Pion-photon transition form factor using light-cone sum rules: theoretical results, expectations, and a global-data fit
A. P. Bakulev, S. V. Mikhailov, A. V. Pimikov, N. G. Stefanis
aa r X i v : . [ h e p - ph ] A ug RUB-TPII-04/2011
Pion-photon transition form factor using light-cone sum rules:theoretical results, expectations, and a global-data fit ∗ A. P. Bakulev, † S. V. Mikhailov, ‡ A. V. Pimikov, § and N. G. Stefanis ¶ Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia Institut f¨ur Theoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany (Dated: November 12, 2018)A global fit to the data from different collaborations (CELLO, CLEO, BaBar) on the pion-photon transition form factor is carried out using light-cone sum rules. The analysis includes thenext-to-leading QCD radiative corrections and the twist-four contributions, while the main next-to-next-to-leading term and the twist-six contribution are taken into account in the form of theoreticaluncertainties. We use the information extracted from the data to investigate the pivotal charac-teristics of the pion distribution amplitude. This is done by dividing the data into two sets: onecontaining all data up to 9 GeV , whereas the other incorporates also the high- Q tail of the BaBardata. We find that it is not possible to accommodate into the fit these BaBar data points with thesame accuracy and conclude that it is difficult to explain these data in the standard scheme of OCD. PACS numbers: 12.38.Lg, 12.38.Bx, 13.40.Gp, 11.10.Hi
I. FORM FACTOR F γ ∗ γ ∗ π IN COLLINEAR QCD
One of the most studied exclusive processes withinQCD, based on collinear factorization, is the pion-photontransition form factor with both photon virtualities beingsufficiently large, see [1] for a review. The transition formfactor is defined by the correlator of two electromagneticcurrents Z d z e − iq · z h π ( P ) | T { j µ ( z ) j ν (0) } | i = iǫ µναβ q α q β F γ ∗ γ ∗ π ( Q , q ) (1)with Q ≡ − q > q ≡ − q ≥
0, and can be reex-pressed in the form [2] F γ ∗ γ ∗ π ( Q , q ) = N Z dx T ( Q , q , µ , x ) × ϕ (2) π ( x, µ ) + O (cid:0) δ /Q (cid:1) , (2)by virtue of collinear factorization, assuming that thephoton momenta are sufficiently large Q , q ≫ m ρ . Here N = √ / f π , f π = 132 MeV is the pion-decay constant,and δ is the twist-four coupling. Then, the quark-gluonsub-processes, formulated in terms of the hard-scatteringamplitude of twist-two, T , can be computed order-by-order of QCD perturbation theory: T = T + a s T + a s T + . . . . The radiative corrections in next-to-leadingorder (NLO), T , have been obtained in [3], the β –part ∗ Presented by the second and third authors at the 5th Joint Inter-national Hadron Structure’11 Conference, Tatranska Strba (SlovakRepublic),June 27–July 1, 2011. † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] of the contribution at the next-to-next-to-leading orderlevel (NNLO β ), encoded in the amplitude T , i.e., β · T β ,was calculated in [4].The binding effects are separated out and absorbedinto a universal pion distribution amplitude (DA) oftwist-two, ϕ (2) π ( x, µ ), defined [5] by the matrix element h | ¯ q ( z ) γ ν γ [ z, q (0) | π ( P ) i (cid:12)(cid:12)(cid:12) z =0 = iP ν f π Z dxe ix ( z · P ) × ϕ (2) π ( x, µ ) . (3)The variation of ϕ (2) π ( x, µ ) with the factorization scale µ is controlled by the Efremov–Radyushkin–Brodsky–Lepage (ERBL) evolution equation [2]; moreover theGegenbauer harmonics { ψ n ( x ) } constitute the leading-order (LO) eigenfunctions of this equation. Therefore, itis useful to expand the pion DA in terms of these har-monics: ϕ (2) π ( x, µ ) = ψ ( x ) + X n =2 , ,... a n (cid:0) µ (cid:1) ψ n ( x ) , (4)where ψ n ( x ) = 6 x (1 − x ) C / n (2 x −
1) and ϕ as ( x ) = ψ ( x ) =6 x (1 − x ) is the asymptotic pion DA [2]. The nonpertur-bative information is contained in the coefficients a n (cid:0) µ (cid:1) with ( n ≥
2) that have to be modeled or extracted formthe data, including evolution effects to account for their µ -dependence. They are usually reconstructed fromthe moments h ξ N i π ≡ R dx (2 x − N ϕ (2) π ( x, µ ) with h ξ i π = 1 that can be determined by employing, e.g.,QCD sum rules (SR)s [6]. We use here the pion DAproposed before in the framework of improved QCD SRswith nonlocal condensates (NLC-SRs) [7] that yield a Gauge invariance is ensured by the longitudinal gauge link[ z,
0] = P exp (cid:0) ig R z A µ ( τ ) dτ µ (cid:1) along a path-ordered lightlikecontour. “bunch” of admissible pion DAs with two harmonics thatfix the coefficients a and a . II. LIGHT-CONE SUM RULES FOR THEPROCESS γ ∗ ( Q ) γ ( q ≃ ) → π The pion-photon transition involving two highly off-shell photons is not easily accessible to experiment. Ex-perimental information is mostly available for an asym-metric photon kinematics, with one of the photons havinga virtuality close to zero q → F γ ∗ γ ∗ π in the vari-able q , taking then q →
0, whereas the large variable Q is kept fixed. Thus, one has F γ ∗ γ ∗ π (cid:0) Q , q (cid:1) = N Z ∞ ds ρ (cid:0) Q , s (cid:1) s + q , (5)with the physical spectral density ρ ( Q , s ) approachingat large s the perturbative one: ρ PT ( Q , s ) = 1 π Im h F γ ∗ γ ∗ π (cid:0) Q , − s − iε (cid:1) /N i . (6)Using quark-hadron duality, we obtain the followingLCSR [11]: Q F γ ∗ γπ (cid:0) Q (cid:1) = Q m ρ Z x dxx exp m ρ − Q ¯ x/xM ! × N ¯ ρ ( Q , x ) + Z x dx ¯ x N ¯ ρ ( Q , x ) (7)with the spectral density ¯ ρ ( Q , x ) = ( Q + s ) ρ PT ( Q , s ),where s = ¯ xQ /x and x = Q / ( Q + s ). Note that thefirst term in (7) is associated with the hadronic contentof a quasi-real photon at low s ≤ s , whereas the sec-ond term reproduces its point-like behavior at the highervalue s > s . We adjust the hadronic threshold in thevector-meson channel to the value s = 1 . , us-ing m ρ = 0 .
77 GeV [12]. We avoid to vary the Borelparameter M in (7) and specify its value by virtue of M = M − pt / h x i Q entering the two-point QCD sumrule for the ρ -meson with M − pt ∈ [0 . ÷ .
8] GeV , where h x i Q denotes some average value of x (at fixed Q ) in theintegration region for the first integral on the right-handside of Eq. (7) [11, 13], i.e., x ( Q ) < h x i Q < III. MAIN INGREDIENTS OF THE LCSRS ANDCONDITIONS OF THE DATA ANALYSIS
It is convenient to invent for each term of the har-monics ψ n , a partial spectral density according to the definition (6) for the twist-two part [14], ρ ( i ) n ( Q , s ) = Im π (cid:2) ( T i ⊗ ψ n ) ( Q , − s − iε ) (cid:3) . The general solution for¯ ρ (1) n in NLO was obtained in [14] and corrected later(third line) in [15]:1 C F ¯ ρ (1) n (cid:18) Q µ ; x (cid:19) = (cid:26) − (cid:2) v b ( n ) (cid:3) + π − ln (cid:16) ¯ xx (cid:17) +2 v ( n ) ln (cid:18) ¯ xx Q µ (cid:19)(cid:27) ψ n ( x ) − n X l =0 , ,... G nl ψ l ( x ) + v ( n ) n X l =0 , ,... b nl ψ l ( x ) − x , (8)with v ( n ) , v b ( n ) being the eigenvalues of LO ERBL equa-tions, whereas G nl and b nl are calculable triangular ma-trices (see [14, 15] for details).The inclusion of the NNLO β contribution to the mainpartial spectral density β · ¯ ρ (2 β )0 , derived from β · T β [4],was realized in [13, 14]. It turns out that, taken togetherwith the positive effect of a more realistic Breit-Wigneransatz for the meson resonance [14] instead of using a δ -function, i.e., δ ( s − m ρ ) F γ ∗ ρπ , as in (7), it is negativeand about –7% at small Q ∼ , decreasing rapidlyto –2.5% at Q ≥ . Here the results are expandedto include the first three harmonics.On the other hand, the twist-six contribution to F γ ∗ γπ was recently computed in [15] using M ∼ . andfound to be very small. Using instead the more moderatevalue M ∼ .
75 GeV [13], it turns out to have almostthe same magnitude as the NNLO β term, but with theopposite sign.As already mentioned, we use here the BMS bunch ofpion DAs with the central point a BMS2 (1GeV ) = 0 . a BMS4 (1GeV ) = − .
14 (termed the BMS model) [7].These pion DAs have their endpoints at x = 0 and x = 1suppressed—even relative to the asymptotic pion DA—and are in good agreement with the CLEO data on thepion-photon transition form factor as well as with thedata for other pion observables [16–18].The key features of our data-analysis are the following:(i) The NLO radiative corrections in the spectral densityare included via the corrected expression (8), emphasiz-ing that this error does not affect our previous resultsin [14, 16–18]. The so-called default renormalization-scale setting is adopted and, accordingly, the factoriza-tion and the renormalization scales have been identifiedwith the large photon virtuality Q . (ii) The twist-fourcontribution is taken into account using for the effec-tive twist-four DA the asymptotic form ϕ (4) π ( x, µ ) =(80 / δ ( µ ) x (1 − x ) [11, 19]. We also admit a sig-nificant variation of the parameter δ = 0 .
19 GeV inthe range 0.15 GeV to 0 .
23 GeV , referring for detailsto [17], and taking into account its evolution with µ .Using a nonasymptotic form for ϕ (4) π would not changethese results significantly [18, 20]. (iii) The evolutioneffects of the coefficients a n are also included in NLO,employing the QCD scale parameters Λ (3)QCD = 370 MeVand Λ (4)QCD = 304 MeV, conforming with the NLO esti-mate α s ( M Z ) = 0 .
118 [12]. (iv) The NNLO β radiativecorrection to the LCSR form factor [13, 14], Eq. (7), isincorporated together with the twist-six term, computedin [15], in terms of theoretical uncertainties. To be pre-cise, the calculation of the NNLO β term involves only theconvolution of the hard-scattering amplitude T β with theDA based on the three lowest harmonics. This treatmentmakes sense due to the fact that for the average value of M ( Q ) ∼ .
75 GeV , these two contributions almostmutually cancel and the net result is small—see Fig.1.
10 20 30 40 - - Q F γ ∗ γπ Tw-6( Q )NNLO β ( Q ) Q [GeV ] FIG. 1: Twist-six contribution (upper solid line in red) andNNLO β contribution (lower solid line in green), obtained withthe BMS model, and their sum (dashed blue line). In fact, it decreases with Q from +0 .
004 at Q =1 GeV —where the twist-six term dominates—down to − .
003 at Q = 40 GeV —where the NNLO β cor-rection starts prevailing. This particular behavior ap-plies only to the moderate value of the Borel param-eter M = 0 .
75 GeV [13], while for the larger value M = 1 . ± . , used in [15], the twist-six termwould be much smaller and the net result would be ev-erywhere negative and almost constant: ≈ − . IV. DATA ANALYSIS
Here we overview our fit procedure of all available ex-perimental data on the pion-photon transition form fac-tor F γ ∗ γπ , within the framework of LCSRs, as workedout in [13]. The main goal of the fit is to extract the pionDA—the main low-energy pion characteristic—best com-patible with all the data. This is done fitting the formfactor by varying the pion DA in terms of the Gegenbauercoefficients a n . To reveal the particular role of the newhigh- Q BaBar data in the fit, we perform our analysisutilizing two different data sets. The first set (set-1) con-tains all available data from CELLO [8], CLEO [9], andBaBar [10] that belong to the Q -window [1 ÷
9] Gev .The second set (set-2) comprises all data in the range[1 ÷
40] GeV . First, we define the optimal number of Gegenbauer harmonics necessary to model the pion DA.Second, we determine the fiducial regions of the corre-sponding coefficients a n . Third, we relate these regionswith the pion DA and its characteristics: profiles, deriva-tives at the origin, and its moments. Finally, we confrontthe obtained results with the data of set-1 and set-2. A. How many harmonics should be taken intoaccount?
To answer this question, we confront the dependenceof the fit quality on the number of the parameters of theinvolved harmonics and the associated statistical errors.The statistical errors in the parameter determination in-crease with their number for statistical reasons, while the χ initially decreases. Therefore, in order to achieve anacceptable compromise, one should use the lowest ac-ceptable number of harmonics. The dependence of thegoodness of fit, χ , on the number n of the involvedharmonics for the two data sets, is presented in Fig. 2.The goodness of fit for set-1 is only slightly decreasing χ ( n ) n FIG. 2: Dependence of the goodness of fit χ ≡ χ / ndf(ndf = number of degrees of freedom) on the number n of Gegenbauer harmonics shown as histograms: set-1—solid(blue) bars; set-2—higher dashed (red) bars. with n and remains almost stable after n = 3. Thus, 2to 3 parameters are actually enough to describe all datain this region with χ ≈ .
5. In contrast, the data de-scription of set-2 is only possible with a χ value 2 or 3times larger—even if we include more harmonics. To fitall the data, we are forced to consider at least χ ≈
1. To have an even better descriptionwith a goodness of fit approximately equal to 0.8, we haveto employ 4 parameters. Further increase of the number n will not provide any improvement.However, for the sake of comparison of the results, oneshould use the same fit model of pion DA, which meansthat the most appropriate number of harmonics may befixed to 3. Best-fit curves for both data sets are shown inFig. 3 as a bunch of form-factor predictions with errorsstemming from the sum of the statistical error and thetwist-four uncertainties. At high values of the momentumtransfer, the fit curve of the set-2 data—long dashed (red)line—exceeds the 68% CL (confidential level) region ofthe set-1 data fit—solid (blue) line. This indicates thatin the framework of LCSRs, the new BaBar data above9 GeV deviate from the low- Q data at the level of a 1 σ deviation and more. Q F ( Q ) [GeV ] Q [GeV ] FIG. 3: Best-fit curves to the experimental data for the tran-sition form factor in the framework of LCSRs: solid (blue)line—best-fit curve of set-1; strip bounded by dashed (blue)lines—68% CL region; long dashed (red) line—best-fit curveof set-2; strip bounded by dashed dotted (red) lines—68% CLregion. Error bars show the sum of the statistical errors andthe twist-four uncertainties. The experimental data are takenfrom the CELLO [8] (diamonds), CLEO [9] (triangles), andBaBar [10] (open crosses) experiments.
B. Data analysis vs pion DA models
Performing the data analysis, we obtain the best-fitvalues of the pion DA in the 68% CL region for a numberof harmonics n = 2 ÷
3. The 3D graphics of the confiden-tial regions for the 3 harmonics analysis were presentedin our recent work in [13], whereas the best-fit valuestogether with the statistical errors and the twist-four un-certainties are given in Table I below. We compare thereour fit results with various pion DA models in terms ofthe goodness of fit χ for the two analyzed sets of ex-perimental data. From the first two lines of Table I, weinfer that the inclusion of the new high- Q BaBar dataaffect only the value of the parameter a , while a and a do not change significantly. Moreover, the good de-scription of the experimental data up to 9 GeV becomesappreciably worse after the inclusion of the high- Q tailof the BaBar data. The BMS pion DA stands out in thesense that it provides the best fit for set-1, while all othermodels cannot reproduce these data good enough.It is worth remarking that one of the models, obtainedfrom fitting the experimental data within the ModifiedFactorization Scheme (MFS), has the value χ = 4 . ≈ . - - - a a - - - a a FIG. 4: (color online). Distorted 1 σ error ellipses for set-1(upper panel) and set-2 (lower panel) from various exper-iments [8–10, 21] using different data-analysis procedures.These ellipses result from merging together the ellipses as-sociated with different values of the twist-four parameter inthe range δ = 0 . ÷ .
23 GeV . The slanted shaded (green)rectangle encloses the area of a and a values determinedby NLC-SRs [7], with the BMS pion DA being marked by ✖ .The middle points of the ellipses ( ✚ and ▼ ), the asymptoticDA ( ◆ ), the CZ DA ( ■ ), and Model III from [15] ( ▲ ) are alsomarked. The range of values of a , restricted by lattice sim-ulations, are indicated by vertical lines: [22]—dashed lines;[23]—dashed-dotted (blue) lines. All results are shown at thescale µ = (2 . , whereas the treatment of the Borelparameter M ( Q ) is explained in the text. Graphics takenfrom [13]. discrepancy indicates that using the same pion DA in theframework of LCSRs and the MFS may lead for the sameobservable to incompatible results—a theoretical bias.Below, we consider in detail the results of the 2D anal-ysis in the ( a , a ) plane, presented in Fig. 4, with the up-per panel showing the results for set-1, whereas the lowerpanel presents those for set-2. To this end, we calculatethe 1 σ error ellipses by allowing the parameter δ tovary by 20% around the value 0.19 GeV . The obtainederror ellipses are then unified into a single (distorted) 1 σ We denote by a 1 σ ellipse (ellipsoid) a 68 .
27% confidence-levelboundary.
TABLE I: Measures of goodness of fit of selected pion DA models (first column) with associated coefficients a n (second column),used in the calculation of the pion-photon transition form factor by means of LCSRs. Note that the coefficients a n are stronglycorrelated and the errors of a n represent the maximal variation in the range of the 1 σ -region. The last two columns show thevalues of χ for the data in set-1 and for the whole set of the data, (set-2), respectively. All values of the coefficients a n aregiven at the scale µ SY = 2 . a , a , . . . ) µ = µ χ , [1 −
9] GeV χ , [1 −
40] GeV
3D fit, [1 −
9] GeV (0 . ± . , − . ± . , . ± .
17) 0 . −
3D fit, [1 −
40] GeV (0 . ± . , − . ± . , . ± . − . . , − . . . . , . , . ≥ . ≥ . . , . . . . , . , . , . . . ) 2 . . . ,
0) 32 . . ,
0) 4 . . ellipse shown in Fig. 4. To be specific, we consider the fol-lowing cases: (i) The result of combining the projectionson the plane ( a , a ) of the 3D (3 parameter) data anal-ysis is represented by the largest ellipse—dashed (red)line with the middle point ▼ . (ii) The analogous resultof the 2D (2 parameter) data analysis in terms of a and a is shown by the smaller ellipse (solid blue line) withthe middle point ✚ having the coordinates (0 . , − . χ ≈ .
5, that almost coincides with the middlepoint ✖ of the parameter area determined by NLC-SRs[7]. (iii) The combination of the intersections with the( a , a ) plane of all 3D ellipsoids generated by the varia-tion around the central value of δ give rise to the smallestellipse (thick line), entirely enclosed by the previous one.For convenience, the locations in the ( a , a ) plane ofsome characteristic pion DAs are also indicated in Fig.4. These are the asymptotic DA ( ◆ ), the CZ model ( ■ ),and the projection of Model III from [15] ( ▲ ). Note thatthe slanted (green) rectangle, containing those values of a and a that have been determined by NLC-SRs [7], ispractically within both larger error ellipses and also over-lapping with the smallest one. Moreover, the BMS modelDA ✖ stands out by lying inside of all 1 σ error ellipses.Thus, the theoretical predictions obtained from the 2Dand the 3D data analyses conform with each other andagree at the level of χ ≤ . σ error ellipses com-ply rather good with the boundaries for a extracted fromtwo independent lattice simulations. The vertical dashedlines denote in both panels the older estimate from [22],while the very recent constraints from Ref. [23] are rep-resented by the dashed-dotted (blue) vertical lines.From the lower panel of Fig. 4 it becomes evident thatthe situation changes significantly when including in theanalysis the high- Q tail of the BaBar data [10]. In-deed, using the same designations as in the upper panel,we display the analogous unified error ellipses and ob-serve that the error ellipsoid has no intersection withthe ( a , a ) plane, whereas the composed error ellipse re-sulting from the 2D analysis (solid blue line) deviates from the region of negative values of a and moves insideits positive domain. At the same time, the fit qualitydeteriorates yielding χ ≈
2, as opposed to the value χ ≈ . σ error ellipse of the 3D projections on the( a , a ) plane (larger dashed red ellipse), its position re-mains unaffected, still enclosing most of the area of the a , a values computed with NLC-SRs—shaded (green)rectangle.The high quality of the data fit parallels the latticefindings, with the 3D error ellipse being almost entirelyinside the boundaries from [22] (dashed vertical lines),while it also overlaps for the larger values of a with therange of values computed in [23] (dashed-dotted verti-cal lines). In contrast, the ellipse from the 2D analysisagrees very roughly with the small a window of [23],sharing also only a small common area with the low endof the a region found in [22]. Obviously, no agreementbetween the 2D and the 3D analysis is found. This dis-crepancy is also reflected in the values of the χ -criterionof the 2D fit model χ ≈ χ ≈
1. Thismeans that a pion DA, based only on 2 harmonics, isnot sufficient to describe all the data on the pion-photontransition form factor. This deviating behavior of the re-sults, associated with the fits for set-1 and set-2, showsup for a larger number of degrees of freedom, i.e., whenincluding into the data analysis the next higher harmon-ics ψ n with n = 6 , ,
10. But, for the case of the set-1 fit,this expansion does not improve any further the value χ ≤ . σ -admissible regions in 2D, 3D, or 4D param-eterizations appear to be each embedded inside the other.In contrast, fitting the set-2 data, these new degrees offreedom lead to a decrease of χ , while the correspond-ing 1 σ -regions in the 2D, 3D, or 4D space, either do notoverlap at all or intersect only marginally.It becomes obvious from Fig. 4 that Model III ( ▲ )from [15] has a projection on the ( a , a ) plane that liesoutside of all considered 1 σ error ellipses of the data. - ϕ π ( x ) x - ϕ π ( x ) x FIG. 5: Left. Comparison of the BMS pion DA bunch (shaded strip in green color) and of the BMS model (black solid lineinside this strip) with the 3D fit to the experimental data on the pion-photon transition form factor. The solid blue line denotesthe best-fit pion DA sample obtained from the analysis of set-1, with the dashed lines indicating the sum of the statisticalerrors of the fit and the twist-four uncertainties. Right. Analogous results obtained with set-2.
However, selecting for the Borel parameter the value M = 1 . , as in [15], the agreement of this modelwith the data improves to the level of χ & . C. Pion DA characteristics
The confidential region of the coefficients { a n } , ob-tained above, can be linked to any other characteristic ofpion DA. The profiles of the pion DA ϕ π ( x ), extractedin the 3D fit procedure, are shown in Fig. 5: left panel—set-1; right panel—set-2. The BMS bunch (shaded greenstrip) and the BMS DA model (black solid line) are alsoshown in both cases. The inclusion into the data fit of thehigh- Q BaBar tail, causes a modification of the shapeof the pion DA—see Fig. 5—giving support to our previ-ous observation that the BMS bunch is within the errorrange of the set-1 fit (left panel), while the best fit toset-2 differs considerably (right panel). In addition, thepion DA becomes endpoint enhanced, as opposed to theendpoint-suppressed BMS pion DA. The endpoint behav-ior can be characterized by its slope at the origin given bythe derivative ϕ ′ π (0) or, more adequately, by the so-called“integral derivative” D (2) ϕ π ( x ), introduced in [27]. Theintegral derivative is the average derivative ϕ ′ π ( x ) definedby D (2) ϕ ( x ) = 1 x Z x ϕ ( y ) y dy with the important property lim x → D (2) ϕ ( x ) = ϕ ′ π (0) . Using a 3D confidential bound on the Gegenbauer co-efficients, we get the values of the derivatives ϕ ′ π (0) and D (2) ϕ (0 . TABLE II: Comparison of the pion DA characteristics forthe data of set-1 and set-2.data set [1 −
9] GeV [1 −
40] GeV ϕ ′ π (0) 20 . ± . ± . . ± . ± . D (2) ϕ π (0 .
4) 6 . ± . ± . . ± . ± . n
2, 3 3, 4 χ . , . characteristics can clearly differentiate the pion DAs gen-erated from set-1 and set-2. D. Combining Lattice constraints withexperimental data
The possibility to extract information on the mo-ment h ξ i π of the pion DA by combining lattice con-straints with the experimental data was first pointedout in [28] and the following range of values was ex-tracted from the 1 σ error ellipse of the CLEO data [9]in conjunction with the lattice constraints for h ξ i π from[29]: h ξ i π ∈ [0 . ÷ . µ = 4 GeV and for M = 0 . . This procedure was refined in [13] inthe following way: first, we expanded the result of the2D analysis to the ( h ξ i π , h ξ i π ) moments. Then, we de-termined the intersection of the confidential region (thearea enclosed by a solid blue line) in Fig. 6 for set-1 (cf.Fig. 4) using the constraints from [22] and [23]. Theintersection of these constraints, evaluated at the typi-cal lattice scale µ = 4 GeV , and the experimentaldata leads to the following moment results, respectively,(i) h ξ i π ∈ [0 . ÷ .
29] and h ξ i π ∈ [0 . ÷ . h ξ i π ∈ [0 . ÷ .
29] and h ξ i π ∈ [0 . ÷ . h ξ i π h ξ i π FIG. 6: (color online). Predictions for the moments h ξ i π and h ξ i π at the lattice scale µ = 4 GeV . The solid (blue)ellipse corresponds to our choice of M , whereas the dashed(red) one results when using M = 1 . . The verticallines show the range of values computed on the lattice: dashedline—[22]; dashed-dotted (violet) line—[23]. Q -dependent Borel parameter—like everywhere in ouranalysis here and in [13]. On the other hand, the value M = 1 . [15], yields only a small intersection ofthe validity region extracted from set-1 (shown in Fig. 6by the dashed red line) with the lattice constraints of [22].This restricts the common region of validity to the value h ξ i π ≃ .
1, whereas there is no intersection at all withthe lattice estimates from [23]. This obvious sensitivityof h ξ i π on the choice of the particular value of the Borelparameter M gives additional support to our choice ofthe value of the Borel parameter. V. CONCLUSIONS
We have presented here a global fit to the data on thepion-photon transition form factor, discussing further ourrecent analysis in [13]. To get a precise measure of theinfluence of the high Q BaBar data on the form fac-tor and the pion DA, we divided the experimental datain two different sets with respect to Q . Set 1 containsall data in the range [1 ÷
9] GeV , whereas the secondset comprises all data in the regime covered by BaBar, i.e., [1 ÷
40] GeV . As a result, we obtained the confi-dential regions of different characteristics of the pion DA(Gegenbauer coefficients, derivatives of ϕ π ( x ) at x = 0,and its moments) by fitting the experimental data withinthe framework of LCSRs. The predictions obtained fromthe CELLO, CLEO, and the BaBar data up to 9 GeV are in good agreement with the previous fits, based onlythe CLEO data [14, 16–18, 25], giving preference to anendpoint-suppressed pion DA [7]. Beyond 9 GeV , thebest fit requires a sizeable coefficient a that inevitablyleads to an endpoint-enhanced pion DA. The data anal-ysis tells us that the inclusion of the high- Q tail of theBaBar data affects mainly the Gegenbauer coefficient a ,while a and a change only insignificantly. The gooddescription of the experimental data up to 9 GeV usingLCSRs becomes considerably less accurate after the in-clusion of the high- Q data but yields an acceptable valueof χ ≈
1. This effect has been discussed before in [30]at a qualitative level. The results obtained with the in-clusion of the high- Q tail of the BaBar data indicate apossible discrepancy between the result of the BaBar ex-periment and the method of LCSRs. Indeed, the high- Q BaBar data require a pion DA with a sizeable number ofhigher Gegenbauer coefficients a n , or alternative theoret-ical schemes outside the standard QCD factorization ap-proach, see, e.g., [31–36]. Similar conclusions were alsodrawn in [37] using Dyson–Schwinger equations and inthe recent works [38], based on AdS/QCD. VI. ACKNOWLEDGMENTS
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