Factorization scheme analysis of $F_2^{gamma}(x,Q^

arXiv:hep-ph/9811455v212Apr1999PRA-HEP98-08FactorizationschemeanalysisofFγ2(x,Q2)andpartondistributionsfunctionsofthephoton1Jiˇr´ıCh´ylaInstituteofPhysics,NaSlovance2,Prague8,CzechRepublicFactorizationschemeanalysisofFγ2(x,Q2)inthenext–to–leadingorderQCDisrevis-ited.Itisemphasizedthatthepresenceoftheinhomogeneoustermintheevolutionequationsforquarkdistributionfunctionsofthephotonimpliessubtlebutimportantdifferenceinthewayfactorizationmechanismworksinphoton–hadronandphoton–photoncollisionsascomparedtothehadronicones.ItisarguedthatnoneoftheexistingNLOanalysesofFγ2(x,Q2)takesthisdifferenceproperlyintoaccount.Thesourceoftheensuingincompletenessistracedbacktothemisinterpretationofthebe-haviourofqγ(x,M)asafunctionofαs(M).Partonmodelinterpretationofthesocalled“constantterms”intheLOphotoniccoefficientfunctionC(0)γ(x)isgivenandsmoothtransitionbetweenthepropertiesofvirtualandrealphotonanalyzed.Finallyphenomenologicalconsequencesofthisanalysisarediscussed.1IntroductionObservedfromalargedistancethephotonbehavesasaneutralstructurelessobjectgovernedbythelawsofQuantumElectrodynamics.However,whenprobedatshortdistancesitexhibitsalsosomepropertiescharacteristicofhadrons.This“photonstructure”isquantified,similarlyasinthecaseofhadrons,intermsofpartondistributionfunctions(PDF),satisfyingcertainevolutionequations.Becauseofthedirectcouplingphotonstoquark–antiquarkpairstheseevolutionequationsare,contrarythecaseofhadrons,inhomogeneous.Thisinhomogeneityhasimportantimplicationsforthewayfactorizationofmasssingularitiesoperatesincollisionsinvolvingphotons,implicationsthathavenotbeenproperlytakenintoaccountinexistingNLOanalysesofFγ2(x,Q2).Theprimaryaimofthispaperistoremovethisshortcoming.Secondly,weshalladdressseveralissuesconcerningthestructureofthevirtualphoton:tran-sitionbetweenthepropertiesofrealandvirtualphoton,propertiesandroleofthelongitudinalvirtualphoton,andpartonmodelinterpretationofthesocalled“constantterms”inLOphotoniccoefficientfunctionC(0)γ(x).Thepaperisorganizedasfollows.InthenextSectionbasicfactsandnotationconcerningPDFofthephotonarereviewedandthepropertiesofthepointlikepartofquarkdistributionfunctionofthephotoncriticallyreanalyzed.InSection3,whichcontainsthemainresultofthispaper,wediscussindetailthefactorizationscaleandschemedependenceofFγ2(x,Q2)attheNLOandpointouttheingredientsthatmustbeincludedtomakethisanalysiscomplete.ThepropertiesofthevirtualphotonandthetransitionofPDFofthevirtualphotontothoseoftherealoneareanalyzedinSection4.PhenomenologicalimplicationsofthepresentanalysisarediscussedinSection5.1SupportedbytheGrantAgencyofASCRundergrantNo.A101060212StructureoftherealphotonDespitetherecentprogressininvestigationofthestructureofthephoton2ourknowledgeofthepropertiesofthephotonstilllagsbehindthatofthenucleon.Weshallbeprimarilyinterestedinstronginteractioneffects,butasthebasicideasandformalismofthepartonicstructureofthephotonhaveacloseanalogyinQED,thelatterwillserveasaguideinsomeofthefollowingconsiderations.2.1NotationandbasicfactsInQCDthecouplingofquarksandgluonsischaracterizedbytherenormalizedcolourcoupling(“couplant”forshort)αs(μ),dependingontherenormalizationscaleμandsatisfyingtheequationdαs(μ)dlnμ2≡β(αs(μ))=−β04πα2s(μ)−β116π2α3s(μ)+···,(1)where,inQCDwithnfmasslessquarkflavours,thefirsttwocoefficients,β0=11−2nf/3andβ1=102−38nf/3,areunique,whileallthehigherorderonesareambiguous.AsweshallstayinthispaperwithintheNLO,onlythefirsttwo,unique,termsin(1)willbetakenintoaccountinthefollowing.Nevertheless,evenforagivenr.h.s.of(1)itssolutionαs(μ)isnotauniquefunctionofμ,becausethereisaninfinitenumberofsolutionsof(1),differingbytheinitialcondition.Thissocalledrenormalizationscheme(RS)ambiguity3canbeparameterizedinanumberofways.Oneofthemmakesuseofthefactthatintheprocessofrenormalizationanotherdimensionalparameter,denotedusuallyΛ,inevitablyappearsinthetheory.ThisparameterdependsontheRSandattheNLOactuallyfullyspecifiesit:RS={ΛRS}.Forinstance,αs(μ)inthefamiliarMSandMSRSaresolutionsofthesameequation(1),butareassociatedwithdifferentΛRS4.InthispaperweshallworkinthestandardMSRSofthecouplant.InQCD“dressed”PDF5resultfromtheresummationofmultiplepartonemissionsoffthecor-responding“bare”partondistributions.AsaresultofthisresummationPDFacquiredependenceonthefactorizationscaleM.Inpartonmodelthisscaledefinestheupperlimitonsomemeasuretoftheoff–shellnessofpartonsincludedinthedefinitionofD(x,M)Di(x,M)≡ZM2tmindtdi(x,t),i=q,q,G,(2)wheretheunintegratedPDFdi(x,t)describedistributionfunctionsofpartonswiththemomentumfractionxandfixedoff–shellnesst.Partonvirtualityτ≡|p2−m2|ortransversemassm2T≡p2T+m2,aretwostandardchoicesofsuchameasure.Becauseatsmallt,di(x,t)=O(1/tk),k=1,2,thedominantpartoftheintegral(2)comesfromtheregionofsmalloff–shellnesst.VaryingtheupperboundM2in(2)hasthereforeonlyasmalleffectontheintegral(2),leadingtoweak(atmostlogarithmic)scalingviolations.ThefactorizationscaledependenceofPDFofthephoton6isdeterminedbyasystemofcoupledinhomogeneousevolutionequationsdΣ(x,M)dlnM2=kq+Pqq⊗Σ+PqG⊗G,(3)2Forrecenttheoreticalandexperimentalreviewssee[1]and[2],respectively.3Inhigherorde