Numerical Approach to CP-Violating Dirac Equation

arXiv:hep-ph/9602269v110Feb1996SAGA–HE–99February10,1996NumericalApproachtoCP-ViolatingDiracEquationKoichiFunakuboa,1,AkiraKakutob,2,ShoichiroOtsukib,3andFumihikoToyodab,4a)DepartmentofPhysics,SagaUniversity,Saga840Japanb)DepartmentofLiberalArts,KinkiUniversityinKyushu,Iizuka820JapanAbstractWeproposeanewmethodtoevaluatethechiralchargeflux,whichisconvertedintobaryonnumberintheframeworkofthechargetransportscenarioofelectroweakbaryogenesis.Bythenewmethod,onecancalculatethefluxinthebackgroundofanytypeofbubblewallwithanydesiredaccuracy.1e-mail:[e-mail]2e-mail:[e-mail]3e-mail:[e-mail]4e-mail:[e-mail]1IntroductionInthescenarioofelectroweakbaryogenesis[1],thechiralchargeflux,whichisconvertedtobaryonnumberbythesphaleronprocess,isoneofkeyingredients.Thatissupposedtobegeneratedbytwomechanisms,bothofwhichdependontheinteractionofthefermionswiththeCP-violatingbackgroundofthebubblewallcreatedattheelectroweakphasetransition(EWPT).Oneofthemechanismsisthechargetransportscenario,whichisquantummechanicalandeffectiveforthinwalls,theotheristhespontaneousbaryogenesis,whichisclassicalandeffectiveforthickwalls.Asforthechargetransportscenario,thefluxisdeterminedbythedifferenceinthereflectioncoefficientsoftheleft-andright-handedfermions,aswellasbythatinthedistributionfunctionsinthebrokenandthesymmetricphasesdividedbythebubblewall.ThereflectioncoefficientsarecalculatedbysolvingtheDiracequationinthebackgroundofthebubblewall,whichisaccompaniedspatiallyvaryingCPviolation.Cohen,etal.[2]firstevaluatedthefluxbysolvingtheDiracequationnumerically,assumingthekink-typeprofileforthemodulusoftheHiggsfieldandafunctionlinearinthekinkforthephase.Ontheotherhand,weformulatedaperturbativemethodapplicableforsmallCP-violatingphaseandprovedvariousrelationsamongthereflectionandtransmissioncoefficients[3,4].Althoughitenablesustoderiveanalyticalrelations,itcanactuallybeappliedonlytothecaseinwhichtheunperturbedDiracequationisexactlysolvedandtheCP-violatingphaseissufficientlysmall.HowevertheprofileofthebubblewallshouldbedeterminedbythedynamicsoftheEWPT.Wepointedout,bysolvingtheequationsofmotionforthegauge-Higgssystem,thattheCP-violatingphasecanbeofO(1)evenifthatissmallinthebrokenphaseatT=0,accordingtotheparametersintheeffectivepotential[5].ForalargerCP-violatingphase,wenaivelyexpectlargerbaryonnumbertobegenerated.Soweneedamethodtoevaluatethechiralchargefluxforarbitraryprofileofthebubblewall.In[2],theone-dimensionalDiracequationissolvedwithinafiniterange[0,z0]includ-ingthebubblewall,imposingtheboundaryconditioninwhichplanewavesareinjectedattheboundary.Strictly,aplanewaveisnotaneigenfunctionatanyfinitez,sothatthechoiceoftherangewillaffecttheprecisionoftheresults.Inthispaper,wepro-poseanotherproceduretosolvetheDiracequationandtocalculatethereflectionandtransmissioncoefficients.Althoughthemethodisnumerical,itsprecisioncanbesocon-trolledthatonecanobtaintheresultswithanyaccuracy.Insection2,wewritedowntheCP-violatingDiracequationandgivetheexpressionsofthereflectionandtransmissioncoefficientsofthechiralfermions.Theprocedureofthenumericalanalysisisgiveninsection3,anditisappliedtosomeprofiles,includingthosewhichwefoundin[5]and2thatin[2]forcomparison,insection4.Thefinalsectionisdevotedtosummaryanddiscussions.2CP-ViolatingDiracEquation2.1TheDiracequationWeconsiderafermioninthebackgroundofacomplexscalar,towhichitcouplesbyaYukawacoupling.Forthescalartohaveanontrivialcomplexvacuumexpectationvalue(VEV),weassumeitisapartofanextendedHiggssectoroftheelectroweaktheory,suchasMSSMormoregeneraltwo-Higgs-doubletmodel.Attheelectroweakphasetransition,theHiggschangesitsVEVfromzerotononzerovaluenearthebubblewallcreatedifthephasetransitionisfirstorder.ThentheDiracequationdescribingsuchafermionisi∂/ψ(x)−m(x)PRψ(x)−m∗(x)PLψ(x)=0,(2.1)wherePR=1+γ52,PL=1−γ52andm(x)=−fhφ(x)iisacomplex-valuedfunctionofspacetime.HerefistheYukawacoupling.Whentheradiusofthebubbleismacroscopicandthebubbleisstaticormovingwithaconstantvelocity,wecanregardm(x)asastaticfunctionofonlyonespatialcoordinate:m(x)=m(t,x)=m(z).Puttingψ(x)=eiσ(−Et+pT·xT),(σ=+1or−1)(2.2)(2.1)isreducedtohσ(γ0E−γTpT)+iγ3∂z−mR(z)−iγ5mI(z)iψE(pT,z)=0,(2.3)wherepT=(p1,p2),xT=(x1,x2),pT=|pT|,γTpT=γ1p1+γ2p2,m(z)=mR(z)+imI(z).IfwedenoteE=E∗coshηandpT=E∗sinhηwithE∗=qE2−p2T,pTin(2.3)canbeeliminatedbytheLorentztransformationψ7→ψ′=e−ηγ0γ5ψ:∂zψE(z)=iγ3h−σE∗γ0+mR(z)+iγ5mI(z)iψE(z).(2.4)3InourworkontheperturbativetreatmentofCPviolation,weusedtheeigenspinorofγ3,sinceunperturbativepartoftheDiracequationcontainsonlyγ3.Hereweshallusethechiralrepresentationoftheγ-matricesasin[2].SotheexpressionoftheDiracequationisthesameasthatin[2],butwewriteitforself-containedness.Inthechiralbasis,theγ-matricesarerepresentedasγ0=0−1−10,γk=0σk−σk0,γ5=100−1.Ifwewritethefour-spinorasψE(z)=ψ1(z)ψ2(z)ψ3(z)ψ4(z),theDiracequationisdecomposedintotwotwo-componentequations:∂zψ1(z)ψ3(z)=iσE∗m∗(z)−m(z)−σE∗ψ1(z)ψ3(z),(2.5)∂zψ4(z)ψ2(z)=iσE∗m(z)−m∗(z)−σE∗ψ4(z)ψ2(z).(2.6)Nowletusintroducedimensionlessparameters.Supposethataistheparamete