Effects Of The Quantity $sigma_{TS}$ On The Spin S

arXiv:hep-ph/9604274v19Apr1996EffectsofthequantityσTSonthespinstructurefunctionsofnucleonsintheresonanceregionZhenpingLi[1]andDongYu-Bing[2]1PhysicsDepartment,Carnegie-MellonUniversity,Pittsburgh,PA,15213-38902CCAST(WorldLaboratory)P.O.Box8730,Beijing100080,P.R.ChinaandInstituteofHighEnergyPhysics,AcademiaSinica,Beijing100039,P.R.China∗February7,2008AbstractInthispaper,weinvestigatetheeffectsofthequantityσTSonthespin-structurefunctionsofnucleonsintheresonanceregion.TheSchwingersumruleforthespinstructurefunctiong2(x,Q2)attherealphotonlimitisderivedforthenucleontreatedasacompositesystem,anditprovidesacrucialconstraintonthelongitudinaltransitionop-eratorwhichhasnotbeentreatedconsistentlyintheliterature.ThelongitudinalamplitudeS12isevaluatedinthequarkmodelwiththetransitionoperatorthatgeneratestheSchwingersumrule.Thenumer-icalresultsofthequantityσTSarepresentedforbothspinstructurefunctionsg1(x,Q2)andg2(x,Q2)intheresonanceregion.OurresultsshowthatthisquantityplaysanimportantroleinthelowQ2region,whichcanbetestedinthefutureexperimentsatCEBAF.PACSnumbers:13.60.Hb,11.50.Li,12.40.Aa,14.20Gk∗Mailingaddress11.IntroductionThequantityσTS,definedinthespinstructurefunctionsofnucleonsg1(x,Q2)=MTK8π2α(1+Q2ω2)σ1/2(ω,Q2)−σ3/2(ω,Q2)+2√Q2ωσTS(ω,Q2)#(1)andg2(x,Q2)=MTK8π2α(1+Q2ω2)2ω√Q2σTS(ω,Q2)−(σ1/2(ω,Q2)−σ3/2(ω,Q2))#,(2)whereKisthephotonflux,xthescalingvariable,andMTthenucleonmass,wasnotfullyinvestigatedduetothefactthatmoststudieswereconcentratedinthedeepinelasticscatteringregion,wherethequantity2√Q2ωσTS(ω,Q2)ing1(x,Q2)vanishes.ThisisnolongerthecaserecentlyastherehavebeengrowinginterestsinstudyingthespinstructurefunctionsinthesmallQ2region,wheretheresonancecontributionsareimportant.Consequently,theinvestigationoftheeffectsofthequantityσTSinthesmallQ2regionhasbecomeincreasinglyimportant.Suchaprogrambeganwiththesuggestion[1]thattheQ2dependenceofthespindependentsumrulemightplayasignificantroleintheinterpretationoftheEuropeanMuonCollaboration(EMC)data[2],whichstartswithanegativeDrell-Hearn-Gerasimov(DHG)[3]sumruleintherealphotonlimitandendswithapositivesumrule[4]inthelargeQ2limit[5]:Z10g1(x,Q2)dx=(−ωth4MTκ2Q2=0ΓQ2→∞(3)whereωth=Q2+2mπM+m2π2M(4)isthethresholdenergyofpionphotoproductions,κtheanomalousmagneticmoment,andΓapositivequantity.Becausethecontributionsfromthequan-tityσTStothesumruleinEq.3alsovanishintherealphotonlimit,mostquantitativestudies[5,6,7]oftheQ2dependenceofthesumruleinEq.3wereconcentratedonthecontributionsfromthequantityσ12−σ32ing1,2(x,Q2).In-deed,theseinvestigationshaveshownastrongQ2dependenceofthesumruleintheQ2≤2.5GeV2region.However,thestudybySofferandTeryaev[8]suggestedthatthequantityσTSplaysasignificantroleinthesmallQ2region,2whichishighlightedbyanothersetofsumrulesforthespinstructurefunctiong2(x,Q2);Z10g2(x,Q2)dx=(ωth4MTκ(κ+eT)Q2=00Q2→∞,(5)inwhichthesamekinematicsinEq.3isused.ThesumrulesinEq.5werefirstderivedbySchwinger[9]intherealphotonlimitandBurkhardtandCummingham[10]inthelargeQ2limit.CombiningEqs.3and5leadstothesumruleforthequantityσTSintherealphotonlimit;limQ2→0Z∞ωthσTSdω√Q2=4π2α4M2TeTκ.(6)ThemagnitudeofthesumruleforthequantityσTSintherealphotonlimitiscertainlycomparabletotheDHGsumrule.Thus,amorequantitativestudyofthecontributionsfromthequantityσTStothespinstructurefunctionsg1(x,Q2)andg2(x,Q2)iscalledfor.SuchastudynotonlyenablesustogiveamorepreciseestimateoftheQ2dependenceofthesumruleforg1(x,Q2),butalsoprovidesaquantitativecalculationofthesumruleforg2(x,Q2)forthefirsttimeinthequarkmodel.ThefocusofthispaperistodevelopaframeworkinthequarkmodeltoevaluatethecontributionsfromthequantityσTS,andpresentthenumericalresultsthatcanbetestedinthefutureCEBAFexperiments.ThesumrulesinEqs.3and5aremoregeneralandmodelindependent,therefore,theymustbesatisfiedinthequarkmodelinordertogiveaconsistentevaluationofthespinstructurefunctionsintheresonanceregion.Thishasbeeninvestigated[6]inthequarkmodelforsumrulesinEq.3,andtheyrequirethattheelectromagneticinteractionforamanybodysystemshouldhastheformHt=Xj{ej~rj·~Ej−ej2mj~σj·~Bj−ej4mj~σj·[~Ej×~pj2mj−~pj2mj×~Ej]+Xjl14MT[~σjmj−~σlml]·(el~El×~pj−ej~Ej×~pl)},(7)andthequantityΓinEq.3isrelatedtothequarkmodelmatrixelementΓ=12hi|Xje2jσzj|iiP−A(8)wherequarkjatpositionrjhasmassmjandchargeej,andA(P)indicatesthatthedirectionsofthepolarizationbetweenphotonsandthetargetarean-tiparallel(parallel).Ontheotherhand,thesumruleforthequantityσTSin3Eq.6hasnotbeeninvestigatedinthequarkmodel.ThederivationofEq.6inthequarkmodelisbynomeanstrivialsinceitwasproved[9]inQEDbyassumingthenucleonasanelementaryparticle.ThesimilartransitionoftheDHGsumrulefromanelementaryparticletoamanybodysystemledtoex-tensivediscussionsinlatesixtiesandearlyseventies[11].Moreover,theproofofEq.6requiresevaluationsofbothhelicityamplitudeA12andthelongitu-dinalamplitudeS12.WhilethehelicityamplitudeA12hasbeencalculated[12]withthetransitionoperatorHtinEq.7,thelongitudinalamplitudeS12hasnotbeentreatedconsistentlyintheliterature.Inparticular,theproblemofthecurrentconservationwasnotfullyunderstood[13],andanadhoccurrentJ′3=−k3J3−k0J0k3wasintroduced[14]toevaluat