Connections Between Lattice Gauge Theory and Chiral Perturbation Theory

In this talk, I address the comparison between results from lattice QCD computations and Chiral Perturbation Theory (ChPT). I briefly discuss how ChPT can be adapted to the much-used quenched approximation and what it tells us about the special role of the

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aConnectionsBetweenLatticeGaugeTheoryandChiralPerturbationTheoryMaartenGoltermanDepartmentofPhysics,WashingtonUniversity,St.Louis,MO63130,USAe-mail:maarten@aapje.wustl.eduAbstract.Inthistalk,IaddressthecomparisonbetweenresultsfromlatticeQCDcomputationsandChiralPerturbationTheory(ChPT).Ibrie ydiscusshowChPTcanbeadaptedtothemuch-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,theserelationsbetweenphysicalquantitiesandquarkmassesinvolvea nitenumberofconstants(the“low-energyconstants(LECs),”whichcannotbedeterminedfromChPTalone.Therefore,ChPTispredictive(atanygivenorder)ifweconsideranumberofphysicalquantitieslargerthanthenumberofLECsneededatthatorder.TheseLECscaninprinciplebedeterminedeitherbycomparisonwithexperimentaldata,orbyatheoreticalcalculationfromtheunderlyingtheory,theStandardModel.Thestrong-interactionpartofsuchcalculationsisnonperturbative,andthisiswhereLatticeQCD(LQCD)comesin.InLQCD,physicalquantities(orre-latedquantities,suchasweakmatrixelements)arecomputedfrom rstprinci-

plesasafunctionofthequarkmasses.By ttingtheresultswiththerelationspredictedbyChPT,onecanthen,inprinciple,determinetheLECs.Thisisverysimilartodeterminingtheseconstantsfromexperimentaldata,withtheaddedadvantagethatinLQCDonecanvarythequarkmasses.

Ofcourse,ingeneral,LQCDresultswillneedtobeofahighprecisioninor-dertoextracttheO(p4)LECs,because,ingeneral,theyshowupintheone-loopcorrectionstothetree-levelpredictionsfromChPT.Thismeansgoodcontroloverbothstatisticalandsystematicerrorsinthelatticecomputations.Forin-stance,weneedthevolumeL3inlatticeunitstobelargeinordertouseasmalllatticespacinga,whilekeepingthephysicalvolumelargeenoughto tthehadronicsystemofinterest.Itisimportanttokeepinmindthatlatticeresults