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hep-lat/9212002 3 Dec 92November,LSUHEP--DiracStringsandtheNonperturbativePhotonPropagatorinCompactQEDKenYeeDepartmentofPhysicsandAstronomy,LSUBatonRouge,Louisiana00-00USAEmail:kyee@rouge.phys.lsu.eduABSTRACT:IntheVillainapproximationofD=+compactQED,themonopolepartofthepartitionfunctionfactorizesfromtheDiracstringpart,whichgeneratesthephotonpropagator.NumericalexperimentsinexactcompactQEDconrmthisresult:photonmasspoleM,originallynonzero,isinsensitivetomonopoleprohibitionbutalmostvanishesifDiracstringsareprohibited.

I.MotivationLatticecalculations[;]haveevaluatedthemasspolesMg;q;;eofgluon,quark,photon,andelectronpropagatorsinLGT(latticegaugetheory).Morerecently,\photon"propagatorsinabelian-projectedSU()LGTwerestud-ied[].RemarkablytheSU()photonmass,asincQED+(compactQEDin+dimensions)[],isnonzerointheconnephaseandvanishesinthedecon-nephase.Sincecomputingthesemasses,whicharegaugevariant[],requiresgaugexingtoeitherLandauormaximalabeliangaugesitisunclearwhethertheyareindicatorsofrealphysics,latticeregularizationartifacts,gaugexingartifacts[],orsomecombinationthereof.Theabelian-projectionpotentiallyprovidesaconcreteframeworkforin-vestigatingtherelationshipbetweenLGTgluonmassesandconnementviaabelian-projectedphotonmasses[].ThispossibilitybeckonsustoclarifytherelationshipbetweennonperturbativephotonmassesandconnementincQED.Tothisend,thisLetterfocusesoncQED+,aQCD-likemodelwithlocalgaugeinvariance,chiralsymmetrybreaking,andarea-lawelectroncon-nement.cQEDconnesbecausequantummonopoleuctuations[]restrictelectricuxtoAbrikosovtubesofwidthinadualMeissnereect[].RecentLGTsimulationsresolveaniteLondonpenetrationdepthwhich,followingLandau-Ginzburgtheory,canbeidentiedastheinversemassofaneectivegaugepotentialAe.Asdescribedbelow,thenonperturbativezero-momentumphotonpropaga-tormasspoleMincQED+isalsoresolvableandnonzero.Theobservationoftwomassscales,andM,raisesthefollowingquestion:How,ifatall,isAerelatedtoprimaryphotoneldAandtoM?Aretheyequivalent?

Indeed,asshownbelow,andMincQED+arenotbloodrelatives.Thegauge-invariantquantityarisesfromquantummonopoleuctuationswhileMarisesfromgauge-variantDiracstrings.II.Monopoles,Gaugexing,andDiracStringsLet@hxhx+^hx,@hxhxhx^,and@@.LinkandplaquetteanglesincQED,A+n;A<;nZ()F[A]@A@A;()dependonlyonphotonA.ThecQEDactionScP<(cosF)isinvariantunderlocalgaugetransformations@!x;A(A@!x)Mod[;)A:()Whileplaquetteexp(iF)isgauge-invariant,agaugetransformationinducingunequalshiftsofnonthefourlinksofFshiftsFbyamultiple.Fdecomposesintoagauge-invariantpart[;)andagauge-variantintegralkinkNZsuchthatF(+N);()F=N=@(A)($):()BecausecosF=cosinSc,requiredsinceNisgauge-variant,FwouldbeamisleadingapproximationtoSc.MorevalidisVillain'speriodicgaussianapproximationSc!SVcwhereeSVcXfNgePx(F[A]N);()

012345 012345012345012345 012345012345FigureA.Monopoles()andDiracstrings(|)onatypicalperiodiclatticebeforegaugexing.

PfNgQ;PN=(N;N),andFP;F.ThesumoverfNgin()isnecessarytomaintaingaugeinvarianceunder().WithinanNpotentialwelltheMaxwellequationis@F=K@N.ConservedcurrentKactsasavirtualsourceofelectromagneticcharge.QuantumNuctuationscanlead,viatheMaxwellequation,toanonzeroeectiveMifhOKiN=MhOAiN+()In()hiNreferstotheBoltzmannaverageoverNandoperatorOservestoabsorbD=+rotationalinvariance.WhileNisgaugevariant,spatialcombinationsofthemformgauge-invariantstructureswhichinuenceasfollows.AccordingtotheHodge-DeRhamtheorem=@+@@;()N=@m+@l@l()where;(;)andm;lZ.andmareinvariantunder(),transformslike,andllike(A)=.(andsimilarlyN)hasindependentpolarizationswhileandarefunctionsbecauseisinvariantunder=@!x.InvectornotationEq. ()becomes~B=~H+~wherethetotal~B,physical~H,andDiracstring~magnetic(actuallyelectromagnetic)eldsare~Br~A;~Hr+r~;~=rm+r~l:(0)Itwillbeadvantageoustorecast~intermsofitsdivergenceandcurlqr~=m;~r~=r(r~l)~l:()Sincer~B=0by(0),r~H=q,thatis,qcausesdislocationsinthephysicaleld~H.Bytautologyqisthemagneticmonopoledensity,gauge

012345 012345012345012345 012345012345FigureB.ThesamelatticeafterLandaugaugexing.Whilemonopolepositionsdonotchange,therearemoreDiracstrings.

invariantsincemisgaugeinvariant.~,acontinuouscurrentwrappingaround~,isgauge-variant.Ingeneral,kinksoccureitherinmonopoles,Diracstringsconnectingamonopoleantimonopolepair,orDiracstringloops.Loopscaneitherbehomo-logicallytrivialortoroidallywindaroundtheperiodicboundaries.Monopolechargedensityqisgauge-invariantbutthenumberofstringloopsandthelengthandshapeofallstringsvarywithgauge.FigureAshowsamonopoleandDiracstringarrangementinarepresentative=:0,periodicnon-gaugexedlattice.FigureBshowsthesamelatticeafter000Landaugauge-xingsweeps.TherearetypicallymoreDiracstringsafterLandaugaugexingthanbefore,asdepicted.III.FactorizationofPhotonPropagatorfromMonopolesUponadoptingtheconditionr~l=0andignoringLaplacianzeromodes,Eqs.()-()constitute-to-variableschangesfNg!fm;~lg!fq;~g!f;~gwhere,ifx;y=x;y,@=0,andx;y=;x;y,then=Pyx;yqyand=APyx;yy.Following(),(),and()ZVcZAeSVc=Zm[0]ZAl[0];()Zm[]XfqgePxxqxPx;yqx(xy)qy;()ZAl[0]=ZePxF[];(;):()Eq.()reliesonthequadraticcharacteroftheVillainapproximationandPxF@=0;O()correctionsfromthecossofScwouldruinfac-torization. ()says~isamasslessnoncompactvectorparticle.

Thedilutegasexpansionandoccupationnumberresummationoverqf0;gofZmin()yields[]Zm[]/ZePx(r())cos()where=e(0).Scalarissemiclassicallyidentiedwithin()because,sincePxxqx=Px@x@x,comparing()to()yieldsh@im=ZmZm@j!0=h@i:()Following(0)thisidenticationyieldshr~HiAlm=hqim=0;()h~Hy~HxiAlm=h(r~)y(r~)xiAl+ZmZmryrx:()If=<<,()and()arereproducedbyanM=freephoton~Aewith~Her~Ae,thatis,r~He=0andh~Hey~Hexieh~Hy~HxiAlm.Thelatterrelationreliesonthemasslessnessof~in(),aconsequenceof(),andcos!=in().~AeisthemassiveLandau-GinzburgphotonandtheLondonpenetrationdepth.The~ApropagatorisgeneratedbyZAl[J],denedbyaddingJAtotheactionin(),whichdoesnotaectfactorization().Thusthe~AmassMhasnothingtodowithmonopolesqand,hence,Misnotdirectlyrelatedto.Aremaininglogicalpossibility,whichwedonotexclude,isthatandMarecoincidentallyrelatedviathestatisticalmechanicsofmonopolesand~loopgasses.~A,unlike~,maybemassivebecauseJAbreaksthepure~-dependentformofZAl[0].Manipulationslikethoseleadingto()yieldZAl[J]=ZAXf~gePx(J+)AFPy:

Figure. (A)cQEDwithactionSc;(B)cQEDwithmonopolesforbidden;(C)cQEDwithkinksforbidden;(D)quadraticapproximationSc!SE,kink-creatinggaugetransformationsallowed;(E)quadraticapproximationSc!SE,kink-creatinggaugetransformationsforbidden.

ZAlisthepartitionfunctionofaCoulombic~loopgas,\loop"sincer~=0.Interestingly~isamixedstateinthegassinceforMtobenonzeroPy;y0x;y0;y0hy;y0;i~musthaveanegativenormmasslessmodetocancelthepoleandanindependentMmode.NotethateventhoughphotonsAareunchargedtheysuerconnementsince,heuristically,the\ad-joint"WilsonloopobeyshQlloopAlihQlloopsinli/RehQlloopeili,wherecrosstermsaresuppressedbygaugeinvariance.IV.NumericalExperimentsWehaveshownthattheApropagatordecouplesfrommonopolesintheVillainapproximationand,accordingly,Misnotaconsequenceofmonopoles.NumericalexperimentssummarizedinFiguresupportthisresultinexactcQED+.M,adimensionlessnumberinD=+,isthelogoftheratioofsuccessive~p=0photonpropagatortimeslices.Thenumer-icalphotonoperatorandgaugeconditionareSsinand@S=0.Thecomputationisbasedon00congurationsonlattices.Therstconguration|independentlyforeachvalue|isthermalizedby00forty-hit,0%-acceptanceMetropolissweepsand000checkerboardgaugex-ingsweeps.Congurationsthereafterareseparatedbyforty-hitMetropolissweepsand000checkerboardgaugexingsweeps.Errorsarejackknifesigmasbasedon00-congurationsubaverages.Congurations0areomittedfromtherstsubaverage,00fromthesecond,andsoforth.\A"inFigurereferstocQEDinLandaugauge;\B"toLandaugaugecQEDwithmonopolesprohibited[];\C"and\E"tocQEDwithkinkspro-hibited.Monopolesprohibitionisimplementedbystartingwiththe=00

congurationandlinkwiseforbiddingmonopole-creatingupdateswiththein-sertionofQfxgq;0intothelinkmeasure.KinksareprohibitedeitherbyinsertingQfNgN;0intothelinkmeasure("C")orbyreplacingcosFinScwithSEF(\E")where()denesF.Landaugaugexing,whichcannotchangeq,proceedsnormallyinB.Sincemaintainingkinkprohibitiondisallowskink-creatinggaugetransformationswhichmaybenecessarytoachieveLandaugauge,agoodLandaugaugeisnotachievedinCandE.Nonthelessthephotonpropagatoriseasilyresolvedinallcasesand,inagreementwithfactorization(),Misrelativelyinsensitivetomonopoleprohibitionbutverysensitivetokinkprohibition.ThesmallresidualMinCisattributabletotheO()termsinScwhichruinfactorization().\D"isalsobasedontheactionSE.UnlikeinE,kink-creatinggaugetransformationsarepermittedduringLandaugaugxinginD.FromtheSEstandpointD,corruptedbyaction-changingkinkcreationandannihilation,isgaugeinequivalenttoE.ThedierencebetweenMinDandE,gaugeequivalentfromtheScviewpoint,isanindicationofhowmuchkinksgeneratedbytheLandaugaugexingalgorithmcontributetoM.ThesmallnessofMinDsuggeststhatthekinksresponsibleforMinAarepresentinthepre-gaugexingcongurationsandnotcreatedduringLandaugaugexing.Mandkink-numberdensity,twogauge-variantquantities,arecorre-lated.IncasesA-Eat=:inLandaugauge,A=:(:0),C0,D=:(:00),andE0.Sincethe=:monopolenumberden-sityis:0(:)0,forbiddingmonopolesdoesn'tchangethekinkdensityandB=A.InthesamerangetheAxialgaugeAissmallerthantheLandaugaugeAby0%andMsmallerby0%.

AcknowledgementsIhavebenettedfromdiscussionsonaspectsofmonopoles,superconduc-tivity,andeectiveactionswithDanaBrowne,VandanaSingh,H.R.Fiebig,Lai-HimChan,ClaudeBernard,andespeciallyDickHaymaker,whoinspiredmetothinkabouttheLondonrelation.KYissupportedbyDOEgrantDE-FG0-ER0;computationwasattheLSUConcurrentComputingLab.REFERENCES.J.MandulaandM.Ogilvie,Phys.Lett.B();C.Bernard,D.Murphy,A.Soni,K.Yee,Nucl.Phys.B(Proc.Suppl.)(0);A.NakamuraandR.Sinclair,Phys.Lett.B(0)..P.Coddington,A.Hey,J.Mandula,M.Ogilvie,Phys.Lett.B()..P.CeaandL.Cosmai,BaripreprintBARI-TH/,unpublished..M.Ogilvie,Phys.Lett.B();C.Bernard,A.Soni,K.Yee,Nucl.Phys.B(Proc.Suppl.)0()0;K.Yee,BNLpreprint#,submittedtoNucl.Phys.B[FS]..Ph.deForcrand,J.Hetrick,A.Nakamura,M.Plewnia,Nucl.Phys.B(Proc.Suppl. )0();S.Petrarca,C.Parrinello,A.Vladikas,M.Paciello,B.Taglienti,Nucl.Phys.B()..K.Yee,workinprogress.

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