On the Extraction of the Neutron Spin Structure Fu

arXiv:nucl-th/9606034v118Jun1996OntheExtractionoftheNeutronSpinStructureFunctionsandtheGerasimov–Drell–HearnIntegralfrom3~He(~e,e′)XdataC.CiofidegliAttiandS.Scopetta1DepartmentofPhysics,UniversityofPerugia,andINFN,SezionediPerugia,viaA.Pascoli,Perugia,I-06100,Italy.AbstractNucleareffectsinpolarizedinelasticelectronscatteringoffpolarized3Hearediscussed;inthekinematicsoffutureexperimentsatCEBAF,Fermimotioneffectsarefoundtobemuchlargerthanindeepinelasticscattering.Itisshownthatimproperlydescribingnucleardynamicswouldleadtotheextractionofunreliableneutronspinstructurefunctions.Ontheotherhandside,asimpleandworkableequationrelatingtheGerasimov–Drell–HearnIntegralfortheneutrontothecorrespondingquantityfor3Heisproposed.PACS:13.60.-r,25.10.+s,11.55.Hx,14.20.GkKEYWORDS:GDHintegral,Fewnucleonsystems,sumrules,nucleonresonances1NowatInstitutf¨urKernphysikderUniversit¨atMainz,Joh.-Joachim-Becher-Weg45,D-55099Mainz,Germany1ThemeasurementofthepolarizednucleonSpinStructureFunctions(SSF)g1andg2intheresonanceregionallowsonetocheckthehelicitystructureofthephoton–nucleoncouplingbetweentheDeepInelasticScattering(DIS)regionandtherealphotonlimit[1].RecentlyithasbeenproposedatCEBAFtostudytheSSFinawiderangeofenergy(0.2GeV≤ν≤3GeV)andmomentum(0.15GeV2≤Q2≤2GeV2)transfers,forboththeproton[2]andtheneutron,usinginthelattercasepolarizeddeuteron[3]and3He[4]targets.Foranyspin12hadronictargetA,theSSFgA1readsasfollows[5]gA1(ν,Q2)=MAK8π2α(1+Q2ν2)σ1/2(ν,Q2)−σ3/2(ν,Q2)+2√Q2νσTL(ν,Q2)#(1)whereσ1/2(3/2)(ν,Q2)isthecrosssectionforphoton–hadronscatteringwithtotalhelicity1/2(3/2),σTL(ν,Q2)istheinterferencebetweenthetransverseandthelongitudinalcrosssections,KisthephotonfluxandMAisthehadronmass.ArelevantquantityrelatedtotheSSFgN1ofthenucleonistheGerasimov–Drell–Hearn(GDH)Integral:IGDH(Q2)=∞Zνthdννσ1/2(ν,Q2)−σ3/2(ν,Q2),(2)whereνth=(Q2+2mπM+m2π)/2Misthethresholdenergyforthepion-electroprodu-ction,mπisthepionmassandMisthenucleonmass.TheintegralIGDHgivesintherealphotonlimittheGDHSumRule[6]:IGDH(Q2=0)=∞Zνthdννσ1/2(ν,Q2=0)−σ3/2(ν,Q2=0)=−2π2αM2κ2≃−0.53GeV−2forprotons−0.60GeV−2forneutrons(3)whereκistheanomalousmagneticmomentofthenucleon.Using(1)and(2),IGDH(Q2)canberelatedinthelargeQ2limittothefirstmomentoftheSSFg1and,byusingtheEMCresultfortheDISSSFoftheprotong1[7],onegets,aroundQ2≃10GeV2,IGDH(Q2)≃0.14/Q2,whichisevidenceoflargechangesinthehelicitystructureoftheγpcouplingbetweentherealphotonlimitandtheDISregion,leadingtoachangeofsignofIGDH(Q2)atsomevalueofQ2.Muchtheoreticalworkhas2beenproduced,bothintherealphotonlimit[8]andatfinitevaluesofQ2[9],inordertounderstandthisbehavior,whichshouldbemainlyduetotheelectroexcitationofnucleonresonances.ExtensiveexperimentalprogramsareplannedtoinvestigatenotonlytheQ2evolutionoftheGDHIntegral,butalso,forthefirsttime,theprotonandneutronGDHSumRule[10];inthelattercase,severalanalysesoftheavailableunpolarizedphotoproductiondatadonotagreewiththeprediction(3)fortheneutron,whilesimilarestimatesseemtogivethecorrectGDHvaluefortheproton;itisforthisreasonthattheneutronmeasurementisofparticularrelevance[1].ItisworthnoticingthatthemeasurementoftheneutronSSFintheCEBAFexperi-mentswillalsocontributetotheinvestigationofthelowQ2evolutionoftheBjorkensumrule,whichrequiresanaccurateknowledgeofgn1for0≤x≤1,beingx=Q2/2MνtheBjorkenvariable.NucleareffectsontheSSFg1ofthedeuteronhavealreadybeenstudied[11]inthekinematicsoftheplannedCEBAFexperiment;theaimofthisletteristopresenttheresultsofcalculationsofnucleareffectsonthespinstructurefunctionof3He.AconvolutionmodelforthenuclearSSFgA1(2)foranyspin12nucleushasbeenobtainedin[12,13];itreadsasfollowsgA1Q2,ν=XN=p,nZdzZdEZd~pMEpMνp·qgN1z,ν,Q2PNk(~p,E)+TN~p,E,Q2δz+M2−p·p2Mν−q·pMν!(4)whereEistheremovalenergy,p≡(MA−q(E+MA−M)2+|~p|2,~p)isthe4–mo-mentumoftheboundnucleon,Ep=qM2+|~p|2andgN1(z,ν,Q2)isthenucleonSSF.NucleareffectsaredescribedbythequantitiesPNk(~p,E)andTN(~p,E,Q2),whicharebothrelatedtotheelementsofthe2x2matrix,representingthespindependentspectralfunctionofanucleoninsideanucleuswithpolarization~SA[14].TheelementsofthismatrixarePNσ,σ′,M(~p,E)=XfA−1h~p,σ;ψfA−1|ψJMihψJM|ψfA−1;~p,σ′iδ(E−EfA−1+EA)(5)3where|ψJMiisthegroundstateofthetargetnucleuspolarizedalong~SA,|ψfA−1iisaneigenstateofthe(A-1)nucleonsystemcorrespondingtoanenergyEfA−1,|~p,σiistheplanewaveforthenucleonN≡p(n).Inparticular,PNk(~p,E)=P1212,12(~p,E)−P−12−12,12(~p,E),whereastheQ2dependenttermTN(~p,E,Q2),whichgetscontributionfrombothgN1andgN2,dependsalsouponP12−12,12(~p,E);thistermhasbeenfoundtogiveaverysmallcontributioninboththeDeepInelasticScattering[12]aswellasinthepresentcalculationoftheresonanceregion:forthisreasonitwillbeomittedhereafter.IntheDISkinematics(Q2→∞,ν→∞),Eq.(4)givesthesimpleconvolutionformuladescribedinRef[12]:gA1(x)=XNAZxdz1zgN1xzGN(z),(6)withthelight–conemomentumdistributiongivenbyGN(z)=ZdEZd~pPNk(~p,E)δz−p+M!(7)wherep+=p0−~p·~q/|~q|isthelight–conemomentumcomponent.Itiswellknown[15]thattheimpulseapproximationusedaboveisatvariancebyabout4%withtheexperimentalresultsonβdecay,whicharerelatedtotheintegralofthedifferenceoftheSSFof3H,g3H1,and