PHYSICSREPORTS(Review Sectionof PhysicsLetters)110, NosI & 2 (1984) 1—162. North-Holland, Amsterdam

SUPERSYMMETRY, SUPERGRAVITY AND PARTICLE PHYSICS

H.P. NILLES*

DépartementdePhysiqueThéorique,Universitéde Genève,1211 Genève4, Switzerland

andCERN,Genève,Switzerland

Received16 February1984

To Elizabeth

Contents:

1. Introduction 3 5.2. Supercolor 622. Global supersymmetry 9 5.3. O’Raifeartaighmodels and the connectionbetween

2.1. The supersymmetryalgebraand its immediate im- supersymmetryandweakinteraction breakdown 66plications 9 5.4. The inversehierarchy 68

2.2. Superfields 11 5.5. How to hide thehiddensector 722.3. Lagrangians 15 5.6. The stability of thesmall scale 742.4. Spontaneousbreakingof supersymmetry 19 6. Low energysupergravity:basics 762.5. Nonrenormalizationtheorems,absenceof quadratic 6.1. The gravitinomass 77

divergenciesand soft breaking 23 6.2. The hiddensector 793. Supergravity 28 6.3. The coupling of the hidden sectorto theobservable

3.1. Local supersymmetryimplies (super)gravity 28 sector 833.2. Couplingof matter andgaugefields to supergravity 31 6.4. The mostgeneralcase 863.3. Spontaneoussymmetrybreakdown 34 7. Supergravity:models 893.4. Thesupertraceof the massmatrix 38 7.1. Early attempts 89

4. Low energysupersymmetry:basics 39 7.2. Models with SU(2)x U(1) breakingat thetree level 914.1. The standardmodel 39 7.3. Can we avoid the introduction of a small mass4.2. Grandunification 42 parameter? 954.3. Supersymmetryand the mass scaleof the weak in- 7.4. Light singlets 98

teractions 45 7.5. Breakdownthroughradiativecorrections 1014.4. The particle content of a supersymmetricstandard 7.6. SU(2) x U(1) breakdown with smaller top quark

model 46 masses 1094.5. Supersymmetricgrandunified models 49 7.7. Rareprocesses 1184.6. Constraintsfrom rare processes 56 7.8. Beyondtheminimal models 121

5. Global supersymmetry:models 58 8. Constraintsfrom cosmologicalconsiderations 1265.1. Fayet—U(1) 59 8.1. Constraintson (almost) stableparticles 126

* Partially supportedby theSwissNationalScienceFoundation.

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SUPERSYMMETRY, SUPERGRAVITY ANDPARTICLE PHYSICS

H. P. NILLES

DépartementdePhysiqueThéorique,Universitéde Genève,1211 Genève4, Switzerland

and

CERN,Genève,Switzerland

NORTH-HOLLAND PHYSICS PUBLISHING-AMSTERDAM

H.P. Nilles, Supersymrnetry,supergravityand particlephysics 3

8.2. The grandunified phasetransition 130 9.3. Wino,zino andHiggsino signals 1448.3. The cosmologicalscenarioandmodels 136 10. Conclusionandoutlook 146

9. Experimentalsignatures 139 11. Appendix:notationsandconventions 1499.1. Thescalarpartnersof quarks andleptons 1443 References 1529.2. Photinosandgluinos 147

Abstract:We give a short introductionto N = 1 supersymmetryand supergravityandreview theattemptsto constructmodels in which thebreakdown

scaleof theweakinteractionsis relatedto supersymmetrybreaking.

1. Introduction

Sinceits discoverysometen yearsago,supersymmetry[288,544, 561] hasfascinatedmanyphysicists.This hashappeneddespitethe absenceof eventhe slightestphenomenologicalindication that it mightbe relevantfor adescriptionof nature.The only fact we know for sureis that supersymmetryhasto bebrokenbadly in the energyrangeexploredby the presentlyavailableaccelerators.

The motivation for supersymmetryis thuspurely theoretical,for good reasons.First of all — it is anew symmetry,and symmetryconsiderationshavein the pastoften led to progressin particlephysics.Itis, moreover,a symmetrywith propertiesnot sharedby anysymmetrydiscussedso far in the context ofhigh energyphysics— it transformsbosonsinto fermions.This symmetrybetweenbosonsandfermionsgives rise to special (sometimescalled miraculous)ultraviolet behaviorof supersymmetricfield theories.Thesimplesttheory,N = 1 supersymmetry(whereN countsthe numberof supersymmetrygenerators),is essentiallyfree of quadraticdivergences.Theoriesof higher N for exampleN = 4 are shown to befinite [410,242,515a].Thefact thatN = 4supersymmetricYang—Mills theory is finite hasled somefamousphysiciststo makestatementswhich I cannotresist the temptationto repeathere.“One is temptedtoarguethat a theory with such uniquepropertiesmust havesomerelevancesomewherefor nature” isone of them, and “Well, now at least we know that N = 4 supersymmetricYang—Mills theory hasnothingto do with nature” is the second.Controversialas thesestatementsmight seem,thereis thehopethat in the local versionof supersymmetry(supergravity)[248,113] it might bepossibleto arrive atan acceptablequantumtheory that includesgravity.If thiswould work out onewould haveamotivationto believe that supersymmetrymight be relevant at energiescomparableto the Planck mass M~1.2x 1019 GeV but could still be broken just below this scale, such that it might be irrelevant for themass scales of present day particle physics. In this paper we will study the question, whethersupersymmetrycould have implications at scalesof the order of 100GeV. For reasonswhich willbecomeclear later we will only consider N = 1 supersymmetry.The main motivationsto considersupersymmetryarethe improvedconvergencepropertiesof supersymmetricfield theories.To makethismorespecific let us look at the statusof presentdayparticlephysics.

With the recentdiscoveryof the intermediatevectorbosonsatthe CERN p~5collider [18,19,26,31]one of the central predictionsof the standardmodel [59, 280, 500, 547, 495, 281, 368] hasbeenconfirmed. This standardmodel is basedon the gauge group SU(3)X SU(2)X U(1) for the strongelectromagneticand weak interactionsof threefamilies of quarksand leptons.Apart from the gaugecoupling constantsthis model needsof the order of twenty parametersused to adjust masses,mixinganglesand Yukawa couplings.The model is believed to describephysics correctly at least up to theenergyregion of (let ussay) 100GeV. This belief is basedon the fact that all experimentaltestsof thepredictionsof this model haveturnedout positive.Of coursethe model is not completelytested;there

4 H.P. Nilles, Supersymmetry,supergravilyandparticle physics

areindeedmanypredictionsof the model which are not checkedat all. Theseincludefor exampletheexistenceof the gaugebosontrilinear and quadrilinearcouplingsas well as the existenceof the Higgssector.Thus thereare still many placeswherephysicsmight hidesurprises,which arenot containedinthe standardmodel. Nonetheless,up to now this model is in accordwith all experimentalfindings.

Theoreticallythe model is also very appealing.It is not just basedon an effectiveLagrangian,likee.g. the Fermi theory of weak interactions,but it is a renormalizablefield theory.Actually we havetobe morecarefulhere. For the model to be an acceptablefield theory,one conditionhasto be fulfilledby the spectrum,which can in turn be regardedas a predictionof the theory: at least one not yetdetectedfermionhasto exist— the so-calledtop quark.Accordingto the model it shouldbefound in anenergyregion between20 and200GeV [538].The fact that the standardmodel is renormalizablecouldhavedrastic consequenceson the rangeof validity of the model; would it be nonrenormalizableitnecessarilywouldonly be definedwith a cutoff A (of dimensionof amass)andonewould expectthat itsregion of validity is boundedfrom aboveby A. Above A one expectsnew things to happenwhich arenot describedby the model.Ultimately onewould think that such a modelwould be an approximationto a morecompletetheory which is alsovalid aboveA. Since the standardmodel is renormalizableitcould howeverbe valid in a muchlargerenergyrange,i.e., scaleswhich are muchlargerthan 100 GeV,the scaleof the intermediatevector bosonmasses.This, however,doesnot prove that nothing newhappensabove 100GeV. Experimentswill check this region in the near future both by directexperimentsabove100GeV as well as by precisemeasurementsat lower energies.It might alsobe thatnew particlesarefoundbelow 100GeV (e.g. the top quark);still many possibilitiesareopen.Whatdowe actuallyknow aboutthe region above100GeV?Well, thereis gravity whosedimensionfulcouplingconstantcan be tradedfor a massscalejust as the weakinteractionscalecamefrom theFermi couplingGF. But this massscale is sufficiently large and probably not of immediateinterestto the 100GeVregion. The very precise measurementsof the KL—KS massdifference and CP violation in the kaonsystemcould indicatesomethingnew at the TeV scale,but theycan alsobe describedby the parametersof the standardmodel without invoking new physicsat higher massscales.The recentefforts to detectproton decay,if positive, would provide us with a new massscaleof iO’~GeV as predictedby grandunified models[271,4731. In such a theory the SU(3)x SU(2)X U(1) gaugegroupis unified in a largergroup,e.g.,SU(5)at this massscale.Thesetheoriesareableto predictoneof theparameters,the weakmixing anglecorrectly. If thesegrandunified modelsare the correct descriptionof naturetheywouldprobably imply the validity of the standardmodel up to 1014_1015GeV (the “great desert”)which interm is only possible since the standardmodel is a renormalizablefield theory. Recentexperimentallimits on proton decay [56], however,havenot confirmed this picture up to now, and havereachedvaluesthat might be alreadyinconsistentwith the simplestgrandunified models.

Let us thereforereturn to the 100GeV region and examinethe standardmodel more closely. Thestandardmodel is probably not the final theory of the world since it containssome twenty freeparameters.There is especiallyone sector in the theory, the Higgs sector [317],which remainsrathermysteriousand which is not checkedat all: not eventhe predictedparticleshavebeenfound. Thissector,however,is very importantfor the model since the massscaleof the W and Z is given by theparametersin this sector.The Higgs particlesarescalar(spin zero)particlesandif foundtheywould bethefirst fundamentalscalarparticlesshownto exist.Scalarparticleshaveseveralnice properties.Theyarethe only particles that can havenonvanishingvacuum expectationvalues without breakingLorentzinvariance.They can induce a spontaneousbreakdownof gauge symmetriesand also allow Yukawacouplingsto thefermionswhich leadto a massgenerationafter thebreakdownof thegaugesymmetries.

Scalarparticleshaveanotherpropertywhich is not consideredasnice: theirmassesaresubjectto quadraticdivergencesin perturbationtheory (fig. 1.1). As long asthesemassesarefree parametersthereis of course

H.P. Nilles, Supersymmetry,supergravityandparticle physics 5

/ ~s/ S

‘ /S IS /5,

Fig. 1.1. Quadraticallydivergentcontribution to themassof a scalarparticle.

nothingwrongwith this, theinputparametersbeingjusttherenormalizedmasses.Suppose,however,thatonewould in thefuturetry tounderstandthemagnitudeof thesemasses.Suchanunderstandingcouldonlysurviveperturbationtheoryif the quadraticdivergencewould be physically cut off at a certainmassscalewhichwouldcorrespondto themassesof thescalarparticles.Thestandardmodelnowrequiresthemassofthescalarparticlesto benotmuchlargerthanaTeV, sinceotherwisethecouplingsin theHiggssectorhavetobechosensolargethatthemodelbecomesatleastill-definedperturbatively[538].Thusamodelin whichthescaleof thebreakdownof theweakinteractionsis understoodin termsof morefundamentalparametersthereis hardly any otherpossibility than to expectnew physics in the TeV region,which cuts off thequadraticdivergenceof thescalarparticles.In thissensetheHiggssectorin thestandardmodelis thesectorthatis mostsensitiveto theregionabove100 GeV.It is, at thesametime thesectorthatis theleasttestedexperimentally.

This “problem” [567,523] of the quadraticdivergenciesof the scalar particleshas led peopletoexploremodelsthat do not containfundamentalscalarparticles:the so-calledtechnicoloror compositemodels [523,551,4751. There the Higgs sector of the standardmodel is replaced by a stronglyinteractinggaugesystem.Fermion—antifermionboundstates(in analogyto quantumchromodynamics)are assumedto condenseand break the weak interactionsymmetry at a scaleof a few hundredGeVandleadto massiveintermediatevectorbosonsat 100GeV. A variety of new compositeparticles,likee.g. technipions,is predictedin the TeV range,as expectedin anymodel in whichthe scaleof the weakinteractionsis understoodin termsof morefundamentalparameters.Much effort hasbeendevotedtothe study of this classof models,especiallywhetherit is possibleto constructa “realistic” model,whichdoesnot contradictknown experimentalresults[1911.These modelscontain a gaugesector stronglyinteractingat 1 TeV, andour knowledge about such theoriesis strongly restricted.One hasto makesimple assumptionsaboutthe dynamicsof suchsystemsmostlyassumingthe technicolorforcesto be ascaledup versionof QCD. With theseassumptionssevereproblemsarise.The modelspredicta varietyof compositepseudo-Goldstonebosonssomeof which might be below 10 GeV in mass,and which arecertainlynot yet experimentallydetected.A secondproblemis the absenceof a satisfactorymechanismto generatethe quark and lepton massesafter the breakdownof the weak interactions. Extendedtechnicolor[128]modelshavebeeninventedto tackle thesequestions.A massmechanismhad beenfound but it led to the predictionof very large flavor changing neutralcurrents, inconsistentwithexperimentalfindings [121].This is the central problem of all thesemodels, and it is not resolved.Maybewe do not know enoughaboutthesestrongly interactinggaugesystemswhich are dealtwith inthesemodels,andthis questioncan only be answeredat a time wherewe havelearntmoreaboutthesesystems.The massgenerationin the standardmodel is done by the Higgs—Yukawacouplings. As aresult the massesare pure input parameters.In an extendedtechnicolormodel theseparametersaredeterminedby thedynamicsof the system.With our limited knowledgeandsimplified assumptionstheyusually turn out to be wrong. We do not know at this time whether thesemodels might give asatisfactoryphenomenologicalprescriptionor not. However, it might very well be that thesemodelswithout fundamentalscalarsdo not work becauseof the absenceof the fundamentalscalarswith whichthe massesof quarksandleptonscan be parametrizedso effectively.

6 H.P. Nilles, Supersymmetry.supergravityandparticlephysics

One might thusbe led to the conclusionto reconsiderfundamentalscalars.Let ussupposethat thestandardmodel is valid up to agrandunificationscaleof 10~GeVor evento thePlanckscale1019GeV.The weak interactionscaleof 100GeV is very tiny comparedto thesetwo scales. If thesethreescaleswere input parametersin the theory the (mass)2of the scalarparticles in the Higgs sectorhaveto bechosenwith an accuracyof

10M comparedto the Planckmass.Theorieswheresuch adjustmentsofincredible accuracyhave to be madeare sometimescalled “unnatural”, since there is no way tounderstandthe smallnessof the scale[531].The only way to rendersucha theory “natural” wouldbe asymmetry that implies the small parametersof the theory to be exactlyzero.The actual valuesof thesmall parameterswould then be relatedto the breakdownof this symmetry.With this definition ofnaturalness,we conclude that the standardmodel is not natural. In the limit of vanishing massparametersin the Higgs sector the symmetryof the model is not enlarged.This manifestsitself in thefact that radiativecorrectionsinduce massesin the scalarsector.This aspectis sometimescalled thegaugehierarchyproblem[274,275] in grandunified theories:it is in generalnot enoughto fine tunethesmall parametersat the tree level, but in general severalordersin perturbationtheory haveto becomputedto consistentlyfine tune the small parametersof the theory. We will call this the technicalaspectof the naturalnessproblem.Oneshouldhoweverkeepin mindthat the reasonfor this behaviouris the existenceof a tiny understoodparameterin the theory,andthat with an “understanding”of thisparameter,all theseproblemswill necessarilyhaveto besolved.

Now the question arises,how onecould render the standardmodel natural?Since the scaleof thebreakdownof the weak interactions is entirely given by the vacuum expectationvalue of scalarparticlesoneneedsa symmetrythat could imply vanishingmassesof scalarparticles.Thereis only oneknown of such symmetries[542,409,569]: supersymmetry.Usually, e.g. chiral, symmetriescannotenforce the masslessnessof a scalar ~, since always ç~p* is a possible mass term that respectsthesymmetry.If one,however,hasa chiral symmetrythatforbids certainfermionmassesthe correspondingscalarpartnersarealsomasslessin a supersymmetricmodel.The motivationto apply supersymmetrytoparticle physics comes from this fact that supersymmetrymight render the standardmodel natural.Notice that this kind of argumentationis in analogyto the situationof spin 1 particles.To havemasslessfundamentalspin 1 particlesin a theory theyareusuallyintroducedasgaugeparticles, i.e., a symmetry[577] is invoked to keep these particles massless.Nonzero massesarise through a spontaneousbreakdownof the correspondingsymmetries.

To renderthe standardmodelsupersymmetrica price hasto be paid.For everyboson(fermion) inthe standardmodel, a supersymmetricpartnerfermion (boson)hasto be introduced,andto constructphenomenologicallyacceptablemodelsan additionalHiggs supermultipletis needed.Comparedto thealternatives,however, the introductionof supersymmetryis the most conservativeway to render thestandardmodelnatural.

In the supersymmetricstandardmodel the (mass)2 of the scalarsare no longer quadratically

divergent, the quadratic divergence is cancelledby the partners (fig. 1.2). The massesof thesescalarsare actually not renormalized at all as a result of certain nonrenormalizationtheorems[562,336,231,296,240] in supersymmetricfield theories.Thesenonrenormalizationtheoremsare actu-ally necessaryfor supersymmetryto survive perturbationtheory in the sensethat radiativecorrectionsdo not inducea spontaneoussupersymmetrybreakdown.

We are, however,not interestedin the exact supersymmetriclimit of the standardmodel. Chargedscalar particles with mass and quantum numbersof, e.g., the electron are not found and thussupersymmetryhasto be brokenin nature.Also, vanishingmassparametersin the Higgs sectorwouldnot induce the breakdownof the weak interactions.The model hasto contain a mechanismfor the

H.P. Nilles, Supersymmetry,supergravityandparticlephysics 7

0Fig. 1.2. Quadraticallydivergent contribution to the mass of a scalar particle (dashedline) that cancels the contribution of fig. 1.1 in asupersymmetricmodel. The solid line representsthefermionic partnerof thescalarparticle.

breakdownof supersymmetrythat splits the massesof thedifferentmembersof the supermultipletsandin addition inducesthe scaleof the breakdownof the weak interactionsof 0 (200)GeV. This breaking,however,shouldnot be arbitrarysincewe still want the scalarmassesnot to be quadraticallydivergent.Suchbreakingshavea name:theyarecalledsoft [525,277]. A particularsoft breakingis a spontaneousbreakdownof supersymmetry.With brokensupersymmetrythe cancellationbetweenthe graphsin figs.1.1 and 1.2 is no longer exact since the different partners in the supermultipletsare no longerdegenerate.If the breakingof supersymmetry,howeveris spontaneous,thiscancellationstill takesplacepartially and leadsto a finite result, the resultbeing relatedto the breakdownscaleM5 of supersym-metry. Thesupersymmetricpartnersof the knownparticlescut off the divergentpartsin the integrals;theyrepresentthat additional sectorneededin anytheory that explainsmassesof scalarparticlesin anaturalway.

One can now imagine a model in which SU(2)x U(1) is unbrokenin the supersymmetriclimit andwherethebreakdownof theweak interactionis inducedby the breakdownof supersymmetry.In suchamodelwe would then understandthe breakdownscaleM~of SU(2)x U(1) in termsof M5. In factonehasreplacedonemisunderstoodscaleby anotherone andonemight ask the questionwhetheronehasgainedsomething.One has. M~and M5 are now related in a natural framework: this has as aconsequencethat thesescalesare stablein perturbationtheory.This for examplesolvesthe technicalaspectof the gaugehierarchyproblem in grandunified theories[569,122, 490,449]. Parametersthat aresmallat the treelevel remainsmallto all ordersin perturbationtheory,their finite radiativecorrectionsbeing governedby supersymmetryand its breakdownscale.Due to this extremestability one mighteven be able to understandM5 in terms of the large scales like the Planck scale and the grandunification scale[4.44].This comesfrom the fact that the scaleof M~at this point is not uniquely fixed,still variousM~can leadto the sameM~.To seethiswe haveto look moreclosely to thisrelationship.Naivelyonemight think that M5 should beof the orderof M~,i.e. a few hundred0eV. This is in facttrue in modelswheresupersymmetryis brokenexplicitly: the splitting in the supermultipletsis denotedby M5 andshouldbe of orderof M~.Suchmodelsof explicitly brokensupersymmetry,however,arearbitraryand lack predictive power. One would thereforeprefer models in which supersymmetryisbroken spontaneously.In thesemodels the splittings of the supermultipletsare proportional to gM5where g is a coupling constant(or a product of coupling constants).This fact has far reachingconsequences.Onecan constructmodelswhereM5 is as largeas 1011GeV, but still sufficiently small toprotectcertainmasssplittingsandthe scaleM~to remain in the 100GeV region. In fact it turnedoutthat modelswith such a large scale of supersymmetrybreaking are actually preferredbecauseofphenomenologicalreasons.The scale M5 = 1011 GeV is actually so large that gravity cannot beneglectedany longer. We can see this as follows. Since all particles participatein the gravitationalinteractionsa supersymmetrybreakingat scaleM~will reflect itself in mass splittings of the super-multiplets of order M~S/MPwhere M~’denotesthe gravitational coupling constant. With M5 -=

1011GeV one arrives at M2~IM~-~ 10~GeV comparableto the breakdownscaleof the weak interactions

[42,4441. The appropriate framework for this type of models is thus local supersymmetry.Local

8 H.P. Nilles, Supersymmetry,supergravityand particlephysics

supersymmetrynecessarilyincludesgravity and is thereforecalled supergravity.In thesemodels thepartnerof thespin 2 gravitonis aspin 3/2 particle: thegravitino. If supergravityis brokenat ascaleM~,the gravitinomasswill be approximatelym312= M

2~/M~[114,100, 101]. The breakdownof supergravitycan induce the breakdownof the weak interactionandM~is typically [97,98, 77] of the orderof thegravitinomassm

312. Thesemodelsbasedon spontaneouslybrokensupergravityactuallyturnedout tobemost promisingon the way to constructphenomenologicallyacceptablemodels,the weak interactionscalegovernedby the gravitino mass.

Onemight beworried becauseN = 1 supergravityis not renormalizable.The supergravitymodelswediscusswill not makesenseunlessoneintroducesa cutoff. Such acutoff will be the Planckscale,andinthiscasethe predictionsof thediscussedmodelswill remainstable[37].In this sensewe arein the samesituation as in all othercases.Wheneverwe havea model, even if it is renormalizable,that doesnotincludegravity, we know that its rangeof validity will terminateat latestat the Planckscale,wheneverit includes gravity it is (up to now) nonrenormalizable.In the modelsunder consideration,the lowenergysector,however,can be describedas a renormalizablefield theory.This sectorhasan explicitlysoftly brokensupersymmetrywith certaindefinite breakingterms.

Within this framework severalclassesof models havebeenconstructedthat result in a low energy(100GeV) sector that still deservethe label “phenomenologicallyacceptable”.Theseseveralclassesofmodelsexist becausetherearestill somefree parameterswhich reflect the ambiguityof the supergravitybreakdown.These parameterinclude, for example, the gravitino mass,the gaugino massesand asupersymmetrybreaking parameterA of the trilinear scalar couplings. The fact, however, thatacceptablemodels can be constructedis nontrivial. Since there are several unconstrainedinputparameters,the phenomenologicalpredictionof thesemodelsarenot unique.This,however,hadto beexpected,sincethe modelsariseessentiallyfrom an extensionof the Higgs sector.This Higgs sectorinthe standardmodeldoesnot makevery definite predictions:a neutralscalarparticle shouldexist withmass somewherebetween 10 and 10000eV. Comparedto this, supergravity models have morepredictivepower.Onecan hardly imaginea model in which theweak interactionbreakdownis inducedby the breakdownof supergravitythat doesnot predict at least one new particle with mass below100GeV. The differentclassesof modelsjust differ in their predictionswhetherthese“light” particlesarephotinos,gluinosor perhapsscalarpartnersof quarksof leptons.Supersymmetricmodelsmightalsomakesome predictionsfor proton decaybut thesepredictionsare very model dependent.The lowenergypredictionsof the supersymmetricmodelsare specific enough to be testedin the foreseeablefuture. Experimentswithin the next five to ten yearswill enableus to decidewhethersupersymmetryasa solutionof the naturalnessproblemof the weak interactionsscaleis a myth or reality.

This far for the 100GeV region.The extensionof the modelsto higher energiesis of courseonlymildly constrained.Supersymmetricgrand unified models are flexible enoughto escapeexperimentalfalsification in the nearfuture. A detectionof proton decayp—* K~could, however,supportthe idea ofsupersymmetricgrand unification.

Also the origin of the mechanismof supersymmetrybreakingremainsrathermysterious.WhetherM~could beunderstoodthrough a dynamicalreasonor througha conspiracyof the large massscalesinthe theory must remain speculationfor a while. Theoretical information might comefrom a betterunderstandingof strongly interactinggaugetheoriesand/orextendedsupergravity.

Needlessto saythat therearemanyproblemsof particlephysicswhich areevennot addressedin thisframework.Oneof them is the questionwhy thereare threegenerationsof quarksandleptonsandtounderstandthe origin of their masses.Supersymmetric(grand unified) models do not improve ourknowledgewith respect to thesequestions.One could arguethat thesequestionsare solved in the

H.P. Nilles, Supersymmetry,supergravisyandparticlephysics 9

context of models based on extended (N> 1) supersymmetry [533] since there gauge and Yukawacouplingsbecomeunified, but at the momentthisremainspure speculation.

Anotherunsolvedquestionremainsthe naturalnessproblemof the cosmologicalconstant[402,536],althoughonewould havethoughtthat supersymmetrymight help in this respect.This hopeis basedonthe fact that for unbrokenglobal supersymmetrythe vacuumenergyis well definedandcan be showntovanish. This remarkableproperty, however,ceasesto hold if the supersymmetryis broken and/orgravity is coupled.In the modelsunder considerationthe only improvement(comparedto the usualtheories)is the vanishingquartic divergenceof the cosmologicalconstant.Again one might hopethisproblem to be solved in the framework of extendedsupergravities.In this report we will, however,restrictourselvesto modelsbasedon N = 1 supersymmetry,sincetheknownlow energyspectrumis notcompatiblewith extendedsupersymmetry,which only allows real particle representationswith respectto the gaugegroupsunderconsideration.

Chapters2 and3 will serveas an introductionto global andlocal supersymmetry,respectively.Theyarenot intendedto fully reviewthissubjectandwewill alsorestrict ourselvesto N = 1 supersymmetry.The content,however,will be sufficient to follow the remainderof this report. For a more exhaustivetreatmentwe referto the existing excellentreviews[217,262, 510, 533, 560].

In Chapters4 and 5 we will review the applicationof global supersymmetryto modelsfor particlephysics.Chapter4 discussesthe basicingredientsof thesemodelsas well as model independentresults,whereasin Chapter5 specific representativemodelswill be presented.Chapters6 and7 discussN = 1supergravitymodelsin this context. Again we will first give therecipeto constructsuchmodelsandwillthen discussspecific models. Chapter8 is devotedto cosmologicalconsiderationsand the resultingconstraintson model building. In Chapter9 we will try to list the most important phenomenologicalpredictionsof the consideredclassesof modelswith specialemphasison theexperimentalpossibilitiesinthe nearfuture. Chapter10 will finally give a summaryandan outlook.

2. Global supersymmetry

2.1. The supersymmetryalgebra and its immediateimplications

Supersymmetryis a symmetrythat transformsbosonsinto fermionsand vice versa.The generatorQof thesetransformationsshouldthereforehavefermionic character.In fact 0. (a = 1, 2) can be chosento havethe transformationpropertiesof a left handedWeyl spinor, a (1/2,0) representationunderLorentz transformations.(For our notationsandconventionsseeAppendix A.) Its Hermiteanadjoint isdenotedby 0,3 aright handedWeyl spinor.Since the supersymmetrygeneratorscarryonehalf unit ofspin they obey anticommutationrelations.The anticommutatorof 0 and0 doesnot vanish as theanticommutatorof anyoperatorwith itsadjointis nonzero.{Q~,Q,~}transformsas(1/2, 1/2)underLorentztransformations.The ColemanMandulatheorem[92] permitsthe conservationof oneoperatorwhichtransformsas(1/2, 1/2): theenergymomentumoperatorP~.Thealgebrathatdefinessupersymmetryisnowgiven by

[Qa,P~]0, (2.1)

{Q,~,Q,~}= ~ (2.2)

where a- denotesthe Pauli matrices. Haag, Lopuszanskiand Sohnius [302] have shown that the

10 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

supersymmetryalgebras,with the possibleextensionsto includecentralcharges,aretheonly gradedLiealgebrasof symmetriesof the S-matrix that areconsistentwith relativistic quantumfield theory.

Equation(2.1) for p~= 0 showsthat 0 commuteswith the HamiltonianH = P°,which leadsto theobservationthat the statesof nonzeroenergyare pairedby the action of 0. Due to the fermioniccharacterof 0 theycontainan equalnumberof bosonicandfermionicstates.

Equation (2.2) has immediateconsequencessince it relatesthe superchargesto the HamiltonianH = P°.Using ~ = 2g~onederives

H = P°= ~(0~Q~ + 0101 + 0202+ 0202), (2.3)

whichimplies that the spectrumof H is semipositivedefinite, H � 0 rememberthat 0 is the Hermiteanadjoint of 0). This is a very strong result. It implies that in supersymmetrictheories,e.g.the vacuumenergy,is well defined.Let us define 0) to be the vacuumstateandsupposethat the vacuumstate issupersymmetric(this means that supersymmetryis not spontaneouslybroken).This implies that thesuperchargesannihilatethe symmetricvacuumstate

Q~IO)=O. (2.4)

SinceH is given by the anticommutatorof the chargesthis leadsto [584]

Evac (OIHIO) = 0. (2.5)

In a supersymmetrictheory the vacuumenergythus is not only well defined,but is alsoboundto vanish.Since H � 0 this implies that if a supersymmetricvacuum state exists it is always at the absoluteminimumof the potential.This is quitedifferent from the situationwith ordinarysymmetrieswheretheexistenceof a symmetricstatedoesnot necessarilyimply that it is a groundstate,as is illustrated[569]in fig. 2.1.

Fig. 2.1. PotentialsV(~)with unbrokensupersymmetry(a andb) and spontaneouslybrokensupersymmetry(c andd). If ~ transformsnontriviallyunder a gaugegroupthegauge symmetrywill be spontaneouslybrokenin casesb andd.

H.P. Nil/es, Supersymrnetry,supergravityandparticle physics 11

Supersymmetricgroundstatesare alwaysat Evac= 0. They, however,might still be degeneratewithother statesthat haveEvac= 0 and the question remainswhetherthesestatesare supersymmetricornot? To answerthis question let us considerthe caseof spontaneouslybrokensupersymmetry,wherethe vacuumstate is not annihilatedby the supercharges

QaIO) � 0, (2.6)

which implies Evac= (OIHIO) � 0 through eq. (2.3). Thus supersymmetryis unbrokenif and only ifEvac= 0. It can bebrokenspontaneouslyonly if the potential is strictly positive.

The spontaneouslybroken case deservesfurther discussion.The right hand side of eq. (2.6) isnecessarilya fermionicstatewhichwe denoteby It//a). We can then write [571]

(IJ~aI0)ftT~pa, (2.7)

whereJ/~ais the (conserved)supercurrentwith

QaJd3XJoa. (2.8)

The supercurrentthuscreatesa fermion out of the vacuumwith couplingf, which is a measurefor thebreakdownof supersymmetry.This situationis analogousto the caseof ordinaryspontaneouslybrokensymmetrieswhere Goldstonebosonscan be createdout of the vacuum. In fact an analogueof theGoldstonetheoremcan be provenfor supersymmetry.It provesthe existenceof a masslessfermion intheories with spontaneouslybroken supersymmetry.This particle is called Goldstone fermion orgoldstino.f is the “goldstinodecayconstant”.Sincethe energymomentumtensorin a supersymmetrictheory can be written as ~ = ü~a{Q~,J~}it can be relatedto the vacuumenergyEvac= f2, andEvac

measuresthe strengthof supersymmetrybreakdown.

2.2. Superfields

In order to construct supersymmetricmodels one would like to have a formalism in whichsupersymmetryis manifest.Such aformalismexists: it is the superfieldformalism introducedby SalamandStrathdee[496,236] whichwe will discussin this section.

Let us first try to construct representationsof the algebragiven in (2.1) and (2.2). To take intoaccountthe anticommutatorin (2.2) we introduce anticommutingparametersO~(a= 1, 2) and O~(,8 = 1, 2) which areelementsof a Grassmannalgebra

{r, o~}= {ö~,~} = {Oa, ~} = 0. (2.9)

The supersymmetryalgebracan then be rewrittenas

[00, ~i] = 2ea-JP~, (2.10)

[00, 0Q]=[~O, c~ë]=0, (2.11)

12 H.P. Nil/es,Supersymmetry,supergravityandparticle physics

wherewe havedroppedthe indices (00= OaQa = OaQ$Ea,3 where Ea,3 is antisymmetricand e12= +1).

A “finite” supersymmetrytransformationdependsnow on the parametersx~,0 and 0 and can bedefinedas follows

S(x,0, 0) = expi(OQ+ 00— x~P~’). (2.12)

The “multiplication” of two successivetransformationscan be computedwith the help of Hausdorf’sformula

S(x,0, 0)S(y,a, a) = S(x + y— iau~0+i0o~,0+ a, 0+ a). (2.13)

A superfield4(x, 0, 0) is nowdefinedas a function of the parameters0, 0 in addition to x,.~suchthat ittransformsas follows undera supersymmetrytransformation:

S(y,a, a)[4(x, 0, 0)] = q5(x + y — iau,L0+ i0o~ã,0 + a, O+ a). (2.14)

To find a representationof the superchargesit is enough to considerinfinitesimal transformationson

the superfieldçb

~5q5= [a -~- + a 2— i(aa-~—0u~ã)~’]~, (2.15)

which leadto

P~=i~-~iä~., (2.16)

Qa ~ (2.17)

& = _+i0,3o~op., (2.18)

a representationof the superalgebrain terms of differentialoperators.One can now definecovariantderivatives. (For an early review see ref. [557].) Theseare derivatives that anticommutewith thevariation

Da(&S4))= ~S(Da4)). (2.19)

Thisgives

Da~~= (3a + io0’~9~)c~, (2.20)

= (~Ôa— i09,~)t/.~, (2.21)

H.P. Nil/es,Supersym,netry,supergravityandparticle physics 13

wherewe havedenoteda/oO’~(o/80”)by 3a(0&). Beforewe proceedto discussspecific superfieldslet usintroducetwo different representationsof the superalgebrabecausetheywill simplify certainformulae.

Insteadof (2.12) onemight alsodefine

SL = exp[i(OQ — xP] exp[iQO], (2.22)

or

SR = exp[i(c~—xP)] exp[iOQ]. (2.23)

Computingthe supersymmetrytransformationS(y, a, ~ SL(R)(x, 0, 0) gives

SL(X + y+ 2i0a-a+ iaa-~ä,0+ a, 0+ a), (2.24)

SR(X + y — 2iao~0—iaa~ã,0+ a, o+ a), (2.25)

respectively.t~L andt/~ superfieldscan now be introducedin analogyto (2.14).They transformunderinfinitesimal supersymmetrytransformationsas

~qSL= (a30+ aa~+2i0o~ät9~)4L, (2.26)

= (a30+ äôë—2iao~0ô~)t~R, (2.27)

andleadto the additionalL(R)-representationsof the superalgebra

OL 89, QL = —ô~+2i0o~t9’~, (2.28)

QRôo—2io~08M; QR—~t9~, (2.29)

where 9,~denotest9/30. The correspondingcovariantderivativesaregiven by

DL=aO+2io~0o~’; DL——Oe, (2.30)

DR—o—2i0o•P.t9~’~. (2.31)

Thereis a simple relationbetweenthe threerepresentationswehaveintroduced:

0, 0) = ~L(X~ + j0a-~0, 0)

= ~/~R(X~ — i0o~0,0, 0) (2.32)

transformidentically underthe supersymmetrytransformation.We havenoweverythingset up to discussspecialexamplesof superfields.Superfieldsthat satisfythe

conditionDçb = 0 or Dçb = 0 arecalledscalarsuperfields.çb(x, 0, 0) with Dçb = 0 (D4 = 0) is sometimesalso called a left- (right-) handedchiral superfield. Let us now consider such a left-handed chiralsuperfieldwith DqS= 0. It is in generala function of x, 0 and 0. It is mostconvenientlydiscussedin the

14 H.P. Ni//es, Supersymmetry,supergravityandparticlephysics

L-representationin which D hasa simple form D = — 8/80. D4 = 0 then implies that 4L is independentof 0. Let usnow expand4L(X, 0) as a powerseriesin 0. Since0 is anticommutingandthereexist onlytwo independentvariables~a (a = 1, 2) this seriesterminatesafter the third term andwe can write

4~L(X,0) = ~~(x)+ 0’~t/Ja(X)+ 0aO$E~F(X) (2.33)

where ~‘pand F are complex scalar fields and ,/‘ is a left-handed Weyl spinor. The effect of asupersymmetrytransformationon these“component”fields can be computedfrom (2.26).With

~L(X, 0) = ~ + 0~t/i+ 00SF (2.34)

oneobtains

= V2at/i, (2.35)

= \/2aF+ iV2u~ã8~, (2.36)

= —iV23’~t//a-Mã. (2.37)

Observethat the supersymmetryvariation of the highest(F) componentof the superfield is a totalderivative, a fact which will becomeimportant when we constructLagrangiansin the next section.Equations(2.35)—(2.37)show that supersymmetrytransformsfermions into bosons.The termswithoutderivativestransformlower componentsinto higher ones~ —* t/t andcit —* F. Let usnowgive someusefulformulae with respect to chiral superfields. In the L(R)-representationone_can easily show thatD3 = D3 = 0. This implies that for a left-handedchiral superfield t/ (with Dçb = 0) the superfieldD2~ DaD,,4 is a right-handedchiral superfieldD(D2~)= 0. Its lowest componentcontains thehighestcomponentof the original superfieldç~.

A scalarsuperfieldcontainsspin 0 bosonsandspin 1/2 fermions.Onewould also like to introducesuperfieldsthat contain spin 1 bosons.This leadsus to the introductionof a vectorsuperfieldV(x, 0, 0)[239,498], a superfieldthat satisfiesthe (reality) condition V~= V. In componentsit can be written as[217]

V(x, 0, 0) = (1 + ~0000I~)C+ (iO + ~00o~0ô~)~+ ~i00(M+ iN)— (2.38)

+ (—iO+ ~000u~8’”)k— ~i00(M—iN)— 0u,~0V1’ + i000A — i000A+ ~0000D,

whereLII = 8~8~’.C, M, N, D are real spin 0 fields, x’ A are Weyl spinorsand V’~’ is a spin 1 field. Thespecialway onedoes the componentdecompositionin (2.38)is hardto understandat this point. It willbecomemoretransparentwhenwe explicitly performsupersymmetrytransformationon the componentfields. With (2.15)

~iV=[a89+ã8~—i(aa~0— 0o~a)8~’]V

weobtain

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 15

FiC=iax—ia~, ~5y~~’=a(M+iN)+a-~ã(8~C+iV~),

= a(M — iN)+ ao~(c9,~C—iV,~), ~N = a(iA — a-,~0~)+ä(—iA+ ã,,.8~x),

~M a(A + io~ô~)+a(A+ iã~35~’), ~ = a0,~X+aô~+iao~A+iãó~A, (2.39)

~iA= aa-~V~+aD, M= aa’~V~~+aD, aD= —acr~o~A+aa-”a~A,

where ó~’~= (1, —o) if a~’= (1, +o) ando~= ~(oJ~~ — o-~’ö~),&‘~“ = ~ — &~~o.M).We havealsointroducedthe field ~ = t9,LVV — 8pVa. Observeagainthat the highestcomponentof the superfieldDtransformsinto a total derivative. One can also see from (2.39) that the general vector superfieldisreducible.The fields V~,,A, A andD form a representationby themselves.As we will see later thesecorrespondto the componentsof a “massless”vector superfield. It will becomeimportantwhen wediscusssupersymmetricYang—Mills theories.The specialchoice of the expansionof V in (2.38) wasusedin orderto makethis reducibilitytransparent.

2.3. Lagrangians[561,562]

Wehavealreadyseenin (2.37) and(2.39) that the highestcomponentsof the superfield(the F-terminthe scalarsuperfieldand the D-term in the vector superfield)transformundersupersymmetrytrans-formationsinto total derivatives.A spacetime integralf d~xof such a quantity is thus invariant undersupersymmetrytransformations(provided,of course,that the fields fall off at infinity fastenough,whichwe will always assume).This is the basic observationfor the constructionof supersymmetricfieldtheories.The Lagrangedensityis a sum of superfields,which arein generalproductsof the elementarysuperfieldsthat we haveintroducedin the last section (seebelow for a discussionof the multiplicationof superfields).There are two types of superfields:scalarand vector. Accordingly we can write theLagrangedensity

(2.40)

where~tF isa sum of scalarsuperfieldsand~tD is a sum of vectorsuperfields.Thehighestcomponents

OOF and 0000D can be projectedwith integrationover 0 and 0. Integration is definedby .f dO= 0 andf d0~0 = 1. An invariant Lagrangianis then obtainedthrough

L_Jd4x[Jd20d2O~[D+Jd2O~EF+h.c.]. (2.41)

Observethat it contains two terms of different nature F- and D-terms, a fact which will becomeimportantwhenwe discussperturbationtheory.

The terms in ~~tD and ~‘~F ariseas productsof “single-particle” superfieldssince ..~ hasto containtermsat leastbilinear in theseelementaryfieldsandwe haveto discussthe multiplicationof superfields.Considera scalarsuperfieldt/ with Db = 0. The product~2 (or 4~’)of such a superfieldis alsoa scalarleft-handedsuperfieldsinceit automaticallysatisfiesD(q52)= 0. ç1 is acomplexfield: 4~the conjugateisa right-handedfield. The product(qY~is thenalsoa right-handedchiral superfield.The productof left-

(right-) handedsuperfieldis thusalways a left- (right-) handedsuperfieldandas such a candidatefor a

16 H.P. Nil/es,Supersymmetry,supergravityand particlephysics

term in the Lagrangedensity .~fFas definedin (2.40). To give an examplelet us computethe squareofthe superfieldgiven in (2.33) ~L(X, 0) = p(x) + Oat/Ia + OOF:

q5~(x, 0) = ~2 + 2çc0’~t,l’a+ (00)[2pF — ltfrat/u] (2.42)

J d2OqY(x,0) = ~[2c~F- lt/ja~/I] (2.43)

an examplefor an F-term in the Lagrangedensity.Actually all F-termsin the Lagrangedensitywillariseas productsof scalarsuperfieldswith the samehandedness.~ is often called the superpotential.

Terms in .~L?Dcan, e.g., ariseif we multiply left- with right-handedsuperfields.As an examplelet usconsider the multiplication of 1/I with its complex conjugate 4~.The superfield ~L(X, 0)=ç~(x)+ Ot/i + 02F is given in the L-representation(where it is independentof 0). Consequentlyitsconjugate~ is given in the R-representationif we just naively takethe complexconjugate

~ q,*+(0ci,)+(eo)F* (2.44)

It is independentof 0 andsatisfiesDRq5~= 0. Beforewe can multiply 4 with /~we havefirst to bringthem in the samerepresentation.This can be done through the shift given in eq. (2.32). Let us bring~ 0) backin the L-representation

4t(x, 0, 0) = ~ — 2i(0o~0),0, 0). (2.45)

We now expandt/4~.andobtainin components

— 2i(0o~0)c9~~q,~K— ~ + (Oçit) — 2i(0a~0)ô~’(~)+ (00)F*. (2.46)

In this form the componentsof ~/i and /~can be multiplied naively,

~ 0, 0) = /(x, O)/~(x — 2i0u0,0) = t/(x, 0) exp[—2i0a-~98~]~(x,0), (2.47)

which can be shownto be a generalvectorsuperfield.With an explicit calculation(usingformulasof theappendix)oneobtainsfor its “highest” (D)-component

(N~)D = FF* — ~ + O~’t~t~ (2.48)

where III = ~ According to (2.39) the quantity f d4x(4K/Y)D is invariant under supersymmetrytransformations.

We arenowin asituationto constructsupersymmetricactionsfor chiral superfields.The expression(2.48)is obviouslyappropriateto describethe kinetic termsfor a complexscalar~ andaWeyl spinor tfi.

Masstermsandinteractionscan be addedthrough F-terms(the superpotential),e.g.,

-~F mçb2+Açb3. (2.49)

The Lagrangian

H.P. Nil/es,Supersytnmetry,supergravityand particlephysics 17

L = (~~)D + m(~2)F+ A(~3)F+ h.c. (2.50)

readsin components

L = (a,~~)(o~*)+ ~it/Io~3t’t/I + FF* + m(2~pF— ~(t//t/i)+ h.c.)+A(3çc2F— ~p(cm/I) + h.c.), (2.51)

up to total divergences.It leadsto the field equations

= 2mF*_~A(tIII/,)+6A~*F*,

ia~ô,.~çIi= 2mt,ti+ 3A~t/i, F* + 2mço + 3A~2= 0, (2.52)

(togetherwith the complexconjugateequations).The field F can be eliminatedvia the equationsofmotion; it is an auxiliary field: F = _2m~*— 3Ac*2. Substitutingthisbackin (2.51) oneobtains

L = I8~12 + ~it/io~8~t/i+ I2m~+ 3A~2I2— m[2~(2m~*+ 3A~*2) + ~(t/jt/,)+ h.c.]

— A[3~2(2mç~* + 3A~*2) + ~ (t/it/i) + h.c.], (2.53)

which describesan interactingcomplexscalarwith aWeyl spinorboth of mass2m. Notice first that themodel containsan equalnumberof bosonicandfermionicdegreesof freedom(namely two) as expectedfrom our discussionof the algebrain section2.1. The interactionsinclude a trilinear ~t/it/i coupling aswell as trilinear and quadrilinearscalar couplings.They are howeverrelatedby supersymmetryandgiven by A and m. The term çiY in the superpotential(2.49) gives rise to quadrilinear scalarselfcouplings. It can be shown that powers of higher than three in the fields contained in thesuperpotentialleads to nonrenormalizablemodels since they imply scalar selfcouplingshigher thanquadrilinear.Thiscan be seeneasilyby a trick whichis alsousefulto eliminatethe auxiliary fields from(2.51). It is theobservationthat the scalarpotential(i.e. the part of the Lagrangianthat doesnot containderivativesor fermions)can be obtainedeasilyfrom the superpotential[569]

V= 8W(c~)/t9ço~2, (2.54)

where W(ç’) denotes(2.49) with the superfieldsreplaced by its scalar component. (Observethat

—8W(~)/ôq’.)In our caseV= I2m~+ 3A~2!2and this leadsto

L = IB,~I2+ ~it/a-~3’~—~m(t/it/I+ tutu) — ~A(~t/n/i+ p*t/n/j) — 2mco+ 3A~2I2, (2.55)

which is equivalentto (2.53).Up to now we haveonly discussedLagrangianswhich describespin 0 and spin 1/2 particles.One

would of coursealso like to considermodelswith spin 1 particles.Since all spin 1 particleswe willdiscuss are “gauge particles” of Yang—Mills gauge theories one hasfor this purposeto constructsupersymmetricYang—Mills theories.We havenot the time andspacehereto do this in full detail. Wewill here restrict ourselves to the Abelian case [563,498] and give the relevant formulas for thenon-Abeliancaseat the end of this section.For a more detailedtreatmentwe refer to the existingreviews [217,560,510].

The centralingredientis the vectorsuperfieldas given in (2.38). It containsa spin 1 field Vt’. In an

18 H.P. Nilles, Supersyrnmetry,supergravityandparticle physics

Abelian gauge theory (e.g. QED) the gauge field V’~ (e.g. the photon) transforms under gaugetransformationsas

VV~+a,.~çc’, (2.56)

where~ is a real scalarfield. —

The field V~’is now a memberof the supermultipletV(x, 0, 0) andthe gaugetransformation(2.56)has to be extendedin a way consistent with supersymmetry.This needs the extensionof ~ to asupermultiplet.Accordingto WessandZumino [5631the transformationof the vectorsuperfield

V-* V+i(A —A~) (2.57)

is the correctgeneralization.A denotesa chiral superfieldcontaininga complexscalarfield ~ as well asa Weyl fermion. An inspectionof (2.57) in componentform showsthat the fields C, x~M and N aregaugeartifactswhile A andD aregaugeinvariant and VM transformsasin (2.56),where~= (~+ ~j

The fields C, x~M and N can be gaugedaway by a particular choiceof the gauge,the Wess—Zuminogauge.In this gaugethe vectorsuperfieldtakesthe form

Vwz(x,0, 0) = —Ou,.~0V’~+ i000A — i000A + ~0000D. (2.58)

It only containsVi’, A andD. The Wess—Zuminogaugedoesnot fix the gaugefreedomcompletely.Westill have~ V,.. = 0~(A andD weregaugeinvariant from the beginning).The Wess—Zuminogaugeis akind of unitary gaugefor supersymmetricYang—Mills theoriesand it is very convenientto constructLagrangians.It is in generallesssuitableif oneactuallywantsto do calculationsin perturbationtheory.

To obtainthe supersymmetricgeneralizationof the electromagneticfield strengthwedefine a spinorchiral superfieldWa by

Wa(DD)DaV (2.59)

whereDa andDa denotethe covariantderivativesas given in (2.19)—(2.21).In componentsit is given

byWa(X, 0) = 4~Aa— 4OaD+ ~ — 19~’V’~)—~ (2.60)

where we have performed the calculation in the L-basis. Wa contains D, A and V~=

~ a~V~(=F,~)as component fields and is gauge invariant. The F-componentof the chiralsuperfieldW’~We,. is thus invariant undersupersymmetrytransformationsandgaugetransformations.Incomponentsit reads

= —~V~V~— ~jAaO~a~0~AY — ~ + ~D2. (2.61)

It is the Lagrangedensity for pure supersymmetricAbelian Yang—Mills theory. The field D is anauxiliary field. It can be eliminatedvia the equationsof motion. Supersymmetryrequiresa fermionicpartnerA of the gaugeboson V5’ which is calledgaugefermion or gaugino.The minimal couplingofmatter fields with chargeg to the supersymmetricYang—Mills theory is given by replacingthe usual

H.P. Nilles, Supersymmetry,supergravityand particlephysics 19

kineticterms(~~4)Dof a chiral superfieldt/ = (~,t/i, F) by

(~/~exp(2gV)q5)~= ID~I2 ~[t/~ a~Etu1~n]+ gco*D~c,+ ig[~*(At/I) (At~’)4]+ 1F1

2 (2.62)

(in the Wess—Zuminogauge)whereD,., = 8,.. + igV,.. is the (gauge)covariantderivative.Thegeneralizationto the non-Abeliancasehasbeengiven by Ferraraand Zumino[237],Salamand

Strathdee[497]. Using V,. = V~T~and Aa = fl ~a where T~are the generatorsof the non-Abeliangaugegroupunder considerationoneconstructsthe spinorchiral superfield

Wa = 1.51.5[exp(—gV)Daexp(gV)], (2.63)

whereg is the gaugecoupling. The gauge transformationsexp(gV)—* exp(—igA~)exp(gV) - exp(igA)transform Wa to exp(—igfl ) W~.exp(igfl). The quantity WaW’~is gaugeinvariant and its F-componentprovidesusagainwith a Lagrangedensityfor puresupersymmetricYang—Mills theory

.II~f=(1/32g2)W”Wa= —~G’”’G~~+ ~D2—~ + ig[V~’, k~])— (0”A~+ ig[V~’, A~])o~asA’~}, (2.64)

where~ = ~ — 3EV,.+ ig[V,~, V~]andA = AaTa is in the adjoint representationof the gaugegroup.The minimal coupling to matteris againgiven by ~ exp(2gV)4.

2.4. Spontaneousbreakingof supersymmetry

From the discussionin section2.1 we know that a nonvanishingvacuumexpectationvalue of theanticommutator{Qa, t/I} signalsa spontaneousbreakdownof supersymmetryin connectionwith theexistenceof a Goldstonefermion tu’ (Q~,denotesthe supersymmetrycharge).The fermions underconsiderationare either “chiral” fermions (i.e. membersof a chiral supermultiplet)or gaugefermions(partnersof the gaugebosons).Recallingthe supersymmetrytransformations

= \/2aF+ iV2o~a3,.~, (2.36)

~3A= aa-~V,.~,+aD, (2.39)

we see that the spontaneousbreakdownof supersymmetrycan be convenientlycharacterizedby theauxiliary fields underconsideration.We have

(0I{Q, ~r}I0)— (OIFIO), (0I{Q, A}I0) — (OIDIO) , (2.65)

since the additional terms in (2.36) 3,.~,and (2.39) ~ contain derivatives of fields which are notsupposedto receivevacuumexpectationvalues(v.e.v.). Thussupersymmetryis spontaneouslybroken ifand only if auxiliaryfieldshavenonvanishingvacuumexpectationvalues.Thefermionic partnerof theauxiliary field that receivesav.e.v. is the masslessGoldstonefermionof brokensupersymmetry.If thereis morethan oneauxiliary field with nonvanishingv.e.v. the goldstinois a suitablelinear combinationoftheir fermionicpartners.

20 H.P. Nil/es,Supersymtnetry,supergravityandparticlephysics

In section2.1 we hadalsorealizedthat the vacuumenergyis an orderparameterfor supersymmetry.This canbe verified hereas well. From(2.54) and(2.61)we knowthat the scalarpotentialcan be writtenas (D is real)

V=FF*+~D2, (2.66)

anda nonvanishingv.e.v. for an auxiliary field implies Evac � 0.In the following we will give somesimple examplesof spontaneouslybroken supersymmetry.The

first is a so-calledO’Raifeartaighmodel[461,199] in which only auxiliary fields of chiral supermultipletsreceivea v.e.v. We considerthreescalarsuperfieldsX = (x, t/i~,Fr), Y = (y, t/i~,F~)and Z = (z, t/i~,F~)with

= (XX*)D + (yy*)D + (ZZ*)D + [AX(Z2— M2) + gYZ]F + h.c. (2.67)

The auxiliary fields aregiven throughthe derivativesof the superpotential(F* = —ÔW(co)I&p)

= —A(z2—M2), F~= —gz, F~= —gy — 2Axz. (2.68)

Supersymmetrywould be unbrokenif (2.68) had a solutionwith F~.= F~= F~= 0. (Since thepotentialV = FXF~X+ F~F+ ~ is (semi)-positivedefinite V = 0 is the absoluteminimum.) However,themodelwaschosenin sucha way that this solution doesnot exist. F~= 0 implies z2 = M2 which is inconsistentwith F~= 0. Thus,supersymmetryis broken.To obtainthe groundstatewehaveto minimize

V A21z2—M212+ g2Iz~2+gy + 2Axzl2. (2.69)

Supposethat M2< g2/2A2. The absoluteminimum is at z y = 0, with F,, = AM2 and consequentlyEva~= A2M4. The vacuumexpectationvalueof x is undeterminedwhich implies a vacuumdegeneracy,thereis a flat direction in the potential,a point to which we will comebacklater.

Accordingto our generalargumentswe know that supersymmetryis brokenandthat t/i,, (the partnerof F,,) is a Goldstonefermion.Let us checkthis explicitly in our example.From(2.67)andy = z = 0 it isobviousthat t/I~ is massless.Therearetwo massivefermionsof massg which arelinear combinationsoft/i~and t/i~.The scalar x is masslessand y hasmass g. This looks still completelysupersymmetric.Splittings,however,occurin the z-scalars.Denotingz = a + ib wherea andb arereal fields, we obtainm~= g2 — 2A2M2 andm~= g2 + 2A2M2andwe seethat the boson—fermiondegeneracyis destroyed.

Severalthings can alreadybe learnedhere.At the treegraphlevel, only the bosonmassspectrumisaffectedby the fact that supersymmetryis broken.In our caseit is actuallyonly the z-scalar.The reasonfor this is that from all the scalarsonly z couplesto the goldstino t/I~ (throughthe term AXZ2 in thesuperpotential).The splittings are thus governedby the coupling to the Goldstonefermion: z~m2=2AFX = 2A2M2.F~is usuallyidentified with the squareof the supersymmetrybreakdownscaleF,, = M~.We see,however,thatthe splittingsof the supermultipletscan be muchsmallerthanM~,providedthatthe coupling to the goldstinois weak enough,a point which will becomeimportant later, when wediscussthe applicationof supersymmetryto particlephysics.

An interestingpropertyof the massspectrumat the tree graph level is the fact that m~+ m = 2g2independentof the supersymmetrybreaking,on averagethe massesremain the sameas in the caseofunbrokensupersymmetry.At the threelevel the fermion massesremainunchangedwhereasthe scalars

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 21

areshiftedby the sameamountin oppositedirections.This resultholdsin general(with oneexceptionwhich we will discussin our next example)as shown by Ferrara,Girardello and Palumbo [227].Theresult is that the supertraceof the squaredmassmatrix

STr.,1~2= ~ (—1)~(2J+ 1).1U~= 0 (2.70)

remainszero in the presenceof spontaneoussupersymmetrybreaking. J denotesthe spin of theparticles in (2.70).We stressagainthat (2.70)only holdsat the tree graphlevel, it is generallyviolatedby radiativecorrections[279].Expression(2.70) implies that the vacuumenergyis not quadraticallydivergentat the one looplevel even in the presenceof spontaneoussupersymmetrybreaking.

So far we have discusseda model where F-fields received vacuumexpectationvalues.Our nextexampleshowssupersymmetrybreakingthrougha D-term [218]andis dueto Fayet[200].It consistsofa chiral multiplet U = (~,t/I, F) with chargee coupledto Abelian supersymmetricYang—Mills theorywith the addition of a Fayet—Iliopoulos [218]term f d

40V~—(V)D —~D where V is the Abelian vectormultiplet.

The Lagrangiandensityreads

= ~fW~Wa]F + [U* exp(2eV)U+ 2~VID. (2.71)

This leadsto the following field equationsfor the auxiliary fields (compare(2.62)):

F=0, D+~+ep*p=0. (2.72)

Eliminating the auxiliary fields leadsto the expression

= ~ — ~iA/A— ~ii Øt/’ + (Dt~q,)(D,.~’*)+ ie(q~*At/t— At/i~)— ~ + e~*~I2, (2.73)

whereD,. = 8,., + ieV,.~. The scalarpotentialis V= ~D2= ~+ e~*~I2.Severalthingscan happenin thismodel, dependingon the parameterse and ~. If ~ and e have oppositesign e~<0 the vacuumexpectationvalue of D is zero and(~*~)= —~/e:supersymmetryis unbrokenbut gaugesymmetry isspontaneouslybroken (fig. 2.2). If, however,e~>0 the equationD = 0 hasno solution and supersym-metry is broken (fig. 2.3). Let us briefly discussthe first case,becauseit gives us the supersymmetricgeneralizationof the Higgs mechanism.We have (p*p)= —~/e>0at the minimum. The formerly

Fig. 2.2. The scalarpotential for the case e~<0.The U(1)-gauge Fig. 2.3. The casee~>0.Supersymmetryis spontaneouslybroken.symmetry is spontaneouslybroken but supersymmetryremainsun-broken.

22 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

masslessboson V’~receivesa massby absorbingone real scalar.The secondreal scalarhasthe samemassV—e~as the massivegaugeboson.This is just the usualHiggs mechanismapartfrom the fact thathereboth massesare equal as a result of supersymmetry.Since supersymmetryis unbrokenthesebosonsmusthave4~generatefermionicpartners.This happ~psthrough the term ep*(Açfr) in (2.73);withthe v.e.v. ~ * = V—~/e(e~<0) thisgives rise to a massV—ge. The two Weyl fermions i/i andA combineto a massiveDirac fermion. This is the‘supersymmetricgeneralizationof the Higgs effect. A chiralsupermultipletcombineswith a masslessvectormultiplet to a massivevectorsupermultiplet.

Let us now consider e~>0. At the minimum of the potential we have (~)= 0 and D = —~

correspondingto Evac = ~2 Supersymmetryis broken through the v.e.v. of D. The partnerof D, thegaugefermion A, is the goldstino.Gaugesymmetry is unbrokenand V,. as well as A remainmassless.Fromthe scalarmultiplet t/s remainsmasslessandthe scalars~ receivea massas a resultof its couplinge to the goldstino

m~=e~. (2.74)

Notice that this is completelydifferent as in the example previously discussed.The two real scalarsreceivea massshift in the samedirection andthe massformula (2.70) is violatedevenat the treegraphlevel. The shift in (2.74) is proportionalto the chargeof the superfield.In the presenceof an unbrokenU(1) symmetryin which supersymmetryis brokenby the D-term(2.70) hasto be modified

STr~2=2TrQ(D), (2.75)

This is the final formula.The only possibleterm on the right handside is the onegiven in (2.75) whereQis the chargematrix of the chiral superfieldsunderconsideration,and(D) is the vacuumexpectationvalueof the auxiliary gaugefield. If we had,e.g., addeda chiral superfieldwith charge—e to the modeldefinedin (2.71), the right handside in (2.75) would vanishsincethe scalarsof this multiplet get shiftedby ~m2=—e~.

So far we havediscussedtwo exampleswherefundamentalauxiliary fields F or D havereceivedvacuumexpectationvalues that leadto a breakdownof supersymmetry.In general,a combinationofthesetwo mechanismsmight occur.The correspondingGoldstonefermion would thenbe a combinationof the respectivegaugeand chiral fermions.One might alsoconsidera situationwherethe Goldstonefermion is not a fundamentalparticleof the theory but a compositeparticlebound,e.g.,by stronggaugeforcesin analogyto the pion as a compositeGoldstonebosonof broken chiral symmetryin quantumchromodynamics (QCD). Such a mechanism has been considered in “supercolor” models[123,569, 137], a supersymmetricversionof technicolormodels[191].Let us considera supersymmetricSU(N) Yang—Mills theory coupled to chiral superuields/, and çlt’ in the N and N representationofSU(N). Supposethat theseinteractionslead to confinementin the infraredregion.The spectrumbelowa certainscaleA will thusconsistof SU(N) singletboundstates.They can be describedsupposedlyin amanifestsupersymmetricway. An example is the compositesuperfieldT = 4~/.With 4 and 4 beingleft-handedchiral superfieldsthe productwill havethe sameproperty;it can bewritten in the L-basisas

T = çc~~+ 0(tp,t/i’ + ~t/j,)+ oo(~~F’+ ç~’F,—~t/i,t~Y). (2.76)

Now assumethat like in ordinary nonsupersymmetricQCD the quantity t/it/I receives a vacuumexpectationvalue. Sincewe believeto be allowed to eliminateF = 0 via the equationsof motionsthis

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 23

would leadto a breakdownof supersymmetrybecausethe highestcomponentof a chiral superfieldhasa nonvanishingv.e.v. We have

{Qa, (‘pçl’+ ~t/,)}—~ çliçli, (2.77)

and the fermionic componentof T would be a Goldstone fermion. Notice that this argumentisrestricted to this chiral superfield, the assumptionsof a supersymmetrybreakdownthrough gaugefermion condensates(AA) are wrong since AA is the lowest-componentof the compositesuperfieldWaW” and not an auxiliary field [443].Equation(2.77) looksquiteconvincingbut the real questioniswhether t/n/i receivesa vacuumexpectationvalueor not. After all we discussa supersymmetrictheoryand we know that supersymmetricvacuum stateshavevanishing energyand are the lowest states(ifthey exist). A statewith (t/It/I) � 0 is nonsupersymmetricand hasas such positive energy.One mightarguethat in a modelwith severalchiral multiplets t/It/i condensationis necessaryto breakcertainchiralsymmetries,but thisrole can aswell be playedby ~ condensatesandunbrokensupersymmetry.Thereare in fact strong argumentsby Witten [573,75] that under some circumstancessuch a dynamicalbreakdownof supersymmetryis not possible.The situation is, however,not clear. Investigationsofthesemodelswith the help of effective Lagrangians[543,530] indicate that such a breakdownmighthappen[475a,448] circumventingthe no-go theoremof Witten. At the moment the questionof thisdynamicalsupersymmetrybreakdownis still open andefforts should be madeto reacha conclusion,aquestion which is also relevant for the study of supersymmetriccomposite models of quarks andleptons.

2.5. Nonrenormalizationtheorems,absenceof quadraticdivergenciesand softbreaking

Supersymmetrictheorieshavethe propertyof vanishingvacuumenergyas dictatedby the algebraand as we haveverified for the examplesdiscussedup to now at the threegraph level. Let us nowconsider the effects of perturbationtheory. One might be worried that radiative correctionsgivecontributionsto the vacuumenergyand as such supersymmetrywould not survive in the perturbativeexpansion.To bemorespecificconsiderthemodeldefinedin (2.67) andlet us supposethat M2 = 0. Thismodel thenhasthe superpotentialAXZ2+ gYZandonemight askthe questionwhethera term A ‘M2Xcould begeneratedin perturbationtheory.If it were, independentof how small the coefficient A’ mightbe, this would induce a breakdownof supersymmetry.For supersymmetryto remainunbrokensuch aterm hasto remainabsent in all ordersof perturbationtheory.Thus supersymmetryrequiresnonre-normalizationof certainparameters[336] (to remain well defined in perturbationtheory). A secondexampleof this potential instability of supersymmetryis the Fayet—Iliopoulosterm in (2.71). If wewould drop it in (2.71) supersymmetrywould be unbrokenin this model, but it could be generatedinperturbationtheory and supersymmetrywould be broken.For this type of breakdown,however,oneadditionalpropertyof this model must be satisfied,therehasto exist a masslessfermion, to becomeaGoldstonefermion in the brokencase[569].

Let us now discussat the oneioop level whethersuch things can happen.Considerfor exampletheM2 term in our first example.It correspondsto a term M2f d204 = F

4, in the superpotentialwhere4 isachiral superfield.At the oneloop level thereis a contributionto F by the graphin fig. 2.4, through thecoupling3A(Fp

2+F*~*2)from (2.51). F and ~ arecomplex fields F=f+ig, ~ = a+ib and we have6A [f(a2 — b2)— 2gab] for the couplingsof the real fields. The massinsertion in the model definedin(2.51)correspondsto 4m2pç,* = 4m2(a2+b2) as canbe readoff from (2.55).The graphof fig. 2.4 for the

24 H.P. N/lies, Supersynimetry,supergravityand particlephysics

(p—

,‘

*C /

N. /— —

‘l’

Fig. 2.4. A potentialcontribution to theauxiliary field F,,. at theone loop level.

externalfield g doesnot exist, thereis no ab mixing term, whereasfor the externalauxiliary field f itsplits into two contributionswith a or b runningaroundthe loop. The couplingsfa2 andfb2, however,haveoppositesigns.Thus thesetwo contributionscancelandas a resulta termf d204 is not generatedat the oneloop level. This is oneof theso-called“miraculous”cancellationsin supersymmetrictheories.As we haveseena wholeclassof thesecancellationshaveto exist to give supersymmetrya well-definedmeaningin perturbationtheory.

A convenientway to discussthesequestionsin general is the supergraph[231,255, 107] formalismwhich we will briefly introduce here. For a more detailed treatmentwe refer to ref. [296], whosenotationwe shall follow. Consider

L = J d~xd4thfr e~’4+~_~Jd’~xd2OW”‘ Wa — Jd4xd20(~m~2+~A~3)+h.c. (2.78)

We will choosethe Fermi—Feynmangauge,which correspondsto adding —~ f d~xd40(D2V)(D2V) to(2.78). Appropriateghost termsshouldbe addedas well. This leadsto the following Feynmanrules forthe effective actionin superspace.The propagatorsare

= _~34(9—0’), = ~ m23~(0- 0’),

2 (2.79)— — m (j., ) 8~~0— 0’)

‘~“~‘ 4p2(p2+m2) “

where D is the covariantderivative and 6~(o) = 0202 The verticescan be directly takenfrom L withadditionalfactors~J~2 (—~D2)acting on the propagatorsfor each çb (q5) line leaving the vertex. Onesuchfactorhasto be omitted if the vertex under considerationcontainsonly a .f d20 (f d20) integrationrather than a f d40 = 5 d20d20 integration. The resulting amputatedone-particleirreducible graphsshouldhaveeachamputatedexternalline multiplied by the correspondingsuperfieldwith —~D2(—~D2)omitted at a vertex for each externalt/(cb) superfield.Thereare the usual combinatorialfactors andintegrations.In addition thereis an integral5 d40 for eachvertex.Sometimesit might beconvenienttolet these factors D2 and D2 act directly on the propagatorswhich then leads to the followingpropagators:

= p2 + m2~(00+00 - 200’)], ~ = p2 +m2 (-~D2)64(0 - 0’), (2.80)

with corresponding5 d20 (5 d20) integrationsat the chiral vertices.

H.P. N/lies, Supersymmetry,supergravityandparticlephysics 25

Grisaru,Rocek andSiegel havenow observed[296]that the 0 structureof thesesupergraphsis verysimple.All but oneof the 5 d40 integrationscan be performedexplicitly andthe resultingtermsin theeffectiveactioncan bewritten as asingle integral5 d’~0.What is most remarkableaboutthisresult is thefact that only the full integration5 d~0occurs,but no termswith 5 d~0or 5d20 alone.As a result theparametersm andA in the superpotentialarenot renormalizedin anyorderof perturbationtheory,andas statedthis holdsfor finite as well as infinite contributions.In componentsthis nonrenormalizationbehaviourmanifestsitself through cancellationsamonga set of graphs. A simple example of thisbehaviouris the one which we discussedearlier wherethe contributionsfrom the a and b particlescancel.We will later seemoreexamplesof thesecancellationswhich, of course,aremost convenientlydiscussedin termsof supergraphs.

Let us now discussthe ultraviolet behaviorof supersymmetrictheoriesandenumeratethe possiblecounterterms.We will confine ourselvesto renormalizabletheories.What makes the Lagrangian in(2.78) renormalizableis the fact that at mostfour factorsof D andD (two D andtwo D) arepresentateachvertex and thiscan be usedas a generalcriterion.

We haveseen alreadythat thereare no counterterms for the superpotential.For possibleotherdivergencieswe havethe following power counting rules. The degreeof divergenceof any graph isgiven by [231,510] (exceptfor contributionsto the superpotential)

d=2—E~—I~, (2.81)

where E~is the number of external chiral lines and I,. is the number of massive internal chiralpropagators.From this one then deducesthe possibledivergentcontributionsto the effective action.They are

J dm044., J d’~0çbV4, Jd’OV, Jd~0VV, J d’~0VVV, (2.82)

wherein the lasttwo termstwo D andtwo D factorsshouldbe includedat arbitrarypositions.We haveassumedthat V appearsas a gaugevector multiplet. The factors4 havedimension1, Da dimension1/2, V is dimensionlessand d40 hasdimensiontwo. As a result all contributionsin (2.82) exceptfor5 d’~0Vare only logarithmically divergent (rememberthat the last two termscontain D2D2). Theycorrespondto the usualwave function andgaugecouplingconstantrenormalizations.The contribution5 d~0Vto the Fayet—Iliopoulosterm is quadraticallydivergent. It is gauge invariant only if the gaugegroup underconsiderationis Abelian (or containsa U(1) factor). We will come back to it later andassumefor the moment that it is absent.We thushaveonly logarithmicdivergencies.In particulartheusual quadraticdivergenceof the massesof scalarparticles is absentin a supersymmetrictheory.Theyusuallycomefrom graphslike fig. 2.5 wherethe couplingconstantis given by 9A2 (6Am) for the modelgiven in (2.55). In a supersymmetrictheory therearenow additionalcontributionsto the ~ * masstermas for examplethe onegiven in fig. 2.6. They cancelthe quadraticdivergentcontributionof the first twographs,confirming the generalresultgiven above.

Let us now comeback to the Fayet—Iliopoulosterm which can only appearin the presenceof anAbelian gaugesymmetry.It receivesa quadraticdivergentcontributionin perturbationtheory by thegraph shownin fig. 2.7, andthereis no othercontributionto this order in the gaugecouplingconstantthat could cancel this contribution. The graph, however, implies a sum of the contributionsof allscalarparticlesin the theory weightedby the chargeswhich define the coupling to the D-field. If one

26 H.P. Nil/es, Supersymmetry.supergravityand particlephysics

(p.—.~.—-...~1’ / \

6m X1/ /

/ \ /

$ip

\ , I__p..____L~____.~._

(p 9)~L p* (p 6mX tp~

a) b)

Fig. 2.5. Quadraticallydivergentcontributionsto m2çoço’ dueto theself-interactionsof thescalarparticles.

Fig. 2.6. Thefermionic contributionsto m2~oco*that cancelthecontributionsin fig. 2.5.

___ (p D

Fig. 2.7. Quadraticallydivergent contributionto theFayet—Iliopoulos Fig. 2.8. A contribution to theD-term at theseven-looplevel.term at theone.looplevel.

would now, for example,impose a parity symmetry thesecontributionswould add up to zero (D is apseudoscalar).More generalthe contributionat the onelooplevel is absentif the traceof the chargesTr 0 of the scalarparticlesvanishes.But this takesonly care of the first order of perturbationtheory.Contributions in higher order, e.g. fig. 2.8, could still lead to quadraticdivergencies.This could forexamplelead to a breakdownof supersymmetryas we havediscussedearlier and it would spoil thenaturalnessof the supersymmetricmodelsas discussedin the introduction.It actuallyturns out that thisdoes not happen.The imposition of the constraintTr Q= 0 is sufficient to forbid the appearanceof5 d~OVin perturbationtheory [569,240]. Independentof Tr 0 no contributionsappearin secondandallhigher ordersof perturbationtheory [240].This resultcan be bestseenby the useof supergraphs(fig.2.9). The two contributionscancel.The blob denotesan arbitrarysupergraphwith external~, 4 and Vexternallinesandit is of courseunderstoodthat the blobsin the two contributionsareidenticalfor thecancellationto hold.

~+ ~aO

Fig. 2.9. The cancellationof thecontributions to the D.term abovethe one.looplevel as given in ref. [240].The wavy line denotesa vectorsuperfieldandthesolid line a chiral superfield.

H.P. Nilles, Supersymmetry,supergravityandparticle physics 27

We can thussummarizethe resultsfor the supersymmetrictheoriesunder consideration.Providedthat Tr 0 = 0, thereareno quadraticdivergencies.The existinglogarithmic divergenciescorrespondtowavefunctionrenormalizationsof the chiral and vector field as well as the gauge coupling renor-malization. The parametersin the superpotential(terms like 5 d20) are not renormalizedat all. Thesameis true for the vacuum energyreflecting the fact that supersymmetryremainswell defined inperturbationtheory. This fact might lead to the speculation that the naturalnessproblem of thecosmologicalconstantmight be solved by supersymmetry[584].This is howevernot the case,at leastnot in the way it is meanthere.We will see this in the next sectionwhen we couple supersymmetrictheoriesto gravity. So let us for the momentput asidetheseconsiderationsfor the vacuumenergy.

Supersymmetry(with Tr 0 = 0) is free of quadraticdivergences.One might now ask the questionwhetherall theoriesthat do not havequadraticdivergencesareautomaticallysupersymmetric(and letus forget for the momentaboutthe divergencesof the vacuumenergy).The answerto this questionisno. Supersymmetrycan be broken explicitly and there are still no quadraticdivergencesin field-dependentquantities[277,122, 309, 490]. The natureof theseallowedbreakingsis howeverrestricted.The breakings that lead to theorieswhere quadratic divergenciesare still absent are called softbreakings.They havebeencompletelylistedby Girardello and Grisaru[277]andareof the form

m2~*, m2(~2+~*2), p~~3+~*3), ,aAA, (2.83)

where~ is a complexscalar and A a gaugefermion. In particularone can see from this list that, ofcourse,spontaneousbreakingis soft: only the first two termsin (2.83) can be obtainedthereas we haveseenin the earlierexamples.It is striking that the term ~AA is soft whereastermslike itt/it/s (wheret/, isthe fermion of a chiral multiplet) are not soft. This can be explainedby gaugesymmetry and the factthat AA is the “lowest” member of the WaW,. multiplet. The fact that a ~st/tçliterm is hard might besurprisingat first sight for anotherreasonsincewe areallowedto addthe soft termsm2~*.Considerfor exampleagaina modelwith superpotential

W= mçb2+A~3 (2.84)

leadingto the masses2m for ~ (4m2pp*)and t/’. Naively onewould think that it shouldbe equivalentfor theultraviolet structureof the theory whetheroneaddsa pt/it/s massterm or subtractsa ~ term.

This, however,is not the case,sinceit is in generalthe relationof the fermion massandthecouplingAthat keepsthe divergenciesundercontrol. This can for examplebe seenif we considerthe potentiallyquadraticdivergentcontributionsto aterm linear in a scalarfield ~ (fig. 2.10).The first graphcontainsthe trilinear coupling ~ * with coupling proportionalto mA whereasthe secondcontains the A~t/it/#couplinganda massinsertion m for the fermion.The two graphscanceleachotherif supersymmetryis

Fig. 2.11. Supergraphfor a contribution F# = f d4OS*4, whereS is aFig. 2.10. Quadraticallydivergentcontributionsto a termlinear in ~. spurionsuperfieldthat representsa soft breakdownof supersymmetry.

28 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

exact.If, however,supersymmetryis brokenby a term /Lt/It/I this cancellationceasesto hold, the firstcontribution is unchanged,whereasthe massinsertion in the secondgetschangedfrom 2m to 2m +

and the resulting contributionto termslinear in ~ is quadraticallydivergent. We havethusexplicitlyseenthat the breakingp44’ is hard. On the otherhand a term ~2~* doesnot disturb the cancellationof the graphsin fig. 2.10. This doesnot, of course,completelyprove the softnessof ~ ‘. The proof ofthe softnessof the termsin (2.83) is againmostconvenientlyperformedwith the useof the supergraphformalism. The explicit breakdownin this context is introduced [277] by x,~-independentbut 0-dependentspurion superfields.They do not obey the translationalinvariancein superspaceand thusbreaksupersymmetry.An exampleis the spurion superfieldS = ~2O2. With it the breakingterm ~2~2

can be introducedin the action through the term 5 d20S4x~ j~2ço2The introductionof thesespurionsuperfieldsleadsto newdivergenciesand alsobreaksthenonrenormalizationtheorems.Now an F-termin the superpotentialcan for example be generatedin perturbationtheory with the help of S.5 d~0S”4 F,,, (fig. 2.11) representedas an integration over the full superspaceoccurs. The natureofthe newly introduceddivergenciesdependson the dimensionalityof the spurionsuperfieldandcan beinvestigatedwith the powercountingrules for superfields.This hasbeendoneexplicitly by Girardelloand Grisaru,with the conclusionthat in generalonly the four termsgiven in (2.83)are soft. With thisexplicit breakingparametersnew logarithmicdivergenciesusuallyappear(e.g. logarithmicallydivergentmasscounterterms)andthereis also a quadraticdivergenceof the vacuumenergy.Oneshould stressthat the breakingsin (2.83) are always soft, other breakingsthat are in generalnot soft, e.g. p44’,however,could be soft in particularmodelswherethe potentially quadraticdivergentcontributionsareforbidden by the absenceof certaincouplingsor by gaugeinvariance.The problematicgraphs in fig.2.10 for the ~tç1n/iinsertionwould for examplenot exist if the theory doesnot containa scalarfield thatis a gaugesinglet.

We will terminatehereour introductionto global supersymmetry.A moredetailed treatment(aswell as a morecompletelist of references)can be found in the existingreviews[217,560, 510]. Thenextchapterwill serveas a short introductionto the local form of supersymmetry.

3. Supergravity

3.1. Local supersymmetryimplies (super)gravity

In the last chapterwe havediscussedglobal (or rigid) supersymmetryin which the parameteratransformedas a two-componentspinor and was space—timeindependent0,.,a= 0. We now want torelax this condition and consider local supersymmetryin which the parametera = a(x) dependsexplicitly on x,.,. That local supersymmetrymight be of particularinterestcan alreadybe readoff fromthe algebra(e.g.eq. (2.10))

[aQ, Oa]= 2ao~&P~’, (3.1)

which says that the product of two supersymmetrytransformationscorrespondsto a translation inspace—timeof which the four momentumP~’is the generator.With space—timedependentparametersa

5(x) and a2(x) the product of two local supersymmetrytransformationsleadsto space—timetrans-lations a1(x)o~ã2(x)8”that differ from point to point, a generalcoordinatetransformation.From thisone might expectgravity to appearnecessarilyin any locally supersymmetrictheory.We will see that

H.P. Nilles, Supersymmetry,supergravityandparticlephysics 29

this expectationwill turn out to be correct. For this reason local supersymmetryis usually calledsupergravity.

Before we go to a localizedversion of supersymmetrylet us recall the transitionfrom the global tolocal casein ordinary symmetries.By “ordinary” we meansymmetriesin which the generatorsarescalarobjects,in contrastto supersymmetrywherethe generatorsarespinorial.As an exampleconsiderthe action of a free masslessDiracspinor

(3.2)

In the last chapterwe haveexclusively used two-componentWeyl spinors.A four-componentDiracspinor can be decomposedin two Weyl spinors t//a (Xx, ~~)•The symbol t/’ denotest/’~y°andA’ = y”ô,. (for our notation and the used representationof the y-matricessee the appendix).TheLagrangedensity in (3.2) has a global symmetry under which t/, transformsinto e~t/iwith scalarparametere. We want to promotethis symmetryto a local symmetryand allow e = e(x)to dependonthe space—timepoint. Now the action is no longer invariant. Under the local transformationçli—*exp(—iE(x))I/i wehave

= ~(0”e(x))t/Iy,.,t/i. (3.3)

Since the density~‘ was invariant for constante the changeis proportionalto the derivative of e. Toarrive at a locally invariant theory we haveto addsometermsto the action.This procedureto arrive ata locally symmetric theory is called the_Noetherprocedure,since it usesthe Noethercurrent of thesymmetry which in our caseis .1,.,. = ~t/iy”t/i.The variation of ..~ under the local transformationis

= (ô”e)J,,. To cancel this term we introduce a gauge field A” with transformationpropertyA” -.A” + 8”e and addthe term ~‘g= —A~J” to the action.Thus we have

IIZ’= ~it/iy”(0~+iA~)t/s, (3.4)

which is invariant under the local transformations.We hadbeenlucky here,the addition of one term(plus the kinetic termsfor the A-field in the usual gaugeinvariant way) were sufficient to arrive at alocally invariant action.This is in generalnot the case.The Noetherprocedureis appliedin generalforinfinitesimal transformationsand becomesan iterative procedurein the parametere. In principle itcould happenthat this proceduremight not leadto an invariant actionafter afinite numberof stepsanddifferentmethodshaveto be used to promotea global to a local symmetry.But in our exampletheprocedurehasworkedafteronestepandhasled to the minimal couplingof the gaugefield to the Diracfermion.

For the local symmetry to exist it requiresthe introductionof new degreesof freedom, the gaugefields: hereA~.Under the local symmetry the changeof thesegaugefields is given by terms whichcontain the derivativeof the symmetryparameter:here8”e. In our examplethe gaugefield is thus aspin 1 field becausethe parametere(x) was scalar. In our minimal formulation only the gaugefieldstransformwith termsthatareproportionalto (r9~e).The transformationof the otherfields containsrbut not r9”e. With this in mind we can now startto considerthe supersymmetriccase.As an exercisewe will do this in a four-componentspinornotation.Insteadof a two-componentWeyl spinor t/’a wework with a four-componentMajoranaspinorx = (tu’a, tfra) We start with the free action of a chiralmultiplet

30 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

S~’=~ (3.5)

wherewe havealreadyeliminatedthe auxiliary field F via the equationsof motion (comparesection2.3.). ~ denotesa complexscalar field ~ = (a + ib)/V2 andx is a Majoranaspinor with ~ = x~’°=(t/,a ti/a) (for our notation seethe Appendix).The action correspondingto (3.5) is invariant underthe

global supersymmetrytransformationsas given in eqs. (2.35)—(2.37). In a four-componentspinornotationwith ~ = (aa, ~ andusingF = 0 theyread

= ~, ~b = —i~y5,y, = —iy”8~(a— iy5b)~,

with 1 0

= = (0 1 —1 _~) (3.6)

Performingexplicitly thesevariationsof the fields in (3.5) we seethat ~ changesby a total derivative

= 0,~(~y”[/(a— iysb)]~), (3.7)

which in addition vanisheson-shell (using equationsof motion). We now would like to promotetheglobal supersymmetryto a local one andallow ~ to be x-dependent.One questionarisesimmediatelyconcerningthe variation of the fermion field since it containsa derivative.According to the remarksmadein the previousexamplewe choose~i,y= —iy”[8~(a — iy5b)]~(x)wherethe derivativeonly actsonthe scalarfields but not on ~(x). We now computethe variation of (3.5) under theselocal trans-formationsto lowestorderin the parameter~(x), andobtain

= 9,~(~y”[/(a— iy5b)]~)+ (ä~~)[y”(/(a— iysb))~], (3.8)

which showsthat ~‘ is not locally supersymmetric.A gaugefield hasto be introduced.Since ~ is aspinorandsince the gaugefield should transforminto 3”~the gaugefield of local supersymmetryis aspinorialvector t/i~. It is called gravitino. We add to the action the Noethercoupling

IN= —kt/i~y”(A’(a—iy5b))~, (3.9)

where we have introduced the dimensionful coupling constantk = 1/massto give ~ the correctdimension.As a consequencet/i” should transformas = (1/k)3~~under the local supersymmetrytransformations.The resultingaction is not yet locally supersymmetric,ascan bealreadyseenfrom themultiplet structure.We haveintroduceda spinorialgaugefield t/I~a andwe know that supersymmetricmodelshavethe samenumberof fermionic and bosonic degreesof freedom.We thus expect newbosonicfields to be necessaryto obtaina locally supersymmetrictheory.

Explicit variationof £~?+ ~N confirms thisexpectation.Considere.g.the variationof x in ZN. It leadsto

~J1N ktJi~y~,.T”~ (3.10)

where the tensor T”~contains two scalarfields and two derivatives (it actually correspondsto the

H.P. N/lies, Supersymmetry,supergravityandparticlephysics 31

energymomentumtensorof the scalarfields). This termcan only be cancelledthroughthe introductionof a tensorfield ~ with local supersymmetrytransformation~ — kt/i~y~~andwe add to the action

= —g~~T”~, (3.11)

which correspondsto the coupling of the graviton to the energymomentumtensor.We thussee thatany locally supersymmetrictheory hasto include gravity; it includesthe masslessspin 2 particle (thegraviton) mediating the gravitational interactions.Its partner is the masslessspin 3/2 gravitino, thegauge field of local supersymmetry.Our argumentsbefore had not uniquely told us that t/I~acor-responds to a pure spin 3/2 state (it could also contain spin 1/2 states) but since tfr~a is thesupersymmetrypartnerof the spin 2 graviton,we know that t/’~,.correspondsto a spin3/2 particle.Thegraviton and the gravitino form thusthe basic multiplet of supergravity,and one expectsthe simplestlocally supersymmetricmodel to contain just this multiplet. Let us thereforeforget the original modelwe startedwith andtry to constructthe Lagrangianfor pure supergravity[248,113]. We startwith theHilbert action

= _(1/2k2)V~g~R. (3.12)

Sincewe know that therewill be fermionspresentin our model it is necessaryto use the vierbein e~insteadof themetric g~,.with the relation~ = e~ei~mn.(Wecannotgive herea detailedpresentationof gravity andwe referthe readerwho is unfamiliarwith the subjectto considerthe standardtextbooks[548].)The expression(3.12) thenreads—(1/2k2)eRwherethe curvatureR dependson the vierbeine~and the spin connectionw~”.Thespin connectiondoesnot representa new degreeof freedom;it canbe eliminated by the equationsof motion. Next we needthe kinetic terms for the gravitino t/I,.,, theRarita—Schwingeraction

ZRS ~ (3.13)

This action hasto be covariantizedwith respectto gravity by replacing8,. in (3.13) by the covariantderivativeD,. = 8,. + ~W~”O~mfl whereOmn = ~[Ym,y~].The sum of the two terms ZG and ZRS (in thecovariantizedform) is locally supersymmetric.The correspondingsupersymmetrytransformationsare

= ~ = (0,. + ~~~Omn)~ ~ D,.~, ~iW~” = 0. (3.14)

The last equation implies the use of the 1.5 order formalism [533]. We see that the first twotransformationlaws contain the termsthat we had foundearlierwith the Noetherprocedure.

3.2. Couplingof matterand gaugefields to supergravity

In global supersymmetrythe most generalcouplingof chiral superfieldmatter andvectorsuperfield

gaugefields can be written as

ReJd20f(S)WW+Jd40~(~e2e%~~,S)+Ref d2Og(S), (3.15)

32 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

whereS(V)denotethe chiral (vector)superfields,andg(S) is the superpotential.f(S) transformslike achiral superfieldunder supersymmetryand as a symmetricproductof two adjoint representationswithrespectto the gaugegroup underconsideration.The theory is renormalizableonly if f(S) is constant.The function 4 transformsas a real vector superfieldand only leadsto a renormalizabletheory if4, = Se~’S.Finally renormalizabilityrequiresg(S) to be a polynomialof degreelessor equalto three.We give up here the strict criterion of renormalizabilitysince we want to couple this matter gaugesystemto supergravity,which in itself is a nonrenormalizabletheory even if the systemcoupledto it isrenormalizable.After the coupling to supergravitywe will however demandthat all the existingnonrenormalizabletermscontainthe gravitationalcouplingconstantsuchthat in the flat limit k—* 0 thetheory becomesrenormalizable.This is the bestwe can do sincethereis no knownsatisfactorysolutionof the problem of the nonrenormalizabilityof gravity. We just assumethat this problemhas no severeimplicationson the physicsat energieslow comparedto the Planckmass.

The task is now to promote the global supersymmetryin (3.15) to a local symmetry which isequivalentto a symmetriccouplingof thefield S and V to the supergravitymultiplet. Thereare severalwaysto do this. Oneis theNoetherprocedurewe havesketchedin the lastsection.Othermoreelegantmethodsusing off-shell formulations are also available.All of thesemethodsrequire rather tediouscalculationsand spaceand time availablehere do not permit us to repeatherea detaileddiscussion.Such a detailed discussioncan be found in ref. [533]. Since the coupling of the action (3.15) tosupergravityis given in the literature in its most completeform we contentourselvesto give herethefinal form of the action in componentfields. For the derivationof the action we refer to the originalpapersby Cremmeret al. [100,101, 97,98, 574] whose notation we also adopt. The membersof thechiral multiplets are denotedby (z1,x~) where i denotesa group index if these fields transformnontrivially underthe consideredgaugegroup.The gaugefermionsarecalled A”‘ and thefield strengthsF~.,where a labels the adjoint representationof the gaugegroup and ~i, v are curvedspace—timeindices.The part of the action that containsexclusively bosonicfields is then given by

e1ZB = exp(—G)(3+ Gk(Gt),kG!) — ~2 Ref~(G~T~zj)(G”T~z,)

— ~Ref~F~F”~+ ~iImf~F~F”~+ G~D,.z~D~~z*)— ~R, (3.16)

here~ denotesthe gaugecouplingconstant,T~”’ the groupgeneratorsin the suitablerepresentationandF,.... is the dual of F,.~.The covariantderivativesD,. are covariantwith respectto gravity andthe gaugegroup.The function G(z1,z~)is a real function of the scalarfields. In termsof the input function 4, andg in (3.15) it can bewritten as

G = 3 log(—4,/3)—log(~gl2), (3.17)

andis called the Kählerpotential.The quantitiesG” = G’ etc. arederivativesof G:

G’7 = 32G/8z~8z*). (3.18)

We will discussthis function later in detail. Next we give the action for the termscontainingfermionsandcovariantderivatives

H.P. Nil/es,Supersymmetry,supergravityand particlephysics 33

e~1ZFkfl= ~Refa~(~A”‘øA’3+ ~A”‘y”’~t/i,.F~ — ~A~y,.A~G’D”z~)— ~iIm fase 1D,.(eA”‘y5y”A’

3)

— ~fa,9XLIU” F,,~,A~— ~t/i,.y5y,,D~t/i~e“~‘e’+ ~e

1e~“~‘t/i,.y~t/i~G’D~z1— G~t//,. LØZ *~y/

2XLj

— XLIØZIXR(GI~+ ~G~<G~)+G~øx’R+ h.c.,

wheref~= Ofa,s/Ozj.The part of the fermionicaction that doesnot contain covariantderivativesreads

e1ZF= ~ ~ + e_a’2[G~~— G1G’ —

....l.f”T”i I ~ —G/2f”IT ,~

2ig , Zj~pL,.y R e ~PR,.Y XLI

+ ~i(Ref)~f~ G’T~”1zJ,j~LkAL~+ 2igG~T~””ZkAaj~.J(’~

+ ~ + ~ + ~

+ ~ (,~1o””A~ti/PLY,.A ~ + it/i,. RY”XLIA ~A~)+ ~ RYdXL1(Cabcdt/JaYbt/ic —

+ ~XL~y”X’RA~Ry,.A L(—2G~Re~ + Ref=~f~f~)

+ ~XL,XLJA?iA~(—4G~(G1)~f~+ 4f~— Re~

~y ,iv~ö D f1 f~ fj16XLi0,.,,~XLj/tLif L ~ ~a$J ayJ13~

+ (—~G~,+ ~G~(G1)~’G~,— ~ G~),~LiXLJ,~RX’R+ h.c. (3.20)

This completesthe action. We haveset the gravitationalcoupling constantequalto unity. It can beinsertedby dimensionalconsiderationsas we will later do in specialexamples.It shouldbe noted thatthe consideredaction dependsonly on the two functions G(z,z*) andfa,g(Z), whereasthe globallysupersymmetrictheory definedby (3.15) hadthreeinput functionsf, 4, andg. Coupledto supergravitythekinetic function 4, andthe superpotentialg loosetheir independentmeaningandenteronly throughthe function G(z,z*) as given in (3.17), expressingthe fact that the consideredscalar field spaceinsupergravityis a Kähler manifold [585,574, 24,25]. G is called the Kähler potential and the Kählermetric G~definesthe kinetic termsof the scalarfields as can be seenfrom (3.16).

The action given above is, in the given form, ill defined at points where the superpotentialgvanishes.This, however, is just an artifact of our notation.The correctaction in the caseg(z)= 0 isgiven by replacingG everywhereby J = 3 log(—4,/3) exceptfor the termse_G, e~’~2which shouldbe setequalto zero.There aresomemorecomplicationsif oneintroducesa Fayet—Iliopoulosterm [42],butwe will not discussthem here.

We finally give the explicit form of the supersymmetrytransformations

= 2ELY tii,.R + h.c.,

= (0,. + ~w,.mn(e,ti,))..~.mneL+ ~(T,.,.ELG~XR~YXL~+ ~Y,.ER e

+ 4t//,.L(G ELXLI — G,~~~’h)_~L(GD,.Z, — G,D,.z*i)+ ~ — LALYA~( Refas,

= —2ELY,.AR+h.c.,

= “~E~V~R+ ~~ERRef~(—~G’T~3’z,+ ~ifsYxLIA~— ~ ~)— ~A~(G’~LXLI —

34 H.P. Nil/es, Supersyinmetry,supergravityand particlephysics

= ELXLI,

~XLi= ~ØZIER — ~EL e’3”2(G 1)~G~— ~ELAA~(G1)~f~k (3.21)+ 1 (r’—l\k r’j! — + ~ (f~ — I —2ELt~~~I)i ‘.~) kXLIXLI 4XLR~~3iERXR‘.-‘ ELXLJ -

3.3. Spontaneoussymmetrybreakdown

In globalsupersymmetrywe had seenthat a spontaneousbreakdownoccursif auxiliary fields F or Dreceive a vacuum expectationvalue (compare (2.65)). The correspondingfermionic partner of theauxiliary field was identified as the Goldstone fermion. As shown, this fact had been a directconsequenceof the algebraandshould thereforebe alsovalid in thecaseof local supersymmetry(withan appropriateinterpretationof the Goldstonefermionas we will seelater). The quantitiesto look forif one wantsto find out whethersupersymmetryis spontaneouslybrokenare thusthe auxiliary fieldsthat arethe partnersof spin 1/2 fermions.We havenot yet discussedthem explicitly in the frameworkof local supersymmetrybut we know how to find them. They are obtained by a supersymmetrytransformationon the spin 1/2 fermions of the theory. More specifically they are the terms in thoseexpressionswhich do not contain space—timederivativesand can be readoff from eqs. (3.29). For thechiral fermionsthe auxiliary field is [98]

F, = exp(—G/2)(G’)~G1+ ~ — (G

t)~~ — ~x1(Gx’). (3.22)

For the auxiliary partnerof the gaugefermionwe obtain

= i Ref~(—gG’T~’z,+ ~ — ~if”~x’A7) — ~Aa(G’~i). (3.23)

As a consequenceof the algebrasupersymmetryis broken if at least one of these auxiliary fieldsreceivesa vacuumexpectationvalue,and the supersymmetrybreakingscaleM~is given by (F~or (D)in completeanalogyto the situationin global supersymmetry.Notice that all the terms but the first in(3.22) and(3.23) containfermion fields explicitly. A vacuumexpectationvaluefor thesetermscan thusonly occur in presenceof a strongly interactinggaugeforce that leadsto a vacuumcondensationofbilinear fermion—antifermionstates.We will discussthesetermsat a laterstageandneglectthem for themoment.This hasthe advantagethat our formulaesimplify substantially.The first termsin (3.22) and(3.23)contain only the scalarfields z~.WhetherF, or D” receiveactuallyvacuumexpectationvaluesornot dependson the functionsG and f and can only be decided after a minimalization of the scalarpotential. For the moment,however, let us assumethat we have done this and supposethat oneauxiliary field, let us say F,, hasa vacuum expectationvalue (0~exp(—G/2)(G’)~G1.I0)� 0. Since thekinetic termsof the scalarfields are given by G~D,.z1D~z*jwe seethat G~hasto be nonzerofor themodel to be well defined.In addition exp(—G/2) is in generaldifferent from zero exceptat the pointswhere the superpotentialvanishes.It is thus the quantity G, that is relevant for a discussionof localsupersymmetrybreaking.This is alsotrueif we wouldhaveconsidereda vacuumexpectationvalue of aD-field as can be seenin (3.23).

In the caseof global supersymmetrythe fermionic partnerof the auxiliary field with nonvanishingv.e.v is thegoldstino; a masslessparticle. Let usnow seewhat happenshere.The relevanttermsfor thisdiscussionare the first, third andfifth in eq. (3.20)

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 35

exp(—G/2)t/’,.cr”~t/ç— exp(—G/2)G’t/i,.y”~~+ exp(—G/2)[G” — G~G1— G’(G1)~’G~]x,x1. (3.24)

We seethat with (G’) � 0 the fermionx~getsmixedwith the gravitino and‘(G’) � 0 is just the criterionfor supersymmetrybreakdown.Whathappenshereis the super-Higgseffect [114,100], the analogueofthe Higgs mechanismin ordinary gauge theories.The spin 1/2 Goldstonefermion exp(—G/2)G’x,combineswith the masslessspin 3/2 gaugeparticleof local supersymmetryto a massivespin 3/2 gaugeparticlewith massexp(—(G)/2).

Before we makethis relationof the gravitinomassto the supersymmetrybreakingmore explicit letus first discussthe scalarpotential.It is given by the first two termsin (3.16)

V = —exp(—G)[3 + Gk(G1)~G’] + ~f~D”‘D’3, (3.25)

where

= gG’T~’z~,. (3.26)

Although the formula is similar to the one in global supersymmetrythereis a striking difference andthis is the relation of the vacuum energy to the supersymmetrybreakdown.In the global case thestatementEvac= 0 was equivalentwith the presenceof unbrokensupersymmetry.This ceasesto be truein supergravity.Let us first considerunbrokensupersymmetry,which implies (Gk) = (D”‘) = 0 andweobtain V= —3 exp(—G) which in generalis nonvanishing.It tells us that in unbrokensupergravitythevacuumenergyis negativesemidefinite (supergravityis always broken if Evac> 0). If supergravityisbrokenthe situationis morecomplicated.Supposewehavefound the minimumof V. We now makearedefinitionof the fields suchthat the kinetic termsarecanonical.We can alwaysdo such a redefinitionif our model is well defined.We then have G~= —3~and fa~= ~ and the potential reads(at theminimum)

V0 = —3 exp(—G0) + exp(—Go)GkoG~+ ~DaoD~. (3.27)

The last two termsarein exactanalogyto the termsin a globally supersymmetricmodel.With G0,. orD~different from zero (i.e. brokensupergravity)everythingcan happen:the vacuumcan be negative,positive or evenzero. What is interestingabout this point is the possibility of broken supergravitywithvanishingvacuumenergy; a situationthat did not exist in the caseof global supersymmetry.As we willseelater this cancellationof the vacuum energyis usually obtainedby an explicit fine tuning. Theproblemwhy the cosmologicalconstantvanishesis not understood,but we haveat leastthe possibilityto fine tuneit to zero,which did not exist in a globally supersymmetricmodel in which supersymmetrywas spontaneouslybroken. Let us again for the moment put Da = 0 and assumethat V0 = 0. Thisimplies that GkG’~= 3 in the brokencase.We can now give a relationbetweenthe gravitino massandthe scale of supersymmetrybreakdown. For this relation between masseswe put back in thegravitationalcouplingconstantk

1/k= M = M0/V8IT= 2.4x 1018GeV. (3.28)

The gravitino massis (3.24)

m312= M exp(—Go/2), (3.29)

36 H.P Nil/es,Supersymmetry,supergravityand particlephysics

andthe supersymmetrybreakingscaleis

M~= —M2 exp(—Go/2)(G~’)~Go,, (3.30)

with G~,= —1 as discussed.V0 = 0 implies G1 = “/3 if only one G’ hasa v.e.v. (if moreof them have

v.e.v.’s onehas just to take the appropriatelinear combinationandredefine the fields). This gives theDeser—Zumino[114]relation

m312 = M~/V3M, (3.31)

in the caseof vanishingcosmologicalconstant.The massm312representsthe splitting in the supergravitymultiplet (the graviton of coursestays massless).The relation(3.31) is similar to thosewe obtainedinthe caseof global supersymmetry(compare(2.74)) wherehere the coupling constantis gravitational.This implies that the masssplittingscan becomevery small comparedto M8 in the caseof M~~ M Wewill lateruse this propertyextensively.

Although it is not of immediateinterestfor applicationsto particlephysics,let us also discussthecaseof a nonvanishingcosmologicalconstant. Formula (3.29) is valid independentof the vacuumenergy. Imagine now a situation with unbroken supergravity G, = 0 but exp(—G/2) � 0: V0 =

—3 exp(—G) and m312= exp(—G/2). Unbrokensupergravitywith m312~ 0 is hard to understand.Theresolutionlies in the ratherdelicateconceptof massin De Sitteror anti-De Sitterspace.A masstermfor the gravitino in anti-Dc Sitterspacedoesnot necessarilyimply that it hasfour degreesof freedom.If the cosmologicalconstantA = V0 is relatedto the gravitino massby

A = —3M2m~,

2, (3.32)

the gravitino hasonly two dynamicaldegreesof freedom(which correspondsto a masslessparticle inthe caseA = 0). This can be seenby the fact that supersymmetryis still unbrokenand as suchcontainsthe “massless”spin 3/2 gauge particle with two degreesof freedom.The transformationlaw of thegravitino,however,changes.The termD,.~becomes(D,. + ~m312y,.)~as can be explicitly seenin (3.21).Thisconsiderationof the y,.~term in the transformationlaw of the gravitino is actuallyan easyway toseewhetherA 0 or not [227].Usuallythis termis written as uy,.~whereu is called thecomplexscalarauxiliary field of Poincarésupergravity.The massterm m312 = M exp(—G/2) can thusonly in the caseofA = 0 be consideredas an unambiguoussignal of supergravitybreakdown.The order parametersofsupergravityarethe auxiliary fields given in (3.22) and(3.23).

Up to now we havediscussedthe scalarpotentialonly in termsof thegeneralfunction G. Let us nowwrite theseformulasmoreexplicitly in order to compareit to the potentialof globally supersymmetricmodelsand to seethe role of the superpotential.In generalthe KählerpotentialGhasto fulfill certainrequirementsfor the resultingmodelto be well defined. It shouldfor exampleleadto thecorrectsignofthe kinetic terms of the scalarfields G~<0. The model is well definedonly in thoseregionsof fieldspacewherethis inequality is satisfied.For the applicationsonewould evenprefer modelswherethisrelationholds globally. Onespecialchoiceis the choiceof minimalkinetic terms[100]

G~= —6’~, fa~= ôa$, (3.33)

whichwe will adoptthroughoutmostof our discussion.It allowsus to seeall essentialfeaturesof these

H.P. Nilles, Supersymmetry,supergravityand particlephysics 37

potentialsin a simplified way. We thus use G(z,z*) = _z,z*l — loglg(z,)12 where g(z1) is the super-

potential.This leadsto

G’ = —z’” — g(z,)/g(z,), (3.34)

with g’(z,) = t9g(z,)/Oz,,and somemanipulationsleadto the scalarpotential

2 3

V= exp(~-)[~g’(z1)+~g(z,)~_~j~IgI2], (3.35)

wheresummationover i is understood.Supersymmetryis brokenif f’ = gi + z*g/M2 receivesa vacuum

expectationvalue. In the globally supersymmetriccasethis role as an order parameterhadbeenplayedby g(z,).The additionof the term z*g(z)/M2 leadsto potentially interestingconsequences.Supposewehavea superpotentialg(z

1) that containsan intrinsic scale~.rsuchthat, e.g., atthe minimumg(z,)= A~r3

where A is a coupling constant. In a globally supersymmetricmodel with spontaneouslybrokensymmetry such a superpotentialwould leadto a breakdownscale

M~—Aj~2, (3.36)

essentiallyof order j.~.This can be different in the local case.Supposethat supersymmetryis broken

fi � 0 andthat A = 0:

M25 = f~= (\/3/M)go(z,)-~ A1i

3/M. (3.37)

Vacuumexpectationvaluesof fields of orderp. leadto a breakdownof supersymmetryat a massscalep.3/M which can besubstantiallysmallerthan p. as long as p. is smallerthanM Moreoverthe splittingsof the multipletsaregiven in generalnot by p. but by m

312 the gravitino mass

m312-~ M~S/M Ap.3/M2. (3.38)

In the breakdownof supergravityit seemslikely that huge ratios betweenmassscalesare generated[446].Accordingto our discussionin Chapter1 it is amongothersthis behaviorof supergravitythatawakenedthe interestof applyingthesetheoriesto modelsof high energyphysics.We will comebackto thesequestionsin moredetail in Chapters6 and7, wherewe will also give explicit examples.

We havehere essentiallylimited our discussionof supergravitybreakdownto the casewhere F’termsreceivea vacuumexpectationvalue. This allowedus to userathersimple formulas.Theinclusionof different mechanismsis straightforward.If a D”‘ term is at the origin of the supersymmetrybreakdownit will leadto a vacuumenergy[97,98]

V0 = exp(—Go)[GOkG o’i — 3] + ~f~~D0aD0I~, (3.39)

asupersymmetrybreakdownscaleM~= D~and the correspondinggaugefermion A”‘ will playthe roleof the “goldstino” which providesthe two additionaldegreesof freedomfor the massivegravitino. If

the breakdownis dynamical,e.g. through a condensationof the gaugefermions~AA)� 0 the potentialis given by [226]

38 H.P. Nil/es, Supersymmeiry,supergravityandparticlephysics

V = —3M4exp(—G)+ [M2 e~’2G1+j, ~16~][M2 e~’

2Gk+ fk AA]*(G.1)I (3.40)

as can be derivedfrom (3.22).The correspondinggoldstinois f~(AaA$)X~.We will comebackto thesequestionsin Chapter6.

3.4. Thesupertraceof the massmatrix

We haveseen in section 2.4 that in spontaneouslybrokenglobally supersymmetricmodelsstrongmass relations hold at the tree level and we will discuss this question here in the case of localsupersymmetry.The relevant quantity to compute is the supertraceof the squared mass matrixSTr Al2 = ~

1(—1)~’(2J+ 1)Al~where J denotesthe spin [227].We will do this here in the caseof Nchiral superfields(z~,Xi) with minimal kinetic terms for a model with vacuumenergyequal to zero[98,574]. More general formulas will be given at the end of this section.The potential is given byexp(—G)[G1Gi — 3] and A = 0 implies G,G~= 3 at the minimum of the potential. The minimumcondition V = t9V/8z

1= 0 gives

GIJG’ = G1, (3.41)

wherewe haveexplicitly used G~G1= 3 and G = —3~.The massof the gravitino is exp(—G/2) and we

get a contribution of —4 exp(—G) to the supertracecoming from J = 3/2. The mass of the scalarparticlesis obtainedthrough the secondderivativeof the potential V~.In our specialcasethis is

V~= exp(~G)[G11G”+ G~’G~]. (3.42)

The situationfor thespin 1/2 particlesis morecomplicated.The massesof theseparticlescanbe readofffrom thethird termin eq. (3.20)but theparticlethat wasabsorbedby thegravitinohasto beremovedsincewe alreadycountedit in the spin 3/2 sector.The goldstinois given by cit = G5~’~.The massmatrix for theremaining(N — 1) chiral fermionsbecomesproportionalto G” — ~G’G~.Summingup thesecontributionsgives for the supertrace

STrAl2 = 2(N — 1)m~,

2, (3.43)

a surprising result. Unlike in the caseof spontaneouslybrokenglobal supersymmetrythe STr Al2 does

not vanish here. If one considerssupergravityin the flat limit M—* ~ (but m312 fixed) spontaneously

broken local supersymmetryresemblesmore closely to explicitly than spontaneouslybroken globalsupersymmetry[98]. In (4.43) the supertraceis positivewhich implies that on theaveragethe bosonsareheavierthantheir fermionicpartners,a propertywhich will be usefulfor later model building[98,177].

Under the given restrictionsA = 0, G) = —~ (3.43) is valid independentlyof the chosensuper-potential, of which we will now considera specialcase.Supposethat g(z) dependsonly on oneof thescalarfields z1. If this would bethe only field we wouldhavevanishingsupertracesinceN = 1. With theaddition of the N — 1 fields z2, - . - , zN the supertracechangesto 2(N — 1)m3/2- Whathappensis that allthe fermionsX2,. . - XN remainmasslessandthe 2(N — 1) real scalarfields receivea commonmassm312as the readercan easilyconvincehimself throughan explicit calculation.Notice that it is the gravitinomassand not the supersymmetrybreakingscaleM5 that correspondsto the splitting of the supersym-

H.P. N/lies, Supersymmetry,supergraviIyand particlephysics 39

metry multiplets. Notice also that in this specialcaseall the 2(N — 1) real bosonsare degenerateinmass.

Formula (3.43) hasbeenderivedunder the restrictionsA = 0 and G~= —6~,and we havenot yetincluded the (spin 1/2, spin 1) multiplets. With thesetaken into account(still A = 0, G~= —6~)thegeneralformulareads[98]

STr Al2 = (N — 1)[2m312 — D”‘Da/M

2] — 2gaD”‘ Tr T”‘. (3.44)

The last termcorrespondsto the onealsopresentin the globally supersymmetriccase.The D2/M2 termis new and gives a negativecontributionto the supertrace.The condition of A = 0 requiresD2/M2<

6m~,2still leaving open the possibility that it might cancel or dominatethe 2m~,2term. No explicit

model, however, is known wherethis happens.The results in (3.43) and (3.44) changeif we relaxtheconditionson A andG~.For A = m~j2M

2[2x— 3] eq. (3.43)generalizesto [446]

STr Al2 = 4m ~,2[(N— 1)(x — 1)— 2(1 — ~x)

2]. (3.45)

It also changesfor nonminimal kinetic terms. (This is actually also true in the case of globalsupersymmetry,but therewe did not considernonminimal kinetic terms.)

Equation(3.43) for nonminimalkinetic termsreads(A = 0) [294]

STrAl2 = 2(N — 1)m312— 2R~F~F

1, (3.46)

with

R~= [logdet(G~)]’~, (3.47)

F, = exp(—G/2)(G1>~G1, (3.48)

compare(3.22).The generalizationof (3.44) to nonminimal kinetic termsis given in ref. [294].We willlater discussthe implicationsof theserelationsin the context of explicit modelsin Chapter6.

This concludesour collection of formulaerelevantin N = 1 supergravity.A moredetailedtreatmentcan befound in refs. [533,97,98,100, 101].

4. Low energysupersymmetry:basics

By low energywe denote the 100 GeV region. In this section we introduce and discuss theingredientsthat everymodelmustpossessin which supersymmetryhasimplicationson this low energyregion.We will alsoencounterthe potentialdangersin thesemodelswhichcould lead to inconsistencieswhen comparedto experimentalresults.We startwith the introductionof the standardnonsupersym-metric model.

4.1. Thestandardmodel[59,280,500,547, 495, 281,368]

The standardmodel is basedon the gaugeinteractionsof the strong and electroweakinteractions

40 H.P. Ni/let, Supersymmetry,supergravityandparticle physics

with gaugegroupSU(3)x SU(2)x U(1). It contains12 spin 1 gaugebosons:eight gluonsof SU(3), threeSU(2)weak gaugebosonsandthe hyperchargegaugebosonof U(1)~.The photonwill be a particularcombinationof an SU(2) gaugebosonand the hyperchargeboson as we will see in a moment.Thefermionsin the theory arethreegenerationsof quarksandleptons

ii ~ (pe) e

“- (:) ~~ (~‘)~i (4.1)

III. (~,) t b (l~) ~

wherewe haveincluded the top quark,which is not yet experimentallyconfirmed. This is a threefoldrepetition of a set of fermions that has the following transformationpropertieswith respect toSU(3)x SU(2)x U(1)

ua = (~)=(3,2,~), u=(~,1,—i), d= ~ a= 1,2

(4.2)

La= (~)=(1,2,_~), e=(1,1,1),

wherethe first two entriesin the bracketsdenotethe dimensionsof the SU(3)x SU(2)representationsand the last entry denotes U(1) hypercharge.The set in (4.2) is anomaly free with respect toSU(3)X SU(2)X U(1). (This is one of the strongargumentsto believe that the t quark exists.) Thefermions in (4.2) denoteleft-handedtwo-componentWeyl spinors.Mass terms are forbiddenby thegaugeinteractions.They can only occur if oneof the gaugesymmetriesis broken.The electricchargeisgiven by 0 = I’~+ Y where T3 is the diagonalgeneratorof SU(2); e.g. we have T3(u)= + 1/2 andQ(u)=2/3.

Let usdenotethe gaugefields for SU(2)andU(1) by ~ andB,. andthe gaugecouplingsby g2 andg1 respectively(i = 1, 2, 3). The field strengthsarethen given by

F~.= ~ — ~ + g2r’~”A~,.A~, B,.~= o,.B~— 3,B,., (4.3)

and the gaugeinteractionsof, for example,the leptonsaregiven by

Zge = ië~o-”(8,.— ig1B,.)e+ iL~o-”(8,.+ ~ig1B,.— ~ig2cr’A’,.)L. (4.4)

The SU(3) interactionswith couplingg3 are given in a similarway.SU(3)X SU(2)~x U(1)~is broken to SU(3)X U(1)em. This is achievedthrough the introduction of

scalarfields: the Higgs sector.The minimal choiceis oneSU(2)doubletof chargedscalarsh = (~)withhyperchargeY = — 1/2, with potential

V”~p.2h~h+ A(h~h)2. (4.5)

H.P. Ni//es, Supersymmetry,supergravityandparticlephysics 41

OnealsointroducesYukawacouplingsfor the interactionsof the scalarswith the fermions,e.g.,

Zy=gdUlzd+gUUh*u (4.6)

in all combinationsthat areconsistentwith the SU(3)x SU(2)x U(1) gaugesymmetry.The multiplication of the doubletsin (4.6) is understoodas Uah~Eabwherea is the SU(2) index and

Cab is antisymmetric,the secondterm is U~2(h*)b~.Next onelets p.2 to becomenegativeso that onecomponent,which we chooseto be the neutralcomponentof h, developsa vacuumexpectationvalue

(h) = ~ (o), v = (_p.21A)u2. (4.7)

This breaksboth SU(2) and U(1)~but leavesU(1)em unbroken.The Higgs mechanismtakesplace andthe chargedbosonsW~= (At. ~ iA~)/V2receive a massM~= ~gv.Oneof the neutralgaugebosons

— ( A3 \I~ 2 2\1/2z,. — ~—g2~-~,.+ gi ,.),~gl + g2,

receivesthe mass~v(g~+ g~”2whereasthe orthogonalcombination

A — I D ..L. A3\/I 2j.. 2\1/2

— ~ U g1Z-1,.),~gl-~-

(the photon)remainsmassless.There is oneremainingreal scalarfield with massV_2p.2, the Higgsparticle. The threeparametersv, g1 and g2 can be determinedexperimentallyby the electromagneticcouplingconstante, the Fermi coupling constantGF and a measurementof the strengthof the weakneutralcurrents.With tan0~= g1/g2 we havethe following relations

GF/V2= g~/8M~=(4.10)

e = g2 Sjfl Ow = g1 cos0~,

whereM~is the massof the chargedweakbosons.This determinesv to be of the orderof 250GeVandM~ (38GeV)/sinO~as well as M~ (38 GeV)/~sin20w, where from the strengthof the weakneutralcurrentsoneobtainssin

20~ 0.21.The consistencyof the (recentlydiscovered)WandZ bosonmasseswith thesepredictionsis a sensitivetestof the model. The massof the Higgs V_2p.2remainshoweveracompletelyundeterminedparameter.

Up to now we haveseenthe implication of the Higgs sector to the gaugebosonsector.The Higgssector fulfills an additional task: the massgenerationfor the fermions. With SU(2)X U(1)~,broken toU(1)em the symmetriesSU(3)x U(1)em do no longer forbid masstermsfor the quarksand leptonsandthe Yukawacouplingslike in (4.6) provide thesemassterms.The massfor the d quark is for examplegiven by gd(v/V2), with v as defined in (4.7). No predictions, however,arisefrom the model: theexperimentallydeterminedmassesof quarks and leptons are obtained by adjusting the Yukawacouplingsto the desiredvalues. In particular it is not understoodwhy the different generationsofquarksand leptonshaveso vastly different masses.Moreover, the massmatrix is not diagonalin theweakbasis.Mixings betweenthe differentgenerationsoccuras given by the Kobayashi—Maskawa[368]matrix:

42 H.P. Niiies, Supersymmetry,supergravityandparticle physics

/ c1 s1c3 —sis3 \ /u\(u, ë, t) ( s1c2 c1c2c3— 5253e c1c2s3+ s2c3e’~j ( c J (4.11)

\ s1s2 c1s~c3+ c2s3e’8 c

1s2s3— c2c3e~/ \ t /

where we havechosenthe (d,s, b) matrix to be diagonal.The symbols s, = sin0~,c1 = cosO~denotethreereal parameters(O~is the Cabibbo angle).The phase6 is CP violating. Three generationsofquarksarenecessaryto arriveat CPviolation through this mechanism.

Much morecould be saidabout the standardmodel whichwe cannotrepeatherein detail.

4.2. Grand unification [271,473,394]

Grandunified modelsaremodelsthat unify the gaugeinteractionsSU(3)x SU(2)x U(1) in a largergroup. To be successfultheyshouldreduceto the standardmodel at low energies.We will not attempthere to give a full accountof what hasbeendonein the context of thesemodels(for a reviewsee thePhysicsReports of Langacker[394]) but will just introduce the simplestSU(5) model [271] both toexplain the ideasandfix the notation.This minimal modelmight at the momenthavesomedifficultiesbecauseof the absenceof protondecayat the predictedrate [56], but we want to discussthe issuesofgrand unification in the simplestmodel.

The gauge group SU(3)x SU(2)x U(1) is unified in the semisimple group SU(5), i.e. the threecoupling constantsg3, g2 and g1 are given by one coupling constantg5. Now at a hundredGeV thecoupling constantsg3, g2 and g1 are very different from each other and this is not a very strongindication for unification. The resolution [272]of thislies in the fact that the couplingconstantsarenotrenormalizationgroup invariant objects.Secondlya grand unified model needsa massscaleas input.This is the scalewherethe grandunified groupbreaksinto the SU(3)x SU(2) x U(1) subgroup.It is atthis scalethat we expectanalgebraicrelationbetweenthe couplingconstantsdictatedby grouptheory.Since the couplingsg3 and g2 are quite different from each other, 100 GeV, one should not be verysurprisedif the scaleof the SU(5) breakdown(the grand unification scaleM~)is very different from100GeV. The discussionof theseissues is somewhat model independentand we will thereforeintroducethe model first andthen discussthesequestions.

As a gaugegroupwe haveSU(5)andthis implies the existenceof 24 gaugebosons,12 of which willcorrespondto the gluons, the W”, Z and photon y. The additional 12 gaugebosons,which underSU(3)X SU(2) transform as (3, 2) + (3,2) are supposedto receive a massof order g5M~through theHiggs mechanismin SU(5) breakdown.The fermions of the standardmodel should be assignedtomultiplets in thegrandunified group.It is the greatbeautyof the SU(5) model that this canbe donein away without predictingnew fermions in the low energyspectrum.In this sensethe SU(5) model is theminimal model. The fifteen Weyl fermionsof one generationof quarksandleptonscan be assignedtotwo irreducibleSU(5) multiplets a 5 (the complexconjugateof the fundamentalrepresentation)and a10 (a two index antisymmetricrepresentation)

/0 ü ü u dI Oüud

(5)a = (d1, d2, d3, e, Ale), (10)’~’= I 0 u d (4.12)Oë

\ 0

H.P. N/lies, Supersymmetry,supergravityandparticlephysics 43

where a = 1, . . . , 5 is an SU(5) index and SU(3)X SU(2)X U(1) is embeddedin an obviousway, thehyperchargegeneratorY correspondingto the generator

~—i 0

if —1—1 (4.13)

0

of SU(5).Formula(4.12)gives the contentof one family, the additionaltwo areobtainedby repetition.The assignmentof quarksand leptonstells us that both baryon (B) and lepton (L) numbercan nolonger be conservedin this grandunified model.Quarkswith B = 1/3, L = 0 andleptonswith B = 0 andL = 1 are membersof the sameSU(5) multiplets andthe SU(5)gaugebosonscausetransitions.Theseare of course those gaugebosonsthat are in the coset SU(5)ISU(3)xSU(2)x U(1) and which willbecomemassiveafter SU(5) breakdown.An examplefor an interactionthat leadsto proton decayisgiven in fig. 4.1, where i, j = 1, 2, 3 denote color SU(3) indices. This graph leads to proton decayp—~.e~ir°.Thereis actually aselectionrule for proton decayin the simple SU(5) model.The reasonforthis is a nonanomalousglobal U(1) symmetrythat occursbecauseone family of quarksand leptonsismade up of two irreducible representationof SU(5). With respectto this symmetry the 5 haschargeX = —3 whereasthe 10 transformswith X = 1. This leadsto

B—L=~(X+4Y) (4.14)

asa globalsymmetry andthedifferenceof baryonandleptonnumberis conservedin the minimal SU(5)model andprotondecayleadsto final statesthat contain anantilepton.This conservationis a propertyof the simple model, it is no longer true in more complicatedmodels.The property of protondecay,however, is in principle common to all grand unified models. The long lifetime of the proton‘r � (1030_1031)yearsleadsto a lower boundof the grandunificationscaleM~of M~� 1015 1016GeV,indeedan enormousscalecomparedto the massscaleof the breakdownof the weak interactions.

So far we haveonly discussedthe gaugebosonandfermion spectrumof the model. Scalarsare alsoneededto trigger the breakdownof SU(5)—* SU(3)X SU(2)X U(1) and of SU(2)X U(1) to U(1)em. A

minimal setincludesan adjoint çb(24) representationof SU(5) (to break SU(5)) anda fundamentalH(S)representationwhich contains the scalardoublet of the standardmodel. The potential of the 24 isarrangedin away that 4, getsa vacuumexpectationvalue v of orderof the grandunificationscale

U”

xt

xL

~45

Fig. 4.1. Proton decaydueto heavySU(5)gaugebosons:p-~e~sr°.

44 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

/1 0/1

v 1 —~ (4.15)

0 —~

which breaksSU(5) to the desiredsubgroup.As a result12 of the 24 gaugebosonswill receiveamassandthe remainingscalarsalsohavea massof the orderof M~.The SU(2)doubletin H5 is usedto breakSU(2)x U(1) at a scale of 100GeV. The Yukawa couplingsof H5 to quarksand leptonsare giventhrough

gy~,.,,Hs~5a10a+ gyabHslOa10b (4.16)

(a, b = 1, . . . , 3) are generationindices.The first term in (4.16) gives massesto the electronandthe dquarks,whereasthe secondtermgivesmassto theu quark throughthe vacuumexpectationvalueof theSU(2)doubletin H

5 after the breakdownof SU(2)x U(i). In addition H5 containsa color SU(3)triplet.SincethroughtheYukawacouplingsin (4.16)thistriplet leadsto protondecaythroughgraphslike theonein fig. 4.2thiscolortriplet hastobesufficiently heavy.TherelevantYukawacouplingsin (4.16)aredictatedby themassesof quarksandleptonsandaresmallerthanthe SU(S)gaugecoupling.As a resultthelowerboundon theH3massesis smallerthantheonesfor thegaugebosons:lO~~_lO~GeV.In theminimalmodelthey receivea massof orderof M~throughtermslike

m2H

5H~+ m’H54,24H~, (4.17)

wherem andm’ areof orderof M~.Noticethattwotermsarenecessarysince(4.17)hasto givealargemass

to thetriplet andatiny massof orderof 100GeV to thedoublet in H5. Replacing4, by its v.e.v. oneneeds

m2— ~m’v4 m2, m’2, v2, (4.18)

which canbe obtainedby carefully adjustingtheparameters.This completesthedefinitionof theminimalSU(5) model. —

SU(5) breaksto SU(3)x SU(2)x U(i) atM~andatM~wehaveg5 = g3 = g~= \/5/3g1.Thefactor(5/3)1/2

occursbecausethehyperchargegeneratorY asdefinedin (4.13)wasnot normalizedtoTr Y2 = 1/2 mainly

for historical reasons.Below M~the gaugecoupling constantsevolve according to the expressions(a,= g~/4ir)

U’

Fig. 4.2. ProtondecaythroughHiggs scalarsin thefundamentalrepresentationof SU(3).

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 45

l/a,(p.)= 1/a~(M~)+(biJ2lT) log(M~Ip.), (4.19)

with the f3-function definedas f3, = b.g~/161T2at the one-looplevel. The b are given in generalby

b = — ‘W + ~N,l,(R)+ ~N~l~(R), (4.20)

where C is the quadraticCasimir operatorof the adjoint representation,N,,. is the numberof Diracfermions(complexscalarbosons)in representationR of which lf.S(R)is thequadraticCasimir(equalto 1/2for the fundamentalrepresentation).For the minimal SU(S)model we obtain

I. — ii 4.~r i. — ~ 4~r lr.r — 4s..r 1~j’U3~11+31VG, V23+31VG+611, Uy3JVG+1QII,

whereNG is the numberof generations(4.3) andH is thenumberof light Higgsdoublets(4.1). As aresultthecouplingconstantsg, evolvedifferentlybelowM~.It is thegreatsuccessof thismodelthatwith anactualcomputationof this evolution oneobtainsthe correctcouplingconstantsg,(M~)in the standardmodelprovidedonechoosesM~—~1014_1015GeV anda

5 -= (41ti—~s)[272].Thusthe two estimatesof M~,onefrom

thelimits of lifetimeof theproton,theotherfromtheevolutionof thecouplingconstants,leadto enormousmassscaleslike 1015GeV. Recentupperlimits on protondecay,however,seemto indicatethat thesetwoestimatesseemto be inconsistentfor the minimal SU(5) model [56]. We will not commenton thesequestionsherebut will comebackto it in theframeworkof supersymmetricmodelswherethingschangequantitatively.

4.3. Supersymmetryand the massscaleofthe weakinteractions

We nowwant to focusour attentionto the sectorof thestandardor grandunified modelsin which themagnitudeof thescalev —~250GeVof thebreakdownof theweakinteractionsis introduced.This isdoneinthe Higgs sectorwhich exclusivelyconsistsof scalarparticles.Noneof theseparticleshasbeendetectedexperimentallysothatthissectoriscompletelyunchecked.In thestandardmodelthev.e.v.of v = 250GeVis insertedby handthroughthe parametersp.2and A (compare(4.7)). The massof the remainingHiggs(_2p.2)u2is completelyundetermined.Of coursep.2 cannotbearbitrarilylargesincewe knowthat —p. 2/Ahasto be (250GeV)2andlargep.2 would imply largeA which would meanthat our perturbativesmallcouplingconstanttreatmentof themodelwouldbeinadequate[538,540].We thereforethink thatp. shouldnot be much largerthanlet ussaya TeV.

In agrandunified model the parametersp.2 andA areput in by handin the sameway. Actually heresomesubtletieswith the definitionof p.2 appearbecausethe Higgs is in a SU(S)multiplettogetherwithacolor triplet that necessarilyhasto be heavy. To havep.2 of the order of (100GeV)2oneactuallyhasto “fine tune” the parametersof the model (compare(4.18)) to an accuracyof p.2/m2~ 1O~.

Independentof this problemwe arein a situationthat in both modelsthe parametersp.2 andA areintroducedby hand andwe do not understandwhy v 250GeV. In particularwe do not understandwhy v is so tiny comparedto the grandunificationscaleM~or moregenerallythe PlanckmassM~.In amore completetheory one would like to understandwhy this cale is sosmall comparedto the otherscales. Looking at models with gaugebosons or fermions we can imagine mechanismsthat keepparameterssmall: gauge symmetriesand chiral symmetries.These mechanismsprovide in generalareason that the correspondingparametersare zero in the symmetric limit. The smallnessof theseparameterscoming from a breakdownof the symmetries,e.g. the SU(2)x U(1) breakdownof the

46 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

standardmodel, is necessaryto allow massesfor the gaugebosonsand fermions.Models in whichextremelytiny quantitiesareunderstoodin thisway areoften called natural[531].

In the caseunderconsiderationwe havev2/M2 ~— iO~,a very small quantity which suggeststhat tofind an understandingof v onefirst shouldtry to understandthe casev = 0 in termsof a symmetry.Ifwe nowtakethe standardmodel with p. = 0 we realize that thisdoesnot increasethe symmetriesof thesystem.Themodel is not naturalin the sensespecifiedabove.This hasas oneconsequencethat p. = 0 isnot stable in perturbationtheory. In fact the parameterp.2 receives quadraticallydivergent con-tributions in perturbationtheory.This is no problemsince p.2 is a free input parameterand it is therenormalizedquantity that is of relevanceandcan be chosenat will.

But let usnow considera naturalmodel,wherea symmetry keepsp.2 = 0. The radiative correctionswill respectthis symmetryandp.2 = 0 will be valid in anyorderof perturbationtheory. If this symmetryis brokenwith parameter~m2 the radiativecorrectionswill be finite andp.2 —~ a~m2will be generatedand still understood.If ~m2becomesinfinite p.2 will becomequadraticallydivergentand the value ofthe bareparameterloosesits meaning.

One might now try to find a symmetrythat could renderthe standardmodel natural.The quantitythat should vanish in the symmetriclimit is the massterm p.2hh* of the Higgs doublet,and the onlysymmetrywe knowof that cando thisis supersymmetry[542,569]. This is thebasicmotivationfor us toconsider the application of supersymmetryto models of particle physics. In the framework of asupersymmetricstandardmodel we would havep.2 = 0 andthis would be stablein perturbationtheoryas we haveseenin Chapter2. This comesfrom the fact that any particular contribution to p.2 iscancelledby anothercontributioninvolving the partnerof thosefields that gave rise to the originalcontribution.If supersymmetryis broken this cancellation is no longer exact sincethe massesof thesupermultipletsare split by ~m and a well defined p.2 — a(~m)2arisesin perturbationtheory. Whatsupersymmetryessentiallydoesfor thesepropertiesof the scalarsis that it relatesthem to fermions forwhich we can have“reasons”why they aremassless.Supersymmetryjust tells usthat the propertiesofthesefermions are also valid for the bosons.In the following we will explore the possibilitiesof anaturalstandardmodel in the frameworkof supersymmetry.

4.4. Theparticle contentof a supersymmetric standardmodel [203]

A supersymmetricmodel hasthe samenumberof bosonicandfermionicdegreesof freedom,as wehaveseenin Chapter2. The standardmodel with threefamilies contains28 bosonicdegreesof freedom(12 masslessgaugebosons,2 complexscalars)and90 fermionicdegreesof freedom(45 Weyl fermions).In order to makethismodelsupersymmetric,new degreesof freedomhaveto be added,onecannotjustdo it through relationsbetweenmassesandcouplingconstants.Let us startwith the spin 1 bosons.Theyare required to havefermionic partners(the gauge fermions or gauginos)with the samequantumnumbers:an SU(3) octet, an SU(2) triplet and a singlet with Y = 0. None of the fermions in thestandardmodelhavethesequantumnumbersandthesegaugefermionshaveto be addedas new fields,and instead of the 12 gauge bosonsin the standardmodel we now have 12 masslessvector super-multiplets. The fermions of the standardmodel haveto be accompaniedby spin zero particles.Theycannotbe partnersof spin 1 bosonsbecausewe haveseenthat the only satisfactoryway to introducespin 1 bosonsis throughgaugesymmetriesand if such bosonswere partnersof the fermionsthis wouldleadto an enlargementof the gaugegroup. Apart from that gaugefermionsare alwaysin the adjointrepresentationsof a gaugegroup, which in particularis a real representation,a propertynot sharedbythe fermionsof the standardmodel.The fermionsarethusenlargedto chiral supermultipletsandevery

H.P. Ni//es,Supersymmetry,supergravityandparticlephysics 47

Weyl fermion hasa complexscalar as supersymmetricpartner.Finally we haveto discusssthe Higgsdoublet.Could it bethat the role of the Higgs is playedby the scalarpartnersof quarksor leptons?Aninspectionof the quantumnumbersshowsthat the Higgs doublet(with Y = — 1/2) transformsin exactlythe same way as the left-handed lepton doublets, e.g. (~“) (compare (4.2)). Unfortunately thisidentification cannotbe made.In the full model the Higgs has to receive a vacuumexpectationvaluethat breaksSU(2)x U(1) and with this identification this would be the partnerof the neutrino. In thisprocess the global lepton number symmetry would be broken and this would lead to disastrousconsequenceslike large Maiorana neutrino masses,a masslessGoldstoneboson, unacceptablylargeneutrino oscillations, etc. We have to introduce the Higgs doublet separatelyand with it itsfermionic partner.So we haveseenthat noneof the particlesof the standardmodel has a supersym-metric partner in this model.For every particlewe haveto introducea supersymmetricpartner.But itcomeseven worse. The fermionic partnersof the Higgs bosonstransformnontrivially under SU(2) xU(1) and render the particle content of the model anomalous.To cancel this anomalyan additionalHiggs doubletchiral superfieldwith Y = +1/2 hasto be introduced.A supersymmetricmodelwith justone complexHiggs doubletcannotexist.Supersymmetryrequiresafter the SU(2)x U(1) breakdowntheexistenceof chargedHiggs scalars,contrary to the caseof the standardmodel which only containedaneutralHiggs scalarto survive the Higgs mehcanism.

Thesupersymmetricversionof thestandardmodel thuscontainsthe vectorsuperfieldscorrespondingto the SU(3)x SU(2)x U(1) gauge symmetry, three (i = I,. . - , 3) families of left-handed chiralsuperfieldsU~,U,, D,, L~,E~(a = 1, 2 is an SU(2) index) and two Higgs left-handedchiral superfieldsH’~and Ha. A supersymmetricaction can be obtainedfollowing the constructionexplainedin Chapter2. This involvesthe waWa kinetic gaugemultiplettermsfor SU(3)X SU(2) X U(l), the chiral superfieldkinetic terms including the minimal gaugecouplings~ *~2~V

4, andthe superpotentialg for the chiralsuperfields.This superpotentialrepresentsthe supersymmetricgeneralizationof all those termsin theaction of the standardmodel that are not kinetic termsand that do not contain a gaugecoupling. Inparticularg shouldcontain the Yukawacouplingswhich generalizeto

g = AijU~”HaU~,+ A~jU~”H”~a&Dj+ A~L~’H”EabEj (4.22)

with the 5 d4x(fd2Og + 5 d2Og*) contributionto the action. (Thesymbol Cab is antisymmetric.)Notice

that we cannotinclude a term like U’1H”‘EabU in (4.22) since this term hashyperchargeY = —1 andwould thusexplicitly breakU(1)~.From (4.22) we can readoff aposteriori an additionalreasonfor theintroductionof H, which we can seeby the considerationof the termsthat are_responsiblefor themasstermsof the quarksandleptons.Taking 5 d2Og andchoosingH = (~),H = (~)for the scalarfields ofthe chiral supermultipletswe obtain

= A~1u1h°ü1+ A~,d,h°d1+ A~e~h°ë3. (4.23)

Without h°we would not be able to give massesto the up quark. In the nonsupersymmetriccasethemassof the u quark was obtainedthroughthe uhO*ü coupling,but in the supersymmetricversionthis isnot possible.The right-handedchiral superfieldH* cannotcoupleto the left-handedU, U superfieldsinthe superpotential.We simply needtwo left-handedsuperfieldH andH. Both h°andh°haveto receivevacuumexpectationvaluesto providemassesfor all quarksandleptons.

The trilinear couplingsin the superpotential(4.22) are not the mostgeneraltermsthat are consistentwith the SU(3)x SU(2)x U(l) symmetry.In fact we couldadd

48 H.P. Nil/es, Supersynimetry,supergravityandparticle physics

~ig= UaLbEa6D+ LaELbEab + UDD. (4.24)

We havenot includedthem in (4.22) becausetheseinteractionsleadto disastrousconsequences.Thefirst two termsin (4.24)violate lepton numberandthe last term violates baryonnumber.The last twotermscould combineto give protondecayvia the graphshownin fig. 4.3, with a ratethat is manyordersof magnitudelarger than the experimentallimits, as long as the couplingsin (4.24) are not extremelysmall. Anotherway out would be a very largemass(e.g. 1015 GeV) for the scalarpartnerof the d quark,but this would imply that supersymmetryis brokenwith masssplittingsof order 1015GeV andcould bedisregardedat low energies.The terms in (4.24) should thereforebe absent.This can be achievedbydemandingthe existenceof global symmetries,which can be done in different ways. One couldintroducethe B, L global symmetriesor discretesymmetries.In supersymmetrythereis in addition apossibility called R-symmetry[200,499]. R-symmetriesare symmetriesthat do not commutewith thesupersymmetrygenerators,unlike the casewith usual global symmetries.In the usualsymmetriesallcomponentfields havethe sametransformationpropertieswith respectto that symmetry.This is nolonger true in the caseof an R-symmetrywherealsothe superspacecoordinates0 get transformed.Asan examplelet us considera symmetry 0—~ e”’O, and a chiral superfield 4, with 4, —~e”‘4,. This impliesthat the scalarcomponentof ~ transformsin ~ e”‘ and the fermioniccomponenti/i is not affectedbythis R-symmetrysince (Ot/i)—*e”‘(Ot/i) and this is alreadyachievedby the 0-transformation.How doesthis affect the terms in the superpotential?Well, we know that 5d

2Og hasto be invariant and d20transformswith e2”, and this implies that the termsin the superpotentialhaveto transformwith e2”‘.For the caseunderconsiderationthis allows only a term quadraticin 4,, g = m4,2: all other termsareforbiddenby the R-symmetry.

In the supersymmetrizedversion of the standardmodel a suitableR-symmetrywould be the onewhereall the particles in the standardmodel (with two Higgs doublets)like gaugebosons,quarksandleptonsand Higgs scalarshavechargezero whereastheir supersymmetricpartners— the newly intro-ducedparticles to render the model supersymmetric.—transformnontrivially under the R-symmetry.Notice that gaugebosonsalwayshaveR = 0 sincetheyappearin the combination0o-,.Ov” in the vectorsupermultipletandif 0 picks up a phaseunderR-transformations0 receivesthe oppositephase,and Vthevectorsupermultipletis real. The abovementionedR-symmetrycorrespondsto R = 0 for the vectormultipletsand theHiggs superfieldsandR = I for the quarkandlepton superfieldsif 0 hasR = 1. Suchasymmetryforbids the termsin (4.24)andallows theterms in (4.22). It is actuallynot necessaryto havethisfull continuousglobal symmetryto do this. In fact a smallersymmetrycalledR-paritycould do thejob as well. R-parity hasa multiplicative quantumnumberR~and all particles in the standardmodelareR even (R~= +1) whereastheir partnersare odd. In componentfields this assignmentcorrespondsto

R~= (— I)38+t.±2S , (4.25)

-i:i

Fig. 4.3. A potentially dangerouscontribution to protondecaythroughthe exchangeof thescalarpartnersof theright-handedd.quarks.

H.P. N/lies, Supersymmetry,supergravityandparticle physics 49

where B, L, S denote baryon, lepton number and spin respectively.R-parity is a subgroupof thecontinuousR-symmetry and could remain valid even after the continuoussymmetry is broken. Inaddition to forbidding the terms in (4.24) thesesymmetriesimply that the newly introducedsupersym-metric partnerscan only beproducedin pairs.This in particularmeansthat the lightest R-oddparticleis stable. We will later come backto thesequestionswhen we discussexplicit models.Notice at thispoint that a continuousR-symmetryof the typeintroducedabovewould forbid Majoranamassesfor allthe gauginoswhereas R-parity does not. As we will see later masslessgluinos would lead to aphenomenologicalproblemandin generalthe continuousR-symmetrieshaveto bebroken in a realisticmodel.

We have not yet discussedthe Higgs potential in the supersymmetrizedversion of the standardmodel. The only renormalizableadditional term in the superpotential(4.22) involving the Higgs fieldsthat is consistent with the gauge symmetriescould be mHaHa. In the presenceof a continuousR-symmetrythis term would be forbidden.Evenin the presenceof such a term, however,this is notsufficient to inducea breakdownof SU(2)x U(1). This term leadsto m2hh* massesfor the Higgsscalarsand it doesnot give us the opportunity to introducem2<0. As a result the minimumof the potentialwill alwaysbe supersymmetricV = 0 and all vacuumexpectationvalues of the Higgs scalarsvanish. Ittells us that additional superfields have to be introduced in the model. One way to induce anSU(2)x U(1) breakdownwould be the introductionof a singlet superfieldY with superpotential

g= Y(HH—m2), (4.26)

which would lead to (h°)= m with unbroken supersymmetry.The question of the breakdownofSU(2)x U(1) in the absenceof supersymmetrybreakdown,however,is not the relevantquestionto askhere.The aim of our approachis to intimately relatethe breakdownof SU(2)x U(I) with the one ofsupersymmetry.In this respecttermslike mHH in the superpotentialareactuallynot desiredsincetheywould correspondto the explicit insertion of a massparameterof order 100 GeV in our model that isnot relatedto the breakdownof supersymmetry.We do not like to insert thesemassparametersin thetheory but would rather try to understandthem from supersymmetrybreaking. The questionof thebreakdownof SU(2)x U(1) shouldthusbe postponedtill we discussthe possiblemechanismsthat breaksupersymmetry.We havenot statedit in this section,but it should be obvious that a breakdownofsupersymmetryis anecessity.As it standsthe modelwould predictfor exampletheexistenceof chargedscalarswith the massof the electronandif theyexist they shouldhavebeendetectedexperimentallyalong time ago. Before we, however, addressthe questionsof supersymmetrybreakdown, let usintroducesupersymmetricgrandunified models.

4.5. Supersymmetric grand unifiedmodels [569, 143, 122, 449,490, 353]

Again we limit our discussionto the SU(5) model, generalizationsto other modelsare straightfor-ward. To render the model supersymmetricwe add gaugefermions in the adjoint representationofSU(5) to completethe vectorsupermultiplet.The threefamilies of quarksand leptonsreceivecomplexscalarpartnersthat form chiral supermultipletsY,(5) and X~(10)(i = 1, 2, 3). Three Higgs multipletsH(S), H(5) and 4,(24) are introducedas well. The action of this model is given by the usualkinetic andgaugecoupling termsand asuperpotential

g = g,1X,XH+g~X~rH+A5Hç6H+A2çb

3+ M4,2+M’HH, (4.27)

50 H.P. Nilles, Supersymmetry,supergravilyand particlephysics

whereSU(5)contractionsareunderstood,andall our remarksaboutB andL variation in the standardmodel apply as well. As in the standardmodel we areforced to introduceH andH. H alonewill notdo, both becauseof anomaliesandthe necessityto give massesto all quarksand leptonsthrough theYukawacouplingswhich representthe first two termsin (4.27).The remainingtermsdefinethe HiggspotentialwhereM and M’ areof the orderof the grand-unificationscalewhich will be specifiedlater.

SU(5) should be broken to SU(3)x SU(2)x U(1) at this scale.Supersymmetryis supposedto bebrokenat a scalesmall comparedto M~to haveany implicationson the low energysector.The Higgspotentialis given by

VH= ~ 2 2 2 (4.28)84, oH OH

and the (supersymmetric)minima of this potentialaregiven by

A1hh+3A2~2+2M~= 0, )t

1tph+ M’h= 0, A1çvh + M’h = 0, (4.29)

where h, Ii and ~ denote the scalar componentsof H, H and 4, and SU(5) indices have beensuppressed.Thereare manydegenerateminima VH = 0 of this potentialoneof which correspondsto abreakdownof SU(S)to SU(3)X SU(2) X U(1) with (h) = (h) = 0 and

0/1

~ I 1 (4.30)I —~

\ 0 —~

wherev is proportional to M/A2. Modifications of the model in which it is possibleto have(4.30) thelowest nondegenerateminimum [146,1491 will be discussedlater in the framework of explicit models.With this breakdownof SU(S)the componentsof H andH will in generalhavea massof orderof thegrand unification scale. In section 4.2 we haveseenthat this is very desirablefor the color triplets(antitriplets) in H(H) in order to suppressproton decay. The SU(2) doublets in H and H shouldhowevernot be supermassivesince theyconstitutea vital ingredientof the low energystandardmodel.As in the ordinary SU(S) model (compare (4.18)) we have to fine tune the parametersin thesuperpotential.To assurethe Higgs doubletsto be masslesswe haveto impose

M’ = ~A1v, (4.31)

which can be seenfrom the last two equationsin (4.29) by inserting (4.30).Two observationsshouldbemadehere.First of all we haveput this fine tuning at the treegraph level andthrough supersymmetrywe know that this relation remainsexact in all ordersof perturbationtheory [569], it hasnot to berepeatedlike it wasthe casein the nonsupersymmetricmodel.This hasled to the notion that sucha finetuning is “technically natural”. This doesnot meanvery muchat the momentbut oneshouldrememberthat wheneverone might be in a situationwhereone hasan argumentfor such a quantity to be smallthis nonrenormalizationof this quantity might be a very useful property. Secondly and more im-portantlyrelation (4.31) is an exactequality andthe Higgs doubletmassesarefine tunedto be exactlyzero, contrary to the casein the ordinary SU(S)modelwherethe two largetermsin (4.18)conspireto

H.P. Ni//es,Supersymmetry,supergravityandparticle physics 51

producea massscalethirteenordersof magnitudesmaller. Onehasthe feelingthat mIM~= 0 might beeasierto explain than m/M~= 10-13. Relation(4.31) might be the consequenceof the Clebsch—Gordan[449,129] coefficientsof the groupsunderconsiderationwhereasit is hard to imaginethis to producethe number1O’~

Severalsuggestionshavebeenmadeto obtainmasslessHiggs doubletswithoutan explicit fine tuningof the parametersof the models,andwe will discussthem here.The first is called the “sliding_singlet”[570,332] mechanism.One introducesa singlet superfieldZ = (z, t/r) and addsa term AHZH to thesuperpotential.The last two equationsin (4.29) readnow

A1çoh+Azh+M’h= 0, A1hço + Ahz+ M’h = 0, (4.32)

wherethe vacuumexpectationvalue (4.30)is fixed by the first equationin (4.29).Supposenow that ina completemodel supersymmetryis broken at a certainscale and that the Higgs doublets receiveavacuumexpectationvalue of 100 GeV, by a mechanismwhich is not yet containedin our model. Thesquareof the left handside of (32) gives a contributionto the potentialand in order to minimize thiscontribution in the case(h2) � 0 the singlet would developa vacuumexpectationvalue (it slides) tomake—~vA1+ A(z)+ M’ = 0, and this would alsoimply that the contributionto the Higgs massof thisterm vanishes.Investigating additional terms that might give a contributionto this massrevealsthatthey can at most be of the order of the (small) vacuum expectationvalue of (h). This mechanism,however,hasonedrawback.Onehasalreadyassumedthat theHiggs v.e.v. is smallandnonzero.In thecontext of explicit models,even if such a local minimum (with (h) � 0 and small) exists, the globalminimumof such a modelmight prefer to have(h) = 0 or large h v.e.v.’s. It hasbeenproven to be verydifficult to makethis mechanismto work in a satisfactorymodelexactly for this reason,as we will seelater [130,442, 148].

A second method to obtain light Higgs doubletswithout fine-tuning is the so-called “missingpartner” mechanism[129,291,416, 70,408]. It is basedon purely group theoreticalarguments.Onelikes to constructasituationwherea direct massterm mH5H5doesnot exist in the superpotentialandwherethe color triplets mix with other SU(S)representationsto becomemassive.Theseother SU(5)representationsshould have the property that this same mass mixing is not possible for the SU(2)doublets.The simplestcasein the frameworkof SU(5)that exhibitsthis mechanism[291,416] requiresthe introductionof a 75 (insteadof 24) and a 50 and 50 representationof SU(5). The 75 can be usedinsteadof the adjoint to break SU(S) to SU(3)x SU(2)x U(1). It is a four index representationwith(75)~,= —(7S)j~,= —(7S)~and(7S)~.= 0. With respectto the SU(2) x SU(3)subgroupit decomposesinto(1,1)+(1,3)+(2,3)+(1,3)+(2,3)+(~~_6)+(2,6)+(1,8)+(3,8). The SO decomposesin (1,6)+(2,8)+(1,1)+ (2,3)+ (3, 6)+ (1,3) and the SO is the conjugateof 50. Notice that the 50 contains a (1,3)representationhut no (2, 1). The superpotentialreads

g = A(75)3+M(7S)2+ A

150 X 75 X 50+ A250X 75 X H+ A35O x 75 X R+ Mso x ~. (4.33)

At the minimum the (1, 1) componentof 75 receivesa v.e.v. The terms with A2 and A3 provide thenmasstermsfor the color triplets in H andH in mixing them with the triplets in the50 and50 leadingtoa massmatrix

(A~v ~ (4.34)

52 H.P. Nil/es, Supersymmetry,supergravityandparticle physics

Nothing happensto the Higgs doubletssince the 50 and50 do not contain SU(2)doublets(which areSU(3)singlets). It wasof coursecrucial that a term MHH was omittedin (4.33). A furthercomplicationof the model can, however,lead to the situationwheresuch a term is forbidden by a global symmetry[291].The missing partnermechanismrequiresthus(at least in SU(S)) the introductionof ratherlargeSU(5) representations.In contrast to the “sliding singlet” mehcanism,however, it is universallyapplicable.

We now haveto discussthe magnitudeof M~in supersymmetricgrand unified models.The particlecontent of the low energysector has changedcomparedto the ordinary SU(5) model discussedinsection4.2 andthis will certainly influencethe evolutionof the couplingconstants.It might evenbe thatthe threecouplingconstantsg1, g2 andg3 do not meet all threeat the samemassscalewhich wouldbedisastrousfor the conceptof supersymmetricgrandunification. The generalformulafor theevolutionofthe couplingconstantshasbeengiven earlier (see(4.19) and(4.20)) and we computein the supersym-metric case[125,154]

b3= —9+2NG, b

2= —6+2NG+H/2, b1= 2NG+~jH, (4.35)

whereNG is the numberof generationswhich we taketo be 3 andH is the numberof Higgs doublets.The introductionof supersymmetricpartnershasincreasedthe coefficientsb,, e.g. b3 haschangedfrom—7 to —3. This slows down the evolution of the coupling constantg3. Notice, however, that theadditionalparticleshaveroughly asimilar effect on all threeof the coefficients,so that it is still possiblefor them to meet at the samepoint. To determinethe grand unification scalewe use the standardformulas

sin2 0~= (1/d)[(b

2 — b3) + (b1 — b2)5a/3a3], (1/2ir) log(M/p.) = t = (1/d)[1/a — 8/3a3],

1/as= (1/d)[((b2 + ~by)/a3)- b3Ia], d = b2 - b3+ ~(b~— b3), (436)

where we shall use a3(p.)= g~(p.)/4ir 0.1 and a~(p.) 128 for p. 100GeV. With the value of0.2� sin

2 0 ~ 0.24 this leads in our minimal supersymmetricSU(5) model to M~ 2 X 1016GeV andg

5 1/25. We see that M~hereis more than an orderof magnitudelargerthan in the ordinarySU(5)model. This has, of course,consequencesfor the predictionsof protondecayin this model. Since theproton decayrate goes with M

4 this model predictsproton decaywith a lifetime of iO~~—iO~~years,beyondthe presentexperimentallimits. In particularthe supersymmetricminimal SU(5)modeldoesnotyet encounterthe problemsof the minimal SU(5)modelwith regardto the experimentalfindings. Twowords of caution should be addedhere. The statementabove is valid for the minimal model and anintroductionof additional particleswith massessmall comparedto M~will changetheseresults. Forexamplewith the introductionof additional5 + 10 Higgs representationsat an intermediatemassscaleitwas shown thatM~can be lowered [335,415]. The statementthata given supersymmetricmodel leadsto a larger M~than its nonsupersymmetriccounterpart(which is obtainedby deleting the partners),however, remainstrue. Secondlyour statementsaboveare only valid for proton decaymediatedbygaugebosons.We hadalreadyseenthat protondecaymight occurdue to the exchangeof Higgs bosonsof mass lO~GeV. The Higgs bosons in our model are sufficiently heavy. In the context of asupersymmetricgrand unified model there is howevera new mechanism: “proton decay throughdimensionS operators”[553,493]. In ordinary SU(5)we haveproton decaythrough a processwheretwo quarksinteractto give an antiquark and antilepton in the final state.This processinvolves four

H.P. Nil/es,Supersymmetry,supergravityandparticle physics 53

fermions: a dimension6 operator,and we concludethat the amplitude is proportionalto 1/M~fromdimensionalconsiderations.These contributions also exist in the supersymmetriccasebut they aresuppressedsinceM~is larger. In supersymmetryonecould in addition havea processwhere two quarksannihilate to the scalarpartnersof an antiquarkand antilepton(fig. 4.4), a processthat involves twofermionsand two bosons:a dimensionS operator.Consequentlyits amplitudeis proportionalto 1/Ms.Of course,the scalars~~ and~ ~havein a secondstepto producetheir partners~ ande but this couldfor examplehappenthrough the exchangeof an SU(2)gauginowithout involving a propagatorof massM~.As a result this decaythroughdimensionS operatorsis suppressedby only one power of M~.Westartour discussionwith the enumerationof the dimension5 operators

(UUL*)D, (UD*E)D, (UUUL)F, (UUUH)F,(4.37)

(UUEH)F, (UUDE)F, (LLHH)F, (LHHH)F,

in our usualnotation where U, L denotethe quark, leptondoublet chiral superfield.D andF denote5 d~0and 5 d20 respectively.We haveto recall now that in the supersymmetricmodel in generalalsodimension4 operatorsoccur(compare(4.24)), but they are forbiddenbecauseof either continuousordiscrete R-symmetrieslike, e.g., R-parity. These symmetriesexcludeautomaticallyalso some of theoperatorsin (4.37) andwe remainwith (UUUL)F and(UUDE)F asthe only alloweddim-S operators.Let us now first consider(UUDE)F which reads

(Ua(J.~DkcEIE~2~)F, (4.38)

wherea, b, c areSU(3) indicesand i, j, k, 1 label the generations.The first two fields areantisymmetricin a andb andobeybosestatisticswhich implies i � j andexpression(4.38) includesat least onecharmor top quarksuperfield.Thesesuperfieldsare SU(2)singletsandcan thereforenot changeflavor in thesecondstepthat is requiredfor proton decay,so that (4.38) can only lead to proton decayin either acharmor a top quark.This leaves(UUUL)F as the only possibility to induce proton decay,writtenexplicitly

(U~U~U~cL~E’~”ErsS~)F (4.39)

with SU(2) indicesr,. . ., u. U is an SU(2)doubletand containsthe superfield a andd, howeverthreeU’s appearin (4.39) which thenvanishesif i = j = k. Again particlesof at leasttwo generationshavetoenter(4.39) to assurea nonvanishingresult [126].Notice further (4.39) correspondsto a processwithfour ingoing left-handedfields. The simplestgraphfor this processcan be written as shown in fig. 4.4

Fig.4.4. Protondecaythroughdimension5 operators.The first stepinvolvesthe exchangeof aheavyHiggs (gauge)fermion andtransformsquarksintopartnersof antiquarksandleptons.

54 H.P. Ni//es,Supersymmeiry,supergravityandparticlephysics

with the exchangeof a fermion of massM~.In principle this could beeithera fermionicpartnerof theHiggs triplets or a superheavygaugefermion. But gaugeparticleshavehelicity conservingverticesandas such cannotlead to a processlike (4.39) and we are left with the contributionof the Higgs tripletfermions(fig. 4.4).

The secondstep involves the transformationof the scalarsto their fermionic partnersthrough theexchangeof a light particle, which again could be either gaugefermions or Higgs fermions (here thedoublet).Since, however,the gaugecouplingsare largerthanthe Yukawacouplingstheformergive thedominant contribution, and we have to consider gluino, wino (W-ino) and bino (B-mo) exchange[126, 179, 169, 11, 492, 76]. Denoting(4.39)by °ijkl oneobtainsfor thecorrespondinglyinduceddimension6 operatorsthrough the gluino [76]

O~= —~[f~(O~1+ OJ,kt) — !~~Ok,~I— f~f2OkJ~I], (4.40)

the wino

= _~[3(f~7)+ f~ iJkI + OJIkI) — (f~i)+ f~“~)(2O~kJj+ Ok~J,)— (f~~)+ f~~)(24k,,+ OkJ,I)]

(4.41)

andthe bino

O~= ~[(f~ — 3f~/Y)(O,Jkl+ O1,k,)— (f~— 3f ~Y1)Ok,Jl — (f~— 3f$’~)Ok~,t] , (4.42)

whereg is the gaugecouplingand m the Majoranamass of the gaugefermion underconsideration.generation.This assumptionallows us to write °ijkl in terms of the SU(3)x SU(2)x U(1) invariantoperators0. The functionsf dependexplicitly on the massesinvolved in the process

g2m F m~ /m2\ m~ 1m2\1= 32ir~(m~—m~)Lm2_ m~log-~)— m2—m~log(\~,)j (4.43)

where g is the gauge coupling and m the Majoranamassof the gaugefermion under consideration.Notice that the assumptionof a mass degeneracyof all partnersof quarks and leptons (even fordifferentgenerations)would lead to vanishingcontributionsin (4.40) and (4.42).Proton decaythroughgluino and bino exchangeis proportionalto the massdifferencesof the scalarpartnersof quarksandleptonsin differentgenerations.

Let us now try to find the decaymodesof protondecayvia this mechanismandtry to estimatetherate. In a first stepwe would like to neglectthe Kobayashi—Maskawamixings (4.11). We hadalreadyseenthat in OijkI not all i, j andk can be equal,particlesoutsidethe first generationhaveto entertheprocess.The c quark is ruled out sinceit is heavierthan the proton.

What remainsare processesthat include the u, d, s quarksandthe v,. neutrino.The muon cannotappearbecauseof the absenceof the cquark.This gives as dominantproton decaymode[126]

p—~K~i,. (4.44)

for examplethrough the graphin fig. 4.5. The rate is determinedby the Yukawa couplingsresponsiblefor the massesof the u and s quarks,two gaugecouplings,the massM~of the Higgs fermion andthe

H.P. Nil/es,Supersym,netry,supergravityandparticlephysics 55

____

(g,B)

____

Fig.4.5. Diagramfor p—~K~i~.

Majoranamassof the gaugefermionwhich we assumeto be of the orderof 100GeV. This leadsto arate of order 1O~°yearsfor the processin (4.44). Up to now we haveneglectedthe Cabbibo mixingsbetweenthegenerations.If weincludethem and assumethat the mixingsbetweenthethird andthe firstgenerationarenot too small, the dominantcontributionmight comefrom graphswherescalarsof thethird family appearin the loop. This could be dominantbecausein the first stepof the processtheYukawa couplingsenter.They are adjustedto give the quarksandleptonstheir massesandsince theparticles in the third generationare heavy thesecouplingsare large. It could thus be that the modep—*K~i’,.is dominant[179].This, however,dependson the mixing anglesbetweenthefamilies whichwedo know only partially. In addition it seemsnot to be too easyto distinguishbetweenthe modesK1,.

and K~experimentally. Nonethelessthe K1 mode is the dominant proton decay mode in thismechanism.This is interestingbecausethereis no other model which predictsKi as the dominantmode. An observationof this mode in absenceof othermodeswould thereforefavor a supersymmetricinterpretation.Observehoweverthat this proton decaythrough dim-S operatorsdoesnot necessarilyhappenin a supersymmetricgrandunified model. It might very well be (andhappensin a lot of specificmodels)that thereexist accidentalglobal symmetrieswhich excludethe dim-S operatorsjust as thedim-4 operatorsareforbidden.In this casethe dominantdecaymodeswould be given by gaugebosonexchangewhich predicte~ir°as the dominantmode in this model. In addition one could haveprotondecayvia Higgs exchangewith mass of lO~GeV and dim-6 operatorswhich seemsto favor the Kp.modes.Here,however,onehasthen to forbid the dim-5 operators.

Let us finally mention two resultsin connectionwith this discussion.The contributionsto protondecayvia dim-S operatorswe havediscussedso far vanishif supersymmetryis unbroken[393].Secondlythereare additional mechanismsof this type that involve the auxiliary D-fields of the gauge super-multiplets [111]which we cannotdiscussherein detail.

4.6. Constraintsfrom rare processes

If supersymmetryis relevant for the solution of our theoreticalproblemsin the mass range of100GeV, we should find deviationsfrom the ordinarymodels in this energyrange.We havenot yetdirect experimentalaccessto this region in the sensethat the particlesof highestmassproducedin thelaboratoriesso far are lighter than 100GeV. But we know alreadysomethingaboutthis energyregionand this knowledge comes from extremely accurateexperimentsat lower energies.One of theseexperimentsis for example the measurementof proton decay which in principle could give usinformation aboutan energyregion of lOls GeV. But therearealso otherexperimentsthat can tell uswhat might happenor not in the region abovehundredGeV. Theseinclude limits on flavor changingneutral currents, (g — 2) of the muon, KL—KS mass difference, e—* p.y measurementsand similarprocesses,We considerheremodelswherenew particlesexist in the region beyond1000eV andin thischapterwe want to explorethe constraintson the massesandcouplingsof thesenew particles imposed

56 H.P. Nil/es,Supersymmetry,supergravityand particlephysics

by the knowledge of the results of the above mentioned rare processes.This we will do in theframework of the most generalmodel that satisfies our theoretical requirementsand as we haveexplainedthis is a modelwith soft explicit breakdownof supersymmetry[122,490]. This model includessupersymmetrybreaking Majorana mass terms for the gauge fermions, mass terms for the scalarpartnersof quarksandleptonsandthe allowedsoft trilinear couplings.We alsoincludean explicit massterm for the Higgs fermionswhich in the context of the modelswe consideris soft, althoughit is notsoft in general. Sucha version of a “supersymmetric”standardmodel or grandunified model can lead(with certain restrictions on the parameters)to a satisfactory model with no phenomenologicalproblems, but it has of course many arbitrary breaking parametersthe origin of which is notunderstood.We will herejust considerthis model to derive the constraintson this modelthat can makeit phenomenologicalacceptablegiven the presentexperimentalinformation, andwith that we will coverthe mostgeneralcasethat can occur in the contextof spontaneouslybrokensupersymmetry.

The boundson the existenceof chargedscalarsin e~eannihilation experimentstell us that all thepartners of quarks and leptons (with exception of the neutrinos) must be heavier than 20 GeV[529,4, 32, 52] and this bound, of course, also applies in the case of the chargedHiggs fermions.Available lower limits on gaugino massesare more model dependent,since they involve indirectmeasurementslike beamdump experiments[27].For thegluino this leadsto mg>(1—5)GeVdependingon the massesof the quarkpartners.No direct limits on Majoranawino, zino andphotinomassesareyet known.

Additional indirect limits on the parameterscan be obtained from the above mentioned rareprocesses[176, 44, 337, 524]. We will restrictour discussionhereto theprocessesthat give the strongestconstraints.These are p. —~ ey for the lepton sector and the KL—KS systemfor the quark sector. Thelimits on KL—*p.p., KL—*p.e, p.—*eëe or p.N-~*eNare in generalnot violated if the parametersof themodels fulfill the requirementsimposedby p. —* ey and KL—Ks. The quantity (g — 2) vanishesin thesupersymmetriclimit [232]andhardly gives constraintsin the brokencase.

We start with the KL—KS system. In the nonsupersymmetricmodel this receivesa contributionfromdiagramsas in fig. 4.6, and onehasto sum over all possible intermediatequark states.Independentofthe KM angles(aslong as thematrix staysunitary),weknow that this contributioncancelsif the quarksunder considerationhavethe samemass.In a more generalcontext flavor changing neutralcurrentsvanishif quarkswith sameisospin andsamethird componentof isospinaredegeneratein mass.In thesupersymmetricmodel thereare now additionalcontributionscorrespondingto the samebox diagramswhere the quarks in the box are substitutedby their partners and the gaugebosonsby the gaugefermions (fig. 4.7). Since the partnersof u, c, t havethe samequantumnumbersas u, c, t again thiscontributioncancelsif ~, ~, q~havethesamemasses.The smallnessof theKL—KS massdifferencewillthus not give constraintson the magnitude of the massesof the scalarsbut will constrain theintergenerationalmassdifferences.Estimatesof thisprocesscan easilybe obtainedin the limiting cases

S u,c,t d s ‘P,.’4’~.4’~ d

1w ~ _ _

S u,c,t d S tp~,tp~,tp~ dFig. 4.7. KL—KS mixing through the exchangeof charged gauge

Fig. 4.6. KL—KSmixing throughw-exchange. fermions.

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 57

mA 4 m~or m~4 mA, where mA (ms)denotethe massof the gaugefermions(scalarpartnersof quarks).Oneobtainsfor m~4 mA contributionsof the form

(a~sin29/m~)(~m~/m~)2, (4.45)

andfor mA ~ rn~

~ sin2 0(~m~)2/m~’, (4.46)

where a~is the weak gauge coupling, sin2 0 denotesthe relevantKM angles and ~ denotesthedifferenceof the squaredmassesof the scalarsunderconsideration.The generalformulaearegiven inrefs. [176,44, 144,390] andwe will not repeatthem here.Using the experimentalvalueof the real partof the K

1—K2 particles in connectionwith the usualvaluesfor the Cabbiboanglesoneobtains

2 2 (~~2 ~ 10~GeV2, (4.47)mAorm. m~j

a ratherstrong constrainton the massdifferenceof the scalars.Notice, however,that this boundonlyappliesto the massdifferencesof the u, c, t scalars.The massdifferenceof, for example,the u anddscalarsis not constrained.The boundgiven in (4.47) can be strengthenedsince in general therearecontributionsto the L~S= 2 processthrough gluino exchangeas shownin fig. 4.8. A nonvanishingresultof this contributionrequiresthe massmatricesof the scalarsto haveoff diagonalelementsin the basiswhere the correspondingquark mass matrix is diagonal,a situation which, however, is possible ingeneral and which will actually be present in various models. This gives constraintson the massdifferencesof the down (d, s, b) scalars.Since herethe strongcoupling constanta~is involved and

1Oa~,at 1000eV one arrives at a bound like (4.47) with 10_8 GeV2 at the right hand side.Assumingfor themoment mA to be 100 GeV thisleadsto ~m~/m~~ ~ as a constraintfrom the realpartof the K

1—K2 massmatrix. The supersymmetricpartnersof the quarks(separatelyin the up andthedown sector)should thereforebe highly degeneratein all realistic models. In addition we have aconstraintfrom the imaginary part of the K1—K2 massmatrix. Potentially this bound is strongerby 3ordersin magnitudecorrespondingto the one in (4.47). It requiresthe massmatrix of the scalarsto becomplex in the basis wherethe correspondingmass matrix of the quarksis real. This can happeningeneralbut it is not likely to happenin the modelsthat havebeenconsideredso far in the literature.We will seethis in moredetail in the following chapters.So far our discussionof the constraintson themassdifferencesof the partnersof quarks.As we havesaid,the constraintsfrom the KS—KL systemgivestrongerboundsthan thoseof otherflavor changingneutralcurrentprocesseslike KL—* p.~p.~

S ~s.14~d’~b d

___________

9

~

Fig. 4.8. i~S= 2 processthroughgluino exchange.

58 H.P. Ni//es, Supersyinmetry,supergravityand particlephysics

Fig. 4.9. Supersymmetriccontributionto ~ -+ ey.

Next we cometo p. —* ey which will give us a constrainton the mass2differencesof the partnersofleptons.A contribution to this processin the frameworkof a supersymmetricmodel is shownin fig. 4.9.A nonvanishingcontributionof this diagramagainrequiresthe massmatrix of the e, p., T scalarmassmatrix to haveoff-diagonal elements,which of coursecan happenin the most generalcase.Assumingthesemixing angles to be of the order of the Cabbiboangleone can derive a bound like the one in(4.47) now for the leptonsector.With the boundof the p. —* ey branchingratio 1.9X 10b0 one obtains[176]

2 2 <1O~GeV2, (4.48)mSormA \ m~/

which leadsto ~ ~ 1O3m~andthusalsorequiresa nearlyperfectdegeneracyin massof thepartnersof the charged leptons. We have thus seen that existing experimental information gives strongconstraintson the massdifferencesof the scalar partnersof quarksand leptons.The separatesectorsshouldbesufficiently degeneratein massto allow fora GIM mechanismto suppressflavor changingneutralcurrents.

This concludesour discussion of the rather model independentconstraintson supersymmetricmodels.Other constraintslike the one of CP violation and the electricdipole moment of the neutroncannotbe discussedin a model-independentway andwill be later discussedin the frameworkof specificmodels. Up to now our discussionhasbeenindependentof a specific breakdownof supersymmetrywhich has to be incorporatedin the models. Our framework till now has been an explicit softbreakdownof supersymmetrywith arbitrary breakingparameterswhich representsthe most generalpossibility.The discussionin this subsectionhas,however,shownthat thesearbitrary parametershaveto be constrainedbecauseof alreadyexistingexperimentaldata.It is hopedthat suchrestrictionscouldbe explained by a more constrainedway of the supersymmetrybreakdown, like a spontaneousbreakdown.One would of course also like to understanda relation between the breakdownofsupersymmetryandthe breakdownof the weak interactionsas a final goal.

5. Global supersymmetry: models

In this section we will discusssupersymmetricextensionsof the standardmodel and grand unifiedmodelsin which supersymmetryis brokenspontaneously.Theoreticallythis is consideredto be moresatisfactorythan just the introductionof explicit breakingparameters.Hereone startswith manifestlysupersymmetricactions.This form of the breakdown,however,is muchmorerestrictivethan an explicitbreakdownas onecan alreadyseefrom the resultson STr Al2 in Chapter2.

H.P. Nil/es,Supersymmeiry,supergravityand particlephysics 59

In our discussionwe will roughly follow the historicaldevelopmentof the field. This implies that westartwith modelsin which supersymmetryis brokenat a scaleM~ 100GeV andwhereall dominantcontributionsto the masses(and coupling constants)are alreadygiven at the tree graph level. We willfinally end up to discussmodels in which M~-~lOb GeV and where the splittings of, let us say, thequarkand leptonsupermultipletsarisethroughradiative corrections.

5.1. Fayet—U(1)

Thesemodelsaredue to the pioneeringwork of Fayet[201—210]andarethe first attemptto give thestandardmodel a supersymmetricextension.Many of the model-independentresultswhich we havedescribedin the last sectionwere discoveredin the framework of this model. Specifically this modeltries to incorporatea spontaneousbreakdownof supersymmetryat a scaleof the order of a hundredGeV in which the scalarpartnersareall heavierthanthe correspondingquark andlepton massesat thetreegraphlevel. Given theserequirementsthe possiblemechanismsof the spontaneousbreakdownarealreadylimited. The reasonfor this is the massformula [227](2.75)

STrAl2 = ~ (~1)11(2J+ 1)Al~= 2TrQaDa, (5.1)

which holdsat the tree graph level in the caseof a spontaneousbreakdownof supersymmetry.Modelsin which supersymmetryis brokenby the vacuumexpectationvalues of the auxiliary fields (F) of chiralsuperfieldsthus alwayshavea vanishingright hand side in eq. (5.1). This implies that on averagethemassesof the bosonsare the sameas the massesof the fermionsin the theory. In the simple caseswediscussedin Chapter2 thisrelation was eventrue separatelyfor the differentsupermultiplets.The two(real)scalarpartnersof (let ussay) the electronsatisfiedthe equality

m~+m~=2m~, (5.2)

which of coursecannot lead to a satisfactorymodel since mA and m8 haveboth to be heavier than

20 0eV as discussedin the last chapter.In generalSTrAl2 = 0 doeson the otherhandnot directly imply

relationslike (5.2) for everymultipletseparately;it only tells us that this is trueon the averageandonecould in principle think of a situationin which the scalarpartnersof quarksandleptonsas well as theHiggs fermions are heavy such that STrAl2 = 0 and there are no problems.This, however,doesnotwork; at leastI am not awareof any modelwith the particlecontentof the standardmodel that exhibitstheseproperties.Onereasonfor this is thefact that the spectrumof the (chiral) fermionsis not affectedby the spontaneousbreakdownof supersymmetryat the treelevel: thereis no Ft/It/I term,andfermionsfeel the supersymmetrybreakdownthrough radiativecorrections.The mass spectrumof the scalarsdependson the coupling of the superfieldsto the goldstinosuperfieldthrough FG~ (whereF

0 is theauxiliary field of the goldstino multiplet and(FG) � 0). For such a model to be satisfactoryall scalarpartnersof quarksandleptonshaveto receivea largemasscancelledby negativem

2 contributionstothe Higgses.There is no known model that fulfills thesecriteria.

Guidedby theseargumentsFayet consideredmodelswith nonvanishingsupertrace.Such a situationcan only occur if supersymmetryis broken by v.e.v.’s of the auxiliary D-fields of the gaugesuper-multiplets.More specificallythisimplies that (beforeor aftergaugesymmetrybreakdown)aU(1) gaugegrouphasto be presentin the theory since D~’carries the index of the adjoint representationof the

60 H.P. Ni//es,Supersymmetry,supergravityandparticiephysics

gauge group and (D’~)� 0 leads to a gauge symmetry breakdownexcept in the casewhere Dcorresponds to a U(1) group. In the standard model the candidates for these U(l) groups are U(1)~ andU(1)em and one could try to break supersymmetry through v.e.v.’s of the D-fields of these groups. Thecorresponding goldstinos would be the bino or the photino. We know now from our discussion inChapter2 that the mass splittings of the scalarparticlesare given through the gaugecoupling to thegoldstino(recall (2.74)). In the case of the photino the partners of the e for example would then receivea negative (mass)2 because the charge of e is negative. This would induce a breakdownof U(1)em

incompatible with present observations. Similar problemsarise if the goldstino were the bino since forexample ü has a negative hypercharge. Thus the D-terms of U(1)~ and U(1)em cannot be used to breaksupersymmetry. In general one could try to have a linear combinationof chiral and theseU(l) gaugefermions as the goldstino, but since this does not lead to a satisfactory model in either of the limitingcases it will also not work in any linear combination. As a result of this discussion we conclude thatunder the given assumptionsa new U(1) gaugesymmetryhas to be introduced. The simplestchoiceis toassign positive charges (Q = + 1) to all quark and leptonsuperfieldsand negativecharges(Q = —2) tothe H, H superfields. One wants now to find a potential in which supersymmetry and SU(2) x U(1) isbroken. To break supersymmetry via a v.e.v. of D of U(1) this requires the introduction of aFayet—Iliopoulos [218]term 5 d40~V— ~D. To discuss the scalar potential we assume for a moment thatthe Yukawa couplings are zero and we have only to consider the Higgses. Later on one must of coursecheck whether the so-found minimum of the potential remains still a minimum after the Yukawacouplingshave been reintroduced.The simplest starting point would then be to have a vanishingsuperpotentialg = 0 and just the ~D term which leads to the scalarpotential

V= ~D2= ~ 2~IhI2—2~hI2I2 (5.3)

(compare(2.72)) whereh, h denote the complex scalars of H, H and ~ is the couplingconstantof U(1)(~> 0). Wehave now two choices for ~. It can be positive implying V= 0 at the minimum and thusunbroken supersymmetry, or it can be negative which leads to Vm = ~2 with unbroken gaugesymmetries. Thus the potential in (5.3) is too simple to achieve what we want. In a next step one mightadd a term mHH in the superpotential leading to additional terms m2Ih~2+m2~h~2in the scalarpotential. For the range of parameters ~> m2/~the desired minimum is obtained with V

0 = (m2/2g)~,

~D)= m2/g and (1h12+ h~2)=(1/2~)(~—m2Ig). The goldstino is a linear combination of the U(1)gaugino and the chiral Higgs fermions with weight m2/~and [m2/2g(~— m2/g)}”2 respectively. Thescalar partners of quarks and leptons receive ~ mass terms proportional to m2andoff-diagonalterms(~,2+ ~*2) with coefficient ~m(1/g)(~—m2/~).To make sure that this leads to mass matrices withpositive determinantoneseeshereexplicitly that the U(1) gauginohasto be the dominantcomponentof the goldstino.The gaugebosonsof the brokengaugesymmetriesbecomemassivethrough the Higgsmechanism and the corresponding gauge fermions combine with the Higgs fermionsto massiveDiracparticles (with exception of the goldstino), as has been shown explicitly in the simpler example ofChapter 2. One potential problem of the model is the existence of the U(1) gauge interactions whichmight disturb the usual SU(2) x U(1) weak interaction phenomenology. Difficulties can however beavoided by special choices of the parameters in the model, such as using a small ~. We will not show thishere in detail, since the model as it stands is not complete becauseof triangle anomaliesinvolving theU(1) current. Before we are allowed to gauge U(1) we must make sure that the fermion representationswe consider are anomaly free. Onemight even like to impose an additional constraint by observing thatin the presentmodel the D-term (i.e. ~) receives quadraticdivergent contributionsin perturbation

H.P. Ni//es,Supersymmetry,supergravityandparticlephysics 61

theory. A ~ of order of 1000eV would thereforenot be natural. This secondconstraintwould justrequire that Tr Q 0 for the representationunder consideration [240]. The cancellation of theanomaliesis much more complicated [553,13, 391,40]. First of all it is not just enough to haveTr Q3 = 0 sincethereare also anomaliesinvolving, e.g., two SU(3)currentsand the U(1) current.Allthe quarkshave Q = + 1 and to cancel theseanomaliesonehas to introducerepresentationswhichtransformnontrivially under SU(3)~and have Q<0. Supersymmetrynow brokenby the v.e.v. of Dwould give negativem2 contributionsto the scalarsof thesesuperfieldswhich indicatesthe dangerof apossiblespontaneousSU(3)~breakdown.Onemight try to avoidthis problemby giving sufficiently largemasstermsto theseadditional fields. But this is not possiblein a gaugeinvariant way. The additionalfields aresupposedto cancelanomaliesof the original model.They thusform an anomalous(reducible)representationandmasstermswould explicitly breakthe gaugesymmetry.As aresult in awide classofmodelsit hasbeen shown that the absoluteminimum of the potentialbreaksSU(3)~spontaneously[13,553].

To giveyou a feelingwhat is requiredto cancelthe anomalieslet us considerthe simplestexample.Inaddition to the quarksand leptonsof one family and the H, H one introducessuperfieldswith thefollowing SU(3)X SU(2)X U(l) x U(1) quantumnumbers

K(3, 1,~,—2), K(3, 1,—~,—2), X(1, 1,0,4), Z(1, 1,0,1). (5.4)

Actually the representationsin (5.4) togetherwith H, H andonefamily of quarksandleptonsarejust adecompositionof a 27 representationin E

6 and thus anomaly free. U(1) correspondsto the cosetE6/SO(10)[129,449]. In fact a model with three27 representationswould be a candidatefor a grandunified model for the unification of SU(3)x SU(2)x U(1) x U(1) but in such a model one hasto makesure that K and K aresuperheavyto avoid proton decayat a disastrousrate. Such a KK masstermrequiresa breakdownof U(1) in agreementwith our earlier expectationsand implies that U(1) gaugeinteractionscan no longer be relevantin the 1000eV energyrange.Independentof this embeddinginE6 it hasbeenshownthat modelswith the addition of the fields in (5.4) do not lead to the desiredvacua[40,391], which shouldbreaksupersymmetryandSU(2)X U~(1)X U(1) but not SU(3)X Uem(1).

Other simplechoicesof anomalycancellationlike the onewith a 0 = —2 color octetsuperfield[553]and singletswere shown to havethe samedisease.Very often supersymmetricminima with brokencolorSU(3) occur. As a resultof this moreand morefields haveto be introduced.This, of course,alsoincreasesthe potentialdangerthat supersymmetricminima exist since the equation

=0 (5.5)

is easierto solve if one hasa lot of fields 4,. someof which havepositive and someof which havenegativecharge. The by now very complicatedscalarpotential hasto be minimized and a judiciouschoiceof parametershasto be imposedto makesurethat supersymmetryis brokenand thatthe desiredminimumis obtained.Nonethelessthereexistmodelsin the literaturethat claim to havesolvedall theseproblems[305,463,256]. They havefor examplea particle contentlike [463]eq. (5.6) which containsthirteennew SU(3)X SU(2)singlet superfields.With all of this in mind, theremight arisethe idea thatthis way of breaking supersymmetryat M5 — 1000eV and whereall scalarsare massiveat the treegraph level is perhapsnot the most economicalmethodto arrive at a supersymmetricversion of thestandardmodel. At the origin of all thesecomplicationsis of course the supertraceformula whichturnedout to be sorestrictive. It is howeveronly valid at the treegraph level [279].

62 H.P. Nil/es,Supersymmetry,supergravityandparticle physics

SU~(3) SU~(2) U~(l) U’(l) SU~(3) SU~(2) U~(1) U’(l)

0, 3 2 1/6 1 P’ 1 1 1 —2U~ 3 1 —2/3 1 ~IC 1 1 —1 —2D~ 3 1 1/3 1 J 1 1 0 4L. 1 2 —1/2 1 JC 1 1 0 —4E~ 1 1 1 1 J1 1 1 0 4N~ 1 1 0 1 J2 1 1 0 4H 1 2 —1/2 —2 J3 1 1 0 4H’ 1 2 1/2 —2 Ra 1 1 0 —2K 8 1 0 —2 R 1 1 0 —2T 1 3 0 —2 X 1 1 0 0P 1 1 1 —2 Y 1 1 0 0pc 1 1 —1 —2

(5.6)

5.2. Supercolor[123, 569, 137, 140]

Sincethe supertraceformula (5.1) ceasesto be valid beyondthe tree graphlevel it is logical to usethis fact to circumventthe restrictions.Models that are basedon this observationusually contain aspecialsectorthat is responsiblefor the breakdownof supersymmetryandit is only thissector thatfeelsthe breakdownat the tree graph level. The partnersof quarksand leptonsremain degeneratewithquarksandleptonsat the treegraph level; theyhaveno tree level couplingsto the goldstino,andtheywill receivetheir massesthrough radiative corrections.This might requirethe supersymmetrybreakingscaleM~to be somewhathigher than in the models previously discussed.A typical scale (which ofcoursedependson the couplingconstant)is M~= 1 TeV. Historicallythe first of this classof modelswasthe supercolormodelby DimopoulosandRaby [123]andDine, FischlerandSrednicki [137].Onemighthavesome reservationswith respectto the mechanismof supersymmetrybreaking in thesemodels(compareour discussionin section2.4) but thesemodelsarevery well suited to describethe mechanismby which quark andlepton scalarsreceivetheir massthroughradiative corrections.

Themodel is basedon a gaugegroupSU(N)scx SU(3)X SU(2)x U(l). The sectorresponsiblefor thesupersymmetrybreakdownconsistsof the particlesthat transformnontrivially underSU(N) supercolor[140]

S=(N,3,1,—1/3), S=(N,3,1,1/3), T=(N,1,2,l/2), T=(N,1,2,—1/2),(5.7)

all chosento be left-handedchiral superfields.In addition to the H, H andquark andleptonsuperfieldsoneintroducesa singlet superfieldN = (1, 1, 1, 0). The superpotentialis chosento be

g = A1SSN+A~TTN+A2HHN+~A3N3+~(g~~U,HLJ+ g

1U,HD~+ g~L,HE,). (5.8)

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 63

tpT___x—.. tP_X~ ,“ I Xg X~ ,“ ~

Fig. 5.1. Contributionsto gaugefermion massesin supereolormodels.

This is the most generalsuperpotentialconsistentwith baryonand lepton numberconservation,gaugesymmetriesand an R-invarianceunderwhich all superfieldstransformwith the samephase.Observethat upto now no massparameterhasbeeninsertedin the theory.It is the R-symmetrythatforbids SS,IT, HH and N2 termsin the superpotential.

SupercolorSU(N) is supposedto becomestrongat a scaleA — 1 TeV and leadto condensation

(5.9)

where~(çli)denotethe bosonic(fermionic)componentsof the respectivesuperfields.Supersymmetryisbrokenby the t/n/I condensates(comparesection2.4) and the correspondinggoldstinois a combinationof the compositefermions cost//~,co.ct/is, ç~-t//f and q~~ç1i~-.None of the gaugesymmetries,however,isbrokensincethe quantitiesin (5.9) areSU(N) x SU(3)x SU(2)X U(I) singlets.The masssplittingsof thesupermultipletsof the model are determinedby the coupling to the goldstino. At the tree graph levelonly N couplesto the goldstino. For the other multiplets this coupling occurs in higher order ofperturbationtheory. The SU(3)x SU(2)x U(1) gaugefermions receive a massthrough the diagramsshownin fig. 5.1. With A = 2.5TeV this leadsto [140]

mAg = 1.2TeV, mA~= 490 0eV, mA,= 2500eV, (5.10)

wherewe haveused the usualvalues for the gaugecouplingconstants.The scalarpartnersof quarksand leptonsas well as the Higgsesreceive a massby the diagramsin fig. 5.2. If supersymmetrywereexact the contributionsof these4 graphswould cancel.This cancellationis incompleteheresince thegauginosaremassivebut the gaugebosonsnot. This thenleadsto masses

m = ~C2(1900eV)2+ ~T2(S40eV)2+ 4Y2(lO 0eV)2, (5.11)

wherewe havecut off the integralsat A. C2 = ~for a color triplet andT2 = ~for a weakdoublet.Noticethat thesecontributionsto the massesof thescalarsariseeffectivelyat the two-loop level: first thegaugefermions gain a massat the one-loop level and then they enterthe contributionto the scalarmasseswhich would be zero if mA would be zero.Notice also from (5.11) that scalarswith the samequantumnumbersreceivethe samecontributionto their massesavoidingpotentialproblemswith flavor changing

64 H.P. Ni//es,Supersymmetry,supergravityand particlephysics

a)

Fig. 5.2. Variouscontributionsto scalarmasses.Dashed(solid, wavy, solid—wavy) lines correspondto scalars(chiral fermions,gaugebosons,gaugefermions). In the limit of exact supersymmetrythesefour contributionscancel.

neutralcurrentsas discussedin section 4.6. Becauseof the strongcouplingconstantg3 coloredscalarsreceivethe highestmasses.The scalarpartnerof the right-handedelectron (p., r) receivesthe smallestmass,sinceit only couplesto hypercharge,hereits mass is 20 0eV andwe now see the reasonwhy Ahasto be largerthan 2.5TeV in this model.

We havenot yet discussedthe breakdownof SU(2)x U(l) in this model. The Higgs scalarshavereceivedpositive m

2contributions from (5.11)and thiscertainlydoesnot leadto such a breakdown.Todiscussthis question we haveto considerthe superpotential(5.8) andwe will find the reasonwhy thesuperfieldN hasbeenintroduced.The scalarpotentialobtainedfrom (5.8) hasto be treatedcarefully.The supercolorbilinearsare replacedby their vacuumexpectationvalues to arrive at a low energyeffectivepotentialand to the tree level potentialwe must add the radiative contributionsfrom (5.11).The qualitativepropertiesof this potentialcan alreadybe seenfrom an inspectionof the auxiliary fieldof N,

F~—AjA2+A2hh+A3cc

2N. (5.12)

The term FNI2 is a contributionto thepotentialandFN = 0 would favor a vacuumexpectationvalueforh, h or 4~N suchthat onewould expecta breakdownof SU(2)x U(1) to be possible.The full potential,however,hasto beconsidered.Neglectingthe Yukawacouplingsthis correspondsto

V= 1A1A

2+A2h°h°+A3~~I

2+2A~NI2+A~NI2[Ih°I2+h°12]+m2(lh°12+Ik°12Y AIA3(cc’N + ~‘),

(5.13)

wherewe_havealreadyminimized the ~D2contributionby assumingthat only the neutralcomponentsof the h, h doubletsreceivea v.e.v. The last termarisesfrom the SSN termwhere(t/’

5t/i~)is replacedbyA

3 and m2 denotesthe mass contribution to the Higgses from (5.11). Potential (5.13) has to beminimized which is hard to do analytically, but specialsolutions(numerical)havebeenfound by Dineand Srednicki [140]. For A

1 = 0.0013, A2 = 0.2 and A3 = 0.31 they found the minimum to be at(çc’N) — 500 0eV and(h°)= (h°)= 1200eV which gives the correctvaluesof the W andZ masses.In anext stepit hasto be checkedthat this minimumremainsthe correctone if one includesthe Yukawacouplingsof the Higgsesto the quarksand leptonsandto checkwhetherthe estimate(5.11) for theirmassesremainsvalid. This correspondsto an inspectionof the auxiliary fields of the H, H multiplets

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 65

—F~=A2hcN+~(g~UDJ+g~.L1E~,), —F~=A2hcc’N+~U~.Uj, (5.14)

andit was found that theseadditionaltermsneitherdestabilizethe minimumnor changethe estimatesin (5.11)substantiallyfor the given choiceof parameters.The bosonicspectrumis thusphenomenolo-gically acceptableand it remainsto discussthe fermionic spectrum.Chargedfermions are the A~,Agaugefermionsandthe Higgs fermionst/’~andt/i~, whichcombineto two Dirac fermionsthe lightestofwhich comesout to be 85 0eV. The neutralfermions t/’~,t/’~,t/’N, A°andA

5’ also becomeheavypartlybecauseof the radiativecontributionsin (5.10). The lightest of theseneutral fermions hasa massof35 0eV with our choiceof parameters[140].

Themodel thus leadsto an acceptablespectrum.Thelightestnew bosonhasa massof 20 GeV; it isshortlived anddecaysinto its partner (e, p., T) and the goldstino.The goldstino itself is sufficientlydecoupledto avoid phenomenologicalproblems.Thenext lightest fermion hasa massof 35 GeV whichdecayspreferablyto the heaviestleptonandthe scalarpartnerof its right-handedcomponent.Carehasto be takenalso of the pseudo-Goldstonebosonsthat occur in the processof the breakdownof globalsymmetriesthrough the supercolorcondensations.The lightest of those are in the region of 40 0eV.Thesearetheonesthat do not receivemassesthroughSU(3)x SU(2)x U(1) gaugeinteractionsbut onlythrough the couplingconstants(e.g. A

1) in the superpotentialwhich also explicitly breaksomeglobalsymmetriesotherwiseunbrokenat the supercolorscale.Massformula (5.11)assuresthe suppressionofflavor changingneutralcurrents,andtherearehenceno phenomenologicalproblemswith this modelatthe moment.The model can be straightforwardly extendedto a grand unified model based onSU(N),.~xSU(5) in the way discussedin the last chapter.

Notice that the only dimensionfulparameterin the theory A is introducedthrough the supercolorcoupling constant.This massscalesets the scale M5 = A of supersymmetryand for A = 2.5TeV; thisinducesthe correctscaleof SU(2)x U(1) breakdown(with thechoicesfor A, as given above).Onemightnow ask: what arethe allowedvaluesof A? We havealreadyseenthat with the given gaugecouplingsg3, g2 and g1 and A = 2.5TeV the lightest bosonsin the theory (the partnersof the right-handedleptons)receivea massof 20 0eV,so A � 2.5TeV. RaisingA would raisethe massesof thesescalars.Itwould howeveralso raisethe radiativecontributionto the mass

2of the Higgsesand thusmakeit moredifficult to induce an SU(2)x U(1) breakdownthrough the terms in the superpotential,which inprinciple would give usan upperboundon A. For a sufficiently largeA the couplingconstantsA- in thesuperpotentialhaveto becarefully adjustedto havethe(h), (h) now small comparedto A. If onewouldinsist that thecorrectpatternof SU(2)x U(1) breakdownis obtainedfor a largerangeof the parametersin the superpotentialonewould beforced to usea A whichis not muchlargerthan 10 TeV. It should bestressedagain that A is the only dimensionfulparameterin this model which determinesall thesebreakdownscales.Anotherpotentiallypossibleterm of this typewould be a Fayet—Iliopoulosterm withparameter~. Unlike for A (which we obtain through dimensionaltransmutation)a naturalvalueof ~would be of the orderof the grandunificationor the Planckscale.~ waschosento be equalto zeroandsinceTr Y = 0 it doesnot receivecontributionin perturbationtheory [240]. A possibleexplanationofthe absenceof ~would be theembeddingof U~(1)in a semi-simplegroup[569](like SU(S))for which agaugeinvariant ED-termcannotexist. Other possibleexplicit masstermslike mHH were forbiddenbyan R-symmetry under which 0 transformswith chargeR = +3/2 and all the chiral superfieldswithcharge R = +1. This R-symmetry is broken by the condensatesin (5.9) spontaneouslyand thisbreakdownwas necessaryfor exampleto allow Majoranamassesfor the gluinos. But a spontaneouslybrokenglobal symmetry usually implies the existenceof a Goldstoneboson.The R-symmetryunder

66 H.P. Ni//es,Supersymtnetry,supergravityandparticlephysics

consideration(though not explicitly brokenby terms in the superpotential)is explicitly brokenby asupercolor anomaly and this would be Goldstoneboson receives a mass of order A through thisanomaly(just like the ~j’ receivesa masscontributionfrom the axial QCD anomaly).The model thusdoesnot give rise to any phenomenologicalproblems.

5.3. O’Raifeartaigh modelsand the connectionbetweensupersymmetryand weakinteractionbreakdown

The previously discussedsupercolor models are based on the assumptionthat a formation offermion—antifermioncondensatesappearsas aresultof the strongsupercolorforce. This then leadsto abreakdownof supersymmetry:the reasonsupercolorwas invented.The assumptionof this condensationhasbeenseriouslyquestionedby Witten [573]on the basisof his index argument[75].This argument,however, is strictly valid only in the case in which one includesmassesfor the nontrivial supercolorrepresentations.Indeed, resultsobtainedin an effectiveLagrangianapproach[543,530] haverecentlygiven backthe hopethat supersymmetrymight be broken in the masslesslimit, leaving this importantquestion still under discussion[475a].At the moment we do now know whether supercolorbreakssupersymmetryor not [104a,448,273].

As a resultof this ambiguity one has tried to constructmodelssimilar to the supercolormodelsinwhich the sectorthat was responsiblefor the supersymmetrybreakdownwas replacedby an O’Raifear-taigh model [461].By similar we meanthat this supersymmetrybreakdownsector is somewhathiddenfrom the normal particlessuch as to forbid massesat the tree graph level for the partnersof quarksand leptons.In the supercolormodel this hiddensectorconsistedof the strongly interactingparticles.The normal particles,of course,did not participate in thesestrong interactionsand in addition theywere not coupledto this sectorin the superpotentialto avoid a direct coupling to the goldstino.Suchaprocedurecan alsobe realizedwith a hiddenO’Raifeartaighsectorandmodels alongtheselines havebeen constructedby Dine and Fischler [132],Nappi and Ovrut [437],Alvarez-Gaumé,ClaudsonandWise [13]. The construction of the hidden sector requires the introduction of several new chiralsuperfields.In the modelof Dine andFischler[132],for example,the following fieldshavebeenadded:

S,S’=(1,2,I/2), S,S’=(1,2,—1/2), T, T’= (3, 1,—1/3),

X,Y=(l,l,O). (5.15)

These are all left-handedchiral superfieldsand the bracketsgive their SU(3) x SU(2)x U(1) represen-tation content.In the_contextof a grand unified model this would correspondto (at least) the additionof two S and two S representations[133]. The superpotentialof this model should contain anO’Raifeartaighsector in which one of the auxiliary fields of the singlets(F~or F~)receivesa vacuumexpectationvalue. Onecan write

g = m1(S’S+S’S)+m2(T’T+ T’T)+ m3SS+m4IT+ Y(A1SS+A2TF—p.~) 16

+ A3X(HH- p.~)+g~~U1H~+ g~U1HD1+ g~L1H~, (. )

and the model possessessix independentdimensionfulparameterswhich are supposedto be in theregion 100 GeV—lO TeV. This model hasno R-invariance.In addition to lepton and baryon numberconservationthe only other conservedquantitiesare the number of S-type and T-type superfields.Actually superpotential(5.16) is not the mostgeneralonecould write consistentwith thesesymmetries,

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 67

which would, e.g., allow terms quadraticand trilinear in Y Thesecouplingsare left out since theirpresencein (5.16) would preventa breakdownof supersymmetry[132]. We know that such termsarenot generatedin perturbationtheoryandit is thus “technically natural” to leavethem out. Onecouldofcourse forbid such terms by imposingan R.symmetrybut such a symmetry in the context of thesemodelsposesproblems.The R-symmetryforbids for examplea Majoranamassfor the gluinos andweconcludethat it is spontaneouslybroken. A spontaneousbreakdownwould lead to an unobservedGoldstoneboson. To generatea mass for this particle one would have to break the R-symmetryexplicitly. In the supercolormodels this occurredbecauseof a supercoloranomaly,here we can inprinciple havea color anomaly but this would make an unwantedPeccei—Quinn—Weinberg—Wilczekaxion [474,552,566] out of the unwantedGoldstone boson. Such an R-symmetry thus has to beexplicitly brokenin the superpotential,equivalentwith the statementthat g is not allowedto possessanR-symmetry.This implies that such a symmetry cannotbe used to forbid the mentionedY3 and Y2terms in the superpotential.Certain terms in g that are consistentwith all the symmetrieshaveto beomitted.This unsatisfactoryproperty is commonto all the modelswewill discussin this section [198].

Apart from thesequestionsg in (5.16) hasthe desiredproperties.The hiddensector, given by thefirst five terms in (5.16) does not couple to the observablesector at the tree graph level and therestrictions of the STr Al2 formula are avoided. Mass splittings in the observablesector will begeneratedin perturbationtheory and since terms in the superpotentialdo not receive radiativecontributionsthis mass generationwill proceedthrough gaugeinteractionswhich in turn assurestheabsenceof problemswith flavor changingneutralcurrents.

Supersymmetryis brokenin the hiddensectorsincethereis no commonsolutionto F~= F’~= F~=F~.= = F.t. = 0, the minimum is obtainedfor F~= p.~and all scalar fields havevanishingv.e.v.exceptfor ~pywhosev.e.v is undetermined.In the observablesectorthe equationF~= 0 forceshh =

andimplies the breakdownof SU(2)x U(1). Observethat in this model SU(2)X U(l) is brokenby handthrough the introduction of the term —p.~Xin the superpotential.In particular the breakdownofsupersymmetryand weak interactions is no longer related. One must actually be careful that theradiative contribution to the Higgs massesfrom the hidden sector are not too large to restoresupersymmetryin the effectivepotential. In this model thisimplies an upperboundof orderof 100TeVon the parametersm

1,. .. ,m~and p.1 and also gives an upper boundof 100 0eV for the lightestpartnersof quarksandleptons.But nonethelesssupersymmetryandweakbreakdownarenot related.Ifp.1 would be zero supersymmetrywould be unbrokenand similarly p.2 = 0 implies unbrokenSU(2)XU(1) independentof the value of p.5. Recalling our discussion about the motivation to considersupersymmetricmodelswe would, however,prefer models in which the supersymmetrybreakdowninducesthe breakdownof the weak interactions.This would meanthat SU(2)x U(l) gets restoredoncethe supersymmetrybreakingparameterM~vanishesand the breakdownof the two symmetriesisintimately related. In the context of the models discussedhere such a possibility exists as hasbeenobservedby severalgroups[332,13, 133, 124]. We will give a simplified versionof this mechanismhere.The full calculation would have to include renormalization group considerationsof the effectivepotential which we will not give here since we will treat this subject in Chapter7 in detail. Theessentialsof the mechanismcan alreadybe seenin the simplified version.

Supposethat supersymmetryis broken in a hiddensectorsuchthat thereareno tree level couplingsof the ordinary sector to the goldstino. The ordinary sector will receive the information of thesupersymmetrybreakdownthrough radiativecorrectionsdue to the gaugeinteractions. Supposenow(which is normally the caseas you canseefrom the examplesgiven so far) that at the onelooplevel theSU(3)x SU(2)x U(1) gauginosreceivea mass.The scalarpartnersof quarksand leptonsreceivetheir

68 H.P. Nil/es, Supersymmesry,supergravityand particlephysics

t~quark

H HFig. 5.3. Contributionto the massof theHiggs bosonthroughitscoupling to thepartnersof quarks.

massbecauseof the presenceof thisnonvanishinggauginomassfrom the graphsin fig. 5.2, the sum ofwhich would vanishif mA would be zero.The graph containingA gives a negativecontributionto mand since this graph becomessuppressedbecauseof mA the full contribution to m~is positive ifsupersymmetryis broken. As a result scalar partnersof quarksand leptonsas well as the Higgsesreceivepositive m2 contributions.With mA = 1 TeV this would leadto a massof order 1000eV for thepartnersof quarksanda massof order 10—SO0eV for theHiggsesandthe partnersof leptons(compare(5.11)). Including thesemasseswe now reconsiderthe graphsin fig. 5.2. The graph with quadrilinearscalarcoupling includesnow a massivepropagatorandsinceit correspondsto a positivecontributiontom2 this contributionis reducedoncethe scalarsaremassive.The effect is strongestif the particle in theloop is a partnerof a quark since they are heaviest.Such a situationcan eitheroccur if the particleunderconsiderationis a partnerof a quark or a Higgs (it cannotoccurfor the partnersof leptons)asshown in fig. 5.3. This two-loop effect of coursecannotdrive the alreadyheavypartnersof quarkstonegativem2but it can havean influenceon m~jgg.providedthat thereis a largequadrilinearcouplingbetweenHiggsesand quarkspartners.A candidatefor such a large coupling is the Yukawa couplingresponsiblefor the massof the top quark.Whetheror not at the endthe Higgsesreceivenegativem2dependsnow on the balancebetweenthe suppressionof the graphsc andd in fig. 5.2. The contributionto the m~jgg.can approximatelybe written as

m~jgg,=a2m~2—g5m~5=a2m~2—gta3m~3 (5.17)

which can becomenegativeif g~(the top quarkYukawa coupling) is large enough.The actual valuesneededfor g~to lead to negativem~jgg.and inducean SU(2)x U(1) breakdownare model dependentandcan only be foundafter a correcttreatmentof the two-loop effectivepotential. In somecasesa topquark mass larger than 25 GeV is sufficient to drive the breakdownof the weak interactions[172].(We will later comebackto thispoint.) Observethat thereis no dangerto inducenegative(mass)

2forthe partnersof leptonssince they do not couple to the partner of quarksand secondlythere is nonegativem2 for the partnersof quarkssincetheyhavereceiveda largemassat the first stage.

This concludesthis section on the hidden sector O’Raifeartaighmodels with M~— I TeV. Theyrequire the introductionof new fields in the hiddensectorwith the introductionin generalof severalmassparameters.Careshould be takento avoid an R-symmetryin the superpotential,which impliesthat the superpotentialhasnot the most generalform consistentwith its symmetries.A sufficiently largevalue of the top quark mass can induce SU(2)x U(1) breakdownas a result of the supersymmetrybreakdown.

5.4. Theinversehierarchy

In this sectionwe want to discussa class of O’Raifeartaighmodelsinventedby Witten [570]which

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 69

aredifferentin spirit thanthe onewe havediscussedso far. In thesemodelsthe scaleM5 is supposedtobe the fundamentalscale and the large scalesM~or M~are generatedin perturbationtheory.Thisapproachmakesuse of a commonproperty of all O’Raifeartaighmodels that the potential has a flatdirection at the minimum. Thiscomesfrom the fact that brokensupersymmetryrequiresthe equationsF, = 3W/34,1= 0 for the auxiliary fields haveno commonsolution.This algebraicinconsistencyimpliesthe F, to be independentof onecombinationof the scalarfield ço~whosevacuumexpectationvalueis inturnundeterminedat the treegraphlevel andleadsto a vacuumdegeneracyalongthis direction.Let usconsideras an examplea model with superpotential[570]

g(A,X, Y)=AX(A2—M2)+gYA, (5.18)

andlet us furthermoreassumethat g/AM ~- 1. The scalarpotentialreads

V= g2Ia~2+A21a2—M212+ gy+ 2AxaI2. (5.19)

For the given rangeof parametersthe minimumis at (a) = 0 and (FX) = AM2 � 0 aswell as (y) = 0. Thevacuumexpectationvalueof x (the scalarwhoseauxiliary field hasav.e.v.) is undeterminedat the treegraphlevel. This degeneracyof the vacuumwill disappearif onegoesbeyondthe tree level. Actuallythis calculation hasto be done since the v.e,v. of x determinesthe massof the a particle. Thesecorrectionsbeyondthe tree graph level havebeendiscussedin detailby Huq [321], Clark, PiguetandSibold [85]. The one-loopcorrectionto the effective potential is determinedby the supertraceof thefourth powerof the massmatrix

~iV= ~ ~ (—1)1tM~(4,)log(M~j/p.2), (5.20)

wherep. is a renormalizationmass.Evaluatingthe leadingtermsfor the model given abovegives

V(x)= A2M4[1 + (A2/8rr2) log(x2/p.2)+ .. ~]. (5.21)

The logarithm in (5.21) can be understoodas a replacementof A by an effective coupling A(x). Intheorieswith spin 0 andspin 1/2 fieldsonly, this couplingalwaysincreaseswith increasingx. As a resultthe coefficient of the logarithm is positive and the one-loop correctionsdeterminethe vacuumexpectationvalueof x to bezero in the model underconsideration.

In theorieswith gauge interactions,however,the sign of the logarithm can becomenegative.Thepotentialthen might look like

V(x)= dA2M4[1 + (bA2 — ce2)log(~x~2/p.2)], (5.22)

where d, b, c are positive numericalconstantsand e is a gauge coupling. One might now have asituationwhereat small values of (x)bA2 <ce2. If the gaugeinteractionsare asymptoticallyfree theeffective coupling ë will decreaseat large x whereasA increases.As a result the effective coefficient(bA2— ce2) of the logarithmchangessign at somex, andthe one-loopeffective potentialwill look likethat shown in fig. 5.4. x will receivea nonvanishingv.e.v. Since the evolutionof the couplingconstantswith the logarithmis ratherslow;the vacuumexpectationvalueof x will belargecomparedto the scale

70 H.P. Ni//es, Supersymmetry,supergravityandparticle physics

Vett

dX2 M4

x=M exp (l/e2)

log (Xl

Fig. 5.4. One.loopeffectivepotential for a model basedon the inversehierarchy.

M, the intrinsic scale of the model and will typically be of the order M exp(+1/e2), the exact valuedependingon the parametersof the model.The modelwill thuscontain two scalesM and(x) with thelatterobtainedby “dimensionaltransmutation”and(x) > M This is the typical situationdescribedbyColemanandWeinberg[93].The nice thing here is that the flat direction of the potentialneednot beobtainedby a fine tuning of parametersbut necessarilyexists in any O’Raifeartaightypemodel.

Having madethis observationWitten constructeda model in which M characterizesthe scaleof thebreakdownof the weak interactionswhereas(x) correspondsto a grand unification scaleM~.Thefundamentalscaleof this model is no longer the big scalebut the small scale.

TheO’Raifeartaighsectornow containstwo adjointrepresentationA and Y~of SU(S) togetherwitha singlet X,

gR=ATrAY+gX(TrA—M), (5.23)

which leadsto brokensupersymmetryand( 2 02

(a)= gM(A2+30g2~”2 2 . (5.24)—3

—3

Onecombinationof (x) and(y) is undeterminedand

/2 0/2

2 . (5.25)A 3

\ 0 —3

The one-loopeffectivepotential that determines(x) and (y) reads

V(x)= M4g2A2 (1~30g

2(29A2—S0e2)1i4~ 526

A2+30g2 ~ 80~2(A2+3Og2)og p.2 ( . )

which leads to large (x) and (y) if 29A2<50e2,where e is the SU(5) gaugecoupling. The potential

H.P. Ni//es, Supersymmetry,supergravityand particie physics 71

decreasesfor increasingx till the region where the approximateformula (5.26) is no longer valid.Asymptotic freedom, however, suggeststhat the potential turns over at some point and a stableminimum develops.The alternativewould be that the potential decreasesmonotonically.Since ourmodel is supersymmetricthe potential is boundedfrom below by zero and one could still have acosmological interpretation of such a model in which the ground state becomessupersymmetricasymptotically.

The fields (x) and (y) now receivelarge vacuumexpectationvaluesand SU(S) is brokenat a scalecharacterizedby M2 exp(1/g2).In addition to Y we needHiggsesin the S+ S representationof SU(S)andwe alsoadd a singlet Z that will playthe role of a sliding singlet (seeour discussionin Chapter4).We addto g~

gs= fHYH + hZ(HH — M’2), (5.27)

whereM’ is a masscomparableto M Its purposeis to induce the breakdownof the weak interactions.As a result the scalarsof H andH will receivea v.e.v. of orderM’. The sliding singlet Z will adjustitsv.e.v. to leavethe color triplets in H andH heavybut keepthe massesof the othercomponentssmallin the rangeof M, M’. A completeinspectionof the modelshowsthat thesev.e.v.’s of H, H disturbthevacuumexpectationvalues of A andmoreimportantly Y which leadsto thefollowing breakingpattern.(y) breaks SU(5)-~SU(3)xU(1)x U(1) at the large scale and at the small scale H and H breakSU(3)X U(1) X U(1) to SU(3)X U(1)em. This is a deseaseof the simplified model which arisesbecausethe Y) couplesboth to A~andHtI~.The problemcan be curedby replacingA~by a 75 representationof SU(5)and the correctpatternSU(S)—~SU(3)x SU(2)xU(1)—~SU(3)x U(1) can be obtained[268].

Independentof thesequestionssuch a modelwith M — 1000eV is problematicbecauseit containstoo many light fields and if one addsthreefamilies of quarksand leptonsSU(3)x SU(2)x U(1) is nolonger asymptoticallyfree andthe couplingsconstantbecomelarge before the grand unification scale[570,2681. It is not clear whetherunification still makes senseunder thesecircumstances.But let uspostponethe discussionof this questionfor themomentandaddthe quarkandleptonsuperfieldsto themodel with the usualYukawa couplingsto H, H in the superpotential.We want to discussthe masssplittings in thesemultiplets that arise through the breakdownof supersymmetry.This breakdownoccurredin the sectordescribedby g~in (5.23) in a mannerthat both F~and F

5’ (the “hypercharge”component)received a vacuum expectationvalue. The goldstino is then the appropriate linearcombinationof the fermions in the X and Y superfields. Again the splittings of the multiplets aredeterminedthrough the coupling to the goldstinoand at the level of the given effectivepotentialonlythe scalarsof the A-multiplet receive correctionsdue to the breakdownof supersymmetry.TheA-scalarsare very heavy due to the large v.e.v. of x and y of orderM~whereasthe supersymmetrysplitting is characterizedby M. The ordinary field quarks, leptonsand Higgsesnow couple to a~viagaugecouplingsandtheir splittingsarisein higher order.Their couplingsto the goldstinoaremediatedthrough loopsinvolving the superheavya-fields whosemassis much largerthan the splittings due tosupersymmetrybreakdown.As a result the splittings in the light sectoraresmall andof orderM

2/M~multiplied by the appropriatecouplingconstants.With M — iO~GeV andM~— 10160eV this impliesfor ~mL the splittings in the light sector to be of order 10’°0eV, a ridiculously small numberwhich,e.g., representsthe splitting of the electronfrom its scalarpartner.The only way out is to raiseMcomparedto M’ whichrepresentstheweakscale.In fact M2/MX shouldbe of the orderof M’ to obtaina reasonablemodel [30,124, 571]. With the usual uncertaintiesof the couplingconstantsof the modelthis would correspondto a breakdownof supersymmetryat a scaleof 109_loll0eV. It is a remarkableproperty of supersymmetry that evenwith such a large breakdownscale the partnersof quarks and

72 H.P. Ni//es, Supersymmetry.supergravityand particlephysics

leptons are kept in the 100GeV range. A price of course has been paid to achieve this, the introductionof a small parameter M’/M. With the now large scale M of coursealsothe problemswith asymptoticfreedom of SU(3)x SU(2) x U(1) in theseWitten-type models disappear.

5.5. How to hidethe hiddensector

Several groupshave independentlyobservedthis possibility to have M5 — lO~~lO~0eV in thecontextof variousmodelsandwecould call this the B

2D2EF2IKNPR2SW[30,40, 124, 133, 171,480, 571]observation. In the model such a situation can usually be realized by inserting appropriatesmallparametersthat render the coupling of the goldstinoto the observablesectorsmall. However, in thedifferentmodelsthis could be achievedin different ways andwe will in thefollowing briefly reviewthepossibilities. -

In the frameworkof the FayetU(l) modelsthis situationhasbeenrealizedby Barbieri,FerraraandNanopoulos[40]. In addition to the quark,lepton andHiggs superfieldsthis modelcontainsfive singletswith SU(3)X SU(2)X U(1) X U(1) transformations

S(l, 1,0,4), R(1, 1,0,4), R(1, 1,0, —4), P(1, 1,0, 1), N(1, 1,0,0), (5.28)

and asuperpotential

g=gy+gH+gN, (5.29)

with

gy = gqUjVH+ g~U,DJH+ g~.L,F,H+ ~LPH,

g~= AHHS + p.RR, g~= i~NSR+ mN(pm+ N). (5.30)

Observethat g hasno R-symmetryand that it is not the mostgeneralsuperpotentialconsistentwith itssymmetries.This property is necessaryto allow gluino masseswithout the appearanceof a Goldstoneboson (or axion) of spontaneously broken R-symmetry. Notice also that the theory is neither anomalyfree nor is Tr Y = 0. This might not be serious in this case since we will see that U(1) is broken at veryhigh energies(Mr) and one might argue that anomaliesat high energydo not necessarilylead toinconsistenciesin the low energy approximation.This argumentation,however, is not necessarilyconvincing.

The potential of the model is discussedin the rangeof patameters

g~>2A2, 4~2~>p.2(1+8g2/A2), (5.31)

and the absolute minimum is found at

(h°)= (j~0) = -~i~_, (F)2 = 4g2A2~— ~~2(8g2+ A2),• (~)= p.2/~2, (5.32)

V2A 16g A

where ~(g) denote the U(1) Fayet—Iliopoulos parameter(coupling constant)respectively.Without

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 73

making the model inconsistent one can- now assume that ~ is very large (or order M~)whereasp. is small p. — 100 0eV, such that ~/p.2— 1O~.With ~ — M~thereare also no problemswithTr Y � 0 since we can hardly imagine the quadraticdivergencesto contribute more thanM~to ~. ~

now determinesthe breakdownof U(1) and this group is broken at M~such that the correspondinggauge boson will certainly not cause phenomenologicalproblems in the low energy theory. TheGoldstonefermion is given by

t/Ig = ~ + (og/3ço~)x~. (5.33)

With ~/p.2~‘ 1 the dominant component of the &oldstino will be the fermioniccomponentof R whoseauxiliary field hasreceiveda v.e.v. of orderp.V~.The vacuumenergyis given by

V0= ~p.2~_ p.

4(A2+ 8g2)/32~2A2 ~p.2~l (5.34)

The observablesector couplesonly to the (tiny) AT-componentof the goldstinoand since(1.5) = p.2/4g2the masssplittings in the low energysectorwill be of orderp., the scaleof the breakdownof the weakinteractions.Thus with the two scalesM~andp. onehasa modelwherethe supersymmetrybreakdownoccursat M~= p.M~— (10100eV)2. The light sector receivesmass splittings of order p. since it onlycouplesto a tiny componentin the goldstino.As it is always the casein the U(1) modelsall the massesandsplittingsarealreadygiven at the tree graphlevel. Thesecontributionsarethe dominantonesandare not upset by radiative corrections as we will see later.

In the frameworkof the O’Raifeartaighmodelsdiscussedin section5.3 a grand unified modelwithM~— 10100eV has been given by Dine and Fischler [133]. They start with the two input scales

— 10’~0eV andp. — 1010GeV. The scaleM~is responsiblefor the breakdownof SU(S) whereasp.governsthe breakdownof supersymmetry.The structureof the model is similar to the onesdiscussedinsection 5.3. Again an R-symmetry has to be avoided for the well-known reasons and the superpotentialis at most technically natural. The difference is now the appearance of the small parameter p./M~.Thegoldstinois not coupledto the light sectordirectly andin higher orderssucha couplingcan only occurthroughheavyintermediatestates.As a resultasmall scaleof orderp.2/Mi — 1O~0eV is inducedby theradiativecorrectionswhich governsthe masssplittings in the light sector.These radiative correctionsalso induce a breakdownof SU(2)x U(1) in this model at scale p.2/Ms very much in spirit of themechanismdiscussedin section5.3. Herethe two scalesp. andM~conspireto give a scalep.2/Ms to thelow energysector. The essentialingredientof the model is p./M~— iO~, and the light mass scaleincluding the breakdown of SU(2) x U(1) is induced radiatively. We will discuss these radiativecorrectionsin a moment.

A different mechanismto raise the breakdownscale of supersymmetryin the context of theseO’Raifeartaighhasbeen proposedby Ellis, Ibanezand Ross [171]. Insteadof introducing two massscalesp. andM with a tiny ratio p./M they introduceexplicit small couplingconstantsthatcommunicatebetweenthe goldstinoandthe light sector.Insteadof onesmall coupling, as theypropose,one couldalso usea successionof hidden sectorswith not so small coupling constants.Insteadof A

1 ~ 1 it isthrough (A5)” ~ 1 that the light sector receivesthe information about the supersymmetrybreakdownandthe separateA, neednot be as small as A1. Themodel is constructedin sucha way that the hiddensectorclosestto thelight sectorcontainsparticlesthat haveSU(3)x SU(2)x U(1) gaugeinteractionsandinducea massfor the gauginosin the light sectorandthe inductionof theweak interactionbreakdowncan then proceedby the mechanismdescribedin section5.3. The supersymmetrybreakdownscalein

74 H.P. Ni//es. Supersymmetry.supergravityandparticle physics

such a model is actually arbitrary. It could be even largecomparedto the Planckmassprovided theappropriatecouplingsare made small enough (or the appropriatenumber of intermediatehiddensectorsis largeenough).We will howeverseelater that the couplingto gravity, an interaction in whichall the fields in all the hiddensectorsparticipate,providesus with an upper limit on M5 if we requirethat the light sectorstill containssupersymmetricpartners.

In the contextof modelsbasedon the invertedhierarchythe necessityof alargebreakdownscalehasbeen observedby Banks,Kaplunovsky[30], Witten [571],Dimopoulosand Raby [124].Unlike in theother modelsthe largescaleM~-~ 10100eV is a necessityrather than apossibility. Along theselines agrandunified model basedupon the inversehierarchyhasbeenconstructed.It hasthe nicefeaturethatonly one scalep. 10~~GeV is introduced,out of which the large scaleM~is inducedout of the flatdirection. Finally the scale in the light sector is induced as p.

2/M~.This model, like the other onesdiscussedso far also has a superpotentialthat is not the most general consistentwith its symmetries.Also care hasto be taken with the color Higgs triplets which in this model turn out to havea massoforderp.. They could introduceproton decayvia dimensionfive operatorsat an unacceptablerate. In themodel this problemis solvedby doubling the numberof 5 and5’s and introducingsomesinglets.Themodel uses the sliding singlet mechanismto keep the Higgs doubletsmassless.Here, however, acomplete treatmentof the effective potential hasto be given to makesure that the theory has thecorrect minimum that breaksSU(2)x U(1). Investigationsof this effective potentialhaveshown thatthere are potential problemswith the use of the sliding singlet [130,442]. These are even morepronouncedwhen the supersymmetrybreakingis very large, especiallyin the context of modelsbasedon theinvertedhierarchy.It might thereforebevery difficult for sucha model to inducethe breakdownof the weak interactionsat the small scale.To answerall thesequestionsa careful examinationof thetwo- (or even three-) loop effective potential of the low energy theory is necessary.Given thecomplexity of such modelsthe searchfor the absoluteminimumof the potentialis not an easytask.

5.6. Thestability ofthe small scale [480,124, 133,41]

We still haveto discussexplicitly the perturbativecorrectionsin modelswith a largesupersymmetrybreakingscale.It seems,at leastto me, highly nontrivial that all the radiative correctionsthat appearinthe low energysectorcorrespondto amassscalethat is so small comparedto M~.A full treatmentofthesequestionshasbeengiven by PolchinskiandSusskind[480]in a toy model that hasall the essentialingredientsof the modelsdiscussedin the last section.We cannotrepeattheir discussionhere in fullcompletenessbut will follow it closelyalso in notation,so that the interestedreadercan completehisknowledge through a study of the original paper. The model contains four chiral superfieldswithsuperpotential

g(B, C, X, L) = gX(B2— p.2/g)+ MB2+ M’BC+ gB3+ gB2L + gBL2+ gL3, (5.35)

whereall dimensionlesscouplingconstantshavebeencalled g. X will be the superfieldthat containsthegoldstino,B, C describeheavy fields and L the light fields. We assumethe parametersof the model tobe arrangedin sucha form that at theminimumb = c = 1 = x = 0 andF~= p.2.M~= p. is assumedto beof the orderof 10100eVwhereasM, M’ correspondto the largescaleM~or M~of order1016_1018GeV.The light field L will be masslessat the tree graphlevel andit will not be split sinceit doesnot coupleto the goldstinoin X at this stage.Thelight field L couplesonly to the heavyfield B. Our taskis now tosee how L is affectedby radiative corrections.The scalarpotentialat the tree graphlevel is given by

H.P. N/lies,Supersymmetry,supergravityand particlephysics 75

gil2

—N--..2 I’ \

b4 4b/

b4 TbA

__~_L_4__

I 3g~ I I gil I

Fig. 5.5. Contribution to the mass of a light scalar 1 through its Fig. 5.6. Contributionto m212 that cancelstheone in fig. 5.5.

coupling to theheavy scalarb. The crossdenotesthesupersymmetrybreakingin theheavy sector.

V = gb2— p.22 + M’21b12 + 2gxb+ Mb + M’c + 3gb2+ 2gb! + gl212 + gb2+ 2gbl + 3g1212. (5.36)

The heavy field B couples to the goldstino and there is a splitting due to the breakdownofsupersymmetry.In addition to the supersymmetricmassterm (M2 + M~2)bb* thereis a term gp.2(b2+

b*2) which splits the two real scalars.It is this splitting that is the sourcefor all splittings in the lightsectorin perturbationtheory sinceL couplesto B. A masscontributionfor the light fields of the formm2l2 could now be inducedin the effective potential through a graph like the one shown in fig. 5.5,which gives a contributionof orderg3p.2 resultingin a massof 10100eV for the scalarparticlesin the“light” sector.We havehoweverto be more careful: in addition to the graph in fig. 5.5, thereis acontributionfrom the graphin fig. 5.6. Again the leadingcontributionof this graphto 12 is g3p.2but thiscontributioncancelsexactly the former one as an explicit calculation of the graphswill show.Thereasonis that thesetwo graphsare containedin the onesupergraphshownin fig. S.7. The supersym-metrybreakingis representedby the v.e.v. of the auxiliary field of X. The contributionto 12 is obtainedas a (LLX)F = 5 d2OLLX contribution of this supergraph.We have seen in Chapter2 that such anF-contributioncannotbe generatedin perturbationtheory;herewe havean explicit examplewhereinthe componentlanguagetwo graphscancel.To obtain a nonvanishingcontributionto 12 we havetoconsidera supergraphwith at leasttwo X field insertionsto constructa D or 5 d40 contributionasforexample(LLX*X)D (seefig. 5.8). This graph,however,hasnow two p.2 insertionsand an additionalheavypropagatorandit resultsin acontribution(g4p.4/M2)!2 which is muchsmallerthanp.2. In analogycontributions to ll*~massterms can be discussed.The most dangerouscontributions come from(LL*D)D as shown in fig. 5.9. It gives g3p.2/M(LL*)~as the scalein the light sector. PolchinskiandSusskindhavegiven a completeconsiderationof all the possiblecontributionsin perturbationtheory.

B~B

Fig. 5.7. Supergraphthat containsthecontributionsgiven in figs. 5.5 and5.6.

76 H.P. Ni/les,Supersymmetry,supergravityand particlephysics

Fig. 5.8. Supergraphwhich givesa contributionto (LLX*X)D allowed Fig. 5.9. Contributionto (LL*X)D.by thenonrenormalizationtheorems.

With the exceptionof onecasetheyhaveshownthat thehighestscalethat could beinducedin the lightsectoris given by g3p.2/MMoreovertheyhaveverified that the low energysector looksas an explicitly(but soft) broken supersymmetrictheory. All the soft breaking termsare in general generatedinperturbationtheoryandnot the hardones.Massesfor chiral fermionscan also be generatedbut only ina supersymmetricway. This meansthat in modelswith M

5— 10~~0eV therecan still exist a light sectorcharacterizedby a scalesmall comparedto M5 and that this separationremainsstablein perturbationtheory.Supersymmetrycan thusprotectparticle massesevenat a scalesmall comparedto M5, withoutfine tunings in any order of perturbationtheory.There is, however,an exceptionto this caseand thisoccurs if the theory containsa light singlet. Such a singlet superfieldwill mix with the goldstinosuperfieldandits typical massscaleas well as the splittings will beof orderp. — M5 [480,451,389]. If itdoes not yet exist at the tree graph level such a masswill be inducedby radiative corrections.If thesinglet then couplesto light as well as heavyfields it will communicatethe masssplittingsof the heavysector to the light sectorandmassesof orderp. will also be inducedfor the other nonsingletscalarsinthe light sector. The separation between the light and the heavy sector is thus potentially unstable in thepresence of a light singlet. This problem with the light singlet is a typical problem of naturalness. Onecan build modelsthat includea light singlet andwherethereis still a strict separationof the light andheavysectorat the expenseof insertingsmallparametersin the theory.If the singlet for exampleonlycoupleswith Yukawacouplingsof order 10-8 to the otherparticlesof the theory it doesof coursenotcauseany problems.In most cases,however,the introductionof such an additionalsmallparameterinthe theory is at leastunsatisfactory.We will discusstheseproblemswith the light singlet in moredetailin the context of the N= 1 supergravitymodelsin the nextchapter.

We shall stop hereour discussionof the globally supersymmetricmodels. The readerwho hasfollowed the discussionup to this point will now be able to constructhis own model and discover(ornot) the potentialproblemsit might contain.

6. Low energysupergravity:basics

In the lastchapterwe haveseena variety of modelsbasedon global supersymmetry.Supersymmetrywas spontaneouslybroken in the so-calledhiddensectorwhich did not coupledirectly to the so-calledobservablesector. In fact the most promisingmodelswere those in which the supersymmetrybreakingscalewas quite largeM~� 1010 GeV which of courseimplied that the hiddensectorwas actually onlyvery weakly coupled. We were then in a situationin which although supersymmetrywas broken at

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 77

M5 � 1010 0eV it could still protectthe scalarmassesof the observablesectorto stay in the 100 0eVregion. In this sectionwe will try to couplethesemodelsto supergravity.In principle onehasfinally tocouplemodels to gravity, it is just a questionwhethergravity hasan influenceat low energies.In thecaseof the standardSU(3)x SU(2)x U(1) modelgravity has no influenceon the physicsat 1000eV andit is believedthat theseeffectscan be neglectedup to energyscalesthat are comparableto the Planckmassof 10190eV. In the caseof supersymmetricmodelsonehasalsoneglectedgravitationaleffectsupto now. To include them it is naturalto do it in the framework of local supersymmetrywhich alreadyincludes gravity. Here as in the nonsupersymmetriccase one has of course now to deal with anonrenormalizabletheory. Although models basedon N = 1 supergravityhave a better ultravioletbehaviourthan usual gravity, it is still nonrenormalizable.To be able to work with such a theory wedefine the theorywith a cutoff A of the order of the Planck mass.This explicitly assumesthat thereexistsa meaningfultheory of supergravity(which we do not know yet) to which N= 1 supergravityis anapproximationat scalessmaller thanM~and that the effectsthat finally render the completetheorymeaningfulhaveno direct influenceat energiessmall comparedto M~.It is sometimesarguedthat sucha mechanismmight be providedby extendedsupergravity.

Having statedtheseassumptions,as in the usual theoriesthe questionremainswhether(and underwhich circumstances)(super) gravity can haveinfluence at all on the low energysector we want todiscuss.

6.1. Thegravitino mass

A quantity relevant for this discussion is the gravitino mass. In supergravity the graviton isaccompaniedby a spin 3/2 fermion, the gravitino. In the unbrokencasegraviton and gravitino aredegenerate and massless. Wehave seen in Chapter 3 that the discussion of the gravitino mass term isquite subtle if the cosmologicalconstantis different from zero. We will for the moment assumethatA ~0 to simplify the discussion.If supersymmetryis broken this manifests itself in a splitting of thesupergravitymultiplet: the gravitonwill stay masslessandthe gravitino will receiveamass.As usualthegravitino mass is not what gives directly the amount of supersymmetrybreakdownbut we havetherelation(3.31) [114,101]

m312= M~/MV3, (6.1)

in which M = M~/V8~r= 2.4x 10180eV andM5 denotesthesupersymmetrybreakdownscaleobtainedthrough the vacuumexpectationvalues of auxiliary fields. Let us now assume,as happensin someglobally supersymmetricmodels, that M~= 10100eV. Through (1) this leadsto a gravitino mass of1000eV. It is this fact that could tell us that gravity effectshaveto be includedif one wants to discussssupersymmetricmodels with a large breakdownscale [42,444,445]. If one would have, e.g., M~=

1000eV the gravitino mass would be tiny and as in the caseof the usual modelswe would safelyassumethat gravitationaleffects can be neglectedin the discussionof the modelsat 1000eV. But inmodelswith M~� 1010 0eV sucheffectsmightbecomeimportant.This is actuallynot sucha big surpriseif onelooksat themodelswith largeM~.The hiddenandthe observablesectorsin suchmodelsareonlycoupledvery weakly, but all the particles under considerationare believed to participate in thegravitational interactions. It is then just a question at which point the gravitational interactionsdominatethe other interactionsbetweenthe hidden and the observablesector.Equation (6.1) mightindicate that this happensat M5 = 10100eV. It is still the question whether the splitting of the

78 H.P. Nil/es, Supersymmetry,supergravityandparticlephysics

supergravitymultiplet is representativefor the supergravityinducedsplittings in the observablesector.It might very well be that the observablesectoris still unaffected.Indeedthe first massformulasfor thecouplingof onechiral superfieldto supergravitygaveus [227,97,98]

STrAI2=0, (6.2)

evenin the presenceof supergravitybreaking(at the tree graphlevel), which if it generalizesto morefields would indicatethat the observablesectormight not be affectedat all by this typeof supergravitybreakingat the tree graph level. Let us now, however,couplemore chiral superfieldsto supergravityand let us for the sake of simplicity assumethat we haveminimal kinetic terms. Formula (6.2) thengeneralizesto formula (3.43) [100,101]

STrAl2 = 2(N — 1)m~12. (6.3)

We could now treatoneof thechiral fields as the hiddensector responsiblefor supergravitybreakdownand the other(N — 1) fields as the observablesector (usualquark, lepton and Higgs superfields)[101].Both sectorsareonly coupledthroughgravitationalinteractions.The hiddensectorincluding onechiralsuperfieldas well as the graviton andthe gravitino couldthen satisfy a massformulalike (6.2) andthe(N — 1) chiral fields in the observablesectorwould havea nonvanishingSTrAl

2 at the tree level. TheSTrAl2 is positiveindicating that the bosonsareon averageheavierthan their fermionicpartnersby anamountof orderof the gravitino mass.In this casethe observablesectorhasmasssplittingsas largeasthe supergravitymultiplets confirming our suspicionthat in modelswith M~� 10100eV gravitationaleffects could influencethe observablesectorand cannotbe neglectedin general.In addition formula(6.3) indicatesthat thesegravitationaleffectsdo everythingin an easyway, what we were trying so hardto realizein globally supersymmetricmodels.They lift the massesof the bosonicpartnersof quarksandleptonsby m

312 alreadyat the treelevel, violating thesupertraceformulasof global supersymmetry.Theconditionsunderwhich (6.3) was derived,however,are not the mostgeneralones,nonminimal kineticterms and gauge multiplets might be included which we will discussin following sections.Having,however,someexperiencewith the supertraceformulas from the last chapterwe expectgravitationaleffects to play a role in the observablesectorif M~is large, barringsomeaccidentalcancellations.Wecould even imagine that in the caseof M~~ 1010GeV the supersymmetricpartnersof quarks andleptonsareno longer protectedto havemassesin the 100GeV rangeandgravitationaleffects raisethemassesto scaleswhich render thesemodelsuninterestingfor our discussionin which wewant to relatethe supersymmetrybreakingscaleandthe breakdownscaleof theweak interactions.In principle, thesegravitationaleffects will then give an upper limit on M5 for the modelsunder consideration,above acertainvaluethe modelswill no longer contain supersymmetricpartnersin the 1000eV range.Beforewe discussthesequestionsin detail, let usmention someotherimprovementsthat occurin the contextof supergravity.

First of all the theory now containsall the interactionswe know of. This includesnow alsothe Planckmass,believed to be a very important mass scale in physics. One motivation for us to considersupersymmetricmodelswas an attempt to understandrelationsbetweenmassscaleslike, e.g., M~andM~.Maybesuchscalescan only beunderstoodif the Planckmassis alsoincluded.It might verywell bepossiblethat we can understandthe magnitudeM~not alonein termsof Mx but only in termsof bothM~andM~.

Secondlysupergravityhas the nice feature that the Goldstone fermion which arises in globally

H.P. Ni//es, Supersymmetry,supergravityandparticlephysics 79

supersymmetricmodels now disappearsvia the super-Higgseffect. There is no longer a masslessessentiallydecoupledfermion in the theory.

Our last commentin this section concernsthe cosmologicalconstant.In global supersymmetrythevacuumenergywasan orderparameterfor supersymmetrybreakdown.Sincewe know that supersym-metry hasto be broken in naturethis leadsin the context of spontaneouslybroken global supersym-metry to a nonvanishingpositive vacuum energy. If this would be true also after the coupling tosupergravitywe would know that thesemodelsarenot relevantfor a descriptionof nature.Fortunatelythis is not true as we haveseenin Chapter3. Unbrokensupergravitycan occurnot only with vanishingbut also with negativecosmologicalconstant. Spontaneouslybroken supergravitycan occur for anyvalue of the cosmologicalconstant.In particular we can havespontaneouslybrokensupergravitywithA = 0. In all caseswe know this is obtainedthrough very specialchoicesof theparametersof the model.We do not know why A is zero but at leastwe can fine tuneit to any valuewe want.

6.2. Thehiddensector

The models we consider contain two sectors:a hidden sector responsiblefor the spontaneousbreakdownof supergravityandthe observablesectorcontainingquarks, leptonsand Higgs superfieldsthe gaugemultipletsand in caseof a grand unified model also the correspondingadditional particles.Thesetwo sectorsonly communicatethrough the gravitationalinteractions.This implies that the fieldsin the hiddensectorare singletswith respectto all gaugeinteractionsin the observablesectorandalsodo not haveYukawa couplingsto this sector.Of courseone could haveothervery weak interactionsbetweenthe two sectorsbut sincethereareno indicationsandno reasonswhy theyshouldexist we donot consider them. In this section we want to discussthe hidden sector and the possibilitiesfor aspontaneoussupergravitybreakdown.We will first do this in the context of what was called minimalkinetic termsin Chapter3 whichimplies for the Kahlermetric G~= —1 everywhere.If thehiddensectorconsistssolely of chiral superfieldsin addition to the supergravitymultiplet the relevantquantity tostudy is the scalarpotential (compare(3.25))

V = exp(—G)[G5G~— 3], (6.4)

andsupersymmetryis brokenif G, hasa nonvanishingv.e.v.,the supersymmetrybreakingscaleis givenby (3.30)

M~=M2exp(—G/2)G

1, (6.5)

with a gravitino massterm

m312 M exp(—G/2). (6.6)

For Evac= 0 onehas IG,12 = 3 and oneobtainsrelation (6.1) betweenM

5 andm312. With theseminimalkinetic termsthe Kählerpotential G for onesuperfield(z,x~F) is given by

G(z,z*) = _zz*/M2— log(g(z)2/M6), (6.7)

andwe havestill the freedomto choosethe superpotentialg(z) at ourwill. In (6.7) we haveindicated

80 H.P. Ni/les, Supersym~neiry,supergravityandparticle physics

that g hasthe dimensionof (mass)3to indicate the connectionwith the global limit, in which V is ofdimension(mass)4andV= g~2. For this specialcase(6.7) the potential(6.4) can now be written as

v= exp(~)[ +~jsg~_~igI2], (6.8)

and supersymmetryis broken if F~= g1~+ (z*!M

2)g hasa nonvanishingv.e.v. To becomeacquaintedwith this potential let us first discussthe simplest case[324]:a constantsuperpotentialg = m3. Thepotential

V = m6exp(zz*/M2)[~z12/M4 — 31M2] (6,9)

has as stationary points z = 0 and z~= MV2 out of which only z = 0 correspondsto unbrokensupersymmetry.For z~= MV2 we have FZ~= \/2(m~/M)andsupersymmetryis broken.The form ofthis potential is shown in fig. 6.1. The supersymmetricstationary point is a local maximum and theabsoluteminimum is obtainedfor Izi = M\/2 with a vacuum energy Evac= e2m6/M2and brokensupersymmetry.Observe that here in contrast to O’Raifeartaigh models in renormalizableglobalsupersymmetry,supergravitycan be broken in the presenceof one field only. Observefurther thatsupergravitybreakingminima can be the lowest oneseven if supersymmetricstationarypoints exist,again in contrast to the globally supersymmetriccase where the existenceof a supersymmetricstationarypoint implied unbrokensupersymmetrywith Eva,,= 0. One might now askthe questionhowone could arrive at Eva,,= 0 in the model given above.The only way to do this is by putting m = 0. Inthis casethe potentialhoweveris identically zero andno well-definedvacuumstateexists.

We areinterestedin a casewherea nonsupersymmetricminimumwith Eva,, = 0 exists.The simplestexample of such a case can be realized in this context through the Polonyi superpotential[483]g = m2(z+ /3) in which /3 is a dimensionfulparameterusedto adjustEvac to vanish.Since this model isa widely used hidden sector in the construction of models we will discussit in detail. The orderparameterof supergravitybreaking is now given by F~= m2+(m2/M2)z*(z+ /3). Solutions to theequationF~= 0 do not exist if /3 <2M andwe aresure that supergravityis broken in this rangeof ~3.We want now to investigatewhetherin this rangethereexist stationarypointswith V = 0. Thesearethesolutionsof the equation

(/3 + z+ z*)[M2+ z*(z+ /3)] = 3M2(z*+ /3). (6.10)

V

Fig. 6.1. Scalarpotential (6.8) in thepresenceof aconstantsuperpotential.

H.P. Nil/es,Supersyminetry,supergravityandparticlephysics 81

Solutionsto theseequationsonly exist for the four values/3 = (±2±V3)M Only the values (2—and(—2 + \/3)M lie in the rangewherenosupersymmetricsolutionsexist. Theycorrespondto v.e.v. ofthe z-field, z = (V3 —i)M and z = (1 — V3)M respectivelyat the minimumof the potential.The othersolutions/3 = ±(2+ V3)M lie outsidethe range /31 <2M. An inspectionof the secondderivativesof thepotentialshowsthat theselatterstationarypointswith V = 0 arenot local minima but saddlepointsandcan therefore not correspondto ground states_ofthe model. For l/~I<(2— \/3)M the potential,however,is positive definite, andfor /31 = (2— \1’3)M the potentialhasan absoluteminimum at V = 0and supergravityis brokensince FZI = m2V3 � 0. The resultinggravitino massis

m312= (m2/M)exp((V3—1)2/2). (6.11)

The model with g = m2(z+ /3) shows thus the desired behavior. The real scalarshave (masses)22\/3m~,

2and 2(2— V3)m~,2and the formula (6.3) is satisfied [43]. The fermionic partner of z isabsorbedby the gravitino.

A hidden sector with one chiral superfield and superpotentialg = m2(z+ (2— \/3)M) is thus

acceptable.Supergravityis brokenat a scalem2 through v.e.v.’s of the z-field which areof orderof thePlanckmassand the vacuumenergycan be fine tunedto vanishfor /3 = (2— V3)M. Observe,however,that the v.e.v. of the scalaris of orderof M~and this is in the rangein which our approximationat thetree graph level might break down. Notice also that the superpotentialg = m2(z+ /3) is not natural inthe strict sense,it is not the most generalconsistentwith its symmetriessincethe termsz2 and z3 havebeenomitted.A superpotentialg = m2zwould be natural sinceit hasan R-symmetrythat forbids thesehigher-orderterms.The introductionof /3 destroysthis symmetry. In generalintroducingtermsz2 andz3 in the superpotentialdestroysthe nice featureof g = m2(z+ /3) to havesupergravitybrokenandtohavethe absoluteminimumat V = 0. In the moregeneralcasessuch a behaviorcan only be obtainedfor local minima at V = 0. Thereare, however,alsoexceptionsto this rule, andwith someeffort otheracceptablesolutions can be found. These investigationsrequire a minimalization of very complicatepotentialsand usuallynumericalmethodshaveto be used to find the minima [378].

Let uscomebackto the questionof naturalness.Doesthereexist at least oneexampleof a hiddensector that is the most generalconsistentwith its symmetriesandthat has its absolutesupersymmetrybreaking minimum at V= 0. The answeris yes but the example I know of has nonminimal kineticterms.It is basedon a Kählerpotential [87]

G = — a (~*)2 — log(Im2z2), (6.12)

andthe vacuumenergy(at the absoluteminimum) vanishesif

a = ~+ ?~[(2+V3)1/3+ (2— V3)L’3]. (6.13)

The superpotentialg = m2zhasan R-symmetrythat forbids additional terms.After the discussionof thesespecialexampleslet us discusssomemore generalpropertiesof the

supergravityscalarpotential[555,446]. Our first remarkconcernsthe role of supersymmetricstationarypoints.Let usconsiderthe generalcasewith

G = K(z,,z~)—logg(z5)I

2, (6.14)

82 H.P. Ni//es, Supersymmetry,supergravityandparticlephysics

wherethe index i labelsthe variousscalarfields. The resulting scalarpotentialis

V = exp(KIM2)[Fj(K_l)~.F*J— (3/M2)~gl2], (6.15)

with

F, = g1 + (l/M

2)K1g; g, ag/az

5. (6.16)

Supersymmetricsolutionsexist if F, = 0. An inspectionof the derivativeof the potential (6.15) showsthat pointswith F, = 0 arealways stationarypoints. We also know that such supersymmetricsolutionscorrespondto negativepotential V= —3 exp(K/M2)(g12/M2).This thenimplies that thetrue vacuumofthe theory whethersupersymmetricor not hasto havenegative vacuum energy,with, of course,theexceptionthat for Evac= 0 thereis a degeneracyof a supersymmetricand anonsupersymmetricgroundstate.With this one exceptionone should look for potentialsin which the F, = 0 haveno commonsolution in order to constructthe desiredsituationwherethe absoluteminimumbreakssupersymmetryand hasEva,, = 0. One might of course also consider situations in which the theory lives in a falsevacuumwhich is stable(enough).In this casemanyof the restrictionsquotedabovedo not apply.

Let us now also discussa special relation betweendifferent supersymmetricminima. Recallthat in a globally supersymmetricmodel different supersymmetricminima are always degeneratewithEva,,= 0. Such a situationoccurred,e.g., in many SU(S) grand unified modelswheredifferent super-symmetric minima with SU(S), SU(4)X U(1) or SU(3)X SU(2)x U(1) or otherswere present.Thisproperty is no longer true in the context of supergravity[555,77]; different supersymmetricminimahave Vo = —3(1g12/M2)exp(K/M2) and are only degenerateif they have the same value of thesuperpotentialwhich is in generalnot the case.The highestof theseminima is the onewith g

0 = 0 andhence V0 = 0. The other supersymmetricminima have in general V0<0. Nonethelessthere areargumentsthat the highest minimum might be stable [555,91]. In general, however, thesecon-siderationsonly apply to the so-calledobservablesector. In the hiddensectorsupersymmetryis brokenand the question of different SU(S) minima can only be answeredin the context of specific models.There are models where only one minimum with the correct breakdown of SU(S)—~SU(3)x SU(2)x U(1) exists [438,149].

Up to now we havediscussedonly the mechanismwhich in the context of globally supersymmetricmodels was called F-type breaking. One could also consider D-type breaking. This requires theexistenceof a U(1) factor in the gaugegroup before or after the gauge symmetry breakdown,andacceptablemodelsare only obtainedof this U(1) cannotbe embeddedin a grand unified group as wehave discussedin Chapters2 and 5. In the global case such models have usually problemswithanomalies.In the caseof supergravitythereis a further restrictionon thesemodelsbecausethey areonly consistentin the presenceof an exactR-symmetry[42]. All theserestrictionsdo not really rule outthe possibility of D-type breakingbut they are so strong that nobody hasreally tried to constructasatisfactorymodel.

Another possibility is a dynamicalbreakdownof supersymmetrydue to condensationprocessesinstrongly interacting gauge theoriesin the hidden sector [111—116].This breakdownhascertain nicefeaturesbut is of coursevery difficult to computequantitativelyapartfrom the fact thatsomedynamicalassumptionshaveto be madethat cannotreally be provenrigorously. We havealreadydiscussedthe(~i~i) condensationin the context of supercolor,which of coursecould also be usedherein the hiddensector.But therearealso otherpossibilities,like gaugefermion condensation.From all our discussion

H.P. Nil/es,Supersymmetry,supergravityand particlephysics 83

we arequiteconfidentthat sucha AA-condensationdoesnot breakglobal supersymmetry[443,543]. Inlocal supersymmetrywe do not know. A specialexamplebasedon an effective Lagrangianapproach[543],however,hasshownthat such abreakdownof supergravitymight be possible[444].This is quiteinterestingin connectionwith the discussionof the massscales.For this rememberthat in the exampleswith a scalarhiddensector,the supersymmetrybreakingscaleof let ussayM5 = 10100eV hasbeenputin by hand, e.g. by choosingm = 1010GeV in the Polonyicaseand the v.e.v.’sof the scalarfields weretypically of the orderof M~.Herethe scalep.

3 of the (AA) condensateis determinedthroughthe stronggaugecouplingin the hiddensector.Knowing that thisprocessdoesnot breakglobal supersymmetrywededucethat M~= f(p., M) hasto vanish in the limit M ‘-~ c~in which we switch off the gravitationalinteractions.The largestsupersymmetrybreakingscalewhich can arise in this caseis then M2

5 = p.3/M

which is also realizedin a specific model. This leadsto a gravitino massof order m312 = p.

3/Ma. Agravitino massof thedesiredmagnitudecanthusbe obtainedevenfor p. ~ 10100eV. Sucha model thenhasthe two input scalesp. andM both large comparedto M

5. Nonethelesstheyconspireto producethetiny scale m312 and could provideuswith an understandingof the small scaleof the breakdownof theweak interactions[446].Of coursesucha model is very hardto discussquantitatively.

The problemwith all the hiddensectorsis of coursethat from their very naturethey arenot (easily)accessiblethrough experimentand are subject to theoreticalspeculationsand taste.The only way inwhich they can be restrictedis through their influenceon the observablesector.This, however,is ingeneralnot very restrictive,sincewidely differenthiddensectorscould still havethe sameimplicationson theobservablesector.Our task thereforehasto beto isolatethe parametersinducedby the hiddensector into the observablesectorin a model independentway.

6.3. Thecouplingof the hiddensectorto the observablesector

Thereare severalwaysto couple thesetwo sectorsand wewill first discussthe onemostcommonlyused[43,77]. Let us denotethe fields in the hiddensectorby z5, the fields in the observablesectorby y,,.andlet uswrite for the superpotential

g(zj,ya)=h(zj)+g(ya), (6.17)

and let us use for this first discussionminimal kinetic termsand postponethe discussionof the mostgeneral case.We assumethat supersymmetryis broken in the hiddensectorwith vacuumexpectationvalues

(z5) = b,M, (h1) = (ah/oz1)= a~mM, (h) = mM2, (6.18)

and that the minimum hasEva,, = 0. We also assumethat the superpotentialof the observablesectordoesnot contain the largescaleM andthat all the vacuumexpectationvalues in the observablesectoraresmall comparedto M. The scalarpotential of sucha model is then given by

v= ~ h,+~g~2~ •~ ~ (6.19)

This is now a very complicatedexpression.Since,however,the scalesin g(y~,)are all small comparedtoM we can simplify our discussionby using a leading order approximation in p./M where p. is a

84 H.P. Ni//es,Supersymmetry,supergravityand particlephysics

characteristicscaleof g(ya). If the p. is of orderof 100 0eV this approximationis fantasticallygood. Ifthe modelalso includesthe grand unificationsomecorrectionscould occur.They, however,affect onlythe quantitativebut not the qualitative result of our approximation as we will see later. Thisapproximationis obtainedin the so-calledflat limit [43] M but m312 fixed. The gravitino massm312through (6.18) is of order m in our model. The low-energy effective potential is then obtainedbyreplacingz5, h, andh by their v.e.v.’s andkeepingonly thosetermswhichdo not vanishwhenM —* (mfixed). It can be obtainedfrom (6.19) by a straightforwardcalculation.With

m312= exp(~Ib5l2)m, (6.20)

and the resealedsuperpotential

g(ya) = exp(~Ibs!2)g(ya), (6.21)

we obtain[43]

V= lgaI2+ m~,2Iya~

2+m3i2[ya~a + (A— 3)g+h.c.}, (6.22)

whereya~a= >-~aya(og/oya)andA is given by [450]

A=b~(a,+b,), (6.23)

with a5 andb~definedin (6.18). For V = 0 at the minimumthesequantitiesarerestrictedto obey

~ Ia~+ b512 = 3. (6.24)

The first term in (6.22) is the usual scalar potential of a globally supersymmetricmodel withsuperpotential~. The secondterm is a commonmassterm for all the scalarparticlesin the observablesectorand is equalto the gravitino mass.In sucha model all the scalarpartnersof quarksand leptonsreceivea commonmassof order m

312.This is a very desiredproperty andit confirms our expectationsfrom the discussionof the supertraceformula that m312 is the scalerelevantfor the low energysector.That they receiveall a commonm~,2~yaI

2contribution is not true in the generalcase.It is due to thechoice of minimal kinetic terms in the observablesector. Nonethelessthis splitting of the scalarsfromthe correspondingfermionicpartners(already at the treegraph level) is exactlywhat we were lookingfor in all the attemptsof supersymmetricmodelbuilding.

The third term in (6.22) gives us new trilinear bilinear and linear couplingsbetweenthe scalars.Itcontainsa single numericalparameterA which dependson the choice of the hiddensector [43,450].For our example h(z)= m2(z+ /3) with /3 = (2— V3)M and (z)= (V3— 1)M one would obtain A =

3 — V3. Observethat this last term in (6.22)breaksall the R-symmetriesof the superpotential~. This isbecausethe superpotential,~(z)necessarily transforms nontrivially under any R-symmetry since5 d2Og(z) hasto be invariant.The modelswe can now considerarethus allowedto haveR-symmetries(which e.g. could forbid someunwantedtermsin ~) andwe havenot to be afraid of Goldstonebosonsthrough the necessaryspontaneousbreakdownof such symmetries,necessaryto allow gauginomasses.This unsatisfactoryfeatureof mostof the globally supersymmetricmodelsis hereabsentbecausethese

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 85

R-symmetriesareexplicitly brokenthroughthe coupling to supergravity.Onesuch R-symmetrywouldbe the one where all superfieldsin the standardmodel have a commoncharge R = 1 and where0-transformswith R = 3/2. In this case ~(Ya) would only contain terms trilinear in the fields. Inparticular this superpotentialthen does not contain explicit mass parametersof lets say 1000eVarbitrarily put in by hand.Such an R-symmetryhasbeenusedas a naturalnessconditionfor modelsinwhich the scaleof the breakdownof the weak interactionscan be understoodin termsof the gravitinomassaloneandwhereSU(2)x U(1) breakdownat the right scaleis inducedthroughthe breakdownofsupergravity.With suchan R-symmetrythe potentialin (6.22) simplifies to [450]

V= ~ (6.25)

andthe R-symmetryis explicitly brokenby the term proportionalto A in the processof spontaneousbreakdownof supergravity.

The potential (6.22) has the form of the potential in an explicitly (but softly) broken globallysupersymmetricmodel. The soft breakingterms are the yay~massterms of the scalarsand the scalarcouplingsproportionalto A. Spontaneouslybrokensupergravitythusmanifestsitself in the low energytheory as soft explicitly broken global supersymmetry[43], a property very convenientfor modelbuilding as we haveseenalreadyfrom our discussionof STr Al2 in the beginningof this chapter.

The potential in (6.22) doesnot representthe most generalcaseand we will discussmore generalcasesin the nextsection.Beforewe do this, let us addsomecommentsto (6.22).The parametersA andm2Iyal2 aredefinedat a very largescaleM. If one now considersthe supersymmetricSU(3)X SU(2)XU(1) model the values of theseparametersat a small scale of 1000eV haveto be obtainedby arenormalizationgroupcalculations[324].To renderour discussionmoretransparent,we will postponetheseconsiderationsto the nextchapter,but thisfact shouldbe kept in mind. If theconsideredmodel isa grandunified model the samestatementapplies.In addition to that, if onewantsto constructthe lowenergypotentialin the region of m

312the heavygrandunification fields haveto be integratedout [307]and this changessomewhatthe parametersof the low energypotential. If one has,e.g., a low energysuperpotential~ = + + g

1 where ~‘ denotesthe termsthat are i-linear in the fields the A termsin(22) would read

Ag3+ (A — 1)g2+ (A — 2)g1. (6.26)

In the casewith heavy fields, this ratio A : (A — 1): (A — 2) will change[307].Theseeffects haveto becomputedin the explicit modelsunderconsiderationand will of coursedependstrongly on the grandunified model used.

Before we discussmore generalcases,let us mention that the usedcouplingof the hiddenand theobservablesectoras given in (6.17) is not unique.An interestingalternative[93]is to addthe logarithmsof the superpotentialinsteadof the superpotentialsh and g themselves.The Kähler potential of thehiddenand the observablesectorareadded

G’ * * ‘~— — ~ YeY�~— I lh(z)12 — 1\Z5, Z51Ya,Ya.’ — M

2 M2 og M6 og M6 ‘

i.e., theresulting superpotentialis a productof the separatesuperpotentials.This is as good_acouplingof the two sectorsas the otherone. In a specialcaseone couldchooseh(z)= m2(z+ 2— V3) and

86 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

g(z, y) = h(z)(l + g(y)/M2m312). (6.28)

The low energysuperpotentialthat arisesin this coupling is

V = ga + m312y~12 + ~D2. (6.29)

Comparingthis to (6.22)we seethat both potentialscoincidefor A = 3. Thecouplingof the two sectorslike (6.27) alwaysgives A = 3 independentof the specialform of the hiddensector.We will later seethat A = 3 is a very specialvalue.

6.4. Themostgenera!case

In the last section we have seen that spontaneouslybroken supergravitymanifests itself at lowenergiesas a theory of soft explicitly broken global supersymmetry.The soft breakingparameterspresentin (6.22) wherescalarmassesm~,

2IyaI2and the terms m

312Ag.As we haveseenin (2.83) therearemorepossibilitiesfor soft breakingtermsandwe will discussin thissectionwhethertheycan occurhereor not.

We start with a gaugino massterm. In a grand unified model with heavy particles that transformnontrivial under SU(3)x SU(2)x U(1) massesfor the gluinos, winos, zino andphotino will be inducedthroughradiativecorrections.This happensalreadyat theone-looplevel sincesupersymmetryis brokenat the tree graph level for theseheavyparticles.The graph in fig. 6.2 gives a finite contributionto thegauginomasses.In the caseof m0~ m312 this contributionis [252,45,112]

= (a5/2rr)m312~ T(R,,,), (6.30)

where T(Ra) is the quadratic Casimir operator of the heavy fields in representationRa. Thecontribution of light particles m0~m312 is of order am~/m312and small. In addition to this one-loopcontributiontherearelogarithmicallydivergent[252]two-loop contributionsfrom the graphsin fig. 6.3,whichcan howeverin generalbe neglectedcomparedto the formerones.The inducedradiativegauginomassesthus dependstrongly on the particle content of a grand unified sector. In SU(5) we haveT(5)= 1/2, T(24)= 5, T(50) = 35/2 and T(75)= 25 and the induced gauginomassesmight reachthemagnitudeof thegravitinomasses(compare(6.30)).In generalonedoesexpectthesegauginomassestobe present[112].

Apart from theseradiative contributionthereis alsothe possibility to havenonzerogauginomassesat the threegraph level [98,324], which requiresthe presenceof nonminimal gaugekinetic terms.Tosee this rememberthe term 5 d

20f~~W”W’~in (3.15). The gauge kinetic terms are nonminimal if

(p _-.*-_

“I __________________________________________K 4 ~ ~

a ma a X a a

Fig. 6.2. Contributionto gauge fermion (A) massesat the one’loop Fig. 6.3. Logarithmically divergent contribution to gauge fermionlevel. 0 denotesa quark and ~ itsscalarpartner. massesat the two-looplevel.

H.P. Nil/es,Supersymmetry,supergravityand particlephysics 87

fall � 8at, wheref is a function of the chiral superfieldsin the theory.The generalformulaof the N = 1supergravityaction containsa term (secondterm in (3.20))

~exp(—G/2)G5(G~)~~àf4A’~”AP, (6.31)

a potentialgauginomassterm.This masscan of courseonly bedifferent from zero if supersymmetryisbroken, comparing(3.22) we can write (6.31) as ~.M2

5f*~AA,but in addition one needsa nontrivialfunction f(z,). Sincef is dimensionlessf’ hasdimension(mass)

1and sincein the scalarsector the zfields havev.e.v.’s of the orderof M suchtermswill leadto gauginomassesof order M~/M— m

312. Inaddition to the two typesof soft breakingsdiscussedin the last sectionm~,2IyaI

2andm312Aga common

massof order m312 for the gauginosthus constitutesan additional soft breakingterm of the effectiveactionof the observablesector,which will be presentin the generalcase.

More complicationsarise if we also allow nonminimal kinetic terms for the chiral superfields[516,79]. We thenhaveto considerthe potential (6.15)

v= exp(1~~)[F,(K_1)~F*i_~lgI2], (6.15)

where~ is to be understoodas g(z5, ye). In generalwe couldwrite

g=~M~f~(z5,Ya), (6.32)

but a necessaryconditionfor the y-fields not to be as heavyas M reducesthis form to

g(z5,y~)= M2h

2(z5) + Mh1(z5)+ f0(z,, ye), (6.33)

where only the last term dependsalso on the observablefields. By the same argumentswe can

parametrizethe mostgeneralform of the kinetic terms

K = M2K

2(z,, z’~)+ MK1(z5,z~)+ Ko(z5,z~,y~,y~a). (6.34)

One,however,would like to constrainthe theory in such a way that the observablesectorin the flatlimit is renormalizablewhich requires

K0 = Aab(Zi, z~)yay~+ F(z5,Z~,Ya) + F*(zs, Z’~’,y*a). (6.35)

Inserting(6.33) and(6.34), (6.35) in (6.15) gives in the flat limit

V= Igal2+ m

312(h(y)+h*(y*))+ ~ (6.36)

with g definedthrough

exp(~r)fo= g(y~)+ f(z5, ye), (6.37)

88 H.P. Ni//es,Supersymmetiy,supergravityand particlephysics

andh(Ya) is an arbitrary additional function containedin f(z~,Ya) completelyindependentof g(ya). Sab

finally is a complicatedfunction of Aab, h2 and K2 [516].The appearanceof such a nontrivial Sab�6ab

implies a nondegeneracyof the m~,2Iyat

2massterms and it is due to the nonminimality of the kinetictermsin the observablesector,andits existenceconfirms the expectationsoriginating from the generalSTr itt2 formula (3.46) [294].

The termsin the potential(6.36) constitute(togetherwith the gauginomasses)the mostgeneralsoftbreakingthat can be inducedthrough a spontaneousbreakdownof supergravity,consistentwith somenecessaryconditions (6.33) and (6.34) arising from the requirementthat the yafields do not acquiremassesof orderM. Whetheror not theseconditionsare also sufficient is not yet provenand is in factvery difficult to decide.Theproblemlies in thefact that before oneconsidersthe potentialone decideswhich fields are“heavy” z

5 andwhich are “light” Ya. In generalthis questioncan only bedecidedafterthe minimization of the potential. In the derivation of (6.36) the necessaryrequirements(6.33) and(6.34) makesurethat therearenot obviousM

2~yaI2terms. Inspecting,however,the first term in (6.35)we see that z-fields and y-fields mix in the kinetic terms which makes it of course difficult andambiguousto decidewho is z~andwho is Ya, andadditional criteria are apparentlynecessary.Only invery specialcasesof g(z

5,Ya) the separationof z~andy~at this level seemsto be consistentafter theminimizationof the potential. It is not known that thesegeneralcasesexist at all; no explicit modelhasbeenconstructedin which the generalsoft breakingin (6.36) are realizedafter a minimization of thecompletepotential,nor is it proven that thesemoregeneralfunction h(y) and Sab(z,)cannotoccur in anontrivial form. To resolve this question is very important and it seemsto me that further study isabsolutelynecessary.

Of course the choice of the hidden sector at this stage is arbitrary; one would, however, in theabsenceof an explicit examplethat inducesthe generalterms in (6.36)be temptedto assumethat not(6.36) but (6.22) constitutesthe most general form of the scalar potential inducedby a spontaneousbreakdownof supergravity.This situationwould actuallybe preferablefor severalreasons.First of all(6.22) is lessgeneralandif it is the mostgeneralthiswould imply that thesetheorieshavea predictivepower independentof the specialchoiceof the hiddensector,which of coursewould be lost if (6.36) hasto be considered.Secondlythe universalityof the m~/2lyaI

2termsis a very desiredproperty in (6.22),which everybodywill agreewith if he has followed our discussionconcerningflavor changingneutralcurrentsin section4.6. If all the general soft breakingterms in (6.36) could be inducedwe actuallywould be forced to adjust a numberof thesetermsin order to avoid phenomenologicalproblemsandthe generalapproachalonewould, e.g.,not explain the absenceof flavor changingneutralcurrents.Wedo not know for certainat the momenthow to answerthis question.We will, however,in thefollowingassumethat Sab = &~b and work with scalarpotentialsof the form of (6.22) andhope that this will becloseto the mostgeneralform that can beobtained.Actually a slight generalizationof (6.22) hasbeenfound in a specialcase[226].Therethe m2lyaI2 termsarestill universalbut are not necessarilyequaltom~,

2.For m2>0 such a change,however,can in generalbe absorbedin a redefinitionof A.

We shouldstressagain that the parameterscontainedin (6.22) as well as the value of the gauginomasses(the otherpossiblesoft breakingterm) are definedat a very high massscaleand the valuesatlow energieshaveto be obtainedthrough renormalizationgroupimproved perturbationtheory. Theywill be different at low energiesjust in the sameway the massesof the bottom quark and r-leptonmassesin grandunified modelsdiffer in valueat low energieseven if they were identical at M~.

In the following chapterwe will discussthe implicationsof (6.22)in the frameworkof variousmodelsand investigateif the threeparametersm

312, A and a commongauginomass are sufficient to renderthesemodelsphenomenologicallyacceptable.

H.P. Nil/es, Supersynimetry,supergravityandparticlephysics 89

7. Supergravity: models

7.1. Early attempts

The first attempt to discuss a low energysupersymmetricstandardmodel in the framework ofsupergravitywas undertakenby Ovrut andWess[470].Their modeldid not containa hiddensectortobreaksupergravityspontaneouslybut just consistedof the low energyobservablefields with a massscalem of the order of 1000eV. Such a model then in generalhas a nonvanishingcosmologicalconstantA. Ovrut andWess now cancelledthis cosmologicalconstantby addinga term to the actionand then took the flat limit. The additional term breakssupergravityexplicitly, it correspondsto theintroduction of an x-independentspurion superfield of the type discussedin section 2.5. As aconsequencethe multiplets in the low energytheory are split. The low energytheory in the flat limitlooksthe sameas an explicitly (butsoftly) brokenglobally supersymmetricmodel.To havethesplittingsinside thesupermultipletsto beof the order m2onerequiresthe introductionof a term A’ = m3M;andthis is exactlythe orderof magnitudeof the cosmologicalconstantin thesemodels.Onecan seethisforexamplein our toy modelwith constantsuperpotentialdiscussedin thelast chapter,whereoneobtaineda vacuumenergyof order M~/M2.Introducing A’ — m3M then correspondsto M~— mM, a quantitywhich representsthe supersymmetrybreakingscale in spontaneouslybrokensupergravity.As in thiscaseherethe relevantscaleis M

8 — 1010GeV.The soft breakingsin the low energytheory inducedthrough the nonsupersymmetriccancellationof

the cosmologicalconstantareof the type

~ ~2 ~3 ~2~* ~~*2 + h.c. , (7.1)

all with coefficientsof order m. The terms linear in p(p*) are irrelevant and can be absorbedin aredefinition of the scalar fields. Observethat the list in (7.1) doesnot include “diagonal” masstermsm

2psp*, thesetermsare not generatedthrough the introductionof A’. As a result, thesemodelsalwaysleadto STr Al2 = 0 at the treegraphlevel, a propertywhich alwayshasmadeit quite difficult to haveitas a basis for a realistic model. The scalarsof the supermultipletsof mass m exhibit the splittingsm2+ ~m2,andmorestructurehasto be addedto the modelsto makethem acceptable.Apart from thatthe fact that supersymmetryis explicitly broken is ratherunsatisfactoryandone would like to replacethis by spontaneousbreakdownfor the usual reasons.It is, however, interesting to see how theintroductionof one breakdownparameterleads to so much structure in the low energytheory.Thistype of model has in addition the nice feature that the usual vacuum degeneracyof globallysupersymmetricgrand unified modelsis lifted exactlyin the sameway as discussedin the last chapter.

The first attempt to investigatethe implicationsof spontaneouslybrokensupergravityto modelsofparticlephysicswasmadein ref. [444].Supergravitywas broken in ahiddensectorhereconsistingof asupersymmetricpureYang—Mills theory.For the descriptionof such a theory an effective Lagrangianapproach,constructedin globally supersymmetriccontext [543],wascoupledto supergravity.The basicpropertyof this Lagrangianwas the occurrenceof gaugefermion condensation(AA) due to the stronginteractionsin the hiddensector.Whereasin the globally supersymmetriccasethis condensationdoesnot lead to a breakdownof supersymmetryin the local casean explicit exampleshowsthepossibilityofa supergravitybreakdown.As a consequenceof theseproperty the breakdownscaleM2

8 can be muchsmaller than the scale (AA) = p.

3 wherecondensationoccurs since supersymmetryis restored in theglobal limit M —~ ~. In fact in the specific modelone obtains

90 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

M2~p.3/M, (7.2)

andconsequently

m312=p.

3/M2. (7.3)

The scalep. is put in through the gaugecoupling in the hiddensector and a gravitino massas low asiO~0eV can be obtainedfrom a p. aslarge as 10140eV by this mechanism[I’ll 116].

The nextstepto discussis the communicationof the hiddensectorandthe observablesector.At thisstagethe resultsof Cremmer,Ferrara,Girardello andVon Proyen[97,98] were not yet availableandwith it the most generalform of the breakingtermsin the low energytheory.

Oneobviousway in which the breakdownof supergravityaffects the observablesectoris a splittingof the gaugefermions from the correspondinggaugebosonsin this sector. It was first speculatedthatsuch contributionsto the low energygauginomassescomesfrom a quadraticallydivergentgraph [42]like the oneshownin fig. 7.1. Later it was howevershownthat this quadraticallydivergentcontributionis exactlycancelled[45] by the onefrom fig. 7.2. But apartfrom thesecontributionsto the low energygauginomassesthereare the finite oneswe havediscussedin section6.4. With thesewe expectthegaugino massesto becomeof order am

312or m312 dependingon the heavy particles in the theory[252,45].

In the flat limit M -~ ~, m3,2fixed, the observablesectorwasglobally supersymmetricexceptfor thesesoft breakingtermsrepresentedby the gauginomasses.We are thusexactlyin the situationwhich wehavediscussedin section5.3. Fromthe gauginomassesalsomassesfor the scalarpartnersof quarksandleptonsandHiggsesareradiatively induced,

mQ=a3mg3, m~,m~—a2m~2, (7.4)

where a3 denotesthe SU(3) couplingconstant,etc. A breakdownof SU(2)x U(1) can be induced ifthereexist largeYukawacouplingsin the theory. In a model in whichthe top-quarkmassis largerthan25 0eV a negative(mass)

2will be inducedfor the Higgsscalarthat couplesto thisheavyquark,while allother scalarsof the theory remain at positive (mass)2. In this way the model has all the desiredproperties[444].The input scalesM andg~(M)lead to a gravitino massof order 1 TeV and in thepresenceof a moderatelylargeYukawacouplingthe breakdownof SU(2)x U(1) is inducedthroughthebreakdownof supergravity. If one would restore supersymmetryby force also the SU(2)x U(1)breakdownwould disappear.Therearealsono problemswith flavor changingneutralcurrentssincetheappropriatescalarparticlesaredegeneratein massas indicatedin (7.4). We will see later that such aninduction of the SU(2)x U(1) breakdownby supergravitythrough gauginomassescan be realized inmany specific modelsand is also the one which is theoreticallymost convincing.Of coursethe value

Fig. 7.1. Contributionto m~throughamassivegravitino(solid line). Fig. 7.2. Contributionto m~that cancelsthequadraticallydivergent

part of fig. 7.1 as observedin ref. [45].

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 91

m1 — 25 0eV indicates only the scale and the actual value is model dependent.It will later bedeterminedthroughthe renormalizationgroupequationsof the relevantparameters.

In the original modelonly the effectsdueto the nonvanishinggauginomasseshad beenconsidered.After the existenceof ref. [98] a more generalevaluationof the soft breakingtermsincluding those inthe scalarpotential in the flat limit was possible.After the compensationof a possiblecosmologicalconstantoneobtains[226]

V = IgaI2+ Bm3/21..VaI + m

312[ y~g.,.+ (A — 3)g+ b.c.], (7.5)

whereA andB dependon the specialform of the hiddensector. In variousexplicit examplesit can beshown that B can be positive negative or evenzero. A model with negativeB doesnot makesensesincein generalthis will alsoimply abreakdownof SU(3),,x U(1)em,andfor positiveB its effectscan beabsorbedin a redefinitionof A. Therelevanceof this additionalparameterB in (7.5) comparedto thecasedescribedin (6.22) can only be seenin relation to the gauginomasses,the othersoft breakdownparameter.With the possibilityB < 1 gauginomassesof orderm312 could be largerthan the coefficientOf IYa 12. In sucha situationit is usually easierto obtain the inducedbreakdownof SU(2)x U(1) in thesensethat this breakdowncan happenwithout the needto chooselargeYukawacouplingscorrespond-ing to largetop-quarkmasses.We will later studythesequestionsin detail.

7.2. Modelswith SU(2)x U(1) breakingat thetree level

Before explicit modelswith a radiative breakdownof SU(2)X U(1) had been investigatedin detailseveral models with a breakdownof the weak interactions at the three graph level had beenconstructed.Such modelswere first proposedindependentlyby Arnowitt, ChamseddineandNath [77]andBarbieri, Ferraraand Savoy [43]. Both models relied on a hidden sectorwith h(z)= m

2(z+ /3)where m was chosento be of order 10100eV to arrive at a gravitino mass in the 100 0eV range.Consequentlythe low energyeffectivepotential [43]was given by (6.22)

v= IgaI2+ m~,2IyaI

2+m312[ y,~.+ (A—3)g+h.c.], (6.22)

with A = 3— V3 andit was implicitly assumedthat the induced SU(3)x SU(2)x U(1) gauginomassesare small comparedto m312. Reference[77] discusseda grand unified model andref. [43] a supersym-metrizedversion of a low energystandardmodel. We will start with a discussionof the low energysectorandlater add the grandunified sector.

Apart from the graviton, the gravitino and the scalarsof the hiddensectorwith masses

= (m2/M) exp(~(V3— 1)2), m = 2V3m

312, m = 2(2 — V3)m312, (7.6)

this modelshould thencontainthe usualHiggs andquarkandlepton superfields.With thesefields alonethe most generalsuperpotentialaccordingto our prejudiceswould be (compare(4.22))

g= p.HH+A51U1HU~+A~UHD1+A~.L1HE,. (7.7)

The quantity g that entersformula (6.22) is obtainedfrom g via (6.21) andwe will in generaldrop theexponentialfactor which is irrelevant for our discussion.The particle content in (7.7) is now not

92 H.P. Ni//es, Supersyinrnetry,supergravityand particlephysics

sufficient to give a breakdownof SU(2)xU(1) at the tree graph level. The p.HH term will lead top.2(hh* + hh*) massesin the scalarpotentialand also the additional terms in (6.22) cannotlead to anegative(mass)2of a Higgsscalar.It is thusnecessaryto introduceadditional fields. The simplestchoiceis to addan SU(3)x SU(2)x U(1) singlet superfieldY anda possiblechoiceof the superpotentialis then[43]

g = AY(HH — p.2)+ Yukawaterms. (7.8)

The scalarpotential of the Higgs sectoris given by

V = m~,2(lyI

2+ hi2 + i/~I2)+ A2Ihh— p.212 + A~yh2 + A21yh12+ m312(3— V3)A(yhh+y*h*h*)

— m312(1— V3)Ap.2(y + y*) + ~D2, (7.9)

wherey, h, /~denotethe scalarcomponentsof thesuperfieldsY, H andH. We havedroppedall factorsexp(V3— 1)2 which implies that we also use m

312 = m2/M insteadof (7.6). The correctmassesin the

theory can then be obtainedat the end by multiplying all masseswith this factor. In a range ofparametersp. > m

3,2/A the minimumof (7.9) is obtainedat [43]

(h)= (UP/A) ~ (p.~)’ (7.10)

wherep andq aredeterminedthroughthe equations

(2p2+ I)q + (\./3_ l)(V3p2+ p.2A2/m~

12)= 0. q2+ (3— \/3)q+ 1 + p2—p.2A2/m~

12= 0.(7.11)

The v.e.v.’s of h and h align and are equal which minimizes the ~D2 contribution in (7.9). At this

minimumSU(2) x U(1) is broken andwe havegaugebosonmasses

= 2g~m~12(p/A)

2, M~= 2(g~+g~)m~12(p/A)

2, M~= 0. (7.12)

The Al2 matrix of the scalarsis in generalgiven by

— (g~g~’ + DaD” + m3126a gabcg + DaDb + m312h~,, 7 13

~ ab — ~gabcg*~ + DaDb + m312h~ ggCb + DaD~+ m3/2db .

where

hab = ay~y[Yc -~-+ (A — 3)g]. (7.14)

For the scalarmassesin the Y, H, H sector this implies in our casefor the two chargedones

(7.15)

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 93

andthe five neutralones

m~=2(1+q2)m~,2+M~,

m~,2= (2p2+~±[~+2p2(3—V3+ 2q)2]1’2)m~,

2,

m~,4= (2p2+ ~ + q2±[(~+ q2)2+ 2(3— V3)2p2]1~’2)m~,

2. (7.16)

The scalarpartnersof quarksandleptonshaveseparatemassmatricesof the form

(m~,2+(ain312p/A)2 oAm~,

2p/A \ (7 17)\, o-Am~,2p/A m~12+(um312p/A)

2)

whereo- denotesthe respectiveYukawacoupling. Sincethe massesof the quarksandleptonsaregivenby m

0= u(m312p/A)the masseigenstatesof (7.17)are given by

ma8—m~12+m~±Am312m0, (7.18)

wherea, b denotethe two (real)scalarpartnersof the fermionof massm0andA = 3— V3 in thecaseunderconsideration.The fermionmassesof the Y, H, H sectorcan beobtainedin the sameway as in aglobally supersymmetricmodel.Theyarenot affectedby thesoft breakingssinceweherehaveassumedthat the Majoranagauginomassesvanishat the treegraph level.

The abovemodel is an exampleof a phenomenologicallyacceptablemodel in the 1000eV rangeprovided that m312 is chosento be largerthan somethinglike 20 GeV. In the presenceof a very heavytop quarkonehasof coursealsoto requirem312+ m — (3— \/3)m312m5to belargerthan20 GeV.To beat thecorrectminimumwe hadalsoto requirep. > m312/A.Fromthis we seethat it is the introductionofthe parameterp. in the low energytheory that causesthe breakdownof the weak interactions.Wouldwe choosep. = 0 SU(2)x U(1) would be left unbroken.TheSU(2)x U(1) breakdownis thusnot relatedto the breakdownof supersymmetryandour goalto explain the onethroughthe otheris not achieved.Theoreticallythis seemsto be unsatisfactoryandwe will later comebackto this point.

Let us first discussthe inclusionof grandunification in this contextfirst proposedin refs. [77,324] inwhich the model contained£(24), H(S), H(5) andthreegenerationsof quarksand leptons Y,(5) andX,(10) wherethe numbersin the bracketsdenotethe dimensionof the SU(5) representations.Thesuperpotentialwas chosento be [77] (compare(4.27))

g = A1(~TrI~+ ~ Tr ~2)+ A2H(~+ 3M’)H + A3UHI~i+ g1X,X1H+ g~1X5Y1H, (7.19)

where U is an SU(5)singlet chiral superfield.In the globally supersymmetriccasethis leadsat the grandscaleto a potentialwith severaldegeneratesupersymmetricminima in which X, Y, H, H, U havevanishingv.e.v.’s and

(i) 1=0,

(ii) ~

(iii) Z.~= I~t[2~— 5(~~+ ~o~)]. (7.20)

94 H.P. Ni//es, Supersymmetry,supergravityand particlephysics

Solution (iii) correspondsto the desiredbreakdownto SU(3)x SU(2)x U(1). To arrive at masslessHiggsdoubletsfor the low energytheory in global supersymmetryone hasto chooseM = M’, which gives azerocoefficient in the secondterm of (7.19)for the 4, 5 componentsof H andH.

This systemis now coupledto supergravityincluding ahiddensectorh(z) = m2(z+ /3) which breakssupergravity spontaneously.The supersymmetricsolutions (7.20) obtained through the equationsag/al = 0 areaffectedby the couplingto supergravity.Forgettingthe light fields for a momentandjustconsideringI the stationarypointsof the potential coupledto supergravityare now obtainednot byag/al = 0 but by F

1 = ag/al + (l*1M2)[h(z) + g(l)] = 0 since supergravityis only broken in the

hiddensector,as we haveexplicitly seenin the last chapterin our discussionfollowing eq. (6.16).As aresultof this the coupling to supergravitywith hiddensectorh(z) inducesa changeof the v.e.v.’s in(7.20). We havenow to solve

F1 = ag/al + (l*1M

2)(m2M+ g(l)) = 0, (7.21)

wherewe haveused /3 = (2— \/3)M and haveinsertedthe v.e.v. (z)= (V3— 1)M of the scalarin thehiddensector.Notice that the secondterm in (7.21) correspondsto l*m

312 andas a consequenceof thisthe v.e.v.’s given in (7.20) will receivea shift of order m312 through the coupling to supergravity[447].This is the leadingcontributionif M ~ M. In the caseof M M thisshift will evenbe larger.

In the modelgiven by (7.19) thereexist nowthreesolutionswith F1 = 0. They will be nondegeneratesince the contribution to the potential is given by (F1)

2 — 3~gI2/M2andg(l) hasdifferent values forthesethreesolutions.In generalthis term inducesa negativecosmologicalconstantwhich, however,canbe cancelledby a slight redefinitionof ~3.Onewould like to do this in a way that the SU(3)x SU(2)xU(1) minimumis at Eva,, = 0. In the model underconsiderationthis thenimplies that the other minimahave Eva,, < 0 and the former is not the absoluteminimum. One now either arguesthat the localminimum with Eva,, = 0 is stable [555], or one complicates g(l) (e.g. by introducing additional24-representations)and arrivesat a situationin which the SU(3)x SU(2)x U(1) minimumwith Eva,,= 0is theabsoluteminimum.An exampleof sucha caseis given in ref. [438]throughthe introductionof anadditionaladjoint representationanda singlet.

Supposenow that we are in the SU(3)x SU(2)x U(1) minimum obtainedthrough superpotential(7.19). With the choice M = M’ to assurevanishing massesof the Higgs doubletsin the globallysupersymmetriccase,we now havethe following effectivesuperpotentialrelevantfor a discussionin the100 0eV range g~= A

2p.HH+ A3UHH+ Yukawa terms (7.22) where H, H now denote the SU(2)doubletsandno heavyfields enter~ The parameterp. is of order m312andappearsasa resultof afinetuning in the globallimit. It is a free parameterin the theory sincea slightly different fine tuningtakinginto accounttheshift of the I v.e.v. dueto the couplingto supergravity(compare(7.21))would allow usto choosep. = 0 [447].

It can now be shownas in the examplediscussedearlier that thescalarpotential derivedfrom (7.22)including the m~12~yi

2and trilinear scalarcouplingsleadsto a breakdownof SU(2)X U(1) for a rangeofparametersof the theory.The existenceof a nonvanishingp. in (7.22) is crucial for thisbehaviour.Thelow energyspectrumis similar to the one in the examplegiven earlier andthereare no phenomenolo-gical problems.This alsoincludesthe gaugefermionsector(e.g.gluino masses)providedthat thereexistsomeheavyparticles in the theory that havenontrivial SU(3)X SU(2) x U(1) transformationproperties,aswe havediscussedin section6.4.

A theoreticallysomewhatunsatisfactorypropertyof the two modelsdiscussedin thissection is the adhoc introductionof a massparameterp. of orderof 1000eV in the theory that is independentof the

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 95

gravitino mass.In the first examplethisparameterwas introduceddirectly, whereasin the secondoneaparticularchoiceof fine tuning led to a nonvanishingp., andas a resultsupersymmetryandSU(2)x U(1)breakdownare not really related, especiallysince the value of this parameterp. was crucial forSU(2)x U(1) breakdown.

Of course,the grand unified model of this section was a specialexampleand one might ask thequestionwhetherin more generalcasesthe situationis different. In all caseswherethe coefficient ofHH resultsfrom a fine tuning the value of p. is arbitrary andunrelatedto m312. In modelswherep. = 0resultsfrom group theory, like the examplewith 75, 50 and50 representations[416,291] discussedinsection4.5, however,the situationmight bedifferent,sincesuchmodelsneedno fine tuning to maketheHiggs doubletslight. In such modelswehavep. = 0 sincethe Higgsdoubletsdo not coupleto fields withlargevacuumexpectationvalues.As a resultp. = 0 will remain true at the threegraph level after thecouplingto supergravitywhich slightly shifts thesev.e.v.’s,anda nonvanishingp. to induceSU(2)x U(1)breakdownhasto be put in by hand.

7.3. Can we avoidthe introduction ofa small massparameter?

The motivation to consider supersymmetricmodels was purely theoreticaland related to somenaturalnessproblemswith the mass scale of the breakdownof the weak interactions.The explicitintroductionof a parameterp. (in p.HH) with the sole reasonto introduceSU(2)X U(1) breakdownisthusa procedurewhich leadsto similarproblemsthat we actuallywould requireto be solved.

Oneis thustemptedto considermodelswhereall the low energymassparametersreally arisefromthe breakdownof supersymmetry.In such modelsSU(2)x U(1) would remainunbrokenin the limitm312= 0. Such models in particular are not allowed to contain explicit small renormalizablemassparametersin the superpotential,as stressedin ref. [450].The only way to achievethis is through thechoiceof a low energysuperpotentialonly trilinear in the fields,

3g(ya) = ~ ya(ag/aya). (7.22)

In a first step we require this only for parametersof order 100GeV. If one discussesgrand unifiedmodelsone wouldallow explicit massparametersof orderM~andpostponeattemptsto understandM~in terms of M~.For our discussionof SU(2)x U(1) breakdownwe disregardthe inclusion of grandunification for the moment.

A superpotentialthat obeys (7.22) automaticallypossessesan R-symmetry,accordingto which allsuperfieldstransformwith the samecharge.Such a superpotentialhasthe nice featurethat it is strictlynatural in thesensethatwe canwrite themostgeneralsuperpotentialconsistentwith its symmetriesandstill arenot forced to choosemassparametersto be zerothat could in principle be nonvanishing.TheR-symmetryforbids the existenceof theseparameters.A superpotentialwith property(7.22) leadsto ascalarpotential of the form [450](compare(6.22))

V= ga~2+Am

3i2(g+g*)+ m~,2jyal2. (7.23)

Whatis remarkableaboutthisform, is thefact that it allows a model independentstatementconcerningthe breakdownof SU(2)x U(1). Notice first that V = 0 at the point where all the light fields havevanishingv.e.v. To havea breakdownof SU(2)x U(1) at the tree graph level this must appearat

96 H.P. Ni//es,Supersym,netry,supergrarityandparticlephysics

minima with V< 0. This actuallywill inducea cosmologicalconstantbut in the full model this can becancelledbyadifferent fine tuning of the parametersin the hiddensector,e.g. for /3 slightly differentfrom (2— \/3)M Since the cosmological constanthas to be fine tuned in all cases this posesnoadditionalproblems.

The first andthe third term in (7.23) are positivedefinite. Only the secondterm can give a negativecontributionand the requirementV� 0 at an SU(2)X U(1) breakingminimumwill give a lower boundon A. To see thiswe take (7.23),use(7.22) andwrite for A = 3

V = i~a+ m312y~2, (7.24)

which is still positive definite. For Al <3 SU(2) X U(1) breakdown cannot occur at the absoluteminimumsincethe absoluteminimumat V = 0 correspondsto vanishingv.e.v.’s for all the scalarfields.For a potentialof form (7.23) breakdownof SU(2)>< U(1) can only occur for [450]

iAi�3. (7.25)

We now see explicitly that in the modelsdiscussedin the last sectionthe introductionof the parameterp. was crucial for the breakdownof the weak interactionssince thesemodelswerebasedon a hiddensectorthat gaveA = 3— ‘~/3<3. With p. = 0 SU(2)x U(1) would not break.

The questionremainswhetherevenwith Al > 3 SU(2)x U(1) can break.We will studythis now in asimple example.We havealreadyseen in the last section that the Higgs sectorhasto be enlargedinorderto haveSU(2) x U(1) breaking.Thesimplestenlargementis the inclusion of a singlet superfieldY.Without it the only possibleterm in the Higgs sectorof the superpotentialwould be p.HH and this isforbiddenby our naturalnesscondition (7.22), andthe completesuperpotentialwould just contain theYukawa coupling terms. Such a superpotentialhas too many symmetries,therewould, e.g., exist aPecceiQuinn type symmetry which after the breakdownof the weak interaction would lead to aphenomenologicalunacceptableaxion, anotherreasonto enlargethe Higgssector.With the inclusion ofY the only choicefor a superpotentialthat fulfills (7.22) is

g= AYHH+~rY3. (7.26)

With A >3 such a superpotentialwill lead to a scalarpotentialwhereSU(2)x U(1) is brokenat theabsoluteminimum. Such a minimumwith nonvanishingv.e.v.’s for y and h andh exist providedthatthe relation[450]

1 <a-/A � j~[A + (A2 — 8)1/212 (7.27)

is fulfilled. Thus for A > 3 a breakdownof the weak interactionscan occur at the tree graph level[450,265].

At the momentwheretheseresultswerefirst derivedthereexisteda problemsincethe only knownacceptableexampleof a hiddensector leadsto A = 3— \/3 andit was not clearwhetherA � 3 could bepossible[519].In the meantime it hasturnedout that this problemis not severe,andwe havealreadyseenin section6.3 that exampleswith A � 3 havebeenconstructed[96,378].

The real problem of this approachturns up when we include the Yukawa couplings in thesuperpotentialas hasbeenfirst pointed out in ref. [252].To see this considerjust the addition of the

H.P. Nil/es, Supersymnietry,supergravityand partic/ephysics 97

electronYukawacoupling to the superpotentialin (7.26)

g = AYHH+~a-Y3+pLHE, (7.28)

wherep is a couplingconstantof order 10_6, responsiblefor the electron massafter the breakdownofSU(2)x U(1). In the caseof p equalto zerothe vacuumexpectationvaluesat the SU(2)x U(1) breakingminimumfor A > 3 are in generalof order (Am

312/A’)and the value of the potentialat this point was(with A’ = A(A/u)U

2)

V0= —(m~,2/A’

2)A4. (7.29)

With p different from zero, additionalminima can occur. They correspondto

(h°)= ~ç~)= (~e) = m312u/p, (7.30)

with

u = ~[lAl+ (A2 — 8)1/2] (7 31)

(compare(7.27)).The value of the potentialatthis minimumis

V0 = —(m~,Jp

2)u2(u2— 1). (7.32)

Sincep is so small, the v.e.v.’s in this casearevery largecomparedto m312 andthe minimumin (7.32)is

very deep.The minimumin (7.32) is of courseunacceptablesince the scalarpartnersof the electronhavea vacuumexpectationvalue andsince theycarry electric chargeU(1)em would be broken.Noticethat minimum(7.32) remainsunchangedevenif oneadds the ~D

2contributionsto the potential,since(7.30) leadsto ~D2= 0. We must now look for a situationwherethe minimum in (7.29) is deeperthanthe onein (7.32).Thiscan be eitherobtainedby a fine tuning u — 1 (which correspondsto A 3) or bychoosingA’ <p. The latter choice, however, leadsto more problems[252].With a very tiny A’ thebreakdownscale of the weak interactionsof order of 1000eV is very large comparedto m

312 since(h)—m3/2/A’ andwith A’— lO

6<p, m312 wouldhaveto be of the order of the electronmassandthis is

of coursenot sufficient to describethe required20 0eV splittings betweenquarks(leptons)and theirscalar partners.In a potential of the form (7.23) the absenceof absoluteminima that breakelectricchargethusrequires

Al ~ 3, (7.33)

which togetherwith (7.25) posessevereproblemsfor modelsof this type. The caseA = 3 is of coursepossible,whereall thesepossibleminima aredegenerateat V0 = 0, andwherewe arefree to chooseourfavorite minimum. There is, however,no reasonto assumeA to be stablein perturbationtheory andthe adjustmentA = 3 is spoiled by radiative correctionsas we will later see explicitly [112].Thisinclusionof radiativecorrectionsgives,however,also the key to solve the problemsof (7.25) and(7.33).The quantity A in (7.23)will afterthe inclusionof radiativecorrectionsno longerbe universalfor all thetermsin the superpotential.Onemight constructcasesin which the effective A~multiplying the third

98 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

term if (7.28) in the potential is smaller than 3 whereasother A constantsare larger than 3 and thedangerousU(1)em breakingminima are avoided.Beforewe, however,discusstheseeffects of radiativecorrectionslet usexplainanotherpotentialproblemin modelswhereSU(2)x U(1) is brokenat the treegraph level.

7.4. Light singlets

This problem has to do with the necessity of enlarging the Higgs sector in models where thebreakdownof the weak interactionsis achievedat the tree graphlevel aswe haveseenin the last twosections.The simplestenlargementis given by the introductionof a singlet Y with, e.g., superpotential

g = ~uY3+ AYHH. (7.34)

One problem with singlets is the absenceof a protectionof a possiblemass term through the gaugeinteractions.This is not a problemin (7.34) sincethereexistsan R-symmetrythat forbids,e.g., a mY2term. This symmetry is broken in the courseof the breakdownof supergravityif A is different fromzero.Sincesupersymmetryis brokenat a largescalethereis the potentialdangerof inducedlargemasstermsif this singlet couplesto heavy particles. In the frameworkof a grand unified model the simplestextensionof (7.34)is to assumethat Y is alsoan SU(5)singlet andassuch it would coupleto the heavyHiggstriplets. As in section5.6 for globally supersymmetricmodels,thestability of the smallmassscaleof order m

312 has to be checkedhere, and as hasbeen pointed out in refs. [480,451,389, 14] theexistenceof a light singlet posespotentialproblemsfor this stability.

Let us discussthesequestionsin a toy model containinga heavy chiral superfieldB and a light Lwith superpotential[451]

g= MB2+p.L2+A

1B3+A

2B2L+A

3L3+A

4BL2, (7.35)

wherep. is comparableto m312 andP~1is large, let ussay M — M~the grandunification scale. Coupled

to spontaneouslybrokensupergravitythis gives rise to a scalarpotential

V = 1g8l2+ gil2 + m~,

2(lbl2+ 112) + Am

312(A1b3+ A

2b2l+ A

3!3 + A

4b12+ h.c.)

+ (A — 1)m312(I~Ib

2+ p.12 + h.c.). (7.36)

Notice that all masssplittingsare small (of order m312) exceptfor the one in the last term which is of

order (A — 1)m312M for the heavy B scalar.The heavyscalarsare thussplit by an amountof orderofthe supersymmetrybreakdownscaleandone hasto worry that thismight inducelargesplittings in thelow energysector through radiative corrections.The light fields arecoupledto the heavyonesthroughthe couplingsA2 and A4. Let usdiscussthe two termsseparately.

The effectsof A4 can be seen in the tadpoleof fig. 7.3, which gives a contributionMm312(12+ 1*2) in

leadingorder.This is howevercancelledby the graphin fig. 7.4. The two graphsarethe componentsofthe supergraphshownin fig. 7.5, wherethe supersymmetrybreakingmasssplittings in the heavysectorare representedby a spurionsuperfieldS = Mm

312(A— 1)02. The contributionto the (12+ 1*2) termsofthis supergraphis a5 d

2OLLScontribution(F-term)andassuch is not generatedin perturbationtheory.In (7.36) thereare, however,additionalbreakingtermscorrespondingto Am

312A1andAm312A4trilinear

H.P. Ni//es,Supersyinmetry,supergravityandparticlephysics 99

Mm3,,2

b

lb

+b 7~1~ _~ 4 ,~ ~ 4I MX4 I X1X4

Fig. 7.3. Tadpolecontributionto themassof the light scalar. Fig. 7.4. Contributionthat cancelstheone in fig. 7.3.

Mm3,,2 Mm

B~B K B6B

Fig. 7.5. Supergraphthat containsthecomponentgraphsgiven in figs.7.3 and7.4. Fig. 7.6. Supergraphwith two supersymmetrybreakinginsertions.

couplings. They also can be representedby spurion superfields, e.g., U = Am312A,0

2 leading to a5 d~0LLU~’Scontribution(D-term) to (12+ 1*2) as shown in the supergraphin fig. 7.6, a contributionnot boundto vanishby thenonrenormalizationtheorem.In fact theseandsimilar contributionasshownin the componentgraphsin fig. 7.7 do not vanishand alsodo not canceleachother.Comparedto theMm

312contributionbeforetheyarehoweversuppressedsinceat leastoneof the trilinear couplings,e.g.MA1 is replacedby Am312A1,and the leading noncancelledcontributionsto (12+ 1*2) are thereforeoforderm3/2~They do not spoil the separationof the massscalesbut theyshow that the light massscaleisdefinedby m312. Evenif we would try to definethe light massscaleto be muchsmallerthan m312 at thetreelevel this adjustmentwould be spoiledby radiativecorrectionsandthe light masseswould be raised

Mm3/2 Mm312

b ç)~b b

I MX~ I i AX4m3,,2 I

Fig. 7.7. Componentgraphsthat givea nonvanishinguncancelledcontribution to (/2 + /*2).

100 H.P. Ni//es, Supersymmetry,supergravityandparticlephysics

- Mm312

Mm31,2b9b _

I mX3 I L mX3 I

Fig. 7.8. Tadpole similar to fig. 7.3 but now with light particle Fig. 7.9. Tadpolewith I-exchangeand two supersymmetrybreakingexchange. insertions.

to m312. All the graphswe havediscussedso far are logarithmicallydivergentandit is understoodthatthey aretreatedvia the usualrenormalizationprocedure.

We now turn to a discussionof the contributions that are proportional to A2, from the A2B2L

coupling of light and heavy fields. Again the most dangerouscontributions are given through thetadpolediagrams,of which an exampleis shown in fig. 7.8, andall the contributionsthat only containone supersymmetrybreakingparametercancel.As in the former casenonvanishingandnoncancelledcontributionsarise from diagramswith at leasttwo supersymmetrybreakinginsertions.The graphin fig.7.9 correspondsto such a leading contribution. Contrary to the former casethis contribution is notsuppressedby the heavyb-propagatorthat mediatesbetweenthe light particlesand the loop and alsothe supersymmetriccoupling mA

2 is comparableto the supersymmetrybreaking trilinear couplingAm312A2.As a result fig. 7.9 gives a noncancelledcontributionof order Mm312 to (12 + 1*2). Since Mm312is much largerthanm~,2this spoilsthe stability of the light massscale.

Is this a serious problem?Actually the contribution to (12+ 1*2) is given by (A — 1)Mm312A2A3.Choosingsmall values for A2 and A3 this contributioncould be madesmall. The choiceA = 1 wouldimply a vanishingone-loopcontribution. SinceA is renormalizedsuch a contributionwould then onlyexist in higher loops.The wholeproblemconstituteswhat is in generalcalled a naturalnessproblem.Alargevalueof 12 can be avoidedby specialchoices(fine tuned)parametersof the theory,but for a widerangeof parameterstheinduced1

2-termwill turn out to be too large.Since thisis a problemof the typethat motivatedour studyof supersymmetricmodelsonewould requireit to be absentin theframeworkof thesemodels.

The problemhasbeenshown to exist in the toy model given by superpotential(7.35) and nextonehasto seeif it is relevantfor realisticmodels.Of course,if the model underconsiderationcontainsnoheavy fields the problemdoesnot exist. But in generalsuch modelsare just a low energydescriptionwhich disregardthe existenceof heavyparticles. In grand unified modelsfor examplesuchheavy fieldsexist. A way out here would be to forbid the coupling of the heavy and light fields (A

2 = 0) which,however,in generalis present,but we haveto be morespecific. The dangerouscontributionarisesfromtadpolediagramsin which a light field connectsthe light fields with the loop, andsuch a contributioncan only exist if this exchangedparticleis a gaugesinglet.Consequentlythe problemoccursonly in thepresenceof a light singlet. Requiring the absenceof such a light singlet would solve the problem.However, in the models discussedso far we had seen that an enlargementof the Higgs sector isnecessaryanda light singlet was the simplestchoice.This light singlethad typically a coupling YH2H2to the Higgs doublets.The extensionto a usualgrandunified modelwould proceedby choosingY to bean SU(5)singletwith a term YH5H5 in the superpotential.The light singlet would then alsocouple to

H.P. Ni//es, Supersymmetry,supergravityandparticlephysics 101

Mm3,2

~~KIIII;•~IIII:~K ~ K

H2 m3,2X H2 H2

Fig. 7.10. Graphgiving rise to the light singlet problem in a simple Fig. 7.11. Graph resulting in a light singlet problem in a moreSU(5) model. complicatedSU(5) model.

the heavyHiggs triplets and as a result the contributionof graph in fig. 7.10 would destroy the smallmass scale,provided that a direct MH3H3 massterm exists. In grand unified models with 75, 50, 50representationssuch a direct massterm does not exist, the massof H3 arisesfrom off-diagonal termswith the correspondingtriplet in the 50. But againthereis a problemas shown in fig. 7.11. It can onlybe avoidedby introducingadditional representationsanda fine tuning of the parametersof the model[230].Evenif this is done the problemcomesback at the two-loop level. It just a normal problemofnaturalness.

Anotherway out would be that Y, althoughan SU(3)X SU(2)X U(1) singlet, is not an SU(5)singletbut a componentof larger SU(5) representation.An inspectionof this possibility reveals that reallylargeSU(5) representationsare necessaryandthenthereis the additionalproblemto give massesto allthe membersof this representationbut the singlet, which leadsto a really bizarreSU(5) sector[418].The cleanestway to solve this problem is to refrain from light singlets. But in a model whereSU(2)x U(1) breakingoccursat the tree graph level we haveto enlargethe Higgs sector.Whataboutan SU(2) triplet insteadanda superpotential

g=TrT3+HTH. (7.37)

In this casewe haveto makesure thatthis triplet receivesa v.e.v. small comparedto the v.e.v.’s of thedoubletsin ordernot to spoil the phenomenologicalconstraintthat the p-parametershouldbe closetoone [541]. It hasbeen shown that the superpotential(7.37) doesnot lead to models that satisfy thisconstraint[14,190]. Additional terms like m Tr T2 are neededwhich in turn violate our naturalnesscondition (7.22). Apart from this, the extensionof such a model to a grand unified theory requiresanenormousenlargementof the grand unified sector. The triplet has to be imbeddedin an SU(5)representationandthen one hasto give largemassesto all the otherparticles.In principal this can bedoneby repeatingthe 75,50, 50 trick designedfor thedoubletsnow for the triplets but thisrequirestheintroductionof severalSU(5) representationswith dimensionsof order of thousand[418].

To concludethis discussionit seemsthat all thesepossibleenlargementsof the Higgs sectorlead todifficulties. It seemsto be worth while to reexamineagainmodelswith a minimal Higgs sector. Ofcoursein such models,SU(2)x U(1) breakdowncannothappenat the tree graphlevel.

7.5. Breakdownthroughradiative corrections

The mechanismin which a SU(2)x U(1) breakdowncan occur through radiative correctionshas

102 H.P. Ni//es. Supersymrnetry,supergravityandparticlephysics

alreadybeenexplainedin section7.1. The driving terms for this mechanismwere largegauginomassesand a large top quark mass.In the context of the supergravitymodels such a mechanismhad beenproposedin refs. [444,324, 180], without giving calculationsin a completemodel. The renormalizationgroup equationsfor a generalexplicitly brokensupersymmetricmodel had been given in ref. [341].Explicit models in the context of spontaneouslybroken supergravityhave been first given in refs.[329,14,170]. The interest in studyingthis mechanismfocusesof coursefirst on a study of a minimalmodel, i.e. a modelwith minimal particlecontentin which the Higgs sectorconsistsonly of H and H.The superpotentialof such a minimal model is given by

g = p.HH + g’~E,L1H+ g~D,Q1H+ g~LJ~QJH (7.38)

where Q denotesthe left-handedquark doublets and i, / = 1,. . . , 3 label the threegenerations.Theparameterp. is neededto avoid thepresenceof an axion. It is of courseunnaturalto insert sucha smallmassparameterin the theory,but we will postponethis discussion.Considernow the modeldescribedby (7.38) coupled to a hiddensector with spontaneouslybroken supergravity.This leads to a scalarpotential given by [14]

V = gal2 + m~,Iyal2+ m

312(A’/~g~E~L1H+ A~g?~15,O1H+ AYjgY~U,Q1iI+ h.c.)+Bp.m312HH+ h.c.,(7.39)

wherewe haveused the samesymbolsfor the chiral superfieldsandits scalarcomponents.Additionalsoft breakingtermscould be presentfor the SU(3)x SU(2) x U(1) gauginosTFL~AaAC,,a = 1, 2, 3. At thePlanckmasstheseparametersarerelatedgiven apotential like (6.22). We have

m~=m~12, A=AE=AD=AU; BA1, thamo. (7.40)

To definetheseparametersat lower energies,renormalizationeffects haveto be included.The changeof theseparametersin the rangebetweenM~andM~is very model dependent.For definitenessweassumethat (7.40) holds at M~andwe supposethat the heavy particleshavebeenintegratedout andstudy the variation of the parameterstaking into accountthe degreesof freedomof the minimal lowenergysupersymmetricmodel.Integratingout theseheavyparticleswill, e.g.,changerelationB = A — 1but we haveto makethis assumptionin order to do definite calculations.The grandunification scaleofthis model is M~— 3 x 10160eV with an SU(5) coupling constant a5= e~/4~1/24 and for theSU(3)x SU(2)X U(1) coupling constantsat M~we have the relation e3 = e2 = (5/3)

112e1= e5. The

renormalizationgroupequations[341,14] which governthe evolutionof the gaugecouplingsat energiesbelow M~aregiven by (comparesection4.5)

a 3 . 1 . 11p.e3 p. e3 = — ~ e3, p.e2 = e2, p.e1 = e1. (7.41)

Integratingthe evolution down to energiesof order M~—100GeV this leadsto a~=128, a3=~O.12

andsin2 0~ 0.23 atM~.The gaugefermion massesevolvesimilarly

ji(tha/e~~,)= 0. (7.42)

H.P. Nilles, Supersymmetry,supergravityand particlephysics 103

For the Yukawacouplingswe have

_______ ~ 2g( 2 3 2 13 2’

16ir2 (g,~)—~-—— ~e

3+~e2+me1),

1 7 2’= ~j3 ~ (g~

3)2g~,— ~ gY~(~e~+ ~ + me1), (7.43)

1 032 32

p.g~= —gE~e2+~el),

whereonly the top quarkYukawacouplingg~is largeenoughto give largeeffects.The A-parametersobey the equations

A~3 332 1 ~

= (3 + 3~t3)~ (ge) — 41T2m312

u 332 ______p.Ai~=ô,3~(g,,) 4i2m3i2~3m3e2m2~~~m1), (7.44)

1IiA’I~= — 4ir

2m (~e~th2+ ~e~th1),

3/2

whereall Yukawa couplingsexceptg~3havebeenneglected.The parameterp. changesaccordingto

33’2 2

~ —e1—3e~], (7.45)

3 33\2 A 33 1 (32- + ~e~th1). (7.46)241Tm312

For the scalar(mass)2termswe have

‘2 21= j~-1~--~[3(g~3)2(m~+ m2 + m~

3+ m~i2lA~3l2)]— ~ [~Ith

2I2e~+ ~lth

1ie1j, (7.47)ü3

‘2 2’jim2 = ~-~--~ [2(g~3)2(m~+ m2 + m~

3+m~,2lA~3l2)]— ~ [~lth

3l2e~+ ~lth

1ie1j, (7.48)(13

U3

/.Lm03 = 82 [(g~3)2(m~+ m2 + m~3+m~,

2lA~3l2)]— ~ [~lth

3l2e~+ ~Irn2Ie2 + ~lmile~]. (7.49)

U3

We neglectthe influenceof the Yukawa couplingsexceptfor (g~3)andall otherscalarmassesevolveaccordingto

p.m~=~ ealmal Gaz (7.50)

104 H.P. Ni//es, Supersymmetry,supergravityand particlephysics

as a consequenceof the gauginomasstermswhich in (7.47)—(7.49)correspondto thesecondterms.Thefirst termsin (7.47)—(7.49)aregiven by the top quarkYukawa couplingandit is important to noteherethat the coefficientsof thesetermsbehavelike 3 : 2: 1 for the threedifferentmasses.This is importantsincethesign of thesefirst terms is positivewhich implies decreasingm2 with decreasingp.. Thefact thatm~ has the largest coefficient implies that it decreasesfaster and this is exactly the behaviorwe arelooking for, sincem~ shouldbecomenegativeat low energiesto induceSU(2)x U(1) breakdownwhilem~

3andm~3shouldstay positive.The secondtermsin (7.47)—(7.49)havenegativesign andtheydo nothelp directly. But this is already clear from our earlier discussion.The gauginomassesinduce firstpositive(mass)

2for the scalarpartnersof quarksand leptonsandHiggsesandespeciallythe partnersofquarksreceivea largecontribution.Thesemassesthen enterthe first termsin (7.47)—(7.49)and speedup the desiredevolutionprovidedthat g~3is largeenough.

Fromthe renormalizationgroupequationsit is clear that if radiative correctionsdrive a (mass)2of ascalarto negative values this will first be the scalarof H. To identify the conditionsfor an inducedSU(2)X U(1) breakdownconsiderthe Higgs potential

V= m~h~2+m~l/~l2+m~(hh+h*h*)+ ~(e~+e~)[lhl2— l/~i2]2, (7.51)

where the last term is the contribution from the ~D2 terms of the gauge interactions.This term isimportantsince it is the only termquarticin_theHiggs fields, which is neededto preventinfinite Higgsv.e.v.’s in caseof negative(mass)2for h or h. From(7.39)we seethat at the grandunification scaletheparametersm~aregiven by

m~=m~=m~12+p.

2, m~=Bp.m312, (7.52)

and below M~thesequantitieswill vary with m~ the candidateto becomenegative.Supposethat thishappens.The last termin (7.51)vanishesfor ~h)= (h) andthe quartic termsareabsentin thisdirection.To makesurethat thepotentialis still well behavedanddoesnot allow the v.e.v.’s to run to infinity theinequality

m~+m~�2lm3l2 (7.53)

hasto be satisfied.In addition a necessarycondition for SU(2)X U(1) breakdownis given by [341]

m~m~<1m31

4, (7.54)

which in generalis fulfilled for m~< 0 and m~still positive.Let us now discussthe behaviorof m~accordingto (7.47)at low energies.To isolatethe influenceof

the various terms on this evolution we considerfirst the casep. = 0 and vanishing gauginomasses:m

0= 0 in (7.40). We know that p. = 0 is problematicsince the model containsan axion but we willdiscussthesequestionslater. The evolution of m~is then given by

31p. ~— m~= 8~2 [m~+ m~3+m~3+m~,2IA~

3l2], (7.55)

wherealso (7.43), (7.44) and(7.48), (7.49) haveto be takeninto accountfor the parametersthat appear

H.P. Nil/es, Supersymmetry,supergravityandparticle physics 105

in the right hand side of (7.55). The evolution of theseparametershasto be determinednumerically.Given (7.40)we havein addition to m312 the two input parametersg~

3andA, and largeg~3andA canlead to afastevolutionof m to negativevalues.From (7.55)onemight eventhink that alargeA alonecould drive the mechanismeven for small top quark masses(given by g~3)but we haveto be careful.From ourdiscussionin section7.3 we haveseen(7.33)that for A > 3 potentiallydangerousminimawithbrokenU(1)em can occur,wherescalarpartnersof theleptonsreceivelargevacuumexpectationvalues.We haveto avoid them hereas well. In the example discussedthere(h), (E) and (L) had vacuumexpectationvaluesof order m3/2/gEandwe hadto chooseA <3 to makesurethat such a minimumwasnot the absoluteminimum of the model. Here A is no longer universaland constraintshaveto beobtainedfor AE, A~,AD separately.In the simplified caseunder considerationthe massesof theparticleswith smallYukawa couplingsarenot much affectedand to avoid theseundesiredminima wehaveto chooseIAEI, lAD! <3. The generalconstraintsgive [112]

m~,+m~+mL>(IAEl2/3)m~/2,

2 ~.. 2 ~ 2>~ A 2/’~\ 2

mH~-mD-,-mQ krIDIJ,ms,2,

m~+m~—mi~l2>(IA~l2/3)m~i

2,

wherethe relevantparametersin (7.56) arethoseat the breakdownscaleof SU(2)X U(1). From(7.44)we seethat in the absenceof gauginomassesA~andAD decreasewith decreasingenergywhereasAE

stays constant.The value of A at M~is thus bound to obey IA! <3 to avoid the U(1)em breakingminima.

The secondparameterin (7.55) is the top-quarkYukawacoupling. It evolvesaccordingto (7.43) andwe see that for large g~

3it decreasesat smaller energies.Actually if we would choosean arbitrarilylargeg~3at M~thisdecreaseis proportionalto (g~3)3andthusarbitrarily large,and g~3at low energiesmight still be small. With (7.43) oneobtainsan upper limit on the Yukawa coupling at low energies,correspondingto m~

0~— 200 GeV. Of course(7.43) is no longer valid for arbitrarily largeg~3at M~and

we notethis upper boundjust for curiosity.The resultsof a numericalanalysisof (7.55)and (7.43)—(7.49)areshown in fig. 7.12. This resultholds

for p. = 0 and m0= 0, a grand unification scale of 3 x 10160eV with a5— 1/24. To obtain the correct

magnitudeof the SU(2)x U(1) breakdownthe condition

m2(Q2 (100 GeV)2)= —~M~ (7.57)

(with M~themassof intermediateZ-boson)hasto befulfilled. Notice that in this caseonly a v.e.v. for his inducedandwe have

42Jt.\2__ -tm

2 2 A,f

2 j’~ 2~ — 1A12— I 2j.. 2\~ m

2— iviwi.~.COS Vw~ 2J.~’~z ,

kel-- e21

but thedown quarksandleptonswill staymasslessbecauseof (h) = 0. Thiswill changeas soonas p. � 0which we haveto chooselater in any caseto avoidan axion. Diagram7.12 showsthe resultsfor a choiceM~— 90 GeVin (7.57), as afunction of A, m312 andg~.The quantity ay denotes(g~)

2/4irat the grandunification scale.Thedashedline gives the limit on A from (7.56) here

106 H.P. Nil/es, Supersyrnmetry,supergravityandparticlephysics

A

1 ~ ° 400

0 O.d05 0.01 0.02 0050.1 —~

100 120 140 160 180 200mt0~(G~V)

Fig. 7.12. RelationsbetweenA, m312 andm,0~for amodel with radiativelyinducedbreakdownof SU(2)x U(1) asgiven in ref. [141.This resultholdsin thecasem0= = 0.

m~+m~—~M~>(lA~l2/3)m~j

2, (7.59)

which in our specialcaseimplies

IA~I<~— M~/4m~,2. (7.60)

With this boundon A the mechanismneedslargetop quark masses

1000eV � m~1~ 1900eV, (7.61)

wherewehaveto rememberthat for ay — 1 ourapproximationbreaksdown. The valueof the gravitinomass m312 can lie anywherebetween70 0eV and infinity but we see that for most of the parameterspaceit lies between80 and4000eV. For typical valueslike A = 3— \/3 andm312= 1000eV oneneedsa top quark massas large as 1800eV to induce the breakdownof SU(2)X U(1). Theseare the resultsfor our specialcasep. = m0= 0. In generalwe will see later that nonvanishingp. andm0 will allow anSU(2)x U(1) breakdownevenin the presenceof smallertop quark massesthan given by (7.61). p. = 0

H.P. Ni//es, Supersymmetry,supergravityandparticlephysics 107

will alsolead to an axion if (h) would be nonvanishingbut with the prescription(7.57) we have(h) = 0and thereforeno down quark and lepton masses.The latter problemcan be removedby demanding

(m~+ m~)IQ2_(1~ GeV)2 = 0 (7.62)

insteadof (7.57) as proposedin ref. [170].In generalif we define (h)/(h)= tan 0 we obtain

sin20=2m~/(m~+m~), (7.63)

andwe seethat for m3= 0 (p. = 0) (7.62)hasto be satisfiedto obtainanonvanishingv.e.v. for h. Models

basedon (7.62) instead of (7.57) lead to similar boundson m~0~as given in (7.61). In this casethebreakdown scale of the weak interactions is related to the gravitino mass through dimensionaltransmutationobtainedthroughthe flat direction in the potential implied throughconstraint(7.62).

The bound (7.61) on m50~requiresvery largetop quark massesbut before we explorethe variouspossibilitiesof lowering this bound,let usdiscussthe spectrumof the modelsdescribedabove. Beforewe do this we haveto include p. � 0 andwe will assumefor the moment that this inclusion doesnotchangethe resultsof fig. 7.12 drastically.We will later seethat for a largerangeof p. this assumptioniscorrect. Equation (7.53) now gives m~+ m~> 2p.Bm312for the stability of the minimum. With thisinclusion andm~((1000eV)

2) = —M~/2not only h but alsoh will receivea v.e.v.

Bp.m3~(k°)+ O(~~), (7.64)m1 — 1m21

wherewe work in the limit p. <m312. p. should not be too small becauseotherwiseone would have(h°)~ (h°)which would imply the presenceof a very largeYukawacouplingof thebottomquark to giveit a mass of 4.5GeV and this certainly had to be takeninto account in the renormalizationgroupequations.We assumethat p. is largeenoughto avoid theseproblems.

The spectrumof the scalarparticlesof the model is given in table7.1, for p. = 0. The scalarh°doesnot receivea v.e.v. and the Peccei—Quinnsymmetry remainsunbroken.With p. � 0 h°will receiveav.e.v. asshown in (7.64). In this caseoneof the neutralHiggs particles(a pseudoscalar)will havea massof order p.. In the absenceof the breaking term Bp.m312onewould havethe massof this particle oforderp.

2m312but the soft trilinear supersymmetrybreakingterm raisesthis massof the “axion” to p..

The gluinos and photinosreceive massesthrough radiative correctionsas discussedin section 6.4.Dependingon the parametersof the modelwe expectthesein the ranges

5 GeV< m3< 100 0eV, 0.50eV � th~< 15 GeV. (7.65)

Theremaininggauginosmix with the Higgs fermions.We havefor the chargedones:

~i-( P112! V2MW’~ (7.66)h\V2Mw!xl p. )

wherex denotestheratioof h to h v.e.v.’swhichisof orderp./m312.With (h) asgivenin (7.64)thelightestofthesefermionswill havea mass

108 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

Table 7.1

Thespectrumof thescalarparticlesin themodel with /1 = 0 asgiven in ref. [14].Partnersof quarksand leptonsare denotedby ~ ~ respectively.The off-diagonal termsAm

312Jm0areexplicitly given for thepartnersof thetop quarkandneglected

in theothercases.

~, m~12+m2(~sin29~_~)(P~ m~12+in~(~~sin2O~)

çcu,çs~ m~12_m~sin2Ow)

~3,~ ~ m~2+m2(~sin29)22 21 1.2~m312+m(3— ~sin O~)( ~ +m2dsin29~—~)+m~ A~

3ms,2m,0~

\A~3m

3i2m,0~~m~12+m~(—°f—~sin29~)+m~0~

~ m~,2+(~_sin2Ow)m~

~ flhl/2+Sli1

2Owfl1~

~ m~12—~m~V2Reh

Th m20 2 12h m312—~m1

in312+ m~_sin20)

V2Mwp.~BIm312/(m~,2~M~)+ C(p.3/m~,

2). (7.67)

For thisparticleto havea masslargerthan20 GeV onemusthave xl >0.17 which implies p. lBl — 0.05—0.17still consistentwith our assumptionthat p. is small.We haveheresucha light particlebecauseofthe asymmetryof the h and h vacuumexpectationvalues.Its left-handedpiecesarepredominantly~‘~+

andc1’h-~

The massmatrix of the neutralfermionsof the Higgs sectoris given by

*0 / m2 0 iM~xcos0~ iM~cos0~\

B° 1 0 iM~xsin 0.,,, iM~sin 0..,, ~ (7.68)~ iM~xcos0,,~, iM~xsin 0.,,, 0 p.

wI \ iM~cos 0.,., iM~sin 0~ p. 0

Thismatrix hastwo largeeigenvaluesof orderM~and two smallones.Oneof the light particlesis thealreadydiscussedphotino

= sin O.,~,1’~”°— cos0~°, (7.69)

and the secondoneis the partnerof the “pseudoscalaraxion” discussedearlier,its main componentis‘/‘b andits massis proportionalto p.. Which of the two is the lightestparticledependson ñia andp.. Ifoneconsidersmodelswith m0 = 0 as we did in this section one in generalexpectsthe photino to belighterthan t//h sincep. hasto be chosenlargeenoughto satisfy the lower boundon the chargedfermionmassgiven in (7.67).

The light supersymmetricparticlesin this class of modelswith m0= 0 and small p. are the photinowhose massis small becauseof m0 = 0 and a chargedand neutral fermion as well as a pseudoscalarbosonwith massproportionalto p.. The othersupersymmetricR-oddparticleshavemassestypically oforderof the massof the weakgaugebosons.Thelightestof the scalarpartnersof quarksandleptonsis

H.P. Nil/es,Supersymmeiry,supergravityandparticlephysics 109

one of the partnersof the top quark provided that A~3is not to small. The mechanismof inducedSU(2)x U(1) breakingrequiresthe existenceof a largeg~3Yukawacoupling and thusa largemassofthe top quark 100GeV~ m

10~,� 1900eV with either choice(7.57) or (7.62) as boundaryconditionsform~ andm~ at energyscale Q

2 — (1000eV)2.Thecasep. = 0 is not possible.It eitherhasan axion or only oneof the Higgs fields receivesa v.e.v.

andleptonsanddown quarkshaveto staymassless.To avoid this anonzerop. hasto beput in by hand.A priori this parameteris unrelatedto the gravitino massandit violatesthe naturalnesscondition (22).It shouldnot betoo small to avoid the predictionof a chargedfermion with masssmallerthan 20 0eV.

7.6. Su(2)x U(1) breakdownwith smaller top quark mass

In the last section we haveconsidereda special class of modelswith m0 = 0 and small p. which

requireda large top quarkYukawa coupling to induce the SU(2)x U(1) breakdown.We now want toaddressthe questionwhetherthis alsoworkswith smallertop quarkmasses.

Oneway to do this is the choice m0 � 0. At first sight this is not obviousas an inspection of (7.47)shows.Nonzerogauginomassesrn2 and th1 give a negativecontributionto the right handsideandwithjust this contribution m~would increasewith decreasingenergy. But we know already from ourdiscussionin section 7.1 that the most important influenceof the gauginomassesis indirect. It firstinduceslarge massesfor the scalar partnersof the quarksvia the d!th3!

2 contributionsin (7.48) and(7.49). Theselarge massesthenenterthe first term in (7.47) andthis acceleratesthe evolution of m~if(g~3)is large enough to dominatethe second term (which does not contain e~Iin

3I2).Our earlier

discussionindicatesthat in such a casethe inductionof the weak breakdownmight be possiblewithsmallervaluesof m

50~thanthe rangegiven by (7.61) andherewe can check this explicitly.Therearetwo ways in which a nonzerom0 can appearin the theory as we havediscussedin section

6.4. The first possibility is to havem0— m3/2 at the tree level and at the grand unification scalethegluinos, winos and bino havethe samemass.Below M~thesemassesevolve differently accordingto(7.42)and thiswill result in

lila = (a,Ja5)mo— (aa/as)m3/2, (7.70)

where a = 1, 2, 3, aa= e~/4irthe runningcouplingconstantsanda5 the grandunified couplingconstantat M~.

A secondpossibility for an appearanceof gaugino massesis through radiativecorrectionsin thepresenceof heavy particles.At M~this will result in

tha = Ca5m312 (7.71)

following our discussionin section6.4. The factor C dependson the spectrumof the heavyparticles.The massesin (7.71) evolveaccordingto (7.42)

tha = Caam3i2. (7.72)

Thus in both caseswe expectthe gluinosto be the heaviestgauginosat low energiesandthe photino

will havea much smallerMajoranamassth7/th3 8aem/3a3. (7.73)

110 H.P. Nil/es, Supersymmetry,supergravityandparticlephysics

With either (7.70) or (7.71) we can now computethe evolution of the various parametersof themodel. We still usesmall p. and demandboundarycondition (7.57) at low energies.The alternativechoice (7.62)will give similar results.The modelsnow dependon the threeparametersA, m312 andm0.Let usfirst considerthe standardvalueA = 3— V3 to investigatethe influenceof m0. Thecalculationofthe evolutionaccordingto (7.41)—(7.50)hasto be donenumerically andwe use the sameM~,a5 inputparametersas in the last section.Theresult is givenin fig. 7.13. For m0= 0 we obtainthe old resultsinthe caseof A = 3— V3. The lower bound on m10~is given by 135 0eV and for m312~ 70 GeV theSU(2)x U(1) breakdownwill not be inducedat all. For m0 � 0 this changesand m10~can be as low as60 0eV. In the casewith generalA andm0 = m312 oneobtainsa lower boundm~0~>65 0eV. A strictlower bound [329,14, 170] is obtainedin the limit m0~ m312,

m50~> 55 0eV (7.74)

which, however,is hard to reachsincewe expect in general m0— m312. In modelswith a low energypotential like the onein (7.5) onecould havem ~ Bm3/2 for small B, but in thesecasesonehasto becareful to makesure that A/VB <3 to avoid unwantedminima [226].

In any casethe presenceof m0 � 0 makesit easierto inducethe breakdownof the weak interactionsas we have expected.The smallest value for the top quark mass is obtained for m312/m0= 0. Thespectrumof thesemodelswith small p. andnonvanishingm0 is similar to the onediscussedin the lastsection.Observethat with m0 � 0 not only the value of m10~but also the value of m312 can be lower

300

m3,,2= 150 6eV m3,,2~—

m31~2=100 A 3—13200 \ \

>w

Ern3,2 ~50

100 -

0 I :z::~EE~EEE:::i:ii:i_.:::;:::v~ I50 100 150 200mt0~ (6eV)

Fig. 7.13. Influenceof nonvanishinggauginomassesm0 on theradiatively induced5U(2) x U(1) breakdownas given in ref. [329].

H.P. Nil/es,Supersymmetry,supergravityandparticle physics 111

than in the previouscase(wherewe neededm3/2� 70 0eV). Here onecan in principle havearbitrarilylow valuesfor m3/2andas a resultthe scalarpartnersof quarksandleptonscould belighter than thosegiven in table7.1. The photinohasstill a fair chanceto be thelightestR-oddparticlesincethe influenceof m0 is not very large(7.73) aslong as m0 is of the orderm312. It might howeverhappenthat oneof theparticleswith massproportionalto p. is lighter than the photino.The massesof the scalarpartnersofquarksandleptonsare affectedby the presenceof the gauginomasses(7.50). A largecontribution isgiven to the partners of quarksbecauseof the presenceof e~!th3I

2in the right hand side of (7.50)(compare(7.48) and (7.49)).The partnersof the right-handedleptonsreceivethe smallestcontributionsincetheyonly coupleto U(1)~.In generalonewouldexpectin this classof modelstheseparticlesto bethe lightest of the partnersof quarksand leptons, and they could very well be in the massrange of20 0eV. Only for m

5>m3/2, m0 and not too small A one of the partnersof the top quark could belighter. Observethat thesecontributionsto themassesof quark/leptonpartnersdo not createproblemswith flavor changing neutral currents, particles of samechargeand same helicity receivethe samecontribution.Masscontributionsthat causeflavor changingneutralcurrentscan only comefrom thosetermsin the renormalizationgroup equationswhich contain Yukawa couplings.We will discusstheseeffectslater in detail.

Gauginomassesalso influencethe evolutionof A. With m0= 0 the sign andphaseof A was actuallyunphysical.It could beabsorbedby aredefinition of the fields. The equations(7.44)contain only termsproportionalto A itself andtheYukawa couplingsandthesign of the termcoincideswith the sign of A.As a result the absolutevalueof A decreaseswith decreasingenergyandstartingwith A <3 at M~onewas surethat all A’s remainedsmallerthan3 at low energies.With m0 � 0 the relativesign andphaseof A andm0 becomeimportant.In (7.44)the gauginomassescontributeto anegativeterm on the righthand side which correspondsto increasingA at low energies.This, however, is in generalnot verydangeroussince the massesof the scalarpartnersof quarksand leptonsalso increasesuch that theupperboundon IA! is in generalnot violatedby radiative corrections,in somecasesonemight chooseA negativeto avoid potentialproblems.A complexphasebetweenm0 andA is in generala dangeroussourcefor CP violation which we will discusslater.

We havenow includednonvanishingm0 and this leadsto the lower bound(7.74) on m10~,which isstill large.Thereareadditionalpossibilitiesto lower thisbound.Oneof them is to considerlargerA. Aswe have pointed out earlier modelswith A > 3 haveseveralproblematicproperties.First of all thedesiredSU(2)x U(1) breakingminimumin which only h°andh°havenonvanishingv.e.v. is no longerthe absoluteminimum (compareour discussionin section 7.3). The absoluteminimumcorrespondstobrokenU(1)em and thereare otherminima with brokenSU(3)~still lower than the desiredminimum.The minima arevery deepbecausefields get v.e.v.’sof orderm312/pwherep is a smallYukawacouplingconstant.On the other hand thesesmall couplingsalso imply that thesedifferent local minima areseparatedby a largebarrier of order m3/2/p from the desiredminimum andit might in fact be thatoncethe desiredminimumwas chosenat a certaintime for certainreasonsthetheory might sit in such aminimumfor a long time. Thequestionof the stability of sucha falsevacuumwasexaminedin ref. [86],andit was found that for A > 3 thereexist certainrangesof the parametersof the model in which thefalse vacuumhasa lifetime largerthan the age of the universe.In particular this rangeof parametersincluded the case in which g~

3is small. In fact it turned out that under these circumstancestheSU(2)x U(1) breakdowncould be inducedfor arbitrarily small top quark masses(for very smallg~3thisis compensatedby A since only the product g~3Aenters (7.47)). In this casea lower bound onmtop~ 20 0eV comessolely from the nonobservationof a toponiumresonancebelow 40 0eV in e~ecollisions. A secondproblemwith largeA concernsthe massesof the scalarpartnersof the top quark.

112 H.P. Ni//es, Supersymmetry,supergravityand particlephysics

They arein generalgiven by a (mass)2matrix of the type

(m~,2+m~0~Am10~m312 775

\. Am~0~,m312m~12+m~0~

wherewe haveneglectedcontributionsfrom radiativecorrectionsand just considerthe simplestmodel.

For IA! <2 the (mass)2eigenvalues

m2= m~12+m~~±Am10~m312 (7.76)

are both positive independent of the values of m10~and m312. For larger A both eigenvaluesare onlypositiveif m10~and m312 havesufficiently different values.Oneof thepossibilitiesis to havesmallg~

3incaseof large A, exactlythe situationdescribedabove,andas shownthis is possiblein a cosmologicallystablefalsevacuum.Equations(7.75)—(7.76)showthat for largeA oneof the partnersof the top quarkmight bethe lightestscalarpartnerof quarksandleptons.If largegauginomassesarepresent,however,thesemassesreceiveadditional contributionsand in such a caseit might alsobe that the partnersofright-handedleptonsare lightest.

The introductionof largeA thuslowersthe boundon m00~to its phenomenologicallimit of 20 GeV

providedoneis willing to live long enoughin a falsevacuum.Anotherpossibility to lower the boundon m50~is given by choosinga larger grandunificationscale

M~.This in generalrequiresthe introductionof new light (1 TeV) fields andonehasto departfrom theso-calledminimal models.One favorite choice is the introductionof new superfieldsin the (8, 1, 0) +

(1, 3, 0)+ (1, 1, 0)SU(3)x SU(2)x U(1) representations,which might also be requiredfor cosmologicalreasonsaswe will discusslater [377,378, 327]. The inclusionof theseparticlesraisesM~to the orderofM~andstill allows an acceptablevaluefor sin

2 0.,.,. In the framework of SU(5) the aboveparticlesarecomponentsof an adjoint representation.With M~aslargeasM~the evolutionof the parametersof thetheory proceedsover seventeenordersof magnitudein energyinsteadof fourteenas before.To reach(7.57) andinduceSU(2)x U(1) breakdownsmallervaluesof g~3aresufficient. With the particlecontentsketchedaboveoneobtainsm

10~>43 GeV insteadof 55 GeV in (7.74) for theminimal model.With thisdeparturefrom the minimal modelonehasof courseopenedthedoor for manymorepossibilitieswhichwe will not discussherein detail. It seemshoweverpossiblethat with someeffort any valueof g~

3caninduce SU(2)X U(1) breakdownin a model especiallyconstructedfor thisvalueof g~3.

Let us come back to the minimal models. We have not discussedyet the caseof larger p.. Aninspectionof the condition (7.54) for the breakdownof SU(2)x U(1)

m~m~<Im3l4 (7.54)

with valuesat M~

m~=m?~=m~12+p.

22 (7.52)m3= Bp.m312

reveals that in the presenceof a p. of order of m312 the SU(2)x U(1) breakdownmight be easiertoobtain sincefor p. 0 we neededto havem ~< 0. Of coursewe cannotchoosearbitrarily largeB on

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 113

rn3,2

~

Fig. 7.14. A possiblecontributionto /L throughradiativecorrections.

one handsinceB is naively relatedto (A — 1) andalso becausewe haveto satisfy (7.53),

m~+m~>2Im3I2, (7.53)

the condition on the potential to be boundedfrom below, and this condition hasto be fulfilled at allmassscales,which in particularat M~implies IBI <2 for p. = m

312.The insertion of the parameterp. violatesthe naturalnesscondition (7.22) and one would like to

havea justificationfor its existence.It would be very desirableto constructaconnectionbetweenp. andm312 and to understanda nonvanishingp. in terms of m312. Rememberthat a lot of effort hasbeendevotedin grand unified modelsto explainwhy p. is zeroand not of theorder of M~,andfinally p. = 0was achieved at the tree graph level. After the breakdownof supersymmetrya nonvanishingp. isgeneratedin perturbationtheory, for example[330],through a graph like the one in fig. 7.14. Thesepossiblecontributions to p. are howeverin general small comparedto m312. To obtain p. -= m312 newsingletsor new “nonrenormalizable”couplingshaveto be introducedin the theory only for the purposeto generatesuch a largep.. Thesemechanismsare no moreand no less artificial than introducingp. byhandand an understandingof the origin of p. is not yet obtained,exceptfor valuesof p. that are smallcomparedto m312. Nonethelessthe phenomenologicalconsequencesof modelswith p. m312 should beanalyzed.

To start this let us first repeatsomekinematics.The Higgs potential in (7.51) for nonzerom~is in

generalminimized for

v2 (h)2+ Kh)2= 2[m~—m~—(m~+m~)cos20]/(e~+d)cos2O, (7.77)

where tan 0 = t~h)/~h~.In the caseof vanishingp. we had instead (7.58) (J)2 = 4m~/(e~+e~).Let usnow write [330]

cos20 = (w2— 1)/(w2+ 1). (7.78)

The earlier condition (7.57) or (7.62) in the caseof p. = 0 is now replacedby

I 2_ 2 2\Jj 2 1\ 2

(0) m2)! ~~(t)— 1) Q2—Mw — 2h5’I Z

to haveSU(2) x U(1) breakdownat the right scale.The previouscaseis obtainedfrom this moregeneralone for w —~x~ In the presentcasewith p. � 0 one doesnot expectsuch an asymmetrybetweenthe

114 H.P. Ni//es, Supersymmetry,supergravityand particlephysics

v.e.v.’s of h and h andw shouldbe closer to one. It hasbeenobservedin refs. [330,374, 343] that thecasew 1 is very special in that it admits the induction of SU(2)X U(1) breakdownwith small topquark Yukawa coupling. We follow here the discussionof the excellent paperof Ibanez and Lopez[330],in which also theeffects of g~in the evolution are takeninto account.

In a model with smallYukawacouplings(including g~3)oneexpectsaccordingto (7.47) and(7.50) asimilar evolutionof m~andm~.Therewill be a slight differencebecauseg~3hasto belargerthang~but thiswill not be significant if g~3is sufficiently small. In thecasesdiscussedpreviouslya largeg~3wasnecessaryto arrive at m~<0at low energiesand now this will no longer be possibleandone obtainsm~— m~>0 at low energiesprovidedthat we havechosenm~= m~at M~which is the usualcase.Buteven with m~—’m~it is possibleto satisfy(7.79), the conditionfor the breakdownof SU(2)x U(1). This,however, requiresw, the ratio of the v.e.v.’s of h and h to be close to one or according to (7.78)cos20 — 0. At first sight one might think that to arrive at such a situationat energiesQ2 — (100 GeV)2one hasto fine tunethe parametersof the theory at M., but this is not necessarilythe case.To see thisrecall

sin20=2m~/(m~+m~), (7.63)

which gives us the evolution of 0 in terms of the m~.From the renormalizationgroupequations(7.45)to (7.50) oneobtainsfor smallg~3

p. -/---(m~+m~)=—12a2th~—4a1th~, p. ~f-m~= (—3a2—ai)m~. (7.80)

A calculationof the evolutionof thesequantitiesshowsthat for a largerangeof m~at M~it will evolveto 2m~— m~+ m~at low energies.For 0.6~ 2m~/(m~+ m~) ~ 1 at M~onewill actuallyobtainsin20 — 1at M~,andthe input parametershavenot beenfine tunedto a largeaccuracyto obtainthe behavior.Ifthereforem = Bp.m3/2 is of comparablesize as m3/2 at the grandunification scalethe breakdownofSU(2)x U(1) can beinducedin absenceof a largetop quark massandthe only lower boundon m10~isgiven by the phenomenologicalboundof 20 0eV. Thus evenin a minimal modelwith A<3 theboundsin (7.61)and (7.74)on m~0~do apply only in the specialcircumstancestheywere derived(p. ‘O).

The relationbetweensin 20(M~)and the massof the top quark is shown in fig. 7.15. The usualinputparametershavebeenused for the integration of the renormalizationgroup equationand the figureshowsthe result for thosecasesin which (7.79) is fulfilled at Q

2— (100GeV)2. We see that for a largeenoughsin 20 at M~the breakdownof SU(2)X U(1) occurs for small values of the top quark massprovided that the gauginomassesm

0 at M~are not to large.Very largegauginomasses(comparedtom3/2) will through(7.80)inducelargevaluesof m~+ m~at low energiesandthis will makeit impossibleto havesin 20 closeto oneat Q

2 (100 GeV)2. A different view of the relationsbetweenthe relevantparametersis shown in fig. 7.16. It showsthat for a choiceof 0.6 ~ sin20(M~)~ 0.7 the SU(2)x U(1)breakdownwith small m

10~and m0= 0 can be obtained.For largevalues of m0 the samebehaviorcanonly occur if sin 20(M~)is chosento be closerto one.

The phenomenologyof the models with p. — m3/2 is different from that of the models discussedpreviously. Of coursethe spectrumof the scalarpartnersof quarksandleptonsis in generalnot verymuchinfluencedby p.. It is mainly given by m3/2andm0. The gauginomassesareessentiallydeterminedby m0 andif this quantity is not too largethe photinois a goodcandidatefor thelightestR-oddparticle.Differencesoccur in the Higgs sector. In modelswith p. 0 we had identified threestateswith mass

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 115

150 A = 1m312= 100 6eV

>

!~

0.8

0.7 0.2=sin2O at M~

50

0.4

0.6

010 1~O 1~0

mt0~(GeV)

Fig. 7.15. The relation betweenm0 and m~0~for various input values of sin29 (defined in (7.64)) asgiven in ref. [330].For sin 20 ~ 0.65 thebreakdownof 5U(2) x U(1) can evenbeinducedfor small valuesof m,0~.

A=1rn3121000eV

200~ o.i

150 70.12

100 0.14

~~20

sin2O at M5

Fig. 7.16. Relation betweensin20(M,)andgauginomassesm0 as given in ref. [330].

116 H.P. Ni//es, Supersymmetry,supergravityand particlephysics

proportionalto p. + (O(p.2/m312); theywill no longer stay light in the presentcasesincep. m312.There

are, however,other stateswhich might be light in thesemodels.To see this, let us considerthe casesin 20(M~)= 1. We know that this cannotbe exactlytrue since it implies m~= m~at M~and this canonly happenif g~

3= g~exactly which in this caseimplies also mbottom= m50~since (h) = (h~,but in a

realistic model sin 20 can be very close to one and the mechanismis easierto explain in this speciallimit. With sin 29 = 1 we havem~+ m =

21m3t2 (compare(7.53)) andsince m = m

V= m2[~h2+ I/~2+ (hi~+ h *j~*)]+ ~(e~+ e~)[Ih2 I/~I2]2. (7.81)

Inequality (7.53) was a condition for the potential to be boundedfrom below.Here the equalityholdsandthis implies that this potentialhasa flat direction andthis in turn implies theexistenceof a masslessscalar particle. Inspectionof (7.81) shows that this particle is the real part of h°+ h°.It is a scalarparticleandshouldnot be confusedwith the pseudoscalar“axion” we haveencounteredin modelswithsmall p.. For the above-mentionedreasonswe haveseenthat sin 20(M~)cannotbe exactly equalto one,but in caseswhere it is close to one andwhereconsequently(h) (h~iwe expectthe existenceof a lightneutral scalarparticle.This particlehassimilar propertiesas the single Higgs in the nonsupersymmetricstandardmodel that survivesthe SU(2)x U(1) breakdown.

In table 7.2 we give sevenexamplesof the low energyspectrumof modelswith moderatelylarge p.and variousvalues of ~ m

312, m0 and A. The previouslydiscussedneutral scalarHiggs correspondsto Hb andit is only very light (asin models I and2) if sin 20(M~)is close to one,The interestedreaderis invited to study this tablecarefully to seehow the variouspossibleinput parametersinfluencethe lowenergyparticle spectrum.We only noteherein addition that modelsof the type6 havea tendencytopredict light scalarpartnersof the neutrinos[330].Theseparticlesare neutraland unlike in the caseofthe other partnersof quarksand leptons they can be lighter than 20 GeV. In model 6 actually the

Table 7.2

Low energyspectrumof seven models for various rangesof the input parametersas given in ref.[3301.The massesaregiven in units of GeV. H1,,,. denotethemasseigenstatesof thethreeneutral

Higgsesthat survive theHiggs mechanism.

1 2 3 4 5 6 7

rn,01, 29 29 30 63 111 111 160rn312 25 50 100 50 200 50 150

rn0 17 5 71 141 108 49 6

A 1 1 1 1 0.5 1 1

sin20(M1) 0.787 0.679 0.779 0.76 0.195 0.038 0.187

sin20(Mw) 0.999 0.999 0.996 0.42 0.43 0.43 0.45

Partnersof quarks 50—52 51—52 215—220 380—396 300—363 120—150 143—160Partnersof leptons 26—28 49—50 105—114 84—122 207—219 67—75 138—156

PL~ 16Gluino 50 15 214 423 324 147 19

Majoranawino mass 14 4 58 115 88 40 5Bino 7 2 29 57 44 20 3ChargedHiggses 86 104 170 136 230 81 163

Neutral H. 96 112 173 118 219 89 147Higgses Hb 5 7 23 80 83 23 81

FI~ 37 69 151 112 217 24 143

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 117

partnersof the neutrinosarethe lightestR-oddparticles,sincewith the choiceof m0— m3/2 — 50 0eVthe photinois relatively heavy.The reasonfor the appearanceof theselight particlescan be tracedbackto a negative contribution of the ~D

2-term(compare (7.13)) to the mass matrix proportional tom~/2cos20 which only occursif ~h) and (h) are sufficiently different. In such a situation the uppercomponentsof the SU(2)doublets(like ~, ~, ~. ~‘,)havesmallermassesthan the lower components(like ‘Pe, ~d) and the singlets (like ~, ~) and the partners of the neutrinos are the lightest of these sincetheyreceivethe smallestradiativecontributionfrom the gauginos.The presenceof nonvanishingm

0 isnecessaryin such a model to avoid a chargedscalar(e.g. thepartnerof the u-quark)to havea masslessthan 20 GeV.

We will stop here for a moment our discussionof minimal models (i.e. models with minimalparticle contentin the low energysector). In addition to the input parametersalreadyneededin thenonsupersymmetriccasefour new parametersenterhere which replacethe parametersof the Higgssectorin ordinary theories.Thesearethe gravitino massm312, gauginomassesm0, the A parameterforthe trilinear couplingsandthe quantity p. for theHH-termin the superpotential.The readermight nowtake his own personalchoice for theseparametersand constructhis favorite model. SU(2)x U(1)breakdownis inducedthrough radiativecorrectionsand in somecasesa largemassof the top quark isneededto achievethis. The parametersm312 andA resultdirectly from the choiceof the hiddensector.Nonvanishingm0 can only occur in the caseof brokensupersymmetryand as a result the naturalrangeof m0 is given by m3/2. Natural valuesfor p. areeitherM~or zero (i.e. p. ~ m3,2) and we still have notfoundasatisfactorymechanismthat explainsp. -= m312. The necessity of p. � 0 in a minimal model is stilla somewhatunsatisfactoryfeature.The alternativeto an explicit insertion of p. in the superpotentialisthe choiceof a nonminimalmodel including a light singlet superfield.Suchamodelhasalsonaturalnessproblems as discussedin section 7.4. In addition we had the apparentcontradiction that for abreakdownat SU(2)x U(1) at the tree level one neededA > 3 but this also implied the existence ofdeeper U(1)em breaking minima after the inclusion of the Yukawa couplings. However, there we hadonly consideredthe parametersas they were given at M~and the relevant parameters at low energiesstill hadto bedetermined.Sucha calculationhasbeendoneby DerendingerandSavoy[112]in a modelbasedon a superpotential

g = AYHH + uY3+ Yukawaterms. (7.82)

In addition to the renormalizationgroupequationsgiven beforewe now haveadditionalequationsforA, o~, AA and A

7. Starting with a universal A at M~one has now to compute the evolution of theseparametersand onehasto find a situationwherethe inequalitiesin (7.56) like

m~+m~+m~.>(IAEI2/3)m~/

2 (7.56)

are satisfied for AE, A~and AD at low energies to avoid the undesired minima. To have SU(2) x U(1)breakdown,however, the correspondingconstraintson AA and A0. should not be satisfied and onerequires

3m~< (lA0.I2/3)m~,

2, m~,+m~+m~< (lAAI2/3)m~/

2. (7.83)

It has been shown that such a situation can be realized for a large range of parameters in these models.In generalit seemsthatasimultaneousvalidity of (7.56) and(7.83) requirestheinitial A to be chosen to

118 H.P. Ni//es, Supersyrnrnetry,supergravityand particlephysics

be less than three.Needlessto say that this breakdownof SU(2)X U(1) is independentof the value of

the top quark mass.7.7. Rareprocesses

We havealready discussedpossibleadditional contributionsto flavor changingneutralcurrentsinsupersymmetricmodelsin section 4.6. They were shown to be absentif all the massesof the scalarpartnersof quarksandleptonswith samehelicity andsamechargeweredegenerate.In the supergravitymodelsdiscussedso far this has beenthe case:all theseparticleshadequal mass:but only at M.,. Atlower energiestheydiffered,however,accordingto the renormalizationgroupequations(7.48) to (7.50)and one has to face this problem again. The contributions to the scalar masses that arise from thegauginomassesare not dangerous:the scalarswith samequantumnumberswith respectto SU(3)XSU(2)X U(1) receive the samecontributions and we have only to worry about those terms in therenormalizationgroup equationsthat contain Yukawa couplings.Usually theseYukawa couplingsarequite small and one might think that theseeffects could be neglected.Some models have a largeYukawacoupling to the top quarkandthis casehasto be investigated[151a,144].

As usualthe mostsensitiveboundsare providedby the experimentalinformationon the KL—KS massmatrix. The most important contribution from supersymmetryin this context is given by gluinoexchangediagramsas discussedearliersimply becausea3 is much largerthana2 anda1 at low energies.It turns out now that such a contribution to the KL—KS massmatrix is presentin the earlier discussedsupergravitymodelsas hasbeenpointedout in ref. [144].To seethis considerthe massmatricesM~forthe u, c, t quarksandMD for thed, s, b quarksandlet us chooseM~to bediagonal.MD will thenbe off

diagonaland contain the usual Kobayashi—Maskawaangles.In a supergravitymodel with parametersdefinedat M~the mass matrices of the scalars will then be

(7.84)

where we have usedA = 0 to renderour formulassimpler.NonzeroA mightalso havean influenceonflavor changingneutralcurrentsbut thesecontributionsaresmall comparedto thosewe will discussin amoment. At M~now we see that ~1l~is diagonaland~11~containsthe angles.Both M~,.Al~and MD,

can be diagonalizedsimultaneously(they are in phase)andthereis no possibility to induceflavorchangingneutralcurrentsvia gluino exchange.This changesafter the evolution of .//~t~Jand .iui~down tolower energies.Apart from contributionsthat are diagonaland universal in flavor spacewe will getadditional contributions from the Yukawa couplings. In particular this will give an additional con-tribution to At~that is proportionalto M~and whose coefficient is of order one,(the small Yukawacouplingsare already absorbedin M~)and we will also havean additional contribution M

2~to ~againwith coefficientof orderone.As a resultof this~ and~ are in phase.To computethe anglesthe first term proportionalto m~,

2in (7.84) is irrelevant.What countsis the ratio of diagonalelementsand the differencesof the diagonalelements.Since thesedifferencesin M~are much larger than thosein MD we thus arrive at a situationin which M~,~tt~and .At~ are essentially diagonal matrices. What isimportant hereis the fact that MD and ~ can no longer be diagonalizedsimultaneously.This thenimplies the existenceof flavor changinggluino vertices like the one shown in fig. 7.17, with couplingconstantse3sin 0 where 0 is the correspondingKobayashi—Maskawaangle. As a result we obtain acontributionto the realpart of the KL—Ks massmatrix dueto the usualbox graphwith gluino exchange.The magnitudeof this contributiondependson the valueof the gluino massandthe massesof the scalar

H.P. Ni//es,Supersymmetry,supergravityandparticle physics 119

d

Fig. 7.17. Flavorchanginggluino quark scalarvertex.

partnersof quarks.The former is determinedby m0 and the magnitudeof the latter onesis determinedpredominantly by the diagonal contributionsfrom m3/2 directly and m0 through the evolution. Thescalarpartnersof quarksarethereforestill pretty degenerateat leastfor the first two generationsandwe can assumeAm

2= m~— m~~ m~.To give you a feeling aboutthe magnitudeof the gluino exchangecontributionto the KL—KS massmatrix let us considera model where the gluino massis equal to themassesof the partnersof quarks.This is approximatelytrue for a wide class of modelsand in modelswhere thesemassesdiffer one in generalobtains similar results. We now consider the effect of thescalarsof the first two generationswherewe can draw conclusionsbecauseof the knowledgeof theCabibboangle.Oneobtains

(M~,~/th~)(Am2)2sin2 O0cos

200~2X10-6, (7.85)

wheresin2 0,,— [/20. The quantity Am2 hasto be estimatedusing the renormalizationgroup equation.

For typical modelswith smallYukawa couplingsthe coefficient of the additional contributionpropor-tional to M~in ~ turns out to be somewhatlarger than one. For the first two generationthedominant mass term comes from mchar,,, and we estimate

Am2 (2GeV)2. (7.86)

From(7.85) we thenobtain the bound

th3 ~ 35 0eV (7.87)

which in generalholdsif the massm2 of the partnersof the d, s quarksis lessor equalto th

3. The boundin (7.87) can only be lowered if m is larger than th3 — 40 0eV. To estimatethe influenceof a largetopquark massis moredifficult becausewe do not know theK—M anglesbetweenthefirst two andthe thirdgeneration.If oneusessin

2 0~,sin 0~,for the anglesbetweenfirst andthird, secondandthird generationand estimatesm~~/th~—1/3 for a largetop quark massone obtains

th3�2000eV, (7.88)

a ratherdramatic number.One should howeverusethe information from the KL—KS massmatrix toobtainrestrictionson the mixing anglesinsteadof deriving (7.88).

In the nonsupersymmetrictheory one canusethe simultaneousknowledgeof the KL—KS massmatrixandtheboundon KL—~p.p. to computean upperboundon the m10~[67].Thisboundchangesif onealsotakes into account the contributions of newly introduced supersymmetricparticles [390]. A fullcalculationof theserestrictionson m10~including the gluino exchangecontributionsis howevernot yetavailable.For the variousaspectsseerefs. [44, 72, 144, 176, 247, 337, 390].

120 H.P. Nilles, Supersymmetry,supergravilyandparticlephysics

IT

14 / \tp

q/T’\\q

Fig. 7.18. Contributionto theelectricdipole momentof theneutronthroughgluino exchange.q and ~qdenotea quarkand itspartner,respectively.

More restrictionson the parametersin the theory could comefrom the imaginarypart of the KL—KS

massmatrix andthe questionof CP violation. In this context,however,the information on the electricdipole moment of the neutron(exceptfor the considerations[250]of E’IE) gives stronger constraintsandwe prefer to discussthisprocess.

The effects of supersymmetry on the value of the electric dipole moment d0 of the neutronhavebeenestimated [161] and computed [66] in the context of a general model with soft supersymmetrybreakdown.The mostdangerouscontributionsto d0 comefrom gluino exchangediagramsas shown infig. 7.18 and they could be as large as d0/e~-4x10

22cm comparedto the experimental limitd

0Ie<6 x 10_25 cm. Gluino exchangecould howeverhe avoidedif in the contextof supergravitymodelsA andm0 havethe samephase.In addition to this sourceof CP violation thereexist othersourceslike aphasein the K—M matrix or a contribution of °OCD (the strong CP problem) [28,103]. If one puts allthesephasesto zeroby hand,problemswith d~can be avoided.The influenceof

0OCD can be naturallysuppressedin the presenceof an invisible axion [360,138, 568,449] which in the context of supersym-metry works in the sameway as in ordinary models. In a supergravitymodelone might now startwithsuch a prescriptionto haveall thesephasesto be equalto zero. On the other handwe know that thereis CP violation in the KL—Ks system.To describethis one includesa phase6 in the K—M matrix. Inadditiononewould like to haveCP violation for a cosmologicalbaryonnumbergeneration.This usuallyinvolves heavy fields and as such it gives no dangerouscontribution to d

0. The only way a sizablecontributionto d0 can occur is throughthe renormalizationof 0 if onehasa phasein themassmatrix oftheseheavyparticles.This, however,can be avoidedby. the choiceof an invisible axion or agenerationof the baryonasymmetryin higher loops.

As a result of all this onehas to insert the phase6 in a supergravitymodel at the grandunificationscale and one chooses0 = 0 (either explicitly or through a Peccei—Quinnsymmetry) as well as avanishingphasebetweenA and m0. In a next stepone has to determinethesephasesat lower energywith the help of the renormalizationgroup equations.Since CP violation is presentthrough 6 a phasebetweenA andm0 will be generatedperturbativelyandat low energiesonehasa gluino contributiontod0. For gluino massesof order of 100 0eV dangerouscontributions to d,, -~ 10

22ecm x CP-violatingphaseswill thenbe generatedat least for a largerangeof parameters[66,482]. To avoid problemswiththe experimentalboundthe parametersof the model haveto be carefully chosen.First of all we havethe choiceof either putting the K—M phasein the up or the down quark sector. In the nonsupersym-metric casethesetwo choiceswereequivalentbut herewith the appearanceof the new phasebetweenA andrn,, this is no longer true. Becauseof thedifferenceof the massesin the two sectorsoneobtainsasmallercontribution to d,, if CP violation originatesin the down quark (d, s, b) massmatrix andwhereM~is real andthereis no phasebetweenA and m

0 at M~.In such casesthe contribution to d0 can bepushedsomewhatbelow the experimentalboundeven in the presenceof gluino massesof order of

H.P. Nil/es,Supersymmetry,supergravityandparticle physics 121

50 0eV [432].With CP violation in the up quark sector this is somewhatharderto achieveand at leastan axion is neededto suppress0. CP violation in the context of supersymmetryis expectedto givecontributionsto dn not far from the presentexperimentallimit, at least oneexpectsthesecontributionsto be larger than in the ordinary case.For additionaldiscussionsseerefs. [8, 55, 66, 83, 105, 106, 151,161, 247, 250, 266, 432, 456, 458, 482]. Other rare processesor experimentallyexactly determinedquantitiesleadto a similarsituation.In the context of supersymmetryone can avoid realproblemswiththe constraintshut sometimesthe contributions are expectednot very far from the presentlimits. Insomemodelswe haveseenthat a largevalue of the massof the top quark is required.This might alsocausesome potentialproblemsin the low energyphenomenologylike a deviationof the p-parameter[537](p = M~/M~cos0) from onebut theyoccuronly for m10~~ 300GeV [14,47]. Strongrestrictionsonthe validity of a largeclassof modelwill howeveroccur if the massof the top quark is experimentallydetermined. The restrictions from g — 2 on the considered models are in general not severe[46,166, 232,289, 373]. Constraintsfrom proton decayhavealreadybeendiscussedearlier [11, 54, 76,108, lii, 126, 169, 179, 334, 393, 406, 436, 487, 491, 492, 493, 501]. They areessentiallythe sameas inthe globally supersymmetriccase.

7.8. Beyondthe minimal models

Up to now we havemainly discussedthe structureof the low energysectorof modelswith minimal(or closeto minimal) particle contentand haveonly occasionallydiscussedmore complicatedmodels.Thelow energyparametersarisefrom the onesat higher energyand the effectivelow energypotentialhasto be computed.These parametersm312, m0, A and p. dependon the structureof the model athigher energiesand through theseparameterswe could learn somethingabout the high energysector.Thereis not muchmorethat constrainsthis highenergysectorexceptfrom cosmologicalconsiderationswhich we discussin the nextchapter.The only constraintsfrom particle physics are given through theproton decay experimentsand they just tell us that M~>i0~~0eV. Consequentlythere is muchfreedomin constructingvariousmodelsfor the high energyandhiddensectorandconstraintscomeonlyfrom theoreticalconsiderations,prejudicesand taste.Onehasof coursealwaysto makesurethat thesevarioushigh energyscenariosdo not lead to wrong predictionsfor low energyexperiments.Therearevery manysuggestionsconcerningthe high energypropertiesof thesetheoriesand we will attempt todescribethe majordirectionsof research(seerefs. [61, 81, 142, 149, 184, 185, 186, 187, 260, 264, 327,333, 375, 377, 378, 379, 397, 399, 401, 426, 428, 433, 452, 457]).

We start with a discussionof M~.In a model with minimal particlecontentfrom M~— 100 GeV upto M~this quantity is fixed to be somewhatabove1016GeV [125,154]. In principle, however,it could belargerandfor exampleas largeas M~.Suchan increaseof M~could be achievedby addingnewdegreesof freedomto the theory betweenM~andM~such that the gaugecouplingconstantse1, e2, e3 join athigher energies.This requiressomecare to preservethe predictionfor sin

2 0,,. but this is in generalnot avery strongconstraint.Suchmodelshavethe nice theoreticalproperty that insteadof the two scalesM~andM~theynow contain only one.On the otherhandwith M~= M,. we haveto face the problemthatin principle we haveto understandgravity in order to makedefinite predictionsfrom the grand unifiedsectorcontraryto the caseM~‘~ M

1, wherewe can neglectgravitationalcorrections.Such modelsdo ingeneralnot needa separatehiddensectorsince thefields that breakSU(5)havealreadyv.e.v.’s of orderM~and they can induce the breakdownof supergravityas well. In a model where this breakdownisachieved through an adjoint representationthe goldstino (absorbedby the gravitino) will thencorrespondto the fermionic SU(3)x SU(2)x U(1) singlet in this representation.To havea supersym-

122 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

metry breakdownscaleat M~— 1011 GeV one hasto introducea small massparameterin the grandunified superpotential(just as one doesusually in the hidden sector),but such a situationmight beactuallydesirablefrom cosmologicalconsiderationsas we will seelater [428].It also implies in generalthe existenceof new low energy fields in the TeV rangebut they are usually phenomenologicallyunproblematicand evenneededto raiseM,. to M~.

A similar situationcan arisewith invertedhierarchymodels[456—467,131, 519]. Hereonestartswiththe inputscaleM1 — 1011 0eV andgeneratesa largescalevia the mechanismdiscussedin Chapter5. Inprinciple one could even imagine that in such a model the scaleM~is generated(with M~= Mr). Itseemshowever hard to induce in such a way gravitationalinteractionsas well. In absenceof such amechanismone couplesthese models to supergravityand also inserts M1,. The slowly decreasingdirectionsin the potentialwill then be turned up through gravitationaleffects andthe naturalscaleforthe vacuumexpectationvalueswill beM~[1311.Thesemodelshaveof coursesomeadditional particlesin the region of 1011 GeV (e.g. more than two Higgs in the (5,5) representationto avoid proton decayvia dimensionfive operators)but againoneneedsthis to raiseM., to M~.A variant of such modelscanalso be obtainedif one just startswith the two scalesM~and m312— 1000eV in a supergravitymodel[264,397]. Taking sucha model in the flat limit certainflat directionsin the potentialmight occurout ofwhich the grand unification scaleM., (not necessarilyequal to M~)could be inducedvia dimensionaltransmutation.Here,however,it seemsin generalto be somewhatdifficult to keepthoseparticleslight(e.g. the Higgs doublets)that shouldremain light. They in fact might evenreceivelargev.e.v.’s. Apartfrom this not much more can be said in this context as the things that were alreadydiscussedin theframeworkof modelsbasedon global supersymmetryandthe cosmologicalconsiderationswhich will bepostponeduntil the nextchapter.

Apart from these considerationsof various grand unified sectors additional possibilities can beobtainedthrough moregeneralterms in the superpotential:so-called“nonrenormalizable”terms[180].Up to now we haveonly discussedsuperpotentialsthat are at most trilinear in the superfieldsto assurethe renormalizabilityof the theory in theflat limit. With nonrenormalizablegravitationalinteractionsatleast in the frameworkof N = 1 supergravityone could now considera moregeneralsituationandaddtermsto the superpotentialthat are quarticand higher in the fields. With our discussionof the N =

supergravitypotentialat the tree graphlevel one might even arguethat we describethe gravity sectorby an effective potential and that one should include all terms consistentwith the symmetriesof thetheory since otherwisethey would be generatedin perturbationtheory.Now we do not really knowwhat happensin suchtheoriesbeyondthe treegraph level andespeciallyif F-termsaregenerated;wedo not even know a satisfactorytheory of gravity. Nonethelesssuch nonrenormalizableterms couldexist and they should be investigated.One should however require that the dimensionful couplingconstantsof thesenonrenormalizableterm are given in powersof K = 1/M. This is obviousfrom ourdiscussionbeforeandalsorequiredby renormalizibility in the flat limit. In the casewhereall fields haveat most vacuum expectationvalues of order M~such terms are irrelevant. If, however,we includegrandunification suchtermsmight becomeimportant,sincetheycan inducerenormalizablecouplingsinthe low energy theory with coupling constants(M0/M)” in terms wheren heavy fields are replacedbytheirv.e.v.’sof orderM~.Suchcouplingsarecomparableto theYukawacouplingsin thelow energytheoryandmightplay asignificant role. ForM = M~this discussionbecomesof courseproblematic.Let us nowdiscusssomeexamples.In a grandunified modelwith matterin the5 and10 representationsof SU(5)wecould addaterm [180]

(A/M)(10 x 10 x 10 x 5) (7.89)

H.P. Nil/es,Supersymmetry,supergravityand particlephysics 123

in the superpotential.Sucha term violatesbaryonnumberandcausesprotondecayin a similarway asthe earlier discusseddimension5 operators.In fact thesetermsare dangerousand to be compatiblewith the experimentallimit on the proton lifetime A hasto be smaller than 10_6 werewe would havenaively expected A to be of order one [169]. We do however not know how to determine Atheoretically.

Butnonrenormalizabletermsdo not alwayscauseproblems,theysometimescando nice things.Theyfor examplecan lead to a simplification of grand unified modelsin which the Higgs doubletsremainmasslessat thegrand unificationscalefor grouptheoreticalreasons[379].In the usualcasethis requireslargeSU(5)representationslike 75, 50 and50. With nonrenormalizablecouplingswe can do without the75. Consider

~ (7.90)

where.~tis an adjoint representationof SU(5).The usualvacuumexpectationvalueof .~at M~breaksSU(5) to SU(3)x SU(2)X U(1). Nonethelessthe Higgs doubletsremain masslesswhereasthe Higgstriplets becomeheavy.

Nonrenormalizabletermscan also be usedto generatethe smallYukawa couplingsandto constructa hierarchybetweenthe massesof fermionsin differentgenerations[431].Oneassumesthat the massesof the third generationoccur directly through the usual Yukawa couplingsand that the massesof thesecondand first family only proceedthrough nonrenormalizableterms

(A/M~)(24)~xH x 5 x 10, (7.91)

with n = 1 for the second and n = 2 for the first generation to arrive at a mass ratioA : A’(M~IM):A”(M~/M)2for the threegenerations.The hard problemremainsan explanationof theabsenceof renormalizableYukawacouplingsfor the first and secondgeneration.Investigationsof theproblemof the fermionmasseshavealso beendoneby inducinglight fermion massesthrough radiativecorrections[323,387, 392, 414,472a].

An additional application of nonrenormalizabletermsis a relation betweenthe variousmassscalesM, M~,M

5 andm312. Onecould for examplechooseashigh energysuperpotential[428]

g=_~X4+~X2Tr(I3)+m2(Z+z~) (7.92)

with singletsX and Z and .~ in the adjoint representationof SU(5). ~t is a constantused to adjust thecosmologicalconstant.In such a model one obtainsM~— Mm and m

312— m2/M out of the two input

scalesM and m. Using variouspowersof X in the first two termsanygrandunified scalem ~ M~~ Mcan beobtained.For sucha mechanismto work, however,renormalizableterms(like Tr(1~3))haveto beabsentin the full superpotential.

Finally one could use nonrenormalizableterms to generatea p.HH term in the low energysuperpotentialwith p. — m

312 [365].One startswith a Peccei—Quinnsymmetry [474] to forbid such aterm for the renormalizablepart of the superpotentialto avoid p. to be as large as M~.ThisPeccei—Quinnsymmetryis brokenspontaneouslyat M~0= 1010GeV to producean acceptableinvisibleaxion [118,2, 125,485] and solve the strongCP problem.Nonrenormalizableterms allowed by the

124 H.P. Nil/es, Supersymmetry,supergravityandparticle physics

Peccei—Quinnsymmetrythen induce a massterm m~(hh+h*h*) in the low energyscalarpotentialwith

m = AAmS/2(M~Q/M)= C(m3/2), (7.93)

afterthe U(1)~0breakingfields havebeenreplacedby their v.e.v.’s.M~0— M5 is crucial to obtain thisresult. Actual models that incorporate this mechanism,however, have a tendency to require acomplicatedgrandunified sector.This seemsinevitablein modelswith an unproblematicinvisible axion[3631.

The sectoreven less constrainedthan the grand unified sector is the hidden sector, as the namealreadyindicates.More generalcasesthan the oneswe havediscussedearlier like for example[378]

g(z)= A(b+ z)exp(cz) (7.94)

have been proposed. A, b, c are constantsused to obtain the desiredcosmological constant andsupersymmetrybreakingscale.One commonproperty of all thesehidden sectorsis the fact that ingeneralz will receivea v.e.v. of orderof M~andthe appearanceof the desiredsmallquantity m312/M~entersthrough the explicit choiceof input parameters.This smallquantity is obtainedeither by explicitintroductionor by the useof the exponentialfunction that allows the generationof this extremelysmallratio m312IM~out of a moderatelysmallparameter[378,184,61].

It is crucialfor our approachto get an ideawhyM~is so small comparedto M~.But thisseemsto bea difficult task. It might come from a dynamicalmechanismor from propertiesof extendedsuper-gravities,andwe would haveto understandgravity to solve this problem.This actuallyseemsinevitableif the fields in the hiddensectorreceive v.e.v.’s which are as large as M~.But maybeone could evenunderstandM~in absenceof a completeknowledgeof the gravity theory.In sucha situationone wouldthink that the v.e.v.’sof the hiddenfields are small comparedto M~andonecould for exampleimaginethat M~and m312 are generatedfrom M~and M~.This ratio M51M0 as well as the value of thecosmologicalconstantremain a puzzle for the time being. One possibility to explore is the questionwhetherthey are related.

Some attemptshavebeenmadeto constructclassesof modelsin which the cosmologicalconstantisboundto vanish [99].They arebasedon the observationthat the Kählerpotential

G(z,z*)= _~log(f(z)+f(z*))2 (7.95)

leadsto a scalarpotential

V=9 exp(~G)~ ~ * exp(-~G) (7.96)

thatvanishesidentically[79].Observethatin thiscasetheKählermanifoldisno longerflat (a2R!azaz*= 0)(compare(3.48)), but obeys

~ = ~ (7.97)

andaccording to the usualterminologymight be calledan Einstein—Kählermanifold. There is a hugevacuumdegeneracyandthe choiceof differentpointsleadsto different gravitino masses

H.P. N/lies,Supersymmetry,supergravityand particlephysics 125

m~12=exp(—G)=1/[f(z)+f(z*)Is. (7.98)

In order to preservethevanishingcosmologicalconstantat the treegraphlevel after the introductionof

the observablesectorone addsthe Kählerpotentialof the two sectors(compare(6.27))G = G(z,2*) + G(ya,Y*a) (799)

wherethe Ya denote the observablefields and onein generalchoosesminimal kinetic termsfor thissector to obtain the usual positive STr~2 As shown in section6.3 one then arrivesat a low energyeffective potential with A = 3 [96], and the cosmologicalconstantvanishesat the tree graph level.RadiativecorrectionschangeA andinduce a cosmologicalconstantin perturbationtheory.Onemightactuallytry to obtainthe gravitino massvia dimensionaltransmutationfrom radiativecorrectionsalongoneof the flat directionsof the scalarpotential.Taking into accountonly nongravitationalinteractionsasmall gravitino mass could be obtainedin this way [173]. If, however,thereexist also gravitationalradiativecorrections(i.e. if they are not forbidden by new nonrenormalizationtheorems)one wouldratherexpect m312 to be of order of M~if it is nonzero,especiallysince thesemodelstend to containfields with v.e.v. large comparedto M~[361].

The Kähler potentialin (7.95) is equivalentto

G = —~ log(z+ z~’f (7.100)

up to field redefinitionsz—*f(z).The scalarLagrangiancorrespondingto (7.100)is given by

3V_gg1~~[a~.z9~.z*I(z+ z*)2], (7.101)

and coincideswith the scalarLagrangianof pure N = 4 supergravity.In particularit hasan SU(1,1)

noncompactsymmetryz -* (az + i/3)/(iyz + 6). (7.102)

The completeN = 4 supergravityLagrangian is fully SU(1,1) symmetric[102]. In the framework ofN = 1 supergravity,however,it is hard to imagine how this SU(1,1) symmetry can survive especiallyafter the introduction of the observablesector. But in principle it could be such a symmetry whichrequiresthe cosmologicalconstantto be absent.It probably hasto be broken in the breakdownofextendedsupergravitiesto simpleN = 1 supergravity.But evenin this caseonecouldimagine asolutionof the problemof the cosmologicalconstantin the sameway as a spontaneouslybrokenPeccei—Quinnsymmetry solvesthe strongCP problem.

The considerationof extendedsupergravitiesmight give us some hints how to constructhiddensectorswith certainnice properties.But the big problemof embeddingN = 1 supergravitywith complexmatterrepresentationsin extendedsupergravityis of coursenot yet addressedin this context.

To close this section let us mention again one additional assumption madeso far in all ourdiscussions.We have implicitly assumedexplicit baryon and lepton number conservation in thesuperpotentialof the low energy (1000eV) sector (comparesection 4.4). This led inevitably to aconservationof R-parity (4.25) in the low energytheory, and as a consequencethe lightest R-oddparticle (e.g. the photino) is absolutelystable.In principle this requirementcould be given up as well

126 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

but one has to be careful. Having baryon violation at this level will usually lead to disastrousprotondecayrates.It is easierto relax the requirementof exact lepton numberconservation,but also thereconstraintsfrom Majorananeutrinomasses,neutrinooscillationsetc. haveto be takeninto accountasdiscussedin ref. [308].Thesequestionsandthe stability of the lightestR-oddparticlewill beimportantin the contextof cosmologicalconsiderationsto whichwe will devotethe nextchapter.

At the time this report is written the discussionof thesenonminimalmodelsis still in progress.Wecannotdiscussall the possibilitieshere.Oneof them is the enlargementof the grandunification group[23,70,84, 246, 322, 362, 398, 407,408, 582, 583] but therearemanyothers.For furtherinformationwerefer the readerto the paperscited at the beginningof thissectionor to theexistingreviewarticlesrefs.[33,34, 35, 38, 130, 157, 158, 159, 160, 193, 207, 212, 214,220, 221, 222, 223, 326, 347, 425, 440, 445, 447,460, 464, 479, 486, 506, 588].

8. Constraints from cosmologicalconsiderations

In this chapterwe want to discusspossibleconstraintson the presentedmodelsthat might arisefromcosmology. This will include constraintson the mass of the lightest R-odd particle predicted bysupersymmetry,the massof the gravitino, the nature of the SU(5) phasetransition in the context ofsupersymmetry,the cosmologicalbaryonasymmetry,monopole abundancesand relatedproblems.Wecannotgive herea review on standardhot big bangcosmologyand refer the readerunfamiliar to thesubject to consult the standardreviews. We will mainly concentrateon those issuesthat are directlyrelatedto the presenceof supersymmetry.

8. 1. Constraintson (almost) stableparticles

In a wide class of the discussedsupersymmetricmodelsone expectsthe presenceof an additionalstableparticle, the lightest R-oddparticle. In the globally supersymmetricmodelsthis is in generaltheexactly masslessgoldstino. In local supersymmetryit will be either the gravitino (in caseof M5’~1010 0eV) or the photino, fermionic partnersof Higgses, scalarpartnersof neutrinosor maybe evenotherparticles.In generalwe expectthoseparticlesto be electrically neutral.If theywerechargedtheyshould be heavierthan 20 GeV for the well-known reasons[4,32, 52,53, 58,529]. The fact that thestable particle is electrically neutral is also favoredcosmologically. Charged stable particles wouldcondensein conventionalmatter with a density far aboveexperimentalupper limits on stableexoticrelics from the big bang[514,575].

Fortunatelywe have not to face theseproblemshere but only the milder restrictionson stableneutralparticles.Such restrictionsarise from the upper limit on the massdensity2 x 10—29 g/cm

3of theuniverse derivedfrom estimatesof the Hubble constantand the decelerationparameter.Given thecosmologicalabundanceandthe massof a neutral stableparticle onehasto checkwhetherthisboundon the massdensity is violatedor not. Suchconsiderationshavefirst beenapplied to obtainboundsonthe massesof heavyneutrinos[396].In the standardbig bandcosmologywe expectthe numberdensityof eachkind of neutrinosto be 6/11 the numberdensityof photonsin the 3K blackbody backgroundradiation, hence300cm3. The upperboundon the massdensityof the universe then gives an upperboundon the neutrinomasses:40 eV.

This bound is true as long as our estimateof the presentcosmologicalabundanceof neutrinosiscorrect. It has beenpointed out by Lee and Weinberg[396] that the estimateon the presentneutrino

H.P. Nilles, Supersymmetry,supergravityandparticlephysics 127

abundanceis incorrect if theneutrinosaresufficiently heavy.We will presentthis argumentheresinceitcan later be carriedover to our discussionof the stableR-oddparticle. Theseparticleswill drop out ofthermalequilibrium when the temperaturedecreasesto a point at which the stable particle collisionratesbecomecomparableto the expansionrate of the universewhich in the caseof neutrinosis 1010K.If the neutrinosarelight (�40eV) this gives an abundanceof 300 cm3 as quotedabove. If, however,the neutrinosareheavy(m...> 1 MeV 10’1K) we expectthis numberto be smallerby the valueof theBoltzmanfactor exp(—mj[1 MeV]) at the time the heavy neutrinosdrop out of thermal equilibrium.Onewould then require

m~exp(—mj[1 MeV]) ~ 40 eV, (8.1)

which leadsto the bounds

~ 40 eV, or mi’ � 13 MeV, (8.2)

accordingto this naive argument.But with theseheavy particleswe haveto be more careful. Theseparticles usually carry a conservedquantumnumber(R-parity or here neutrino number)which is ofcoursethe origin of their stability. As a result their numberdensitycan relax to the equilibrium valueonly throughparticle-antiparticleannihilation.At the temperatureof 1 MeV the heavyneutrinosaresorare that their annihilation rate is already much less than the cosmic expansionrate, althoughthereenergydistribution is still thermalized.Consequentlythe heavy neutrinosgo out of chemicalequili-brium (their numberdensityexceedstheequilibrium value)at a temperatureT

1 muchlargerthan 1010K.

The condition (8.1) hasthen to be replacedby

m~exp(—m~.IkT1)� 40eV. (8.3)

Since kT~>1 MeV the naivelower boundof 13MeV is raisedif onetakesinto accountthis fact. T1 nowdependson the propertiesof the particlesunder consideration.For the heavy neutrinosit hasbeenestimatedto be about5% of the heavy neutrinomass,andoneobtains

m~40eV, or m~2GeV, (8.4)

considerablylarger than the boundof 13 MeV obtainedfrom the naiveestimate.Suchstable particlesare thereforecosmologically unproblematicas long as their massesare not in the forbidden mass“window”: here40 eV� m~~ 2 0eV.

We can now applythis cosmologicalconstraintto our supersymmetricmodels.In the global casethelightestR-odd particleis the masslessgoldstinoandno constraintarisesprovidedthat the otherR-oddparticles decay sufficiently fast into goldstinos and R-even particles. But even these models haveeventually to be coupled to gravity and in the context of supersymmetryone would expect thesuper-Higgseffect to occur in which this goldstinois absorbedby the now massivegravitino m312—M

2S/M whereM~denotesthe supersymmetrybreakdownscale.As beforewe now expecta “window”of forbiddengravitino massesandconsequentlya rangeof M

5 that is not allowed. Taking into accountthe expectedprimordial gravitino abundanceand assumingit to survive until presenttimes for lightgravitinos one obtains m3/2~1 keV as was pointed out in ref. [472],which in turn implies ~2 x 1060eV. This is the boundthat correspondsto m,, <40 eV in the caseof neutrinos.

128 H.P. Nil/es, Supersymmetry,supergravityandparticle physics

This bounddoesnot apply if the gravitino massis heavy enough,asin the neutrinocasewhereabove2 0eV no problem occurred.This lower boundnow dependson the decouplingtemperatureand thegravitino annihilation rate: it hereimplies m312> 0.3Mg, andthe window of forbiddengravitino massesis quite large [554]. Can we now concludethat values 1060eV~ M5 ~ M~are cosmologicallyforbid-den?Therearetwo reasonsto believethat the answerto this questionis in generalno. First of all wehaveseen that in the consideredmodelswith large m3/2 (think of m3/2 — 1000eV)we no longer expectthe gravitino to be the lightest R-oddparticle andconsequentlyit will decayand this will decreaseitscosmologicalabundance.In thesecaseswe will of coursehaveanotherstableR-odd particle whosemass boundswe will discusslater. The decayof the gravitino in lighter R-odd particleswill typicallyproceedwith a rate F— m~/2IM~andthe questionremainshow heavy the gravitino hasto be to decayfastenoughsuch that no cosmologicalproblemsarise.If theydecaytheywill give rise to an increaseofthe cosmic entropydensity. If cosmic nucleosynthesiswere over by the time this occurredthis wouldhave led to too much helium production. It seemstherefore necessaryto require m312 to be largeenough to havethem decay before nucleosynthesis,or that after there decay the reheatingof theuniverseis largeenough(>0.4MeV). Theseconsiderationsrequire m3/2~ 10~0eV as estimatedin ref.[554].This is still largeand correspondsto a supersymmetrybreakingscaleM~>~1011 0eV.

But thereis anotherway out of the problem, which hasto do with the fact that on the one handgravitinosare producedat temperaturesKT — M~in the veryearlyuniversewith the normalabundance,but on the otherhandthe constraintson m3/2 arisebecausethe gravitino decayrate is much less thanthe expansionrate of the universeuntil the temperaturehasdroppedway below m3/2.This is a problemwhich typically can be solved by inflation [298].If the universewould haveundergonean inflationaryepochthis would havedilutedthe primordial gravitino density[174].Above m3/2 theproductionrateofthe gravitinos is alsovery smallandthe primordial densitywill thereforenot be recreatedandhencenoproblemswith the cosmicmassdensitywill occur. An actualestimateof the amountof inflation neededto solve the “gravitino massproblem” showsthat oneneedslessinflation thanrequiredfor thesolutionof the “monopole” and “flatness” problem [298].Taking into accounttheseconsiderationswe canconcludethat no seriouscosmologicalproblemsconstrainthe gravitino mass.In the case1 keV ~ m312~i0~0eV someamountof inflation is neededto dilute the primordial gravitino density [174].

In the conventionalsupergravitymodelsthe gravitino is not the lightest R-oddparticle. In a largeclassof models the photino plays this role. In somecasesthis lightest particle can also be a neutralHiggs fermion or a scalar neutrinoas we havediscussedin the last chapter.Potential cosmologicalproblemscan in thesecasesnot be cured by inflation in the very early universe,since after inflationtheir equilibrium densitycan be recreated(beforethey drop out of equilibrium) dueto their strongerinteractions.

Let us thereforefirst assumethat the photinois the lightestR-oddparticle andderivecosmologicalconstraintson its mass.Such a photino shows similar behavioras a neutrino, but there are somedifferences.First of all its interactions are not so well known as those of the neutrino, which arecharacterizedby the SU(2)x U(1) couplingconstantsandthe massesM~andM~.The interactionsofthe photinodependon the way supersymmetryis broken.A crucialparameteris the massof the scalarpartnersof quarksand leptonssince n-creationand annihilation will proceedthrough exchangeoftheseparticles.Thecosmologicalboundon m~will thereforedependon thesemasses.Secondly,unlikethediscussedheavy neutrino,~ is not aDiracbut aMajoranafermion andas aresultn-annihilationissuppressedby p-wave phasespaceat low temperatures.Consequentlythe boundson heavy neutrinoscannotbe directly carriedover to the photinobut the whole calculationhasto be reexamined.Suchacalculationhasbeendone in ref. [286].The photinoscannotbe as light as 40 eV, sincethey receiveat

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 129

2O~j7

m5 (6eV)

Fig. 8.1. Lower boundon photinomass(m~)if thephotinois the stableR-odd particleasgiven in ref. [286].The boundincreaseswith increasingmassrn, of the scalarpartnersof quarksandleptons.

leastradiativecontributionsfrom the quarksandleptonsto their massandas discussedearliereven inthe absenceof superheavyparticlestheyreceivea massof at least100MeV. Consequentlywe haveonlyto concentrateon the second cosmological (lower) bound on the photino mass.The result of thecalculationis given in fig. 8.1, wherethe lower boundon the photinomassis plottedagainstm the massof the scalar partners of quarks and leptons here assumedto be degenerate.For small m (theexperimentallimit is given approximatelyby 20 0eV) m~ hasto be largerthan2 0eV, and this boundstaysconstantup to m — 50 0eV. This is dueto the fact the s-waveannihilationof photinosinto heavyfermions(like r and c) is effective for m s50 0eV whereasthe annihilation into masslessfermionsisp-wavesuppressed.For massesm > 50 0eV this annihilation of the cool photinosbecomesmore andmoreincompleteandthe lower boundon m .~ increaseslinearly with m. Above m = 60 0eV the bounddependssomewhaton the choiceof T* relatedto thefreeze-outtemperatureof the photinos[286,381].For m = 1000eV the lower boundon m~ is of the orderof 20 0eV which is a relatively large value.Looking back to the presentedmodels,however,we see that in general theseboundson m can beeasilyfulfilled. So far our discussionon the photinomass.

Another candidatefor the lightestR-odd particle is a massiveneutral Dirac fermion formed as acombinationof a left-handedHiggsfermion anda gauginoof SU(2) (compareour discussionin section7.5). The cosmologicallower boundon the massof this fermion (if it is the lightest R-oddparticle) isusuallyhigher thanthe oneon m.~. This comesfrom the fact that the gaugecouplingsareusuallylargerthanthe Higgscouplingsandas a result the cool gauginosannihilatemore efficiently. Thesequestionshavebeenstudiedin detail in ref. [168].In a typical modelwith small p. and m3~2and a light photinoand Higgs gaugino(H) the results can be presentedin diagramslike given in fig. 8.2. m~is plottedagainstm~ andthe cosmologicallyallowedregion is thestradedarea.The missingedgein the upper lefthandcorneris forbiddenbecauseof e~eannihilationexperimentalconstraints.In everymodel underconsiderationit shouldbe checkedwhetherthesecosmologicalconstraintsarefulfilled. Usually theycanbesatisfiedeasily if the photinois the lightestR-oddparticle.Sincein mostof the discussedmodelsthephotino is the lightest R-odd particle thesecosmologicalconstraintsfrom the mass density of the

130 H.P. Nil/es, Supersymmetry,supergravityandparticlephysics

m~(6eV)

Fig. 8.2. The shadedarea givesthecosmologicalallowed region in the“massplane”of thephotino (m ~)andHiggsino (mci) asgiven in ref. [168].Itis assumedthat the lightestof theseparticlesis stable.

universeare not very strong. In addition they could becomecompletelyirrelevant in modelswhereR-parity is not an exactsymmetryandhencethe formerly stableR-oddparticlescould decay[308].Thecasein which the scalarpartnerof the neutrino is the lightest R-oddparticle hasalsobeendiscussedinthe literature— with similar conclusions[328].Finally in modelswherethe strongCP problem is solvedwith the introductionof an invisible axion, we expectthe partnerof this axion [528](the axino; not tobeconfusedwith the saxino[363])to be the lightestR-oddparticle[361].Sincethis particleis very lightno cosmologicalproblemswill arise, provided that the photino decaysfast enough. This lowers theboundon m~comparedto the boundgiven above[361].

8.2. Thegrand unifiedphasetransition

In the first sectionwe haveconsideredprocessesin which the relevantfreeze-outtemperatureswereof the orderof KT — MeV or 0eV. We will now discussprocessesthat happenedat earlier timesandhigher temperatures.In the framework of grand unified modelsthis implies a discussionof the SU(5)phasetransition, the creationof a baryon asymmetry[494,578, 127] and the productionof magneticmonopoles[298].

Let us begin with an SU(5) gaugetheory with one chiral superfieldA in the adjoint representationand a superpotential

g = Tr(~mA2+ ~AA3). (8.5)

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 131

In aglobally supersymmetricmodel this leadsto a scalarpotential

V = Tr(F*F) + ~DaD,., (8.6)

with

F*=_ogIoA. (8.7)

The peculiarpropertiesof suchsupersymmetricpotentialsis theexistenceof severaldegeneratevacuumstateswith V = 0, here

A=0, A= (m/3A)diag(1,1,1, 1,—4), A= (m/A)diag(2,2,2,—3,—3), (8.8)

correspondingto unbrokengaugegroupsSU(5), SU(4)x U(1), SU(3)x SU(2)x U(1) respectively.m is amasscomparableto the grandunification scaleM~.In a locally supersymmetricmodel thesestateswillget split but theywill still remainapproximatelydegeneratesincetheir differencein energywill be verysmall comparedto m4. Let usnow discusssthis potentialat high temperaturesas done by Srednicki[517] and Nanopoulosand Tamvakis[434].At temperatureslarge comparedto m one hasto add atemperaturedependentterm to the scalarpotential [141]

A V(T) = ~T2I02gI3A~i9A~I2+ ~T2g2C(R,)A~A1, (8.9)

where g is the gaugecouplingand C(R1) is the eigenvalueof the quadraticCasimir operatorfor therepresentationsunderconsideration.In our casethis correspondsto [517,434]

A V(T) = ~.iiT2(21A2+ 25g2)Tr(A*A). (8.10)

In eachof the threegroundstates(8.8) onehasF = D = 0. At finite temperaturethesestatessplit as aconsequenceof theadditionalterm(8.10)in thepotential.A V(T) isobviouslyminimizedby (A) = 0 andathigh temperaturethe theory will favor the SU(5) symmetricphase.The approximationleadingto (8.9) isonly valid when T is large comparedto m. At T — m we cannotmakedefinite statementsbut at lowtemperaturesT ~ m wecan usethemasslessquantumgasapproximation.The effectivepotentialis givenby the free energyof the systemandonewrites [434,517]

V = —(ir2/90)T4(N8+ ~NF) + Tr(F*F) + ~D

2, (8.11)

whereNB(NF) is thenumberof masslessbosonic(fermionic)degreesof freedomat energiesKT. F1omthe

threepossiblephasesthemostsymmetricone(herewith SU(5)symmetry)hasthemostmasslessdegreesoffreedom.Sincethefirst termin (8.11)comeswith anegativesignagaintheSU(5)phasehaslowestenergyassketchedin fig. 8.3. Asfor T ~ m alsofor T ~ m thesymmetricphaseis favored.This is differentfrom thecaseof standardnonsupersymmetricSU(5)wherethecritical temperaturefor the SU(5)phasetransitionisof the orderof m [298].

In the supersymmetriccasethe universestays in the SU(5) symmetricphasestill for temperatures

132 H.P. Nil/es, Supersymmetry,supergravityand particlephysics

V

~~i3~2~1

T>O

~M~2~1’

Fig. 8.3. Sketchof thescalarpotential in a supersymmetricSU(5) model at zero andfinite temperature.

that are small comparedto M~.One might now ask whetherthereis a possibility at all for a phasetransitionto the SU(3)x SU(2)x U(1) phaseat lower temperatures.According to (8.11) this can onlyhappenif the numbersNB andNF of masslessdegreesof freedomchangein the SU(5) phaseif we go tolower energiesandthereis in fact such a possibility. To see this considerthe running gaugecouplingconstantsin eachof the threephases.The SU(5)phasehasthe most masslessdegreesof freedomandthe SU(5) coupling constantwill therefore increasefaster with decreasingenergyas the respectivecouplingconstantsin the otherphases.Let usnow considera modelwith A, threegenerationsof quarksandleptonsand two light Higgs doublets.With a5— 1/24 andM~— 1016 0eV a5 increasesfast in theSU(5) phaseand becomesof order unity at A — iO~—iO~°0eV while in the other phasesthe gaugecouplingconstantsarestill small. At A we expectconfinementeffectsto occur in the SU(5)phase.If thestronginteractionsbreaksupersymmetryonewouldexpectan additionalcontributionof orderA

4 to thevacuum energydensity. If they do not break supersymmetrywe would still expect the temperatureindependentpart to have zero vacuum energy. The confinementeffects do however decreasethenumberof the masslessdegreesof freedomat T — A. Masslessparticleshaveto be SU(5)singletboundstatesand there are not very many of those one would expect.As a result of this the temperaturedependentpart of the effective potential in the SU(5) phasechangesdramaticallyat A. Due to theconfinementeffectsthe vacuumenergyin the SU(5)phaseincreasesandwill in generalbe higher thanthoseof the otherphaseswhich fulfills thusa necessarycriterion for a phasetransitionfrom SU(5) toSU(3)x SU(2)x U(1) [517,434].

With the specific modelunderconsiderationat A — 10~0eV, however,the phasewith lowestvacuumenergyis the SU(4)x U(1) phase.It hasthe mostmasslessdegreesof freedom.Consequentlythe SU(4)coupling constantincreasesfast down to lower energiesand will becomeof order unity a

4— 1 atA’ — 1060eV and confinementeffects will also occur in this phasewhile the coupling constantin the

H.P. Ni//es,Supersymmetry,supergravityandparticlephysics 133

SU(3)x SU(2)x U(1) phasearestill small.Below A’ — 1060eV we thus expectthe SU(3)x SU(2)x U(1)phaseto havethe lowestvacuumenergy.The last part of our argumentis modeldependent.Onecouldvery well havesituationsin which the SU(3) x SU(2)x U(1) phaseis the lowestat A — 10~0eV suchthata direct phasetransitionfrom SU(5) to SU(3)x SU(2)x U(1) could occur there.For this to happenit isof coursenecessaryto include moredegreesof freedomin the theory [517].This, however,is usuallythe casefor other reasonsas we will see later. Independentlyof this argumentwe haveseenthat at acertain critical temperaturethe effective potential evolves in a way that the desiredSU(3)X SU(2)XU(1) phaseis energeticallyfavored.This critical temperaturelies somewherebetween106 and 1010 0eVand is thereforemuch smaller than in the nonsupersymmetriccasewhereit is usually of the order ofM~.

This of coursehascertainimplications, two of which we want to discussnow. The first one is theproductionof magneticmonopoles.Thesegrandunified monopoleshavea massof orderM,./a5 andareexpectedto be producedin a phasetransition wherea semisimplegroup breaksinto a group withexplicit U(1) factors[484].

Let usnow supposethat the phasetransitionfrom SU(5)happensdirectly to SU(3)X SU(2)X U(1) atTcrit < 10~0eV and let us supposethat the bubblenucleationrate is sufficient for percolationand arapid phasetransition, an assumptionthat is supportedby the standardestimatesif Tcrit is so small[517]. The differentbubblesare initially uncorrelatedand arbitrary orientationsof the Higgs fields inthe adjoint representationoccur. The bubble collision will then produce topological knots whichbecomemonopoles.The numberdensityof the monopolesso producedis expectedto be within a fewordersof magnitudeof the numberdensityof bubblesand strongly dependson the unknownbubblenucleationrate. In the standardnonsupersymmetricgrand unified models thereappeareda problemsince no bubble nucleationrate was compatiblewith both percolationand a sufficient suppressionofmonopoleproductionto satisfy the observationalbounds.This changesin the supersymmetriccaseandthe main reason for that is the low critical temperature.If Tcrjt < 10~0eV for a large range ofnucleationratesthe transition is completedquickly and bubblespercolate.The thermalproductionofmonopolesis heavily suppressedby the Boltzmann factor exp(—M,.IasTcrjt)and is compatiblewithobservationalboundsfor the small values Tcrit < i0~0eV. This is actually a nice property of thesupersymmetricscenario[517,434]. The SU(5) confinementscaleleadsto an upper boundon Tcrjt10~0eV and this is sufficient to suppressmonopoleproduction.In the nonsupersymmetricmodel oneusuallyhasto imposeinflation to solve this problem[298].

But such a low critical temperaturealso has its problemsand they show up if one considersthepossiblecreationof a baryonasymmetry.Usually such abaryon—antibaryonasymmetryis generatedbyB and CP-violating decaysof superheavy(—Mi) particles that go out of thermal equilibrium as thetemperaturedropsbelow their mass[494]. In the supersymmetriccasethe universe staysvery long inthe SU(5) symmetricphaseand these“heavy” particlesare usually massless(e.g. SU(5)gaugefields)andstay in thermalequilibrium. Theenergyreleasedduring the phasetransitionis of orderA

4 and thisis not enoughto createa sufficiently largenumberof thesesuperheavyparticles(in the SU(3)x SU(2)X

U(1) phase)to createan acceptablebaryon—antibaryonasymmetrythrough their “out of equilibrium”decays.

To createan acceptablebaryonasymmetrynew particleswith massesof order 10100eV (in theSU(3)X SU(2)x U(1) phase)haveto be introduced[436]. Someof them could be the Higgs triplets inthe (5 + 5) representationsof SU(5). The absenceof proton decayrequiresthem to be heavybut sincethey only couple with small Yukawa couplingsto the particles that participate in proton decaythepresentobservationalboundson the protonlifetime do not really requiretheir massesto be as largeas

134 H.P. Nilles, Supersymmetry,supergravityand particlephysics

M,.. The boundis somewherein the rangeof 1010 0eV andit is amazinghow all thesescalescoincideinthis rangeof iO~to 1011 0eV (including the supersymmetrybreakdownscale). In a model with suchsmallHiggs triplet massesonewould thenalsoexpectprotondecayto, e.g.,p.K becauseof the Yukawacouplings.One must howeverwatch out in such a model with M~— 10100eV for proton decayviadimension5 operatorssince this would lead to a far too small proton lifetime. This decayhasto beforbiddenandthis can usuallybe achievedby the introductionof additionalHiggs (5 + 5) multiplets andsomeadditional symmetries.One usually should makesure that thereis only one set of light Higgsdoublets(—1000eV) in order to keep an acceptablevalue for sin20~(although this also could beachievedin differentways).This increasein the numberof Higgs particlesis alsonecessaryto introducethe requiredCP violation in this sector.This CP violation is usuallyput in through the couplingsin thissectorand not directly in the massmatrix [436].If one would introduce it alreadyat the level of themassmatrix this would lead to nonnegligibleradiative contributionsto the 0-anglewhichin turn wouldgive unacceptablevaluesfor the electric dipole moment of the neutronas discussedearlier,unlessaninvisible axion is present.

During the phasetransitionat 10~0eV a sufficient numberof theseparticlesof 1010 0eV can now beproduced.They are not thermalizedand decay out of equilibrium. With a correct insertion of theCP-violating phasesone can then obtain an acceptablebaryon—antibaryonasymmetry.We shouldmentionhere that one can alsoconstructothermechanismsto createa baryon asymmetrybut the onediscussedaboveseemsto be the onewhich is the moststraightforwardandlike the oneabovealsotheotheronesneedthe introductionof additional particleswhoseonly reasonof existenceis the needforthe creationof the baryonasymmetry[436,371].

Let us now look moreclosely at the SU(5)phasetransition.Up to now we havealwaysassumedthatit happensbut a closer inspection of the effective potential shows that this is not necessarilytrue[518,175]. The intrinsic scale of the grand unified potential is given by M~.Let us now considertheeffective potential at 10~0eV andassumethat the SU(3) X SU(2)x U(1) phasehasthe lowest energy.The energiesbetweenthe different phasesare of order of 10~0eV, but thereis a barrier betweenthedifferent minima still given in magnitudeby M~— 10’~0eV and huge. It is now the questionwhethersuch a high barrier will not preventthe phasetransition from SU(5) to SU(3)x SU(2)x U(1), and todecide this we have to estimatethe bubble nucleationrate. In the semiclassicalapproximation thisnucleationrateis given by [90,71.5,403,300]

A(T)= Kexp(—S11), (8.12)

where K is a determinantalfactor and S0 is the action of the “bounce” solution for the adjointrepresentationof the Higgs field

lIT

S0= J dtJd3x[~3~A~2+V(A)], (8.13)

and V(A) is the free energy.An estimateof A in the thin wall approximationgives [518]

A(Tcr~t= 10~0eV)= 10~1062(0eV)4. (8.14)

This numbershouldbe comparedto the fourth power of the expansionrateof theuniverseat thesame

H.P. Nil/es, Supersymmesry,supergravityandparticlephysics 135

temperatureT which is of the order of (1 0eV)4 at Tcrjt = iO~0eV. One might now arguethat theestimatein (8.14) was obtainedunder specialassumptionswhich might be questionablebut one canhardly imaginethat it is wrong by 1062 ordersof magnitude(observethat this is not 62 but 1062 ordersofmagnitude).Consequentlya rapid phase transition in which many bubbles form and percolate isimpossible in the presentmodel. The universewill supercoolin the SU(5) phaseand will stay thereessentiallyfor ever[518].

Onemight now arguethat in modelsbasedon supergravitythe temperatureindependentpart of thepotential hasalready nondegeneratevacua and that the splitting of the vacua in the different phasesmight be largerthan just the thermalsplitting at T — 10~0eV, but thesesplittingsare proportionalto~iE~ac (MXm

3/2)2 and hencetoo small to enhanceA in (8.14) sufficiently. One might also consider

examplesfor grandunified models in which the phasewith unbrokengaugesymmetrydoes not exist.But therein addition onehasto makesurethat the SU(3)x SU(2)x U(1) phasewe want to endup withhasthe most masslessdegreesof freedomat all temperaturesbelow M~.This then also implies theabsenceof the SU(4)x U(1) phaseand other phaseswith largergaugesymmetry than SU(3)x SU(2)X

U(1) and this seemsto be in generalimpossible(at leastno exampleis known up to now). This meansthat we haveno examplewherewe understandwhy the SU(3)X SU(1)x U(1) phasemight be chosenaboveM~.

The only way to get out of this problemis to decreasethe barrier that separatesthe moresymmetricphasesfrom the SU(3)x SU(2)x U(1) phase[429].Unfortunatelythis alwaysrequiresa fine tuning ofthe parametersof the theory or the insertion of very small parametersin the theory. Let us look atpotential (8.6) to seewhat one can do:

V = Tr(~rnA+ AA212) + ~g2(A*TaA)2, (8.15)

where P are the groupgenerators.The A’s should receivev.e.v.’s of order M~which in generalaregiven by the ratio rn/A. Thus we cannot changern/A but we can changem and A separatelyprovidedrn/A stays fixed. The secondterm (~D2)is fixed by the gaugecouplingandwe cannotdo anything.Thesimplestway to suppressthebarrier is to multiply the first term by a couplingconstanto- andto adjustits value in a way that the phasetransitioncan occur. The secondterm is still large, but therearedirections in which D = 0 and tunnelling can occur there. The phasetransition can occur at T~~

11—

10~0eV if the barrier is not larger than let us say 1010 0eV. With M~— 1016 0eV we have thus tochooseu 10-12, indeeda very small numberwhich is introducedby handartificially. Becauseof thissmall o onemight also arguethat the massesof the whole adjoint representationsare small (althoughthey havelargev.e.v.’s)andhaveto be included in the computationof the runningcoupling constants.Such effectswill be discussedin a moment.The introductionof a- — 1012 seemsvery artificial. One hastherefore enlargedthe model and has introduced additional singlet fields and more complicatedsuperpotentialsto suppressthe barrier [429], but in these casesthis needs some adjustmentstoincredible accuracyin the couplingconstantsof thesesinglets.Finally one could try to usenonrenor-malizabletermsas discussedin the last chapterto suppressthe barrier [428].This, however,in generalonly works with superpotentialsin which all terms that contain particles with large v.e.v. arenonrenormalizable.The alsopossiblerenormalizableterms(like thosein (8.5)) haveto besuppressedbyextremelysmall couplingconstantlike ci in our previousdiscussionand a solution to this fine-tuningproblemcan only be found if we find a reasonwhy theserenormalizabletermsare small.

But independentlyof the way one achievesthis fine tuning it has consequencesfor the low energytheory sinceit predictsnew particleswith massesin the TeV range.The potential(to be specific let us

136 H.P. Nilles, Supersymmetry,supergravityand particlephysics

take (8.15) in which the first term is multiplied by ci) has relatively flat directionswherethe ~D2 termvanishes.This correspondsto particle massesof order uM~if we considerthe third solution of (8.8)correspondingto the SU(3)X SU(2)x U(1) phase[428].Since ci hadto be smallerthan 10_12 to suppressthe barrier sufficiently thesemasseshaveto be smaller than 10 TeV. Not all of the directionsof thepotentialareflat only thosewhere~D2 vanishesand as a consequencenot all of the componentsof Ahaveto be that light. In the case in which A is the adjoint representationof SU(5) the light particlesincludean SU(3)octet,an SU(2) triplet andasinglet (exactly thosewe havealreadydiscussedin section7.6). The presenceof theselight particles changesthe value of M,. (it increasesM~by two ordersofmagnitude)andalso the predictionfor sin2 0,.,. Additional light fields haveusuallyto be introducedandwith thesecomplicationsthe modelcan be madeto work.

This situation changesif the representationA is larger. One common choice for A is the 75representationof SU(5). With such a choice the above describedcosmological scenario becomesinconsistent[428].To suppressthe barriermanyof the membersof the 75 have to remain light. Theselight particleshaveto be included in the calculationof the evolution of a

5 in the SU(5)phase.Thecoupling cr5 had to becomestrong at energiesbelow M~to allow the SU(3)x SU(1)X U(1) phaseto beenergeticallyfavored. With A ~ 75 this no longerworks: a5 decreasesbelowM~and the only possiblechoicefor A remainsthe adjoint representation.As we havediscussedin the last chapterone can alsoachievea satisfactorysplitting of the Higgs triplets and doublets by introducing nonrenormalizableterms.For furtheraspectsof the SU(5)phasetransitionseerefs. [375,377, 378, 379, 383, 425, 428,460].

8.3. Thecosmologicalscenarioand models

Most cosmologicalscenariosof the early universetoday involve inflation, i.e., an epochof exponen-tial expansion[298].For inflation in the contextof supersymmetryseerefs. [10,174, 178, 178b, [81,263,267,298, 313, 383, 404, 405, 427, 469, 535]. Sucha mechanismallows us to explain the so-calledflatness,isotropy, rotation and monopole density probelms.To introduce inflation some parametersin thepotentialsunderconsiderationhaveto bechoosenwith extremeaccuracyto havethe inflationary epochlong enoughto solveall theseproblemsin a satisfactoryway. Such a choicehasalsoto be done in thesupersymmetriccase,with the difference that here one might claim such choicesto be technicallynatural [178](i.e. not upsetby radiative corrections).But thereis also anotherdifference.In the usualnonsupersymmetriccase inflation is supposedto solve the grand unified monopoledensity problem.Since thesemonopolesare producedin the grand unified phasetransition the inflationary period todilute the monopoledensityhasto happenat a time wherethe temperatureof the universewassmallerthan M~.This is not necessaryin the supersymmetriccosmological scenariosince therethe SU(5)transition is supposedto happenat T ~ iO~0eV (with only minor reheating)and the production ofmonopolesis suppressedby a Boltzmannfactor. In this caseinflation could occuras well at timeswherethe universewas still hotterthan M,~.Such a primordial inflation [178b]solvesall theotherproblemsaswell including the earlier discussedgravitino densityproblem(providedthat inflation takesplacebelowM’~).Primordial inflation hassomeadditional nice properties,the fine tuning of the parametersin thepotential becomesless acuteand it might also be easierto avoid [178a]an overproductionof densityperturbations[310,579, 48, 299, 312, 520] ~pIp in connection with galaxy formation. As a result thisprimordial inflation is usually usedin the context of supersymmetirccosmologicalscenarios.

Such a scenariothen startswith a singlet field 4 (the inflaton) [178b]whoseonly role in life is toproduceinflation. The parametersin the potentialarecarefully adjustedto havean inflationary periodand to have it last long enoughto solve the mentionedproblems.In the contextof supersymmetryone

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 137

hasin thiscaseto rely on nonrenormalizableinteractionsin the superpotential.The massscalern of thescalar potential of the inflation field should be of order 102Mg to have sufficient inflation andacceptabledensityperturbations.This then leadsto a vacuumenergydensityin the falsevacuum

V(4 = 0) = A(m6/M2) 10~4M4 (8.16)

for a special choiceof A. During the rollover of the 4 to its global minimum at (/) — M the universeinflates. The vacuumenergydensity at the completionof the phasetransition is convertedinto theproductionof inflatonswhichdominatethe universe.The massof the inflaton field in this model is givenapproximatelyby

rn~= rn3/M2= 10130eV. (8.17)

It is actually crucial for the model that rn~hasthis value[426].A smallervaluewould causeproblemswith the generationof a baryon—antibaryonsymmetrywhereasa largervalue would causea problemwith too large a gravitino production after reheating.The inflatons eventually decay with rateproportionalto rn ~/M2 preferablywith çb —~ 3Y where Y is a newly introducedsinglet. This happensthrough gravitationalinteractionsandtakesplace at temperaturesof the orderof 1010 0eV. The singletfield Y was introducedto serveas a decayproduct for 4. It will also be coupledto the Higgs tripletsand decayto thesefields eventually. Y is supposedto havea massof 1010 0eV andthe parametersinthe modelarearrangedin such a way that the Y—t. H

3H3 decayhappensat 10~0eV [460].The universeis meanwhilein the SU(3)X SU(2)X U(1) phaseandthe out of equilibrium decayof the Higgs tripletswith mass of order 10’°0eV producesthe baryon asymmetryand this completesthe cosmologicalscenario[460].The SU(5) phasetransitionoccurredas describedin the last section and one had tosuppressthe potentialbarrierto makeit possiblewhich in turn implies the existenceof additionallightparticles.As we haveseenthisscenarioneedscareful adjustmentsof theparametersin variousstagestomakeit work. In addition alsothe introductionof the singlets4 and Y seemsad hoconly justified bythe reasonsthey were invented for. Potentially additional problemscould occur before primordialinflation [433]but theydo not seemto be very serious.Recentlyalsoa potentialentropy“crisis” [94]ofsupersymmetrictheorieshasbeenfound. This “crisis” is relatedto the existenceof a simple hiddensectorwith just onefield (like the Polonyi example).The crisis can be avoidedby minor complicationsof the hiddensector [136].We stophereour descriptionof possiblecosmologicalscenariosand theirproblems.This field is still in progressand it is by nowprematureto drawdefinite conclusions.We havenot discussedcosmologyin modelsbasedon the inverted hierarchy.Such questionsare addressedinrefs. [9,29,320, 367, 476].

At the endlet usinvestigatethe influenceof cosmologyon the low energysector in the frameworkofthe supergravitymodels. We have seen in section 8.1 that cosmologicalconstraintsfrom the massdensityof the universefavor the photino to be the lightest R-odd particlealthoughotherpossibilitiesare not ruled out. The main constraintsfrom the grand unified phasetransition in the scenarioasdescribedin section8.2 imply the existenceof newparticlesin the TeV range.They might howeverstillbeheavyenoughandthereforehaveno immediateconsequencesfor the physicsat 1000eV. This set ofparticlescontain usually a light singlet. Since it is essentiallydecoupledfrom the otherparticlesin thelight sectorthis posesno additionalnaturalnessproblemsas the onedescribedearlier.

Other influencesof the cosmologicalscenarioson the low energysectorof modelsarelessdirect anddependon the parameterschosento obtainacosmologicalacceptablescenario[375].They can only be

l38 H.P. Nil/es,Supersymmetry,supergravityandparticle physics

Table8.1Light particle spectrum(in units of GeV) for cosmologicallyacceptablemodels (CAM) and

minimal models(MIM) for variousvaluesof the top quark massas given in ref. 375).

CAM MIM CAM MIM CAM MIM

A 3 3 2.8 2.8 2.0 1.6m

312 15 15 15 15 15 15m0 42 33 48 47 53 29

15 17 16 18 11 7

mtop 25 25 35 35 50 50

All families29 27 32 36 35 25

21 20 22 23 23 19

First two families58 77 66 108 72 67

1 61 104 6760 103 65

Thirdfamily58 76 64 106 68 6654 74 60 104 66 65

81 96 95 132 112 106

26 54 23 78 21 37

ChargedHiggses 96 93 95 94 88 83106 104 105 105 100 95

NeutralHiggses 3 3 4.4 5.3 6 551 46 49 48 35 19

Gluinos 42 84 47 118 52 72Photino 11 4.6 9.4 7.3 4 3

89 87 87 94 84 9078 82 79 79 82 75

winos, zinos 99 108 99 116 101 106Hlggslnos 92 85 91 80 88 83

26 23 24 24 17 9

revealed after the solution of the renormalizationgroup equations.Such a calculation has beenperformedin modelswhere the earlier discussedparameterp. is of order of m312 andwhere m312 isrelatively small. In table8.1 we show the comparisonof a minimal model (MIM) with a cosmologicallyacceptablemodel (CAM) for comparableinput parametersin this classp. — m312 [375].There is nostriking differencebetweenthe low energyspectraof thesemodels.Apart from that it might be possiblethat thelow energyspectrumin a given minimal modelwith certaininputparametersA, m312, m0 andp.can be reproducedin a cosmologicallyacceptablemodelwith somewhatdifferent input parameters.I donot know whetherthis happensin generalbut it seemsvery likely. A similar behavioris of courseexpectedin models with small p. and/or small rn0. In the modelsshown in table 8.1 it seemsthat arather largemO/m3/2is needed.It might very well be that such a large m0 is neededin all cosmologicalacceptablemodelsbut this is not necessarilythe case.In the modelspresentedin the tableone needsalarge m0 becauseone hasuseda smallvalue rn312— 15 0eV and hasto arriveat a massm � 20 0eV for

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 139

the partnersof the right-handedleptons.It seemsthereforethat the constraintsfrom cosmologyon thespectrumof the low energytheory are not very restrictive—exceptfor the massof the lightest R-oddparticle— although a completeanalysisis only availablein modelswith p. — m312. The lower boundonthe massof the lightest R-oddparticle can usually be satisfiedeasily if this particlecoincideswith thephotino.For a moredetaileddiscussionseerefs. [375]and[167].

9. Experimentalsignatures

We want to devotethis chapterto a discussionof the potentialexperimentalsignaturesthat couldeither verify or falsify the discussedmodels. One relevantprocessis proton decay which we havediscussedearlier in detail. The predictionof thesupersymmetricmodelsdependstrongly on the chosengrand unified sector.There is howeverone modep—* K~r’ which seemsto be only predictedin thecontextof supersymmetryalthoughthereit needsnot necessarilyto be present.A detectionof such amodeas the dominantmode of proton decaywould thereforestrongly supportsupersymmetricideas.On the otherhandsuch a modeis not easily accessibleexperimentally.All other proton decaymodescan probably be accommodatedboth in conventionaland supersymmetricmodels, and thereremainsthe questionin which modelsthiscan be donemoreeasily. A dominantdecayp—* ir°e~would certainlyfavor conventionalmodels.Absenceof proton decayto a level of 10~~yearson the otherhandwouldfavor supersymmetricmodelssince we in generalexpect there a larger value of M~,But whateverhappensit will not be conclusive(with exceptionof p—*Kv).

The other potentialexperimentalpredictionsof the supersymmetricmodelsare concernedwith thelow energyspectrumof the models,i.e. the region of orderof 100 0eV. But beforewe discusssthis wehaveto warn the reader.Thereareno real specific predictionsof theseclassesof models.Somemodelspredict the existenceof certainparticles in a certainmassrangebut other equally acceptablemodels(from a theoreticalpoint of view) do predictcompletelydifferent things.As alwaysit would be easiertoverify than to falsify the whole approach.Sucha falsification would requirecompletemeasurementsupto an energyof let us say 10000eV proving the nonexistenceof supersymmetric(R-odd)particlesandpossiblya detectionof conventionalHiggses.If this is done oneprobablywould think that the idea of apossibleconnectionof supersymmetryand the breakdownscale of SU(2)x U(1) is wrong, althoughsomepeople might argue the boundaryof 1 TeV could be increasedsomewhat.One should comparethe situationin the supersymmetriccontextto the predictionof theconventionalstandardmodel Higgssector. It predictsa neutralHiggs particlesomewherein the rangebetween7 0eV andlet ussay I TeV.We can certainly constructsupersymmetricmodelsthat look like this model and where all the newstructureis pushedup to higher energies.In such modelsit is howeverhardto link supersymmetrytothe breakdownof the weak interactionsandthesemodelslook very contrived.

On the other hand supersymmetrymight be relevant at much lower energies(as happensin thetheoreticallymost promising models) and this hasto be checked experimentally.We will concentratemainly on the predictionsof the supergravitymodels which usuallycontain less low lying statesandthereforehave more predictive power. This discussionthen usually also covers the most relevantsignaturesof globally supersymmetricmodelswith the differencethat the lowestlying statein the globalcase is the goldstino. The supergravitymodels at low energieslook like a special type of explicitlybrokensupersymmetricmodelsandas such coverall relevantmodels.The low energyspectrumof thesemodelsis characterizedby four parametersaswe haveseenin Chapter7. Theseare the gravitino massrn312, the trilinear coupling A, the gauginomassesm0 and the parameterp.. The gravitino massandm0

141) H.P. Ni//es, Supersymmetry,supergravityand particlephysics

(at M~)arein generalconsideredto be of the sameorderof magnitudealthoughonemight alsoimaginern0 ~ m312— 20—400 0eV. A is a number with A~<3. For theoretical reasonsone might think thatp. ~ rn3,2 but such a restriction might just comefrom our ignoranceaboutthe high energysectorandshould not be posed in this context where we want to discusspossibleexperimentalsignaturesof allmodels.Specific modelsnow correspondto specific choicesof theseparametersand as such a wholevariety of experimentalconsequencesmight occur. In the casesin which m312— m0 (i.e.. rn0 not largecomparedto m312) one expectsin generalthe photino to be the lightest newly introduced supersym-metric particle (R-odd particle) although otherpossibilitiesalso might occur as we havepointed outearlier.Usually such a particleis consideredto be stableunlessone breaksR-parity andallows explicitlepton numbernonconservation.If it is stablethereis a cosmologicalboundof rn ~ 2 0eV. One canavoid this boundonly through a breakdownof R-parity, which is usually hard to achieve,or throughthe constructionof modelsin which thelowestR-oddparticleis essentiallymassless(i.e. massesat mostof the orderof eV). In the globally supersymmetricmodelsthereis a candidatefor such a particle: thegoldstino.In modelsbasedon supergravitywhere rn312 is of the orderof 1000eV thereis usually notsuch a candidate.The photino will in generalreceive a radiative mass much larger than eV even if

= 0. To haveotherR-odd particles, like for examplethe scalarpartnersof neutrinos,masslesssomevery exact adjustmentsarenecessarysincethey in generalreceivemasscontributionsof order m312 atthetreelevel. The only possibleexceptionmight bethe partnerof the invisible axion. In such a casethephotino could decayandthe massboundon ~ could be lowered[361].

Let us now discussthe possibleexperimentalsignatures.These investigationshaverecently drawnvery much attention.They includethe searchfor gluinos [73,74, 153, 183, 283, 304, 318, 348, 349, 355,441], photinos[211,164, 261, 355, 489] andfor the scalarpartnersof quarksandleptons[259,282, 284,285, 290, 311, 315, 316, 388, 488, 502] including scalarneutrinos[50,253, 328,350]. Also gluino boundstateshave been considered[80,194,420]. The possible production mechanismsfor such particleshave been considered to be eke, ee reactions as well as deep inelastic scattering[12,15, 120, 344,356,380] andpp collisions,wherespecialemphasishasbeengiven to W andZ decays[556,36, 49, 51, 78, 95, 119, 162, 168, 215, 292, 411, 556]. Recently also some review articleson thesubjectareavailable[506,158, 347]. We cannotdiscusshereall thesepossibilitiesin detail andwill justdescribethe major directionsof research.Let us startwith

9.1. Thescalarpartnersof quarksand leptons

Theseare, with the exceptionof the scalar neutrinos,charged spin 0 particles. Boundson theirmassesare alreadyavailablefrom e~edata and they imply thesemassesto be largerthan somethinglike 20 GeV, just given by the upper limit on the available centerof mass energyof presente~emachines[4,32,52,53,58,529]. To avoid problemswith flavor changingneutralcurrentsparticlesof thistype with samechargeandsamehelicity haveto be almost degeneratein mass(aswe havediscussedearlier in detail) which alsois a presentlyavailableexperimentallimit. In the standardmodelsin whichthe parametersarefixed at a largescaleandthenevolvethroughrenormalizationgroupequationflavorchangingneutralcurrentscan also give an absolutemasslimit on the partnersof the down quarks(d, s, b) of m � 35 0eV. This limit arosefrom gluino contributionsto the real part of the KL—KS massmatrixas discussedin Chapter7, andthusgives no restrictionson the partnersof the up quarks(u, c, t).Such restrictionscould come from D°—D°mixings but therethe experimentalresults are less precise.There are also no suchrestrictionson the absolutemassvalue of partnersof leptonssince there theusuallydominantgluino contribution is necessarilyabsent.

H.P. Nilles, Supersymmetry,supergravityandparticlephysics 141

Themassesof theseparticlesaredeterminedprimarily by m312 and m0 andtheir splittingsby A. Letusfirst discussthe casem0 = 0. At the largescaleM~one startswith degeneratemassesgiven by m312.The two real scalarscorrespondingto the original left-handedfermionswill be split if A doesnot vanish(compare(7.76))

m~,2=m~,2+m~±Am312m0, (9.1)

wherem0 is the massof the correspondingquark.With m0= 0 thiswill essentiallybe the situationatlow energies.This might bedifferent for the partnersof the top quarkdueto the largeYukawacouplingwherealsothe restrictionsof flavor changingneutralcurrentsare lessstringentdueto our ignoranceofthe top quark K—M angles.With m0= 0 all theseparticlesaredegenerateat the two massvaluesin (9.1)with the exceptionof the top scalars.Indeed in this caseone expectsone of the partnersof the topquark to be the lightest partnerof quarksandleptons.With m0 � 0 this situationchangessincetheyreceivedifferent radiative contributionsto their massesat low energies.With m0 comparableto rn3,2one then expectsthe partnersof quarksto be heaviestand the lightest of theseparticleswill be thepartnersof the right-handedleptonssincetheyonly coupleto hypercharge.In somemodelswith largep.one could as well arrive at small m � 20 0eV massesfor the scalar neutrinos[330] but let us firstconcentrateon the chargedscalars.

The experimentallycleanestway to detecttheseparticlesis a studyof e~eannihilation. Sincetheyalways appearin groupsof at least threeessentiallydegeneratein massa first sign of their existenceshouldshow up in the total crosssectionwhereeveryparticlegives a contributionof ~Q

2to R and Qisthe electric charge.Once produced,thesechargedscalarswill decay rapidly since for cosmologicalreasonstheycannotbestableas we havediscussedin the last chapter.The heavieronescould decayincascadesto the light ones and the light ones will then decay in qg or q’~,where q denotesthecorrespondingquark and ~(j’) the gluino (photino).The former decaymodeof courseoccursonly inthose models where the gluino mass is smaller than the one of the decaying particle. The gluinosubsequentlydecaysinto ~ whereq, ~ denotelight quarks.In globally supersymmetricmodelsonewould in generalhave~—~gGwhereg is a gluon andGthe goldstino.Independentlyof the specialwaythesedecaysproceedwe will finally haveat least one of the lightestR-odd particlein the final state.Such particles are neutral stable particles which interact very weakly and will probably only bedetectableas “missing energy” very similar to the neutrinosin ordinary decaysof heavy leptons.Atypical decay of thesepartnersof quarksand leptonswill thus proceedin this way; the final decayproductswill contain oneor severalof the lightestR-oddparticles(to be specific let usassumethat thisis 5) which leavethe detectorunobserved.The signatureis thecleanestif ~ is heavierthanthe partnersof quarksand leptons. In this case~ or � will decayin q(e)+ ~ and one will just observeq(t) andmissing energyout of which one could in generaleasilyreconstructthe massof the decayingparticle.Independentlyof the fact whether~ is the lightestR-oddparticleor not onewould expectthis decaytooccur, since one can hardly constructa modelalong the standardlines in which ~ is heavierthanthechargedpartnersof quarksandleptons.Theseparticles, if they exist,arethereforeeasilydetectableine~e-annihilationexperiments.This detectionis of coursenot possiblefor the scalar partnersof theneutrinosand alsowe haveto wait sometime for the e4e machinesof the nextgeneration(SLC andLEP) to comeinto operation[506].

In the meantime one would therefore also like to investigatedifferent ways to producetheseparticles.The highestcenterof massenergiestodaycanbe obtainedat thep~collidersin which recentlythe weak gaugebosonsW and Z havebeendetected.If allowed by phasespacethe scalarpartnersof

142 H.P. Nil/es,Supersymmetry,supergravityand particlephysics

quarksandleptonscan beproducedas decayproductsof W andZ°.This alsocan nowgive informationon the scalarpartnersof the neutrinosin the possibleprocesses[36]W—~ë1 or Z-+ i~’i.Productionanddecayratesfor theseprocesseshavebeenstudiedextensivelyin the literatureandsuch processeswereshownto be observableif theyarekinematicallyallowed. Outsidethe W andZ resonanceswe can alsoexpectthesescalarpartnersof quarksandleptons~(e~’)to be producedprovidedtheir massesaresmallenough.This could eitherhappenthroughthe productionq~-~ gluon —* or throughq~-* ~ where~ isa componentin the “sea” of the proton [380].Ratesfor theseprocesseshavebeencomputedand incertainfavorablecasescouldbeevenlargerthanpp—~W—~ei + X [506]. We thus can expect that withan increaseof the luminosity in the pp collider new boundson the massesof ~ and � will becomeavailablealthough the signalsare not as clean as the ones in e~emachinesat comparableenergies.Similar conclusionscan also be drawn for productionof ~ and ~‘ in ep colliders (like projectedinHERA) wherethey can either be produceddirectly in chargedand neutralcurrent reactionsor byassociatedproduction. Rates for these processeshave also been computed. The signaturesarecomparableto thosein pp reactions.

9.2. Photinosand gluinos

In mostmodelswe expectthe photino(‘p) to bethe lightestR-oddparticle. It is thena stableneutralMajoranafermionwith rn~� 2GeV. With otherparticlesit caninteractmainly throughthe exchangeofscalarpartnersof quarksand leptonswhich are believedto be heavierthan 20 0eV. It thushasonlyfeebleinteractionsjust like the neutrinoin conventionalmodelswhere its interactionsaredescribedbythe exchangeof W or Z bosons.We thus cannotexpect its direct observationin presentlyavailabledetectors.The photino will be a final decayproductin any decayof heavier R-oddparticlesandwillleavethe detectorsas missingenergy.It couldalsobe producedin pp —~ + X but a direct detectionisnot possible.In e~eannihilationit could be produced[211,164] via e~e—~ y~ (fig. 9.1) similar to theprocesse+ e -+ yvi. In fact all that hasbeensaid aboutneutrinodetectioncan be carriedover to thephotino. This includesthe influenceon the decaywidth of theZ°the tfr, Y or toponium [73] andalsotoponium to yyy decays.For a very heavy top quark, the decay top—~tap+ ~ might also be possibleand proceedwith a large branchingratio [183].All theseprocessescould give indirect information onthe photino.

This is different for the gluino which also has strong interactions. The ~ and ~ massesgiveinformation on the input parameterm0 in the theory.We haveseenin our earlierdiscussionthat evenifm0 were smallat the tree-levelan effectivem0 will in generalbe generatedthroughthe heavyparticlesin the theory. The gluino—photino mass ratio is in generalgiven by mg/rn~— 3a3/8aemand it wouldthereforebe surprising if rn g wouldbe smallerthan let us say 5 0eV.

le

Fig. 9.1. Photinosearchin e~eannihilation similar to neutrino counting. The cross sectionheredependscrucially on themassesof the scalarpartnersof theelectron.A careful measurementof this processat presentlyexistinge~e~machinescouldgive new informationon m~.

H.P. Nil/es, Supersymmetry,supergravityandparticlephysics 143

Fig. 9.2. Sketchof thegluino searchin beamdump experiments. Fig. 9.3. Gluinosearchin quarkoniumdecay.This graphcorrespondsto aprocessthat is independentof themassof the Q partner.

Thereexist alreadysome(model-dependent)experimentallower boundson the massof ~from beamdump experiments[27,82] as sketched in fig. 9.2. To recoveran information on mg from such ameasurementdependson the parametersof the model under consideration.The crucial parameteristhe massof the partnersof quarks(~).The absenceof any eventas the oneshown in the figure givesrn g ~ 1 0eVfor rn4 = 1000eV or rn � 4 0eV if rn = 40 0eV. Other informationon gluino massescancomefrom quarkonium(QQ) decay. There is the possibility of (QQ)_s.gg~as shown in fig. 9.3, aprocessthat apartfrom the usual couplingconstantssolely dependson the gluino mass[73].In such aprocessit seemshoweverto be hard to isolatethe producedgluinos.They will in generaldecayto lightquarksand aphotino ~—* q~~. To isolate~ in the aboveprocessonethushasto look at four jet eventsin which two of them havemissing energy.According to the decay ~—.q4j the gluino appearsin itssignaturesas aheavyneutralleptonandall the methodsto searchfor suchparticlescan alsobe usedforthe gluino.

With respect to quarkonium decaysone can have additional gluino decay modes like the onesdisplayedin fig. 9.4, QQ—~g~or ggg. They leadto cleanersignatures(especially~) than the decaypreviously discussed,but their ratedependson the massof the scalarpartnerof 0. Theserateshavebeencomputedfor wide rangesof mg andm~ andit hasbeenshownthat the branchingratioscould besubstantialat leastfor the toponium case.Onecan thereforeexpectinformationon mg oncetoponiumhasbeendiscovered[73,74, 153, 183, 283, 304, 318, 348, 349, 355, 441].

Other than that gluino productioncould occur in thee~e~continuumin ep andp~scatteringalongthe lines already discussedearlier.The gluino contribution could play a major role [380,15] in theevolution of the nucleon structurefunctions at large Q

2. This would then also lead to the earlierdiscussedprocessq+g—s.~in p~scattering,where~ is in the “sea” of p. We cannotdiscussall theseprocessesin detail andrefer the readerto the original literature[164,211, 261, 355, 489, 259, 282, 284,285, 290, 311, 315, 316, 388, 488, 502].

a - a

a g — I

a

Fig. 9.4. Gluino productionin quarkoniumdecay.Therateof theseprocessesdependson themassof thescalarpartnersof Q.

144 H.P. Nil/es,Supersymmetry,supergravityandparticlephysics

9.3. Wino,zino and Higgsinosignals

A lot of work hasbeendoneto studyproductionanddecayof suchparticlessincein a largeclassofmodelssomeof theseparticlesmight belighter thanthe W andthe Z [49,51, 78, 95, 119, 162, 168, 215,292, 411, 556].

Let us start with the chargedgauginosand Higgsinos. In the breakdownof SU(2)X U(1) gauginosandHiggsinoswill combineto form Dirac fermionsin the supersymmetricHiggs effect. If we considerthe minimal modelwe haveto considerthe left handedWeyl fermionsH, H~andthe partnersA~orW~with massmatrix (compare(7.75))

A~ I-f

~ e2v2~~~ (9.2)\e2v1 p. / H

in which v1, v2 denotethe vacuumexpectationvaluesof the scalarHiggses(M~= ~e~(v~+v~),e2 theSU(2)gaugecouplingconstant,rn2 the A~A massterm obtainedfrom rn0 andp. the massparameterinfront of HH in the superpotential.With m0 and p. large,theseparticleswill of courseall be heavy, butlet usdiscussherethe caseswhere rn2 and/orp. aresmall comparedto M~.

Let us first considertñ2 = 0. We compute~ andobtainthe eigenvalues

= ~(e~(v~+ v~)+ p.2)±~[e~(v~— v~)2+ 2p.2e~(v~+ v~)+ p.4]112, (9.3)

which impliesthat oneof the chargedfermionsis lighter thanM~.This light particlein the caseth~= 0haspredominantlyA~,A components.Similarly one could considerthe case~ —~0, in

2 � 0 and oneagainobtainsa fermionwith masssmallerthanM~,it now haspredominantlyH~Hcomponents.Thelimit in which both rn2 andp. aresmall (the Wiggsinolimit) [292]implies theexistenceof alight chargedfermionwhose left-handedcomponentsare predominantlyA~and H-. In all of thesethreecaseswehavea chargedfermion which could be found if W decaysvia W—~A~[556].Let usherestressagainthat this must not be the casein general.It is just a possibility in a (although)wide classof models.p.and rn2 could both be larger than M~and in such a case no such light chargedfermion exists.Nonethelessthis is an interestingpossibility.The branchingratio for W—* A~hasbeencomputedanditwas shown that it can well be comparableto the W—~e~mode if A and ‘~ are light enough.Thesignature,however,is not so clean as that of the ci.’ mode becauseof the subsequentA-decaywhichusually is expectedto occur in severalsteps.The final decayproductswill be usualquarksandleptonsaswell as photinos(the lightestR-oddparticle).They give rise to so-calledZen events[168]as shown infig. 9.5, several hadronicjets recoiling against missing momentum.The decay chain of the A starts

Fig. 9.5. Zen signatureof W-decayin a cascadeinvolving R-odd particlesas given in ref. [1681.The one clappinghandrepresentsthe hadronicshower.

H.P. Nil/es,Supersymmetry,supergravityandparticlephysics 145

usuallywith A —* A°+ (e~or q~)whereA°is a neutralgauginowhich itself could decayinto ~+ q~.If thescalarpartnersof quarksand leptonsare lighter than A even more complicateddecay chains couldoccur.

In this decaychainwe expectthe productionof neutralgauginosandHiggsinosthe massesof whichwewill discussnow. Apart from the photinowhich is a pureMajoranafermion that doesnot mix withthe Higgsinosin the supersymmetricHiggs effect, there are threeadditional spin 1/2 neutralWeylfermionsin the minimal model. The massmatrix is given by (compare(7.68))

/ 1112 0 e2v1/V2 e2v2/V2 \ A°1 0 ~(a1/a2)th2 e1v1/V2 —e1v2/\/2 1 B° (9.4)I ~2Vi/V2 e1v1/\/2 0 p. ) i:i~\ e2v2iV~ —eiv2/V~ p. 0

whereA°andB°are theneutralwino of SU(2)andthe binoof U(1)~respectively.The photinois given bythe linear combination

= (e1A°+ e2B°)/(e~+ e~)”2, (9.5)

with mass

m~= ~[e~/(e~+ e~)]in2, (9.6)

andwhichusually is thelightestparticle.In the casethat eitherin2 or p. or both aresmall, oneadditionallight neutralfermion is present.As in our discussionof the chargedfermionsthisadditional light particlewill predominantlybeagauginoif in2 is small,or aHiggsinoif p. is small, ora mixtureif p. andin2 arebothsmall.A closerinspectionof the massmatrix in (9.4) comparedto thatin (9.2)revealsthat thelight neutralfermionhasa smallermassthan the light chargedfermionsuch that a decayA —~x°+ all is possible(,y°denotesthe light neutralfermionin addition to thephotino).Let usherepointout againthat alight x°is apossibility in a certainclassof modelsandis not a generalpredictionof supersymmetry.

Thevarious branchingratiosof the decaychain W-~~A~—~A°—~~ havebeencomputedand foundtobe nonnegligable.Theydependon the parametersof the specialmodelsunder considerationandfordetailswe refer the readerto the original literature[49,51, 78, 95, 119, 162, 168, 215, 292, 411, 556]. Todetect theseparticles one has to be aware of the conventionalbackgroundsin W-decay. The mostdangerousone is W-decayinto heavy leptons(like T) andneutrinoswhich hasthe samesignature.Thedifferencebetweenthesetwo processesis the Michel parameterwhich in the caseof gauginoswill ingeneralnot correspondto pure V — A or V + A interactions.These points have to be clarified todistinguishbetweennewheavyleptonsandgauginos.Somuchfor gauginoproductionin W-decays.Theneutralfermionx°can of coursealsobe producedin Z-decays.Theproductionof theselight fermionsin e~eannihilation havealso been studied[489].We, of course,know from therealready that thechargedgaugino—Higgsinoshaveto beheavierthan20 0eV andtheseparticleswill be easilydetectablewith e~emachinesat higher centerof massenergies.The crosssection for e~e—~x°~is of coursesmaller than that for A~A,more comparableto e~e-+ z.~i.Nonethelessfor small massesof x°and ~sucha reactioncould betestedwith high luminosity measurementsat presentlyavailablee~emachinesandcould give valuableinformationof m~°and m

Let usclosethis chapterwith somegeneralremarks.We havenot discussedeverythingthat could be

¶46 H.P. Ni//es, Supersymmetry,supergravityandparticle physics

said aboutpossibleexperimentalsignaturesin supersymmetricmodels.This hasits reasonprimarily inspacelimitationsand alsoin the fact that manyof the “predictions”arevery modeldependentandcanonly be presentedincluding explanationsof the very specific models under consideration.We havethereforefocusedon thosepredictionsthat appearin wide classesof modelsandwe wereof coursealsoguidedby theoreticalprejudices.The investigationson the potentialexperimentalsignaturesare stillvery muchin progressat the time thispaperis written andwe canexpectmoreinformation in the nearfuture.

The most promisingprocessesto find supersymmetricparticles are in e~eprocesses,W, Z andtoponium decaysand we havemainly concentratedon thosereactions.At the presenttime we expectnew information mainly from measurementsat the pp collider with higher luminosity. The e~e—*x°~or y~modeshouldbe investigatedwith the presentlyavailablee~emachinesPETRA andPEP.Theprojectedpp, e~eand ep machinesat higher energywill certainly increaseour knowledgeabout the(non)existenceof theseparticlestremendously.

Let usclosewith a repetitionof the warninggiven earlier.It might very well bethat supersymmetryis relatedto the breakdownscaleof the weak interactionsand doesnot showtracesof its existenceinthe massregion below 100 0eV. Once,however,the boundson the massesof the R-oddparticlesarepushedup to higher andhigher energiesthis possibilitybecomesmoreandmoreunlikely. In this sensewe expectenoughexperimentalinformation within the next5 to 10 yearsto makeup our mind.

10. Conclusion and outlook

Our aim was to constructphenomenologicallyacceptablemodelsin which the breakdownscaleM~of the weak interactionswas stabilizedby supersymmetry.In addition we requiredthe absenceof smallparametersin the theory that were artificially put in by hand. This then required a link betweenthesupersymmetrybreakdownandthe breakdownof the weak interactions.Up to now thereis no modelwhich is theoreticallycompletelysatisfactoryin thisrespect.We do not really knowwhy supersymmetryis brokenandhide our ignoranceby constructingcomplicatedadditional sectorsin the theory whoseonly role in life is to providea breakdownof supersymmetry.It seemshoweverthat thesequestionscanonly be answeredat energyscalesmuchlargerthanM~— 1000eV whereexperimentalinformation isvery limited. Of more immediateinterestat the momentis the physicsat scalesof orderof M~and letus recapitulatefirst the implications of supersymmetrythere before we go to higher energieswherethingsbecomemuchmore unconstrained.

Supersymmetryrequiresthe introduction of many new degreesof freedom.Every particle of thestandardmodel receivesa supersymmetricpartnerand in addition we haveto introducea secondHiggschiral superfield. This is the particle content of the minimal supersymmetricstandardmodel, ifsupersymmetrywould remainunbroken.To breaksupersymmetryone needsusually to introduceanadditional sector,but let usdiscussthis later.

With this minimal set of superfieldsthereexists thepossibilityof B andL violationsin the low energysector,contraryto the casein the nonsupersymmetricstandardmodelwheresuchviolationshaveto beabsentbecauseof the symmetriesand the requirementof renormalizability.To avoid theseprocessesoneusuallyintroducesan R-symmetryor R-paritywhich explicitly forbids suchtermsanddistinguishesbetweenthe Higgs H and the left-handedlepton doublet superfield. Once such an R-symmetry ispresentit also forbids a massfor, let us say the gluino. This in turn then implies a breakdownof thisR.symmetry.Models with supersymmetrybreakdownscale M~~ 1 TeV have usually problems to

H.P. Nil/es, Supersypnmetry,supergravityandparticlephysics ¶47

incorporateall theserequirements(with the possible exceptionof supercolorwhere the R-symmetrycould be brokenby a supercoloranomaly)andthis requiresthe investigationof modelswith breakingscaleas high as M5 — 1011 0eV. It is nontrivial that evenwith such a largeM~the massscaleM~ofsomeof the particlesof the theorycould remainstable. Onehasof courseto be careful andhide thesupersymmetrybreakingsectorsufficiently: a necessitypartially alreadyrequiredfrom the constraintsofthe supertraceformula at the tree graph level. The particles in the hidden sector, that feel thebreakdownof supersymmetrydirectly couple only very weakly (e.g. through the exchangeof super-heavyparticles)to the low energyobservablesector.The fact thatM~cannotbe largerthan 1011 0eVcomesfrom the existenceof gravity in which all particles(whetherhiddenor not) participate.In theframework of supersymmetrygravity is incorporatedby the introductionof local supersymmetry.InsuchmodelsM~is thencomparableto rn312 the gravitino massandkeptstablein a massrangeof orderof 1000eV. The supersymmetricpartnersof the particles in the standardmodel also havemassesoforder m312 andtheir existenceprovidesa physicalcutoff for the otherwiseexistingquadraticdivergencesof scalarparticles.This existenceof various new degreesof freedomin the 100—10000eV rangeseemsto be a necessityin all attemptsto understandMw; comparethe technicolormodels[191]or modelsinwhich even quarks and leptons are bound states [475]. The question remains now whether thebreakdownof SU(2)x U(1) can be relatedto the breakdownof supersymmetry.In the framework ofsupergravity,this can be achievedin aminimal model. Radiativecorrectionsdrive oneof the rn

2 of theHiggsesto negativevaluesandSU(2)x U(1) is brokenas a resultof the breakdownof supersymmetry.What is nice aboutthis mechanismis the fact that it is oneof the Higgs scalarsthat receivesa negativem2. We do not haveto maketremendousefforts to avoid negativem2 for the scalarpartnersof quarksandleptons,apotentialdangerin many othermodels.Onecan achievethis breakdownin variousways.Most straightforwardit proceedsif the theory includesa large Yukawa coupling responsiblefor themass of the top quark. If one allows for a p.HH term in the superpotentialwith p. — m

312 it can,however,work for arbitraryvaluesof rn10~.

This parameterp. actually posesa slight problemin the low energytheory. It hasto be nonzerotoavoid the existenceof an unacceptableaxion. Our whole motivation to study supersymmetryin thiscontext on the otherhandwas to explainwhy p. is small comparedto M~.To do this in a satisfactoryway one first lookedfor a situationin which p. = 0. Having p. = 0 at the tree level it is then not easytogeneratep. — m312 in higher orders.Thisquestion,however,can only be answeredif onealso considersthe models at high energieswhich allow for many possibilitiesand it remainsopen for the moment[365].

Theminimal low energymodelthendependson four new input parameters:the gravitino massm312,the scalartrilinear couplingA, gauginomassesm0 andtheparameterp. in p.HH. Nonminimalmodelsofcoursehavemore new particlesand input parametersandin generallesspredictivepower.Thesefourparametersshouldbe explainedthrough the theory at higher energies.There,however,thingsbecomearbitrary andunconstrained.We can constructat the moment various hidden sectorsthat serve asexamplesin which certainvaluesof parametersm312, A, m0 andp. areobtained.Usuallythis is theonlything the consideredhidden sectorhasto do and the whole situationremainsarbitrary. Moreoverinmost casescertainparametershaveto be chosenvery carefully to arrive at m312~ M~andwe do notreally havea reasonwhy M~is small comparedto M~.

The inclusion of grand unification is straightforward.The minimal extensiondoesnot explain theabsenceof p.. Models with p. = 0, however,tendto be very complicated[291].With the experimentalabsenceof proton decay one, however,might postponethe question of grand unification for themoment.

148 H.P. Nilles, Supersymmetry,supergravityand particlephysics

The constructionof models beyondthe TeV rangeis thus at the moment a place for relativelyunconstrainedtheoreticalspeculation.The main questionto studyis an explanationof M5/M~— 10_8.Onemight think that such aratio is easierto understandthanthe correspondingratioM~/M~— 10’~inthe standardmodelbut this neednot necessarilybe the case.The differencehereof courseis that M~andconsequentlyM~arestablein perturbationtheory.

Suggestionsin this direction arealreadypresentin the supercolormodels[123,137] whereonecouldunderstandM~in the sameway as one understandsthe massesof hadrons(via AQCD) in the usualmodels.This could work for variousvaluesof M~betweenio~and 1011 0eV but our understandingofthe dynamics of thesemodels is very limited and still controversial. Another suggestion is thatnonperturbativeeffects (like instantons)might not respect the nonrenormalizationtheoremsandthereforeinduce a supersymmetrybreakdownat a small scale[569] (i.e. small comparedto an inputscale like M~or Mi). Attempts havebeenmadein this direction [6, 7] but in four dimensionsuch abreakdownof supersymmetryhasnot yet beenestablished[104a].

Other models havebeen constructedin which the fundamentalinput scaleis not M~but ratherM~— 1011 0eV. This scaleplaysavery specialrole. Apart from thefact that supersymmetryought to bebrokenthereit is also the scaleof the SU(5)phasetransitionand the generationof the cosmologicalbaryonasymmetry.Moreover in modelswherethe strong CPproblemis solvedthrough the presenceofa Peccei—Quinnsymmetryone expectsthe breakdownscaleof this symmetry somewherebetween10~and10120eV. Invertedhierarchymodelsthenstartwith sucha scaleandgeneratelargerscales.For themodelto becompletethe generationof M~is necessary.This could in principle work in theframeworkof somekind of inducedgravity and remainsan unsolvedproblem.

A satisfactorymodelshouldprobablyexplainM~,M~,M5 andM~in termsof eachother.The abovedescribedmodelsrelate M5 andM~even if M~/M~— 108, but the relation betweenM~,M~andM5still remainsa mystery.Possibledescriptionshavebeengiven in variousframeworks[444,428,264] buttheycan at mostserveas toy models.

To shed some light on thesequestionsone could also go beyond N = 1 supersymmetry.N = 1supergravityis certainly not the final theory for gravitation; it is nonrenormalizableand thereis thehope that extendedN> 1 supergravitiesshow an improved ultraviolet behavior. An application ofextendedsupersymmetriesto particle physics hashoweverto face one centralproblem. The particlespectrumof thesetheoriesis always real with respectto the gaugesymmetriesunder consideration,contraryto the spectrumof the particlesin thestandardSU(3)x SU(2)x U(1) or SU(5)models.This is aserious problemsince there is no way to removesome particles (give them a large mass)withoutbreakingthe gauge symmetriesandarrive at chiral spectraat low energies,it seemsto be a majorstumbling block for generalizations of Kaluza—Klein ideas to extended supergravities[150,188,346, 572]. Onecouldtry to avoid this problem by assumingthat the low energyspectrumdoesnot contain the fundamentalparticlesbut dynamicalboundstates[163,110]. Evenin this case,however,the problemsseemsto exist. Our knowledgeof the structureof thesemodels,however,is still limited atthe momentandinvestigationsof thesemodelswill go on. What would be of particularinterestis thestudy of the breakdownchain N—~ N’ —* 1 —*0 in supergravityand to find examplesin which the foundbreakdownfrom N = 1 to N = 0 happensso late at M5 — 1011 0eV ~ M~. Maybe progressin thisdirection would improve our situationconcerningan understandingof M~[109].The correspondingquestionin the caseof global N-extendedsupersymmetryis alreadyanswered:the breakdownof onesupersymmetryimplies the breakdownof all N of them. Interestingresultscould also be expectedinextendedsupergravitywith stronggaugeinteractionsthat could providea supercolormechanism.In allthesecasesthereremainsthe problemof the reality of the representationsandso at the momentone

H.P. Nil/es,Supersymmeiry,supergravityandparticlephysics 149

could usethesemore generalmodelsdirectly only for constructionof the hidden sectorsand try tounderstandthe problemof understandingthe valueof M5. Apart from thatonecould try to investigatethe valueof the cosmologicalconstantin thesemodelsandperhapsfind a connectionto the breakdownof supergravity.At the momentall thesequestionshaveto remainopenandwe can only havethe hopethat the situationimproveswith amorecompleteunderstandingof thesemodels.

Fromtheexperimentalsideit is theregion of 1000eV that is of immediateinterest.Experimentscaneither falsify or confirm in the nearfuture our theoreticalprejudiceswhethersupersymmetryis relevantin this energyrange.If no tracesof supersymmetryarefound thereonewould ratherbelievethat rn3,2 isas big as M~and this questionof M5 will becomeirrelevant for a long time. Supersymmetrythenprobably will be relevant only at this high scale(if at all). If, however,someof the supersymmetricparticlesarefound in the 100GeV rangewe would haveof coursea renewedmotivation to investigatethesequestionson the origin of M5.

The supersymmetricextension of the standardSU(3)x SU(2)x U(1) model has to be checkedexperimentallyand I am sure that thiswill bedonein the nearfuture. This extension,of course,is morecomplicatedthan the original standardmodel,but this seemsto be a necessityin everymodel in whichthe breakdownof SU(2)x U(1) is explainedby a morefundamentalconcept.

11. Appendix:notationsandconventionsWe follow closely the notation of ref. [296].The transformationparameters0,. (a = 1, 2) are

two-componentanticommutingobjects.Indicescan be raisedby applyingthe two-dimensionale tensor

= ~ (A.1)

where ~ = e’~is antisymmetricin a and/3 (a, /3 = 1, 2) and s12 = +1. The complexadjoint of 0 isdenotedby

= (r)~. (A.2)

As beforewe have0,. = ~ with e,~= r°’~antisymmetricand e12= +1. The squaresof 0 and 0 aredefinedthrough

02O~0a’Ea$0’~O~, 020a~~~ë$. (A.3)

Observethat

0~0aOa0”. (A.4)

For the ~ symbolwe have

= ~, (A.5)

where~ denotesthe Kroneckersymbol. The Pauli matricescarry a space—timeindex p. as well as adottedandundottedindex

150 H.P. Nilles, Supersymmetry,supergravityand particlephysics

cr~’I~= cr~= (1, ~ (A.6)

where

/0 1\ /0 —i\ /1 0\

U1 ~ o)’ ~ o)’ ~3— ~o —i)’ (A.7)

andwe choose

g~ (1 i (A.8)

It is sometimesworthwhile to know the following relations:

= 2g~, r~o~= 288~, (A.9)

as well as

(0a~0P)(07U~,y~0ô)= ~g~~0202, (A.10)

O~$payO~’ = ~ + g~’pg~Lcr— g~Tg).LP + iE ~r~] (A.11)

with e~ totally antisymmetricand ~ = +1.Differentiation with respectto 0 is definedthrough

(A.12)

With theserules onehas, e.g.,

9,,.0220a, apo~(0a0a)=_4. (A.13)

Integrationis definedthrough

JdO=0, JOdO=1. (A.14)

The matricesa aredefinedthrough

= (11, ~L.aí~ (A.15)

A left- (right-) handedtwo-componentWeyl-spinortransformsas 0(0). A four-componentDiracspinorcan be constructedwith two Weyl spinors

H.P. Nil/es, Supersymmetry,supergravityand particlephysics 151

(A.16)

wherea = 1, 2, 3,4, a = 1, 2 and/~= 3,4. For the Dirac matriceswe have

~ ~) (A.17)

i.e.,

/0010\ /0 001\ /00 Oio(0 0 0 11 1( 0 0 1 01 2.40 0 —i 0

y ~1000J )‘ ~0—100) ~‘ ~0—i 00\o 1 0 0/ \—i 0 0 0/ \i 0 0 0

/001 0\y3— ( 0 0 0 1 (A.18)

~—1 0 0 0\o 10 0

and

~ (~~ j~ (A.19)

The symbol ~/i for aDirac spinordenotes

(A.20)

which in the notation of (A.16) reads

= (~,~. (A.21)

The massterm for a Dirac spinor is written as m~frt/iwhich in terms of the two-componentspinorsisgiven by

m(~~a+ ~ (A.22)

A Majoranaspinorcan be written in termsof a Weyl spinoras

= (~). (A.23)

152 H.P. Nilles, Supersymmetry,supergravityandparticle physics

Acknowledgements

I am indebtedto SergioFerrara,Stuart Raby, Lenny Susskindand Daniel Wyler for numerousdiscussions,suggestionsand help. I also enjoyed fruitful collaborationswith J.F. Donoghue,W.Fischler,J.M. Gerard,L. Girardello.J.E. Kim, J. PolchinskiandM. Srednicki.VariousdiscussionswithR. Arnowitt, J. Bagger, R. Barbieri, T. Banks, R.M. Barnett, M. Claudson,E. Cremmer, J.-P.Derendinger,M. Dine, S. Dimopoulos,M. Duff, M.B. Einhorn,J. Ellis, U. Ellwanger,G.R. Farrar,P.Fayet, J.M. Frère,R. Gatto,H.E. Haber,L.J. Hall, I. Hinchliffe, L. Ibanez,J. Iliopoulos,D.R.T. Jones,G.L. Kane, C. Kounnas,A.B. Lahanas,J. Lykken, A. Masiero, D.V. Nanopoulos,H. Nicolai, H.Nishino, K. Olive, B.A. Ovrut, M. Peskin,0. Piguet, H.R. Quinn, N. Sakai, A. Salam, C.A. Savoy,M.G. Schmidt, K. Sibold, M. Sohnius, K. Tamvakis, P. van Nieuwenhuizen,A. van Proeyen,M.Veltman,0. Veneziano,S. Weinberg,J. Wess,T. Yanagida,S. Yankielowicz andB. Zuminowereveryhelpful. I would alsolike to thank FrancineNicole for her efficient typing of the manuscript.

This report hasbeenpreparedin partial fulfilment of the requirementsto obtaina “Habilitation” atthe University of Bonnandis submittedfor publicationin PhysicsReports.

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