Nuclear Shadowing in DIS Numerical Solution of the

arXiv:hep-ph/0301043v25Jun2003NuclearShadowinginDIS:NumericalSolutionoftheEvolutionEquationfortheGreenFunctionJ.NemchikInstituteofExperimentalPhysicsSAS,Watsonova47,04353Kosice,SlovakiaAbstractWithinalight-coneQCDformalismbasedontheGreenfunctiontechniqueincorpo-ratingcolortransparencyandcoherencelengtheffectswestudynuclearshadowingindeep-inelasticscatteringatmoderatelysmallBjorkenxBj.CalculationsperformedsofarwerebasedonlyonapproximationsleadingtoananalyticalharmonicoscillatoryformoftheGreenfunction.WepresentforthefirsttimeanexactnumericalsolutionoftheevolutionequationfortheGreenfunctionusingrealisticformofthedipolecrosssectionandnucleardensityfunction.Wecomparenumericalresultsfornuclearshadowingwithpreviouspredictionsanddiscussdifferences.11IntroductionNuclearshadowingindeep-inelasticscattering(DIS)offnucleiisintensivelystudiedduringthelasttwodecades.Itcanbetreateddifferentlydependingonthereferenceframe.Intherestframeofthenucleusthisphenomenonlookslikenuclearshadowingofthehadronicfluctuationsofthevirtualphotonandisoccurredduetotheirmultiplescatteringinsidethetarget[1,2,3,4,5,6,7,8,9,10].Intheinfinitemomentumframeofthenucleusitcanbeinterpreted,however,asaresultofpartonfusion[11,12,13,14]leadingtoareductionofthepartondensityatlowBjorkenxBj.Althoughthesetwophysicalinterpretationsarecomplementary,wewillworkintherestframeofthenucleus,whichismoreintuitiveandiswellsuitedalsoforthestudyofthecoherenceeffects[15].ImportantphenomenonwhichcontrolsthedynamicsofnuclearshadowinginDISiseffectofquantumcoherence.Itresultsfromdestructiveinterferenceoftheamplitudesforwhichtheinteractiontakesplaceondifferentboundnucleons.Itcanbetreatedalsoasthelifetimeofthe¯qqfluctuationandestimatedbyrelyingontheuncertaintyprincipleandLorentztimedilationas,tc=2νQ2+M2¯qq,(1)whereνisthephotonenergy,Q2isphotonvirtualityandM¯qqistheeffectivemassofthe¯qqpair.Itisusuallycalledcoherencetime,butwealsowillusethetermcoherencelength(CL),sincelight-conekinematicsisassumed,lc=tc.CLisrelatedtothelongitudinalmomentumtransferqc=1/lc.TheeffectofCLisnaturallyincorporatedintheGreenfunctionformalismalreadyappliedinDIS,Drell-Yanpairproduction[15,16]andvectormesonproduction[17,18](seealsothenextSection).ThenuclearshadowinginDISwasstudiedin[15,16]usingcorrectquantummechanicaltreatmentbasedontheGreenfunctionformalism.TheGreenfunctioncontrolsthennotonlytherelativetransversemotionofthe¯qqpairbutalsoanimportanceofthehigherordermultiplescatteringsinthenucleus.ThesolutionoftheevolutionequationfortheGreenfunctionwasperformedsofaranalytically.Thisanalyticalsolutionrequires,however,toimplementseveralapproximationsintoarigorousquantum-mechanicalapproachlikeaconstantnucleardensityfunction(seeEq.(21))andaspecificquadraticformofthedipolecrosssection(seeEq.(20)).Consequently,obtainedinasuchwaytheharmonicoscillatorGreenfunction(seeEq.(22))wasusedforcalculationofnuclearshadowing.However,thefollowingquestionnaturallyarises;howaccurateistheevaluationofthenuclearshadowinginDISusingthisGreenfunction?InordertoclarifythisoneshouldsolvetheevolutionequationfortheGreenfunctionnumerically.Itdoesnotbringanyadditionalassumptionsanddoesnotforceustousesupplementaryapproximations,whichcausethetheoreticaluncertainties.ThereforethemaingoalofthispaperistopresentforthefirsttimethepredictionsofnuclearshadowinginDISatmoderatelysmallxBjbasedonexactnumericalsolutionoftheevolutionequationfortheGreenfunction.Inaddition,applyinganalgorithmdescribedintheAppendixAwepresentalsocalculationsofnuclearshadowingwithintheharmonicoscillatorGreenfunctionapproachusingquadraticformofthedipolecrosssection(Eq.(20))andaconstantnucleardensityfunction(Eq.(21)).Wecheckwhethertheycorrespondtotheresultsalreadypresentedin[15].Finallyweanalyzeanddiscussthedifferencesbetweentheexactandapproximatepredictionsfornuclearshadowing.Advantagesofanexactnumericalsolutionofthetwo-dimensionalSchr¨odingerequationfortheGreenfunction(seeEq.(17))presentedinthispaperprovideabetterbaselineforthefuture2studyoftheQCDdynamicsnotonlyinDISoffnucleibutalsoinfurtherprocessesoccurredinlepton(proton)-nucleuscollisions.Calculationsofnuclearshadowingpresentedinthepaper[15]wereperformedassumingonly¯qqfluctuationsofthephotonandneglectinghigherFockcomponentscontaininggluonsandseaquarks.Performingrealisticcalculations,weincludetheeffectsofhigherFockstatesastheenergydependenceofthedipolecrosssection,σ¯qq(~r,s)1.Weusetworealisticparametrizationsofσ¯qq(~r,s)(seethenextSectionandEqs.(5)and(6)),wheretheenergydependenceisnaturallyincluded.However,wewillneglecthigherFockstatesleadingtogluonshadowing(GS)[19]assumingonlylowandmediumvaluesofthephotonenergyνaswasdonealsoin[15].Thepaperisorganizedasfollows.InthenextSectionwepresentthelight-conedipolephenomenologyfornuclearshadowinginDIStogetherwiththeGreenfunctionformalism.TheSection3supplementedbyAppendixAisdevotedtodescriptionofanalgorithmfornumericalsolutionoftheevolutionequationfortheGreenfunction.NumericalresultsbasedonrealisticcalculationsandacomparisonwithpredictionswithinharmonicoscillatorGreenfunctionapproacharepresentedinSection4.Fi