Pion-Photon Transition Form Factor and Pion Distribution Amplitude in QCD: Facing the Enigmatic Behavior of the BaBar Data

0.15 MB·English

by  N. G. Stefanis

HEP-version(tobepublishedin“Phi-to-Psi2011”Proceedings)RUB-TPII-07/2011
Pion-Photon Transition Form Factor and Pion Distribution Amplitude in QCD:
Facing the Enigmatic Behavior of the BaBar Data
N.G.Stefanisa,b,,A.P.Bakuleva, ,S.V.Mikhailova,A.V.Pimikova
∗ ∗∗
1
1 aBogoliubovLaboratoryofTheoreticalPhysics,JINR,141980Dubna,Russia
0 bInstitutfu¨rTheoretischePhysikII,Ruhr-Universita¨tBochum,D-44780Bochum,Germany
2

v
o
N
Abstract

0
3 Wepresentanextendedanalysisofthedataforthepion-photontransitionformfactorfromdifferentexperiments,
CELLO,CLEO,andBaBar,anddiscussvarioustheoreticalapproacheswhichtrytoreasonfromthem. Wefocuson

] thedivergentbehavioroftheBaBar dataforthe pionandthoseforthe η(η)pseudoscalarmesonsandcommenton
h ′
p recentlyproposedexplanationsforthisdiscrepancy. Wearguethatitisnotpossibleatpresenttoaccommodatethese
- datawithinthestandardQCDframeworkself-consistently.
p
e Keywords: Mesontransitionformfactors,piondistributionamplitude,QCD(light-cone)sumrules
h
[

1 1. Two-photonprocessesforπ0andη(η′)inQCD the longitudinal and intrinsic transverse momenta car-
v
riedbythepartons.Theyhavetobedeterminedbynon-
7 Atthe basisof applicationsofQuantumChromody-
perturbative methods (models), lattice calculations, or
3 namics(QCD),whichgovernstheinteractionsofquarks
1 fromthedata.
and gluons, is the property of factorization of corre-
7 The (spacelike) transition form factors (TFFs) of
sponding amplitudes in hard processes. Proving the
. pseudoscalarmesons, in particularthe pion, havebeen
1
property of factorization means that a hadronic pro-
1 extensivelystudiedwithinQCD,becauseinleadingor-
cess at large momentum transfer can be dissected in a
1 der they are purely electromagneticprocesses with the
1 productofdistinctquantitieseachassociatedwithsep- bindingQCD effectsbeingfactorizedoutintothepion
: arate regimes of dynamics. Then, subprocesses devel-
v distribution amplitude (DA)—see [1] for a review and
opingatshortdistancesandinvolvingpartons—quarks
i [2] for recentreferences. This meansthat for a highly
X and gluons—can be accurately described in a system-
virtualphotonwiththefour-momentumtransferQ2and
r aticwaywithinperturbationtheory. Ontheotherhand,
a aquasi-realphotonwithq2 0,theTFFcanbecastas
the dynamical(mainlynonperturbative)featuresof the →
the convolutionofthe twist-two hard-scatteringampli-
hadronbindingeffectsareencodedincorrelators,which
tudeT(Q2,q2 0,x) = Q 2(1/x+O(α )),describing
contain quark and gluon field operators, and can be → − s
the elementary process γ γ qq¯, with the twist-two
parameterized in terms of light-cone wave functions pionDAϕ(2)(x;µ2)sotha∗t →
and parton distribution functions. These quantitiesare π
process-independentanddescribetheuniversalinterpo- 1
√2
lgarteioesnobfetfwreeeednomhadasrodnisstarinbduttihoenisr qovuearrkthaendfragclutioonnsdoef- Q2Fγγ∗π(Q2) = 3 fπZQ2T(Q2,x)ϕ(π2)(x;Q2)dx
0
δ2
Correspondingauthor +O , (1)
∗ Q2!
Speakerofthetalkatthe“Psi-to-Phi”Conference
∗∗
Emailaddress:[email protected]
(N.G.Stefanis) where δ2 is the scale of the twist-fourterm taking val-
N. Stefanisetal./HEP-version(tobepublishedin“Phi-to-Psi2011”Proceedings)RUB-TPII-07/2011 2
uesin the rangeδ2 = 0.15 0.23GeV2. Notethatwe (iii) Beyond 10 GeV2, an acceptable statistical de-
÷
have omitted for simplicity variables irrelevantfor our scriptionofthedata(χ2 > 1)demandstheinclusionof
discussion(see[3]and[4]formoredetails). a sizeable and positive Gegenbauer coefficienta with
6
Several experimental groups have measured a 1.8a 1.7a (at the scale µ = 2.4 GeV
6 4 2 SY
Q2Fγ∗γπ0(Q2,q2 0) and Q2Fγ∗γη(η′)(Q2,q2 0) in [10]≃). −Stillhigh≃ercoefficientshavebeentakenintoac-
→ →
two-photon processes e+e e+e γ γ e+e X, countin[11] butfoundto havelittle effect. Moreover,
− − ∗ −
→ →
where X = π0 [5–7], η and η [6, 8]. The itisdifficulttoselecttheoptimalnumberofharmonics
′
range of probed photon momentum varies from needed. Thus,theendpointenhancementprovidedthis
0.7 2.2 GeV2 (CELLO), to 1.64 7.90 GeV2 way is not sufficient to reproduce the steep rise of the
÷ ÷
(CLEO), to 4.24 34.36 GeV2 (BaBar). A recent pion TFF observed by BaBar [7]. The range of varia-
÷
compilation and discussion of the experimental data tion of the associated pion DAs, conforming with this
with the focus on the BaBar data can be found in 3D(a ,a ,a )analysis,wasworkedoutforbothsetsof
2 4 6
[9]. The current situation from the experimental data, i.e., [1 9] GeV2 —set1and[1 40]GeV2 —
÷ ÷
point of view can be summarized like this: all data set2,in[11],seealsoFig.1. Thisfigureshowsbest-fit
for the TFFs pertaining to the non-strange part of results forboth data sets in the formof bandsof TFFs
the η and η conform with the asymptotic QCD witherrorsstemmingfromthesumofthestatisticaler-
′
limit [3] limQ2 Q2Fγ∗γπ0(Q2) √2fπ = 0.185 rorsandthetwist-fouruncertainties. Thebest-fitcurve
and are best d→es∞cribed by endpo−i→nt-suppressed DAs, toset1inFig.1isrepresentedbythesolid(blue)line,
while there is remarkable contrast to the high-Q2 whereasthebest-fitcurvetoset2athighQ2isdenoted
tail of the BaBar data for the π0 TFF [8, 9]. The bythedashed(red)line. Thisbehaviormakesitappar-
rising trend of the BaBar data can be reproduced entthatintheframeworkofLCSRsthefittotheBaBar
by the fit [7] Q2F(Q2) = A(Q2/10 GeV2)β with data above9 GeV2 deviatesfromthatatlow-Q2 at the
A=0.182 0.002GeVandβ=0.25 0.02.Thelatter levelof1σ.
± ±
value differs significantly from 0 predicted by QCD
perturbationtheory,cf. Eq.(1).
Fromthetheoreticalside,theconclusionsdrawnfrom 0.25 Q2F(Q2) [GeV2]
different approaches are at odds with each other and
cannot accommodate all available data simultaneously
0.20
toresolvethepuzzle. Inourrecentdataanalysisin[2]
(see also [11]), we used Light-Cone Sum Rules (LC-
SRs) [12, 13] and performed an extended calculation
0.15
of the pion TFF which takes into account the next-to-
leading(NLO) orderradiativecorrections[14] and the
twist-fourcontribution[12,15],whilealsoincludingin
0.10
termsofuncertaintiesthemainnext-to-next-to-leading- Q2 [GeV2]
order(NNLO)correction[4,16]andthetwist-sixterm,
computed in [17]. The evolution of ϕ(2)(x;µ2) with 1.0 1.5 2.0 3.0 5.0 7.0 10.0 15.0 20.0 30.0
π
µ2 > 1 GeV2 was also taken into account at the NLO
level with Λ(3) = 370 MeV and Λ(4) = 304 MeV. Figure1: Best-fitcurvestotheexperimentaldatafortheTFFinthe
QCD QCD frameworkofLCSRs,insidecorrespondingbandsof68%CLregions
Themainobservationsarelistedbelow.
assumsofstatistical errorsandtwist-fouruncertainties. Bluelines
(i)AlldataforQ2Fγ∗γπ0(Q2)intherange[1 9]GeV2 refertoset1andredlinestoset2ofthedatafrom[5–7],withdesig-
can be described at the level of χ2 < 1 with÷only two nationsasinFig.2.Thehorizontaldashedlinemarkstheasymptotic
QCDlimit.
Gegenbauercoefficientsa anda withnegativea ,sat-
2 4 4
isfying a .a .
4 2
| |
(ii) The error ellipses of the data (CELLO, CLEO, Moreover,following[20]withsomerefinementsex-
BaBar)inthe(a ,a )planeoverlapwiththeallowedre- plained in [2], one obtains the following windows for
2 4
gion for these two parameters determined before [18] the values of the moments: ξ2 [0.23 0.29],
π
h i ∈ ÷
using QCD sum rules with nonlocal condensates [19]. ξ4 [0.102 0.122] and ξ2 [0.26 0.29],
π π
h i ∈ ÷ h i ∈ ÷
ThesepionDAsarecharacterizedbyastrongsuppres- ξ4 [0.11 0.122]. These estimates were derived
π
h i ∈ ÷
sionoftheirkinematicalendpoints x = 0,1,where xis from ourLCSR analysis [2, 11] by combiningall data
thelongitudinalmomentumfractionofthequarkinside of set 1 with the lattice results obtained in [21] and
thepion. [22],respectively. Allvalueswereevaluatedatthetyp-
N. Stefanisetal./HEP-version(tobepublishedin“Phi-to-Psi2011”Proceedings)RUB-TPII-07/2011 3
ical lattice scale µ2 = 4 GeV2, using the definition question is: What kind of mechanism underlies such
Lat
ξN 1ϕ(2)(x,µ2 )(2x 1)Ndx,withϕ(2)(x,µ2)be- a strong flavor-symmetry breaking in the pseudoscalar
h iπ ≡ 0 π Lat − π mesonsectorofQCD?
ing normRalized to unity. The “window” for ξ2 , ob-
π
h i
tained for the data with Q2 < 10 GeV2 (set 1), has in
terms of the 1σ error ellipse a large intersection with
themostrecentlatticeestimatesfrom[21–23]. Includ- 0.35 Q2F(Q2) [GeV2]
ingintothefitthehigh-Q2BaBardata,thisintersection 0.30
significantlydeteriorates—see[2,11]forfurtherdetails.
0.25
(iv) These findings indicate a significant discrep-
ancy between the BaBar data for π0 and the method 0.20
of LCSRs—and in more general terms the QCD
0.15
factorization—athigh-Q2 values,anindicationthatthe BaBarγ∗γ→π0
analysis in [17] is possibly no turning point in un- 0.10 BaBarγ∗γ→η,η′
Tdehrussta,noduirngantahleyshisigdho-eQs2nπo0t cBoanBfiarrmdtahteaowpipthoisniteQcCoDn-. 0.05 BCCaELBELaLOrOγe+∗γγ∗e−γ→→→ππ0γ0η,γη′Q2 [GeV2]
0.00
clusions drawn in [17] which uses the same calcula-
1 10 100
tionalscheme, butalargervalueoftheauxiliaryBorel
parameter M2, notably 1.5 GeV2, instead of values Figure2:Theoreticalpredictionsforthescaledtransitionformfactors
M2 < 1GeV2 asinourapproach,(whichfollows[12]) forπ0andthenon-strangepartoftheηandη′fromvarioustheoretical
andmorecoefficientsintheconformalexpansionofthe approaches. Thegreenstripcontainstheresultsobtainedin[2]using
themethoddescribedinthetext. Thetwosolid(blue)linesrepresent
pionDA.
thefindingsof[17],whereasthedottedandthedouble-dotted-dashed
(v)ConformitywiththeincreasingtrendoftheBaBar linesdenotetwoindependentpredictionsderivedfromAdS/QCDin
data can be actually achieved only with a flatlike pion [29] and [30], respectively. The experimental data are taken from
DA, like that proposed in [24] and in a different con- variousexperiments,referencedinthetext.
text in [25]. However, the use of such a pion DA re-
In the next section, we will discuss some proposed
duces the accuracy of the predicted TFF at lower val-
ues of Q2 [2, 26]. Moreover, it was emphasized in explanationsoftheBaBardatawithoutattemptingtobe
comprehensiveorconclusive.
[2, 11, 27] that then one becomes unable to reproduce
the BaBar data [8] for the nonstrange part of the η:
n = 1/√2 uu¯ + dd¯ .1 Indeed, employing the
| i | i | i 2. DataExplanations
mixing(cid:16)scheme(cid:17)(cid:16)of[28],one(cid:17)canrelatetheTFFof n to
| i
thatoftheπ0multipliedbyafactor5/3duetothequark
Let us start by recalling some facts related to the
charges (also assuming that f = f ). Hence, it ap-
n π pionTFF.TheCLEOdatafavorapionDAclosetothe
pearsthattheTFFfortheπ0andthe n followantithetic
| i asymptotic (as) form [31, 32] excluding pion DAs of
trends that correspond to DAs with distinct endpoint
the Chernyak–Zhitnitsky (CZ) [33] type. Later, more
characteristics: extreme endpointenhancementfor the
detailed analyses of these data [10, 34] have excluded
firstvs. endpointsuppressionforthe second[27]. Be-
ϕCZ andϕas atthe4σandthe3σlevel,respectively.To
cause the properties of n are not so sensitive to the π π
| i achieveanagreementwiththeCLEOdatawith1σac-
choiceofthemixingangle,suchastrongantitheticbe-
curacy,onehastouseendpoint-suppressedpionDAsof
haviorcannotbeascribedtothisuncertainty. Thesere-
theformderivedin[18]withthehelpofQCDsumrules
sultsareshowninFig.2usingalogarithmicscaleforQ2
with nonlocal condensates. In view of these findings,
andcomparingthemwithalltheavailableexperimental
it was thought that more precise measurements which
data. Our predictions [2] are represented by a (green)
would extend the range of Q2 to much higher values,
stripwhosewidthisameasureoftheinvolvedtheoreti-
would confirm this trend because for large momentum
caluncertainties,whilethesolid(blue)linesreproduce
transferstheapplicationofperturbativeQCDontheba-
andextendfartheroutthepredictionsof[17]. Theother
sisofcollinearfactorizationshouldworkbetterandbet-
lines will be explained shortly in Sec. 2. The crucial
ter. Surprisingly,therapidgrowthofthehigh-Q2BaBar
dataisatoddswiththeseexpectations,thoughthiswas
1The separation of the nonstrange part of the η and the η is
′ not immediately recognized. But in coincidence with
modeldependent. Moreover,thedecayparameters fηand fη′ arenot thepublicationoftheBaBardatainthearXiv,twoofus
wellknownandstronglydependonthemixingbetweenthetwoeta
mesons.WehereemployedthemixingschemeofRef.[28]. haveclearlystatedthatnoagreementbetweenthestan-
N. Stefanisetal./HEP-version(tobepublishedin“Phi-to-Psi2011”Proceedings)RUB-TPII-07/2011 4
dardQCDschemeandarisingTFF(scaledbyQ2)can with each other and with the BaBar data for the η(η),
′
beachievedintermsofendpoint-vanishingπDAs[4]. though they somewhat overestimate all data at lower
Interestingly,otherauthorsproposedatthesametime values of Q2. Moreover, one observes that both pre-
explanations of such an “anomalous” behavior by ap- dictionsoverlapwith the bandofthe resultsderivedin
pealing to flatlike pion DAs [24, 25, 35], outside the [2]andindicateconformitywithendpoint-vanishing(or
standard QCD context. Such approaches suffer from even endpoint-suppressed)pion DAs, like in the BMS
thepointofviewthattheydependinasensitivewayon formalism[18,34].
specific,i.e.,contextualnonperturbativescalesthatcan- The chiral anomaly plays an important role also in
not be linked to the standard QCD scales. Therefore, anotherapproach,proposedin[41],whichcombinesan
the meaning of these scales, with their particular val- exactnonperturbativesumrule,followingfromthedis-
uesneededtoexplaintheBaBardata,remainsobscure. persiverepresentationoftheaxialanomaly,andquark-
Consequently,thoughthe BaBardatacanbeexplained hadronduality. Within thisapproachitisclaimed[42]
withinsuchschemes,becausetheyprovidelogarithmic that the increase of the BaBar data for the π0 is due
enhancementtotheTFF, itisnotpossibletoidentifya to small corrections to the continuum (i.e., an infinite
singleunderlyingphysicalmechanismwhichyieldsen- numberofhigherresonances)whichentailastrongen-
hancementof the pion TFF. Moreover, it is difficultto hancementofthepionTFF,whereasthesameeffectfor
understandwhy the pion and the η should behave like theηturnsouttobeseveraltimessmaller.3 Depending
“pointlike”particles(see,forinstance,[36])asimplied on the particularmixing scheme adopted, the obtained
byflatlike(orflat-top)DAs,whiletheDAsoftheη and predictions[41]fortheηandη mesonsagreewiththe
′ ′
theη shouldbeclosetotheirasymptoticforms(even- gross of the BaBar data for the associated TFFs. The
c
tually with additionalendpointsuppression).2 The as- usefulnessofthedispersiveapproach[42]—whichdoes
sertionin[35]thatthisisduetothelargermassofthese notrelyuponfactorization—hastobefurthertestedby
particles is not very convincing,given that the DAs of extending it to the singlet channel. Another approach
theπ0andtheηareverysimilarinshapeinthenonlocal thatrelatestheBaBareffecttothechiralanomalyisdis-
chiral quark model of [35], though the corresponding cussedin[43].
massesm =140MeVandm =548MeVareverydif- The light pseudoscalar meson-photon TFFs for the
π η
ferent. Nevertheless, it would be prematureto dismiss pion and the η and η mesons were also discussed in
′
thecorrectnessand/orrelevanceofsuchexplanations. [44], using a quark-flavor mixing scheme with only
Thecalculationoftheπ0,η,andη TFFshasbeencar- one mixing angle [45]. The associated DAs of these
′
ried out within holographic approaches of AdS/QCD, mesons are obtained from a light-cone wave function
e.g.,[29],[30],and[38].InFig.2thetwobroken(blue) [46]bytuningamasterparametertoappropriatevalues,
lines represent the independent findings of two such whereas the constituent quark masses and the mixing
approachesfor Q2Fγ∗γπ0(Q2). The dotted line denotes angle play a relatively minor role. The main message
the prediction derived in [29], using a dressed elec- fromthisanalysisisthatthe(logarithmic)growthofthe
tromagnetic current and taking into account the twist- scaledTFFQ2Fγ∗γπ0(Q2),indicatedbytheBaBardata,
twoandthetwist-fourhadronicAdScomponentsofthe cannot be reproduced, while it remains unexplained
pion wave function (Eq. (43) in [29]) within a soft- why Q2Fγ∗γπ0(Q2) and Q2Fγ∗γη(Q2) should behave so
wall holographic approach. The combination of the differently.Infact,withintherangeofthemodelparam-
nonperturbativebound-statedynamics,predictedbythe eters,theBaBardataonQ2Fγ∗γη(Q2)andQ2Fγ∗γη′(Q2)
holographic AdS/QCD correspondence, with the per- can be explained simultaneously in the whole Q2 re-
turbative Efremov–Radyushkin–Brodsky–Lepage evo- gion. So far, no single mechanism with a physical in-
lution[3,39]wastreatedindetailin[40]. Thedouble- terpretation within the standard QCD framework has
dotted-dashedlineshowsan analogousresult[30], ob- beenidentified to explainthe increase of the pionTFF
tained by an extension of the hard-wall AdS/QCD as a result of particular quark-gluon interactions. Ef-
modelwhichincludestheChern–Simonsterm,required fectsassociatedwiththetransverse-momentumdegrees
toreproducethechiralanomalyofQCD.Notethatthe of freedom, intrinsic, i.e., inside the pion wave func-
showncurve beyond10 GeV2 was generatedby us. It tion, and resummed in terms of Sudakov factors, have
is obvious that both displayed predictions are incon- alsobeendiscussedwithrespecttotheBaBardata,see,
gruent with the BaBar data for π0, while they agree
3Notethatuntilnow,allcalculated corrections, perturbative and
2NoteparentheticallythattheTFFfortheηcapproachestheQCD nonperturbative,amountintotaltoanegativecontributionsothatitis
predictionfrombelow—see[37]foradiscussion. notclearhowthisscenariocanberealized.
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