Connections Between Lattice Gauge Theory and Chira

arXiv:hep-ph/9710468v212Nov1997ConnectionsBetweenLatticeGaugeTheoryandChiralPerturbationTheoryMaartenGoltermanDepartmentofPhysics,WashingtonUniversity,St.Louis,MO63130,USAe-mail:[e-mail],IaddressthecomparisonbetweenresultsfromlatticeQCDcomputationsandChiralPerturbationTheory(ChPT).IbrieflydiscusshowChPTcanbeadaptedtothemuch-usedquenchedapproximationandwhatittellsusaboutthespecialroleoftheη′inthequenchedtheory.Ithenreviewlatticeresultsforsomequantities(thepionmass,pionscatteringlengthsandtheK+→π+π0matrixelement)andwhatquenchedChPThastosayaboutthem.1IntroductionChiralPerturbationTheory(ChPT)(Weinberg(1979),GasserandLeutwyler(1985),Gasser(1997))givesusinformationaboutthefunctionaldependenceofquantitiesassociatedwithlow-energyGoldstoneBoson(GB)physicsonthelight-quarkmasses.ExamplesofsuchquantitiesaretheGBmasses,decaycon-stantsandscatteringamplitudes.Atanygivenorder,theserelationsbetweenphysicalquantitiesandquarkmassesinvolveafinitenumberofconstants(the“low-energyconstants(LECs),”whichcannotbedeterminedfromChPTalone.Therefore,ChPTispredictive(atanygivenorder)ifweconsideranumberofphysicalquantitieslargerthanthenumberofLECsneededatthatorder.TheseLECscaninprinciplebedeterminedeitherbycomparisonwithexperimentaldata,orbyatheoreticalcalculationfromtheunderlyingtheory,theStandardModel.Thestrong-interactionpartofsuchcalculationsisnonperturbative,andthisiswhereLatticeQCD(LQCD)comesin.InLQCD,physicalquantities(orre-latedquantities,suchasweakmatrixelements)arecomputedfromfirstprinci-plesasafunctionofthequarkmasses.ByfittingtheresultswiththerelationspredictedbyChPT,onecanthen,inprinciple,determinetheLECs.Thisisverysimilartodeterminingtheseconstantsfromexperimentaldata,withtheaddedadvantagethatinLQCDonecanvarythequarkmasses.Ofcourse,ingeneral,LQCDresultswillneedtobeofahighprecisioninor-dertoextracttheO(p4)LECs,because,ingeneral,theyshowupintheone-loopcorrectionstothetree-levelpredictionsfromChPT.Thismeansgoodcontroloverbothstatisticalandsystematicerrorsinthelatticecomputations.Forin-stance,weneedthevolumeL3inlatticeunitstobelargeinordertouseasmalllatticespacinga,whilekeepingthephysicalvolumelargeenoughtofitthehadronicsystemofinterest.Itisimportanttokeepinmindthatlatticeresults2MaartenGoltermanneedtobeextrapolatedtothecontinuumlimitbeforetheycanbecomparedwithChPT.Also,forChPTtobevalid,GBmassesneedtobesmallcomparedtothechiralsymmetrybreakingscale.Again,thisleadstotherequirementoflargeenoughvolumes.(FinitevolumeeffectscanbestudiedwithinChPT(GasserandLeutwyler(1987));wewillseesomeexamplesinthefollowing.However,LQCDpractitionersarenotonlyinterestedinthecomparisonwithChPT!)Thetempo-ralextentofthelatticeneedstobelargeenoughsothattheloweststateinanychannelcanbereliablyprojectedoutbytakingthelarge-timelimitofeuclideancorrelationfunctions.ThishasledtothewidespreaduseoftheQuenchedApproximation(QA)(Parisietal.(1981),Weingarten(1982)),inwhichthequarkdeterminantisomittedfromtheLQCDpath-integral.Thisisequivalenttoomittingallcontri-butionstocorrelationfunctionsthatinvolvesea-quarkloops(valence-quarkloopsresultingfromcontractionsofquarksincompositeoperatorsofcourseremainpresent,astheyhavenothingtodowiththedeterminant).Thereasonisthat,ifthefermiondeterminantisincluded,thenecessarycomputertimeincreasesbyordersofmagnitudeifwekeepthephysicalparameters(latticespacing,volume,quarkmasses)thesame.QuenchedQCDisadifferenttheoryfromQCD(aswewillsee,it’snotevenahealthytheory!),andthereforethepredictionsofChPTdonotapplytolatticeresultsobtainedintheQA.Fortunately,itturnsoutthataquenchedversionofChPT(QChPT)canbedevelopedsystematically,anditisinthisframeworkthatwecancomparequenchedLQCDwithChPT.Unfortunately,thisalsoimpliesthattheLECspredictedbyquenchedQCDarenotnecessarilyequaltothoseoffullQCD,andassuch,knowledgeofthemisofsomewhatlimitedvalue.Inthistalk,IwilldescribehowQChPTworks,anddiscusssomeexamplesofthecomparisonoflatticeresultswithChPT.Ishouldnoterightawaythat,aswewillsee,currentlylatticedataarenotpreciseenoughyettounambiguouslyseeChPTone-loopeffects.Beforewegetintothis,letmeendthisintroductionwithafewremarks.First,recently,morelatticeresultswith“dynamicalfermions”(i.e.includingsea-quarkloops)arebecomingavailable.However,sincetheoverheadincomputingthefermiondeterminantissolarge,theseresultsareoftenforoneortwovaluesofthesea-quarkmass,whilemanyvaluesofthevalencequarkmassesareconsidered.Themethodsdescribedinthistalkcaneasilybeadaptedtothese“partiallyquenched”theorieswithseaquarkswithamassthatdiffersfromthatofthevalencequarks(BernardandGolterman(1994),Sharpe(1997b)).Second,thestudyofQChPTshedsmuchlightonthenatureoftheQA,and,assuch,hasbeenveryhelpfulforLQCD.2QuenchedChPT,theη′,andthePionMassEuclideanquenchedQCDcanbedefinedbytakingtheordinaryQCDlagrangian,andaddinganewsetofquarks{˜qi}toitwhichcarryone-by-oneexactlytheLatticeGaugeTheoryandChPT3samequantumnumbersasthenormalquarks{qi},butwhichhaveoppositestatistics(Morel(1987)):L=qi(/D+mi)qi+˜qi(/D+mi)˜qi,i=u,d,s.(1)Wewillrefertothesenewwrong-statisticsquarksas“ghost-quarks.”Thereasonforth