Non-perturbative Renormalization in Lattice Field

arXiv:hep-lat/0011081v127Nov20001Non-perturbativeRenormalizationinLatticeFieldTheory∗StefanSint†aaUniversit`adiRoma“TorVergata”,DipartimentodiFisica,ViadellaRicercaScientifica1,I-00133Roma,ItalyIreviewthestrategieswhichhavebeendeveloppedinrecentyearstosolvethenon-perturbativerenormalizationprobleminlatticefieldtheories.Althoughthetechniquesaregeneral,thefocuswillbeonapplicationstolatticeQCD.Idiscussthemomentumsubtractionandfinitevolumeschemes,andtheirapplicationtoscaledependentrenormalizations.Theproblemoffiniterenormalizationsisillustratedwiththeexampleofexplicitchiralsymmetrybreaking,andIgiveashortstatusreportconcerningSymanzik’simprovementprogrammetoO(a).1.INTRODUCTIONAQuantumFieldTheoryisrenormalizedbydefiningtheinfinitecutofforcontinuumlimitoftheregularizedtheory.Theprocedureisfamil-iarinperturbationtheory,butismoregenerallyapplicable.Inparticular,therenormalizationofasymptoticallyfreetheoriesatlowscalesrequiresagenuinelynon-perturbativeapproach.Unfortu-natelythereexistveryfewanalyticaltools,andnumericalsimulationsofthelatticeregularizedtheoryareoftentheonlymethodtoobtainquan-titativeresults.Oneisthusledtodiscussrenormalizationofthelatticefieldtheoryandtotakethecontinuumlimitbasedonnumericaldata.Ingeneralthisisonlypossiblebymakingassumptionsaboutthecontinuumapproach.MotivatedbySymanzik’sanalysisofthecutoffdependenceinperturba-tiontheory,oneusuallyassumesthatthecon-tiuumlimitisreachedwithpowercorrectionsinthelatticespacinga,possiblymodifiedbyloga-rithms.Inthiscontext“unexpectedresults”havebeenpresentedrecentlybyHasenfratzandNie-dermayer[1,2].TheyfindthatthecontinuumapproachofarenormalizedcouplingintheO(3)non-linearsigmamodeliscompatiblewithbeinglinearina,ratherthanquadratic,asonewouldexpectforabosonicmodel.Whilethestatistical∗basedonaplenarytalkpresentedattheInternationalSymposiumonLatticeFieldTheory,August17–22,2000,Bangalore,India.†addressafterOctober1,2000:CERN,TheoryDivision,CH-1211Geneva23,Switzerlandprecisionofthedataisquiteimpressive,itseemsprematuretodrawconclusions.Wealsonotethattherearemanyresultsin(quenched)QCDwhicharewellcompatiblewithexpectations,andsomeofthemwillbepresentedbelow.Finally,Iwillas-sumethe“‘standardwisdom”concerningasymp-toticfreedomandtheoperatorproductexpan-sion(OPE)tobecorrect.Whilethishasneverbeenestablishedbeyondperturbationtheory,anumericalcheckoftheOPEintheO(3)modelhasrecentlybeenpresentedinref.[3].Foranun-conventionalpointofviewregardingasymptoticfreedomIrefertoref.[4].DespitethemoregeneraltitleIwillfocusonlatticeQCD.Forone,non-perturbativerenormal-izationtechniquesinQCDhavereachedamaturestage,sothatsystematicerrorscanbediscussed.Second,QCDisatypicalexample,andthetech-niquescarryoveressentiallyunchangedtoothertheories.Thisarticleisorganizedasfollows.Insect.2,Isketchthenon-perturbativerenormalizationprocedureforQCD.Thenthetwoclassesofschemesarereviewedwhichhavebeenproposedtosolvetheproblemofscale-dependentrenor-malization,namelythemomentumsubtractionschemes(RI/MOM,sect.3)andfinitevolumeschemesderivedfromtheSchr¨odingerfunctional(sect.4).Thisisfollowedbyadiscussionoffiniterenormalizations(sect.5),andO(a)improvement(sect.6).Iconcludewitharemarkconcerningpowerdivergences.22.NON-PERTURBATIVERENORMALIZATIONOFQCD2.1.Determinationoffundamentalparam-etersThefreeparametersoftheQCDactionarethebarecouplingandthebarequarkmassparame-ters.Atlowenergiesthetheoryisusuallyrenor-malizedinahadronicscheme,i.e.onechoosesacorrespondingnumberofhadronicobservableswhicharekeptfixedasthecontinuumlimitistaken.Oncethetheoryhasbeenrenormalizedinthisway,nofreedomisleftandtherenormalizedpa-rametersinanyotherschemearepredictedbythetheory.Ofparticularinterestistherelationtotherenormalizedcouplingandquarkmassesinaperturbativescheme.ThisisequivalenttoadeterminationoftheΛ-parameterandtherenor-malizationgroupinvariantquarkmasses.For,giventherunningcoupling¯gandquarkmassesmiatscaleμ,onefindstheexactrelationsΛ=μ(b0¯g2)−b1/2b20expn−12b0¯g2o(1)×exp−Z¯g0dxh1β(x)+1b0x3−b1b20xi,Mi=mi(2b0¯g2)−d0/2b0×exp−Z¯g0dxhτ(x)β(x)−d0b0xi,(2)whereilabelsthequarkflavours.Here,βandτarerenormalizationgroupfunctionswithasymp-toticexpansions,β(g)∼−b0g3−b1g5+...,(3)τ(g)∼−d0g2−d1g4+....(4)Notethattherenormalizationgroupinvariantquarkmassesareschemeindependent,whereastheLambdaparameterchangesaccordingtoΛ′=Λexp{c(1)/b0}.(5)Here,c(1)istheone-loopcoefficientrelatingtherenormalizedcouplingconstantsg′=g+c(1)g3+O(g5).ΛintheMSschemeandMiarereferredtoasthefundamentalparametersofQCD.Inordertorelatethehadronicschemetotheseparametersonemayintroduceaninter-mediaterenormalizationschemeinvolvingquan-titieswhichcanbeevaluatedbothinperturba-tiontheoryandnon-perturbatively.Anobviouspossibilityistotakearenormalizedcouplingandrenormalizedquarkmasseswhicharedefinedbe-yondperturbationtheory.Insuchaschemealsotherenormalizationgroupfunctionsβandτaredefinednon-perturbatively,sothattheformulae(1,2)canbeappliedatanyscaleμ,oncethematchingtothehadronicschemehasbeenper-formed.2.2.Renormalizationofcompositeopera-torsIntheStandardModelcompositeoperatorsof-tenarisewhentheOPEisusedt