Establishment of the Coulomb law in the layer phas

arXiv:0801.3722v1[hep-lat]24Jan2008EstablishmentoftheCoulomblawinthelayerphaseofapureU(1)latticegaugetheoryK.Farakos∗andS.Vrentzos†February3,2008PhysicsDepartment,NationalTechnicalUniversity,15780ZografouCampus,Athens,GreeceAbstractInthisarticleweexaminetheLayerphaseofthefivedimensional,anisotropic,Abeliangaugemodel.Ourresultsaretobecomparedwiththeonesofthe4DU(1)gaugemodelinanattempttoverifythatfourdimensionalphysicsgovernsthefourdimensionallayers.Themainresultsarei)FromtheanalysisofWilsonloopsweverifiedthe1Rbehavior,inthelayeredphase,forthepotentialbetweenheavycharges.Therenormalizedfinestructureconstantinthelayerphaseisfoundtobeequaltothatof4DCoulombphase,αlayer=α4D.ii)BasedonthehelicitymodulusanalysisweshowthatthelayersareintheCoulombphasewhilethetransversebulkspaceisintheconfiningphase.WealsocalculatedtherenormalizedcouplingβRandfoundresultscompatiblewiththoseobtainedfromtheCoulombpotential.Finallywecalculatedthepotentialinthe5DCoulombphaseandfound1R2behaviorforthestaticq¯qpotential.Fromthestudyofthehelicitymoduluswehaveapossibleestimateforthefivedimensionalrenormalizedfinestructureconstantintheregionofthecriticalvalueofthebaregaugecoupling.1IntroductionTheideathatweliveonahypersurfaceembeddedinahigherdimensionalspacehasattractedtheinterestofparticlephysicistsandcosmologistsinconnectionwiththehierarchyandthecosmologicalconstantproblems.Theseideasaremotivatedbystring-andM-theorythatareformulatedinmulti-dimensionalspaces.Oneversionistoconsidertheextradimensionsflat,compactifiedtoalargescaleMKK∼1RvaryingfromthePlanckscaletoafewTeVdependingonthenumberofextradimensions[1,2].Analternativeconceptisthesocalledbraneworldscenario[3],wherealloftheparticlesarelocalizedonathree-dimensionalsubmanifold(brane),embeddedinamultidimensionalmanifold(bulk),whilegravitonsarefreetopropagateinthebulk.Thismodelinfivedimensions,impliesanon-factorizablespacetimegeometryoftheformAdS5aroundthebrane,assuminganegativefive-dimensionalcosmologicalconstant.Thefour-dimensionalparticlesareexpectedtobegravitationallytrappedonthebrane.IndeedgravitonsandscalarsforthesecondRSmodel[3]havealocalizedsolutiononthebrane,plusacontinuumspectrum.Gravitons,scalarsandfermionsalsoexhibita∗E-mail:[e-mail]†E-mail:[e-mail]1normalizablezeromodelocalizedonthebraneifweassumeanontrivialscalarbackgroundinthebulk,suchasakinktopologicaldefecttowardtheextradimension[4].Themostdifficulttaskisthemasslessnon-abeliangaugefieldlocalizationonthebrane,asanyacceptablemechanismmustpreservethechargeuniversality[5].Alocalizationmechanismthatmaybetriggeredbytheextradimensionalgravityisproposedbytheauthorsofrefs[6,7,8].Itisbasedontheideaoftheconstruc-tionofagaugefieldmodelwhichexhibitsanon-confinementphaseonthebraneandaconfinementphaseonthebulk,thatistosay,ahiggsmechanismdrivenbythecouplingwiththegravity.Thusgaugefields,andmoregenerallyfermionsandbosonswithgaugecharge,cannotescapeintothebulkunlesswegivethemenergygreaterthanthemassgapΛG,whichemergesfromthenonperturbativeconfiningdynamicsofthegaugemodelinthebulk.Inapreviouswork[9]wehavestudiedthe4+1dimensionalpureAbelianGaugemodelonthelatticewithtwoanisotropiccouplings,independentfromeachotherandthecoordinates,focusingourattentiontothestudyofthephasediagramandtheorderofthephasetransitions.Withthismodelwewantedtoexplorethepossibilityofagaugefieldlocalizationschemebasedontheobservationthattheanisotropyofthecouplingsproducesanewphase,thelayerphase,whichmimicstheCoulombbehaviorinfourdimensionsbutconfinesalongtheremainingone.Thismodelisknownsincethemid-eightieswhenFuandNielsenproposeditasanewwaytoachievedimensionalreduction[10].ItisdefinedonaD-dimensionalspacecontainingd-dimensionalsubspaces.Ifthed(d−1)2couplingsinthed-dimensionalsubspacesareidentical(β)andtheremainingD(D−1)−d(d−1)2couplingcoefficientsarealsotobetakenidentical(β′),thenforacertainrangeofparameters(typicallyβ′≤1dandβ≥O(1))∗thisnewphaseemerges.Theconfinementalongthe(D−d)directionsandtheresultingdetachmentofthed-dimensionallayersleadstothefollowingphysicalpicture.Chargedparticlesinthelayerphasewillmainlyrunonlyalongthelayerssinceiftheyattempttoleavethelayersinwhichtheybelongtheywillbedrivenbackbyalinearpotential,analogoustotheoneresponsibleforquarkconfinement.Alsogaugeparticleswillfollowthelayerssincethereisnomasslessparticle(photon)movingacrossthelayers.WemustnoteherehoweverthatastablelayerphaseexistsonlyifD≥5andd=D−1fortheU(1)model.Ford≤3wecannothavealayerphasesincelatticegaugetheoryinlessthanfourdimensionsexhibitsconfinementforallfinitevaluesofthecouplingconstant,renderingthe3-dimensionalsubspacesofthemodelincapableofrealizingaCoulombphase.Also,duetotheasymmetricroleofβandβ′intheaction,thereisnolayerphasewiththeirrolesreversed.ManynumericalinvestigationsofthemodelhavebeenmadeusingMonte-Carlotechniques,ver-ifyingthestructureofthephasediagramandthepropertiesofthedifferentphases[11,12,13].Inoneofthem[11]thepresenceoffermionsinthemodelwasinvestigated.Theanalysis,bothanalytical(meanfieldapproximation)andnumerical(MonteCarlosimulations)revealedthatthequalitativecharacter