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DESY-0MayANewWaytoSettheEnergyScaleinLatticeGaugeTheoriesanditsApplicationtotheStaticForceandsinSU()Yang{MillsTheoryR.SommerDeutschesElektronen-SynchrotronDESY,HamburgAbstractWeintroduceahadronicscaleR0throughtheforceF(r)betweenstaticquarksatintermediatedistancesr.ThedenitionF(R0)R0=:amountstoR0'0:fminphenomenologicalpotentialmodels.SinceR0iswelldenedandcanbecalculatedaccuratelyinaMonteCarlosimulation,itisanidealquantitytosetthescale.InSU()puregaugetheory,weusenewdata(andR0tosetthescale)toextrapolateF(r)tothecontinuumlimitfordistancesr=0:fmtor=:fm.ThroughR0wedeterminetheenergyscaleintherecentlycalculatedrunningcoupling,whichusedtherecursivenitesizetechniquetoreachlargeenergyscales.Alsointhiscase,thelatticedatacanbeextrapolatedtothecontinuumlimit.TheuseofoneloopSymanzikimprovementisseentoreducethelatticespacingdependencesignicantly..IntroductionInthelimitwhenthemassofaquarkhbecomeslargecomparedtotypicalQCD{scales(suchastheproton'smass),boundstateshhareexpectedtobedescribedbyaneectivenonrelativisticSchrodingerequation[].Thenonrelativisticpotentialisgivenbytheenergyofstaticfundamentalchargesseparatedadistancer.Intherealworld,suchadescriptionshouldbeapproximatelyvalidforthebbandmaybeevenfortheccspectra.Infact,boththesespectracanbedescribedbyoneeectivepotential.Since(asoneknowsaposteriorifromasuccessfulmodel)thebbstateshaverms{radiiof0.fmto0.fmandtheccstateshaveasizeof0.fmto.0fm,thespectradeterminetheeectivepotentialintherangeofr'0:fmtor'fm.AlthoughwemustrememberthattherelationshipbetweenthestaticQCDpotentialandtheeectivepotentialusedinphenomenologyisnotwellunderstood,thisdiscussionsuggeststhatthedistance

rangewherewehavethebestinformationontheforceF(r)betweenstaticquarksisatdistancesofaround0.fm.Inlatticegaugetheorycalculations,weneedtoxonedimensionfulquantityinordertosettheoverallscale.Inthepuregaugetheory,thisisusuallydonethroughthestringtension,denedasK=limr!F(r).Thelimitingprocedureisnoteasytodobecausethestatisticalandsystematicerrorsontheforceincreasewiththedistance.Itisthereforesuperiortousetheforceatintermediatedistancestodeterminethescale:onecalculatesR(c)suchthatF(R(c))R(c)=c.Becauseoftheabovementionedarguments,wechosec=:,whichcorrespondstoR0R(:)=0:fmintheCornell[]andtheRichardson[]modelandverysimilarvaluesintheothersuccessfullmodels.ThestatisticalandthesystematicerrorsofR0arequitemoderate.Thisopensthepossibilityofextrapolationstothecontinuumlimit.Insection,wedeterminetheforceinthecontinuumlimitoftheSU()YangMillstheory:thedimensionlessquantityH(x)=F(r)rjr=xR0iscalculatedatvedierentvaluesofthelatticespacingandthenextrapolatedtolatticespacingzero.Itiswelldeterminedjustaboutinthesamerangeasthephenomenologicalpotential.WecompareH(x)tothephenomenologicalpotential.theshortdistanceperturbativeformandtheexpectedlargedistanceform.Thelatterisgivenbyaneectivebosonicstringtheoryincludingitsuniversalleadingcorrections.Ourmainmotivation,however,wastosettheenergyscaleinthecalculationoftherunningcoupling.Inref.[]aphysicalcouplingwasdenedforthepuregaugetheoryinnitevolume,thatdependsonlyononescalewhichisthesizeoftheboxL.Onecancalculatethestepscalingfunctionwhichgivesthechangeofthecouplingg(L)whenthescaleLischangedbyafactors.Thiscomputationwasdoneinref.[]andthelatticeresultscouldbeextrapolatedtothecontinuum.Asaresultg(L)isknownforL=L=,0.00(),0.(),0. ()and0.00().HereLisdenedbyg(L)=:.ThelaststepistocalculatetherelationofLtosomephysicalscale.Tothisendwepreviously[]usedthestringtensionbutwecouldonlygiveacrudeestimateofLpK.InsectionwewillinsteadcalculateL=R0andextrapolateittothecontinuum.Inthepartofthecalculationthatinvolvesg(L),weusedtheSymanzikimprovedaction[]wheretheO(a=L)latticeartifactsareremovedtoonelooporder.Comparingthistotheresultswithoutimprovement,weobserveaverysignicantreductionofthelatticeartifacts.

.R0{aPreciseLowEnergyScaleInordertosetthescaleinalatticegaugetheorycalculation,weneedtodetermineaphysicaldimensionfulquantity.Astotoday'sknowledge,theforceF(r)betweenstaticquarksisthequantitywhichcanbecalculatedmostprecisely.Inordertoavoidanextrapolationinthedistancer,wedeneahadroniclength{scaleR(c)whichdependsonadimensionlessparametercthroughrF(r)jr=R(c)=c:(.)Asdiscussedintheintroduction,choosinge.g.c=:correspondstoR(:)R0'0:fm:(.)Theadvantagesofthischoiceare:R(c)isdenedprecisely,bothinthepuregaugetheoryandinthetheorywithdynamicalfermions.Inparticular,itdoesnotrefertothequenchedapproxima-tion.Incontrasttothestringtensionitdoesnotcontainresidualsystematicerrorsthatoriginatefromassumptionsonsubleadingtermsintheforceatintermediatedistances.Itcanbecalculatedwithgoodstatisticalprecision.Thereforeitsusetosetthescaleforotherphysicalobservableswillnotintroducesignicantadditionaluncertaintiesintotheobservablesofinterest.Thisisofspecialimportancewhenonewantstoextrapolatelatticeresultstothecontinuum.Examplesaregiveninsections.and..RelationshiptootherObservablesAsinthecaseofthestringtension,onehastousetheeectivepotentialinordertorelateR0(ormoregenerallyR(c))toexperimentalobservables.UsingtheRichardsonpotential,weobtainR0=0:fm.Soe.g.therelationshiptotheP{Ssplittingintheccsystemmandtotheproton'smassmpisgivenbymR0=:;mR()=0:(.)mpR0=:;mpR()=:(. )VerysimilarnumbersareobtainedwhenoneusesotherQCD-inspiredphenomenolog-icalpotentials,whereasthelogarithmicpotentialgivesvaluesthatarelowerbyabout0%.

OnceonecancalculateinfullQCDitwillbeinterestingtocheckeq.'s(.,.)andonecangetacertainmeasureoftheprecisionofthenonrelativisticdescription..NumericalDeterminationofR(c)Itisofvitalimportancethatthequantitythatoneusesasareferencescalecanbecomputedaccurately.WedemonstratethatthisisthecaseforR(c)inthefollowing.ThreestepshavetobediscussedinthecomputationofR(c)inlatticeunits:)thecalculationofthepotentialatacertaindistance~r=a,)thedenitionoftheforcethroughanitedierencefromthepotentialand)theinterpolationoftheforcewhichisnecessarytodetermineR(c)=athrougheq.(.).Step)isimportantbutrathertechnical.Ithasalreadybeendiscussedindetailin[].Ourspecicmethodisdescribedintheappendix.Here,wesummarizeonlytheconclusion:usingavariationalprincipleandalargeenoughrangeoftforthesmearedWilsonloops,oneobtainsanupperboundVu(~r)andalowerboundVl(~r)forthepotential.Wepointoutthatitisadvantageoustohavelargervaluesoftthantheonesusedin[],sincethenVu(~r)andVl(~r)areclosetoeachotherandhavemoderatestatisticalerrors.Step):giventhepotential,theforceatdistancerIandalongtheorientation~discomputedfromF~d(rI)=j~dj[Vl(~r)Vl(~r~d)];(.)rI=[j~dj(G(~r)G(~r~d))]=(.)G(~r)=aZdk()Qj=cos(rjkj=a)Pj=sin(kj=)(.)Here,theargumentrIischosensuchthatF~d(rI)isatreelevelimprovedobservable,i.e.toorderg0wehaveF~d(rI)g0=(rI).ThisdenitionremovestheO((a=r))latticeartifactsthatwouldbepresentinthenaivechoicer=j~r~d=jasargumentofF.Notisnecessarytoachievethis.Noteinaddition,thattheexactchoiceoftheargumentisirrelevantwhentheforcebecomesconstant.Eq.(.)isaconvenientdenitionthatgreatlyreducesthelatticeartifactsintheforce.Westress,however,thattheremaininglatticeartifactshavetobecontrolledbyextrapolatingsimulationresultswithvarious(small)atothecontinuum.Thiswillbedescribedinthefollowingsections.WecalculatedthecentralvaluesoftheforcefromVl.Inordertocoverthesystematicerroroftheforcethatisduetousingeq. (A)atnitevaluesoft,wealwaysrepeatedthecalculationoftheforcewithVl(~r)!Vu(~r)andaddedtheresultingdierenceofthe

centralvaluestothelargeststatisticalerrorsoftheforce.ThelatteroriginatedfromVl(~r).AlthoughthereisnoreasonthatinsertingVu(~r)insteadofVl(~r)intoeq..willalwaysresultinalargervaluefortheforce,thisisgenerallythecaseinthedata.Thisreectsmainlythefactthatthegapeq. (A)decreaseswithgrowingj~rjandthereforeVu(~r)Vl(~r)increaseswithj~rj.Step)consistsofinterpolatingtheforce.HereonecansimplyusethetwoneighboringpointsandtheinterpolationfunctionF(r)=f+fr.Asthisislocallyanexcellentapproximationtother{dependence,theinterpolationerrorissmall:itcanbeestimatedfromthechangethatarisesfromaddingatermfrandtakingathirdpoint.Intheapplicationsdiscussedbelow,thischangewasaddedasasystematicerror.Itiswellbelowthestatisticaluncertainty..ReconstructionoftheForceintheContinuumAsarstapplicationofsettingthescalethroughR0,wedescribeherethecalculationoftheforceinthecontinuumlimit..TheMonteCarloSimulationInordertobeabletoextrapolatetothecontinuumlimit,weperformedcomputa-tionsatdierentvaluesofthelatticespacing.WesimulatedLlatticeswithperiodicboundaryconditionsandthestandardWilsonaction[].Accordingtothestateoftheart,weusedahybridalgorithmperformingNexactlymicrocanonicaloverrelaxationsweepsfollowedbyoneCreutzheatbathsweep.TablesummarizestheparametersL=aSitmax=aMNFsweeps.0,,0k.0,00k.0,00k.0,,0kTable:Parametersofoursimulations.MeasurementsweretakeneveryF(N+)sweeps.Inordertoreducetheautocorrelationsbetweenmeasurements,wecycledthroughtheeightsublattices~b(cf.theappendix)frommeasurementtomeasurement.Sigivesthenumberofsmearingiterationsasdiscussedintheappendix.tmaxdenotesthemaximumtime{extentoftheWilsonloopsandthemaximumspatialseparationisrmax=Mpa.usedinthesimulations.ThelargestsystemwassimulatedontheCERN-IBM,whereaspeedofaboutMop/scouldbeachieved.

Integratedautocorrelationtimeswereestimatedforallsmearedloopsusingjacknifebinning.Theyaree.g.roughly0sweepsat=:andsweepsat=:forthesmallestsmearedWilsonloopsanddecreasewithincreasingloopsize.Intable,wegivetheresultsfortheforceonallourfourlattices.Theerrorsquotedarethejacknifeerrors[]oneq.(.)plusthechangeinthecentralvalueoftheforcewhenwereplaceVl(~r)!Vu(~r).For=:,thecorrespondingnumberswerereadofromtableofref.[].Anumberofour=:0resultscanbecomparedwiththeonesin[0].Ourcentralvaluesareallsmaller.Thisdierenceissomewhatoutsideoftheerrorbars.Althoughitispossiblethatthelargestdierenceisonlyatypingmistakein[0],itappearsthatthegeneraldierenceisduetothesmallerrangeintusedin[0].Therelativelylargeerrorbarsat=:reectthefactthatthegapinlatticeunitsissmall.HenceVu(~r)Vl(~r)becomessignicantatsuchasmalllatticespacingandthecalculationoftheforceisdicultat=:..TakingtheContinuumLimitOncewehavesetthephysicalscalethroughR0,theforceisentirelydescribedbyadimensionlessfunction:H(x)=F(r)rjr=xR0:(.)Atdierentvaluesofthelatticespacinganddierentorientations~di,weobtainthelatticeapproximationsH~di(x;a)toH(x).FromSymanzik'sdiscussionofthecutodependenceofloopintegrals[],oneexpectsthatthelatticeapproximationsconvergetothecontinuumwithcorrectionsthatareroughlyproportionaltoa:H~di(x;a=R0)=H(x)+O(a=R0)(.)Thelatticeresultsareplotteding.forvesamplevaluesofxasafunctionofthesquarelatticespacing[a=R0].Theextrapolationsaccordingtoeq.(.)areshownaswell.Withinourprecision,nosignicanta{dependenceisseenandtheextrapolationstothecontinuumarestableandclearlydonotdependonthefunctionalformoftheassumeda{dependence.Toalargeextentthisisduetothetreelevelimproveddenitionoftheforce(cf.section. ).Extrapolationsweredoneforarangeof0:x:.Atsmallervaluesofx,wehavenotperformedtheextrapolationsincetherewehaveonlytwopointsina.Forx>:thereisdatawith:only.

Figure:ContinuumextrapolationforH(x)withx=:()x=:(),andx=0:;0:;;0:;0:()fromtoptobottom.Thecirclesarefortheforcealongorientation~d,thesquarescorrespondtoorientation~dandthetrianglestoorientation~d..DiscussionoftheForceThecontinuumresultsforH(x)areplotteding..Itisdeterminedaccuratelyinaboutthesamedistancerange,wheretheuniversalityofthephenomenologicalforceholds[].WecomparethisSU()Yang{MillsforcewiththephenomenologicalmodelbyRichard-son[].Ascanbeseene.g.ing.ref. [],otherphenomenologicalmodelsdonotdiermuchfromtheRichardsonmodelintherangeofxdiscussedhere.Wenote,however,thattheforceoftheMartin[]andthelogarithmicpotential[]diersfromtheRichardsonforceasmuchastheSU()Yang{Millsforce.Inthissense,evenwithoutfermionsandforgaugegroupSU(),thestaticforcecomparesquitewellwiththephenomenologicalforces.

Figure:H(x)=F(r)rjr=xR0asafunctionofx.Symbolsasing..TheRichardsonmodelisshownasfulllineandthebosonicstringmodel(normalizedbyH():)asdashedcurve.Inthegure,wealsoshowthepredictionofthebosonicstringmodelnormalizedatx=.Itincludesbesidesthestringtensiontheuniversal=(r)correctiontotheforce.Thiscorrectionshouldbethereatlarger[].Asalreadynotedearlier[],thisformgivesanexcellenteectiverepresentationoftheforcealsoforrathersmallvaluesofx.Notethatpreviousclaimsintheliterature[],thattheYang-MillsWilsonloopsareeectivelydescribedbyafermionicstringmodelarebasedonthesubtractionofperturbativecontributions..TheRunningCouplingqq(r)ThequantityH(x)maybeusedtodeneaphysicalrunningcouplingqq(r)=H(x);rxR0:(.)

Itisofinteresttosee,inhowfarwehavereachedtheperturbativeregioninx.Tocheckthis,westartwithourvalueofqq(r)atthesmallestvalueofr,andintegratethe-loopandthe-loopperturbativerenormalisationgroupequationstowardslargerr.Thisisshowning..Wesee,thatthe-loop-functiondescribestheevolutionFigure:qq(r)togetherwith-loop(dashed)and-loop(dotted-dashed)renormalisationgroupevolutionstartingatthesmallestvalueofthecoupling.ofthecouplingforonlyachangeinscaleuptoafactortwo.Inaddition,the-looprunningisverydierent,sincethe-loopterminthe-functioncontributes%andmoreinthisrangeofr.Weconcludethatr'0:fmisoutsideoftheuniversalperturbativedomain.Wemayalsoconsidertheperturbativerelationbetweenqq(r)andthenitevolumecouplingg(L)thatwillbediscussedfurtherinthefollowingsection.Thisreadsqq(r)=(L)+0:0(L)+:::;(L)g(L=r)=().Usingtheresultofthefollowingsection,wegetqq(r=0:R0)=0:+O(),whichissignicantlybelow

thenonperturbativeresultqq(r=0:R0)=0:().However,atthisvalueofr,theO()termis0%oftheO()termandwearriveagainattheconclusionthatristoolargetouseperturbationtheory.Withinourapproachofextrapolatingthelatticenumberstothecontinuum,wecouldnotreachsignicantlysmallervaluesofr..SettingthePhysicalScaleintheComputationoftheRun-ningCoupling(L)Inref.[],arenormalisedcouplingg(L),thatrunswiththebox{sizeLwascalculatedfordierentlengthscalesL.Atthelargestscaleinvestigatedinref.[],thecouplinghasthevalueg(L)=:.Atthisvalueonecanmakecontactwiththelowenergyscalesofthetheoryininnitevolume:onecalculatestheproductLE,withEsomeenergyscaleofthetheoryininnitevolume.Thechoiceofref.[]wasE=pK,withKthestringtension,sincedatafortwosmallvaluesofthelatticespacingexistedforthisquantity.ThiscanbeimprovedbyusingE==R(c)instead.WedescribenowthecomputationofL=R(c),extrapolatedtothecontinuumlimit.Inarststep,wedeterminethebarecouplingasafunctionofthelatticesizeL=aforxedg(L)=:.Then,weuseR(c)=aatdierent{valuesinthesamerange,formtheratioL=R(c)andextrapolateittothecontinuuma=L=0.L=aimpr.().00().().0(0).(0).().().()0.().().().().0()Table:Barecouplingfromref.[]andbarecouplingimprforthe{loopimprovedactionvs.thelatticesizeatxedg(L)=:.Atthelargestvaluesofthecoupling,asignicantlatticespacingdependenceofthestepscalingfunctionwasfoundinref.[](cf.g.inthatreference).Itwasthereforetobeexpected,thatL=R(c)showslatticeartifacts.Weattemptedtoreducetheseartifactsbyusingthe{loopSymanzikimprovedactiongiveninref. [].Wedenotetheresultsthatareobtainedusingthisactionwithasubscript"impr".The{valuesFordetailsonthedenitionandthecomputationofg(L),wereferthereaderto[,].0

arelistedintabletogetherwiththeoneswithoutimprovementthatwerealreadyobtainedinref. [].Both(L=a)jg(L)=:andimpr(L=a)jg(L)=:arealmostlinearfunctionsoflog(L=a).Therefore,the(noninteger)valuesL=aandLimpr=aforthevaluesofwhereweknowtheforce(table)areeasilydeterminedbyinterpolation.Atthese{valuesvalues,wecalculatedR~di(c)=aasdiscussedinsection..ExamplesofthedataforL=R~d(c)areplotteding..ItiswellworthnotingthatFigure:L=R~d(c)(),Limpr=R~d(c)(),L=R~d(c)()andLimpr=R~d(c)(lled)asfunctionofthelatticespacinga=Landc=0:;:0;:and:0fromtoptobottom.Thelinesarethelinearextrapolationtothecontinuumusinga=Limpr=.Ata=Limpr=0theextrapolatedvaluesareplottedtogetherwiththeirerrorbars.g.isthersttimethatthesuccessoftheSymanzikimprovementprogramcanbedemonstratedin{dimensionalpuregaugetheories:thedependenceoftheratiosonthelatticespacingismuchweakerwhenthe{loopimprovedactionisused.Infact,

duetotheweakcutodependence,thedatawithimprovementcanbeextrapolatedtothecontinuumwithcondence.Thephysicallymostinterestingcaseisaroundc=:andweuseR(:)=R0'0:fminthefollowing(atnitevaluesofthelatticespacing,R~d(c)dependsontheorientation~d.Inthecontinuumlimit,thisdependencedisappearsandwedropthecorrespondingindex.)OurextrapolationtothecontinuumyieldsL=R0=0:()(.)or|usingR0=0:fmasanillustrationalsoforSU()Yang{Millstheory|L=0:()fm.Forcomparison,wealsogivethenumbersthatwereobtainedfromtheforcealongtheothertwoorientations:L=R0=0:()for~dandL=R0=0:()for~d.Thelatterextrapolationshadtobeperformedwithoutthe=:points.Theseextrapolationsarestable:Withintheerrorbars,weobtainthesameresultsifweincludeonemorepointatlargeraorifweremovethelastpointfromtheextrapolation.Furthermore,thedierenceoftheextrapolatedvaluefromthepointatlargestathatwasincludedintheextrapolationisonlyabout%.Therefore,theabovecalculationgivesthemomentum{scalefortherunningcouplingg(L)toaprecisionofabout%inthecontinuumlimit.ItismoreeasytocontrolsystematicerrorsinthecalculationofR0thaninthede-terminationofthestringtension.Nevertheless,itisinterestingtocheck,inhowfaroneobtainsadierentresultifthescaleissetbythestringtension.Wehavethere-foredeterminedthestringtensionfromF~di(r)=K+=(r)forrK:.Thisrangeinrwaschosensuchthatthe=:datacouldbeincluded.The=(r)correction[]islessthan%andtheresultsofsection.providesomeevidencethattheuncertaintyonthecorrectionmaybeneglected.ExtrapolatingLimprpKtothecontinuumasshowning.,givesLpK=0:(0)(. )or(withpK=MeV)L=0:()fm.ThesmallchangecomparedtothedeterminationthatusesL=R0isduetothefactthattheforceinSU()Yang{MillstheorydeviatessomewhatfromtheRichardsonmodelintherelevantr{range.ThestatisticaluncertaintyinbothdeterminationsofLinphysicalunitsisquitesimilar,becausetheyoriginatefromtheforceinaboutthesamerangeofr.However,thestringtensiondeterminationusestheassumptiononthesubleadingcorrectiontotheforce.Althoughg.providesreasonableevidencethatwemayusethecorrectionderivedfrombosonicstringvibrations,anuncertaintyduetothisassumptionisnoteasytoquantify.

Figure:TheextrapolationofLimprpKtothecontinuumlimita=Limpr=0..ConclusionsIthasalreadybeenpointedoutin[],thatagoodwaytosetthescaleinthepuregaugetheoryisthroughtheforceatsomenitedistancewellinsidethenonperturbativeregion.WehaveshownherethatthisproposalcanbecarriedoutinpracticeandthatthequantityR0providesalowenergyscalethatcanbecalculatedprecisely.Comparedtothestringtensionthisavoidstheextrapolationinrandisthereforeindependentofparametrisationsoftheforce.TherelationtoexperimentisbasedontheassumptionofidentifyingthestaticQCDpotentialandthephenomenologicalpotential.UsingR0andthe{loopimprovedactionforthesimulationdeterminingg(L),wewereabletocomputethescaleintherunningcouplingof[,]towithin%inthecontinuumlimit.Theforceitselfcouldbeextrapolatedtothecontinuumlimitwithallsystematicerrorstakenintoaccount.Inordertodothis,itwasmostimportanttohaveprecisedataatseveralrathersmallvaluesofthelatticespacing.Inaddition,latticeartifactscouldbesuppressedbychoosingatree{levelimprovednitedierenceforthelatticeforceintermsofthelatticepotential.R0calculatedinthewaydescribedherewillclearlybeausefulreferencescalealsointheSU()puregaugetheory.Itshouldbementioned,however,thatonceoneapproachesfullQCD,thereareotherquantitiesthataredirectlymeasuredinexperimentsthat

canplaythisrole.Neverthelessitwillbeinterestingtoreconstructtheforcealongthelinesofsection.andcomparetophenomenologicalmodels.Inaddition,acheckoftherelationseq.'s(.,.)willbeofinterest.Acknowledgement:IthankU.WolforthecollaborationcalculatingthedatainTable.IhaveprotedfromanumberofdiscussionswithM.LuscherandIthankhiminparticularforcriticallyreadingthemanuscript.ThemostimportantnumericalcomputationsoftheforcewereperformedontheIBMatCERN.IwouldliketothanktheCERN-CNstaaswellasPierreAubryfromIBMfortheirexcellentsupport.ATheCalculationofthePotentialTheforcebetweenstaticquarksismostecientlycalculatedfrom\smeared"Wilsonloops.Wehaveappliedavariationofsmearingbasedontheoriginalideasofref.[,].ThesmearingandvariationaltechniqueemployedherdeviatesonlyslightlyfromtheoneusedbyMichaeletal.[0,].Adetaileddiscussionofthemeritsofthemethodcanbefoundin[].Here,wedescribetheexactimplementationthatwasusedanddiscusstheresults.WestartfromlatticegaugeeldsU(x)SU(),thatarethegaugeconnectionsbe-tweenpointsxandx+^a,=0;;;ofahypercubiclatticewithlatticespacinga.Inarststep,thespace{componentsofthegaugeeldare\smeared"byoneiterationUk(x)!U0k(x)=PfUk(x)+!Xj=k=[Uj(x)Uk(x+^ja)Uyj(x+^ka)(A)+Uyj(x^ja)Uk(x^ja)Uj(x^ja+^ka)]g;k=;;;HerePdenotestheprojectionintoSU()andwehaveused!==.Nextweblocktoasub{lattice~bcharacterizedbyavector~bwithcomponentsbi=0;.Thesublatticehas/thenumberpointsperspace{dimensionbyrestrictingthecomponentsof~x=atobeeven(bi=0)orodd:xi=a=bimod,i=;;forx~b.Thegaugeeldsonthesublatticearesimply~Uk(x)=Uk(x)Uk(x+^ka);x~b:(A)Onthisblockedlatticeweapplyeq. (A)Sitimes(witha!aandU!~U).Inordertobeabletocalculatethepotentialalongthethreeorientations~d=a(;0;0);~d=a(;;0);~d=a(;;),wethenconstructgeneralizedgaugeeldsV(i)thatconnectx=(x0;~x)with(x0;~x+~di);i=;;.V(i)areobtainedbyaveragingtheparalleltransportersoverthedierentshortestlatticepathson~b.

WecomputedsmearedWilsonloopswithspaceextent~r=ni~di,ni=;;:::;MusingFigure:log[C(t;~r)=C(ta;~r)at=:.Thedatapointsarefromtoptobottomfor~r=~d;~d;~dand~d.Thelongdashesrepresentsthettoeq.(A)andthefulllineisthettoeq.(A),plottedintheirrespectivetranges.Asanillustration,thetstoeq. (A)arecontinuedtotvaluesoutsideofthetrangesasshortdashedcurve.theappropriateproductofV(i)forthespace{likepartsandthe-linkintegral[0]forthetime{likepartsoftheloops.InsertingVaftersmearinglevelSiattime0andSjattimet,oneobtainsamatrixcorrelationfunction.Thelattercorrespondstomatrixelements(^Tn(~r))ij;nt=aofpowersofthetransfermatrix^TinthecorrespondingchargedsectoroftheHilbertspace.ThegeneralizedeigenvalueequationXj[^Tn+(~r)]ijv(~r)(k)j=k(~r)Xj[^Tn(~r)]ijv(~r)(k)j;(~r)(~r);:::(A)Inthisdiscussionweneglectcorrectionsduetoanitetimeextentofthelatticeandusetheconventionthattheenergyofthevacuumiszero.

givesestimatesk(~r)thatconverge,forn!,(exponentially)tothelowesteigen-valuesofthetransfermatrix[].Inoursimulations(cf.section(.)),wefoundthatthersttwoeigenvaluesarefairlystablewithrespectton,oncen.Highereigenvaluesarediculttodeterminesince{apparently{thedierentstatesgeneratedbytheabovesmearingprocedurearenotlinearlyindependenttoasucientdegree.WeusedthisvariationaltechniquetoconstructacorrelationfunctionC(t;~r)=Xi;jv(~r)()i[^Tn(~r)]ijv(~r)()j(A)thatobtainsonlysmallcontributionsfromexcitedstatesbecauseitisprojectedontotheapproximategroundstate.Furthermore,weestimatedthegapinlatticeunitsbya(~r)=log((~r)=(~r)):(A)WethendeterminedthepotentialfromtsC(t;~r)=C(~r)exp(Vu(~r)t)and(A)C(t;~r)=C(~r)exp(Vl(~r)t)[+C(~r)C(~r)exp((~r)t)]:(A)Here(~r)wasxedtothevaluedeterminedfromthevariationalmethodeq.(A).Vu(~r)isanupperboundforthepotentialsincetheexcitedstatecontributionsareneglectedineq.(A).Ontheotherhand,Vl(~r)givesalowerboundforthepotentialprovided(~r)asdeterminedfromeq.(A)isareasonableestimateforthegap.Thisistrue,becauseeq.(A)parametrizesallcorrectionstotheloweststatebyaneectivecontributionfromtherstexcitedstate.Itthusoverestimatesthecorrectionsatlarget(i.e.t{valuesoutsideofthetrange).Astheyarelowerandupperbounds,onecan{inprinciple{usethecombinationofeq. (A,A)foranytranges(int),irrespectiveofwhetherthetsarestatisticallysatisfactory.Ifthisisdoneallowingtoosmallvaluesoft,however,Vu(~r)andVl(~r)becomesignicantlydierentandonehasadominantsystematicerrorVu(~r)Vl(~r).Inthepresentcalculation,wehavearelativelylargerangeint.Wechosethetrangessuchthattheyyieldasatisfactory(includingthecorrelationsofthedata).ThisresultedinVu(~r)=Vl(~r)withinonestandarddeviation.WegiveexamplesofthedatatogetherwithtsintheformofthetimederivativeofC(t;~r)ing..Thevaluesof~rchosenforthegureareintherangethatismostrelevantfortheanalysisoftheforcediscussedinthiswork.ItisevidentfromthegurethatC(t;~r)providesacorrelationfunctionwhichreceivesonlysmallcontributionsfromexcitedstatesintheaccessiblerangeoft.Thereforewecanextractthegroundstatepotentialwithmoderateerrorbars.

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rI=aF~d(rI)arI=aF~d(rI)arI=aF~d(rI)a.0.0.0().000.00().00.0().0.0.0().0.0().0.00().0.0.0().00.0(0).00.0()..0.0().000.00().00.0()..0.0().0.00().0.00()..0.0().00.0().00.0(0)..0.0(0).0.0().00.0().0.0.0().0.0(0).0.0().0.0.00().000.0().00.00().0.0.0().0.00().0.00().0.0.0().00.00().00.0().0.0.0().0.0().00.0().00.0.00().0.0().0.0().0.0.0().0.0().0.0().0.0.00().000.0().00.0().0.0.00().0.0().0.0().0.0.0().00.0().00.0().0.0.0().0.0().00.00().00.0.0().0.00().0.00().0.0.0().0.00().0.00().0.0.00().0.00().0.00()..0.0()..0.0()..0.00()..0.00().0.0.00()..0.00()..0.000()..0.00()..0.00(0).0.0.00()..0.00()Table:Forceandimprovementradiusinlatticeunits.Forconveniencewehavealsoincludedtheresultsof[].InphysicalunitsrIcoversroughlythesamerangeforallve.

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