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hep-lat/9310018 18 Oct 1993CU-TP-0QuarkMassRenormalizationontheLatticewithStaggeredFermionsWeonjongLeeDepartmentofPhysicsPupinPhysicsLaboratoriesColumbiaUniversityNewYork,N.Y.00,U.S.A.Oct.,AbstractTheQCDlightquarkmassrenormalizedataGeVscaleintheMSschemeisobtainedfromthenumericalresultsofthelatticeQCDsimulationwithstaggeredfermions.Theprimaryemphasisisgiventotheconnectionbetweenthelatticeandcontinuumparameters.TheresultsarecomparedwiththosefromtheQCDsumrule.

IntroductionAconnectionbetweenlatticeandcontinuumrenormalizedparametersisnecessarytocomparethelatticeobservableswiththecontinuumobservables(orexperimentaldata)andtocheckwhetherthenumericalsimulationoflatticeQCDmakesphysicalsense.ThebridgefortheQCDcouplingbetweenthelatticeandthecontinuumwasdonethroughtheweakcouplingexpansiontoone-looporder[,,,].ThebridgefortheQCDquarkmasswith-avorstaggereddynamicalfermionswasdoneinRef. [,].Buildingonthatwork,wewillattempttorelatethebare,lightquarkmassof.MeVobtainedfrom-avorstaggeredfermionsimulations[,,]withtherenormalizedlightquarkmassofbetween.andMeVdenedataGeVscaleintheMSschemeanddeducedfromtheQCDsumrules[0].Inordertomakethiscomparisonwemustassumethatthecouplingconstantusedinthelatticecalculation(==g=:)iswithintheper-turbativeregion[,].Weuseandcompareanumberofpossibleperturbativeapproachesin-cludingthemeaneldmethod(tadpoleimprovement)suggestedbyLepageandMackenzie.LatticeQCDwithStaggeredFermionsLatticeQCDhasalotofdicultiesinimplementingquarkavordy-namics.Therearetwopopularmethodstoputthefermionsonthelattice:oneistheWilsonfermionformalismandtheotherthestaggeredfermionformalism[].Inthelimitofzeroquarkmass,thestaggeredfermionschemehasremnantsofchiralsymmetryandmaybepreferredforthenumericalcalculationofthemesonandhadronspectrum..LatticeQCDActionThecurrentnumericalsimulationoflatticeQCDisbasedonthefollowingaction[,,].S=Xx;(x)(x)Uy(x)(x+a)(x+a)U(x)(x)

maXx(x)(x)+Sgluon=Xt(D=+ma)X+Sgluon()whereSgluon=X(Nc)ReTrUandthedenitionof(x)isgiveninRef. [,].Thecurrentapproachtoavordynamicsisasfollows:hOi=Rd[U]Oexp(Sgluon)[det(D=+ma)]NfRd[U]exp(Sgluon)[det(D=+ma)]Nf()forNfdegenerateavorsofmassm.Itisimportanttonoticethatmisthebarequarkmassandaninputparametricmassforthenumericalsimulation.Wewillletm=M0(a)representthephysicalvalueforthisbarelatticequarkmassforagivenvalueofthelatticespacinga..BareQuarkMassontheLatticeFromthenumericalresults[],at=:,Ma=0:0()+:()ma(=:)Ma=0:()+:()ma(=0:0)MNa=0:()+:0(:)ma(=:)fa=0:0()()Wecanchooseourconventionsinthefollowingway.ThelatticeQCDscaleisobtainedsuchthatMhasthephysicalvalueinthelimitofm!0.Thebarequarkmass,M0(a)isobtainedsuchthattheratio,MMhasthephysicalvalueinthelimitofm!M0(a).At=:,a=M0:=:GeVM0=0::aMM=:MeV()

SinceMN=M=:inthecalculationofRef. [],wewillgetdierentvaluesof=aandM0(a)ifwechooseMNtosetthescale;a=MN0:=:00GeVM0=0::aMMN=:MeV()Wemayalsochooseftosetthescale=a.a=f0:0=:GeV()MassRenormalizationintheContinuumThemostcommonrenormlizationschemeincontinuumQCDistheMSschemeinFeynmanngauge.Usingdimensionalregularization,theone-loopself-energycontributioncanbecalculatedasfollows:(p;M0;g0;)=g0R(Nc)Zddk()di(p=+k=)+M0k=ip=+M0()andd=;whereM0andg0arethebarequarkmassandcouplingrespectively.Therenormalizedcouplingsarerelatedasfollows:g0== msexp(E=)p!=ZmsggmsZmsg=gmsNfThequantitiesinEq.

()aregivenbyR(Nc)=NcNc=gmsR(Nc) Z0dx(x)ln[ms]! ()

=gmsR(Nc) Z0dxln[ms]! ()=x(x)p+xM0:(0)Therenormalizedquarkpropagator,Gisrelatedtothebarequarkpropaga-tor,G0asfollows:G(p;Mms;gms;ms)=Zms G0(p;M0;g0;)jM0=ZmsmMms;g0==msZmsggms()whereG0(p;M0;g0;)=ip=+M0+(p;M0;g0;)()IntheMSscheme[],therenormalizationconstantsZms andZmsmaregiventothelowestorderingmsbyZms =gmsR(Nc)Zmsm=gmsR(Nc);sotherenormalizedquarkpropagatorforNc=isG(p;Mms;gms;ms)= gms"Z0dx(x)ln(ms)#!

ip=+Mms"gms Z0dx(+x)ln(ms)+!#! ()Inperturbationtheory,weinterpretthelocationofthepoleintherenor-malizedquark-propagatorasthephysicalmassoftheparticle,regardlessoftherenormalizationscheme[0].Soifwedenethepolelocationasp=Mphy,Mphyshouldsatisfythefollowingcondition.Mphy=Mms"gms Z0dx(+x)ln(ms)+!#p=Mphy()Sincethepolelocationisindependentoftherenormalizationscheme,itwillbeimportantforconnectingthecontinuumandlatticeschemesbelow.

MassRenormalizationontheLatticeSincetheultra-violetdivergencesarealreadyregularizedbythelattice,onlyasubtractionprescriptionisneededforrenormalization.OneofthemostimportantcharacteristicscommontobothMSandMSschemesisthattherenormalizationconstantsareindependentofanydimensionfulla-grangianparameterssuchasthemass[].Wewillfollowasimilarprocedureindeningarenormalizationschemeonthelattice,choosingrenormalizationconstantsinthelattice-regularizationformalismthatareindependentofanydimensionfullagrangianparameterssuchasthemass..RenormalizationPrescriptionInordertoimplementtheminimalsubtractionideainthelatticeregu-larizedtheory[,],onlydivergenttermssuchas(ln[aL]Constant)iwithinwillbesubtractedconsistentlyinthelatticeregularizedFeynmanndiagramsofgnLorder.Thearbitraryconstantwillbechoseninaphysicallyreasonableway,asisdescribedindetaillater.Insuchaminimally-subtractedtheory,itiswellknownthattherenormalizationgroupfunctions,(g)andm(g)areindependentofthecovariantgaugechoice[].Lettherenormalizationconstants,ZLg,ZLmandZL bedenedasfollows[]:g0(a)=ZLg(t;gL(L))gL(L)()M0(a)=ZLm(t;gL(L))ML(L)() 0(a)=qZL (t;gL(L)) L(L)()wheret=ln(aL)andListhelatticerenormalizationscaleintroducedbyoursubtraction.Forthecouplingconstantrenormalization,wechooseaprescriptionsothatL,givenbytheconventionalformula[,,]L=aexp 0g0(a)! [0g0(a)]0f+O(g0(a))g;canbeexpressedbytheidenticalformulawheng0(a)isreplacedbygL(L).

ThisrequiresthatourchoiceofZLgsatisfy:ZLg(t=0;gL(L=a))=:()Eq.()insuresthattherenormalizedcoupling,gL(L)andthebarecoupling,g0(a)agreeforL==aL.Bychoosingtheprescriptionasabove,wehavedenedLintheconventionalway[,,].HavingmadeachoiceforLasabove,weknowthattheratio,L=msisequaltoL=msinordertokeepthecouplingcommoninbothrenor-malizationschemes.Inthissense,arenormalizationscaleonthelattice,L==aGeV(Eq.(),Eq.()andEq. ())correspondstoaboutms0GeV[]forthe=:numericalsimulations[,,].ThenaturalquestioniswhatchoiceofLisbesttodoperturbation.Experimentsuggeststhattheperturbativeexpansionconvergesmostrapidlyifwechoosetherenormalizationscaleequaltotheenergy-momentumoftheactualphysicalprocessintheMSschemeratherthaninotherschemes(ms=ps[0]).Henceweknowthatms=physicalenergy-momentumintheMSschemeisthebesttodoperturbation[0].Whenworkingwiththisquarkmassparameter,M0(a)initiallydenedatthelatticescalea,itisreasonabletousems==atosetthephysicalenergyscale.SomsGeVischosen,whichcorrespondstoL0MeVonthelattice.Thepointisthatitisbettertochoosems==athanL==ainordertoimprovetheconvergenceoftheperturbationseries.Forthemassandwavefunctionrenormalizationconstants,wewillleaveourprescriptionquiteexiblerequiringonly:ZLm(t=C;gL)=;ZL (t=C;gL)=()wheretheuncertainties,nowtransferredtoCandCwillbedeterminedlaterinSection..One-LoopSelf-EnergyandMassRenormalizationTheFeynmannrulesandgauge-xing(Feynmangauge)weusearecon-sistentwithRef.

[,,].OnlythetwoFeynmanndiagramsinFigurecancontributetotheone-loopself-energy,asfollows:(p;M0(a);g0;a)

=g0aR(Nc)XZ+dk()(+cos(ap+k))Pisin(ap+k)+aM0Psin(k)g0aR(Nc)Xisin(ap)Z+dk()Psin(k)=XipL+M0(a)L(0)HereNcisthenumberofcolorsandR(Nc)isgiveninSection.ThearethenormalEuclideanDiracmatricesandtheexpressioninEq.(0)isindependentofcolorandavor.ItisthesameforNf=asthepropagatorderivedinRef.[].LandLinEq. (0)areL=g0R(Nc) ln(aL)Z0dx(x)ln"L#!+O(a)()L=g0R(Nc) ln(aL)Z0dxln"L#!+O(a)()and=0:0;=0:[]:Intheaboveone-loopcalculation,itisassumedthat=ap.AsinEq.()andEq.

(),therenormalizedpropagator,GLisrelatedtothebarepropagator,G0,asfollows:GL(p;ML;gL;L)=ZL G0(p;M0;g0;a)jM0=ZLmML;g0=ZLggL()whereG0(p;M0(a);g0;a)=Xip+M0(a)+(p;M0(a);g0;a)()Fromthepresciptionsinsection.,wecandeterminetherenormalizationconstantsuptogLorder,asfollows:ZL =+gLR(Nc)ln(aL)+CZLm=+gLR(Nc)ln(aL)+C;()sothattherenormalizedquarkpropagatorforNc=isGL(p;ML;gL;L)=(+gL +CZ0dx(x)ln"L#!)

ip=+ML(+gL ()+CZ0dx(+x)ln"L#! )!Ifwesupposethatthepoleislocatedatp=Mphy,Mphyshouldsatisfythefollowingcondition:Mphy=ML(+gL ()+CZ0dx(+x)ln"L#!)()where=(p=Mphy):Eq.

()willbeusedlatertorelateMLandMmsdenedinthetwodierentschemes.ConnectionbetweentheLatticeandCon-tinuumParametersThebridgeconditionwillbeconstructed,exploitingthesimilarformtherenormalizationgroupequationstakeinthetwoschemesthroughone-looporderandchoosingthesamepolestructureforthequarkparpagatorinbothschemes..TheBridgeConditionIntheframeworkofamass-independentrenormalizationscheme,therenormalizationgroupdeterminestherenormalizedcouplingasafunctionofonly=andtherenormalizedmassastheproductoftherenormaliza-tiongroupinvariantmass,Mandafunctionofonly=[0],whereisanintegrationconstantchosenbystandardconvention.Thenthefollowingtwoconditionscanbechosen:Couplingrelation:Therenormalizedcouplingsarethesamefortheintroducedrenormalizationscales.Forexample,gL(L)=gms(ms)=g()LL=msms()becauseusingtheconventionaldenitionoftheparameterinbothschemeswehavegL(L)=f(LL)andgms=f(msms);

forthesamefunctionf()throughtwoloops.Massrelation:Therenormalizationgroupinvariantmassiscommontoanyrenormalizationschemethatisindependentofdimensionfulla-grangianparameters.Forexample,LL=msmsandcommonM()ML(L;M)=Mms(ms;M)()againbecauseML(L;M)=Mh(LL)andMms(ms;M)=Mh(msms);throughone-looporder.Inparticular,for(g)=0ggg,itisawell-knownfactthatthersttwocoecients(0and)areindependentoftherenormal-izationschemewhilealltheothercoecientsdependontherenormalizationscheme.Howeverform(g)=0g+g+,onlytherstcoecient0isindependentoftherenormalizationscheme[0].Intheframeworkofthebridgecondition,onlythescheme-independentpartsoftheandmfunc-tionsareconsideredandthescheme-dependentpartsignoredsothatf()andh()areuniversali.e.independentoftherenormalizationschemeuptotheorderofourpresentcalculation.TheconstantCintroducedinEq.()willbedeterminedonlyuptoone-looporder.Lmsisestimateduptoone-looporder[,,]andhasnohigherordercorrections[].Thelocationofthepoleinthetwo-pointGreen'sfunctionshouldbeindependentoftherenormalizationscheme[]anddeterminesthephysicalmassoftheparticle.Thelocationofthepoleinthequarkpropagatorisindependentoftherenormalizationscheme.TheconstantCcanbechosenexplicitlytoconformtotheconventionsofEq.(),bydemandingthatthepolelocationinthequarkpropagatoristhesameforbothMSandlatticerenormalizationscheme.ThereforefromEq.(),Eq.(),Eq.()andEq. (),ML(L;M)Mms(ms;M)=gms ()++C+ln"Lms#!=;soC=()+lnmsL;()

whereandaregiveninEq.().VariousvaluesofCforNf=0,,,,,and,aregiveninTable..RenormalizedCouplingConstantThereareanumberofwaystoobtaintherenormalizedcouplingcon-stant,g().OnceweknowtheQCDparameter,weknowthecouplingconstantg().Forexample,onemayusetheexperimentalvalueoftheparameterintheMSscheme[].Foragivenrenormalizationscalems,wecanobtaintherenormalizedcouplingconstant,gL(L)sincetheratio,L=MSiswell-known[,,,].Anothermethodusesthebarecouplingonthelatticeinordertoobtaintherenormalizedcoupling.Inthisarticle,thelattermethodischoseninordertodoeverythingconsistentlyintermsofthelatticeparametersandthelatticeobservables.FromEq. (),therenormalizationgroupequationcanbederived,asfollows: @@t+(gL)@@gL!gLZLg(t;gL)=0(0)where(g)=0gg::::and0=(Nf);=(0Nf)Eq.(0)canbesolved,withEq.()asaboundarycondition[],uptotheleadingorderingL,ZLg(t;gL)=q+t0gLgms(ms)=gL(L)=g0(a)t0g0(a)()wheret=ln(aL)=[ln(ams)+lnLms]()FromEq.

(),forvariousNfandms,gmscanbeobtained.0

.RenormalizedMassWehavealreadyobtainedanexpressionforthemassrenormalizationconstant,ZLminEq.().Thusthenaiverelationshipbetweentherenormal-izedquarkmassinMSandthebarequarkmassinthelatticeformulationfollowsfromEq.(),Eq.(),Eq.(),Eq.()andEq. ():Mms(ms)=ML(L)=ZLmM0(a)Mms(ms)="gLln(ams)+lnLms+C#M0(a)Mms(ms)="gms ln(ams)+()!#M0(a)()Asdescribedinsection.,itisreasonabletousems==aasconditionfortheoptimalperturbativeexpansion.FromEq.

(),Mms(=a)="gms ()!#M0(a)()Solvingtherenormalizationgroupequation,wecanobtaintherenormalizedquarkmassatanyotherscaleinthefollowingway.m(g)=ms@ln(Mms)@ms=0g+g+::::()where0=WecansolveEq.()suchthattherenormalizationscalerunsfrom=atoanarbitraryscale,ms.ThenthesolutiontoEq. ()isMms(ms)= gms(ms)gms(=a)!00Mms(=a)= gms(ms)gms(=a)!00"gms ()!#M0(a)()OtherauthorsquotevaluesforMms(ms)evaluatedatms=GeV.FromEq.

(),wecanobtaintherenormalizedquarkmassatthatGeVscalein

theMSscheme(Table),althoughweareenteringaregionwhereperturba-tionisunlikelytobeaccurate.Equation()isjustthelowestorderrelationshipandcanbeimprovedthroughleadinglogarithmicapproximation.FromEq. (),therenormaliza-tiongroupequationisderivedasfollows: @@t+(gL)@@gLm(gL)!ZLm(t;gL)=0()ThesolutionofEq.()withEq.()asaboundaryconditionis,uptotheleadingorderingL[],ZLm(t;gL)=h+0gL(tC)i00andfromEq.(),Eq.()andEq.(),ML:L:A:ms(ms)=ZLmM0(a)=[+0gms(tC)]00M0(a)()wheretherenormalizedcouplingandtcanbeobtainedfromEq.()andEq.().Asonecanseeintheabove,uptogmsorder,Eq.()isidenticaltoEq.().FromEq.()andEq.(),forvariousNf,for=:andforms=GeV,MmsandML:L:A:msaregiveninTable..RenormalizationGroupInvariantMassUsingtheconventionsofRef.[0]andexpandingthefunction,m(g),therenormalizationgroupinvariantmassisdened,uptoleadingorderingL,byM=ML(0gL)00:()FromEq.(),Eq.(),Eq.(),Eq.()andEq.

(),MisrelatedtoM0(a),asfollows:M= 0g0(a)C!00M0(a)(0)

Eq.(0)isindependentoftherenormalizationscale,Lorms,whichmeansthatMislesssensitivetotherenormalizationscheme.Itisreasonabletochoosetherenormalizationgroupinvariantmassasthephysicalquantitytoextractfromalatticecalculation,sincewedon'tneedtointroducetherenormalizationscale,Lorms.TherenormalizationgroupinvariantquarkmassisgiveninTable,fromEq.(0)..LightQuarkMassforNf=;=:Inthissection,itisexplicitlyexplainedhowtherenormalizedmassatthegivenscale(GeVinMSscheme)andtherenormalizationgroupinvariantmass[0]canbeestimatedinthemannerdescribedaboveusingthenumericalresultsofthelatticeQCDsimulation.Letusstartwiththerenormalizedcoupling,gms(ms)orgL(L).Weknowthat=:correspondstothebarecouplingconstant,g0(a)=:0sinceg0(a)==.FromRef.[]andTable,itisknownthatL=ms=0:0forNf=.SinceL=(L=ms)ms,weknowthatms=GeVcorrespondstoL=0:0GeV.FromEq.(),Eq.(),Eq()andRef.[],a=.GeVandt=ln(aL)=:0.ForNf=,0=,whichisdenedinEq.(0).Sinceweknowg0(a),tand0,therenormalizedcouplingconstantatascaleofGeVisfromEq.(),gms(ms=GeV)=gL(L=MeV)=::()Inasimilarway,wecanobtaintherenormalizedcouplingconstantatascaleof=a:gms(ms==a)=gL(L=MeV)=:0:()Nowletusgoaheadtoobtaintherenormalizedlightquarkmassatascaleof=a.Weknowgms(=a)and()fromEq.().ThebarequarkmassM0(a)=:MeVatmostfromEq.(),Eq.()andRef.[,].Sotherenormalizedlightquarkmassatascaleof=aisnaively,fromEq. (),Mms(=a)M0(a)=:Mms(=a)=ML(MeV)=:MeV:()

WecanalsoobtaintherenormalizedquarkmassatascaleofGeVfromEq.(),Eq.()andEq.().Mms(GeV)M0(a)=:Mms(GeV)=ML(MeV)=:MeV:()TherenormalizedlightquarkmassatascaleofGeV,improvedbyleadinglogarithmicapproximationis,fromEq.(),ML:L:A:ms(GeV)=ML:L:A:L(MeV)=:M0(a)=:MeV:()Nowletusobtaintherenormalizationgroupinvariantmass.KnowingL=ms=0:0fromTableandRef.[],wecandetermineC=:0fromEq.().Sinceweknow0,0,g0(a)andC,therenormalizationgroupinvariantmassis,fromEq.(0),M=:M0(a)=:MeV:()Eq.()isjustacruderesultofone-looprenormalization,whileEq.(0)ismorereliablesinceitincludesalltheleadinglogarithmiccontributionsandisindependentofalltherenormalizationscalesintroduced,whichmeanslessdependenceontheconventions.The0%dierencebetweenthenumbersinEq.()andEq. ()suggeststhesizeoftheomittedhigherordercorrections.MeanFieldTheoryLepageandMackenzieshowedthattadpolediagramsarethemainsourceofthelargedierencebetweenthebarelatticecoupling,g0(a)andtherenor-malizedcoupling,gMS(MS=a)[].Theysuggestameaneldmethodforremovingthedominanteectoftadpolediagrams.Hereweapplytheirmean-eldmethodtothestaggeredfermionformalism,inordertoimprovetheestimationoftherenormalizedquarkmassonthelatticenon-perturbatively..TadpoleImprovementfortheCouplingConstantThematchingoflatticeoperatorswithcontinuumoperatorsisbasedontheexpansionU(x)eiagA(x)!+iagA(x)()

whenthelatticespacingaissmall.ButhigherordertermsintheexpansionofUcontainaddtionalfactorsofagAandthecontractionofA(x)'switheachothergeneratesultra-violetdivergence(i.e./an)whichcanceltheaddionalpowersofa(i.e./an).Thesetermsareoftheorderofgn,notsuppressedbythepowersoflatticespacinga,andareverylarge.ThesearecalledtheQCDtadpolecontributions[].TheselargetadpolecontributioncausespoorperturbativeexpansioninlatticeQCD.LepageandMackenziesuggestedthemeaneldmethod[]inordertorenenaiveperturbativeexpansionbyremovingtadpolecontribu-tion.Theynoticedthatthevacuumexpectationvalueofthelinkmatrixissmallerthan.TheysuggestedthattheappropriateconnectionwiththecontinuumgaugeeldisU(x)!u0(+iagA(x))()whereu0representsthemeanvalueofthelink.Theychooseu0inagaugeinvariantway:u0Re()Otherchoicesofu0arepossibleandequallygood.ItistherescaledlinksUu0thatshouldbeexpandedaroundunity.ThestaggeredfermionactiongiveninEq. ()canberewrittenintermsof pu0S=Xx;(x)[ (x)Uy(x)u0 (x+a) (x+a)U(x)u0 (x)]Mu0aXx (x) (x)+Sgluon(0)Sgluon=XMFNcReTrUu0()whereMF=NcgMF;gMF=g0(a)Re()TheperturbativeexpansionforTrUgiveninRef.

[]isRe=g0(a)+O(g0(a))()

WecanrelategMFtog0perturbativelyasfollows:gMF=g0(a)Re=g0(a)[+g0(a)+O(g0(a))]()FromEq.(),Eq.()andEq.()wecanalsorelategMFtogms(ms)per-turbatively:gms(ms)=gMF[0gMFfln(ams)+lnLmsggMF+O(gMF)]()forNf=,gms(ms)=gMF[0gMFfln(ams):g+O(gMF)]()forms=a,gms(a)=gMF[0:00gMF+O(gMF)]()AsonecanseeinEq.(),gMFisextremelyclosetogms(a),wherethedierencebetweenthesetwocouplingconstantsisonlyafewpercentfromthestandpointofpertubativeexpansion.ThismeansthatbyrunningtherenormalizedcouplingintheMSschemeatascaleofa,onecanabsorbthelargetadpolecontributionintotherenormalizedcouplingconstant..TadpoleImprovementfortheQuarkMassAsthelatticespacing,agoestozero,theactiongiveninEq. (0)willbeS=Zx (x)[D+Mu0] (x)+Sgluon()Heretheclaimisthat isbettermatchedtothecontinuumquarkeldthan.Fromtheaboveaction,wenoticethatMMFM0(a)u0takestheplaceofthebarequarkmassonthelattice.Hencetherenormalizedquarkmasscanbeobtainedasfollows:Mms(ms)=Zm(ams;gms(ms))M0(a)=Zmu0MMF(a)~Zm((ams;gMF)MMF(a)()

where~ZmZmu0.FromEq.(),Eq.(),Eq.(),Eq.()andEq. (),theperturbativeexpansionof~Zmcanbeobtainedasfollows:~Zm((ams;gMF)=Zm(ams;gms(ms))u0="gms ln(ams)+()!#gms=gMF+O(gMF)gMF="gMF ln(ams)+()+!#(0)Thenumericaldataforu0isobtainedbytheColumbialatticegroupat=:forNf=[]:u0=Re=0:()u0=0::()Puttingeverythingaltogether,wecanobtaintherenormalizedquarkmass:MMF(a)=:MeV()~Zm(;gMF=:)=:()Mms(a)=:MeV()Thisnalresult(Eq.())isingoodagreementwiththatofEq.

()inthesection..Thisgoodagreementmeansthatinfactthroughperturbativeex-pansionwithrespecttogms(=a),onecanremovethetadpolecontributions,whichisverylargewhenonedoesthepertubativeexpansionwithrespecttothelatticebarecoupling,g0.LightQuarkMassfromQCDsumruleTherearevariousmethodstodeterminethelightquarkmassincontin-uumQCD[0,,,].Theactualdeterminationofthelightquarkmass

intheframeworkofQCDcanbedonereasonablythroughtheQCDsumruleformalismbyShifman,VoloshinandZakharov(S.V.Z. )[,].Inthatformalism,theystudythetwopointcorrelationfunctionofthedivergenceoftheaxialcurrent(isospinsector).Theydeterminetherunningquarkmassat.GeVscalewithNf=i.e.threedynamicalquarksmovingaround.IntheMSscheme,theychooserenormalizationscalemstobethesameastheenergy-momentumscaleofthephysicalprocess.Theirchoice,ms=:GeVmeansphysicallythatthecharmandbottomquarksaredecoupled[0]andthatonlythreelightquarkscontributetothedynamicsoftheQCD.Inotherwords,atms=:GeV,wecannotdecouplethestrangequarkfromthedynamicsoftheQCD.NowwehaveaproblemthatlatticeQCDissimulatednumericallywithdynamicalquarksbuttheQCDsumruleassumesdynamicalquarks.Fromtherecentexperimentaldata[],wehavethefollowingQCDparameter:ms(Nf=)=00MeVwheretheerrorispartlyacombinationofstatistialandsytematicerrorsandpartlyduetothescaleuncertainty.Weshouldusems(Nf=)atthescalebelowthecharmquarkmass.Usingms(Nf=)=00MeV,theQCDsumrulegivesthefollowinglightquarkmass[0].ForNf=,Mms(ms=GeV)=Mu+Md=::0MeVM=:0:0MeV;wherethelargervaluecorrespondstothesmallerms<0MeVandthesmallervaluecorrespondstothelargerms>00MeV.TheQCDsumrulestakeintoconsiderationtheresonancecontributionbutnotthecontinuumcontributiontotheimaginarypartofthetwopointcorrelationfunctionoftheaxialcurrentdivergence.Omittingthiscontin-uumcontributionfromthenon-perturbativelow-energyregioncanintroducearound0%errorstotheaboveexpectationvalueofthelightquarkmass[].Eventhoughthereisadierenceinthenumberofdynamicalquarks,thelatticeQCDexpectationvalueofthequarkmasswithNf=giveninSection.agreeswiththatoftheS.V.Z.QCDsumrulewithNf=intheabove.

Butinordertodotheexactcomparison,thelatticeQCDsimulationwiththreedynmicalquarks(threeseaquarks:oneofthemis00timesheavierthantheothertwolightseaquarks)isnecessarysincethestrangequarkcannotbedecoupledfromtheQCDdynamicsattheenergy-momentumscaleofGeV[0].ConclusionTheQCDdynamicalquarkmassisrenormalizedintwodierentrenor-malizationschemes(MSandlattice-regularizedminimalsubtraction).Thebridgeconditionsarechosentomakeconnectionbetweenthetwoschemes.TheratiosofthecontinuumrenormalizedquarkmasstothelatticebarequarkmassaregiveninTable.Ref.[]containsanearlierattempttocomputetherenormalizedquarkmass.Theirvalueof.MeVissomewhatdierentfromthatofthisarti-cle(.MeV).Fukugitaetal.usetheone-looprenormalizationandEq.(.)inRef.[]isthesameasEq.()exceptforthecouplingconstantthatap-pears.This%dierencecomesfromthedierentchoiceofthecoupling:thebarecouplingonthelatticeisusedinRef. []whiletherenormalizedcouplingisusedinthisarticle.Ithasbeenpointedoutthatthelatticebarecouplingconstantmaybeapoorchoiceasanexpansionparameterandthattheuseoftheimprovedcouplingconstantincludingrenormalizationduetogluontadpolecontributionsimprovesthereliabilityofthelatticeperturba-tiveexpansion[].UsingthemeaneldmethodsuggestedbyLepageandMackenzie,thetadpole-improvedrenormalizedquarkmassisobtained,whichisextremelyclosetothatobtainedbytheuseoftherenormalizedcouplingintheMSschemeatascale.Ithasbeenproposedinothercontexts[0]tousetherenormalizedcoupling(intheMSschemewiththerenormalizationscaleequaltothephysicalenergy-momentum)insteadofthebarecoupling.Attheleast,thedierencesbetweentheresultsobtainedhereandthoseofFukugitaetal.representthesizeoftheperturbativeerrors.TheQCDsumrulepredictsthattherenormalizedquarkmass(ataGeVscaleintheMSscheme)is::0MeVandthattherenormalizationgroupinvariantmassis:0:0MeVforNf=,with0%uncertainty[0,,].Therenormalizedquarkmass(ataGeVscaleintheMSscheme)obtainedfromthelatticeQCDsimulationis.MeVandthe

renormalizationgroupinvariantmass.MeVforNf=.Thedierencebetween-avordynamicsand-avordynamicswithoneofthethreeavorsmuchheaviermaybepresumedtobesosmallthatthecomparisonbetween-avorand-avordynamicsmaymakephysicalsense.TherearetwoothersourcesoferrorinthelatticeQCDsimulation:oneisnite-temperatureeectandtheothernite-volumeeect.Inthehadronmasscalculationson,andshowsapproximately0%eect[].Thesesmalleectsarecompletelyneglectedinthisarticle.Butthesystematicanalysisofnitesizeeectsshouldbedonetolookintophysicsonthelatticemoreprecisely.AcknowledgementsIamindebtedalottoProf.NormanH.Christ.Thisworkcouldnothavebeendonewithouthisconsistenthelpandencouragement.HelpfulconversationswithProf.R.Friedberg,Prof.RobertD.Mawhinney,Prof.A.MuellerandProf.V.P.Nairareacknowledgedwithgratitude.0

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NfLmsCMms(=a)M0Mms(GeV)M0ML:L:Ams(GeV)M0MM000.0-.....0.0-.....0.0-.00....0.0-.0....0.00-.0....0.000-....0.0.00-0....0.Table:Hereweuse=:,a=:GeV.Theratio,L=msisobtainedfromRef. [].

(a)(b)Figure:One-LoopContributiontotheSelf-Energy(a)gluonexchagedia-gram(b)gluonbubblediagram

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