Center vortex model for the infrared sector of SU(

arXiv:hep-lat/0512015v112Dec2005CentervortexmodelfortheinfraredsectorofSU(4)Yang-Millstheory:StringtensionsanddeconfinementtransitionM.Engelhardt1PhysicsDepartment,NewMexicoStateUniversityLasCruces,NM88003,USAAbstractArandomvortexworld-surfacemodelfortheinfraredsectorofSU(4)Yang-Millstheoryisconstructed,focusingontheconfinementpropertiesandthebehav-ioratthedeconfinementphasetransition.AlthoughthecorrespondingdatafromlatticeYang-Millstheorycanbereproduced,themodelrequiresamorecomplexactionandconsiderablymoretuningthantheSU(2)andSU(3)casesstudiedpreviously.Itspredictivecapabilitiesareaccordinglyreduced.Thisbehaviorhasadefinitephysicalorigin,whichiselucidatedindetailinthepresentwork.AsthenumberofcolorsisraisedinYang-Millstheory,thecorrespondinginfraredeffec-tivevortexdescriptioncannotindefinitelycontinuetorelyondynamicsdeterminedpurelybyvortexworld-surfacecharacteristics;additionalcolorstructurespresentonthevorticesbegintoplayarole.Asevidencedbythemodelingeffortreportedhere,definitesignaturesofthisbehaviorappearinthecaseoffourcolors.PACS:12.38.Aw,12.38.Mh,12.40.-yKeywords:Centervortices,infraredeffectivetheory,confinement1email:[e-mail]1IntroductionThevortexpictureofthestronginteractionvacuumwasoriginallyconceived[1–5]asanexplanationforconfinement.Morerecentwork[6–23]hasestablisheditasacom-prehensive,andinmanyrespectsquantitativeinfraredeffectivemodelofthevacuum,generatingnotonlyconfinement,butalsothespontaneousbreakingofchiralsymmetryandtheaxialUA(1)anomaly.Theevidenceforthevortexpicturerestsontwopillars.Ontheonehand,methodshavebeendevelopedwhichpermittheidentificationandisolationofvortexstructuresinthegaugeconfigurationsoflatticeQCD[6,7,19].Basedonthesetechniques,thephenomenologyinducedbythevorticespresentinlatticegaugeconfigurationscanbestudied.Highlightsoftheresultsobtainedinthismannerincludevortexdominanceofthestringtension[6,7,19],theabsenceofnonperturbativeeffects(stringtension,chiralcondensate,topologicalsusceptibility)whenvorticesareremovedfromlatticegaugeconfigurations[18,19],andanaturalexplanationofthedeconfiningphasetransitionasavortexpercolationtransition[16].Ontheotherhand,aninfraredeffectiverandomvortexworld-surfacemodelhasbeendeveloped[21–27]whichdemonstratesthatthemainnonperturbativefeaturesofthestronginteractionvacuumcanbedescribed2onthebasisofaweaklycoupledvortexdynamics.Thisfurtherbuttressesthenotionthatvorticesconstitutetherelevantinfraredgluonicdegreesoffreedominthatvacuum.Throughitsreducedsetofdegreesoffreedomandthecorrespondinglysimplifieddynamics,therandomvortexworld-surfacemodelhasallowedtheinvestigationofawiderrangeofscenariathanhasbeenaccessibleusingthevorticesextractedfromlatticegaugeconfigurationsdiscussedfurtherabove.Intherandomvortexworld-surfacemodel,notonlyhastheSU(2)casebeeninvestigatedcomprehensively[21–23],butalsotheconfiningpropertiesoftheSU(3)casehavebeenstudiedindetail[24–26],andthepresentworkextendsthemodeltoSU(4)color.2Motivation2.1VortexcolorstructureToclarifythemotivationforthepresentinvestigation,itisusefultobeginbyreviewingthedifferentcolorstructureswhichcanoccuroncentervortices,specificallyinvariousAbelianformulations.Aswillbediscussedbelow,vorticesarenotonlycharacterizedbytheirlocationinspace-time,butpossessconsiderableadditionalstructureincolorspace.Ingeneral,itcannotbeexcludedthatthisstructuremaysignificantlyinfluencevortexdynamics;besidesgeometricalworld-surfacecharacteristics,thevortexeffective2IntheSU(2)andSU(3)modelsstudiedtodate,adjustmentofasingledimensionlesscouplingcon-stantsufficestoarriveataphenomenologicallyviablemodel,anumberofnonperturbativeobservablesbeingquantitativelypredictedasaresult.TheSU(4)caseinvestigatedhereturnsouttobehavelessfavorablyinthisrespect,andthereasonsbehindthisrepresentaprincipalfocusofthepresentwork.1actionpotentiallymayalsoincludeadependenceontheaforementionedcolorstructure.Investigationofthepossibilityorevennecessityofsuchadependenceisaprincipalfocusofthepresentinvestigation.Thus,areviewofvortexcolorstructureisinorder.Centervorticesareclosedtubesofquantizedchromomagneticfluxinthreespa-tialdimensions.Accordingly,theyaredescribedby(thickened)world-surfacesinfour-dimensional(Euclidean)space-time.ThequantizationisdefinedbythepropertythataWilsonloopencirclingavortex(moreprecisely,linkedtothelatter)acquiresafactorcorrespondingtoacenterelementofthegaugegroup.InSU(N)gaugetheory,therearethusN−1possiblevortexfluxes(thetrivialunitelementcorrespondstonofluxbeingpresent).Inparticular,thisentailsthatvorticescanbranchforN≥3,asdescribedingreaterdetailbelow.Thisdescriptionofvorticesintermsoftheirspace-timelocationandtheirinflu-enceonWilsonloopsisinsomerespectsincomplete.WhileitisadequateforthediscussionoftheconfinementpropertiesencodedinWilsonloops,tofullydescribethetopologicalpropertiesofvorticesandtherelatedchiralsymmetrybreakingphenom-ena,itisnecessarytotakeintoaccountthedirectionofthevortexfieldstrengthincolorspace[12–14,22,23,28,29].Whilethisdirectioncanberotatedbygaugetrans-formations,certainpropertiesofitcarrygauge-independentsignificance.Forinstance,genericvortexwo