Perfect Lattice Perturbation Theory A Study of the

arXiv:hep-lat/9711054v128Nov1997PerfectLatticePerturbationTheory:AStudyoftheAnharmonicOscillatorW.BietenholzaandT.StruckmannbaHLRZc/oForschungszentrumJ¨ulich52425J¨ulich,GermanybPhysicsDepartmentUniversityofWuppertalD-42097Wuppertal,GermanyPreprintHLRZ1997-67,WUB97-35Asanapplicationofperfectlatticeperturbationtheory,weconstructanO(λ)perfectlatticeactionfortheanharmonicoscillatoranalyticallyinmomentumspace.Incoordinatespaceweobtainasetof2-spinand4-spincouplings∝λ,whichweevaluateforvariousmasses.Thesecouplingsneverinvolvevariablesseparatedbymorethantwolatticespacings.TheO(λ)perfectactionissimulatedandcomparedtothestandardac-tion.WediscusstheimprovementforthefirsttwoenergygapsΔE1,ΔE2andforthescalingquantityΔE2/ΔE1indifferentregimesoftheinteractionparameter,andofthecorrelationlength.11IntroductionTheonlynon-perturbativeaccesstocomplicated4dquantumfieldtheories,suchasQCD,whichprovedsuccessful,areMonteCarlosimulationsonthelattice.TheynecessarilytakeplaceatafinitelatticespacingaandinafinitesizeL.Inordertorevealinformationaboutcontinuumphysicsinaninfinitevolume,wehavetorequireaξL,whereξisthecorrelationlength.Inparticularthefinitenessofξ/acausesserioussystematicerrorsinpracticalsimulations.Itisnowveryfashionabletofightsuchartifactsbyusing“improvedlatticeactions”[1].Thesearediscretizationsofthecontinuumaction,whicharesupposedtodisplaythecorrectscalingbehaviordowntoamuchshortercorrelationlengthinlatticeunits,thanitisthecaseforthestandardlatticeaction.Intheliterature,therearemainlytwostrategiestoconstructimprovedlatticeactions,inparticularforQCD.ThefirstoneiscalledSymanzik’spro-gram[2].Onetriestoeliminatethelatticespacingartifactsorderbyorderina–similartotheRungeandKuttaprocedureforthenumericalsolutionofordinarydifferentialequations.Thisisachievedbyaddingirrelevantoper-ators.Ontheclassicallevel,andintheframeworkofon-shellimprovement[3],thestandardWilsonactionforQCDcouldbeimprovedtoO(a)ana-lyticallybyaddingtheso-calledcloverterm[4].Onthequantumlevel,thecoefficientofthistermgetsrenormalized,andthequantumcorrectionwasfirstestimatednumericallybyameanfieldapproach[5].ThecompleteO(a)improvementwasfinallydeterminedbytheALPHAcollaborationbasedonextensivesimulations[6].However,itseemshardlyfeasibletocarryonthisprogrambeyondO(a).Thealternativemethodusesrenormalizationgroupconceptstoconstructquasi-perfectactions.Theseareapproximationstoperfectactions,i.e.toactionswhicharecompletelyfreeofcutoffartifacts[7].AsafundamentaldifferencefromSymanzik’sprogram,thismethodisnon-perturbativewithrespecttoa.Asafirststep,thisprogramcanberealizedperturbatively(intheinteraction),whichyieldsanalyticexpressionsfortheperfectquark-gluonand3-gluonvertexfunctions[8,9,10].1ThusoneeliminatesallartifactsofO(an)andO(gan),suchthattheremainingartifactsareofO(g2a)andbeyond(gisthegaugecoupling).ThisisopposedtotheactionofRef.[6],1PerturbativelyperfectactionshavealsobestudiedfortheSchwingermodel,[11,12].2whichisfreeofartifactsinO(gna),butplaguedforinstancebysystematicerrorsinO(a2).Anextensionofthisprogramistheconstructionof“classicallyperfectactions”[13].Thisapproach,whichisdesignedparticularlyforasymptoti-callyfreemodels,isnon-perturbativealsowithrespecttothecouplingg.Usingamultigridprocedure,oneidentifiesthefixedpointactionofanrenor-malizationgrouptransformation.Thiscanbedonesolelybyminimization–thefunctionalintegralreducestoaclassicalfieldtheoryproblem–andthefixedpointactionthenservesasanapproximativelyperfect(“classicallyperfect”)actionatfinitecorrelationlengthtoo.Inasequenceoftoymodels,itturnedoutthatclassicallyperfectactionsareexcellentapproximationsto(quantum)perfectactions,inthesensethattheydrasticallysuppresslatticespacingartifacts.TheimprovementachievedinthiswaygoesfarbeyondfirstorderSymanzikimprovement.Thishasbeenobservedforthe2dO(3)andCP(3)model[13,14],theSchwingermodel[15]andthe1dXYmodel[16].Inprinciplethatprogramcanbeextendedalsobeyondclassicalperfection,ifoneperformse.g.onerealspaceMCRGstepatfinitecorrelationlength,startingfromaclassicallyperfectaction.Theconstruction,whichisnon-perturbativeinaanding,ispresum-ablytheclimaxoftheimprovementprogram.However,inperfectandalsoinclassicallyperfectactionsthecouplingstendtoinvolveinfinitedistances,andwecanatbestachievelocalityinthesenseoftheirexponentialdecay.Forpracticalpurposesatruncationisneeded,whichdoessomeharmtothequalityoftheimprovement.Thisisthemainreasonwhythesecond,moresophisticated,improvementprogramcouldnotbeappliedyetinasatisfac-torywaytoQCD.Herewefocusontheperturbativelyperfectaction.Ithaspotentialappli-cationswithtworespects:itcaneitherbeuseddirectly,orasastartingpointofthenon-perturbativemultigridimprovement[9].Adirectapplicationofatruncatedperfectquark-gluonvertexfunction–togetherwithtruncatedperfectfreequarks–toheavyquarksispresentlyunderinvestigation.Pre-liminaryresultsforthecharmoniumspectrumaregiveninRef.[17].Thepurposeofthispaperistotestspecificallysuchadirectapplicationinaverysimplesituation.Ourmodelisthe1dλφ4model,oranharmonicoscillator.Asatoymodel,theanharmonicoscillatorhasanumberofvirtues:we3canachieveane