Simulating the Electroweak Phase Transition in the SU(2) Higgs Model

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

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aDESY94-159SimulatingtheelectroweakphasetransitionintheSU(2)HiggsmodelZ.Fodor ,J.Hein,K.Jansen,A.Jaster,I.Montvay DeutschesElektronen-SynchrotronDESY,Notkestr.85,D-22603Hamburg,GermanyAugust,1994AbstractNumericalsimulationsareperformedtostudythe nitetemperaturephasetran-sitionintheSU(2)Higgsmodelonthelattice.InthepresentlyinvestigatedrangeoftheHiggsbosonmass,below50GeV,thephasetransitionturnsouttobeof rstorderanditsstrengthisrapidlydecreasingwithincreasingHiggsbosonmass.Inordertocontrolthesystematicerrors,wealsoperformstudiesofscalingviolationsandof nitevolumee ects.

1Introduction

ThemassesofelementaryparticlesintheStandardModelaregeneratedviatheHiggsmecha-nismbythenon-zerovacuumexpectationvalueofthescalarHiggs eld.Athightemperatures,abovethescaleofthevacuumexpectationvalue,theHiggsmechanismisnotoperative,thesymmetryofthevacuumgetsrestored[1].Infact,intheearlyuniverse,accordingtothebigbangcosmology,matter rstexistedinthesymmetryrestoredphase.Asaconsequenceofex-pansionandcooling,anon-zerovacuumexpectationvalueofthescalar eldwasdevelopedinthephasetransitionbetweenthesymmetricphaseathightemperaturesandtheHiggsphaseat

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

lowertemperatures.Thepropertiesofthiselectroweakphasetransitionmighthaveasubstan-tialin uenceonthelaterhistoryoftheUniverse.Forinstance,sincethenumberofbaryonsisnotconservedintheminimalstandardmodel[2],thesmallbaryonasymmetryoftheUni-versecouldperhapsbecreatedinnon-equilibriumprocessesduringastrongenough rstorderelectroweakphasetransition[3,4].Thiso ersthepossibilitythatthebaryonasymmetrycanbeexplainedwithintheminimalstandardmodel.Theresolutionofthisquestionisthereforeamajorchallengeforelementaryparticlephysics.

Thestandardcalculationalmethodforthestudyofthesymmetryrestoringelectroweakphasetransitionisresummedperturbationtheory[5,6,7,8].IntheHiggsphaseperturbationtheoryisexpectedtoworkwellfornotveryhighHiggsbosonmasses,sincethecouplingsaresmall.Inthehightemperaturesymmetricphase,however,thesituationissimilartohightemperatureQCD:irreparableinfraredsingularitiesoccurwhichpreventaquantitativecontrolofgraphresummation[9].Sincethecalculationofphysicalquantitiescharacterizingthephasetransitionrequirestheknowledgeofbothphases,thereisapriorynoreasonwhyperturbationtheorycouldprovideaquantitativetreatmentoftheelectroweakphasetransition.Indeed,theresultsofperturbationtheoryshowbadconvergence.

Forabetterunderstandingseveralnon-perturbativemethodshavealsobeentried.Forsimplicity,fermionsandtheU(1)gauge eldareoftenomitted.Thiscanbeexpectedongeneralgroundstobeareasonable rstapproximation.InthiswayoneisleftwiththeSU(2)Higgsmodeldescribingtheinteractionofafour-componentHiggsscalar eldwiththeSU(2)gauge eld.Possiblenon-perturbativeapproachesincludeablockspinprocedureleadingtoevolutionequationsforaverageactions[10],the -expansionat4 spatialdimensions[11]and,ofcourse,numericalsimulations.Afterpioneeringworks[12,13],recentnumericalsimulationsconcentratedontheunderstandingofthe nitetemperaturebehaviouroftheSU(2)HiggsmodelatlargeHiggsbosonmassesnearandabovetheW-bosonmass[14].Anothernon-perturbativeapproachisbasedondimensionalreduction,studyingthethree-dimensionale ectiveHiggstheory,whichisobtainedinthehigh-temperaturelimit[15,16,17].Afurthersimpli cationleadstoane ectivescalartheory[18],whichhasalsobeenstudiednumericallyinthereducedmodel[19].

Thenon-perturbativeinvestigationsoftheelectroweakphasetransitiondidnotyetleadtoaconvincinguniquepicture.Therefore,wedecidedtoperformalargescalenumericalsimulationofthesymmetryrestoringphasetransitionintheSU(2)Higgsmodel.Westayintheoriginalfour-dimensionaltheorywithoutreduction.Thishastheadvantageofkeepingthenumberofbareparameterssmallandnotintroducinganyfurtherapproximationsbeyondthelatticeregularization.Firstresultshavebeenpublishedinarecentletter[20].Herewegiveadetaileddescriptionofthetechniquesusedandincludeadditionalresults.Asitisknownfrompreviousstudies[14],forHiggsbosonmassesnearandabovetheW-bosonmassthenumericalsimulationsintheoriginalfour-dimensionalmodelaretechnicallydi cult.ThereforewerestrictthepresentcalculationstosmallerHiggsbosonmassesbelow50GeV.Sincethisregionofparametersof

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

theminimalstandardmodelisalreadyexcludedbyexperiments,ourpresentscopeismerelytheoreticalbecausewewouldliketocheckthevalidityofsomeothertheoreticalapproximationschemes,e.g.resummedperturbationtheory.WeplantoextendthisinvestigationtoheavierHiggsbosonmassesinfuturepapers.

1.1Latticeaction

ThelatticeactionoftheSU(2)Higgsmodelisconventionallywrittenas

S[U, ]=β 1 1

pl

2Tr( +x x)+λ 1

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

l2forlowλandLt=2;

l3forlowλandLt=3;

h2forhighλandLt=2;

h3forhighλandLt=3.

Inthenextsectionthenumericalsimulationmethodswillbediscussed.Animportanttoolfortheorientationinbareparameterspacewillbetheinvariante ectivepotentialintroducedinsection3.Thendi erentgroupsofnumericalsimulationresultswillbediscussed:thelocationofthephasetransitionpointsinsection4,massesandcorrelationlengthsinsection5,therenormalizedgaugecouplingandtherenormalizationgrouptrajectoriesinsection6,thelatentheatinsection7and nallytheinterfacetensioninsection8.Thelastsectionisdevotedtothediscussionofresultsandtoasummary.

2MonteCarlosimulation

InthissectionsomeaspectsoftheappliedMonteCarlosimulationtechniquesarediscussed.Thiscanbeskippedbyreadersnotinterestedintechnicaldetails.

ThesimulationshavebeenperformedontheAleniaQuadricscomputersofDESY.TheQuadricsQ16isamassiveparallelmachinewithSIMD1architecturewhichconsistsof128processors(nodes).Dependingonthegoalsandfeaturesoftherespectivesimulation,weusedi erentstrategies:

Alatticeisassignedtoeachnode.Notimeiswastedforthecommunicationsbetweenthenodes.Limitationsofmemoryallowthisonlyforsmallenoughlatticeextensions. TheQ16maybeswitchedtoconsistof16independent23tori.

Thewholemachineisarrangedasathree-dimensionaltorus.

Inthiswaythelatticeisdistributedover1,8or128nodesandoneobtains128,16or1independentdatasetsfromonerun,respectively.Ofcourse,thelatticeextensionshavetobemultiplesofthecorrespondingtori.

TheQuadricso ers32bit oatingpointarithmetics.Thisissu cientformostofourpurposes,exceptforbuildingglobalaverages,whenweuseasimplevariantofsoftwarebaseddoubleprecisionarithmeticsforsummation.

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

2.1Updating

InaMonteCarlosimulationtheautocorrelationofsubsequentcon gurationsisoneofthemainproblemsonehastodealwith.Theautocorrelationisusuallyworseatphasetransitions.Thisphenomenoniscalledcriticalslowingdown.Unfortunately,thecontinuumlimithastobeperformedinthisregionofparameterspace(inourcaseκ,βandλ).

Duetothesmallλ-valuesweuse,large uctuationsofthesquaredHiggs eldlengthρ2x≡1

bx,m≡ 42Tr+ νUxν+Ux ν ,ν x ν ν=1iκ +x ,

ζb

xx,j 2 +λ ρ2x 1

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Thiscubicequationcannotbesolvedfastenoughinanupdating.Startingfromtheobservationζx=1for|bx|=1,wesplitζxinζx=1+εxandgetthefollowingapproximateexpressionfortheoptimalζx:

ζx=1 2λ+2λ·|bx|2+O(ε2(6)x,εxλ).

Thisapproximationworksverywellinasu cientrangeof|bx|.Inpracticetheaverageaccep-tanceofthisalgorithmturnedouttobelargerthan98%.

FortheSU(2)-variablesUxµandαx≡ x/ρxweusestandardoverrelaxationmethods[22,24].Becausetheabovedescribedheatbathalgorithmfor xo ersalsoanergodicupdatefortheangularpartαxofthescalar eld x,weneedanergodicupdateforUxµonly.Forthispurposeweusetheheatbathalgorithmdescribedin[25,26].

Inallupdatings,therandomnumbergeneratorproposedandimplementedbyMartinL¨uscher[27]isapplied.ItisbasedonanalgorithmofMarsagliaandZaman[28].ThelatteralgorithmisknownbythenameRCARRY,iftheparametershavebeenchosenappropriately.RCARRYo ersanextremlongperiod>10171,butunfortunatelyitownssomeshortrangecorrelations.Asithasbeenshown[27],onlongrange,achaoticnatureofthealgorithmcomestolight.Skippingfromtimetotimesomehundredsofnumbersinthesequencethecorre-lationispracticallyeliminated.Duetotheskipthisrandomnumbergeneratorisrelativelyslow.Inordertobene tfromtheparallelarchitecture,therandomnumbergeneratorhastobeinitializedindependentlyonthenodesofthemachine.

Forupdatingthe eldcon gurationsacombinationoftheabovedescribed vealgorithmsisused.Wechoosesomebasicsequenceofelementaryupdatingsforthedi erentsetsof eldvariables,whichisrepeatedperiodicallymanytimes.Thewholesequence,whichvisitseveryvariableatleastoncebutusuallymanytimes,willbecalledsweep.Theoptimizationofthisbasicsequencemakingupasweepisadi cultbutimportanttask,whichwillbediscussedinthenextsubsection.

2.2Autocorrelations

LetusconsidertheautocorrelationofaquantityQ[Uxµ, x]measuredonasequenceof eldcon gurations.QnisthevalueofQ[Uxµ, x]measuredonthen-thcon gurationand Nn=1Qnistheaverageoverthecon gurations.TheautocorrelationfunctionforthisquantityN

isde nedas: 1QQn ΓQ(t)≡limN→∞

∞ ΓQ(t)

2t= ∞

2

N

(Q).(9)

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure1:AnexampleofautocorrelationfunctionintheHiggsphaseatT=0.

(t)andΓCf(t)coincide.ThecurvesforΓρ2x

Weinvestigatedtheintegratedautocorrelationtimeforfourcharacteristicquantities:ρ2x,

2ρ2(x+Lt/2)ρx,UplandthelargestcalculatedWilsonloop.Thesecondquantitycharacterizesthecorrelationfunctionofρ2atthelargestdistance.InzerotemperaturesimulationsthelargestWilsonloophadthesizeLs/2 Lt/2,withLsandLtdenotingtheextensionsofthelatticeintimeandspacedirection,respectively.At nitetemperatureonlythe1 1Wilsonloopwasconsidered.ForWilsonloopsnoteveryorientationwastaken:twosideswerealwaysinthedirectionofthelargestlatticeextension.Furtheron,werefertothecorrelationfunctionasCfandtotheWilsonloopasWl.Ifequation(8)isevaluatedona nitesequenceofcon gurations,onehastodecidewheretotruncatethesumovert.Asmentionedbefore,thelargestτint-values

(t).werefoundforρ2x,sowetruncatedatthe rstzeroofΓρ2x

Onlargerlatticesweusuallyhad16independentcon gurationsinthecomputer.Theywereevaluatedseparatelyandanestimateforthestatisticalerroroftheintegratedautocorrelationtimewasobtainedfromthevariance.

Wemadesomeinvestigationshowtooptimizetheautocorrelationbychangingthenumberofcallsofthevariousupdatingsinthecompletesweepsbutwedidnottrytooptimizetheorderingofthealgorithms.Ofcourse,weoptimizedtheautocorrelationinCPUtimesincetheabovechangesa ectthetimerequirementsofthecompletesweeps.Itshouldbementionedthattheoptimalnumberofcallsdependsonlatticesizeandparameterrange.Themeasurementroutineswerecalledaftereachsweep.

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure2:Autocorrelationfunctionsatthephasetransitionlineona3·242·96lattice.Theleftpicturereferstothesymmetricphase,therightonetotheHiggsphase.Evaluatedsweeps:32000sweepsinthesymmetricphaseand80000sweepsintheHiggsphase.Inbothcasesthesameupdatingschemewasapplied.Inthe

=217±66=21±4sweepsandintheHiggsphaseτint,ρ2symmetricphaseτint,ρ2xx

sweeps.

TheautocorrelationfunctionwasinvestigatedintheHiggsphasebothforT=0andfor nitevaluesofT.Inthesymmetricphaseonly niteTwasconsidered.Atypicalexampleof

(t)isautocorrelationintheHiggsphaseisshownby g.1.TheautocorrelationfunctionΓρ2x

toagoodapproximationasingleexponential.Thereisnosigni cantdi erencebetweenΓCf(t)

(t).TheautocorrelationfunctionsforquantitiesdependingonlyonUxµshowafastfallandΓρ2x

o fortvaluesverysmallcomparedtoτint,ρ2.Forlargervaluesofttheexponentialdescentof

.Wealwaysfoundtheinitialfallo tobelargerforΓUpl(t)andΓWl(t)isthesameasofΓρ2x

ΓWl(t)thanforΓUpl(t).

Acomparisonofautocorrelationsinthetwophasesisgivenin g.2.Inthesymmetricphaseat niteTwefoundtheautocorrelationtimeforthequantitiesUplandWltobelessthan1.Thelefthandsideof gure2displaystheextremelyfastdescentoftheseautocorrelation

(t)di ersfromthebehaviourintheHiggsphase,becausetherefunctions.ThebehaviourofΓρ2x

isastrongcurvatureinthelogarithmicplot.Thiscouldbeasignalforadensespectrumofstates.TheintegratedautocorrelationtimeinthesymmetricphaseisusuallymuchsmallerthantheoneintheHiggsphaseatthesameparametersandlatticeextensions.

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

ofρ2Table1:Integratedautocorrelationtimeτint,ρ2xfor5di erentupdatingx

schemes.Thelatticesizeis163·32,theparametersareβ=8andλ=0.0005.The rstitemisatκ=0.12885andalltheothersareatκ=0.1289.Foreachitemmorethan32000sweepswereevaluated.

heatbath

Uxµ

1

3

1

1

16333160.5±6.228.0±2.1τint,ρ2xαx3133.3±2.319.6±1.3insweeps11.4±0.4

Ona163·32latticewithparametersβ=8,κ=0.1289andλ=0.0005wecomparedfourdi erentcompositionsofthecompletesweep.TheseparametersgiveapointintheHiggsphase.

aregivenintable1insweepsandinCPUTheresultsforthelargestautocorrelationtimeτint,ρ2x

secondsofQ16,assumingthatthelatticeisdistributedonthewholemachine.AcomparisonofthethirdandfourthrowsshowsthatmoreworkontheSU(2)variableshasnoin uenceontheautocorrelation:theautocorrelationtimeinsweepsisaboutthesameforbothupdatingschemes.Thefactthatmoreoverrelaxationforρxdoesnotleadtoabetterautocorrelationisplausible[22].

Theautocorrelationmeasuredinsweepsdecreasessigni cantlyifmoreheatbathiscalledbut,duetothetimeneededfortheheatbathalgorithms,thereisnosigni cantdi erencebetweenthesecondandthethirdrowoftable1,iftheautocorrelationtimeismeasuredinCPUtime.

Theupdatingschemeinthe rstrowoftable1isthebestcombinationwefound.Itwasthereforeusedinmanypoints.BecauseofthelargeautocorrelationsintheHiggsphaseat nitetemperatures,whichweretypicallyabout10timeslongerinsweepsthanatT=0,wecouldnotcomparedi erentupdatingschemesthere.Anothercomparisonofupdatingschemeswasperformedona183·36latticewithparametersβ=8.15,κ=0.1281andλ=0.00011.Theconclusionswereverysimilar.

2.3Multicanonicalsimulation

AnimportantproblemofMonteCarlosimulationsofasystemwith rstorderphasetran-sitionisthesupercriticalslowingdown.Atthetransitionpointthetunnelingratebetweenthetwophasesisexponentiallysuppressedforanylocalupdatealgorithm(e.g.overrelaxation,

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

heatbath).Toovercomethisproblemthemulticanonicalalgorithmwasdeveloped[29].Thebasicideaisanenhancementofthemixedstates,whicharesuppressedduetotheadditionalfreeenergyoftheinterfaces.Thisenhancementisreachedbyanextratermintheaction,i.e.S→S+f(O).ThistermcanbeafunctionofanyorderparameterO.Theeasiestwayistousetheactionandacontinuousfunctionf(S)=βkS+αkwithconstantβk,αkforSintheintervalIk=(Sk,Sk+1].Infact,insteadofthelatticeactionineq.(1),weusedasanorderparameterthemodi edaction

Slog≡S[U, ] 3 xlog(ρx),(10)

whichisnaturaltotakeifρxisusedasanintegrationvariableinthepathintegral.Thischoiceisparticularlyconvenient,sincealltheoverrelaxationalgorithms(ρx,αx,Uxµ)canbeusedwithoutchanges.TheintervalsIkandtheparametersαkandβkarechoseninsucha

mcisnearly at.ThisisachievedifwaythatthemulticanonicalprobabilitydistributionPL

f(Slog)≈log(PL)betweenthetwomaximaandisconstantelsewhere.HerePListhecanonicalprobabilitydistributionoftheactionSlog.ThedistributionPLisobtainedinamulticanonical

mcsimulationbyreweightingPLwithexp(βkSlog+αk).

Inpracticea rstchoiceforthemulticanonicalparametersismadeandtheyareoptimized

mcafterwards.Ifnecessary,theprocedureisrepeateduntilPLbecomes at.A rstguesscan

beobtainedfromsmallerlattices.InthiswaythedistributionoftheactionSlogandthelinkvariableL ,de nedineq.(24),wasmeasuredon2·42·64and2·42·128latticesatthe“low”valueofthequarticcoupling(λ=0.0001).

Forlargerlatticestwoproblemsarise.Theparametershavetobetunedverypreciselyandtheautocorrelationtimesbecomeevenforoptimallytunedvaluesverylarge,oftheorderofO(10000)sweeps.Tosolvetheseproblemswecombinedthemulticanonicalmethodwiththeconstrainedsimulationmethod[30].Inwhatfollowswecallthiswayofsimulationconstrained-multicanonicalmethod.

WedividetheintervalbetweenthetwomaximaofPLintosubintervals.Thesearechosentohaveanoverlapwiththeirneighbours.Startinginonephasewetunethemulticanonical

mcparameterssuchthatPLis atinagivensubintervalandsuppressedelsewhere.Thismeans

thatf(Slog)isapproximatelyequaltolog(PL)inthissubintervalandincreasesrapidlybeyondtheboundaries.Bymovingthesubintervaloneisgoingfromonemaximumtotheother.Attheendeverysetisreweighted.Incaseoflargeoverlapsbetweenneighbouringintervalstheabsolutenormalizationcanbeobtainedwithsmallerrors.

Toensurethatthismethodyieldsthesameresultasthepuremulticanonicalone,weperformedsimulationsusingbothmethodson2·42·128lattice.Theresultscoincidewithinstatisticalerrors.Onlytheconstrained-multicanonicalalgorithmhasbeenusedforsimulationson2·82·128lattice.

ThetechnicalrealizationofthemulticanonicalapproachbytheMetropolisalgorithmisstraightforward.Asithasbeenemphasizedabove,duetoourspecialchoiceoftheorder

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

parameter,theoverrelaxationalgorithmscanbeusedwithoutchanges.Themodi cationsfortheheatbathalgorithmsaremoreinvolved(seee.g.[31]).AdescriptionofourimplementationofheatbathalgorithmsforUxµand xisgivenintheappendix.Sincetheheatbathalgorithmsaremoree cient,wealwaysusedtheminsteadoftheMetropolisalgorithms.

Inoursimulationstheacceptancerateforthemulticanonicalheatbathalgorithmswasverygood:forthemodi edgaugealgorithmatleast99%andforthe -algorithmatleast96%.Theoverlapsfortheneighbouringintervalswerechosentobeapproximately40%.Thenumberofsubintervalswas5forthe2·42·128latticeand13forthe2·82·128lattice.Theautocorrelationtimeswereonaverageabout500sweepsfortheconstrained-multicanonicalsimulations.Forthetwosmallerlatticeswemeasuredabout2000and7000sweepsasautocorrelationtimeswiththepuremulticanonicalalgorithm.

TheeasiestwaytoparallelizethemulticanonicalalgorithmontheQuadricsQ16machineistosimulateseverallatticesindependentlyoneachnode.Foranyotherimplementationthereisaneedforcommunicationbetweenthedi erentnodesforeachupdatingstep,sincefisafunctionoftheglobalaction.Anotherdisadvantageofpartitioningthelatticewouldbeadecreaseintheacceptancerateduetosimultaneouschangeofseveralvariables.

3Invariante ectivepotential

Intheperturbativeapproachtotheelectroweakphasetransitionthemostimportantquantitytocomputeisthee ectivepotential.Interestingphysicalobservableslikelatentheat,surfacetensionormassescanbeextractedfromit.Ofcourse,theselatterquantitiescanalsobeobtainedfromthenon-perturbativeapproachofnumericallatticesimulationsbymeasuringsuitableobservables.Nevertheless,adirectcomparisonofthee ectivepotentialitselffrombothmethodswouldobviouslybedesirable.Onthelattice,however,theactionisgaugeinvariantandsoaretheobservables,asdemandedbyElitzur’stheorem.Inperturbationtheorythee ectivepotentialiscalculatedindi erentgauges.(ThemostpopularoneistheLandaugauge.)Inordertocomparethiswithlatticeresultsoneoughtto xthegaugeonthelattice,anotoriouslydi culttaskinparticularfornon-abeliangaugegroups.

Awayoutisthestudyofthegaugeinvariante ectivepotentialthathasbeeninitiatedrecently[32,33]2.InthisapproachoneconsiderscompositegaugeinvariantoperatorsinthestandardLegendretransformationframework.Theobviousadvantageofthisapproachisthatthepotentialcanbeevaluatedperturbativelyand,sinceitisgaugeinvariant,itcanbedirectlycomparedtolatticesimulations.Ito ersthereforeaconceptuallycleananddirectlyaccessibletoolofconfrontingresultsobtainedinperturbationtheorywithnumericaldata.Inthispaperwewanttoreportaboutour rstexperienceswiththegaugeinvariantpotential.Wewilluseitmainlyforthedeterminationofthetransitionpoints.Wepostponeadiscussionofits

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

renormalizationandtheextractionofphysicalquantitiestoafuturepublication.

Thestartingpointforthegaugeinvariante ectivepotentialforthelengthsquareoftheHiggs eldisthefreeenergyF(J)inthepresenceofaconstantexternalsourceJ

e F(J)= [dU][d ]e S+J xρ2x,(11)

withStheactioneq.(1)and thelatticevolume.Fromthisthee ectivepotentialisobtainedbyaLegendretransformation

V(¯ρ2)=F(J(¯ρ2)) ρ¯2J,

where

ρ¯2= (12)

+1.2λ

ThelastequationiseasilyinvertedforJ(¯ρ2)andthee ectivepotentialtotreelevelis

Vtree(¯ρ2)=(1 8κ)¯ρ2+λ(¯ρ2 1)2.(16)(17)

Togettheone-loope ectivepotentialwehavetoconsider uctuationsaroundthestationarypoint(16).The uctuationstoone-loopconsistofagaugepartandaHiggspartandatthislevelnomixingappears.Oneobtains

V1 loop=Vtree+ πd4k

2 π 2+m2)+ln(kg1

2κg2ρ¯2,m2φ=4

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Thesolutionofthee ectivepotentialgivenabovecorrespondstothebrokenphaseoftheSU(2)-Higgsmodel.In[33]itwasemphazisedthatthereexistsanotherstationarypointwhichbelongstothesymmetricphaseofthemodelandwhichisgivenbyρ¯=0.Inthiscasethetreelevelpotentialistriviallyzeroandwehavetostartwiththeone-loopformulaforthefreeenergy1 2+m2),ln(k(20)F(J)=04(2π)

withm20=(1 8κ 2λ)/κ+J.Toobtainthee ectivepotentialonehastosolveeq.(13)forJ(¯ρ2).In[33]thesolutionhasbeengivenforthethreedimensionalHiggsmodelinaclosedform.Adescriptionofthee ectivepotentialinthesymmetricphasehasbeenfoundwithaquitecharacteristicasymmetricshape.

Inourcasewehavetoworkwithlatticeintegralsor nitelatticesums.Thenthesolutioncannolongerbegiveninaclosedform.However,onecanperformtheLegendretransformationandsolveeq.(13)numerically.Theresultofthisprocedureforthepointswhereoursimulationsareperformedcon rmthegeneralshapeofthepotentialinthesymmetricphaseandarequalitativelyinagreementwiththepotentialextractedfromthedistributionsofρ2valuesfromthesimulations.However,wedonothaveaquantitativeunderstandingofthesymmetricphaseyet.Wehopetocomebacktothisquestioninafuturepublication.

A nalremarkconcernsthelatticesimulationswherethegaugeinvariante ectivepotentialisobtainedfromadistributionoftheoperatorunderconsideration.Thepotentialcomputedinthiswayistheso-calledconstrainte ectivepotential[35].Inthein nitevolumelimitthispotentialcoincideswiththeonede nedbymeansoftheLegendretransformationabove.Inperturbationtheorybothapproachesdi erinthetreatmentofthezeromodes.Fortheone-loopresult(18)thisamountstoleavingoutthek=0modeinthe nitelatticesumsfortheconstrainte ectivepotential.

4Phasetransitionpoints

AnumericalsimulationoftheSU(2)-Higgsmodelshouldstartby rstdeterminingthephasetransitionpoints.Physicallythetransitionistriggeredbyatemperaturechange.Keepingallotherparameters xed,thiscouldonlybeachievedonthelatticebyasymmetriccouplings,whichispossiblebutcumbersome.Itwillbecomeclearlater(seesections5and6)thatintheparameterrangeweareinterestedinβandλare xing,toagoodapproximation,therenormalizedparameters.Achangeofκisre ectedmainlyinachangeofthelatticespacinga.Thereforeifonecrossesthetransitionat xedβ,λbychangingκ,theessentialchangeisinthephysicaltemperatureT=1/(aLt).(Thephysicalvolumeisassumedtobelargeenoughsuchthatitschangewitha3isnotimportant.)Thuswearelookingforthephasetransitioninthehoppingparameteratκ=κc,for xedβ,λ.

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure3:Thermalcycleexhibitingahysteresisinρ2.Thesolidlineindicatesthevaluesoftheabsolutminimafromthegaugeinvariante ectivepotential,theshortdashedlinetheonesofthefalseminima.Thelongdashedlineonlyconnectsthedatapointstoguidetheeye.

4.1One-loope ectivepotentialandtransitionpoints

Wefoundthatforsearchingthetransitionpointthegaugeinvariante ectivepotentialcanbeveryhelpful.Itcanserveasatooltoprovidequiteaccurateinformationaboutκcwhichhelpstoselecttheκ-valueswheresimulationsarethenperformed.Wede nethetransitionpointκcastheκ-valuewherethesymmetricandbrokenminimaofthegaugeinvariante ectivepotentialaredegenerate.Forthecomputationweusedtheone-loopformulaeq.(18)forthebrokenphaseandthetrivial,ρ2=0,minimumforthesymmetricphase.Althoughthisiscertainlynottheexactvalueforthesymmetricminimum,wewillseeinthefollowingthatforsmallλthetransitionκ′sareinverygoodagreementwithnumericaldata.

Forthecomputationofthegaugeinvariante ectivepotentialona nitelatticeofsizeLx·Ly·Lz·Lttheintegralsin(18)havebeenreplacedbythecorrespondinglatticesums.FollowingtheexperienceinQCD[36],wealsousedthemean eldimprovedgaugecoupling g→g/

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

sise ectswhenthesystemgetsstuckinthewrongminimum.Thereforehysteresise ectsinthermalcyclesareoftentakenasanindicationfora rstorderphasetransition.Thee ectivepotentialallowstocomputethevaluesofρ2alsointhefalsevacuumandshouldhencerepro-ducethehysteresis.In g.3weshowathermalcycleatβ=8,λ=0.0001ona2·42·32lattice.Thedatapoints,connectedbyalongdashedlinetoguidetheeye,showaclearhysteresis.Thesolidlinerepresentsthevaluesofρ2intheabsolutminimumobtainedfromthegaugeinvariante ectivepotential.Theshortdashedlineindicatethevaluesofρ2inthesecondminimum.The guredemonstratesthatthenumericaldataareverywelldescribedbytheone-loopgaugeinvariante ectivepotential.Theagreementgetsworseforthehighpointatλ=0.0005.Herethetransitionpointsfromtheperturbativelyevaluatedpotentialandthesimulationsshowsomediscrepancy,seetable2.

Insection7wewillcomputethelatentheat.Forthisweneedtransitionκ′sforLt=2,...,5.ForthehigherLt=4,5-valuesnumericalsimulationsareverydemandingasonewouldhavetoscaletheotherextensionsofthelatticeaccordingly.Thereforewewillresorttheretothevaluesofκcasobtainedfromthee ectivepotential.Forthispurposeweperformeda nitesizescalinganalysisofκLconvarioussizelattices

νκL+κ∞

c=aVc,(21)

whereV=Lx·Ly·LzandLtiskept xed.WecomputedκLcforvariousVfromthegaugeinvariante ectivepotentialand tteditto(21).Inallthe tsperformed,wefoundavalueofν=1.00(2).Thisagaincon rmsthe rstordernatureoftheelectroweakphasetransitionforHiggsmassesbelow50GeV.Theobtainedresultsforκ∞

cforvariousLt,βandλareplotted

in g.12wherewediscussthelinesofconstantphysics.Thenumericalvaluesinourfour

∞∞basicpointsare:l2:κ∞

c=0.128290(1);l3:κc=0.128082(1);h2:κc=0.128625(1);

h3:κ∞

c=0.128273(1).

4.2Two-couplingmethodandtransitionpoints

Providedhysteresise ectsinthermalcyclesareseenata rstorderphasetransition,thetwo-couplingmethodisusefulforaprecisedeterminationofthepositionofthephasetransitionpoint.

Letusconsiderthelargestextensionofthelatticetobethez-direction.Inthisdirectionthelatticeisdividedintotwohalves.Theideaistochoosedi erentcouplingparametersinbothhalves,toenforceoneparttostayinthesymmetricphaseandtheotheroneintheHiggsphase.Weassumethatthez-directionislongenoughforaccomodatingapairofinterfacesbetweenthephases.(Ifthez-directionisnottoolong,theappearanceofmoreinterfacepairshasanegligibleprobability.)Bywarmingupthecon gurationfarenoughinbothdirectionsfromthetransitionpointonecanachievethatatthebeginningofthesimulationattheenvisagedpairofparametersbothphasesandtheinterfacepairarepresent.Inthiscon gurationthesystemcansensitivelyreacttofreeenergydi erencesbyshiftsoftheinterfacepositionsinthe

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Table2:Hoppingparameterκcatthetransitionpoint,obtainedwiththetwocou-plingmethod.Aslatticesize,thetotalsizeofbothpartsisgiven.Forcomparison,theestimatesobtainedonlatticeswithhalfthesizefromtheone-loopinvariante ectivepotentialarealsogiven.

Lattice

8.0

8.0

8.15

8.15

8.0

8.15λκc0.12840(5)0.12828(2)0.12811(2)0.12811(1)0.12862(1)0.12826(1)

z-direction.Ifthecon gurationstaysformanyautocorrelationtimesinthemixedstatethefreeenergiesofthetwophasesatthechosenparametersaresuchthatthissituationisstableagainsttransitionstoauniquephase.Thustheparametersetsofthetwophasesgivealowerandanupperboundonthetransitionparameters.Infact,duetotheadditionalfreeenergyassociatedwiththeinterfaces,thephasetransitionfromatwo-phasesituationtoauniquephaseoccurssomewhatearlierthantheequalityofthefreeenergiesofthetwophases.Thisparametershiftgoes,however,exponentiallytozeroforincreasinglatticevolumes.

Ofcourse,thetwo-couplingmethodforthedeterminationofthetransitionpointworksonlyifoneisabletotellonephasefromtheother.Apossiblewayistoperformhysteresisrunsonlatticeswiththesameextensionsashalfofthelatticeforthetwo-couplingmethod.Thehysteresisplotsareusedforthedistinctionofthetwophasesatagivenκ.

Asstatedintheintroductionofthissection,weareinterestedinthepositionofthephasetransitioninthehoppingparameteratκ=κcfor xedβandλ.Thustheonlyparameterchosendi erentlyinbothpartsofthelatticeisthehoppingparameter.Ateachpairofκthesystemwasobservedforatleast10autocorrelationtimes.κcwasde nedasthemeanvalueofthebestlowerandupperbounds.Thebestestimatesforthetransitionpointκcobtainedwiththismethodaregivenintable2.4.3Multicanonicalmethodandtransitionpoints

Themostprecisewaytodeterminethetransitionpointhasbeenprovidedbyacombinationofthemulticanonicalmethodandκ-reweighting[37].Inthismethod rstanorderparameterdistributionisgeneratedthroughaMonteCarlosimulation.Thisdistributionisthenextrap-olatedtonearbyκ’sto ndthetransitionpoint.Theactionwiththeadditionallogarithmicterm(Slogineq.(10))andthelink-variable(L ineq.(24))havebeenconsideredasorder

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure4:Thedistributionoftheactiondensityobtainedfromaconstrained-multicanonicalsimulationon2·82·128lattice.Notethatthelefthandpeakcorre-spondstotheHiggsphaseandtherighthandonetothesymmetricphase.

parameters.Thetransitionpointisdeterminedbytheequalheightsignal:κcisthehoppingparametervalueforwhichtheheightsofthetwopeaksinthedistributionareequal.Thesystematicuncertaintyofκccanbeestimatedbycomparingitsvaluesobtainedbyequalheightsignalandequalareasignalfordi erentorderparameters.

Themulticanonicalsimulationhasbeenperformedatκ=κv,closetotherealtransitionpoint(|κv κc|=O(10 5)).Theapproximatetransitionpointκvwasdeterminedbytheone-loopinvariante ectivepotential.Thedistributionofanorderparameteratanearbyκ′,withallothercouplings xed,canbeobtainedbyattachingaweight

w(L )=e8 L (κ κv)′(22)

tothedi erentcon gurations.

Theactiondensitydistributionatthelowpointforκv=0.1283onalatticewith =2·82·128sitesisshownin g.4.Thedistributioninneighbouringpointsisobtainedbyeq.(22).Thetwopeaksareofequalheightatκc=0.128307( g.5).Asitcanbeseen,notonlytheheightsareequalbutalsothewidthsofthetwopeaksarequitesimilar,thustheequalheightconditionforκcisroughlyequivalenttotheequalareacondition.Atthesametimethe atregimebetweenthepeaksisalmostconstant.Thismeansthatinamulticanonicalsimulationthetwophasescanmixwitheachotherwithanarbitrarymixingratio.Thesupressionisduetotheinterfacesbetweenthephases.TakingL asanorderparameter,thetransitionpoint

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure5:Thereweighteddistributionoftheactiondensityobtainedfromthedataoftheprevious gureatκ=κc=0.128307.

correspondingtotheequalheightsignalisgivenbyaslightlyhigherκ=0.128308.Thisleadstoatransitionpointatκc=0.128307(2+1),wherethe rstnumberinbracketsdenotesthesystematicthesecondonethestatisticalerrorestimate.Similarlyfor =2·42·128andfor =2·42·64thetransitionpointsareatκc=0.128366(3+1)andκc=0.128367(3+1),respectively.

5Massesandcorrelationlengths

Importantcharacteristicfeaturesofanystatisticalphysicalsystemarethecorrelationlengths.Atzerotemperaturetheirinversesgivethemassesofthelowlyingparticles.Particularlyinterestingarealsothe(inverse)correlationlengthsonthetwosidesofthephasetransition.

5.1Zerotemperaturemasses

1ThephysicalHiggsmassMHcanbeextractedfromcorrelatorsofquantitiesasthesitevariableRx≡

2+Tr(αx+ µUxµαx),L ,xµ≡1

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure6:CorrelationfunctionintheHiggsbosonchannelinpointh2[16/90].Thecurveshownisthebest tfortimeslicesbetween1and11withMH=0.2663andaconstantfactorfH=4.08110 2.Theχ-squareofthis tisχ2=0.54.Thestatisticalerrorofthecorrelationfunctionatdistance1isabout0.5%,atdistance11about6%.

Table3:Theparametervaluesofnumericalsimulationsfordeterminingzerotem-peraturemassesandWilsonloops.

index

123·24

183·36

123·24

163·32

163·32

183·36β0.000100.000110.000500.000500.000500.00051κ1500001250001600006000016000050000

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

Figure7:CorrelationfunctionintheW-bosonchannelinpointh2[16/90].Thecurveshownisthebest tfortimeslicesbetween3and15withMW=0.4522andaconstantfactorfW=1.94610 5.Theχ-squareofthis tisχ2=1.30.Thestatisticalerrorofthecorrelationfunctionatdistance3isabout0.2%,atdistance15about26%.

Table4:TheW-bosonmassMWandHiggsbosonmassMHinthepointsde nedbytheprevioustable.TheirratioisRHW≡MH/MWwhichleadstotheHiggsbosonmassinphysicalunitsgiveninthelastcolumn.

index

l2

h2[12]

h2[16/90]MW1.059(24)0.144(4)0.427(8)0.253(8)0.453(3)

0.175(7)0.587(12)1.153(16)0.614(32)1.171(19)47RHW0.222(12)0.464(2)49MH(GeV)18

Numerical simulations are performed to study the finite temperature phase transition in the SU(2) Higgs model on the lattice. In the presently investigated range of the Higgs boson mass, below 50 GeV, the phase transition turns out to be of first order and

TheW-bosonmassMWcanbeobtainedsimilarlyfromthecompositelink elds(r,k=1,2,3)

Wxrk≡1

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