandsmall scales. ItispossibletoadjustornetunetheloopcorrectionssuchastokeeptheHiggslight, butthiswouldrequireadjustmentstomanydecimal guresoneachorderofperturbationtheory. ThisnetuningisconsideredunnaturalandanexplanationofwhytheHiggsmass(andthewholeelectroweakscale)canbenaturallymaintainedtobehierarchicallysmallerthanthePlanckscaleoranyotherlargecutoscaleisrequired.InSUSY, bosons have thesamemasses as fermions, sonoproblemabout hierarchyfor all squarks andsleptonssincethefermionshavetheirmassprotectedbygaugeinvariance.Secondly,wehaveseeninsubsection4.1.2thatexplicitcomputationofloopdiagramscancelbosonagainstfermionloopsduetothefactthatthecouplingsdeningtheverticesoneachcasearedeterminedbythesamequantity(gintheYukawacouplingoffermionstoscalarandg2inthequarticcouplingsofscalarsaswas mentioned in the discussion of the WZ model). These miraculous cancellations protect the Higgs massfrombecomingarbitrarilylarge.Another waytosee this is that eventhoughamass termis still allowedfor the Higgs bythe couplinginthe superpotential H1H2, the non-renormalizationof the superpotential guarantees that, as longassupersymmetryisnotbroken,themassparameterwillnotbecorrectedbyloopeects.Thereforeifsupersymmetrywasexactthefermionsandbosonswouldbedegenerate,butifsupersymmetrybreaksatascaleclosetotheelectroweakscalethenitwillprotecttheHiggsfrombecomingtoolarge. Thisisthemainreasontoexpectsupersymmetrytobebrokenatlowenergiesof order102 103GeVtosolvethenaturalnesspartofthehierarchyproblem.76 CHAPTER6. THEMSSM-BASICINGREDIENTSFurthermore, thefact that weexpect supersymmetrytobebrokenbynon-perturbativeeects (of ordere1/g2) is verypromisingas awaytoexplainthe existence of the hierarchy(rst part of the hierarchyproblem). Thatisthatif westartatascaleM Mew(e.g. M MplinstringtheoryorGUTs), thesupersymmetrybreakingscale canbe generatedas MSUSYMe1/g2, for asmall gauge coupling, sayg 0.1,thiswouldnaturallyexplainwhyMSUSY M.6.5 ThecosmologicalconstantproblemThisisprobablyamoredicultproblemasexplainedinsection1.4. Therecentevidenceofanacceleratinguniverseindicatesanewscaleinphysicswhichisthecosmological constantscaleM, withMMewMewMpl1015. ExplainingwhyMissosmallisthecosmologicalconstantproblem. Againitcanexpressedintwoparts,whytheratioissosmalland(moredicult)whythisratioisstableunderquantumcorrections.Supersymmetry could in principle solve this problem, since it is easy to keep the vacuum energy to be zeroinasupersymmetrictheory. Howeverkeepingitsosmall wouldrequireasupersymmetrybreakingscaleoforder m M 103eV. But that would imply that the superpartner of the electron would be essentiallyof thesamemass as theelectronandshouldhavebeenseenexperimentallylongago. Thereforeat bestsupersymmetrykeepsthecosmologicalconstantsmalluntilitbreaks. IfitbreaksattheelectroweakscaleMewthatwouldleadtoM Mewwhichisnotgoodenough.Can we address both the hierarchy- and the cosmological constant problem at the same time?Some attemptsarerecentlyputforwardintermsofthestringtheorylandscapeinwhichouruniverseisonlyoneofasetof a huge number of solutions (or vacua) of the theory. This number being greater than 10500would indicatethatafewoftheseuniverseswillhavethevalueofthecosmologicalconstantwehavetoday,andwehappento live in one of those (in the same way that there are many galaxies and planets in the universe and we justhappentoliveinone).Thisisstillverycontroversial, buthasleadtospeculationsthatifthisisawayofsolvingthecosmologicalconstant problem, it would indicate a similar solution of the hierarchy problem and the role of supersymmetrywould be diminished in explaining the hierarchy problem. This would imply that the scale of supersymmetrybreakingcouldbemuchlarger. Itisfairtosaythatthereisnotatpresentasatisfactoryapproachtoboththehierarchyandcosmological constantproblems. Itisimportanttokeepinmindthateventhoughlowenergysupersymmetrysolvesthehierarchyprobleminaveryelegantway,thefactthatitdoesnotaddressthecosmologicalconstantproblemisworrisomeinthesensethatanysolutionofthecosmologicalconstantproblemcouldaectourunderstandingoflowenergyphysicstochangethenatureofthehierarchyproblemand then the importance of low energy supersymmetry. This is a very active area of research at the moment.Chapter7ExtradimensionsItisimportanttolookforalternativewaystoaddresstheproblemsthatsupersymmetrysolvesandalsotoaddressothertroublespotsoftheStandardModel. Wementionedintherstlecturethatsupersymmetryandextradimensionsarethenatural extensionsofspacetimesymmetriesthatmayplayanimportantroleinourunderstandingofnature. Herewewillstartthediscussionofphysicsinextradimensions.7.1 BasicsofKaluzaKleintheories7.1.1 History In1914Nordstromand1919-1921Kaluzaindependentlytriedtounifygravityandelectromag-netism. Nordstromwasattemptinganunsuccessfultheoryofgravityintermsofscalarelds, priorto Einstein. Kaluza usedgeneralrelativityextendedto ve dimensions. His concepts were basedonWeylsideas. 1926Klein: cylindricuniversewith5thdimensionofsmallradiusR after1926,severalpeopledevelopedtheKKideas(Einstein,Jordan,Pauli,Ehrenfest,...) 1960s: deWittobtaining4dimensional YangMillstheoriesin4dfromD>5. AlsostringswithD = 26. In1970s and1980s: Superstrings requiredD=10. Developments insupergravityrequiredextradimensionsandpossiblemaximumnumbersof dimensionsforSUSYwerediscussed: D=11turnedouttobethemaximumnumberofdimensions(Nahm). WittenexaminedthecosetG/H =SU(3) SU(2) U(1)SU(2) U(1) U(1)dim(G/H) = (8+3+1)(3+1+1) = 7whichalsoimpliedD=11tobetheminimum. 11dimensions, however, donotadmitchiralitysinceinodddimensions,thereisnoanalogueofthe5matrixinfourdimensions.7778 CHAPTER7. EXTRADIMENSIONSFigure7.1: Exampleof avedimensional spacetimeM4 S1whereS1isacircularextradimensioninadditiontofourdimensionalM4. 1990s: SuperstringsrevivedD = 11(Mtheory)andbraneworldscenario(largeextradimensions).Exercise7.1: ConsidertheSchrodingerequationforaparticlemovingintwodimensionsxandy. Theseconddimensionisacircleorradiusr. Thepotential correspondstoasquarewell (V (x) = 0forx (0, a)andV= otherwise). Derivetheenergylevelsforthetwo-dimensional Schrodingerequationandcomparetheresultwiththestandardone-dimensional situationinthelimitr a.7.1.2 Scalareldin5dimensionsBeforediscussinghigherdimensional gravity, wewill startwiththesimplercasesof scalareldsinextradimensions,followedbyvectorelds andotherbosonic elds ofhelicity 1. This willillustrate the eectsof havingextradimensions insimpleterms. Wewill bebuildinguponthelevel of complexitytoreachgravitationaltheoriesinveandhigherdimensions. Inthenextchapterweextendthediscussiontoincludefermionicelds.Consideramassless5Dscalareld(xM), M= 0, 1, ..., 4withactiono5D=_d5xMM.Settheextradimensionx4= ydeningacircleofradiusrwithy y + 2r.Ourspacetimeisnow M4S1. PeriodicityinydirectionimpliesdiscreteFourierexpansion(x, y) =

n=n(x)exp_inyr_.Notice that the Fourier coecients are functions of the standard 4D coordinates and therefore are (an innitenumberof)4Dscalarelds. Theequationsof motionfortheFouriermodesare(ingeneral massive)waveequationsMM = 0 =

n=_n2r2_n(x)exp_inyr_= 07.1. BASICSOFKALUZAKLEINTHEORIES 79= n(x)n2r2n(x) = 0.Thesearethenaninnitenumberof KleinGordonequationsformassive4Delds. ThismeansthateachFouriermodenisa4Dparticlewithmassm2n=n2r2 . Onlythezeromode(n=0)ismassless. Onecanvisualizethestatesasaninnitetowerofmassivestates(withincreasingmassproportional ton). Thisiscalled KaluzaKleintower and the massive states (n ,= 0) are called KaluzaKlein- or momentumstates, sincetheycomefromthemomentumintheextradimension:01/r2/rFigure7.2: TheKaluzaKleintowerofmassivestatesduetoanextraS1dimension. Massesmn= |n|/rgrowlinearlywiththefthdimensionswavenumbern Z.Inordertoobtaintheeectiveactionin4Dforalltheseparticles,letusplugthemodeexpansionofintotheoriginal5Daction,o5D=_d4x_dy

n=_n(x) n(x)n2r2[n[2_= 2 r_d4x_0(x) 0(x)+..._= o4D+....This means that the5Dactionreduces toone4Dactionfor amassless scalar eldplus aninnitesumof massive scalar actions in4D. If we are onlyinterestedinenergies smaller thanthe1rscale, we mayconcentrateonlyonthe0modeaction. If werestrictourattentiontothezeromode(likeKaluzadid),then(xM)=(x). Thiswouldbeequivalenttojusttruncatingallthemassiveelds. Inthiscasespeakofdimensional reduction. Moregenerally, ifwekeepall themassivemodeswetalkaboutcompactication,meaningthattheextradimensioniscompactanditsexistenceistakenintoaccountaslongastheFouriermodesareincluded.7.1.3 Vectoreldsin5dimensionsLet us now move to the next simpler case of an abelian vector eld in 5D, similar to an electromagnetic eldin4D.WecansplitamasslessvectoreldAM(xM)intoAM=___A(vectorin4dimensions)A4=: (scalarin4dimensions).80 CHAPTER7. EXTRADIMENSIONSEachcomponenthasadiscreteFourierexpansionA=

n=Anexp_inyr_, =

n=nexp_inyr_.Considertheactiono5D=_d5x1g25DFMN FMNwitheldstrengthFMN:= MANNAMimplyingMMANMNAM= 0.Chooseagauge,e.g. transverseMAM= 0, A0= 0 = MMAN= 0,then this obviously becomes equivalent to the scalar eld case (for each component AM) indicating an innitetowerofmassivestatesforeachmasslessstatein5D.Inordertondthe4Deectiveactionweonceagainplugthisintothe5Daction:o5D o4D=_d4x_2rg25DF(0)F(0)+2rg25D00+..._,Thereforeweendupwitha4Dtheoryofagaugeparticle(massless),amasslessscalarandinnitetowersofmassive vector and scalar elds. Notice that the gauge couplings of 4- and 5 dimensional actions (coecientsofFMNFMNandFF)arerelatedby1g24=2rg25.InDspacetimedimensions,thisgeneralizesto1g24=VD4g2Dwhere Vnis the volume of the n dimensional compact space (e.g. an n sphere of radius r). Higher dimensionalelectromagneticeldshavefurtherinterestingissuesthatwepasstodiscuss.Electric(andgravitational)potentialGausslawimpliesfortheelectriceld

EanditspotentialofapointchargeQ:_S2

Ed

S = Q = |

E| 1R2, 1R4dimensions_S3

Ed

S = Q = |

E| 1R3, 1R25dimensionsSoinDspacetimedimensions|

E| 1RD2, 1RD3.7.1. BASICSOFKALUZAKLEINTHEORIES 81Ifonedimensioniscompactied(radiusr)likein M4S1,then|

E| ___1R3: R < r1R2: R r.Analoguesargumentsholdforgravitationaleldsandtheirpotentials.CommentsonspinanddegreeoffreedomcountingWe know that a gauge particle in 4 dimensions has spin one and carries two degrees of freedom. We may askaboutthegeneralizationoftheseresultstoahigherdimensionalgaugeeld.RecallLorentzalgebrain4dimension_M, M_= i_M+MMM_Ji= ijk Mjk, J M23.FormasslessrepresentationsinDdimensions,O(D 2)islittlegroup:P= (E, E, 0, ..., 0..O(D2))TheLorentzalgebraisjustlikein4dimensions,replace,,... byM,N,...,soM23commuteswithM45and M67for example. Dene the spin to be the maximum eigenvalue of any Mi(i+1). The number of degreesof freedomin4dimensionsis2(A Aiwithi=2, 3)correspondingtothe2photonpolarizationsand(D 2)inDdimension,AM Aiwherei = 1, 2, ..., D 2.7.1.4 DualityandantisymmetrictensoreldsSofarweconsideredscalar-andvectorelds:scalar vector index-rangeD = 4 (x) A(x) = 0, 1, 2, 3D > 4 (xM) AM(xM) M= 0, 1, ..., D 1We will see now that in extra dimensions there are further elds corresponding to bosonic particles of helicity 1. Theseareantisymmetrictensorelds, whichin4Darejustequivalenttoscalarsorvectoreldsbyasymmetryknownasduality. Butinextradimensionsthesewill benewtypesof particles(thatplayanimportantroleinstringtheoryforinstance).In4dimensions,deneadualeldstrengthtotheFaradaytensorFviaF:=12

F,thenMaxwellsequationsinvacuoread:F= 0 (eldequations) F= 0 (Bianchiidentities)82 CHAPTER7. EXTRADIMENSIONSTheexchangeFF (theelectromagnetic duality) correspondingto

E

Bswaps eldequations andBianchiidentities.In5dimensions,onecoulddeneinanalogyFMNP= MNPQRFQR.Onecangenerallystartwithanantisymmetric(p + 1)-tensorAM1...Mp+1andderiveaeldstrengthFM1...Mp+2= [M1AM2...Mp+2]anditsdual(withD (p + 2)indices)FM1...MDp2= M1...MD FMDp1...MD.Considerforexample D = 4F= [B]=F= F= aThedual potentialsthatyieldeldstrengthsF Fhavedierentnumberof indices, 2tensorB a(scalarpotential). D = 6FMNP= [MBNP]=FQRS= MNPQRS FMNP= [Q BRS]HerethepotentialsBNP BRSareofthesametype.Antisymmetrictensorscarryspin1orless,in6dimensions:BMN=___B: ranktwotensorin4dimensionsB5, B6: 2vectorsin4dimensionsB56: scalarin4dimensionsToseethenumberofdegreesoffreedom,considerlittlegroupBM1...Mp+1 Bi1...ip+1, ik= 1, ..., (D 2).Theseare_D2p+1_independentcomponents. Notethatunderduality, couplingsgaremappedto(multiplesof)theirinverses,L =1g2([M1BM2...Mp+2])2 g2([M1BM2...MD(p+2)])2.pbranesElectromagneticeldscoupletotheworldlineofparticlesviao _Adx,7.1. BASICSOFKALUZAKLEINTHEORIES 83This canbeseenas follows: theelectromagneticeldcouples toaconservedcurrent in4dimensions as_d4xAJ(withDiraccurrentJ=foranelectroneldforinstance). Foraparticleof chargeq,thecurrentcanbewrittenasanintegralovertheworldlineoftheparticleJ= q_d4(x )suchthat_J0d3x = qandsothecouplingbecomes_d4xJA= q_dA.Wecanextendthisideaforhigherdimensionalobjects. ForapotentialB[]withtwoindices,theanalogueis_Bdx dx,i.e. needastringwith2dimensionalworldsheettocouple. Furthergeneralizationsare_Bdx dx dx(membrane)_BM1...Mp+1dxM1 ... dxMp+1(pbrane)Thereforewecanseethatantisymmetrictensorsofhigherrankcouplednaturallytoextendedobjects. Thisleadstotheconceptofapbraneasageneralizationofaparticlethatcouplestoantisymmetrictensorsofrankp + 1. Aparticlecarrieschargeunderavectoreld, suchaselectromagnetism. Inthesamesense, pbranescarryanewkindofchargewithrespecttoahigherrankantisymmetrictensor.Exercise7.2: ConsiderthefollowingLagrangiano =_d4x_1g2HH+a H_.Solvetheequationof motionfortheLagrangemultiplieratoobtainanactionforapropagatingmasslessKalb-RamondeldB. Alternatively, solvetheequationof motionfortheeldH, toobtainanactionforthepropagatingaxionelda.Whathappenstothecouplinggunderthistransformation?7.1.5 GravitationinKaluzaKleintheoryAfterdiscussingscalar-,vector-andantisymmetrictensoreldsspin deg. offreedomscalar 0 1 + 1vectorAM0, 1 D 2antisymmetrictensorAM1...Mp+10, 1_D2p+1_wearenowreadytoconsiderthegravitonGMNofKaluzaKleintheoryinDdimensionsGMN=___GgravitonGnvectorsGmnscalarswhere, = 0, 1, 2, 3andm, n = 4, ..., D 1.Thebackgroundmetricappearsinthe5dimensionalEinsteinHilbertactiono =_d5x_[G[(5)R,(5)RMN= 0.84 CHAPTER7. EXTRADIMENSIONSOnepossiblesolutionisthe5dimensionalMinkowskimetricGMN= MN,anotheroneisthatof4dimen-sionalMinkowskispacetimeM4timesacircleS1,i.e. themetricisofthe M4S1typeds2= W(y) dxdxdy2where M3S1S1is equally valid. In this setting, W(y) is a warped factor that is allowed by the symmetriesofthebackgroundandyisrestrictedtotheinterval[0, 2r]. Forsimplicitywewillsetthewarpfactortoaconstantbutwillconsideritlaterwhereitwillplayanimportantrole.ConsiderexcitationsinadditiontothebackgroundmetricGMN= 13___g2AA_AA__inFourierexpansionGMN= (0)13___g(0)2(0)A(0)A(0)_(0)A(0)(0)A(0)(0)__. .KaluzaKleinansatz+ towerofmassivemodesandplugthezeromodepartintotheEinsteinHilbertaction:o4D=_d4x_[g[_M2(4)plR14(0)F(0)F(0)+16(0)(0)((0))2+..._Thisistheuniedtheoryofgravity,electromagnetismandscalarelds!Exercise 7.3: Show that the last equation follows from a pure gravitational theory in ve-dimensions,using(5)R =(4)R2e2e14e2FFwhereG55= e2.RelatethegaugecouplingtotheU(1)isometryofthecompactspace.Symmetries general4dimensionalcoordinatetransformationsx xt(x), g(0)(graviton), A(0)(vector) ytransformationy yt= F(x, y)Noticethatds2= (0)13_g(0)dxdx(0)_dyA(0)dx_2_,so,inordertoleaveds2invariant,needF(x, y) = y +f(x) = dyt= dy +fxdx, A

(0)= A(0)+1fxwhicharegaugetransformationforamasslesseldA(0)! Thisisthewaytounderstandthatstan-dardgaugesymmetriescanbederivedfromgeneral coordinatetransformationsinextradimensions,explainingtheKaluzaKleinprogrammeofunifyingalltheinteractionsbymeansofextradimensions.7.2. THEBRANEWORLDSCENARIO 85 overallscalingy y, A(0) A(0), (0)12(0)= ds2 23ds2(0)isamasslessmoduluseld,aatdirectioninthepotential,so (0)andthereforethesizeofthe5th dimension is arbitrary. (0)is called breathing mode, radion or dilaton. This is a major problem forthesetheories: Itlookslikeallthevaluesoftheradius(orvolumeingeneral)oftheextradimensionsareequallygoodandthetheorydoesnotprovideawaytoxthissize. Itisamanifestationof theproblemthatthetheorycannotpreferaat5DMinkowski space(inniteradius)over M4S1(orM3 S1 S1, etc.). Thisisthemoduli problemof extradimensional theories. Stringtheoriessharethisproblem. Recentdevelopmentsinstringtheoryallowstoxthevalueofthevolumeandshapeoftheextradimension,leadingtoalargebutdiscretesetofsolutions. Thisistheso-calledlandscapeofstringsolutions(eachonedescribingadierentuniverseandoursisonlyoneamongahugenumberofthem).Comments The Planck mass M2pl= M3 2ris a derived quantity. We know experimentally that Mpl 1019GeV,thereforewecanadjustMandrtogivetherightresult. ButthereisnootherconstrainttoxMandr. GeneralizationtomoredimensionsGMN=___g2AiAj hij_mnKniAimnKmiAimn__TheKmiareKillingvectorsofaninternalmanifold /D4withmetricmn. ThetheorycorrespondstoYangMillsin4dimensions. NotethatthePlanckmassnowbehaveslikeM2pl= MD2VD4 MD2rD4= M2(Mr)D4.Ingeneral weknowthatthehighestenergiesexploredsofarrequireM>1TeVandr CCsinceTisamodulus(T , = 0)andCisamattereld(C = 0)andsotheminimumof thepotential isV =0withDSW=0andC=0. Thevacuumenergyof thepotential vanishesatitsminimum. Tiscalledamoduluseldsincewecanvaryitfreelyandstill remainattheminimumof thepotential. Moduli elds denote at directions of the potential. To check whether SUSY is broken we need todeterminetheF-terms:FS= DSW = a eS1S +S_C3+a eS+b_.C.5. CHAPTER7 123SadjustssuchthatDSW= 0attheminimum,whereC= 0.FC= DCW = 3 C2+3CT+TCC(C3+a eS+b)C=0= 0.FT= DTW = 3T+TCC(C3+a eS+b) = 3T+T(a eS+b) ,= 0.AttheminimumFT ,= 0andhencesupersymmetryisbroken. Thisisanotherfeatureofno-scalemodels.2C.5 Chapter7Exercise7.1WestartwiththepotentialV (x) =___0 x (0, a) otherwiseTheydirectionisidentiedundery y + 2r.TheSchrodingerequationis_V (x)

22m_2x2+2y2__n,m(x, y) = En,m n(x, y)Werequiren(x, 0) = n(x, 2r),whichallowsustowriten(x, y) =

m=n,m(x) eimyr.Thesquarewellpotentialrequiresn,m(a) = n,m(0),whichleadstothefollowingansatzn,m(x) = An,msin_nxa_.Intotalthewavefunctionsaren,m(x) = An,msin_nxa_eimyr,whereAn,mis suitablynormalised. Wedeterminetheenergylevels byapplyingtheHamiltoniantothissolution:_ 22m_2x2+2y2__n,m(x, y) =

22m_n22a2m2r2_n,m(x, y)SoweobtainEn,m=

22m_n22a2+m2r2_In the limit r a the energy of any excited Kaluza-Klein state m ,= 0 is very much larger than the ordinarysquare-wellstates,whichdecouplefromthephysicsatlow-energy.2Unfortunately,thisdoesnotsolvethecosmologicalconstantproblem: higherordercorrections(e.g. loops)alwaysbreaktheno-scalestructureandregenerateacosmologicalconstant.124 APPENDIXC. SOLUTIONSTOTHEEXERCISESExercise7.2o =_d4x_1g2HH+a H_VariantA:Assumingalleldsvanishatinnity,wecanrewritetheactionaso =_d4x_1g2HHa H_Varyingthisactionw.r.t. Hgives2g2Ha = 0SoweobtainH=g22a , H=g22a Substitutingthisintheaction,wecanwriteo[a] =_d4x_1g2_g22_(a) (a)(a )g22(a )_=_d4x_g24_(a ) (a )=_d4x_g24_(6 a a)=_d4x3g22a aVariantB:Thee.o.m. fora(intheoriginalaction)gives

H= 0.Inotherwordstheexteriorderivativeof HisvanishingdH=0. Since R4istopologicallytrivial, wecanwriteH= dB,withallpropagatingdegreesoffreedomembeddedinB.ThismeansthattheoriginalactionisequivalenttoaLagrangiano[B] =_d4x1g2(dB) (dB)Both actions o[a] and o[B] describe the same physics. Therefore in 4D a 2-form potential is dual to a scalar,andcanberewrittenintermsof one. Inthisduality, thecouplingconstanttransformsasg 1g(uptonumericalfactors).Exercise7.3Westartwithapurelygravitationaltheoryin5Do =_d5x_[G[(5)R.The5dmetricisGMNds2= GMN dxMdxN= g dxd x+2 G5 dxdx5+G55dx5dx5.C.5. CHAPTER7 125Weusethefollowingdecompositionofthe5-dimensionalRicciscalarR :(5)R =(4)R2 e2e14e2F F.Nowinthevacuum A = 0,thisimpliesG =g4e,sowecanrewritethe5Dactionasfollowso5D=_d5x_[g[_e(4)R2 2e14e3F F_Allquantitiesarecalculatedusingg.WenowwanttorescalethemetricsothattheEinstein-Hilberttermiscanonical. Letusmakethefollowingansatzforrescalingg=: e g.ThischangesRasfollows(4)R = e_(4) R+3232_Onetermintheactionabovecanbediscardedasatotalderivative2_d5x_[g[ () e= 2_d5x__[g[ _e= 0Nowwecancollectallourresultso =_d5x_[ g[__(4) R+ 32. .=0asatotalder.3214e3FF__=_d5x_[ g[_(4) R3214e3FF_.Recallthat_[g[ggFF=_[ g[ g gFF.WeendedupwiththeEinstein-MaxwellLagrangianthatwewerelookingfor.Notethatthegaugecouplingdependsonthesizeof theextradimensions: Largegaugecouplingimpliessmallextradimensions.Inhigherdimensions, everyU(1)isometryof theextradimensionsgivesrisetoaU(1)gaugeeld. If forinstanceanSU(2)isometryexists,thenthelower-dimensionaltheoryhasanSU(2)gaugetheory.Exercise7.4Wehavea5dgravitytheorywithanegativecosmological constant, compactiedonaninterval (0, ).Each end of the interval corresponds to a 3-brane with tension /k. The bulk metric is GMN. The inducedmetriconthevisibleandhiddenbranesisg,visandg,hidden.Wewouldliketoshowthatthewarpedmetricds2= e2W(y) dxdx+r2dy2satisesEinsteinsequations. Weusetheresultthat,forthismetricansatz,Einsteinsequationsreduceto6 Wt2r2= 2 M33 Wttr2=2k M3r_(y ) (y)126 APPENDIXC. SOLUTIONSTOTHEEXERCISESLetustryW(y) = r [y[_12 M3.ThenWt2= r2 12M3issatisedandwell-denedatr= 0sinceWt(y= 0)iswell-dened. WeregardW(y)asaperiodicfunctioniny,onlydenedfor < y< andthendenedthroughperiodicity. Nearx = 0ddx [x[ = 1 +2H(x) =___1 x > 01 x < 0 .d2dx2 [x[ = 2 (x).ForperiodicW(y)andW(y) = r_12M3[y[for0 < y< Wtt(y) = 2r_12M3_(y) (y )Thisimplies_12M3=12kM3 k2=12M3WecanwriteW= rk[y[andsothemetricbecomesds2= e2rk]y] dxdx+r2dy2The5-Dactionis2M3_d5x_[g[ R = 2M3_d4x_dy_[g[ R.The 4d curvature term is determined by the volume of the extra dimension in dimensional reduction. Usingthatg= r2e2kr]y]g4and gR = re2rk]y]g4R4,weseethatthefollowingrelationforthe4-dimensionPlanckmasshastohold:2M2pl= 2M3r_dye2kr]y](C.1)Afterintegrationweobtain:M2pl=M3k(1 e2kr)TheHiggsLagrangianisovis_d4x_[gvis[_gvisDHDH([H[2v20)2_gvis=e2krg4_d4x_[g4[ e4kr_g4e2krDHDH([H[2v20)2_HekrH=_d4x_[g4[_g4DHDH([H[2e2krv20)2_InthelaststepwecanonicallynormalisedtheHiggseldsuchthatthekinetictermsarecanonical. TheHiggsmassthenisgivenbymH= ekrv0anddependsonthewarpfactor. Thenaturalscaleforv0isthePlanckscale. ToobtainaHiggsmassattheweakscaleweneedkr 50 :mH ekrMplThissolvesthehierarchyproblemthroughwarping. The4-dimPlanckscaleandthe5-dimscaleMareherecomparablease2kristiny.C.6. APPENDIXA 127C.6 AppendixAExerciseA.1:Antisymmetryofthe bilinear: = ( ) =

() =

() = () = () Invertingtheproduct: = () ( ) = ()

() = ( )

() = ( )() = () ( )= From=i2[ ],iteasilyfollowsthat =i2 [ ] =i2 [ ] = .ExerciseA.2:Firstly() ( )!= 12( ) ( )= 12() ( )= 12 () ( ). .=2 = = () ( )= () ( ),128 APPENDIXC. SOLUTIONSTOTHEEXERCISESandsecondly12() ()!= () ()= () ()= () () = () () =14() ()

() ()=14() ( ) () ()=12() ().Bibliography[1] Alistofbooksandonlineavailablelecturenotesinalphabeticalorder:I.Aitchison,SupersymmetryinParticlePhysics: AnElementaryIntroduction,CUP(2007).H.Baer,X.Tata: WeakScaleSupersymmetry,CUP(2006).D.Bailin,A.Love: SupersymmetricGaugeFieldTheoryandStringTheory,IOP(1994).P.Binetruy,Supersymmetry: Theory,experimentandcosmology,OUP(2006).I.Buchbinder,S.Kuzenko,IdeasandMethodsofSupersymmetryandSupergravity,IOP(1998).C. Csaki, TASI lectures onextradimensions andbranes, publishedin*Boulder 2002, Particlephysicsandcosmology*605-698,hep-ph/0404096.C. Csaki, J. HubiszandP. Meade, TASI lecturesonelectroweaksymmetrybreakingfromextradimensions,publishedin*Boulder2004,PhysicsinD 4*703-776,hep-ph/0510275.S.P.deAlwis,Potentialsforlightmoduliinsupergravityandstringtheory,hep-th/0602182.M. Dine, SupersymmetryPhenomenologywithaBroadBrush, published in *Boulder 1996, Fields,stringsandduality*813-881hep-ph/9612389.G. Gabadadze, ICTPlecturesonlargeextradimensions, published in *Trieste 2002, Astroparticlephysicsandcosmology*77-120,hep-ph/0308112.S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, Superspace,oronethousandandonelessonsinsupersymmetry,Front.Phys.58(1983)1,hep-th/0108200.G.D.Kribs,Phenomenologyofextradimensions,publishedin*Boulder2004,PhysicsinD 4*633-699,hep-ph/0605325.PLaBelle,SupersymmetryDeMYSTiFied,McGraw-HillProfessional(2009).M.A.Luty,2004TASIlecturesonsupersymmetrybreaking,publishedin*Boulder2004,PhysicsinD 4*495-582,hep-th/0509029.S. P. Martin, ASupersymmetryPrimer, in *Kane, G.L. (ed.): Perspectives on supersymmetry II*1-153,hep-ph/9709356.H.P.Nilles,Supersymmetry,SupergravityAndParticlePhysics,Phys.Rept.110,1(1984).R. Rattazzi, Cargeselecturesonextradimensions, publishedin*Cargese2003, Particlephysicsandcosmology*461-517,hep-ph/0607055.R.Sundrum,Tothefthdimensionandback.(TASI2004),publishedin*Boulder2004,PhysicsinD 4*585-630,hep-th/0508134.J.Terning,ModernSupersymmetry: DynamicsandDuality,OUP(2006).129130 BIBLIOGRAPHYS.Weinberg,Thequantumtheoryofelds,VolumeIIISupersymmetry,CUP(2000).J.Wess,J.Bagger,SupersymmetryandSupergravity,PUP(1992).P.C.West,Introductiontosupersymmetryandsupergravity,WorldScientic(1990).[2] H.J.W.M uller-Kirsten,A.Wiedemann,Supersymmetry,anintroductionwithconceptual andcal-culational details,WorldScientic(2010).[3] S.Weinberg,Thequantumtheoryofelds,VolumeIFoundations,CUP(1995).[4] M. T. Grisaru and H. N. Pendleton, Some Properties Of Scattering Amplitudes In SupersymmetricTheories,Nucl.Phys.B124(1977)81.[5] V.Gates,E.Kangaroo,M.RoachcockandW.C.Gall,Stuperspace,Physica15D(1985)289.[6] A.SalamandJ.A.Strathdee,SupergaugeTransformations,Nucl.Phys.B76(1974)477.[7] A. Salam and J. A. Strathdee, On Superelds And Fermi-Bose Symmetry, Phys. Rev. D 11 (1975)1521.[8] F.A.Berezin,A.A.Kirillov,D.LeitesIntroductiontosuperanalysis,Reidel(1987).[9] B.deWitt,Supermanifolds,CUP(1992).[10] N. Seiberg, Naturalness Versus Supersymmetric Non-renormalizationTheorems, Phys. Lett. B318(1993)469[arXiv:hep-ph/9309335].[11] S.Weinberg,Thequantumtheoryofelds,VolumeIIISupersymmetry,CUP(2000).[12] N.SeibergandE.Witten,MonopoleCondensation,AndConnementIn A= 2SupersymmetricYang-Mills, Nucl. Phys. B 426 (1994) 19 [Erratum-ibid. B 430 (1994) 485] [arXiv:hep-th/9407087].[13] V.KaplunovskyandJ.Louis,Fielddependentgaugecouplingsinlocallysupersymmetriceectivequantumeldtheories,Nucl.Phys.B422(1994)57[arXiv:hep-th/9402005].[14] Z. Komargodski andN. Seiberg, Comments ontheFayet-Iliopoulos TerminFieldTheoryandSupergravity,JHEP0906(2009)007[arXiv:0904.1159[hep-th]].[15] L. Randall andR. Sundrum, Alargemasshierarchyfromasmall extradimension, Phys. Rev.Lett.83(1999)3370[arXiv:hep-ph/9905221].[16] P.C.West,Supergravity,branedynamicsandstringduality,arXiv:hep-th/9811101.[17] J.Polchinski,SuperstringTheoryandbeyond,VolumeII,CUP2005.[18] J.A.Strathdee,ExtendedPoincareSupersymmetry,Int.J.Mod.Phys.A2(1987)273.