Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

8

2

ay

M

8

2

]

h

t

-

ep

h

[

3

v

9

9

2

.

5

8

:0v

i

XraInvestigatingtheUltravioletPropertiesofGravitywithaWilsonianRenormalizationGroupEquationAlessandroCodello Institutf¨urPhysik,Johannes-Gutenberg-Universit¨at,Staudingerweg7,D-59099Mainz,GermanyRobertoPercacci InstituteforTheoreticalPhysics,UtrechtUniversity,Leuvenlaan4,NL-3584,TheNetherlandsandSISSA,viaBeirut4,I-34014Trieste,ItalyChristophRahmede SISSA,viaBeirut4,I-34014Trieste,Italy,andINFN,SezionediTrieste,Italy

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

Abstract

Wereviewandextendinseveraldirectionsrecentresultsontheasymptoticsafetyapproachtoquantumgravity.ThecentralissueinthisapproachisthesearchofaFixedPointhavingsuitableproperties,andthetoolthatisusedisatypeofWilsonianrenormalizationgroupequation.Webeginbydiscussingvariouscuto schemes,i.e.waysofimplementingtheWilsoniancuto procedure.Wecomparethebetafunctionsofthegravitationalcouplingsobtainedwithdi erentschemes,studying rstthecontributionofmatter eldsandthentheso–calledEinstein–Hilberttruncation,whereonlythecosmologicalconstantandNewton’sconstantareretained.Inthiscontextwemakeconnectionwitholdresults,inparticularwereproducetheresultsoftheepsilonexpansionandtheperturbativeoneloopdivergences.WethenapplytheRenormalizationGrouptohigherderivativegravity.Inthecaseofageneralactionquadraticincurvaturewerecover,withincertainapproximations,theknownasymptoticfreedomofthefour–derivativeterms,whileNewton’sconstantandofthecosmologicalconstanthaveanontrivial xedpoint.Inthecaseofactionsthatarepolynomialsinthescalarcurvatureofdegreeuptoeightwe ndthatthetheoryhasa xedpointwiththreeUV–attractivedirections,sothattherequirementofhavingacontinuumlimitconstrainsthecouplingstolieinathree–dimensionalsubspace,whoseequationisexplicitlygiven.Weemphasizethroughoutthedi erencebetweenscheme–dependentandscheme–independentresults,andprovideseveralexamplesofthefactthatonlydimensionlesscouplingscanhave“universal”behavior.

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

I.INTRODUCTIONItiswellknownthatgeneralrelativitycanbetreatedasane ectivequantum eldtheory

[1,2,4].Thismeansthatitispossibletocomputequantume ectsduetogravitonloops,aslongasthemomentumoftheparticlesintheloopsiscuto atsomescale.Forexample,inthiswayithasbeenpossibletounambiguouslycomputequantumcorrectionstotheNewtonianpotential[2].Theresultsareindependentofthestructureofany“ultravioletcompletion”,andthereforeconstitutegenuinelowenergypredictionsofanyquantumtheoryofgravity.Whenonetriestopushthise ective eldtheorytoenergyscalescomparabletothePlanckscale,orbeyond,well-knowndi cultiesappear.Itisconvenienttodistinguishtwoordersofproblems.The rstisthatthestrengthofthegravitationalcouplinggrowswithoutbound.Foraparticlewithenergypthee ectivestrengthofthegravitational couplingismeasuredbythedimensionlessnumber

1Strictlyspeakingonlytheessentialcouplings,i.e.thosethatcannotbeeliminatedby eldrede nitions,needtoreachaFP.See[3]forarelateddiscussioninagravitationalcontext.

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

InordertoaddressthesecondproblemwehavetoinvestigatethesetofRGtrajectoriesthathavethisgoodbehaviour.WewanttousetheconditionofhavingagoodUVlimitasacriterionforselectingaQFTofgravity.IfalltrajectorieswereattractedtotheFPintheUVlimit,wewouldencounteravariantofthesecondproblem:theinitialconditionsfortheRG owwouldbearbitrary,sodeterminingtheRGtrajectoryoftherealworldwouldrequireinprincipleanin nitenumberofexperimentsandthetheorywouldlosepredictivity.Attheotherextreme,thetheorywouldhavemaximalpredictivepoweriftherewasasingletrajectoryendingattheFPintheUV.However,thismaybetoomuchtoask.AnacceptableintermediatesituationoccurswhenthetrajectoriesendingattheFPintheUVareparametrizedbya nitenumberofparameters.Atheorywiththesepropertiesissaidtobe“asymptoticallysafe”[5].

Tobetterunderstandthisproperty,imagine,inthespiritofe ective eldtheories,ageneralQFTwithallpossibletermsintheactionwhichareallowedbythesymmetries.Wecanparametrizethe(generallyin nitedimensional)“spaceofalltheories”,Q,bythedimensionlesscouplingsg i.Weassumethatredundanciesinthedescriptionofphysicsduetothefreedomtoperform eldrede nitionshavebeeneliminated,i.e.allcouplingsare“essential”(suchcouplingscanbede nede.g.intermsofcrosssectionsinscatteringexperiments).WethenconsidertheRenormalizationGroup(RG) owinthisspace;itisgivenbythebetafunctions

βi=kdg i

dk

where=Mij( gj g j ),

Mij= βi(3)

2RGtransformationsleadtowardslowerenergies,andthetrajectorieslyinginCarerepelledbytheFPunderthesetransformations.Forthisreason,Cisalsocalledthe“unstablemanifold”.SinceweareinterestedinstudyingtheUVlimit,itismoreconvenienttostudythe owforincreasingk.

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

TheattractivitypropertiesofaFParedeterminedbythesignsofthecriticalexponents i,de nedtobeminustheeigenvaluesofM.Thecouplingscorrespondingtonegativeeigenvalues(positivecriticalexponent)arecalledrelevantandparametrizetheUVcriticalsurface;theyareattractedtowardstheFPfork→∞andcanhavearbitraryvalues.Theonesthatcorrespondtopositiveeigenvalues(negativecriticalexponents)arecalledirrelevant;theyarerepelledbytheFPandmustbesettozero.

Afreetheory(zerocouplings)hasvanishingbetafunctions,sotheorigininQisaFP,calledtheGaußianFP.IntheneighborhoodoftheGaußianFPonecanapplyperturbationtheory,andonecanshowthatthecriticalexponentsarethenequaltothecanonicaldimen-sions( i=di),sotherelevantcouplingsaretheonesthatarepower–countingrenormalizable3.Inalocaltheorytheyareusually niteinnumber.Thus,aQFTisperturbativelyrenor-malizableandasymptoticallyfreeifandonlyifthecriticalsurfaceoftheGaußianFPis nitedimensional.PointsoutsideC owtoin nity,ortootherFP’s.Atheorywiththesepropertiesmakessensetoarbitrarilyhighenergies,becausethecouplingsdonotdivergeintheUV,andispredictive,becauseallbuta nitenumberofparametersare xedbytheconditionoflyingonC.Asymptoticsafetyisaformofnonperturbativerenormalizability.Itgeneralizesthispicture,replacingtheGaußianFPbyanarbitraryFP.Anasymptoticallysafetheorywouldhavethesamegoodpropertiesofarenormalizableandasymptoticallyfreeone:thecouplingswouldhavea niteUVlimitandtheconditionoflyingonCwouldleaveonlya nitenumberofparameterstobedeterminedbyexperiment.Ingeneral,studyingthepropertiesofsuchtheoriesrequirestheuseofnonperturbativetools.IfthenontrivialFPissu cientlyclosetotheGaußianone,itspropertiescanalsobestudiedinperturbationtheory,butunlikeinasymptoticallyfreetheories,theresultsofperturbationtheorydonotbecomebetterandbetterathigherenergies.

Inordertoestablishwhethergravityisasymptoticallysafe,severalauthors[5,6,7]ap-pliedthe expansionaroundtwodimensions,whichisthecriticaldimensionwhereNewton’sconstantisdimensionless.ThebetafunctionofNewton’sconstantthenhastheform

2βG = G+B1G,(5)

=Gk andB1<0[5,6],sothereisaFPatG = /B1>0.UnfortunatelythiswhereG

resultisonlyreliableforsmall anditisnotclearwhetheritwillextendtofourdimensions.

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

Inthee ective eldtheoryapproach(ind=4),Bjerrum-Bohr,DonoghueandHolsteinhaveproposedinterpreting

aclassofoneloopdiagramsasgivingthescaledependenceofNewton’sconstant[4].Theycalculate

G(r)=G01 167

r2 ,

whereristhedistancebetweentwogravitatingpointparticles.Ifweidentifyk=1/ar,withaaconstantoforderone,thiswouldcorrespondtoabetafunction

2167 βG=2G a

.ThiscalculationwasbasedonperturbativemethodsandsincetheFPoccurs

,itisnotclearthatonecantrusttheresult.WhatwecanatanotverysmallvalueofG167a2

saywithcon denceisthattheonsetoftherunningofGhastherightsignforasymptoticsafety.Clearlyinordertomakeprogressonthisissueweneeddi erenttools.

InthispaperwewilldiscusstheapplicationofWilsonianrenormalizationgroupmethodstotheUVbehaviorofgravity.InsectionIIwewillintroduceaparticularlyconvenienttool,calledthe“ExactRenormalizationGroupEquation”(ERGE)whichcanbeusedtocalculatethe“betafunctional”ofaQFT.Renormalizabilityisnotnecessaryandthetheorymayhavein nitelymanycouplings.InsectionIIIweillustratetheuseoftheERGEbycalculatingthecontributionofminimallycoupledmatter eldstothegravitationalbetafunctions.Inthissimplesetting,wewillreviewthetechniquesthatareusedtoextractfromthebetafunctionalthebetafunctionsofindividualcouplings,emphasizingthoseresultsthatare“schemeindependent”inthesensethattheyarethesameirrespectiveoftechnicaldetailsofthecalculation.InsectionIVweapplythesametechniquestothecalculationofthebetafunctionsforthecosmologicalconstantandNewton’sconstantinEinstein’stheoryinarbitrarydimensions,extendinginvariouswaystheresultsofearlierstudies[8,9,10,11,12].WealsoshowthattheFPthatisfoundinfour–dimensionalgravityisindeedthecontinuationfor →2oftheFPthatisfoundinthe2+ expansion.Wecomparevariouswaysofde ningtheWilsoniancuto and ndtheresultstobequalitativelystable.InsectionsVandVI

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

wemakeconnectionwitholdresultsfromperturbationtheory.InsectionVwerederivethe’tHooft–VeltmanoneloopdivergencefromtheERGEandweshowittobescheme–independent.WealsodiscusswhytheGoro –Sagnottitwoloopdivergencecannotbeseenwiththismethodandwediscussthesigni canceofthisfact.InsectionsVIandVIIweconsiderhigherderivativegravity.InsectionVIwederivetheexistenceoftheFPinthemostgeneraltruncationinvolvingfourderivativesatoneloop,andwehighlightthedi erencesbetweentheWilsonianprocedure[13]andearliercalculations.InsectionVIIweconsiderhigherpowersofcurvaturebutrestrictingourselvestopolynomialsinthescalarcurvature.Wegivemoredetailsofourrecentcalculations[14]andextendthemtopolynomialsofordereight.InsectionVIIIweassessthepresentstatusoftheasymptoticsafetyapproachtoquantumgravityanddiscussvariousopenproblems.

II.THEERGEANDITSAPPROXIMATIONS

ThecentrallessonofWilson’sanalysisofQFTisthatthe“e ective”(asin“e ective eldtheory”)actiondescribingphysicalphenomenaatamomentumscalekcanbethoughtofastheresultofhavingintegratedoutall uctuationsofthe eldwithmomentalargerthank

[15].Atthisgenerallevelofdiscussion,itisnotnecessarytospecifythephysicalmeaningofk:foreachapplicationofthetheoryonewillhavetoidentifythephysicallyrelevantvariableactingask4.Sincekcanberegardedasthelowerlimitofsomefunctionalintegration,wewillusuallyrefertoitastheinfraredcuto .Thedependenceofthe“e ective”actiononkistheWilsonianRG ow.

Thereareseveralwaysofimplementingthisideainpractice,resultinginseveralformsoftheRGequation.Inthespeci cimplementationthatweshalluse,insteadofintroducingasharpcuto inthefunctionalintegral,wesuppressthecontributionofthe eldmodeswithmomentalowerthank.Thisisobtainedbymodifyingthelowmomentumendofthepropagator,andleavingalltheinteractionsuna ected.Wedescribeherethisprocedureforascalar eld.WestartfromabareactionS[φ],andweaddtoitasuppressionterm Sk[φ]thatisquadraticinthe eld.In atspacethistermcanbewrittensimplyinmomentumspace.Inordertohaveaprocedurethatworksinanarbitrarycurvedspacetimewechoose

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

asuitabledi erentialoperatorOwhoseeigenfunctions n,de nedbyO n=λn n,canbetakenasabasisinthefunctionalspaceweintegrateover:

φ(x)= n n n(x),φ

naregeneralizedFouriercomponentsofthe eld.(Wewilluseanotationthatiswhereφ

suitableforanoperatorwithadiscretespectrum.)Then,theadditionaltermcanbewrittenineitherofthefollowingforms:

Sk[φ]=1

2 n 2Rk(λn).φn(7)

ThekernelRk(O)willalsobecalled“thecuto ”.Itisarbitrary,exceptforthegeneralrequirementsthatRk(z)shouldbeamonotonicallydecreasingfunctionbothinzandk,thatRk(z)→0forz kandRk(z)=0forz k.Theseconditionsareenoughto

ncorrespondingguaranteethatthecontributiontothefunctionalintegralof eldmodesφ

toeigenvaluesλn k2aresuppressed,whilethecontributionof eldmodescorrespondingak-dependentgeneratingfunctionalofconnectedGreenfunctionsby

e Wk[J]=Dφexp S[φ] Sk[φ] dxJφtoeigenvaluesλn k2areuna ected.Wewillfurther xRk(z)→k2fork→0.Wede ne

andamodi edk-dependentLegendretransform

Γk[φ]=Wk[J] dxJφ Sk[φ],

where Sk[φ]hasbeensubtracted.ThefunctionalΓkissometimescalledthe“averagee ectiveaction”,becauseitcanbeinterpretedasthee ectiveactionfor eldsthathavebeenaveragedovervolumesoforderk d(dbeingthedimensionofspacetime)[17].The“classical elds”δWk/δJaredenotedagainφfornotationalsimplicity.Inthelimitk→0thisfunctionaltendstotheusuale ectiveactionΓ[φ],thegeneratingfunctionalofone-particleirreducibleGreenfunctions.ItissimilarinspirittotheWilsoniane ectiveaction,butdi ersfromitinthedetailsoftheimplementation.

Theaveragee ectiveactionΓk[φ],usedattreelevel,givesanaccuratedescriptionofprocessesoccurringatmomentumscalesoforderk.Inthespiritofe ective eldtheories,weshallassumethatΓkexistsandisquasi–localinthesensethatitadmitsaderivative

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

expansionoftheform∞

Γk(φ,gi)=

whereg(n) g(n)(n)i(k)Oi(φ),(8)n=0i(n)

i(k)arecouplingconstantsandO iareallpossibleoperatorsconstructedwiththe eldφandnderivatives,whicharecompatiblewiththesymmetriesofthetheory.Theindexiisusedheretolabeldi erentoperatorswiththesamenumberofderivatives.Fromthede nitiongivenabove,itiseasytoshowthatthefunctionalΓksatis esthefollowing“ExactRenormalizationGroupEquation”(orERGE)[18,19]

kdΓk

2Tr Γ(2)+Rk 1kdRk

k

δφδφfortheinversepropagatorofthe eldφde nedbythefunctionalΓk.Ther.h.s.of(9)canberegardedasthe“betafunctional”ofthetheory,givingthek–dependenceofallthecouplingsofthetheory.Infact,takingthederivativeof(8)onegets

kdΓk

(n)dk=dgi

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

TheERGEcanbeseenformallyasaRG–improvedoneloopequation.Toseethis,recallthatgivenabareactionS(forabosonic eld),theoneloope ectiveactionΓ(1)is

1Γ(1)=S+.δφδφ

LetusaddtoSthecuto term(7);thefunctional

Γk=S+(1)(12)1

δφδφ+Rk, (13)

maybecalledthe“oneloopaveragee ectiveaction”.Itsatis estheequation

k(1)dΓk

2Tr δ2S

dk,(14)

whichisformallyidenticalto(9)exceptthatinther.h.s.therenormalizedrunningcou-plingsgi(k)arereplacedeverywherebythe“bare”couplingsgi,appearinginS.Thusthe“RGimprovement”intheERGEconsistsinreplacingthebarecouplingsbytherunningrenormalizedcouplings.Inthisconnection,notethatingeneralthecuto functionRkmaycontainthecouplingsgiandthereforethetermkd

dkRkinther.h.s.goestozeroformomentagreaterthankand

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

makestheintegrationconvergent.So,wecanregardthederivationgivenaboveasmerelyformalmanipulationsthatmotivatetheformoftheERGE,butthentheERGEitselfisperfectlywellde ned,withouttheneedofintroducinganUVregulator.IfweassumethatatagivenscalekphysicsisdescribedbyarenormalizedactionΓk,theERGEgivesusawayofstudyingthedependenceofthisfunctionalonk,andthebehaviorofthetheoryathighenergycanbestudiedbytakingthelimitofΓkfork→∞(whichneednotcoincidewiththebareactionS).

Inmostcasesitisimpossibletofollowthe owofin nitelymanycouplingsandacommonprocedureistoconsideratruncationofthetheory,namelytoretainonlya nitesubsetoftermsinthee ectiveactionΓk.Forexampleonecouldconsiderthederivativeexpansion(8)andretainalltermsuptosomegivenordern.Whateverthechoice,onecalculatesthecoe cientsoftheretainedoperatorsinther.h.s.of(9)andinthiswaythecorrespondingbetafunctionsarecomputed.IngeneralthesetofcouplingsthatonechoosesinthiswaywillnotbeclosedunderRGevolution,sooneisneglectingthepotentiale ectoftheex-cludedcouplingsontheonesthatareretained.Still,inthiswayonecanobtaingenuinenonperturbativeinformation,andthisprocedurehasbeenappliedtoavarietyofphysicalproblems,sometimeswithgoodquantitativeresults.Forreviews,see[22,23,24].

Ifwetruncatethee ectiveactioninthisway,thereisusuallynosmallparametertoallowustoestimatetheerrorwearemaking.Oneindirectwaytoestimatethequalityofatruncationreliesonananalysisofthecuto schemedependence.Thee ectiveactionΓkobviouslydependsonthechoiceofthecuto functionRk.Thisdependenceissimilartotheschemedependenceoftherenormalizede ectiveactioninperturbativeQFT;onlyphysicallyobservablequantitiesderivedfromΓkmustbeindependentofRk.Thisprovidesanindirectcheckonthequalityofthetruncation.Forexample,thecriticalexponentsshouldbeuniversalquantitiesandthereforecuto –independent.Inconcretecalculations,usuallyinvolvingatruncationoftheaction,criticalexponentsdodependonthecuto scheme,andtheobserveddependencecanbetakenasaquantitativemeasureofthequalityoftheapproximation.Ultimately,thereisnosubstituteforperformingcalculationswithtruncationsthatcontainmoreterms.Notethatagoodtruncationisnotnecessarilyoneforwhichthenewtermsaresmall,butoneforwhichthee ectofthenewtermsontheoldonesissmall.Inotherwords,insearchofanontrivialFP,wewanttheadditionofnewtermsnottoa ecttoomuchtheFPvalueofthe“old”couplings,northe“old”criticalexponents.

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

III.MATTERFIELDSANDCUTOFFSCHEMES

Inthissectionweillustratethemethodthatisusedtocomputethetraceinther.h.s.of

(9)inagravitationalsettingandtoevaluatethebetafunctionsofthegravitationalcouplings.Quitegenerally,wewillconsiderthecontributionof eldswhoseinversepropagatorΓ(2)isadi erentialoperatoroftheform = 2+E,where isacovariantderivative,bothwithrespecttothegravitational eldandpossiblyalsowithrespecttoothergaugeconnectionscoupledtotheinternaldegreesoffreedomofthe eld,andEisalinearmapactingonthequantum eld.Ingeneral,Ecouldcontainmasstermsortermslinearincurvature.Forexample,inthecaseofanonminimallycoupledscalar,E=ξR,whereξisacoupling.Apriori,nothingwillbeassumedaboutthegravitationalactionandalsothespacetimedimensiondcanbeleftarbitraryatthisstage.

InordertowritetheERGEwehavetode nethecuto .Fortheoperatortobeusedinthede nitionof(7),severalpossiblechoicessuggestthemselves.LetussplitE=E1+E2,whereE1doesnotcontainanycouplingsandE2consistsonlyoftermscontainingthecouplings.Wecallacuto oftypeI,ifRkisafunctionofthe“bareLaplacian” 2,oftypeIIifitisafunctionof 2+E1andoftypeIIIifitisafunctionofthefullkineticoperator = 2+E.Thesubstantialdi erencebetweenthe rsttwotypesandthethirdisthatinthelattercase,duetotherunningofthecouplings,thespectrumchangesalongthe ow.Forthisreasonthesecuto saresaidtobe“spectrallyadjusted”[25].6

LetusnowrestrictourselvestothecasewhenE2=0,i.e.thekineticoperatordoesnotdependonthecouplings;thenthereisonlyachoicebetweencuto softypeIandII.ThederivationofthebetafunctionsistechnicallysimplerwithatypeIIcuto .InthiscasewechoosearealfunctionRkwiththepropertieslistedinsectionIIandde neamodi edinversepropagator

Pk( )= +Rk( ).(15)

IftheoperatorEdoesnotcontaincouplings,using(A10)thetraceinther.h.s.oftheERGE

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

reducessimplyto:

∞

Tr tRk( )

(4π)d/2 QdB2i( )(16)

i=0Pk

whereB2i( )aretheheatkernelcoe cientsoftheoperator andtheQ-functionals,de nedin(A14,A15)aretheanalogsofmomentumintegralsinthiscurvedspacetimesetting.Wehavewritten tRktodenotethederivativewithrespecttotheexplicitdependenceofRkonk;whentheargumentofRkdoesnotcontaincouplingsthiscoincideswiththetotalderivatived

Pk( 2)+E.SinceEislinear

incurvature,inthelimitwhenthecomponentsofthecurvaturetensorareuniformlymuchsmallerthank2,wecanexpand

tRk

P +1.

k

Eachoneofthetermsonther.h.s.canthenbeevaluatedinawayanalogousto(A10),sointhiscasewegetadoubleseries:

Tr tRk( 2)

(4π)d/ ∞2 ∞Qd

i=0 =0Pk +1 dx√

Pkm

appearingin(16)and(18)willbeequaltok2(n m+1)timesanumberdependingonthepro le function.AsdiscussedinAppendixA,theintegralswithm=n+1areindependentoftheshapeofRk.Thus,ineven-dimensionalspacetimeswithacuto oftype

thecoe cientoftheterminthesum(16)withi=d

Pk II,andusing(A19),Bd( )=2Bd( ).On

theotherhandwithatypeIcuto ,using(A18),(A19)and(A5)thetermswith =d

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

addupto

d/2 =0Q

=2

=2Bd( 2+E) tRkg( 1) trE b2i( 2)Ed/2b0( 2) dx√(d/2)!

Therefore,inadditiontobeingindependentoftheshapeofthecuto function,thesecoef- cientsarealsothesameusingtypeIortypeIIcuto s.

Asanexamplewewillnowspecializetofour-dimensionalgravitycoupledtonSscalar elds,nDDirac elds,nMgauge(Maxwell) elds,allmasslessandminimallycoupled:

√¯µ µψ+1 µφ µφ+ψγΓk(gµν,φ,ψ,Aµ)=d4x2

d;E(M)=Ricci;E(gh)=0.(20)

Here“Ricci”standsfortheRiccitensorregardedasalinearoperatoractingonvectors:Ricci(v)µ=Rµνvν.Forthegauge eldswehavechosentheLorentzgauge,and (gh)istheoperatoracting

onthescalarghost.(Itcanbeshownthattheresultsdonotdependonthechoiceofgauge[26].)

WithatypeIIcuto ,foreachtypeof eldwede nethemodi edinversepropagatorPk( (A))= (A)+Rk( (A)).Then,theERGEreducessimplyto

(S)dΓk tRk( ) tRk( (D))Tr(S)Tr(D)22 tRk( (M))Tr(M)2Pk( (gh))

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

=1

(4π)2

+

+11 d4x√ Pk Pk

32π2 d4x√

dt=nS

2Tr(M)Pk( 2) tRk( 2) nDPk( 2)+R

Pk( 2) .(22)

Expandingeachtraceasin

(A10),collectingtermswiththesamenumberofderivativesofthemetric,andkeepingtermsuptofourderivativesweget

dΓk1 tRkg(nS 4nD+2nM)Q22

tRk tRkQ1Q1nD263Pk 1 tRk+ Q2Pk 180

+5nSR2+12(nS+nD 3nM) 2R +.... (3nS+18nD+36nM)C2 (nS+11nD+62nM)E(23)

Weseethatthetermslinearincurvature,whichcontributetothebetafunctionofNew-ton’sconstant,havechanged.However,thetermsquadraticincurvaturehavethesame

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

coe cientsasbefore,con rmingthatthebetafunctionsofthedimensionlesscouplingsarescheme–independent.

Inordertohavemoreexplicitformulae,andinnumericalwork,oneneedstocalculatealsothescheme–dependentQ-functionals.Thisrequires xingthepro leRk.Inthispaperwewillmostlyusetheso–calledoptimizedcuto (A21)inwhichtheintegralsarereadilyevaluated,seeequations(A22,A23,A24).Thiscuto hastheveryconvenientpropertythatQ n tRk

dt= tRk+ R′ E

k

i

dt=a(n)

ik4 n,

wherea(n)areconstants.Then,thebetafunctionsofthedimensionlessvariablesg (n)

ii=

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

kn 4gi(n)are

dg i(n)

4 n,

1(25)inparticularwritingg(0)=2ZΛandg(2)= Z=

dt dG +8πa(0)G +16πa(2)G Λ ,= 2Λ

4nS 4nD+2nM

dependingonwhethertherearemoreNotethattheFPoccursforpositiveornegativeΛ

,ontheotherhand,willbebosonicoffermionicdegreesoffreedom.TheFPvalueofG

positiveprovidedtherearenottoomanyscalar elds.Forn=4,(24)givesalogarithmicrunning

gi(k)=gi(k0)+ailn(k/k0),

implyingasymptoticfreedomforthecouplings1/gi.ThisisthesamebehaviorthatisobservedinYang–Millstheoriesandisinaccordancewithearlierperturbativecalculations

[29,30].Asnoted,itfollowsfrom(A24)thatwiththeoptimizedcuto ,forn>4,g i =0.ThecriticalexponentsatthenontrivialFPareequaltothecanonicaldimensionsoftheg(n)’s,soΛandGareUV–relevant(attractive),1/gi(4)(n)(4)(4)(4)(4) nS+2nD+4nM.(27)aremarginalandallthehighertermsareUV–irrelevant.NotethatinperturbationtheoryGisirrelevant.AtthenontrivialFPthequantumcorrectionsconspirewiththeclassicaldimensionsofΛandGtoreconstructthedimensionsofg(0)andg(2).Thismusthappenbecausethecriticalexponentsforg(0)andg(2)areequaltotheircanonicaldimensionsandthecriticalexponentsareinvariantunderregularcoordinatetransformationsinthespaceofallcouplings;thetrasformationbetween andgG (2)isregularatthenontrivialFP,butitissingularattheGaußianFP.

Thissimple owisexactinthelimitN→∞,butisalsoaroughapproximationwhengravitone ectsaretakenintoaccount,asweshalldiscussinsectionsIV-GandVI.Itisshownin gure1.

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

G

FIG.1:Thegenericformofthe owinducedbymatter elds.

IV.EINSTEIN’STHEORY

Asa rststeptowardstheinclusionofquantumgravitationale ects,wediscussinthissectiontheRG owforEinstein’sgravity,withorwithoutcosmologicalconstant.Thistruncationhasbeenextensivelydiscussedbefore[8,10].Herewewillextendthoseresultsinvariousdirections.Sincethedependenceoftheresultsonthechoiceofgaugeandpro lefunctionRkhasalreadybeendiscussedin[10,11]hereweshall xourattentiononaparticulargaugeandpro lefunction,andanalyzeinsteadthedependenceoftheresultsondi erentwaysofimplementingthecuto procedure.Thesimplicityofthetruncationwillallowustocomparetheresultsofdi erentapproximationsandcuto schemes,aluxurythatisprogressivelyreducedgoingtomorecomplicatedtruncations.

ThetheoryisparametrizedbythecosmologicalconstantΛandNewton’sconstantG=1/(16πZ),sothatwesetg(0)=2ΛZandg(2)= Zinequation(8).Allhighercouplingsareneglected.Thenthetruncationtakestheform

√Γk=dx

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

ofthetype:

SGF(g(B),h)=

whereZg(B)χµg(B)µνχν,1+ρ(29)χν= µhµν

2

includingthegauge xingterm,canbewrittenintheform

1

containingtheminimaloperator:

Γk

where8(2)µνρσ 1,whichleadstoconsiderablesimpli cation.Theinversepropagatorofhµν,ghµνΓk(2)µνρσhρσ µν µν=ZKρσ( 2 2Λ)+Uρσ,

;µνδρσ=(30)µνKρσ=1

2

1µνPρσ 1dgµνgρσ;µνµνUρσ=RKρσ+

gh.dµνUsingthatK=1

2

equation(30)ineitherofthefollowingforms:

Γk(2) P,ifd=2wecanrewrite=ZK( 2 2Λ1+W)Z=2 P 2 2Λ1 4

2(d 2)(Rρσgµν+gρσRµν Rgρσgµν).

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

Notethattheoverallsignofthesecondterminthesecondlineof(31)isnegativewhend>2.ThisisthefamousproblemoftheunboundednessoftheEuclideanEinstein–teron,wewillneedthetraces:

tr1=d(d+1)

RµνRµν+3

d 22d 5d2+8d+4;trW=d(d 1)

Onthed-dimensionalspherewecanwrite

U=1 νµ¯µ 2δµgC RννC.R Pd 2

dR .d(d 1)

Then,usingthesecondlineof(31),wehave

Γk=(2)Z

d(d 1)R d 2

dR .

(33)

Wewillnowdiscussseparatelyvarioustypesofcuto schemes.

A.Cuto oftypeIa

Thisistheschemethatwasusedoriginallyin[8].Itisde nedbythecuto term

1√ Sk[hµν]=ghµνRk( 2)µνρσhρσ dx

dZ

Z

22=ZK tRk( )+ηRk( ).

dt(37)

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian r

Thecalculationin[8]proceededasfollows.ThebackgroundmetricischosentobethatofEuclideandeSitterspace.Themodi edinversepropagatorisobtainedfrom(33)justreplacing 2byPk( 2).Usingthepropertiesoftheprojectors,itsinversionistrivial:

Γ(2)+R 121

kk=P2

k 2Λ+d 3d+4d 2P (38)

dR

Decomposinginthesamewaythetermd

tRk+ηRk

dt=1

Pδ tRk

k 2Λ+d2 3d+42TrxLPν.

dR TrxLµ

d

Onecannowexpandto rstorderinR,usethetraces(32)andformula(A10)toobtain:

dΓk

(4π)d/2 dx√

Pk

d(d+ 2Λ dQd+1)tRk+ηRk

2 16Qd

d(d 1)

2 4Qd tRk+ηRk2 Pk Pk tRk

本文来源：https://www.bwwdw.com/article/1r0e.html