Local-Duality QCD Sum Rules for Pion Elastic and (p 0,h ,h ) gg Transition Form Factors Revisited ′ ∗ → 2 1 0 Irina Balakireva 2 ∗ D.V.SkobeltsynInstituteofNuclearPhysics,MoscowStateUniversity,119991,Moscow,Russia n E-mail: [email protected] a J WolfgangLucha 0 1 InstituteforHighEnergyPhysics,AustrianAcademyofSciences,Nikolsdorfergasse18,A-1050 Vienna,Austria ] h E-mail: [email protected] p - Dmitri Melikhov p InstituteforHighEnergyPhysics,AustrianAcademyofSciences,Nikolsdorfergasse18,A-1050 e h Vienna,Austria, [ FacultyofPhysics,UniversityofVienna,Boltzmanngasse5,A-1090Vienna,Austria,and 1 D.V.SkobeltsynInstituteofNuclearPhysics,MoscowStateUniversity,119991,Moscow,Russia v E-mail: [email protected] 9 4 0 Thelocal-dualityformulationofQCDsumrulesallowsforthepredictionofhadronicformfactors 2 withoutknowledgeofthesubtledetailsoftheirstructure.Withtheaidofthisformalism,wetakea . 1 freshlookatthebehavioursofthecharged-pionelasticformfactorandoftheformfactorsentering 0 2 inthetransitionsoftheground-stateneutralunflavouredpseudoscalarmesonsp 0,h ,h ′toonereal 1 andonevirtualphotonwithinabroadrangeofmomentumtransfersQ2.Theuncertaintiesinduced : v bytheapproximationsinherenttothislocal-dualityapproachareestimatedbystudying,inparallel i X toQCD,quantum-mechanicalpotentialmodels,wheretheexactformfactors,obtainedbysolving r theSchrödingerequation,maybecomparedwiththecorrespondinglocal-dualitysum-ruleresults. a ForQ2 5–6GeV2,wejudgethepredictionsofthesimplestlocal-dualitymodeltobereliableand ≥ expecttheiraccuracytoimproveveryfastwithincreasingQ2.Thelarge-Q2predictionforthepion elasticformfactorshouldbeapproachedalreadyatmoderatemomentumtransferQ2 4–8GeV2; ≈ largedeviationsfromitslocal-dualitybehaviourforQ2=20–50GeV2,predictedbysomehadron- structuremodels,seemratherunlikely.The(h ,h ) gg formfactorsdeducedfromthesimplest ′ ∗ → local-dualityapproachexhibitexcellentagreementwithexperiment.Instartlingcontrast,BABAR measurementsofthep 0 gg formfactorimplylocal-dualityviolationswhichevenrisewithQ2. ∗ → TheXXthInternationalWorkshopHighEnergyPhysicsandQuantumFieldTheory September24–October1,2011 Sochi,Russia Speaker. ∗ (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva ′ ∗ → 1. Introduction: Motivationand Incentive forReconsidering a Long-Standing Issue QCDsumrulesaimtopredictthecharacteristicfeaturesofground-statehadrons(theirmasses, decayconstants,formfactors,etc.) fromtheunderlyingquantumfieldtheoryofstronginteractions, quantumchromodynamics(QCD),byevaluatingmatrixelementsofsuitablychosenoperatorsboth onthelevelofhadronsandontheleveloftheQCDdegreesoffreedomquarksandgluons. Wilson’s operatorproductexpansionallowsfortheconversionofthesenonlocaloperatorsintoseriesoflocal operators. BythisprocesstheQCD-levelmatrixelementsreceivebothperturbativecontributionsas wellasnon-perturbativecontributionsinvolvinguniversalquantitiescalledvacuumcondensates. In ordertosuppressthecontributionsofhadronicexcitationsandcontinuumandtoremovesubtraction terms,Boreltransformationstonewvariables,dubbedastheBorelmassparameters,areperformed. RepresentingtheperturbativecontributionstoourQCD-levelmatrixelementsinformofdispersion integralsovercorrespondingspectraldensitiesallowsustobypassourignoranceabouthigherstates byinvokingtheconceptofquark–hadronduality: beyondsomeeffectivethresholdstheperturbative QCDcontributionsandtheexpressionsofhadronexcitationsandcontinuumareassumedtocancel. TheoutcomeofthesestepsaresumrulesrelatingQCDparameterstoobservablehadronproperties. InthelimitofinfinitelylargeBorelmassparameters,allnon-perturbativeQCDcontributionsvanish andweareleftwithwhatisknownaslocal-duality(LD)formofQCDsumrules,renderingpossible toderivefeaturesofground-statehadronsfromperturbativeQCDandoureffective-thresholdideas. Recently,weappliedtheLDsum-ruleformalismtoreanalyzeboththeelasticformfactorofthe pion[1]andtheformfactorthatdescribesthetransitionP gg ofsomelightneutralpseudoscalar ∗ → mesonP=p 0,h ,h toarealphotong andavirtualphotong [2]. Oneparticularlyattractivefeature ′ ∗ oftheLDsum-ruleapproachisthepossibilitytoextractpredictionsforhadronformfactorswithout knowledgeofallsubtledetailsofthestructureofthehadronicboundstatesandtoconsiderdifferent hadronsonanequalfooting. Here,wetakearetrospectivelookfrombird’seyeviewatourfindings: Afterrecalling,fortheexampleofthepion,theratherwell-knownbasicfeaturesoftheLDsum-rule approachtopseudoscalar-mesonformfactors,inordertogetanidea(orevenroughestimate)ofthe accuracytobeexpectedforreal-lifemesonsdescribedbyQCDsumruleswemakeabriefandinthe meanwhilewell-establishedsidesteptotheirquantum-mechanicalanaloguesasameanstoexamine theuncertaintiesinducedbymodelingtheimpactofhigherhadronicstatesinarathernaïvefashion. Then,equippedwithsufficientconfidenceinthereliabilityoftheadoptedLDapproximationforthe effectivethresholds,wediscuss,inturn,thep elasticand p 0,h ,h gg transitionformfactors. ′ ∗ → (cid:0) (cid:1) 2. DispersiveThree-Point QCDSum Rulesinthe LimitofLocal Duality[3] ThebasicobjectsexploitedherefortheinvestigationofthebehaviourofformfactorsF(Q2)as functionsoftheinvolvedmomentumtransferssquared,Q2= q2 0,arethree-pointfunctions,the − ≥ vacuumcorrelatorofonevectorandtwoaxialvectorcurrents,withdoublespectraldensityD ,for pert theelasticformfactorFp (Q2)andthevacuumcorrelatorofoneaxialvectorandtwovectorcurrents, withsinglespectraldensitys pert,forthetransitionformfactorFpg (Q2),satisfyingtheLDsumrules seff(Q2) seff(Q2) s¯eff(Q2) 1 1 Fp (Q2)= fp2 Z ds1 Z ds2D pert(s1,s2,Q2), Fpg (Q2)= fp Z dss pert(s,Q2). (2.1) 0 0 0 2 PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva ′ ∗ → Here, fp isthecharged-piondecayconstant: fp =130MeV.Nowalldetailsofthenon-perturbative dynamicsareencodedintheeffectivethresholdss (Q2)ands¯ (Q2)thatenterasupperendpoints. eff eff Wetakethelibertyofintroducingthenotionofanequivalenteffectivethreshold,definedbythe requirementthattheuseofthisquantityaseffectivethresholdintheappropriatedispersivesumrule —suchastheLDrepresentativesofEq.(2.1)—reproducesfortheformfactorunderconsideration eithergivenexperimentaldataoraparticulartheoreticalpredictionexactly. Withsuchpowerfultool atourdisposal,weareabletoquantifyourobservationsandmakeourconclusionsmuchmoreclear. Withinperturbationtheory,thespectraldensitiesD (s ,s ,Q2)ands (s,Q2)arederivedas pert 1 2 pert seriesexpansionsinpowersofthestrongcouplinga byevaluatingtherelevantFeynmandiagrams: s D (s ,s ,Q2) = D (0)(s ,s ,Q2)+a (Q2)D (1)(s ,s ,Q2)+O(a 2), pert 1 2 pert 1 2 s pert 1 2 s s (s,Q2) = s (0)(s,Q2)+a (Q2)s (1)(s,Q2)+O(a 2). (2.2) pert pert s pert s Asfarastheiraspectsrelevantforourpresentpurposesareconcerned,thetheoreticalstatusofthese spectraldensitiesmaybesummarizedasfollows. InthedoublespectraldensityD (s ,s ,Q2),for pert 1 2 fixeds andlargemomentumtransfersQ2,theone-loopcontributionD (0)(s ,s ,Q2)vanisheslike 1,2 pert 1 2 D (0)(s ,s ,Q2)(cid:181) 1/Q4andthetwo-loopcontributionD (1)(s ,s ,Q2)approachesthebehaviour[4] pert 1 2 pert 1 2 1 D (1)(s ,s ,Q2) ; pert 1 2 −Q−2−−→¥ 2p 3Q2 → inotherwords,inthelimitQ2 ¥ thelowest-ordertermdecaysfasterthanthenext-to-lowestterm. → Inthesinglespectraldensitys (s,Q2),thetwo-loopcorrections (1)(s,Q2)hasbeenproven[5]to pert pert vanishidentically: s (1)(s,Q2) 0.Higher-orderradiativecorrectionshavenotyetbeencalculated. pert ≡ Withtherequiredspectraldensitiesavailableatleastuptosomeorderofperturbationtheory,as soonasthedependenciesoftheeffectivethresholdss (Q2)ands¯ (Q2)onthemomentumtransfer eff eff Q2havebeenfound,theformfactorsofinterestcanbeeasilyextractedfromtheLDsumrules(2.1). Factorizationtheoremsforhardformfactors[6],allowingforseparationofthedynamicsintoshort- andlong-distancecontributions,establishtheasymptoticbehaviouroftheformfactorsforlargeQ2: Q2Fp (Q2) 8pa s(Q2)fp2 , Q2Fpg (Q2) √2fp . −Q−2−−→¥ −Q−2−−→¥ → → Thesumrules(2.1)withthespectralfunctions(2.2)reproduce,atO(a 2)accuracy,thisbehaviourif s limseff(Q2)= lims¯eff(Q2)=4p 2 fp2 0.671GeV2 (2.3) Q2 ¥ Q2 ¥ ≈ → → holds. Theremainingtaskistodeterminethebehaviouroftheeffectivethresholdsatfinitevaluesof Q2.Unfortunately,asanalyzedindetailinRefs.[7],theformulationofareliablecriterionforfixing athresholdposesasomewhatdelicateproblemas,forfiniteQ2,theeffectivethresholdss (Q2)and eff s¯ (Q2)cannotbeassumedtobeequaltotheirasymptotes(2.3);rather,theywilldependonQ2and, eff generally,differfromeachother[8]. Averysimpleideaistoassumethattheuseoftheirasymptotic valuesprovidesameaningfulapproximationalsoatmoderatebutnottoosmallmomentumtransfer: seff(Q2)=s¯eff(Q2)=4p 2 fp2.ThischoicedefinesastraightforwardalbeitrathernaïveLDmodel[3]. Itgoeswithoutsayingthatsuchcrudeapproximationstotheeffectivethresholdsmaybewellsuited toreproducetheoveralltrendbutcanhardlyaccountforanysubtledetailofconfinementdynamics. 3 PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva ′ ∗ → 3. Exactand Local-DualityFormFactors inQuantum-Mechanical PotentialModels The(quantum-field-theoretic)LDsum-ruleapproachtobound-stateformfactorsmaybeeasily carriedovertoquantummechanics. Withinthelatterframework,thefeaturesofanyboundstatecan beobtainedwith,inprinciple,arbitrarilyhighprecisionfromtherelatedsolutionoftheSchrödinger equationfortheHamiltoniangoverningthedynamicsofthesystemunderconsideration. Therefore, quantum-mechanicalpotentialmodelsconstituteanidealtestgroundforestimatingthesignificance ofLDmodelsthatemployfortheeffectivethresholdsenteringintheadoptedsumrulestheconstant limitsfixedbysomeasymptoticbehaviouratexperimentallyaccessiblelowermomentumtransfers. Forthisveryreason,weexaminequantum-mechanicalpotentialmodelsdefinedbyHamiltoniansH whichmustincorporate,forthestudyoftheelasticformfactor,confiningandCoulombinteractions (h =1)but,fortheinvestigationofthetransitionformfactor,merelyconfininginteractions(h =0): k2 a H = +V (r) h , V (r)=s (mr)n , r x , n=2,1,1/2 . conf conf n 2m − r ≡| | Weensurearealisticdescriptionofmesonsbyadoptingforournumericalanalysisparametervalues appropriateforhadronphysics: m=0.175GeVforthereducedmassoflightconstituentquarksand a =0.3forthecouplingstrengtha oftheCoulombinteractionterm. Fortheconfininginteractions, weconsiderseveralpower-lawpotentialshapesV (r),adjustingtheassociatedcouplingstrengths conf s suchthatineachcasetheSchrödingerequationpredictsthesamevaluey (0)=0.078GeV3/2for n theground-statewavefunctiony attheorigin: s =0.71GeV,s =0.96GeVands =1.4GeV. 2 1 1/2 Then,thesizeofthelowest-lyingboundstateisabout1 fmandthusoftypicalhadronicdimensions. (cid:143)(cid:143) keffHQL@GeVD keffHQL@GeVD 0.9 0.9 Vconf~r2 Vconf~r Vconf~r1(cid:144)2 0.8 kLD=H6Π2RgL1(cid:144)3 0.8 0.7 0.7 kLD=H6Π2RgL1(cid:144)3 0.6 V ~r2 0.5 conf 0.6 0.4 k Q@GeVD 0.3 Q@GeVD 2 4 6 8 10 0.5 1 1.5 2 2.5 3 Figure1:Exactquantum-mechanicaleffectivethresholdsforelastic(left)andtransition(right)formfactors. WiththenumericallyexactsolutionoftheSchrödingerequationathand,weareinapositionto confronttheformfactorsarisingthereofwithcorrespondingpredictionsofthequantum-mechanical counterpartsoftheLDQCDsumrules(2.1),whichinvolveeffectivethresholdsk (Q)andk¯ (Q), eff eff respectively. AsintheQCDcase,theasymptoticbehaviouroftheelasticandtransitionformfactors inthelimitofinfinitelylargemomentumtransferQmaybederivedfromfactorizationtheorems[6]. Intermsoftheground-statedecayconstantR y (0)2,thisasymptoticbehaviourisguaranteedif g ≡| | theeffectivethresholdsfulfillk (Q ¥ )=k¯ (Q ¥ )=(6p 2R )1/3.Figure1showsthattheLD eff eff g → → modelk (Q)=k¯ (Q)=(6p 2R )1/3approximatesindependentlyoftheconfiningpotentialinuse eff eff g theexacteffectivethresholdsyieldingthetrueformfactorswithimprovingaccuracy,startingforthe elasticformfactoratQ2 5–8GeV2andforthetransitionformfactoratsomeevenlowerQ2value. ≈ 4 PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva ′ ∗ → 4. The PionElasticFormFactor[1] Thepionbelongs,beyonddoubt,tothebest-studiedmesons. Nevertheless,theoneortheother ofitsmostimportantpropertiesstillcannotseriouslybeclaimedtobesufficientlywellunderstood.1 Figure2displaysasnapshotofthepresentstatusofthepion’selectromagneticorelasticformfactor Fp (Q2)fromboththeexperimental[9]andthetheoretical[1,10]pointsofview. Obviously,thereis ampleroomforcontroversy,butnoconsensusonFp (Q2)formomentumtransfersQ2 5–50GeV2. ≈ Q2F HQ2L@GeV2D Π 0.6 BPS’2009 BT’2008 0.5 0.4 GR’2008 0.3 BLM’2008 0.2 pQCDasymptotics 0.1 Q2@GeV2D 2.5 5 7.5 10 12.5 15 17.5 20 Figure2:PionelasticformfactorFp (Q2):experimentaldata[9]andsomerecenttheoreticalfindings[1,10]. Inordertocastsomelightontothesedisquietingpuzzles,Figure3depictsourtranslationofthe findingssummarizedinFig.2toequivalenteffectivethresholdss (Q2)calculatedbackfromeither eff experimentaldataortheoreticalpredictionsforFp (Q2): theexacteffectivethresholdextractedfrom thedataiscompatiblewiththeassumptionthattheLDlimitisapproachedatratherlowQ2whereas, contrarytoquantumphysics,theoryseemsnottocareaboutlocalduality,atleastforQ2 20GeV2. ≤ seffHQ2L@GeV2D seffHQ2L@GeV2D 0.9 1 BT’2008 0.8 0.9 BPS’2009 0.7 seffH¥L=4Π2fΠ2 0.8 0.6 BLM 0.7 seffH¥L=4Π2fΠ2 Exact s HQL eff GR’2008 0.5 0.6 Exact Q2@GeV2D Q2@GeV2D 0 2 4 6 8 0 5 10 15 20 Figure3: Parametrizationoftheeffectivethresholds (Q2)byanimprovedLDmodel[1](labelledBLM) eff vs.exactbehaviour(red)oftheequivalenteffectivethresholdextractedfromexperimentaldata[9](left),and equivalenteffectivethresholdscorrespondingtothetheoreticalresultsforFp [1,10]depictedinFig.2(right). RatherprecisemeasurementsmaybeexpectedfromJLabafterthe12 GeVupgradeofCEBAF. 1Ofcourse,wheneversomeprobleminthetreatmentofanyoftheground-statepseudoscalarmesonsisencountered, asakindofautomaticreflex-likeresponseonemaybetemptedtoblamewithinQCDthepseudo-Goldstone-bosonnature oftheparticleforpreventingusfromacquiringasatisfactoryunderstanding.Nevertheless,allcomprehensiveapproaches shouldbeexpectedtobeabletodealwiththissortof“inconvenience”andtoultimatelyincorporatesuchcrucialfeatures. 5 PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva ′ ∗ → 5. The (p 0,h ,h ) gg TransitionFormFactors[2] ′ ∗ → Inordertoconsolidateourconcernsandtosubstantiateourconfusions,wediscusstheh andh ′ transitions(h ,h ) gg beforeturningtothecontroversialissueofthepion’stransitionp 0 gg . ′ ∗ ∗ → → 5.1 FormFactorsfortheTransitions(h ,h ) gg ′ ∗ → Thetwoisoscalarmesonsh andh ,havingthesameJPCquantumnumbers,aremixturesofall ′ lightquarks. Intheflavourbasis,themixingofthenon-strangeandstrangecontributionsisgivenby u¯u+d¯d u¯u+d¯d h = cosf s¯s sinf , h = sinf + s¯s cosf , ′ | i (cid:12) √2 (cid:29) −| i | i (cid:12) √2 (cid:29) | i (cid:12) (cid:12) withmixingangle(cid:12)(cid:12)f 39.3◦;see,e.g.,Refs.[11,12]. Thef(cid:12)(cid:12)ormfactorsreflectthisflavourstructure: ≈ Fhg (Q2)= 5Fng (Q2)cosf Fsg(Q2)sinf , Fh g (Q2)= 5Fng (Q2)sinf +Fsg(Q2)cosf . 3√2 − 3 ′ 3√2 3 Here,thenon-strangeand(s¯s)componentsFng (Q2)andFsg(Q2)oftheLDformfactorsaregivenby s¯(n)(Q2) s¯(s)(Q2) eff eff 1 1 Fng(Q2)= f Z dss p(enr)t(s,Q2), Fsg (Q2)= f Z dss p(es)rt(s,Q2), n s 0 4m2 s wheres (n) ands (s) labelthesinglespectraldensitys ofEq.(2.1)withthecorrespondingquark, pert pert pert n=u,dors,propagatingintheloop;eachcomponentutilizesaneffectivethresholdofitsown[12]: s¯(enff)=4p 2 fn2 , fn≈1.07fp , s¯e(sff)=4p 2 fs2, fs≈1.36fp . Inournumericalcalculations,weadoptm =m =0andm =100MeVforthelight-quarkmasses. u d s Q2F HQ2L@GeVD Q2F HQ2L@GeVD ΗΓ Η' Γ 0.25 0.3 LD 0.25 0.2 LD 0.2 0.15 0.15 0.1 Η®ΓΓ* 0.1 Η'®ΓΓ* 0.05 0.05 Q2 @GeV2D Q2 @GeV2D 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 Figure4:TransitionformfactorsF(h ,h )g(Q2):forh andh ′theLDmodelfitstheexperimentaldata[13,14]. ′ Accordingtoallourexperiencegainedbyin-depthinvestigationsoftheLDsum-ruleapproach withinquantummechanics,thisstraightforwardbutadmittedlynottoosophisticatedLDframework maynotperformreallywellforlowmomentumtransfersQ2,where,asabrieflookatFig.3reveals, theexacteffectivethresholdisbelowtheconstantLDeffectivethresholdinferredfromthelarge-Q2 form-factorbehaviour. However,forlargermomentumtransferthesimplequantum-mechanicalLD modelentailsaccuratepredictionsforformfactors. Figure4showsthat,forbothh andh transition ′ formfactors,wefindtheanticipatedoverallagreementbetweentheLDpredictionsandexperiment. 6 PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva ′ ∗ → 5.2 FormFactorfortheTransitionp 0 gg ∗ → InviewoftheundeniablesuccessesoftheLDmodelinthecaseofthep elasticformfactorand oftheh andh ′transitionformfactors,itsfailureinthecaseofthep 0transitionformfactorFpg (Q2) isallthemoresurprising. Figure5displayshowmarkedlytheLDpredictionforFpg (Q2)missesthe BABARdata[15]. ThisbecomesevenmoremanifestbythelinearrisewithQ2ofthecorresponding equivalenteffectivethresholds¯ (Q2),which,atleastintheregionuptoQ2 40GeV2,exhibitsno eff ≈ tendencyofapproachingitsLDlimit(2.3). Thisintriguingpuzzlestillawaitsacompellingsolution. Q2FΠΓHQ2L@GeVD seffHQ2L@GeV2D 1.1 0.3 Linear 1 0.25 0.9 0.2 0.8 0.15 LD 0.7 4Π2fΠ2 0.1 0.6 0.05 0.5 Q2@GeV2D Q2@GeV2D 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 Figure5:FormfactorFpg (Q2)forthepiontransitionp 0 gg ∗:someexperimentaldata[13,15],atleastthe → BABARdata(reddots),apparentlydivergefromourLDprediction;thisunexpectedbehaviourisreflectedby theequivalenteffectivethresholds¯ (Q2)exhibitingalinearrisewithQ2insteadofapproachingitsLDlimit. eff 6. Summary: Findings andConclusions Byreconsideringthedependenceofthepionelastic[1]andp 0,h ,h transition[2]formfactors ′ onthemomentumtransferQ2usingQCDsumrulesinLDlimit,wegainhighlyinterestinginsights: Pionelasticformfactor: Transferringtheoutcomesofourquantum-mechanicalanalysistoQCD, weexpectthesimpleLDmodeltobeapplicablewithincreasingaccuracyforQ2 4–8GeV2 ≥ irrespectiveoftheadoptedconfininginteractions. Forrealisticconfininginteractions,thisLD modelreproducestheelasticformfactorforQ2 20–30GeV2withhighprecision. Accurate ≥ measurementsofthisformfactoratsmallQ2suggestthatassumingfortheeffectivethreshold itsLDlimitalreadyatratherlowQ2=5–6GeV2mayconstituteareasonableapproximation. Hence,largedeviationsfromthisLDlimitatQ2=20–50GeV2mustberegardedasunlikely. Transition formfactorsforp 0,h ,h : Ourobservationsinquantummechanicscanbeunderstood ′ ashintsthat,forboundstatesoftypicalhadronextensions,theLDapproachshouldworkwell forQ2largerthanafewGeV2,anditindeeddoesfortheh gg andh gg formfactors. ∗ ′ ∗ → → However,arecentmeasurementoftheformfactorfortheneutral-piontransitionp 0 gg by ∗ → theBABARexperiment[15]impliesaviolationoflocaldualitywhichevengrowswithQ2,at leastuptoQ2ashighas40GeV2.WithintheLDsum-ruleformalism(2.1),suchbehaviourof atransitionformfactorcannotbeaccommodatedbyaconstantequivalenteffectivethreshold butmustbedescribedbyalinearQ2-dependenceofs¯ (Q2);aconvincingexplanationofthis eff hasyettobefound. 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