giudice.pdf

Jan 27, 2023

by GF GIUDICE · Cited by 17 — G. F. GIUDICE. Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: gian.giudice@cern.ch. I review recent developments in extensions

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THEORETICALPREDICTIONSFORCOLLIDERSEARCHESG.F.GIUDICETheoreticalPhysicsDivision,CERN,CH-1211Geneva23,SwitzerlandE-mail:gian.giudice@cern.chIreviewrecentdevelopmentsinextensionsoftheStandardModelthataddressthequestionofelectroweaksymmetrybreakinganddiscusshowthesetheoriescanbetestedatfuturecolliders.Intheirsearchforunderstandingthefundamen-tallawsofelementaryparticles,highenergyphysi-cistsareeagertoprovethatoneofthemostsuc-cessfulandelegantscienti¯ctheorieseverformulated{theStandardModel(SM){actuallyfailsatdis-tancessmallerthanaboutahundredzeptometers(10¡19m).Agenerationofhighenergycolliders,culminatingwiththeLHCunderconstructionatCERN,hasbeendesignedtoachievethisgoal.HereIwillbrie°yreviewthepresentstatusofthetheoret-icalspeculationsfornewphysicswithinreachoftheLHC.Inthespiritofthisconference,andbecauseoflimitedspace,Iwillconsideronlytopicsinwhich,inmyopinion,therehavebeenrecentdevelopments,leavingasideotherpossibilitieswhich,althoughin-teresting,haveknownonlylimitedprogress.Thelistofreferencesisalsolargelyincomplete.1.\Big”and\Little”HierarchyProblemsThehierarchyproblem1isnotnecessarilythemostpuzzlingopenquestionoftheSM,butitiscertainlythemostrelevanttocolliderexperiments,sinceitsresolutionlies{mostlikely{intheTeVenergyrange.Itsformulationiswellknown.TreatingtheSMasane®ectivetheoryvaliduptoascale¤SM,andcuttingo®momentainloopintegralsatthesamescale,we¯ndthatthedominantradiativecorrectiontotheHiggsmassisgivenby±m2H=3GF4p2¼2¡2m2W+m2Z+m2H¡4m2t¢¤2SM=¡µ200GeV¤SM0:7TeV¶2:(1)Therequestofno¯ne-tuningbetweenthetree-levelandone-loopcontributionstomHimplies¤SM<»TeV.Inotherwords,theSMcannotbevalidbeyondtheTeV,andnewphysicsshouldappeartomodifytheultravioletbehavior.Iwillrefertothisresultasthe\big"hierarchy,sinceotherfundamen-Table1.90%CLlimitsonthescale¤LH(inTeV)ofdimension-sixoperatorsOinthee®ectiveLagrangianL=§¤¡2LHO,inbothcasesofconstructiveanddestructiveinter-ferencewiththeSMcontribution(§).Thelimitsontheop-eratorsrelevanttoLEP1arederivedundertheassumptionofalightHiggs.¡+LEP13Hy¿aHWa¹ºB¹º109.7jHyD¹Hj25.64.6iHyD¹H¹L°¹L9.27.3LEP24¹e°¹e¹`°¹`6.14.5¹e°¹°5e¹b°¹°5b4.33.2MFV212(¹qL¸u¸yu°¹q)26.45.0Hy¹dR¸d¸u¸yu¾¹ºqLF¹º9.312.4talscales,suchasthePlanckmassMPl,areknowntobemuchlargerthan¤SM.Asawordofcaution,Ishouldrecallthat,ifIapplythesamenaturalnessargumenttothequarticdivergences,thenthepresentupperboundonthecosmologicalconstantimpliesthatthecorrespond-ingcut-o®hastobesmallerthan10¡3eV.Althoughwecannotfullyruleoutthepossibilitythattheultra-violetbehaviorofgravityisprematurelymodi¯edatabout10¡3eV,thisembarrassingresultundoubtedlycastsagrievousshadowoverthehierarchyproblem.LetmecomebacktotheSMasane®ectivetheory.Unknownnewphysicsatthecut-o®isparametrizedintermsofnon-renormalizableopera-tors.Tobemostconservative,Iwilladdonlyopera-torsthatpreservealllocalandglobalSMsymmetriesandsatisfythecriterionofminimal°avorviolation.2AsshowninTable1,typicallimitson¤LH,de¯nedasthee®ectivescaleofthenewdimension-sixoper-ators(L=§¤¡2LHO),are¤LH>5{10TeV.Iwillrefertothe\little”hierarchy(LH)prob-1

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2lemasthetensionbetweentheconstraintson¤LH{whichimplythatnew-physicsvirtuale®ectscouldonlyemergeatenergieslargerthan5{10TeV{andtheno-¯ne-tuningcondition{whichrequiresthepresenceofnewdynamicsatthescale¤SM,belowtheTeV.Tosomereadersthiscouldseemlikeamarginalproblem,butIbelieveitprovidesaveryusefulguidelineinthesearchforthecorrecttheorybeyondtheSM.ItisatypicalLEPheritage,whereanynew-physicstheoryhastobeconfrontedwithveryprecisedata.The\little”hierarchybetween¤SMand¤LHisalreadygivingussomelessonsonthehypotheticaltheorybeyondtheSM.Firstofall,newphysicsat¤SMismostlikelyweakly-interacting,orelsee®ec-tiveoperatorswouldbegeneratedwith¤LH’¤SM.Strongdynamics,ifany,canonlyappearatscaleslargerthan¤LH.Moreover,evenforweakdynamics,physicsat¤SMshouldnotinduce(sizable)tree-levelcontributionstoe®ectiveoperators.Recentlytherehasbeengrowinginterestinconstructingnewtheo-riesthatdescribethephysicsbetween¤SMand¤LH,asIwillreportinSecs.4,5,and6.2.SupersymmetrySupersymmetry5providesanelegantsolutiontothe\big”hierarchyproblem,byeliminatingquadraticdivergenceswithasymmetryprinciple.Insoftly-brokensupersymmetry,thesuperpartnermasses(generatedbygauge-invariantterms)e®ectivelyplayther^oleof¤SM.Becauseoftheabsenceofquadraticdivergences,thevalidityofthetheorycanbeex-tendedtoenergyscalesaslargeasMPl,withouten-counteringnaturalnessproblems.Thisextensionhasseveralwelcomeconsequences.Ito®ersthepossibil-itytolinktheSMtospeculativeideasaboutquan-tumgravityatMPl.ItprovidesaviablesettingforGUT’s(withsuccessfuluni¯cationofgaugecouplingconstants),aframeworkforneutrinomasses,andforsuppressingprotondecay.Finally,itallowsanim-plementationofcosmologicalmechanisms,suchasin°ationorbaryogenesis,whichbene¯tfromextrap-olationsofparticlephysicsmodelstohighenergies.Incidentally,theneedforanextrapolationoftheSMuptosuper-heavyscaleshasbeenchallengedbyrecentresearchaimingatconstructingnewmodelswhosevaliditycut-o®isnotveryfarfromtheelec-troweakscale.Inthesescenarios,at¯rstsight,theadvantagesoftheenergy\desert”arelost.Never-theless,theoreticianshavesuggestednewwaysofre-coveringsomeofthepositiveaspects.Justasanex-ample,letmeconsidergaugecouplinguni¯cation.Intheorieswithextradimensions,Kaluza{Kleinexcita-tionsofSMparticlescanacceleratetherunning,pos-siblyleadingtoaprecociousgaugeuni¯cation,6al-thoughpredictivityislostinthepowerrunning.Al-ternatively,onecanconsideranN-foldreplicationoftheSMgaugegroup,wheretheuni¯cationscalebe-comes1013=NTeV.7Yet,onecanabandontheusualtree-levelexpressionfortheweakmixingangleinGUT’s:sin2µW=TrI23=TrQ2=3=8(wherethetraceisoveranyGUTirreduciblerepresentation).TakinganelectroweakgroupSU3£SU2£U1bro-kentoSU2£U1,inthelimit~g2;~g1À~g3(where~giarethecouplingconstantsofthehighenergygroup),one¯ndssin2µW=1=4.8Thisismuchclosertotheexperimentalvaluesin2µW=0:231than3=8,andthereforelittlerunningisneeded.Asidefromthisconnectiontosuper-highener-gies,supersymmetryhasseveralinterestingandsuc-cessfulfeaturesinthelow-energydomain.(i)Gauge-couplinguni¯cationleadstoapredictionofaSMpa-rameter:®s(MZ)=0:124(fortypicalsupersymmet-ricthresholdsat1TeV)and0.130(forthresholdsat250GeV).GiventheintrinsicuncertaintyofGUTthresholds,thisisconsistentwiththePDGvalue®s(MZ)=0:117§0:002.9(ii)Electroweaksymme-tryistriggeredradiativelybythetopYukawain-teraction,withthebrokengroupdynamicallycho-sen(colorSU3ispreservedsincesquarksdonotde-velopnegativesquaremasses,whileweakSU2isbro-ken).(iii)TheHiggsbosonispredictedtobelighterthanabout130GeV,compatiblywiththeindica-tionsfromelectroweakprecisiondata.(iv)The\lit-tle”hierarchyissatis¯ed,ifoneassumesR-parityconservation.Inthiscase,supersymmetricvirtuale®ectsareloop-suppressedand¤LH»4¼¤SM.(v)Thereisanaturalcandidateforcolddarkmatter,ifR-parityisconserved.Thesepositiveaspectsoflow-energysupersym-metryarewellknown,butletmespendafewwordsonthenegativeaspects.The¯rstobviousdrawbackoflow-energysupersymmetryisthatnosuperpartnerhasbeenobserved.Thisisincontradictionwiththerequirementoflessthan10%¯ne-tuning,whichpre-dictedadiscoveryatLEP2.11Inconventionalmod-elsoflow-energysupersymmetry,themoststrin-

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3101001000330300lightestcharginomassinGeV020406080100120lightesthiggsmassinGeVexcludedbyLEP1excludedbyLEP2Figure1.ScatterplotofthecharginoandHiggsmassob-tainedbysamplingtheparameterspaceofminimalsuper-gravity.TheregionsexcludedbyLEP1andLEP2searchesareshown.This¯gurewasproducedbyA.Strumia,updatingearlierresults.10gentdirectboundscomefromchargino(mÂ+>103:5GeV),12chargedslepton(m~eR>99:4GeV),13andHiggssearches.TheSMHiggsmassboundof114.4GeV14appliesonlyinthelimitofaheavypseu-doscalar,whilethegeneralboundis91.0GeV15(the¯nalcombinedLEPanalysisonthesupersymmetricHiggshasnotappearedyet).Theselimitsimplya¯ne-tuningoftheorderofthepercent.Thisisil-lustratedinFig.1,whichshowsthevaluesofHiggsandcharginomassesobtainedbyarandomsampleofsupersymmetricmodelswithuniversalboundaryconditionsattheGUTscaleandcorrectelectroweakbreaking.10Lessthan1%ofthepointshavesimul-taneouslymÂ+>103GeVandmH>114GeV(forpseudoscalarHiggsmassesclosetomH,theboundbecomesweakerbuttheallowedregionofFig.1re-mainsessentiallythesame).Thisisaverydisappointingresultofsupersym-metry,leadingtothreepossibleexplanations.(i)Supersymmetryisnotrealizedattheweakscale.(ii)Low-energysupersymmetryiscorrect,butpa-rametersare¯ne-tunedatthelevelofthepercent(andLHCwilldiscoverthesuperpartners).Inthiscase,youmayberelievedtoknowthatthereisoneexampleinnatureofanaccidentalcorrelationofthisorderofmagnitude.TheapparentsizesoftheSunandtheMoonintheskyareroughlyequal.Ifyouappliedthenaturalnesscriteriontoconcludethattheiractualsizesareroughlyequaltoo,youwouldbequitewrong.TheirradiusanddistancetotheEarthare\¯ne-tuned”togive,onaverage,(µSun¡µMoon)=µSun»3%,whereµSun;Moonarethesolarandlunarangularsizes.(ii)Low-energysuper-symmetryexists,butitsrealizationisdi®erentfromwhatwehadimagined.Letmeexpandonthis.Theoriginofthe¯ne-tuningislinkedtothesuccessofsupersymmetrytotriggertheelectroweakbreaking.Indeed,fromtheHiggsmechanismweknowthatM2Z=¡g2¹2H=¸,where¹2Hand¸arethecoe±cientsofthequadraticandquartictermsintheHiggspotential.Insupersymmetry,¸»g2,andthereforeM2Z»¡¹2H.Thee®ective¹2Hhasatree-leveldirectcontributionfromthesofttermsandaone-loopnegativecontributionroughlyproportionaltothestopmass(which,inturn,isusuallydeter-minedbythecontributionproportionaltothegluinomass).Intheconventionalsupergravityapproach,theone-loopcontributionhasalargelogarithmanditovercomesthetree-levele®ect.Thereforej¹Hjisoftheorderofthesupersymmetricmasses,whichareexpectedtolieclosetoMZ.Onemaythinkthatthesituationimprovesingauge-mediatedmodels,16sincethelogarithmcanbemadesmallwithsu±cientlylightmessengers.However,thisisnotthecase,be-causegaugemediationgivesaboundaryconditionforthestopmassatthemessengerscale,whichissigni¯cantlylargerthanthetree-levelcontributionto¹2H.Actually,ingaugemediation,the¯ne-tuningisusuallymoreacute,becausetheright-handedselec-tronisafactorofabout2lighterthantheHiggsmassparameter,andbecauseofthelimitonthevisiblede-cayofthelightestneutralino.10Thesituationcouldimproveinmodelswherethestopandgluinomassesarenotlarge(withrespecttotheothersoftmasses)orinmodelswithalowsupersymmetry-breakingscale,17orinmodelswhere¹2Histrulyaone-loopfactorsmallerthanallsupersymmetricmasses(seeforinstanceref.18).Theseconddrawbackoflow-energysuper-symmetryisthelackofpredictivityinthesupersymmetry-breakingsector,whichtranslatesintothe\supersymmetric°avorproblem”(i.e.thelarge°avorviolationspresentwithgenericsoftterms).Wearenowawareofvariousschemes,al-

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4ternativetotheconventionalsupergravityscenario,whicharemorepredictiveandaddressthe°avorproblem:gaugemediation,16anomalymediation,19gauginomediation.20However,atpresent,wecan-notsaythatoneschemeispreferabletotheothers.Forinstance,inoperator-basedschemes(likesuper-gravity)theHiggsmixingmass¹canbeproperlyaccountedfor,21whileinotherschemeswherethesofttermsarecalculable,theHiggsmassparameters¹andB¹typicallyappearatanundesiredorderinperturbationtheory.Luckily,thedi®erentschemesusuallyhavequitedistinctfeaturesbothinthemassspectrumandinthenatureofthelightestsuper-symmetricparticles,leadingtodistinctsignalsatcolliders.Therefore,inthecaseofadiscovery,thequestionofthecorrectschemeofsupersymmetry-breakingtermscanbesettledexperimentally.Recentlymanynewmodelshavebeenproposed,wheresupersymmetryattheweakscaleisimple-mentedinunconventionalways.Theseproposalsmakeuseofnewingredients,comingfromextradi-mensions.Beforediscussingthem,letme¯rstintro-duceextradimensions.3.ExtraDimensionsThehierarchyproblemcanmotivatetheexistenceofexperimentally-accessibleextradimensions.22;23Thescenariosarebynowwellknown:ourthree-dimensionalspaceisembeddedinalargerD-dimensionalspace-time.WhileSM¯eldsarecon-¯nedonthethree-dimensionalbrane,gravityisde-scribedbythegeometryofthefullspace.Thecru-cialassumptionisthatthefundamentalPlanckscaleoftheD-dimensionaltheoryMDisoftheorderoftheweakscale,avoidinganyhierarchy.AnydistancescaleshorterthanTeV¡1(associatedwithNewton’sconstant,withGUT’s,right-handedneutrinos,etc.)hastobeexplainedbysomegeometricalproperty.For°atandcompactextradimensions,theob-servedweaknessofgravityistranslatedintoalargevalueofthecompacti¯cationradiusR,sincetheordi-naryPlanckmassisgivenby22MPl»R±=2M1+±=2D,whereD=4+±.Theinterpretationisthatweob-serveaweakgravitationalforcenotbecausegravityisintrinsicallyweak(MD»TeV),butbecauseofthesmalloverlapbetweenourworldandthegravi-tonwavefunction,whichisspreadinaverylargevolume.Forspaceswithnon-factorizablemetrics,thelargehierarchyMPl=MWcanbereproducedwithmuchsmallervaluesofR,exploitingthestrongfunc-tionaldependenceinthewarpfactor.Inthiscase,theinterpretationisthat,becauseofthenon-trivialgeometry,thezero-modegravitonwavefunctionispeakedonabranefarfromoursandonlyitsexpo-nentialtailoverlapswithourworld,leadingtotheobservedweaknessofgravity.Thehierarchyistheresultofthefamiliargravitationalredshiftingen-eralrelativity.Aphotonthatclimbsoutofagravi-tationalpotentialseesitsmeasuredenergyreduced.Itisknownthatfortime-independentmetricswithg0j=0,theproductEpjg00jisconserved,ratherthantheenergyE.Therefore,aphotonemittedatadistancerfromasphericallysymmetricgravita-tionalsourceisobservedatin¯nitytoberedshiftedbyanamount(Eobs¡Eem)=Eem=pjg00j¡1’¡GNM=r,sincetheSchwarzschildmetricsgeneratedbyamassMhasg00=1¡2GNM=r.Similarly,intheRandall{Sundrumset-up,massesinthehid-den(Planck)braneareblueshiftedbyawarpfactor,whenmeasuredontheobservable(TeV)brane,asaresultofthenon-trivialgravitational¯eld.Althoughthescenarioswithaquantum-gravityscaleattheTeVhavevarioustheoreticalandcosmo-logicaldi±culties,theyrepresentawonderfulpos-sibilityfornew-physicssearchesatfuturecolliders.AttheLHC,gravitonemissionisobservedinjetplusmissing-energyevents24;25(for°atdimensions)orasresonancesinDrell{Yan26(forwarpeddimen-sions).Tree-levelgravitonexchangeleadstoane®ectivedimension-8operatorT¹ºT¹º,whichpre-dictscorrelatedsignalsindiphotonanddilepton¯nalstates.24;27Gravitonloopsgiverisetodimension-6operators,witha°avor-universalaxial-vectorcon-tactinteractionplayingaspecialr^ole.28Becauseofthelowerdimensionalityoftheinducedoperators,loope®ectsaregenerallymoreimportantthantree-levele®ects,unlesstheshort-distancebehaviorofgravityismodi¯edatanenergyscalesigni¯cantlysmallerthanMD.Theradion,ascalar¯eldcon-tainedintheextra-dimensionalgraviton,canmixwiththeHiggsboson,givingrisetointerestingvari-ationsintheHiggssearches.29Thetheoreticaldescriptionoftheexperimentalsignalsdescribedabovereliesuponlinearizedgravity,whosevalidityendsastherelevantenergyofthepro-cess(ps)approachesthefundamentalgravityscale

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