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hep-lat/9212016 13 Dec 92Submittedto:DOE/ER/0-PhysicalReviewDU.MDPP#-0U.KYPP#UK/-0TRIUMFPP#TRI-PP--0BaryonOctettoDecupletElectromagneticTransitionsDerekB.LeinweberDepartmentofPhysicsandCenterforTheoreticalPhysicsUniversityofMaryland,CollegePark,MD0TerrenceDraperDepartmentofPhysicsandAstronomy,UniversityofKentucky,Lexington,KY00R.M.WoloshynTRIUMF,00WesbrookMall,Vancouver,B.C.CanadaVTANovember,AbstractTheelectromagnetictransitionmomentsoftheSU()-avorbaryonoctettodecupletareexaminedwithinalatticesimulationofquenchedQCD.ThemagnetictransitionmomentfortheN$channelisfoundtobeinagreementwithrecentexperimentalanalyses.Thelatticeresultsindi-catep=p=0:().IntermsoftheParticleDataGroupconvention,fM=0:()GeV=forp$+transitions.LatticepredictionsforthehyperonMtransitionmomentsagreewiththoseofasimplequarkmodel.Howeverthemannerinwhichthequarkscontributetothetransi-tionmomentsinthelatticesimulationisdierentfromthatanticipatedbyquarkmodelcalculations.Thescalarquadrupoleformfactorexhibitsabe-haviorconsistentwithpreviousmultipoleanalyses.TheE=Mmultipoletransitionmomentratiosarealsodetermined.ThelatticeresultssuggestREMGE=GM=+%forp$+transitions.OfparticularinterestaresignicantnonvanishingsignalsfortheE=Mratioinandelectromagnetictransitions.TypesetusingREVTEX

I.INTRODUCTIONOneofthegreatpromisesofthelatticegaugeapproachtoQCDistorevealthequarksub-structureanddynamicsofhadrons.LatticecalculationsofSU()-avoroctetanddecupletbaryonelectromagneticformfactors[,]havemadesignicantstepsinthisdirection.Cal-culationsofquarkchargedistributionradiihavedescribedthemannerinwhichthequarksaredistributedwithinbaryonsandhowthesedistributionschangefromonebaryontothenext.Themagneticpropertiesofquarkswithinbaryonshavealsobeenexamined.Astrongsensitivitytotheenvironmentinwhichaquarkresidesisseeninthequarkcontributionstotheoctetbaryonmagneticmoments.Evidencesuggestsdynamicalquarkmasseects,nonperturbativegluoninteractionsandrelativisticdynamicsarethemechanismsunderlyingthisphenomena.ThelatticecalculationshavealsogivenusaccesstomanyQCDobserv-ablesthatotherwisearenotavailableatpresentfromlaboratoryexperiments.Thisnewinformationwillbeindispensableinboththedevelopmentandtestingofmodelhypothesesforlow-energyhadronphysics.InthispaperwecontinueourexplorationofbaryonelectromagneticstructurethroughcalculationsoftheelectromagneticmultipoleformfactorsdescribingtheN$transitionaswellasotheroctettodecupletbaryonelectromagnetictransitions.TheelectromagnetictransitionsofN$havebeenthesubjectofintensestudysincethepreliminaryanalysesofthephotoproductiondata[{]inwhichanonvanishingvaluefortheE=Mratioofformfactorswasfound.Thenitevalueofthisratioindicatessomedeviationfromsphericalsymmetryinthenucleonand/orgroundstatewavefunctions.Thecalculationandpredictionoftransitionmomentsisanintegralpartofthedevelop-mentandtestingofmodelhypotheses.Bycalculatingthetransitionmomentsoftheentirebaryonoctetwehopetodiscoverthedependenceofthetransitionmomentsonthequarkmassandmakemoreextensivecomparisonswithmodelcalculations.Itwillbeinterest-ingtolearnwhetherthequarkelectromagneticpropertiesresemblethoseofquarksinoctetbaryonsordecupletbaryons.Acomparisonoftheeectivequarkmomentsdeterminedfromthetransitionmomentswiththosedeterminedinourpreviousanalyseswillgiveagreatdealofinsightintotheconceptofconstituentquarksandintrinsicmoments.Moreover,thelat-ticeresultswillprovideaccesstomanymoretransitionmomentswhichmaybeusefulformodeldevelopment.WhilemostofthepresentattentionisdirectedtoadeterminationoftheE=Mtran-sitionmomentratioitshouldbenotedthattheactualvalueoftheMtransitionmomentisitselfnotwellknown.Factorscontributingtothisarethelongstandingandwellknownproblemsassociatedwithdeningthepropertiesofahadronunstabletostronginteractions.Inthislatticecalculation,theseissuesdonotpresentaproblemuntilweattempttomakeacomparisonwiththeexperimentalmultipoleanalyses.Forthequarkmassesconsideredonthelatticetheisstable,sinceenergyconservationpreventsitfromdecayingtoamoremassiveNstate.Anobviousapproachtothedenitionofpropertiesonthelatticeistocalculatewheretheisstableandsmoothlyextrapolatethevaluestothephysicalquarkmasses.Inessence,thisdenesamodelforwhatwemeanbythepropertiesofthe.Suchadenitioniscommontomanymodelsofhadronstructure,suchasthesimplequarkmodel,whichhaveexcitationswhicharestabletostronginteractions.However,theshouldberegarded

asdressedasthepossibilityofZ-graphsinthequarkpropagatorsallowintermediatestatesofthewithmultipleqqstates.TheseintermediatestatesareexpectedtohaveoverlapwithmesonicdressingsofincludingvirtualNintermediatestates[].ThequarkmassdependenceoftheMtransitionmomentisextremelyweak.Anyproblemsassociatedwiththeextrapolationtothephysicalquarkmassesarenotapparent.EarlylatticecalculationsofelectromagneticformfactorsfocusedonthepionwithSU()color[{]andlaterwithSU()color[0{].Calculationsoftheprotonelectricformfactorfollowed[].Electromagneticformfactorsof,andNwerecalculated[]fromwhichmagneticmomentsandelectricchargeradiiwereextracted.Ouranalysisoftheentirebaryonavor-octetfollowed[]inwhichelectromagneticpropertieswerereportedforbothbaryonsandthequarksectorcontributions.Theqdependenceofthenucleonelectromagneticformfactorswasexaminedusingamethodwhichcharacterizesoneofthenucleoninterpolatingeldsasazeromomentumsecondarysource[].Finally,ourexaminationofthebaryondecuplet[]revealedunderlyingquarkinteractionsinwhichmasseectsandspin-dependentforcescounteracteachother.Thesignicantbaryondependenceofquarkelectromagneticpropertiesseenintheoctetwasnotreproducedindecupletbaryons.Thestudyofhadronicwavefunctionshasalsoreceivedsomeattentionandhasconrmedthepresenceofspin-dependentinteractionsinlatticesimulationsofQCDthatcausethedistributionofthedquarksintheneutrontobelargerthanthatoftheuquark[].Similarspin-dependentinteractionswereseeninourcalculationsofoctetbaryonelectromagneticformfactorsandareresponsibleforthenegativesquaredchargeradiusoftheneutron.Thewavefunctionanalysesalsoindicateanabsenceofsubstantialscalardiquarkclustering,inagreementwithourcalculationsofoctetanddecupletbaryonchargeradii[].AcomparisonofwavefunctionscalculatedinquenchedandfullQCD[]indicatesthereislittledierencebetweenthetwocalculationsoutsideofasimplerenormalizationofthecouplingconstant.Thissupportstheusefulnessofthequenchedapproximation,atleastforthequarkmassescurrentlyinvestigatedonthelattice.Thequalitativebehaviorofchargedistributionsinthenucleonhavebeenconrmedinwavefunctionanalyses[]whichemployaverydierentapproachinobtaininginformationonquarkdistributions.Unfortunatelyaquantitativecomparisonisnotpossibleaswavefunctionsarenotdenedinagaugeinvariantmanner.Wavefunctionscalculatedindierentgaugeshavedierentshapesanddistributionsizes.Incontrasttoourlatticeformfactoranalyses,itisnotpossibletocalculate,forexample,achargeradiusfromthewavefunctionsthatmaybedirectlycomparedwithexperimentalmeasurements.Theformatofourpaperisasfollows.InsectionII,wereviewtheinterpolatingeldsusedtoexcitetheoctetanddecupletbaryons.Thetwo-andthree-pointcorrelationfunctionscorrespondingtotheseinterpolatingeldsarepresentedatthequarklevel.Thetransitionmomentsfor$0arenotreportedhereasthecorrelationfunctionsforthistransitiondierfromthoseoftherestoftheoctet.ThesetransitionmomentswillbeexaminedinafuturelatticeinvestigationofHbaryonsandheavyquarksymmetry.SectionIIIreviewsthecovariantmatrixelementthatdenesthebaryonmultipoletransitionmoments.Theformalismdevelopedtoisolateandextracttheformfactorsfromthecurrentmatrixelementispresentedindetail.LatticetechniquesarebrieysummarizedattheendofsectionIII.Calculationsofthethree-pointcorrelationfunctionsusedtoextracttheformfactorsfromthelatticesimulationsareillustratedinsectionIV.Toprovidesomebackgroundtothestudy

oftransitionmoments,wehavegeneralizedthequarkmodelofDarewych,HorbatschandKoniuk[]forandtransitions,toincludenucleonandtransitions.ThesequarkmodelpredictionsaresummarizedinsectionV.SectionVIpresentsthelatticedeterminationsofthetransitionmomentsandcomparesthelatticeresultswithmodelexpectations.SectionVIIpresentsanoverviewoftheresultsandanoutlookonfuturecalculations.II.CORRELATIONFUNCTIONSATTHEQUARKLEVELA.InterpolatingeldsFortheoctetbaryoninterpolatingeldsweusethefollowingstandardforms.Fortheproton,p(x)=abcuTa(x)Cdb(x)uc(x):(.)Thisinterpolatingeldhastheadvantageofexcludingcomponentswhichvanishinanon-relativisticreductionandwhichotherwiseacttoincreasestatisticaluncertaintiesinlatticesimulations[].Unlessotherwisenoted,wefollowthenotationofSakurai[0].TheDiracgammamatricesareHermitianandsatisfyf;g=,with=i[;].C=isthechargeconjugationmatrix,a;b;carecolorindices,u(x)isau-quarkeld,andthesuperscriptTdenotestranspose.Asinourdecupletbaryonanalysis,weutilizethefollowing+interpolatingeld+(x)=pabchuTa(x)Cdb(x)uc(x)+uTa(x)Cub(x)dc(x)i:(.)Otherbaryoninterpolatingeldsareobtainedwiththeappropriatesubstitutionsofu(x)ord(x)!u(x);d(x)ors(x).Forthetransitionmomentsof0$0weusetheoctetinterpolatingeld0(x)=sabchuTa(x)Csb(x)dc(x)+dTa(x)Csb(x)uc(x)i;(.)andthedecupletinterpolatingeld0(x)=sabchuTa(x)Cdb(x)sc(x)+dTa(x)Csb(x)uc(x)+sTa(x)Cub(x)dc(x)i:(.)

B.CorrelationfunctionsCorrelationfunctionsatthequarklevelareobtainedthroughthestandardprocedureofcontractingoutpairsofquarkelds.Considerthep$+two-pointcorrelationfunctionatthequarklevel.hG+p(t;~p)i=X~xei~p~xhjT+(x)p(0)ji;(.)=sX~xei~p~xabca0b0c0nSaa0uCSTbb0dCScc0uSaa0dCSTbb0uCScc0u(.)+Saa0utrhCSTbb0dCScc0uiSaa0utrhCSTbb0uCScc0dio;wherethequark-propagatorSaa0u=Saa0u(x;0)=Tua(x);ua0(0)andsimilarlyforotherquarkavors.HererepresentstheQCDvacuum.Similarly,the+!pcorrelationfunctionhasthefollowingformatthequarklevelhGp+(t;~p)i=X~xei~p~xhjTp(x)+(0)ji;(.)=sX~xei~p~xabca0b0c0nSaa0uCSTbb0uCScc0dSaa0uCSTbb0dCScc0u(.)+Saa0utrhCSTbb0uCScc0diSaa0utrhCSTbb0dCScc0uio;Bothofthesetwo-pointcorrelationfunctionsvanishunderSU()-isospinsymmetryasrequiredbytheisospininvarianceofstronginteractions.HoweverwithSdreplacedbySsasin+thereremainssomeoverlapbetweentheinterpolatingelds.Unfortunately,thisoverlapisinsucienttoextractanyusefulinformationonthespin-/componentofthespin-/interpolatingeld.Thecorrespondingconnectedthree-pointfunctionmaybeconstructedbyreplacingeachofthethreepropagatorsS,oneatatime,bybSdenotingthepropagationofaquarkinthepresenceoftheelectromagneticcurrent.Thethreepointfunctionanalogousto(.)ishG+jp(t;t;~p0;~p)i=X~x;~xei~p0~xe+i(~p0~p)~xhjT+(x)j(x)p(0)ji;(. )=sX~xei~p0~xabca0b0c0(

Saa0uCSTbb0dCbScc0uSaa0dCSTbb0uCbScc0u+bSaa0utrhCSTbb0dCScc0uibSaa0utrhCSTbb0uCScc0di+bSaa0uCSTbb0dCScc0uSaa0dCbSTbb0uCScc0u(.0)+Saa0utrhCSTbb0dCbScc0uiSaa0utrhCbSTbb0uCScc0di+Saa0uCbSTbb0dCScc0ubSaa0dCSTbb0uCScc0u+Saa0utrhCbSTbb0dCScc0uiSaa0utrhCSTbb0uCbScc0di);wherebSaa0u=bSaa0u(x;0;t;~q;)and~q=~p0~p.Thethree-pointcorrelationfunctionanal-ogousto(.)followsasimilarpattern.Usingtheinterpolatingeldsof(.)and(.),itisstraightforwardtoverifytheSU()-isospinsymmetryrelationshipforthree-pointtransitioncorrelationfunctions0=++:(. )Itisusefultoexaminethesymmetriesmanifestinthesecorrelationfunctions.First,itisapparentthatoneofthedoublyrepresentedquarkscannotcontributetothetransitionmomentswhenthemassesoftheremainingtwoquarksareequal.In(.0)thenetcon-tributionoftherstfourtermsvanishesunderSU()-isospinsymmetry.Thecontributionsoftheremainingtwoquarkstotheelectromagnetictransitionmomentsdieronlybythechargesofthequarksandaminussign.Animmediateconsequenceofthisisthatallthetransitionmomentsforp$+areequalandoppositeinsigntothetransitionmo-mentsforn$0.Dierencesbetweenthemagnitudesofnandptransitionmomentsreectisospin-symmetrybreakingintheu-d-quarksectorandthecontributionsofdiagramsinwhichthephotoncouplestoadisconnectedquarkloopinteractingviagluonswiththenucleonand.UnderSU()-avorsymmetry,theelectromagnetictransitionmomentsfor$andfor$vanish.ThispointwasrstdiscussedbyLipkin[].BreakingSU()-avorsymmetrythroughtheintroductionofthemoremassivestrangequarkallowsanontrivialresultfortheandtransitions.However,themagnitudesofallthetransitionmomentsforandaregovernedbythesizeoftheu-s-quarkmasssplitting.Furthermore,thetransitionmomentsof$and$willdisplayanapproximatesymmetryanalogoustotheexactisospinsymmetryofp$+=(n$0).Thetransitionmomentsof+$+and0$0willalsodisplayasimilarapproximatesymmetry.Thespin-/componentofthespin-/hyperoninterpolatingeldsisasourceofpossiblecontaminationinandtransitionmoments.However,ourlatticeresultsfordecupletbaryontwo-pointfunctionsgivenoevidenceofalow-lyingspin-/excitationfromthespin-/componentofthehyperonspin-/interpolatingelds.Thebaryonsarethelowestlyingbaryonsinthemassspectrumhavingtheappropriatequantumnumbersandthereforeanyspin-/excitationshavealargermassandwillbeexponentiallysuppressed.Thesmallnessofthetwo-pointcorrelationfunctionsdescribingtheoverlapbetweenoctetanddecuplethyperoninterpolatingeldsfurthersupportstheabsenceofanysignicantoctetbaryoncontaminationsinthehyperoncorrelationfunctions.

Whileitmaybedesirabletousethespin-/projectionoperator[]P=(p)=gp(pp+pp);(. )onemustcalculateadditionalelementsofthematrixinDiracandLorentzspacesofthethreepointcorrelationfunctions.SuchacalculationwouldexceedourcurrentanalysisofLorentztermsandDiractermsfortwocomponentsoftheelectromagneticcurrentbyafactorof.III.CORRELATIONFUNCTIONSATTHEBARYONLEVELA.CurrentMatrixElementsInthissectiondiscussingcorrelationfunctionsatthehadroniclevel,theDiracrepresen-tationofthe-matricesasdenedinBjorkenandDrell[]isusedtofacilitatecalculationsofthe-matrixalgebra.Itisthenasimpletasktoaccountforthedierencesin-matrixandmetricdenitionsinreportingthenalresultsusingSakurai'snotation.InthefollowingwewilllabeloctetanddecupletbaryonsbyNandrespectively.However,theresultsmaybeappliedtoanyoftheoctettodecupletbaryontransitions.TheelectromagnetictransitionmomentsofN$havebeenthoroughlyexaminedandasaresultthecurrentmatrixelementforevenparitytransitionsiswellestablished.InthisinvestigationweadopttheformwhichexpressesthecurrentmatrixelementdirectlyintermsoftheSachstransitionformfactors[,].Thismatrixelementisthemostgeneralformrequiredforthedescriptionofon-shellnucleonandstateswithbothrealandvirtualphotonmomentumtransfers.ThecurrentmatrixelementforN!transitionshasthefollowingformh(p0;s0)jjjN(p;s)i=is MMNE0EN!=u(p0;s0)Ou(p;s);(.)withO=GM(q)KM+GE(q)KE+GC(q)KC;(.)whereKM=f(M+MN)qgPq(M+MN)MN;(.a)KE=KM(q)Pq(P+q)qi(M+MN)MN;(.b)KC=(q)qqPqPqi(M+MN)MN;(.c)and(q)=n(M+MN)qon(MMN)qo(.)

Momentumisdenotedbyp,p0,spinbys,s0,andu(p;s)isaspin-vectorintheRarita-Schwingerformalism[].Hereq=p0pandP=(p+p0)=.TheformfactorsGM,GEandGCarereferredtoasthemagneticdipoleM,electricquadrupoleE,andelectricchargeorscalar(timecomponentoftheelectromagneticcurrent)quadrupoleCtransitionformfactors[].Thecurrentmatrixelementfortheinversereaction!NisdenedbytheHermitianconjugateof(. ),hN(p0;s0)jjj(p;s)ih(p;s)jjjN(p0;s0)iy;=is MMNEE0N!=u(p0;s0)Ou(p;s);(.)wherewehaveusedthesymmetry0(O)y0=O:(.)B.CorrelationFunctionsToisolateandextracttheformfactorsweconsiderthefollowingensembleaveragesoftwo-andthree-pointGreenfunctionsatthehadroniclevelhGNN(t;~p;)i=X~xei~p~xhjT(x)(0)ji;(.)hG(t;~p;)i=X~xei~p~xhjT(x)(0)ji;(.)hGjN(t;t;~p0;~p;)i=X~x;~xei~p0~xe+i(~p0~p)~xhjT(x)j(x)(0)ji;(.)andhGNj(t;t;~p0;~p;)i=X~x;~xei~p0~xe+i(~p0~p)~xhjT(x)j(x)(0)ji:(.0)Here,isamatrixinDiracspaceand;areDiracindices.Thesubscript(and)istheLorentzindexofthespin-/interpolatingeld.AtthehadronicleveloneproceedsbyinsertingacompletesetofstatesjB(p;s)ianddeninghj(0)j(p0;s0)i=ZsME0u(p0;s0);(.)

whereZrepresentsthecouplingstrengthof(0)tothebaryonwhichmaybeanybaryonresonancehavingthequantumnumbersofthe.E0=(~p0+M)=andDiracindiceshavebeensuppressed.Similarly,theoverlapbetweentheoctetinterpolatingeldandthephysicalstatesisdenedashj(0)jN(p;s)i=ZNsMNENu(p;s):(.)UsingtheRarita-Schwingerspinsum,Xsu(p;s)u(p;s)=p+MM(gppM+ppM);(.);theDiracspinorspinsum,Xsu(p;s)u(p;s)=p+MNMN;(. )ourusualdenitionsfor,j= j000!

;0== I000!;(.)and~p=(p;0;0),theoctetanddecuplettwo-pointfunctionstakethefollowinglargeEu-clideantimelimits:hGNN(t;~p;)i'ZNMNENeENttrp+MNMN;(.)=ZNEN+MNEN=eENt;(.)andhG(t;~p;)i'ZBMEeEttr[]:(.)Similarly,forlargeEuclideantimeseparationsttandtthethree-pointfunctionatthehadroniclevelhasthelimithGjN(t;t;~p0;~p;)i=Xs;s0eE0(tt)eENthjj(p0;s0)ih(p0;s0)jjjN(p;s)ihN(p;s)jji:(.)wherethematrixelementoftheelectromagneticcurrentisdenedin(.)through(.),andthematrixelementsoftheinterpolatingeldsaredenedby(.)and(. ).FurthermorehGNj(t;t;~p0;~p;)i=Xs;s0eE0N(tt)eEthjjN(p0;s0)ihN(p0;s0)jjj(p;s)ih(p;s)jji;(.0)

wherethecurrentmatrixelementisdenedin(. ).ToisolateandextracttheSachsformfactorsweconstructthefollowingratioR(t;t;~p0;~p;;)= hGjN(t;t;~p0;~p;)ihGNj(t;t;~p;~p0;y)ihgG(t;~p0;)ihGNN(t;~p;)i!=;(.

)'EN+MNEN= E0+ME0!=R(~p0;~p;;)(.)wherewehavedenedthereducedratioR(~p0;~p;;).Notethatthereisnoimpliedsumoverin(.).ForlargetimeseparationsR(~p0;~p;;)becomesconstantandindependentoftime.InourdecupletbaryonanalysiswestressedtheimportanceofmaintainingthelatticeWardidentitywhenselectingthetwo-pointfunctionstobeusedintheanalogousratio.Inthiscase,however,thereisnoidentitytomaintainandsowearefreetochooseanycombinationofdecuplettwo-pointfunctionsintheratio.Inpractise,weusethesumoftwo-pointfunctionswhichprovidestheminimalstatisticaluncertainties.Theoptimumcombinationofdecuplettwo-pointfunctionsusedinplaceofgG(t;~p0;)in(.)throughoutthisanalysisisgG(t;~p0;)=nG(t;~p0;)+G(t;~p0;)o;(. )=Z E0+ME0!=eE0t:(.)IndeterminingtheappropriateformssuitableforcalculationsusingSakurai'sconventionsthedenitionsofthe-matricesusedintheinterpolatingeldsandelectromagneticcurrentaretakenintoaccount.Thechargeformfactorisassociatedwiththetimecomponentoftheelectromagneticcurrentandthereforedoesnotcontributetophotoproductionprocesses.Byselectingthetimecomponentofthecurrentand~q=(q;0;0)theelectricchargetransitionformfactormaybeextractedinthefollowingthreewaysGC=pEN+MNM+MNMMN~qR(~q;0;i;);(.a)GC=pEN+MNM+MNMMN~qR(~q;0;i;);(.b)GC=pEN+MNM+MNMMN~qR(~q;0;i;);(.c)wheretheGreenfunctionsatthequarklevelappearinginRiaredenedintermsofSakurai'snotation.Whileeachof(.

)producesvaluesforGCwhichareinagreement,thestatisticaluncertaintiesarerelativelylarge.WewillreportvaluesforGCtakenfromattothesumofthesethreeratios.Similarly,byselectingthespatialcomponentofthecurrentand~q=(q;0;0),theMandEtransitionmomentsmaybeisolated.ThemagneticMtransitionmomentmaybeextractedinthefollowingtwoways:0

GM=pEN+MNM+MNMNj~qjR(~q;0;i;);(.a)GM=pEN+MNM+MNMNj~qjnR(~q;0;;)R(~q;0;;)o:(.b)ValuesforGMtakenfromeitheroftheseratiosareinagreementwithinstatisticaluncer-tainties.Optimumresultsareobtainedfromattothesumoftheseratiosandarereportedinthefollowing.FinallytheelectricEtransitionmomentmaybeobtainedfromGE=pEN+MNM+MNMNj~qjnR(~q;0;;)+R(~q;0;;)o:(.)C.LatticeTechniquesHerewebrieysummarizethelatticetechniquesusedinthefollowingcalculations.Ad-ditionaldetailsmaybefoundinRef.[].Wilson'sformulationisusedforboththegaugeandfermionicaction.SU()-isospinsymmetryisenforcedbyequatingtheWilsonhoppingpa-rametersu=d=.Weselectthreevaluesof,whichwedenote=0:,=0:and=0:.Tomakecontactwiththephysicalworld,theformfactorscalculatedatourthreevaluesofarelinearlyextrapolatedtocrwhereanextrapolationofthesquaredpionmassvanishes.Dierencesbetweenlinearextrapolationstom=0asopposedtothephysicalpionmassaresmallandareneglectedinthefollowing.Toaccountfortherelativelyheavystrangequarkwexs=,thesmallestofthethreevaluesofconsidered.Thisallowsanacceptableextrapolationofthelightquarkstothechirallimitthroughvaluesofquarkmasslessthanorequaltothestrangequarkmass.Ourcalculationsofoctetanddecupletbaryonmassesindicatethatthisselectionforsgivesareasonabledescriptionofthestrangequarkdynamics.TheconservedelectromagneticcurrentisderivedfromthefermionicactionbytheNoetherprocedure.ThelatticeWardidentityguaranteesthelatticeelectricformfactorreproducesthetotalbaryonchargeatq=0.Thequarkpropagatorscoupledwithxedmomentum~q=(q;0;0)tojarecalculatedusingthesequentialsourcetechnique(SST)[{0].TominimizenoiseintheGreenfunctions,weexploittheparitysymmetryofthecorrela-tionfunctions,andtheequalweightingoffUgandfUggaugecongurationsinthelatticeaction.DeningsPasG(~p0;~p;~q;)=sPG(~p0;~p;~q;);sP=;(.)andsCas=sCeCeC;sC=;(. )whereeC=C,thecorrelationfunctionsarerealprovided

sC=sP:(.0)Thisconditionissatisedwiththeselectionsforindicatedin(.a),though(.).WhilethisapproachrequiresanextramatrixinversiontodetermineanadditionalSSTpropagatorwithmomentum~q,inclusionofbothfUgandfUgcongurationsinthecalculationofthecorrelationfunctionsprovidesanunbiasedestimateoftheensembleaveragepropertieswhichhassubstantiallyreduceductuations[].Twenty-eightquenchedgaugecongurationsaregeneratedbytheCabibbo-Marinari[]pseudo-heat-bathmethodonaperiodiclatticeat=:.Dirichletboundaryconditionsareusedforfermionsinthetimedirection.Congurationsareselectedafter000thermalizationsweepsfromacoldstart,andevery000sweepsthereafter[].Timeslicesarelabeledfromto,withthe-functionsourceatt=.Asymmetriccombinationofthecurrent(j(xb)+j(x))=iscenteredattimeslicet=.Thespatialdirectionoftheelectromagneticcurrentischoseninthez-direction.Thefollowingcalculationsaredoneinthelabframe~p=0;~p0=~q=(=;0;0),theminimumnonzeromomentumavailableonourlattice.Usingthenucleonmass,thelatticespacingisdeterminedtobea=0:()fm,a=:()GeV.Statisticaluncertaintiesarecalculatedinathird-order,singleeliminationjackknife[,].Athirdorderjackknifeprovidesuncertaintyestimatesforthecorrelationfunctions,tstothecorrelationfunctions,andquantitiesextrapolatedtothechirallimit.Ideally,wewouldliketocalculatetheformfactorsatorveryclosetoQ=0,allowingadirectcomparisonwiththemorecommonlyreferredtotransitionmoments.Forthedecayofourlatticeatrest,energy-momentumconservationrequiresaphotonmomentumofapproximately0MeV,whereastheminimummomentumavailableonourlatticeis0()MeV.Sincetheminimummomentumavailableonthelatticeisinverselyproportionaltothelongestphysicalspatialdimension,calculationsatQ=0willrequirelatticeswithmuchlargerphysicalvolumes.Thisdicultyisfurthercompoundedbyproblemsassociatedwithtuningthephysicallatticesizetoreproducethedesiredmomentumoralternatively,usingaverynelatticespacingandmuchlargerlatticevolumestoreducetheneedfortuningthelatticelength.Themomentumtransferatwhichtheformfactorsarecalculatedisapproximatelyin-dependentofthebaryonunderinvestigation.Forp,andtransitionstheQis0.(),0.()and0.()GeVrespectively.Asimilarindependenceisseenoverthethreevaluesofandcrunderconsideration.Thereforevariationoftheformfactormomentumtransferintheextrapolationsisnotasourceofconcern.TomakecontactwiththetransitionmomentsatQ=0,wewillfollowtheusualprocedureofdescribingtheQdependenceofthethreetransitionformfactorsbyacommonfunction[].InadditionwewillassumethatthemomentumtransferdependenceofthetransitionformfactorsissimilartothemomentumdependenceofthedecupletbaryonchargeformfactorGE.Fortunately,thedecupletbaryonelasticformfactorsaredeterminedatasimilarQof0. ()GeV.ThisallowsasimplescalingofthetransitionformfactorstoQ=0withoutspecicreferencetothefunctionaldependenceonQ.Asinouroctetanddecupletbaryonanalysesthescalingisdoneseparatelywithineachquarksector.TheQdependenceoftheindividualquarksectorcontributionscanbequitedierent,particularlyinhyperons.Consider,forexample,themagnetictransitionform

factorforhyperons.ThestrangeandlightquarksectorsarescaledseparatelybyGsM(0)GsM(q)=GsE(0)GsE(q);(.)andsimilarlyforthelightquarks,suchthatthemagnetictransitionmomentofahyperonisgivenbyGM(0)=GlM(0)+GsM(0);(.)wherellabelsthelightquarks.ForN$transitionsitisnotnecessarytoseparatetheu-andd-quarksectorsduetotheSU()-isospinsymmetryofthecorrelationfunctions.Thisapproachwasusedinourpreviousoctetanddecupletbaryonanalysesandispreferredoverextrapolationsinqtoq=0whichsuerfromlargestatisticalerrors.OnemightarguethatanaverageoftheoctetanddecupletbaryonchargeformfactorsshouldbeusedinscalingthetransitionmomentstoQ=0.HoweverdierencesinthequarkchargedistributionradiiinoctetbaryonsdependingonwhetherthequarkissinglyordoublyrepresentedwouldinduceanasymmetryinthequarksectorcontributionstothetransitionmomentsatQ=0.Thiscontradictsthesymmetrymanifestinthethreepointcorrelationfunctionof(.0)whichdemandstheu-andd-quarkcontributionstothep$+transitionmomentstobeequalandoppositeinsignforequallychargedquarks.Asaresultwechoosetousethedecupletbaryonchargeformfactorsalonewhichpreservethissymmetry.Thedierenceinthescaledtransitionmomentsusingthetwodierentapproachesissmallastheu-quarkdistributioninpisapproximatelyequaltothatin+,whilethed-quarkdistributionradiusisonlyslightlylargerin+.Inourdecupletbaryonanalysis,wefoundthatthechargeradiusof+mayactuallybesmallerthanthatoftheproton[,].ThismightseemtocontradictevidencefromtheQdependenceoftransitionformfactorswhichsuggeststhesizeoftheresonanceislargerthantheproton[{].Wenotehowever,thatthepredominantdierencebetweenthequarkchargedistributionsoftheprotonandisthebroaderdistributionofthedquarkin.Forchargeradiithenegativechargeofthedquarkactstodecreasethechargeradiusof.However,forthetransitionmomentsofp$+thedquarkcontributesinapositivemanner.Thereforethedistributionradiusassociatedwiththetransitionmomentislargerthanthatassociatedwithelectricchargein+.IV.CORRELATIONFUNCTIONSInthissectionweexaminethelatticecalculationsofthecorrelationfunctionratiosusedtoextracttheelectromagnetictransitionformfactors.Letusrstconsiderthecorrelationfunctionratiosusedtoextractthechargetransitionformfactor,GC.FiguredisplaysthesumofratiosPi=Riforu-andd-quarkcontributionstop$+chargetransitions.TheargumentsofRiareasindicatedinequations(. ).Quarkchargefactorshavenotbeenincluded.Atallquarkmassesareequalandthereforegureillustratesthequarkscontributionstoanyoftheoctettodecupletbaryonchargetransitionsprovideddoublyrepresentedquarksareidentiedwiththeu-quarkcontributionandthesinglyrepresented

FIG..TheratiosumPi=Riforu-andd-quarkcontributionstothep$+chargetransitionformfactor.ArgumentsofRiareasindicatedinequations(. ).Quarkchargefactorshavenotbeenincluded.Thed-quarkcontributionshavebeenosetintimeforclarity.quarkwiththed-quarkcontribution.Thechargetransitionformfactorissmallandbytimeslicethesignalislostinthenoise.Correlationfunctionsforlighterquarkmasseshavelargeruncertainties.QuarkcorrelationfunctionratiosforthemagnetictransitionformfactorGMofp$+areillustratedingure.TheratiosdisplayedcorrespondtothesumoftheratiosRiof(.a)and(.b).Takingresultsfromasumofallthreeratiosreducesthestatisticaluncertaintiesparticularlyforlargervaluesof.Thetimeevolutionofthecorrelationfunc-tionisasfollows.Attimeslice,abaryonwiththequantumnumbersoftheprotonoriscreated.Afterexcitedstatesareexponentiallysuppressedrelativetothebaryongroundstate,thequarksinteractwiththeelectromagneticcurrentattimeslice.Afteranumberoftimesteps,thealternatebaryonisannihilated.Forlargetimeseparationsbetweentheelectromagneticcurrentinteractionandannihilation,thecorrelationfunctionratiosaretobecomeconstantandindependentoftime.Thecorrelationfunctionratiosillustratedingureforourintermediatevalueofdisplaysmallstatisticaluncertainties.However,thecentralvaluesdonotformasataplateauasinourelasticformfactoranalysesofoctetanddecupletbaryons.Similarresultsareseenforeachof(.a)and(.b).Figuredisplaystheanalogousratiosforthemagnetictransitionformfactorofatthelightestu-ord-quarkmassesconsidered.Thestrangequarkcorrelationfunctionratioformsaconvincingplateaufortimeslicesgreaterthanorequalto.Thecentralvaluesofthelightquarkcorrelationfunctionratiodonotformasimilarplateau.Howeverfortimesslicesgreaterthanorequaltoitiseasytottheratiowithahorizontalline.Thisdrift

FIG..QuarkcorrelationfunctionratiosumforthemagnetictransitionformfactorGMofp$+at.ArgumentsofRiareasindicatedinequations(.).inthecentralvaluesofthelightquarkcorrelationfunctionratiosistypicalofotherbaryontransitionssuchasN$.Theformfactorsaredeterminedbyttingthecorrelationfunctionratiosbyahorizontallinefortimestandt.Weconsidertsoftheratiosfromtimeslicethroughinintervalsincludingtopoints.Theresultsareselectedfromthese0tsbasedontheatnessofthecorrelationfunctionsandthestatisticaluncertainties.Fitsoftheorpointsinthetimesliceintervalto0orprovidetheoptimumbalancebetweenthesesystematicandstatisticaluncertainties.Thiscontrastsourstudyofoctetanddecupletbaryonelasticformfactorswheretheoptimumintervalwasthrough0.ElectricquadrupoletransitionformfactorsaredeterminedfromthesumofcorrelationfunctionratiosRandRasindicatedin(. ).FiguredisplaysR,RandthesumR+Rfortheu-quarkcontributiontothep$transitionatu=.IndeterminingGEonecantbothRandRandcombinetheresultoralternativelytthesumR+R.Theextractedvaluesagreewithinstatisticaluncertainties.SincebothRandRshouldbecomeconstantandindependentoftimewechoosetoenforcethisconditionbyttingbothRandRandwerefertotheseresultsinthefollowing.Correlationfunctionratiosforlargervaluesofhavelargerstatisticaluncertaintiesandthereforetheextractionofaclearnonzerovaluefortheelectricquadrupoletransitionmomentinthisanalysisisnotpossibleformostbaryons.Howeveracombinationofavorsymmetrybreakingandthesymmetrymanifestinthethree-pointcorrelationfunctionsallowsapredictionforthenegativelychargedhyperonelectricquadrupolemomentsthatdiersfromzerobytwostandarddeviations.

FIG..QuarkcorrelationfunctionratiosforthemagnetictransitionformfactorGMof$atu=d=.ArgumentsofRiareasindicatedinequations(.).FIG..Correlationfunctionratiosforu-quarkcontributionstotheelectricquadrupoletran-sitionformfactorofp$+atu=s=.TheargumentsoftheratiosRandRareasindicatedin(. ).

Thenegativechargebaryonsareuniqueinthatthechargefactorsmultiplyingthequarkthree-pointcorrelationfunctionsareequal(=).MoreoverthequarkcorrelationfunctionsareequalandoppositeinsignintheSU()avorlimit.Thusuctuationsinthecorrelationfunctionsareanticorrelatedandtoalargeextentcancelwhenaddedtoconstructthenegativechargebaryons.Figuresanddisplaythecorrelationfunctionsfors-andu-quarkcontributionstothe0Etransitionmomentatu=.Acomparisonwithgureforthep$+transitionintheSU()-avorregimewhereu=s=revealsthatSU()-avorsymmetrybreakingin0$0hascausedthestrangequarkcontributiontolargelyvanishwhiletheu-quarkcontributionremainsnite.Additionofthequarkcontributionswithanticorrelateductuationsgivesaniteresultdierentfromzerobytwostandarddeviations.Similarresultsareseenforthetransitionmomentsof$wherethesinglyrepresentedstrangequarkcontributionisonceagainseentolargelyvanish.TheniteEtransitionmomenthasitsorigininthelightquarkwhichislesslocalizedandmoresensitivetotheperiodicboundaryconditionsandspatialasymmetriesofourlattice.V.SIMPLEQUARKMODELPREDICTIONSBeforeproceedingtothelatticeresultswepresentherethetransitionmomentpredictionsofasimplequarkmodel.WehavegeneralizedthemodelcalculationsofDarewych,HorbatschandKoniuk[]toincludetheentirebaryonoctettodecuplettransitions.Themodelissimpleinthatnoattempthasbeenmadetoaccountforcongurationmixing[]inthespin-avorwavefunctionsortheinclusionofexplicitpiondressingsofthenucleon[0].Forthebaryontransitionsunderexaminationhere,thegeneralresultoftheirmodelmaybewrittenGM=pMM=eK(DS)(.)whereMandMareoctetanddecupletbaryonmassesrespectively,andDandSaretheintrinsicmomentsofthedoublyandsinglyrepresentedquarksrespectively.TheparameterKisdenedasK=q=hwhereq=(MM)=Mandhistheharmonic-oscillatorstrengthparametertakentobe0.GeV.Formoredetailsofthemodelcalculation,theinterestedreaderisreferredtotheoriginalpublication[].Itisinterestingtonotethesimilaritybetween(. )and(.0).IntheSU()-avorlimit,bothindicateanequalandoppositeweightingofthesinglyanddoublyrepresentedquarks.Howeverwithadditionalinformationfrom(.0)ithasbecomeapparentthatitispossibletoidentifyoneofthedoublyrepresentedquarkswhosenetcontributiontothetransitionmomentvanishesintheavorsymmetriclimit.For0,theeectivemomentforDis(u+d)=.Inthespiritoftheoriginalpaper[]wetakeu=d=p=ands=d=0:.Sincecongurationmixinginthebaryongroundstatehasnotbeenincluded,theEtransitionmomentsinthissimplemodelarezero.ThemagneticdipoletransitionmomentsGMaresummarizedintableIinunitsofnaturalmagnetons.Helicityamplitudes,decaywidthsandtransitionmomentsusingthe

FIG..Correlationfunctionratiosfors-quarkcontributionstotheelectricquadrupoletransi-tionformfactorof0$0atu=.Thestrangequarkcontributionhaslargelyvanishedastheu-quarkhasbecomelight.FIG..Correlationfunctionratiosforu-quarkcontributionstotheelectricquadrupoletransi-tionformfactorof0$0atu=.

TABLEI.QuarkModelPredictionsforTransitionMomentsTransitionGMfMA=A=(B)Unitsof(0GeV=)(keV)p$+.000n$0.000+$+.000$0.0$0.00.0$0.$0..morewidelyusedconventionsoftheParticleDataGroup[]arealsogiven.RelationshipsamongthesequantitiesaresummarizedintheAppendix.Fortheprotontransitionmoment(.)reducestothewellknownrelation[]p=pp(. )providedoneneglectsthekinematicalfactors.Fortransitionmomentsthisisnotalwaysagoodapproximation.VI.RESULTSA.MagneticDipoleTransitionFormFactors.BaryonTransitionMomentsMagneticdipoleformfactorscalculatedatQ'0:GeVarereportedintableIIateachvalueofconsideredalongwiththevaluesobtainedfromalinearextrapolationtocr.QuarksectorcontributionsarereportedintableIII.ExtrapolationsoftheMtransitionmomentsforafewrepresentativebaryonsareillustratedingure.ThedependenceonisparticularlyweakforN$transitions.SU()-avorsymmetrybreakingisclearlyevidentintheandextrapolations.However,thesymmetryofthethreepointfunction(.0)holdstoagoodapproximationeveninthebrokenavorsymmetryregimeasthetransitionmomentsofandareroughlyequalandoppositeinsign.ExtrapolatedtransitionmomentsfortheotherbaryonsmaybefoundintableIV.Intheoctetbaryonanalysisitwasfoundthatthemagnitudesofthelatticeresultsformagneticmomentswereconsistentlysmallerthantheexperimentalmeasurements.Itwasarguedthatat=:somedeviationsfromasymptoticscalingmayoccur.Amorerecentanalysis[]determinesnucleonformfactorsat=:0onacubiclatticewithphysicalspatialdimensionsroughlyequaltooursmalleryandzdimensions.Someimprovementisseeninthemagnitudesofthemagneticmomentswhicharestill0to%lowcomparedtoexperiment.Chiraldressingsofthenucleonmaycauseourlinearextrapolationin=tounderestimatethemagnetictransitionmomentsinthephysicalregime[].Finitevolume

TABLEII.Baryonmagneticdipole(M)transitionformfactorsinunitsofnaturalmagnetons(Be=MB).Transition=0:=0:=0:cr=0:()p$+.().().().()n$0.().().().()+$+.().().().()0$00.()0.0()0.()0.()$0.000.0()0.()0.()0$0.().().().()$0.000.0()0.()0.()TABLEIII.QuarksectorcontributionstotheMtransitionformfactor.Quantitiesarenormalizedtounitchargeandarereportedinunitsofnaturalmagnetons(Be=MB).TransitionQuark=0:=0:=0:cr=0:()p$+u.().().().()d.().().().()+$+u.().().().()s.().().0().(0)0$0u.().().().0()s.().().().()TABLEIV.BaryonMtransitionformfactorsatQ=0.Transition=0:=0:=0:cr=0:()p$+.().().().(0)n$0.().().().(0)+$+.().(0).0().()0$00.()0.()0.()0.()$0.000.0()0.0()0.()0$0.().0().0().()$0.000.0()0.0()0. ()0

FIG..ExtrapolationofMtransitionmomentsforafewbaryonsrepresentativeofbaryonoctettodecuplettransitions.ThedependenceonisparticularlyweakforN$transitions.eectsmayalsogiverisetotheunderestimationofthemagnetictransitionmomentsasthebaryonisrestrictedbyitsperiodicimages.Theprotonrmselectricchargeradiusatindicatestheprotonlargelyllsthelatticeinoursmalleryandzspatialdimensions.Photoninteractionswithdisconnectedquarkloopsandothernon-quenchedcorrectionsmayalsoprovideadditionalcontributions[].Toreducetheeectsoftheseuncertainties,ratiosofthelatticeresultstothelatticeprotonresultareusedwhenmakingcomparisonswithexperimentalmeasurementsormodelcalculations.TableVreportstheratiooftheextrapolatedbaryonmagneticdipoletransitionmomentstotheprotonmagneticmomentscaledtoreproducetheprotonmagneticmoment.ValuesaregivenfortheSachsformfactor,GM==B,whereBistheunitofnaturalmagnetons(e=MB,MBisthemassoftheoctetbaryon).ValuesarealsoreportedfortheParticleDataGroup[]convention,fM,calculatedfromGMusingthephysicalbaryonmasses[].RelationshipsfortheSachsformfactorsandtheconventionsoftheParticleDataGrouparesummarizedintheappendix.TheSU()-avorsymmetryrelationships($)=($)and(+$+)=(0$0)areseentoholdtoagoodapproximation.Thissuggeststhequarkcontributionstothetransitionmomentsdonotdependstronglyonthebaryoninwhichthequarksreside.IngurethelatticepredictionsoftheSachsformfactorGMarecomparedwiththoseofthesimplequarkmodelreviewedintheprevioussection.Remarkableagreementisseenthroughoutthebaryonoctettodecuplettransitions.Asimilaragreementwasseenbetweenthelatticeresultsandthesimplequarkmodelinourdecupletbaryonanalysis.Furtherexaminationrevealedthattheagreementwaslargelyduetoanapproximatebaryoninde-

FIG..LatticepredictionsfortheSachsformfactorGM.ThedashedlinesarepredictionsbasedonthesimplequarkmodelreviewedinsectionV.Remarkableagreementisseenthroughoutthebaryonoctettodecuplettransitions.pendenceofthequarkeectivemagneticmoments.Thiswasincontrasttoouroctetbaryonanalysiswhereitwasfoundthattheelectromagneticpropertiesofaquarkhaveastrongdependenceonthebaryoninwhichitresides.TABLEV.BaryonMtransitionmoments.Thelatticeresultshavebeenscaledtoreproducetheprotonmagneticmoment.TransitionGMfM(B)(GeV=)p$+.()0.()n$0.()0.()+$+.()0.(0)0$0.0()0.0()$0.()0.0()0$0.()0.()$0. ()0.0()

.EectiveQuarkMomentsItisinterestingtoexaminetheelectromagneticcontributionsofthequarksectorsindivid-ually.Thebaryontransitionmomentsarecomposedofasumofquarksectorcontributionsandinterestingphenomenamayremainhiddenintakingthesum.Todeneaneectivequarkmomentweturntothequarkmodelreviewedintheprevioussection.DeningGDM(GSM)tobethedoubly(singly)representedquarksectorcontributiontothetransitionmoment,wedenetheeectivequarkmomentstobeBD=+pMBMB=MNMBGDM;(.a)BS=pMBMB=MNMBGSM;(.b)whereMBandMBarethemassesoftheoctetanddecupletbaryonsundertransition.ThesecondmassratioMN=MBexpressestheformfactorGMinunitsofnuclearmagnetons.Forthecentralvaluesofthelatticemasses,thefactorexp(K)appearingin(.)takesthevalues0:0:00.Sincethisfactorisapproximatelyoneandhasanobviousmodeldependencewehavedroppedthisfactorfromthedenitionoftheeectivequarkmoment.Thethree-pointcorrelationfunctionof(.0)indicatesthequarksectorcontributionstothetransitionmomentareequalandoppositeunderSU()symmetry.Asimilarsymmetrywasseeninthedecupletbaryonthree-pointcorrelationfunctions.HenceaninterestingquestiontoaskiswhetherSU()-avorsymmetryisbrokeninthesamemannerinthemagnetictransitionmomentsasinthemagneticmomentsofthedecupletbaryons.Thisquestionmaybeansweredwithminimalmodeldependencebytakingaratiooftheeectivequarktransitionmomentsandcomparingtheresultwithasimilarratioofeectivequarkmagneticmomentsindecupletbaryons.Inthiswaythemassratiosappearingin(.)andotherfactorsneglectedintheeectivequarktransitionmomentdenitionsareeliminatedfromtheSU()avorsymmetrybreakingmeasure.TocompareSU()-avorsymmetrybreakingwecalculatethefollowingratio(s=u)transitions,(s=u)decuplet:(.)Ofcourseasimplequarkmodeldenesthisratiotobe.ThelatticeresultsindicatethatSU()-avorsymmetryisbrokeninadierentmannerforthetransitionmomentscomparedtothedecupletelasticmoments.Forthetransition$theratioof(.)is0.(0).Similarresultsholdfor$transitionswheretheratiois0.().Todiscoverwhetheritisasuppressionofthestrange-quarkoranenhancementoftheu-quarkcontributionstothetransitionmomentsthatisresponsibleforthedeviationsfromthesimplequarkmodeldescriptionofavorsymmetrybreaking,weturntotheactualvaluesoftheeectivequarkmoments.Figureillustratestheeectivequarkmomentsdenedin(. )normalizedtounitcharge.Toagoodapproximation,theeectivequarkmomentsareindependentoftheenvironmentinwhichthequarkresides,withthepossibleexceptionoftheuquarkin.Theratioofeectiveu-quarktransitionmomentsforandpindicatea

FIG..Eectivequarkmomentsdeterminedfromthequarksectorcontributionstoradiativetransitionsofoctetanddecupletbaryons.Thequarkmomentsaredenedin(.),andarenormalizedtounitcharge.Toagoodapproximation,theeectivequarkmomentsareindependentoftheenvironmentinwhichthequarkresides,withthepossibleexceptionoftheuquarkin. ()%suppressionoftheeectiveu-quarkmomentin0$0transitions.Asimilarratioforthes-quarksinandtransitionsindicatessmaybesmallerby0(0)%.Inourdecupletbaryonanalysisasimilargentleenvironmentsensitivityofthequarkmomentswasseen.Figure0summarizestheeectivequarkmomentsforquarksindecupletbaryons.Here,thequarkmomentsdecreaseasstrangenessisadded.Thismaybeattributedtotherolethebaryonmassplaysinsettingthescaleatwhichthequarkscontributetothemagneticmoment.Similarconclusionscouldbedrawnforthetransitionmomentsifitwerenotforthelackofsuppressionintheu-quarkmomentin+transitions.Howevertheu-quarkin+isdoublyrepresentedandSU()-avorsymmetryisbroken.Termsofthethree-pointtransitioncorrelationfunctionof(.0)whichcannotcontributetonucleontransitionscannowprovideadditionalcontributionstotheu-quarksectorof+transitions.Similareectsmaybeoccurringintheeectivequarktransitionmoments.However,theinuenceofanadditionalstrangequarkinthebaryonappearstobeplayingastrongerrole,anddecreasesthemagnitudeofthemagneticmomentcontribution.Figuredisplaysratiosoftheeectivequarkmomentsfromtransitionstotheeec-tivequarkmomentsfromdecupletbaryons.Thelightquarktransitionmomentsaremoreconsistentwiththevaluesdenedbydecupletbaryonmagneticmomentsthanthestrangequarktransitionmoments.Hence,itisasuppressionoftheeectivetransitionmomentsofstrangequarksrelativetotheirvaluesindecupletbaryonsthatislargelyresponsiblefor

FIG.0.Eectivequarkmomentsdeterminedfromthequarksectorcontributionstodecupletbaryonmagneticmoments.Approximatebaryonindependenceofthequarkmomentsisdisplayed.However,somedecreaseisseenintheeectivequarkmomentsasstrangenessisadded.thedeviationsfromthesimplequarkmodelpredictionsofavorsymmetrybreaking.Theeectivestrange-quarktransitionmomentsaresuppressedby0()%and(0)%relativetotheirdecupletbaryonvaluesforandtransitionsrespectively.Inourpreviousanalyses,wehavefoundtheeectivemomentoftheuquarkintheprotontobeequal,toagoodapproximation,totheu-quarkmomentinthe.Similarlytheeectiveu-quarkmomentdeterminedfromtransitionsofN$isinagreementwiththeeectiveu-quarkmomentsinpand.Giventhesimilarityofuquarkpropertiesinpand,thisisasonemightexpect.Thereforethedriftinthecentralvaluesofthethree-pointcorrelationfunctionsmaysimplybeanindicationoftheneedforbetterstatisticsasopposedtoasignatureofthecorrelationfunctionsfailingtoreachtheplateauregionbeforethelatticeboundaryisencountered.FinallyitisworthcommentingonwhythelatticeresultssuggestvaluesfortheMtransitionmomentsofN$thataresomewhatlargerthanthatanticipatedbythesimplequarkmodel.Themainsourceofthedierencestemsfromthemannerinwhichthequarkscontributetothemagneticmomentoftheproton,whichhasbeenusedtosettheoverallscaleofthemagneticmoments.WhileSU()-spin-avorsymmetrypredictsp=ud;(. )ouroctetbaryonanalysisindicatesthatthed-quarkcontributionissuppressedinthelatticeresultsbyafactorofapproximatelyfromthatanticipatedbySU().Sincetheeective

FIG..Ratiosoftheeectivequarkmomentsfromtransitionstotheeectivequarkmomentsfromdecupletbaryons.Thelightquarktransitionmomentsaremoreconsistentwiththevaluesdenedbydecupletbaryonmagneticmomentsthanthestrangequarktransitionmoments.u-quarkmomentintheprotonisapproximatelyequaltotheeectiveu-quarkmomentinp$+transitions,thedierenceinlatticeandquarkmodelpredictionsforthep=pratiomaybeattributedtothesmallnessofthed-quarkmomentcontributiontotheprotonmagneticmoment.B.ElectricChargeTransitionFormFactorsElectricchargeformfactorscalculatedatQ'0:GeVarereportedintableVI.Thestatisticaluncertaintiesarelargeandpreventusfromdrawinganystrongconclusions.However,itinterestingtonotethatthecentralvalueofthechargeformfactorforthep$+transitionhasthesignandmagnitudeanticipatedbymultipoleanalyses[,]atabout0%ofthemagneticformfactor.Thelatticeresultssuggestthechargeformfactormaybelargeforthenegativelychargedhyperontransitions.ThisisduetoanadditionofthequarksectorcontributionswhichareillustratedintableVII.Examinationofthecorrelationfunctionsforthehyperonsrevealsthatthecorrelationfunctionsaresomewhatnoisyanddonotformaconvincingplateau.Afuturehighstatisticsanalysismayprovidesomeinterestinginsightsintothechargeformfactor.

TABLEVI.ScalarQuadrupole(C)transitionformfactors.Transition=0:=0:=0:cr=0:()p$+0.0()0.0()0.0()0.()n$00.0()0.0()0.0()0.()+$+0.0()0.0()0.()0.0()0$00.0()0.0()0.0()0.()$0.000.()0.(0)0.()0$00.0()0.0()0.00()0.0()$0.000.0()0.()0.()TABLEVII.QuarksectorcontributionstotheCtransitionformfactor.Quantitiesarenormalizedtounitcharge.TransitionQuark=0:=0:=0:cr=0:()p$+u0.0()0.0()0.0()0.()d0.0()0.0()0.0()0.()+$+u0.0()0.()0.()0.()s0.0()0.()0.()0.()0$0u0.0()0.0()0.()0.()s0.0()0.0()0. ().0()C.ElectricQuadrupoleTransitionFormFactorsAsdiscussedinsectionIV,onlytwooftheoctettodecupletEtransitionformfactorsarestatisticallydierentfromzero.FigureillustratestheextrapolationoftheseEtransitionformfactorsforandtransitions.Atthelargestvalueofquarkmassconsidered,SU()-avorsymmetryisexactandthetransitionmomentsvanishforthesebaryons.Thesymmetryofthethree-pointcorrelationfunctioncontinuestoholdeveninthebrokenavor-symmetryregionastheEtransitionformfactorsareseentobeapproximatelyequalandopposite.ValuesfortheEtransitionformfactorsateachvalueofconsideredaswellasatcraresummarizedintableVIII.QuarkcontributionstotheseformfactorsaregivenintableIX.TABLEVIII.Baryonelectricquadrupole(E)transitionformfactors.Transition=0:=0:=0:cr=0:()p$+0.0()0.0()0.0()0.0()n$00.0()0.0()0.0()0.0()+$+0.0()0.0()0.0()0.0(0)0$00.0()0.0()0.0()0.0()$0.0000.00()0.0()0.0()0$00.0()0.0()0.0()0.0()$0.0000.00()0.0()0.0()

FIG..ExtrapolationoftheEtransitionformfactorsforandtransitions.Thesym-metryofthethree-pointcorrelationfunctioncontinuestoholdeveninthebrokenavor-symmetryregionastheEtransitionformfactorsareseentobeapproximatelyequalandopposite.ThequantitythathascapturedtheattentionandexcitementoftheeldistheE=MratioofelectromagneticformfactorsREMfE=fM=GE=GM.Analysesofexperi-mentaldataplacethisratioat(:0:)%,[];(:0:)%,[];:%,[];and+(:0:)%,[].Thereaderisdirectedtotheoriginalpapersforacompletediscussionoftheuncertaintiesreectedinthenumericalerrorbarsgivenhere.DetailsofthelatticedeterminationsofREMarereportedintableX.AsummaryofthelatticeratiosREMisgiveningure.Unfortunatelythestatisticaluncertaintiesinthelatticeresultsarerelativelylarge.Alltheresultsfromtheexperimentalanalysesliecomfortablywithinourdeterminationfromrstprinciplesof+()%.Theinterestingfeature,however,isthatthelatticeapproachcanmakepredictionsforTABLEIX.QuarksectorcontributionstotheEtransitionformfactor.Quantitiesarenor-malizedtounitcharge.TransitionQuark=0:=0:=0:cr=0:()p$+u0.0()0.0()0.0()0.0()d0.0()0.0()0.0()0.0()+$+u0.0()0.00()0.0()0. (0)s0.0()0.0()0.0()0.0(0)0$0u0.0()0.0()0.0()0.0()s0.0()0.0()0.00()0.0()

FIG..AsummaryoflatticecalculationsoftheE=MratioREM.Statisticaluncertaintiesinthelatticeresultsaretoolargetofavoranyparticularmodelcalculation.theEtransitionmomentsofandtransitionswhichwillprovidevitalinformationtothosedevelopingmodels.Still,thepresentvaluesmustbetakenwithsomecaution.Thenon-vanishingcontributiontotheseEmomentshasitsorigininthelightd-quarksector.Thisquarkhasabroaddistributionradiuswhichmaybesensitivetotheasymmetriesourelongatedlattice.InteractionswithperiodicimagesmayinduceanEmoment.ItisclearthatafuturecalculationonacubiclatticewillprovidemuchneededinsightintotheEtransitionformfactor.TABLEX.TheE=MratioREMatQ=0.Transition=0:=0:=0:cr=0:()p$+0.0()0.0()0.0()0.0()n$00.0()0.0()0.0()0.0()+$+0.0()0.0()0.0()0.0()0$00.0()0.0()0.0()0.0()$0.0()0.0()0.0()0$00.0()0.0()0.0(0)0.0()$0.0()0.0()0.0(0)

VII.MODELCOMPARISONSLatticepredictionsforratiosofoctetbaryonmagneticmomentstothelatticeprotonmomentareinexcellentagreementwithexperimentalmeasurementswhenthebaryonmo-mentispositive.Ratiosforbaryonswithnegativemagneticmomentsaremoresensitivetocontributionsfromdisconnectedquarkloopcontributionswhicharenotincludedinpresentformfactorcalculations[].Sincethetransitionmomentforp$+ispositive,thelatticeapproachshouldprovidereliableestimatesofthemagnetictransitionmomentratiop+=p.Figuresummarizesmanycalculationsofthemagnetictransitionmoment,fM.Theneedforacalculationofthisquantityfromrstprinciplesisreectedinthewiderangeofvaluesforthisquantity.Thecalculationshavebeencategorizedintosixdierentapproachesincludinganalysesofexperimentalpionphotoproductiondata(Expt.),ourlatticeQCDcalculation(Latt.),nonrelativisticquarkmodeldeterminations(Q.M.),hedgehogmodelsincludingtheSkyrmeandHybridmodels(Hedge.),bagmodels(Bag),andaBethe-Salpetercalculation(B.S. ).Inthefollowingwediscusseachoftheseapproachesinrelationtoournewlatticedetermination,andgivespecicreferencestothemodeldeterminations.Theseresultsarebynomeansexhaustive,butarerepresentativeandindicatethebreadthofinterestinthedeterminationofthetransitionmoment.Theanalysesofexperimentaldatainclude(fromtopdown)Davidson,Mukhopadhyay,andWittman'sinvestigationusinganeectiveLagrangianwithanumberofdierentuni-tarizationmethods[];Arndt,Workman,LiandRoper'senergy-independentpartialwaveanalysis[];Nozawa,Blankleider,andLee'scalculationinvokingoshellmodelingofNinteractions[];andasimilarapproachbyTanabeandOhtawhereadditionalparametersareoptimizedbyachi-squaret[].TanabeandOhta'sresultfortheMtransitionmomentisrelativelysmallcomparedtotheothercalculationsreportedingure.Theynotehowever,thattheyhavecalculatedthebarecouplingand,assuch,theirresultshouldbecomparedtotheMITbagmodel(thelowerofthetwoentriesintheBagcolumnofgure),asopposedtotheChiralbagmodelforexample.Theyarguethattheirmodelexplicitlytakesintoaccountthepioncloudeectseparately.SimilarargumentsholdforthecalculationofNozawa,Blankleider,andLee.Inthislatticecalculationtheisstableandasaresulttheproblemsassociatedwithdeningthepropertiesofabaryonunstabletostronginteractionsdonotarise.Inthelatticesimulations,threequarksarecreatedandallowedtopropagatealongpathsinspace-timedeterminedbytheQCDLagrangianbeforetheyarelaterannihilated.ThepossiblepathsthequarkscantakeincludepathssuchasZ-graphswhereaquarkemitsagluonandscattersintoanegativeenergystateetc.Atintermediatetimestherearequarksandanynumberofquark-antiquarkpairsinthewavefunction.Presumably,thesequarkwavefunctionsmaybedescribedinaFock-spaceexpansionofmeson-baryonintermediatestates,includingNintermediatestates.Baryon-antibaryonpairsmayalsoplayarole.Inessence,thebaryonsimulatedonthelatticeisdressed.However,itisnotcompletelydressedsincesomediagramscorrespondingtopiondressingsarenotincludedinmakingthequenchedapproximation.Thequarkmodelcalculationsinclude(fromtopdown)GuiasuandKoniuk'scalculationinwhichmesonicdressingsofthenucleonareexplicitlyincluded[0];Capstick'scalculation0

FIG..Calculationsofthemagnetictransitionmoment,fM.Thecalculationshavebeendividedamongsixcategoriesincludinganalysesofexperimentalpionphotoproductiondata(Expt.),ourlatticeQCDcalculation(Latt.),nonrelativisticquarkmodeldeterminations(Q.M.),hedgehogmodels(Hedge.),bagmodels(Bag),andaBetheSalpetercalculation(B.S. ).inwhichcongurationmixinginthebaryonSU()wavefunctionsisaccountedfor[];andacalculationbasedonthesimplequarkmodelofDarewych,Horbatsch,andKoniuk[].Itisoftenarguedthatthesimplequarkmodeldoesnotincludethephysicsofmesonicdressings.Howeverwehaveseenremarkableagreementbetweenthelatticecalculationsandsimpleconstituentquarkmodelsinthedecupletbaryonanalysisandnowinthetransitionmomentsunderinvestigationhere.Itisimportanttoaskwhatphysicsisrepresentedbytheconstituentquark.Insimplequarkmodels,nonperturbativegluoninteractionswithcurrentquarks(whichnaturallyincludesphysicsassociatedwithquark-antiquarkpairsandthusmesons)areapproximatedthroughtheuseofaconstituentquarkwithaneectivemass.Infact,theconstituentquarkmassesaredeterminedpredominantlybyreproducingnucleonpropertiessuchasthemagneticmoment.Ofcourse,protonpropertiesreectphysicswhichmaybeascribedtotheassociatedpioncloud.Thereforeconstituentquarkmodelpredictionsofmagneticmomentsandmagnetictransitionmomentsimplicitlyincludethephysicsofmesonicdressings.Asaresult,itisnotappropriatetodirectlycomparetheresultsofTanabeandOhtaorNozawa,Blankleider,andLeewiththequarkmodel.GuiasuandKoniukhaveattemptedtoexplicitlyincludethephysicsofthepioncloudinaquarkmodelcalculationofthehelicityamplitudesofN$transitions.Ofcourse,ifonewishestoexplicitlyincludepiondressingsinthequarkmodel,onemustrecalculatetheconstituentquarkparameters.Todothis,GuiasuandKoniukrecalculatedtheoctetbaryonmagneticmomentsintheirnewmodel.Perhapsit'snottoosurprisingthattheirnewresult

islargelyunchangedfromthesimplestquarkmodel.Theirapproachsimplytooksomeofthepioncloudphysicsimplicitlycontainedintheconstituentquarksandmovedittoanexplicitpioncloud.HedgehogmodelsappeartopredictvaluesforfMwhicharegenerallylargerthanthelatticeprediction.Theresultspresentedhereareobtainedbytakingtheratiooftransitiontoprotonmomentsandscalingtheresultsuchthatthemodelcalculationsreproducetheprotonmoment.Thisapproacheliminates,tosomeextent,thesensitivityofthehedgehogresultstodierencesintheparametersetsofdierentauthors.Fromtopdown,thecalculationsincludethehybridmodelofCohenandBroniowski[];anearlySU()SkyrmemodelcalculationbyAdkins,NappiandWitten[0];SU()SkyrmemodelcalculationsbyKunzandMulders[];andanSU()SkyrmemodelcalculationbyChemtob[].ThebagmodelsincludeanoldMITbagcalculationofDonoghue,Golowich,andHolstein[]andachiralbagcalculationofKalbermannandEisenberg[].TheuncertaintyregionforthechiralbagcalculationreectsthesensitivityoffMonchangingthebagradiusfrom0.to.0fm.FinallytheBethe-SalpeterdeterminationindicatedinthenalcolumnofgureisthatofMitraandMittal[].AQCDsumruleresulthasnotbeenincludedherebecausetheapproachisnotabletoprovideapredictionforfMthatisfreeofunknownparameters[].LatticepredictionsoftheresonantcontributionstothehelicityamplitudesandradiativedecaywidthsaresummarizedintableXI.ThevaluesaredeterminedfromtheSachsformfactorsusingtherelationshipsreviewedintheAppendix.Valuesforradiativebranchingratiosestimatedusingheavybaryonchiralperturbationtheory[]spanarangewhichisconsistentwithourlatticepredictions.ExperimentalestimatesoftheradiativedecaywidthsarealsogivenintableXI.Theexperimentallimitforthe0!0decaywidthisrelativelyclosetoourprediction.Anonvanishingexperimentalmeasurementofthisdecaywidthmaybepossibleinthenottoodistantfuture.TABLEXI.Latticepredictionsforresonantcontributionstothehelicityamplitudesandradia-tivedecaywidths.Latticeresultshavebeennormalizedtoreproducetheprotonmagneticmoment.Experimentalestimatesoftheradiativedecaywidthsarealsogiven.LatticePredictionsExperimentTransitionA=A=(GeV=)(GeV=)(MeV)(MeV)p$+0.()0.()0.()0.()an$00.()0.()0.()+$+0.0()0.(0)0.00()0$00.0()0.0()0.0()<.b$0.0()0.0()0.00()0$00.0()0.0()0.()<0.b$0.0()0.0()0.00()aValuecalculatedfromtheanalysisofDavidsonetal. [].bValuescalculatedfromradiativebranchingratios[].

VIII.SUMMARYANDOUTLOOKWehavepresentedafullyrelativisticformalismforisolatingandextractingtheelectro-magneticmultipoleformfactorsofspin-/tospin-/transitionsinlatticeeldtheory.ResultsoftherstlatticeQCDanalysisofSU()-avoroctettodecupletbaryontransitionshavebeensystematicallyexaminedtorevealnewaspectsoftheunderlyingnonperturbativequark-gluondynamics.TheMcorrelationfunctionsforbaryontransitionsshowstatisticaluncertaintiessimilartothatseeninouroctetbaryonanalysis.Thecentralvalueswereseentodrifttosomeextentintheplateauregionforthelightu-andd-quarkthree-pointcorrelationfunctions.However,thisismorelikelyanindicationoftheneedforbetterstatisticsasopposedtoasignatureofthecorrelationfunctionsfailingtoreachtheplateauregionbeforethelatticeboundaryisencountered.LatticecalculationsofthebaryonoctettodecupletMtransitionmomentsagreewithsimplequarkmodelpredictionswhenthelatticeresultsarescaledtoreproducetheprotonmoment.Surprisingly,themannerinwhichthequarkscontributetothemomentsinthelatticecalculationsisquitedierentfromthatanticipatedbythequarkmodel.Ultimately,ahighstatisticscalculationwillrevealdierencesbetweenthelatticeandquarkmodelresults.However,dierencesintheresultsofthetwoapproachesaresmallrelativetothemoredramaticeectsseeninouroctetbaryonanalysis.Heretheeectivequarkmomentsindicatecorrectionstothesimplequarkmodeldescriptionoftransitionmomentstheorderof0to0%.Incontrast,0%correctionswereseeninouroctetbaryonstudy.WelookforwardtoanexperimentaldeterminationofthehyperonMtransitionmomentswhichwilltestthesepredictions.QuenchingoftheeectivequarkmomentsinhyperonsisseenintheMtransitions.Thequenchingislargerinthan.Thisissimilartoourresultsfordecupletbaryons.However,eectivestrangequarkmomentsdeterminedfromtransitionmomentsarefoundtobesuppressedrelativetothevaluesdeterminedfromdecupletbaryonmagneticmoments.Thelatticeresultsprefervaluesfortheratiop=pwhicharelargerthansimplequarkmodelpredictions.Thisislargelyduetodierencesinthemannerinwhichthequarkscontributetotheprotonmoment.ThelatticepredictionfortheMtransitionmomentofN$isinagreementwithmultipoleanalyses,nonrelativisticquarkmodels,andaBethe-Salpetermodelapproach,andsuggestsvaluessmallerthanthatofthechiralbagmodelandtypicalvaluesproducedinhedgehogmodels.ThestatisticaluncertaintiesinthelatticeresultsforREMarerelativelylarge.Alltheresultsoftheexperimentalanalysesliecomfortablywithinourpredictionfromrstprinciplesof+()%forN$transitions.TheCcorrelationfunctionsbecomenoisyatlargetimeseparationsandathoroughexaminationofthisformfactorwillhavetowaitforhigherstatisticscalculations.Thelatticeresultsareconsistentwithexpectationsofmultipoleanalyses.ThespatialasymmetryofourelongatedlatticepreventsusfromdrawinganystrongconclusionsontheEformfactors.However,animportantdiscoveryisthemannerinwhichstatisticaluctuationsarecompensatedincombiningthequarkcontributionstonegativechargehyperontransitionmoments.Calculationsonacubiclatticewillprovideprecise

estimatesoftheEtransitionmomentsforandhyperonswhichwillbeindispensableinmodeldevelopmentandtesting.Ahighstatisticsanalysisofhadronicelectromagneticformfactorswouldprovidecon-siderableinsighttohadronicstructure.Ourpresentunderstandingsoftheessenceoftheunderlyingquarkdynamicsmaybeconrmedandrened.Forexample,theEmomentofisparticularlyinterestingsinceitprovidesaglimpseintotheshapeofthebaryongroundstate.Ingeneral,hedgehogmodelssuchastheSkyrmionpredictalargeEmoment.Ourlatticeresultsagreewithhedgehogmodelpredictionsmainlyduetothepresenceoflargestatisticaluncertaintiesinthelatticepredictions.Thecentralvalueofthedistributionsug-gestsasmallerEmoment.AhighstatisticslatticecalculationwouldbeabletoconrmorrejectthehedgehogSkyrmiondescriptionofbaryons.Statisticallysignicantpredictionsforallthehigherordermultipolemomentswouldbeusefulinthedevelopmentofmodelhypothesesandevaluationofmodelpredictions.AnonperturbativeQCDdeterminationoftheE=Mratiofromrstprinciplesfortheelectromagnetictransitionmomentsofp$+withstatisticaluncertaintiesonparwiththeexperimentallybaseddeterminationsisanxiouslyawaitedbythoseworkinginthiseld.ItisencouragingthatabinitiolatticeQCDcalculationsofelectromagneticmultipolemomentsarealreadycompetitivewithhadronicmodelswhichuseadjustableparameters.Withfurtherinvestigationstoreducestatisticalandsystematicerrors,latticestudiesofhadronicelectromagneticformfactorswillcontinuetoprovidenewinsightintononpertur-bativeQCD.Thestrongsignalsformagneticdipoletransitionsforalloctetbaryonsandfortheelectricquadrupoletransitionsofandbodewellforfuturelatticecalculationsofthesequantitiesbeingabletofurtherdiscriminateamongmodelsofhadronicstructure.ACKNOWLEDGMENTSThecomputingresourcesforthisstudywereprovidedbytheComputingScienceDe-partmentandtheCenterforComputationalSciencesattheUniversityofKentuckyontheirIBM00-00Jsupercomputer.D.B.L.thanksSatoshiNozawaandThomasCohenforanumberofhelpfuldiscussions.T.D.thanksKeh-FeiLiuformanyusefulconversations.ThisworkissupportedinpartbytheU.S.DepartmentofEnergyundergrantnumbersDE-FG0-ER-0andDE-FG0-ER-0,theNationalScienceFoundationundergrantnumberSTI-0andtheNaturalSciencesandEngineeringResearchCouncilofCanada.APPENDIX:TRANSITIONMOMENTPHENOMENOLOGYInthissectionwemakecontactwithotherobservablesandformalismsassociatedwiththephenomenologyofelectromagnetictransitionmoments.Thefollowingrelationshipsarewellestablishedinthecontinuum.However,theserelationshipsmaynotstrictlyholdforquenchedlatticeQCD.SincetherearegoodreasonsforcalculatingtheSachsformfactors(GM;GE)inthelatticeapproach[]asopposedtofMandfE,wewillusethecontinuumrelationshipswiththeirparametersdeterminedbyexperimentalvalues.Withthisapproach,thefollowingrelationshipsmaybesimplyregardedasaformof\unitconversion".

TherelationshipsbetweentheParticleDataGroup's[]electromagnetictransitionam-plitudesfMandfEandtheSachsformfactorsinvestigatedhereare[],fM=eMN j~qjMMN!=GM;(Aa)fE=eMN j~qjMMN!=j~qjMMMNGE;(Ab)wheree=p.Intherestframeofatq=0,energy-momentumconservationsetsj~qjM=MNNand,fM=eMN MMNMN!=GM;(Aa)fE=eMN MMNMN!=GE:(Ab)TheratioofEtoMformfactorsisdenedbyREMfEfM=GEGM:(A)ResonantcontributionstothehelicityamplitudesaregivenbysimplelinearcombinationsoffMandfE:A==(fM+fE)=;(Aa)A==p(fMfE)=:(Ab)PartialwidthsmayalsobeinferredfromtheSachsformfactorsassumingcontinuumdis-persionrelations:M=(MMN)MNMGM;(Aa)E=(MMN)MNMGE:(Ab)

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