Hierarchy problem in lattice Electroweak theory

arXiv:0804.0140v3[hep-ph]15Jul2008HierarchyprobleminlatticeElectroweaktheoryITEP-LAT/2008-09A.I.Veselov,M.A.ZubkovITEP,B.Cheremushkinskaya25,Moscow,117259,RussiaAbstractWediscussHierarchyprobleminlatticeElectroweaktheory.WeinvestigatenumericallylatticeWeinberg-SalammodelforrealisticvaluesofthefinestructureconstantandtheWeinbergangle.WealsoanalyzethedataofthepreviousnumericalinvestigationofSU(2)GaugeHiggsmodel.Theindicationisfoundthatthefinetuningisnotpossibleonthelattice.ThismeansthatmovingalongthelineofconstantphysicswhentheultravioletcutoffΛ(theinverselatticespacing)isincreased,oneshouldleavethephysicalHiggsphaseofthetheoryatacertainvalueofΛthatisabout430±40Gev.1IntroductionItiswell-knownfromthepointofviewofperturbationtheorythattheenergyscale1TeVappearsintheHierarchyproblem[1].Namely,themassparameterμ2forthescalarfieldreceivesaquadraticallydivergentcontribu-tioninoneloop.Therefore,theinitialmassparameter(μ2=−λcv2,wherevisthevacuumaverageofthescalarfield)shouldbesettoinfinityinsuchawaythattherenormalizedmassμ2Rremainsnegativeandfinite.Thisisthecontentoftheso-calledfinetuning.Itiscommonlybelievedthatthisfinetuningisnotnatural[1]and,therefore,oneshouldsetupthefiniteultravi-oletcutoffΛ.Fromtherequirementthattheone-loopcontributiontoμ2islessthan10|μ2R|onederivesthatΛ∼1TeV.However,strictlyspeaking,thepossibilitythatthementionedfinetuningtakesplaceisnotexcluded.InthispaperweconsiderlatticerealizationofElectroweaktheory(with-outfermions).Thephasediagramofthecorrespondentlatticemodelcon-tainsphysicalHiggsphase,wherescalarfieldiscondensedandgaugebosonsZandWacquiretheirmasses.Thisphysicalphaseisboundedbythephase1transitionsurface.Crossingthissurfaceoneleavesthephysicalphaseandentersthephaseofthelatticetheorythathasnothingtodowiththecon-ventionalcontinuumElectroweaktheory.Inlatticetheorytheultravioletcutoffisfiniteandisequaltotheinverselatticespacing.Thephysicalscalecanbefixed,forexample,usingthevalueoftheZ-bosonmassMphysZ∼90GeV.Thereforethelatticespacingisevalu-atedtobea∼[90GeV]−1MZ,whereMZistheZbosonmassinlatticeunits.Withinthephysicalphaseofthetheorythelinesofconstantphysics(LCP)aredefinedthatcorrespondtoconstantrenormalizedphysicalcouplings(thefinestructureconstantα,theWeinbergangleθW,andHiggsmasstoZ-bosonmassratioη=MH/MZ).ThepointsonLCPareparametrizedbytheUl-travioletcutoffΛ∼[90GeV]M−1Z.Ingeneral,therearetwopossibilities:eitherLCPcorrespondenttorealisticvaluesofα,θW,andη,remainsinsidethegivenphasewhenΛisincreased,oritcrossestheboundaryatacertainvalueofΛ=Λc.InthesecondcaseΛcisthemaximalpossibleultravioletcutoffinthelatticeEletroweaktheory.InthispaperweinvestigatenumericallylatticerealizationofWeinberg-Salammodel.AlsoweanalyzeexistingdataofthenumericalinvestigationoftheSU(2)Gauge-Higgsmodel.WefindtheindicationsthatthereexiststhemaximalpossibleultravioletcutoffΛc.WeexpectΛctobeabout430±40Gev.Itisworthmentioningthatthereexistsalsotheso-calledtrivialityprob-lemintheStandardModel.FromthepointofviewofperturbationtheoryitisrelatedtoLandaupoleinscalarfieldselfcouplingλandinthefinestructureconstantα.TheLandaupoleinfinestructureconstantisrelatedtothefermionloopsand,therefore,isoutofthescopeofthepresentpaper(weneglectdynamicalfermionsinourconsideration).Asforthetrivialityproblemrelatedtoλ,thecorrespondentenergyscaleismuchlargerthan1TevforsmallHiggsmasses(lessthanabout350Gev).Theconsideration,however,becomesnontrivialwhenλ→∞,andtheperturbationexpansioninλcannotbeused.InthiscaseHiggsmassapproachesitsabsoluteupperbound1andbothtrivialityandHierarchyscalesapproacheachother.1AccordingtothepreviousinvestigationsoftheSU(2)Gauge-Higgsmodelthisupperboundcannotexceed10MW.22LatticeWeinberg-SalammodelBelowweusethefollowinglatticevariables:1.ThegaugefieldU=(U,θ),whereU=U11U12−[U12]∗[U11]∗!∈SU(2),eiθ∈U(1),(1)realizedaslinkvariables.2.AscalardoubletΦα,α=1,2.(2)TheactioncanbeconsideredinthefollowingformS=βXplaquettes((1−12TrUp)+1tg2θW(1−cosθp))+−γXxyRe(Φ+UxyeiθxyΦ)+Xx(|Φx|2+λ(|Φx|2−1)2),(3)wheretheplaquettevariablesaredefinedasUp=UxyUyzU∗wzU∗xw,andθp=θxy+θyz−θwz−θxwfortheplaquettecomposedoftheverticesx,y,z,w.Hereλisthescalarselfcoupling,andγ=2κ,whereκcorrespondstotheconstantusedintheinvestigationsoftheSU(2)gaugeHiggsmodel.θWistheWeinbergangle.BarefinestructureconstantαisexpressedthroughβandθWasα=tg2θWπβ(1+tg2θW).(4)TherenormalizedWeinbergangleistobecalculatedthroughtheratioofthelatticemasses:cosθW=MW/MZ.Therenormalizedfinestructurecon-stantcanbeextractedthroughthepotentialfortheinfinitelyheavyexternalchargedparticles.Latticemodelwiththeaction(3)wasinvestigatednumericallyinthenumberofpapers.MostofthepapersdealtwiththeSU(2)Gauge-Higgsmodel,i.e.withthecaseθW=π/2.ThesystemwitharbitraryθWhasbeeninvestigatednumericallyatunphysicallylargeαin[2].HerewelistsomeofthepapersthatconsiderSU(2)Gauge-Higgsmodelatrealisticvaluesofβaroundβ=8:[3,4,5,6,7,8,9,10,11,12,13,14,15,16].Implyingthatthehyperchargefieldistobeincludedintoconsiderationperturbatively,onecanuseexpression(4)withsin2θW=0.23andestimateα=1110thatisnotfarfromitsphysicalvalueα(MW)=1128.33NumericalresearchofthemodelatθW=π/6He