Local-Duality QCD Sum Rules for Pion Elastic and (pi^0, eta, eta') to gamma gamma* Transition Form Factors Revisited

0.14 MB·English

by  Irina Balakireva

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. ThisconclusionenjoysfullagreementwiththefindingsofRefs.[16,17].
7
PionElasticand(p 0,h ,h ) gg TransitionFormFactors IrinaBalakireva
′ ∗
→
Acknowledgments
D.M.isgratefulforfinancialsupportbytheAustrianScienceFund(FWF),projectno.P22843.
References
[1] I.Balakireva,W.Lucha,andD.Melikhov,arXiv:1103.3781[hep-ph];arXiv:1110.6904[hep-ph],
Phys.Rev.D(inpress);
I.Balakireva,PoS(QFTHEP2010)059(2010)[arXiv:1009.4140[hep-ph]];
V.Braguta,W.Lucha,andD.Melikhov,Phys.Lett.B661(2008)354.
[2] W.LuchaandD.Melikhov,arXiv:1110.2080[hep-ph].
[3] V.A.NesterenkoandA.V.Radyushkin,Phys.Lett.B115(1982)410;
A.V.Radyushkin,ActaPhys.Polon.B26(1995)2067.
[4] V.V.BragutaandA.I.Onishchenko,Phys.Lett.B591(2004)267.
[5] F.JegerlehnerandO.V.Tarasov,Phys.Lett.B639(2006)299;
R.S.PasechnikandO.V.Teryaev,Phys.Rev.D73(2006)034017.
[6] S.J.BrodskyandG.P.Lepage,Adv.Ser.Direct.HighEnergyPhys.5(1989)93.
[7] W.Lucha,D.Melikhov,andS.Simula,Phys.Rev.D76(2007)036002;Phys.Lett.B657(2007)
148;Phys.Atom.Nucl.71(2008)1461;Phys.Lett.B671(2009)445;
D.Melikhov,Phys.Lett.B671(2009)450.
[8] W.Lucha,D.Melikhov,andS.Simula,Phys.Rev.D79(2009)096011;J.Phys.G:Nucl.Part.Phys.
37(2010)035003;Phys.Lett.B687(2010)48;Phys.Atom.Nucl.73(2010)1770;J.Phys.G:Nucl.
Part.Phys.38(2011)105002;Phys.Lett.B701(2011)82;
W.Lucha,D.Melikhov,H.Sazdjian,andS.Simula,Phys.Rev.D80(2009)114028.
[9] C.J.Bebeketal.,Phys.Rev.D17(1978)1693;
JeffersonLabFp Collaboration,T.Hornetal.,Phys.Rev.Lett.97(2006)192001;
JeffersonLabFp Collaboration,V.Tadevosyanetal.,Phys.Rev.C75(2007)055205;
JeffersonLabFp Collaboration,G.M.Huberetal.,Phys.Rev.C78(2008)045203.
[10] S.J.BrodskyandG.F.deTéramond,Phys.Rev.D77(2008)056007;
H.R.GrigoryanandA.V.Radyushkin,Phys.Rev.D78(2008)115008;
A.P.Bakulev,A.V.Pimikov,andN.G.Stefanis,Phys.Rev.D79(2009)093010.
[11] V.V.Anisovich,D.I.Melikhov,andV.A.Nikonov,Phys.Rev.D52(1995)5295;55(1997)2918.
[12] T.Feldmann,P.Kroll,andB.Stech,Phys.Rev.D58(1998)114006;Phys.Lett.B449(1999)339.
[13] CELLOCollaboration,H.J.Behrendetal.,Z.Phys.C49(1991)401;
CLEOCollaboration,J.Gronbergetal.,Phys.Rev.D57(1998)33.
[14] BABARCollaboration,P.delAmoSanchezetal.,Phys.Rev.D84(2011)052001.
[15] BABARCollaboration,B.Aubertetal.,Phys.Rev.D80(2009)052002.
[16] H.L.L.Robertsetal.,Phys.Rev.C82(2010)065202.
[17] A.P.Bakulev,S.V.Mikhailov,A.V.Pimikov,andN.G.Stefanis,Phys.Rev.D84(2011)034014.
8