Theory of Small x Inclusive Photon Scattering, I

arXiv:hep-ph/9604263v16Apr1996FTUAM96-12March,1996THEORYOFSMALLxINCLUSIVEPHOTONSCATTERING,IF.J.Yndur´ain*DepartamentodeF´ısicaTe´orica,C-XIUniversidadAut´onomadeMadridCantoBlanco,28049-MadridABSTRACT.-Intheearlyeighties,L´opez,Gonz´alez-Arroyoandthepresentauthorprovedthat,ifatagivenQ20largeenoughforperturbativeQCDtobevalid,structurefunctionsbehaveasapowerofxforx→0,thenforalllargerQ2onehasF2(x,Q2)≃BS[αs(Q2)]−d+x−λ+BNS[αs(Q2)]−D11x0.5,FG(x,Q2)≃BG[αs(Q2)]−d+x−λR(x,Q2)=r0αs(Q2)π,withD11,d+,BG,r0calculableintermsofBS,λ.Moreover,itwassuggestedthatthe“hard”partofthescatteringcrosssectionforrealphotons(Comptonscattering)obeysasimilarlaw,sothatσγp≃Bγpsλ+AγpˆσP,withavalueofλcomparabletothatintheexpressionforthestructurefunc-tions,andwhereˆσP∼log2sisauniversal,Pomeron-typecrosssection,andAγp,Bγpareconstants.InthepresentpaperitisshownthattherecentHERAmeasurementsmaybedescribedbytheseformulas,withachi-sqared/d.o.f.substantiallylessthanunity,andwithvaluesoftheparameterscompatiblewiththoseoftheoldfitsofthe’80s.Moreover,furtherdiscussionsarepre-sentedbothonthelowQ2limit,andthetransitionbetweenComptonanddeepinelasticscattering,inparticularinconnectionwithpossiblesaturationofthecouplingconstantαs(Q2)atsmallQ2;andontheultrahighenergylimit,andhowonemighttesttheso-calledBFKLconjecture,limx→0Q2→∞F2(x,Q2)∼x−c0αs.Withrespecttothelastwefindsomeevidenceagainstit,atleastattheHERAenergies.*e-mail:[e-mail]11.-INTRODUCTIONWeconsiderinthisnotehighenergyinclusivescatteringof(virtualorreal)photonsoffprotons:γ∗(q)+p(p)→all.(1.1)InthecaseofrealphotonswewillcalltheprocessComptonscattering;forvirtualphotonswehaveDIS(=deepinelasticscattering).Inthislastsituationweconsidertheregionofverysmallxwith*x=Q2/ν,ν=p·(p+q),Q2=−q2.(1.2)ComptonscatteringmaybethoughtofasthelimitofDISwhenQ2→0,x→0insuchawaythatQ2/x≃s,s=(p+q)2.(1.3)ToalargeextentthepresentworkmaybeconsideredasanaggiornamentooftheoldanalysisofC.L´opez,A.Gonz´alez-Arroyoandthepresentauthor,applyingittotherecentHERADISandComptondata.InthisrespectitisshownthatthenewdataarefittedverywellbytheformulasderivedfromQCDinthesmallxregion,andthattheexpressionsF2(x,Q2)≃BS[αs(Q2)]−d+x−λ+BNS[αs(Q2)]−D11x0.5,FG(x,Q2)≃BG[αs(Q2)]−d+x−λ(1.4)R(x,Q2)=r0αs(Q2)π,withBG,D11,d+,r0calculableintermsofBS,λ,giveanexcellentapproximationtothedataforx∼10−2(Sect.2).Likewise(Sect.3)wefindthattheveryhighenergyComptoncrosssectionisstillcorrectlydescribedbytheformulaσγp≃Bγpsλ+AγpˆσP,(1.5)withavalueofλsimilartotheλin(1.4),andwhereˆσP∼log2sisauniversal,Pomeron-typecrosssection.AγpandBγpareconstants.Thequalityofthefitsissogood,withchi-squaredoflessthanonebyd.o.f.,thatonemayconsiderstudyingthecorrectionsto(1.4),andtheinterpolationbetween(1.4)and(1.5);thiswedoinSect.4.InSect.5wediscusstheultrahighmomentumlimitofourformulas,inparticularinconnectionwiththeBFKLapproachwhichsuggeststheasymptoticbehaviourF2(x,Q2)∼xc0αs.(1.6)Westudytheregionofvalidityof(1.4),limitedforfixedxbyacertainQ2(x).Wegetindicationsthat,forx∼10−2,Q2(x)∼200−300GeV2.Inwhatregardsthetransitionfrom(1.4)to(1.6)weget(notconclusive,however)evidenceagainstitatleastintheregioncoveredbytheHERAdata,i.e.,belowQ2≃800GeV2.*Moredetailsonnotationmaybefoundinref.122.-DEEPINELASTICSCATTERINGATx→02.1.-GeneralconsiderationsForDIStherelevantquantitiesarethestructurefunctions,F2(x,Q2),FG(x,Q2);R(x,Q2)=F2(x,Q2)−F1(x,Q2)F2(x,Q2).(2.1)FGisthegluonstructurefunction,andRis(proportionalto)thelongitudinalone.ItisconvenienttousethesingletfunctionFS,normalizedsothatthemomentumsumrulereadsZ10dx[FS(x,Q2)+FG(x,Q2)]→Q2→∞1.(2.2)TherelationbetweenF2andFSisasfollows:onehasF2(x,Q2)=he2qiFS(x,Q2)+FNS(x,Q2),(2.3a)whereFNSistheso-callednonsingletstructurefunction.Sinceinthelimitx→0itdecreasesveryfastcomparedtoFS,wemayconsidertheapproximaterelationF2(x,Q2)≃x→0he2qiFS(x,Q2).(2.3b)Theaveragechargeoftheexcitedflavoursin(2.3)ishe2qi=518fornf=4orhe2qi=1145fornf=5.ForthevaluesofQ2,xwith8GeV2∼Q2∼65GeVweareinamixedsituationinwhichbottomisexcitedinthesvariablebutnotintheQ2variable.Wewillthuspresentresultsforbothnf=4and5:aswillbeseentheydifferverylittle.ForQ2∼9GeV2onlythreeflavoursareexcitedintheQ2variable.Bothforthisandotherreasonsthisenergyregionrequiresaspecifictreatment.Theevolutionequationsforthestructurefunctionsmaybewrittenintwoways.OnehasthesocalledAltarelli-Parisi,orDGLAPequations[2,3],thatwewriteonlyforthesingletstructurefunctions,∂Fi(x,Q2)∂t=XjZ1xdzPij(z)Fj(x/z,Q2),(2.4)i,j=S,G;t=logQ2,andtheexplicitformofthesplittingfunctionsPijmaybefoundinrefs.1,3.Alternatively,onemaydefinethemomentsofthestructurefunctions,μS(n,Q2)=R10dxxn−2FS(x,Q2),μG(n,Q2)=R10dxxn−2FG(x,Q2)(2.5)andthentheintegro-differentialequations(2.4)maybesolvedfortheμi:μS(n,Q2)μG(n,Q2)=αs(Q20)αs(Q2)D(n)μS(n,Q20)μG(n,Q20),(2.6a)wheretheanomalousdimensionmatrixD(n)is[4]3D(n)=1633−2nf×12n(n+1)+34−S1(n)3nf8n2+n+2n(n+1)(n+2)n2+n+22n(n2−1)94n(n−1)+94(n+1)(n+2)+33−2nf16−9S1(n)4.(2.6b)ThefunctionS1isrelatedtothedigammafunction,S1(z)=ψ(z+1)+γE,ψ(z)=dlogΓ(z)/dz,γE=0.5772....Itshouldperhapsbestressedthateqs.(2.4)and(2.6)arestrictlyequivalent,