Formation of Topological Defects in a Second Order Phase Tra

The classical evolution equations of the Abelian Higgs model are studied at temperatures below the Ginsburg temperature of a phase transition which is assumed to be second order. It is shown that the initial thermal fluctuations provide a domain structure

BROWN-HET-934

DAMTP94-5

March1994

arXiv:hep-ph/9403269v1 11 Mar 1994FORMATIONOFTOPOLOGICALDEFECTSINASECONDORDERPHASETRANSITIONRobertH.Brandenberger1)andAnne-ChristineDavis2)1)PhysicsDepartment,BrownUniversity,Providence,RI02912,USA

2)DepartmentofAppliedMathematicsandTheoreticalPhysics

andKingsCollege,UniversityofCambridge,CambridgeCB39EW,U.K.

Abstract

The classical evolution equations of the Abelian Higgs model are studied at temperatures below the Ginsburg temperature of a phase transition which is assumed to be second order. It is shown that the initial thermal fluctuations provide a domain structure

TheclassicalevolutionequationsoftheAbelianHiggsmodelarestudiedattemperaturesbelowtheGinsburgtemperatureofaphasetransitionwhichisas-sumedtobesecondorder.Itisshownthattheinitialthermal uctuationsprovideadomainstructurewhichisstableagainstlatetime uctuations.ThisresultlendssupporttotheKibblemechanismfortheformationoftopologicaldefects.

The classical evolution equations of the Abelian Higgs model are studied at temperatures below the Ginsburg temperature of a phase transition which is assumed to be second order. It is shown that the initial thermal fluctuations provide a domain structure

1.Introduction

Topologicaldefects1)areplayinganincreasingroleinvariousbranchesofphysics.Inparticular,cosmicstringsandglobaltexturesgiverisetoattractivescenariosfortheoriginofstructureintheearlyUniverse(forrecentreviewsseee.g.,Refs.2and3).Itisthereforeimportanttoobtainadetailedunderstand-ingoftherateofformationoftopologicaldefectsinphasetransitionsfromahotsymmetricphasetoacoldphasewithbrokensymmetry.

TheoriginalmechanismofadefectformationisduetoKibble1).Hearguedthatatthephasetransition,inanytheorywhichadmitstopologicaldefects,anetworkofsuchdefectswithcorrelationlength(i.e.,typicalseparation)ξwillbefrozeninattheGinsburgtemperatureTG.Here,ξisthecorrelationlengthattG,thecosmictimecorrespondingtotemperatureTG.

StartingpointofKibble’sargumentwastheassumptionthatonscaleslargerthanξ,theorientationoftheorderparameterinthevacuummanifoldisrandom,thattheorderparametersmoothlyinterpolatesbetweentheserandomvalues,andthattherethusisa niteprobabilitytohavenontrivialwinding.ThisprobabilitydependsonthetopologyofthevacuummanifoldandhasbeencalculatedinseveralinterestingcasesinRef.4.

TheKibbleargumenthasbeenwidelyusedincosmology.Forexample,ithasbeenusedtogeneratetheinitialstringcon gurations5)forcosmicstringevolutionstudies,tocalculatetheabundanceofmagneticmonopoles6),andtojustifytheoccurrenceoftextures7)inmodelswithnonvanishingπ3(M),Mbeingthevacuummanifold.AnotherapplicationoftheKibblemechanismisthepredictionofvorticesinapressurequenchofsuper uidhelium8).

ThetwomainassumptionsoftheKibblemechanismarethattheorderparam-etertakesonrandomvaluesinMonscaleslargerthanξ,andthatitsmoothlyinterpolatesbetweenitsvaluesatdi erentpointsinspace(the“geodesicrule”).Recently9),thevalidityofthegeodesicrulehasbeenchallenged,inparticular

The classical evolution equations of the Abelian Higgs model are studied at temperatures below the Ginsburg temperature of a phase transition which is assumed to be second order. It is shown that the initial thermal fluctuations provide a domain structure

forgaugetheories.Sincetheenergydensityingaugetheoriesisproportionalto(Dµφ)2andnot( µφ)2(whereDµisthecovariantderivativeoperator),theor-derparameterφneednotinterpolatesmoothlyinordertominimizethegradientenergy.Hence,ithasbecomeimportanttoinvestigatethevalidityoftheKibblemechanismmorecarefully.

Fortheorieswithaglobalsymmetry,thegeodesicruleiswelljusti ed.Inthiscase,theKibblemechanismhasbeentestedbothinnumericalsimulations10)andinthelaboratory11,12).

Fortheorieswithalocalsymmetrythesituationislessclear.Inmodelswitha rstorderphasetransition,arecentanalysis13)ofthedynamicsoftheclassical eldshasprovidedstrongsupportfortheKibblemechanism.Themainideaoftheanalysis,however,hingedonthephasetransitionproceedingviathenucleationandsubsequentcollisionofbubblesofthebrokensymmetryphase.Themethodsarethereforenotdirectlyapplicabletomodelswithasecondorderphasetransition.

InthisletterweinvestigatethesolutionoftheclassicalequationsofmotionforanAbelianHiggsmodelwithasecondorderphasetransition.Weconsidera eldcon gurationsetupbythermal uctuationsattheGinsburgtemperatureandstudyitsstabilityagainstthermal uctuationspresentatlatertimes.Weconcludethattheinitialdomainstructureispreserved,althoughnaturallytheamplitudeoftheorderparameterincreases.OurresultslendsupporttothehypothesisthattheKibblemechanismappliesalsotogaugetheorydefectsproducedinasecondordertransition.

The classical evolution equations of the Abelian Higgs model are studied at temperatures below the Ginsburg temperature of a phase transition which is assumed to be second order. It is shown that the initial thermal fluctuations provide a domain structure

2.SystemandBasicEquations

AsatoymodelweconsidertheAbelianHiggsmodelwithacomplexscalar eldφandaU(1)gaugeconnectionAµ.ItsLagrangeanis

L=(Dµφ) Dµφ V(φ) 1

|φ|2

and

νφ. µ µAν 2e2Aν|φ|2= ieφ φ=0(2.4)(2.5)

ItisconvenienttoseparateEq.(2.4)intoequationsfortheamplitudeρandphaseαofφ.Inserting

φ=ρeiα

into(2.4)weobtain

2ρ ( α eA)2ρ e2A2ρ+2 V(2.6)

The classical evolution equations of the Abelian Higgs model are studied at temperatures below the Ginsburg temperature of a phase transition which is assumed to be second order. It is shown that the initial thermal fluctuations provide a domain structure

and

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