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aBU-HEP-98-7DOE/ER/40561-358-INT98-00-6UW/PT-97/24hep-th/9803132E ectiveFieldTheory,BlackHoles,andtheCosmologicalConstantAndrewG.Cohena,DavidB.KaplanbandAnnE.Nelsonc(a)DepartmentofPhysics,BostonUniversity,Boston,MA02215,USA(b)InstituteforNuclearTheory,1550,UniversityofWashington,Seattle,WA98195-1550,USA(c)DepartmentofPhysics1560,UniversityofWashington,Seattle,WA98195-1560,USABekensteinhasproposedtheboundS≤πMP2L2onthetotalentropySinavolumeL3.Thisnon-extensivescalingsuggeststhatquantum eldtheorybreaksdowninlargevolume.Torecon-cilethisbreakdownwiththesuccessoflocalquantum eldtheoryindescribingobservedparticlephenomenology,weproposearelationshipbetweenUVandIRcuto ssuchthatane ective eldtheoryshouldbeagooddescriptionofNature.Wediscussimplicationsforthecosmologicalconstantproblem.We ndalimitationontheaccuracywhichcanbeachievedbyconventionale ective eldtheory:forexample,theminimalcorrectionto(g 2)fortheelectronfromtheconstrainedIRand

UVcuto sislargerthanthecontributionfromthetopquark.

Itisgenerallyassumedthatparticlephysicscanbeaccuratelydescribedbyane ective eldtheorywithanultraviolet(UV)cuto lessthanthePlanckmassMP,providedthatallmomentaand eldstrengthsaresmallcomparedwiththiscuto http://wendang.chazidian.computationsperformedwithsuche ective eldtheories,forexampletheStandardModel,havebeenextraordinarilysuccessfulatdescribingpropertiesofelementaryparticles.Nevertheless,considerationsinvolvingblackholessuggestthattheunderlyingtheoryofNatureisnotalocalquantum eldtheory.InthisLetterweattempttoreconcilethisconclusionwiththesuccessofe ectivequantum eldtheorybydeterminingtherangeofvalidityforalocale ective eldtheorytobeanaccuratedescriptionoftheworld.WeaccomplishthisbyimposingarelationshipbetweenUVandinfrared(IR)cuto s.WewillarguethatthisproposedIRbounddoesnotcon ictwithanycurrentexperimentalsuccessofquantum eldtheory,butexplainswhyconventionale ective eldtheoryestimatesofthecosmologicalconstantfailsomiserably.

Forane ectivequantum eldtheoryinaboxofsizeLwithUVcuto ΛtheentropySscalesextensively,S～L3Λ3 .Howeverthepeculiarthermodynamicsofblackholes[1,2]hasledBekenstein[1]topostulatethatthemaximumentropyinaboxofvolumeL3behavesnon-extensively,growingonlyastheareaofthebox.ForanyΛ,thereisasu cientlylargevolumeforwhichtheentropyofane ective eldtheorywillexceedtheBekensteinlimit.’tHooft[3]andSusskind[4]havestressedthatthisresultimpliesconventional3+1dimensional eldtheoriesvastlyover-countdegreesoffreedom:asthese eldtheoriesaredescribedintermsofaLagrangedensity,theyhaveextensivityoftheentropybuiltin.TheBekensteinentropyboundmaybesatis edinane ective eldtheoryifwelimitthevolumeofthesystemaccordingto

22L3Λ3<～SBH≡πLMP(1)

whereSBHistheentropyofablackholeofradiusL[2,1].ConsequentlythelengthL,whichactsasanIRcuto ,cannotbechosenindependentlyoftheUVcuto ,andscalesasΛ 3.

AsstartlingastheBekenstein-motivatedconstrainteq.(1)seems,thereisevidencethatconventionalquantum eldtheoryfailsatanentropywellbelowthisbound.’tHoofthasstressedthatordinary eldtheoriesshouldfailonlargescalesifnearthehorizonofablackhole[3].Inthepresenceofevenaverylargeblackhole,alowenergydescriptionofparticlephysicsisexpectedtobeinadequate,sinceinfallingparticlesexperiencePlanckscaleinteractionswithoutgoingHawkingradiationnearthehorizon.Furthermore,ithasbeenshowninstringtheorythatlocalobservablesdonotnecessarilycommuteatspace-likeseparationinthepresenceofablackhole[5].Theseproblemsariseevenintheabsenceofanylarge eldstrengthsormomenta.Localquantum eldtheoryappearsunlikelytobeagoode ectivelowenergydescriptionofanysystemcontainingablackhole,andshouldprobablynotattempttodescribeparticlestateswhosevolumeissmallerthantheircorrespondingSchwarzschildradius.

Ane ective eldtheorythatcansaturateeq.(1)necessarilyincludesmanystateswithSchwarzschildradiusmuchlargerthantheboxsize.Toseethis,notethataconventionale ectivequantum eldtheoryisexpectedtobecapableofdescribingasystematatemperatureT,providedthatT≤Λ;solongasT 1/L,suchasystemhas2thermalenergyM～L3T4andentropyS～L3T3.Wheneq.(1)issaturated,atT～(MP/L)1/3,thecorrespondingSchwarzschildradiusLSforthissystemisLS～L(LMP)2/3 L.Toavoidthesedi cultiesweproposeanevenstrongerconstraintontheIRcuto 1/LwhichexcludesallstatesthatliewithintheirSchwarzschildradius.Sincethemaximumenergydensityinthee ectivetheoryisΛ4,theconstraintonLis

HeretheIRcuto scaleslikeΛ 2.Thisboundisfarmorerestrictivethaneq.(1):wheneq.(2)isnearsaturation,theentropyis

Smax SBH.3/42L3Λ4<～LMP.(2)(3)

Weproposethatane ectivelocalquantum eldtheorywillbeagoodapproximatedescriptionofphysicswheneq.(2)issatis ed.Thisboundismorerestrictivethaneq.(1)becauseweareexplicitlyconsideringonlythosestatesthatcanbedescribedbyconventionalquantum eldtheory .

Cansuchadramaticdepletionofquantumstatesberelevanttothecosmologicalconstantproblem ?IftheStandard

Model

isvalidinanarbitrarilylargevolumeuptoatleastLEPenergies,thenthequantumcontributiontothevacuumenergydensitycomputedinperturbationtheoryis～(100GeV)4.Theempiricalboundon 2.5thecosmologicalconstantcorrespondstoavacuumenergydensity<eV)4.Conventionallythisdiscrepancyis～(10

explainedbyeitherunknownphysicsathighenergieswhichconspirestocancelthisvacuumcontributiontoenormousprecision,orelsenewphysicsat～10 2.5eVwhichadjuststocancelthevacuumenergywhilebeingdeviousenoughtoescapedetection[11,12].

Thereishoweverathirdpossibility—thattheusualperturbativecomputationofthequantumcorrectiontothevacuumenergydensity,whichassumesnoinfraredlimitationtothequantum eldtheory,isincorrect.Thereisinfactnoevidencethat eldsatpresentexperimentalenergiescan uctuateindependentlyoveraregionaslargeasourhorizon.Infact,ifwechooseanIRcuto comparabletothecurrenthorizonsize,thecorrespondingUVcuto fromeq.(2)isΛ～10 2.5eVandtheresultingquantumenergydensityofΛ4requiresnocancellationtobeconsistentwithcurrentbounds.Thisobservationdoesnotpredictthecosmologicalconstant’svalue,asonecanalwaysaddaconstanttothequantumcontribution.Howeveritdoeseliminatetheneedfor ne-tuning.

ThepeculiarrelationshipbetweenIRandUVcuto seq.(2)is,inprinciple,testableasitlimitsthesuccessfulapplicationofquantum eldtheorytoexperiment.Forinstanceifwewishtosearchfornewphysics(comingfromnewinteractionsorparticlesathighenergieswhichdonotviolatelowenergysymmetries)usinghighprecisionexperimentsatlowenergiesp,thereisamaximalenergyscalethatcanbeprobedwithoutincorporatinge ectsbeyondconventionalquantum eldtheory.Surprisingly,thisscaledependsonp,andcanbemuchlowerthanMP.Inordertoperformane ective eldtheorycalculationwesimultaneouslyimposeaUVandanIRcuto consistentwitheq.(2).Therewillbesmalldiscrepanciesbetweensuchacalculationandaconventionaloneperformedinanin nitebox.Suchadiscrepancycanbeofinterestwhentryingtodiscovernewphysicsthroughradiativecorrections.Forexample,consider(g 2)fortheelectron.TheUVandIRcuto sthatwemustimposeeachleadtocorrectionstotheusualcalculation,whosetotalsizeis

δ(g 2)～α

Λ 2+ 1

π me

meMP 2 .(5)

1/3ThisuncertaintyinourcalculationisminimizedbychoosingtheUVcuto tobeΛ～(m2～14TeV,sothateMP)

δmin(g 2)～α

MP 2/3～α

Bankshasalsoarguedthatadrasticallyreducednumberoffundamentaldegreesoffreedommaybepartofanexplanationofthesmallsizeofthecosmologicalconstant[9].Hisexplanationdi ersfromoursasheusestheweakerBekensteinboundontheUVcuto ,andassumesbothanIRcuto whichismuchlargerthanthepresenthorizon,aswellassubstantialcancellationsofthezero-pointenergiesofthefundamentaldegreesoffreedomduetosupersymmetryina2+1dimensional“holographic”description.Horava[10]hasproposedamodelwithnonextensivefundamentaldegreesoffreedomwhichalsogivessuppressionofthecosmologicalconstant.§TheconventionalcontributionofanewheavyleptonofmassMto(g 2)is[13] (g 2)=(1/45)(α/π)2(me/M)2,wheremeistheelectronmass.

2

222(α/π)(p/Λ)(D 4).TherequiredIRcuto L<～MP/Λleadstoadditionalcorrections～(α/π)(1/(Lp)),whichare

atleastasbigas(α/π)(Λ2/(pMP))2,accordingtoourconstrainteq.(2).MinimizationofthistheoreticaluncertaintyoccursforaUVcuto Λ～p(MP/p)2/D.Thusinagivenexperimentthereisamaximumenergyscalethatcanbeprobedandamaximumaccuracythatcanbeachievedusingconventionalquantum eldtheory,withtheenergyscale2/5dependingonMPtoaremarkablysmallfractionalpower.Foroperatorsofdimension5thisscaleisMP,whilefor

1/3operatorsofdimensionsixitisMP.

Notethattherelativesizeofthesee ectsgrowswithp.Whenpistheweakscaleandthee ectivetheoryistheStandardModel,newphysicsatshortdistancesappearsinthee ectivetheorythroughdimensionsixoperators,andthemaximumenergyscalethatcanbeconventionallyprobedis108GeV,withacorrespondinguncertaintyof10 13.Ifthenewhighenergyphysicsappearsthroughdimension veoperatorsthemaximumenergyscalewouldbe109GeVwithanuncertaintyof10 9.

Wemightworrythatthelowscalewhichcanbeprobedbyelectroweakphysicseliminatesthepossibilityofcom-putingcouplingconstantuni cation,whichinvolvesanenergyscaleMGUT～1016GeV.HoweveritisstillpossibletoconsiderrunningofdimensionfouroperatorsuptoenergiesashighasMP.Inordertocomputecouplingcon-stantrunninginthepresenceofourIRandUVcuto s,wemayusearenormalizationgrouptreatment,matchingtheS-matricesoftwotheorieswithparameters{L,Λ}and{L′,Λ′}(eachofwhichobeyeq.(2))intheircombineddomainofvalidity.Thereisaninherentuncertaintyinthebetafunctionatagivenenergyscaleduetothee ectsofthesecuto s.Choosing{L,Λ}ateachenergyscaletominimizetheuncertainty ,leadstocorrectionsoftherelationbetweentheuni edcouplingatMGUTandtheStandardModelgaugecouplingsatMZ:

4π

αGUT(MGUT)+bilnmz

λ,inordertoavoidspurious nitevolumee ects.Thisimplies

thatonecannotperformtheRGscalingtoUVcuto slargerλ(1/4),andthatconsequentlyoneneverseesa netuningproblemforthevacuumenergy.

Itisconceivablethatblackholesandtheirinteractionswithparticlescanbedescribedbysomee ective eldtheory,eliminatingthemotivationfortheboundofeq.(2).Itremainsdi culttounderstandthenecessarynon-extensivebehavioroftheentropywithoutsomeinfraredlimitationofe ective eldtheoryatleastasstrongastheBekenstein-motivatedboundofeq.(1).Howevereventhislatterboundleadstoconclusionsqualitativelysimilartothoseabove.ForexampleexperimentsatascalepsensitivetonewphysicswhicharisesthroughdimensionDoperators(D>4) 41/(D+2)wouldbelimitedtoprobingenergiesbelowΛ～pD 2MP,andthemaximumtheoreticalaccuracywouldbe～(α/π)(p/MP)4(D 4)/(D+2).

Bothboundsgiverelativelylargecorrectionstoe ective eldtheorycomputationscomparedtoconventionalcom-putablequantumgravitationale ects.ThelatteraregenerallyexpectedtobesuppressedbyintegralpowersofMP;suchexpectationsarebornoutbyexplicitconstructionsofe ective eldtheoriesfromstringtheory[14].

Itistemptingtoconsideralessdrasticsolution:patchingupconventionale ective eldtheory(withaPlanckscaleUVcuto andnoIRcuto )byeliminating“byhand”thosestatescorrespondingtoblackholes.WedonotknowhowtopruneaHilbertspaceinthismanner;theresultwouldlikelybeabizarre,nonlocaltheory.Still,onecouldimaginethateventhoughmostofthedegreesoffreedominane ective eldtheoryinanarbitrarilylargeboxhavenosensiblephysicalinterpretation,forsomereasonthetheoryaccuratelydescribesthepropertiesoffewparticlestates.Thiswouldleaveconventionalcalculationswhichcontainnointermediatestatesapproachingblackholeformation

unchangedtolowordersinperturbationtheory,whilerejectingthenumerousstatespredictedbythesametheorywhichliewithintheirownSchwarzschildradius,howevertherewouldbedrastice ectsonthermaldistributionsevenattemperaturesT Λ.Instead,ourmainassumptionisthatalocale ective eldtheorywhichcorrectlydescribesallsingleparticlestateswithmomentauptop～Λ,shouldalsodescribemulti-particleexcitations,andwouldhaveanormaldensitymatrixforthermaldistributionswithT Λ.Whileconventional,thisassumptionmaynotbevalidwhentheunderlyingtheoryisnotlocal.Thealternativethatane ective eldtheorycanbevaliduptoascaleΛforcertaincalculations,butfailtocorrectlydescribeathermalsystemattemperatureT Λ,seemsatleastasstrangeasourassumption.

Inconclusion,manydi erentresultsaboutthephysicsofblackholesimplythat,inthepresenceofquantumgravity,therearenofundamentalextensivedegreesoffreedom.Furthermore,considerationsofthemaximumpossibleentropyofsystemswhichdonotcontainblackholessuggestthatordinaryquantum eldtheorymaynotbevalidforarbitrarilylargevolumes,butwouldapplyprovidedtheUVandIRcuto ssatisfyaboundgivenbyeq.(2).Theexperimentalsuccessofquantum eldtheorysurvives,aslongasthise ectivetheoryisnotappliedtocalculationswhichsimultaneouslyrequirebothalowinfraredcuto andanoverlyhighUVcuto .ThesimultaneousUVandIRsensitivityofcomputationsrelevantforcurrentlaboratoryexperimentsnevercomesclosetorequiringcuto swhichviolateeq.(2).Incontrast,thecomputationofthequantumcontributiontothevacuumenergyofthevisibleuniversewithinquantum eldtheoryrequiresaUVcuto oflessthen10 2.5eV.Withthiscuto ,no ne-tunedcancellationofthecosmologicalconstantisrequired.Recognitionthatquantum eldtheoryvastlyovercountsstatescanhelpresolvetheenormousdiscrepancybetweenconventionalestimatesofthevacuumenergyandtheobservedcosmologicalconstantandeliminateacelebrated ne-tuningproblem.

Acknowledgements.WegratefullyacknowledgeTomBanksforveryusefuldiscussions,andtheAspenCenterforPhysics,wherethisworkwasinitiated.WealsothankfullyacknowledgecriticalcorrespondencefromFinnLarsen,SteveGiddings,PetrHorava,JuanMaldacena,YossiNir,JoePolchinskiandananonymousreferee.A.G.C.issupportedinpartbyDOEgrant#DE-FG02-91ER40676;D.B.K.issupportedinpartbyDOEgrant#DOE-ER-40561;A.E.N.issupportedinpartbyDOEgrant#DE-FG03-96ER40956.

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