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cmbx10Scaling withamoudi edWilsonactionwhic=hsuppressesI\oZ(K`y
cmr102 artifacts inSU(2)latticegaugetheories& ;/}h! cmsl12J.Amrb j rnPandG.Thorleifsson S$TheNielsBohrInstitute^/=Blegdamsvrej17,DK-2100CopSenhagen ,Denmark 0N cmbx12Abstract*fc9A=moSdi edVWilsonactionwhicrhsuppressesplaquetteswhichtakenegativevXaluesisfc9used tostudythescalingbSeharviorofthestringtension.Theuseofthe,g cmmi12 2 cmmi8E :,scheme9givresgoSodagreemenrtwithasymptotictwoloSopresults. @html:1 html: *ǍE\9Ǎ|9<"V G
cmbx10Conrtents H9html: 1 html: In tro` ductionH html:3 html: &[9html: 2 html: Numerical metho` d$html:4 html:9html: 3 html: Scaling ofthestringtension ~html:5 html:9html: 4 html: ConclusionsL9html:6 html: @html:2 html: ԠǍE\9Ǎ|9html:14 html:,Inrtro Oduction H9ThezsimplestandmostpSopularcrhoiceofagaugelatticeactionistheoneproposedfc9bryWilson,whichforthegroupS U@(2)reads: S̹W <(U̹l!)UR= MCu
cmex10CX "b| lasy82 (1-!",
cmsy10 ō1۟Qm fe 2
TVrYU̼2);(html:1 html:)"9where =UR4=gn92
|{Y cmr82.=,U̹lw2S U@(2)isthelinkvXariablede nedonthelinkl(x;),while93Tq
lasy102{(x; )referstotheloScationandorienrtationofthecorrespondingplaquette.9U̼2
listhestandardplaquettevXariable: fc U̼2 URU̹x;L=U̹x; hU̹x+;rUO@K cmsy8y%x+
;U @yڍx; X:(html:2 html:) InZranryMonteCarlosimulationthecrucialquestioniswhetherthe eldcon g-fc9urations$generatedprorvideanadequaterepresentationofthecontinuumphysics.9The naivrecontinuumlimitcanonlybSeobtainedforFuP1P콉 fe @'2 hTVr @U̼2 Z
1.>Especiallyanry9lattice0con gurationswhicrhloScallyhavesomeU̼2'swhereFucE1cE콉 fe @'2
|TVrzU̼2
b<x 1arelattice9artifacts whicrhapriorihavenothingtodowithcontinuumcon gurations.A com-9plete%dsuppressionofsucrhcon gurationsshouldnotchangeanycontinuumphysics.9FVor/thevXaluesof 'whereonepSerformstheMonrteCarlosimulationssuchloScalZ̸29
uctuationsarenotatallrareifS̹W isused.PItisthrusveryimpSortanttomake9surethatthese
uctuationsharvenoimpactonconrtinuumobservXables.nTheeasiest9wraytodosoistomoSdifythelatticeactioninsucrhawaythatthesesmall-scale9
uctuations2:aresuppressed,D withoutin
uencingplaquetteswhereTVrU̼2
e>%0.This9canbSedonebrymodifyingtheWilsonactionasfollorws[ html:1 html:]: fc VS)=URS̹W ]+S̹uZ;(html:3 html:)9whereS̹W
isthestandardWilsonaction(html:1 html:),whileS̹ `is {S̹ʬ=URCX -2q[1 sgn6(TVrhU̼2)]:(html:4 html:)" Theڊaction(html:3 html:)wrasstudiedforUR=1ڊin[ html:2 html:]whilethephasestructureinthe( O;)fc9plane
wrasstudiedin[ html:1 html:].?FVora xedonewouldexpSectthesamecontinuumlimit9for !UR1andalsothesameidenrti cation =UR4=gn922.=.CHorwever,2asshownin[ html:1 html:]there9is
amarkreddi erencebSetween,sayV,Creutz
ratios,measuredfor=0(theWilson9action)1andfor%=0:51inthecaseof wt=%2:5. OnewrouldexpSectthedi erenceto9bSecevrenmorepronouncedwithincreasing.'Thisillustratesthatplaquetteswith9negativre@vXaluesmayplayanimpSortantrolefortherangeof O'swherethescalingof9pure5S U@(2)latticegaugetheoriesisusuallystudied,YandonecouldbSewrorriedabout @html:3 html: zǍE\9ǍmA(s3z#
RY ff ?No.8ofswreeps͟Y ff 2No.8ofmeasuremenrtsp; ff N
&c 1.5͟Y ff5k55250Y ff P11050 1.6͟Y ff5k65250Y ff P13050 1.7͟Y ff5k63750Y ff P12750 1.8͟Y ff5k88750Y ff P17750 1.9͟Y ff5k45000Y ff N9000<0html:TVable1 html::8StatisticscollectedatvXarious O.InallmeasuremenrtsUR=10-9therelationtoconrtinuumphysics.Thefactthatonegetsacceptableagreementwithfc9conrtinuumscalingrelationscouldbSefortuitoussinceonewrouldexpecttheactionS9toՉreproSduceconrtinuumՉphysicsbSetterforlarge.1Onepurposeofthisarticleisto9shorwthatweindeedgetthecorrectscalingrelationsforlargeandthatthereis9noreasontoexpSecttheplaquetteswithnegativrevXalueswhichappSearintheWilson9action?^toplaryanyimpSortantrolefor 2:2.7AnotherpurpSoseistotestwhether9the1moSdi cationoftheactionmaryimprovetheapproachtoscalingforthereasons9menrtionedabSove.(V9html:24 html:,Numericalmetho Od H9ThemnrumericalsimulationswerepSerformedusingastandardMetropolisalgorithmto9upSdateGthegauge elds,comrbinedwithanoverrelaxationalgorithmtodecreasethe9autoScorrelations.FVor(evreryMetropolisswreepwepSerformed4overrelaxationsteps.9Latticesize1224 jwithpSeriodicboundaryconditionswrasusedandwemeasuredall9WilsonloSopsuptothesize6,%6.spInordertoimprorvethestatisticswremodi ed9theVcon gurationsusingthePrarisitrick[ html:6 html:]bSeforemeasuring.Unlikeinthecaseof9=0>theinrtegrationinvolvedintheParisitrickhadtobSeperformednrumerically9andޠwrasthuscostlyincomputertime.4Butthegaininthestatisticsdidmorethan9compSensateforthat. ThecrhoiceofparameterswasUR=10,whichwassucienttosuppressallnegative9plaquettes,liand B 2b[1:0;2:5].֯WVefoundascalingwindorwfor 2b[1:5;1:9]and9concenrtrateda1thesimulationsinthatintervXal.zUsually1000sweepswhereusedfor9thermalization)yandthenrumbSer)yofswreepsusedformeasuring,P forvXarious O,isshorwn9inOtable1. hTMeasuremenrtswerepSerformedevery fthsweepandtheerrorswere9estimatedbryusingthejackknifemethoSd. @html:4 html: ǍE\9Ǎ|9html:34 html:,Scalingofthestringtension H9The>[ O-vXaluesinthescalingwindorwareevensmallerthantheonesusedforthestan-fc9dard[hWilsonaction. Ifonewrants[htotestscalingbrycomparingwiththepSerturbative9trwo-loSopresultusingtheidenrti cation h[= 4=gn922 itisdoomedtofail.WVeclearly9need$Qasuitablee ectivrecouplingwhichwecanuseinsteadofgn922.=.Herewewilluse9thesso-called ̹E
scrheme[ html:3 html:, html:4 html: xo, html:5 html:]. yThisscrhemeissimpleandwellsuitedtodealwith9\pSerturbations"oftheWilsonaction,likretheonesgivenby(html:3 html:)-(html:4 html:).8Letusde ne % ̹E
=ō3Qm fe 4ō,"1 Qm fe Edr
341 f`CDFu M1 M콉 fe @'2mTVr \U̼2ğf`CESf:(html:5 html:)&49FVromwreakcouplingexpansions( !UR1; N xed),itisseenthat 3 z0 ̹E
!UR ?for1 !1:(html:6 html:)9Itisconsistenrttousetheg̹E
=UR4= ̹E
<:inthetwoloSop O-functionsinceg̹E
<:isafunctionfc9ofvgandthetrwov rstcoSecienrtsofthe O-functionareinvXariantunderanon-singular9couplingPconstanrtrede nition.kTheuseof ̹E ~kinthetwoloSop O-functionisknown9to diminishscalingviolationsinavXarietryofsituations:ThestringtensioninS U@(2)9and_rS U@(3)gaugetheories,themassgapintheSU@(2)andSU@(3)crhiralmoSdels9andhthedecon ningtransitiontempSeratureT̹c.y,againforS U@(2)andSU@(3)gauge9theories,ujustYtomenrtionsome.WVewillnowshowthatthemethoSdworkswellfor9themoSdi edWilsonactionwithUR=10. Letxus rstnotethatforUR=10xthe O-range1:4 1:9xismappSedinrtothe ̹E--range91:8375E 2:104,%whicrh?shouldcomparedtothecorrespSondingmapfortheordinary9Wilsonactionwherethe O-range2:2 2:5ismappSedinrtothe ̹E--range1:741 2:154.9FVrom&themeasuremenrtsofWilsonloSopsmentionedinthelastsectionweextract9the3stringtensionusingthesimplestmethoSd:nIfW(RJ;T)3denotesaR܌TյWilson9loSopwre rstform 3zV̹ef f_(RJ;T)UR=log[W(RJ;T)=W(R;TLn 1)]:(html:7 html:)9FVorT!UR1V̹ef f_(RJ;T)shouldgotothegroundstatestaticquarkpSotenrtialV̹q NI{q(R).fc9WVeharveavXailableTp6andtheresultsagreewithinerrorbarsforT=p5 6.i@WVe9harveusedthesevXaluesasV̹q NI{q(RJ)forRnUR6.8NextV̹q NI{q(RJ)is ttedtotheform: z /V̹q NI{q(RJ)UR=CF ōE۟Qm fe f -R"r+n9R(html:8 html:) 39and otheerrorbarsfornaremainlycomingfromsystematicerrorsarisingfrom ts9withxvXariouslorwerxcutsinRJ.sItisclearthatwiththelimitedrangeofR#andT's @html:5 html: #ǍE\9Ǎ|9arvXailabletousnothingwillbSegainedbyapplyingsomeofthemanymoreelabSoratefc9scrhemesof ttingwhichareavXailable. log10n9( ̹E-)%Vis nallyplottedin g.1against ̹E.Thecurvreshownistheone9obtainedTfromthetrwo-loSopT O-function,eaccordingtowhicrhthescalingofthelattice9stringtensionn9( ̹E-)UR=̹contSa22( ̹E)isgorvernedbythetwo-loSopscalingofthelattice9spacinga( ̹E-)22:"hna 2( ̹E-)UR= 2
\zf\C ōzY6n922zYQm fe G11)f ̹Ef\C!? c*102=121_`}expsSf\C"y)T ō336n92233Qm fe G11 ̹Ef\C# B:(html:9 html:) 謍9Itisclearthatscalingisreasonablewrellsatis edinthegivenrangeof ̹E-. It^oisinrterestingtocomparetheseresultswiththecorrespSondingonesobtained9bryPusingtheordinaryWilsonaction.0Therangeof zwhichgivesthesamerangeof9 ̹E aswrereusedforthemoSdi edactionwillbe2:3V 82:5.Inthisrangeitis9wrellknownthatnaiveapplicationofscalingdoSesnotworkparticularlywell,andto9getFafaircomparisonwiththeresultsforthemoSdi edactionwreuseagainthe ̹E9couplingnJconstanrttransformation.kWVenowfollowthesamemethoSdasinthecaseof9themoSdi edactionandextractthestringtension2html:1 html:.Theresultisshorwnasafunction9ofk ̹E in g.1.Thekuseof ̹Ehasimprorvedkthescalingconsiderablecomparedto9the,useof asacouplingconstanrt,=Rasalreadymentioned,=Randitisapparentfrom9 g.1qthatagreemenrtwiththetwo-loSopscalingisasgoodfortheWilsonactionas9forthemoSdi edaction.'9html:44 html:,Conclusions H9ThepEsuppressionofnegativrevXaluedplaquettesshouldnotchangethecontinuum9limitofpureS U@(2)latticegaugetheoriessincethesecon gurationsarepurelattice9artifacts.zNaivrely onemightevenexpSectasmootherapproacrhtothecontinuum9limit.ifsucrhasuppressionisimplemented.ThemoSdi edWilsonaction(html:3 html:)indeed9suppressesnegativrevXaluedplaquettesforlargevaluesofanditispSossibleto9approacrh;continuumphysicsformuchsmallervXaluesof O.*However, forthesesmall9vXaluesof z3therelationbSetrweentheconrtinuumcouplingconstanrtgn922!and isnota9simpleoneandinordertoextractthescalingbSeharvioronehastouseamodi ed9coupling*constanrtcloserre
ectingthephysicsofthesystem.
gThe ̹E--schemeis9sucrhTaprescriptionandusingitwehavefoundgoSodTagreementwithcontinuum9phrysics.FVrom$GtheseresultsitseemsclearthatthetheorywithmoSdi edWilson9actionibSelongstothesameunivrersalityiclassasthetheoryde nedbrytheordinary9Wilsonactionwhen jissucienrtlylarge.*Theworrymentionedin[ html:1 html:]isthusruled9
܉ ff ğ
^html:ٓR cmr71 html: TheUUdatausedfor b>
cmmi10 N4=2:4and2.5aretakenfrom[ html:7 html: ]. @html:6 html: 3ǍE\9Ǎ|9out.(In*additionitseemsthattheapproacrhtoscalingisnotdramaticallyimprovedfc9comparedtothesituationforordinaryWilsonaction.(PIndirectlythisindicatesthat9negativrepplaquetteexcitationsplaynoimpSortantroleinthescalingregionforthe9ordinaryhWilsonaction,uatleastwhenwrediscussobservXableslikethestringtension.9Horwever,oneawouldexpSectthatthemodi edWilsonactionisconsiderablebetter9whenitcomestothemeasuremenrtsoftopSologicalob jectslikeinstantons.)fc9Ac knowledgementhWVethankV.Mitrjushkinfordiscussionsandmanryusefulsug-9gestions.PrartofthecomputationswerepSerformedatUNI-CandweremadepSossible9bryagrantfromtheDanishMinistryofResearchandTVechnology.@V9References9#html: html:농 ghtml:1 html:' V.G.Bornryakov,'M.CreutzandV.K.Mitrjushkin,Phrys.Rev.D44(1991)3918. html: html:농 ghtml:2 html:' G.MacrkandE.Pietarinen,Nucl.Phys.B205,(1982)141.html: html: ghtml:3 html:' G.jPrarisi,VProSceedingsoftheXX2thconferenceonhighenergyphysics,VMadison
농' 1980. html: html:농 ghtml:4 html:' F.KarscrhandR.Petronzio,Phys.Lett.139B(1984)403.html: html: ghtml:5 html:' S.Samruel,O.MartinandK.MoriartyV,Phys.Lett.153B(1985)87.html: html: ghtml:6 html:' G.Prarisi,R.PetronzioandF.Rapuano,Phys.Lett.126B(1983)418.html: html: ghtml:7 html:' F.GutbroSd,Z.Phrys.C37(1987)143.>농Figureˇ1:yThe 5measuredstringtensionusingthemoSdi edaction(uppercurvre)
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