The pion transition form factor revisited

Model analysis of the world data on the pion transition form factor S. Noguera∗ and V. Vento† arXiv:1205.4598v3 [hep-ph] 7 Nov 2012 Departamento de Fisica Teorica and Instituto de Fı́sica Corpuscular, Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain. (Dated: September 23, 2018) We discuss the impact of recent Belle data on our description of the pion transition form factor based on the assumption that a perturbative formalism and a nonperturbative one can be matched in a physically acceptable manner at a certain hadronic scale Q0 . We discuss the implications of the different parameters of the model in comparing with world data and conclude that within experimental errors our description remains valid. Thus we can assert that the low Q2 nonperturbative description together with an additional 1/Q2 term at the matching scale have a strong influence on the Q2 behavior up to very high values of Q2 . PACS numbers: 12.38.Lg, 12.39.St, 13.40.Gp, 13.60.Le New data of the pion transition form factor (πT F F ) from the Belle collaboration have just appeared [1]. These data, above 10 GeV2 , are smaller in magnitude than the previous BABAR data [2], which generated considerable excitement. The question to unveil is the scale of asymptotia. BABAR data, taken at face value, implied that asymptotic QCD behavior lies at much higher Q2 than initially expected [3, 4]. Belle data seem to lower that scale. We show here that our scheme can accomodate easily all data without changing the physical input. At the time of the BABAR data we developed a formalism to calculate the πT F F [5], which consists of three ingredients: i) a low energy description of the πT F F ; ii) a high energy description of the πT F F ; iii) a matching condition between the two descriptions at a scale Q0 characterizing the separation between the two regimes. For the low energy description we took a parametrization of the low energy data to avoid model dependence at Q0 . The high energy description of the πT F F , defined by the pion Distribution Amplitude (πDA), contains Quantum Chromodynamic (QCD) evolution from Q0 to any higher Q, a mass cut-off to make the formalism finite, and an additional 1/Q2 term which leads to modifications of the matching condition. Let us recall some aspects of the formalism. The high energy description, to lowest order in perturbative QCD, for the transition form factor in the process π 0 → γ γ ∗ in terms of the pion distribution amplitude (πDA), is given by √ Z 2fπ 1 dx Q F (Q ) = φ (x, Q2 ). M2 π 3 0 x + Q2 2 2 (1) We follow the proposal of Polyakov [6] and Radyushkin [7] and introduce a cutoff mass M to make the
expression finite. Q2 = −q 2 , qµ is the momentum of the virtual photon, φπ x, Q2 is πDA at the Q2 scale and fπ = 0.131 GeV. In this expression, the Q2 dependence appears through the QCD evolution of the πDA. Despite the fact that several models reproduce the low energy data, in order to have a model independent expression for the form factor at low virtualities, we adopted a monopole parametrization of the ∗ Electronic † Electronic address: [email protected] address: [email protected] 2 πT F F in the low energy region as F (0)
F LE Q2 = . 2 1 + a mQ2 (2) π0 Asymptotic limit 0.3 Belle BaBar CLEO CELLO 0.2 2 2 Q ·F(Q ) (GeV) with F (0) = 0.273(10) GeV−1 and a = 0.032 (4) [8], determined from the experimental study of π 0 → γ e+ e− [9]. 0.1 0 10 20 Q 2 30 40 2 (GeV ) 0.15 2 2 Q ·F(Q ) (GeV) 0.20 0.10 2 4 6 Q 2 8 10 12 2 (GeV ) FIG. 1: We show the result for the transition form factor in our formalism for M = 0.690 GeV, a = 0.032 and C3 = 2.98 10−2 GeV3 and defining the matching point at Q0 = 1 GeV (solid line). The band region results from the indeterminacy in ∆a = ±0.004. The lower plot shows the detailed behavior for low virtuality. Data are taken from CELLO [10], CLEO [11], BABAR [2] and Belle [1]. Additional power corrections can be introduced in Eq. 1 by adding to the lowest order calculation a 3 Belle BaBar CLEO CELLO 0.2 2 2 Q ·F(Q ) (GeV) Asymptotic limit 0.3 0.1 0 10 Q 2 20 30 40 2 (GeV ) 0.15 2 2 Q ·F(Q ) (GeV) 0.20 0.10 2 4 6 Q 2 8 10 12 2 (GeV ) FIG. 2: We show the result for the transition form factor in our formalism for M = 0.620 GeV, a = 0.032 and the value of C3 = 1.98 10−2 GeV3 corresponding to 20% of the contribution at the matching point at Q0 = 1 GeV (solid line). The band region gives the variation of the results due in ±10% in the contribution of higher twist. The lower plot shows the detailed behavior for low virtuality. Data are taken from CELLO [10], CLEO [11], BABAR [2] and Belle [1]. term proportional to Q−2 , 2 2 Q F (Q ) = √ 2fπ 3 Z 0 1 C3 dx φ (x, Q2 ) + 2 . M2 π Q x + Q2 Using a constant π DA the matching condition becomes [5], (3) 4 √ F (0) Q20 2fπ Q20 + M 2 C3 = , + ln Q2 3 M2 Q20 1 + a m20 (4) π0 with Q0 = 1 GeV. This equation allows to determine M, once we have fixed the value of C3 . 0.2 2 2 Q ·F(Q ) (GeV) 0.3 Asymptotic limit 0.1 Belle BaBar CLEO CELLO 0.0 0 10 Q 2 20 30 40 2 (GeV ) 0.1 2 2 Q ·F(Q ) (GeV) 0.2 0.0 2 4 6 Q 2 8 10 12 2 (GeV ) FIG. 3: We show the result for the transition form factor in our formalism for M = 0.690 GeV, a = 0.032 and the value of C3 = 2.98 10−2 GeV3 corresponding to 30% of the contribution at the matching point at Q0 = 1 GeV (solid line).The lower plot shows the detailed behavior for low virtuality.The dotted curve represents the higher twist contribution.Data are taken from CELLO [10], CLEO [11], BABAR [2] and Belle [1]. We analyze here the sensitivity of the data to the various parameters involved. We keep as close as possible to our previous fit analyzing the data with respect to small variations in the low virtuality parameter a and in the higher twist parameter C3 . In Fig.1 we show the effect of the precision in the 5 determination of the monopole parametrization. We see that as a increases from 0.032 to 0.036, i.e. within the error bars, the πT F F decreases. The sensitivity to C3 is shown in Fig. 2 and we note that as the value of C3 increases from C3 = 0.99 10−2GeV3 , which corresponds to a 10% contribution to the form factor at Q0 , to 2.98 10−2 GeV3 , which corresponds to a 30% contribution, again the value of the πT F F decreases. Thus a small increase in a and C3 moves our result toward the Belle data. Finally, in Fig. 3 we plot the better fit (χ2 /dof = 1.21) taking into account all the world data which corresponds to a = 0.032 with the C3 term at the 30% value. We stress that there is no strong correlation between a and C3 as long as a is kept within its experimental error bars. Thus the fit is quite stable with respect to the parameters of the low energy model. The fit to the data is excellent with a very small variation of the 1/Q2 contribution at Q0 from previous fit, i.e. from 20% to 30%. It must be said, before entering the discussion of this fit, that in our previous work [5] we pointed out that the average value of the highest energy data points of BABAR were too large, a conclusion reached also by other analyses [12, 13]. In Fig. 3 we show not only the fit for 30% contribution of C3 /Q2 at Q0 , but its behavior for higher values of Q2 . As can be seen, also stressed in our previous work, this contribution is small in size. However, and this an important outcome of our analysis, it is instrumental in fixing the initial slope at the matching point, which determines, after evolution, the high energy behavior of the form factor. In our opinion the Belle data confirm the BABAR result that the πT F F crosses the asymptotic QCD limit. This limit is well founded under QCD assumptions, but nothing is known of how this limit is reached, if from above or from below. BABAR and Belle data suggest that the limit is exceeded around 10 − 15 GeV. Our calculation is consistent with this result. The necessary growth of the πT F F between 5 − 10 GeV to achieve this crossing is in our case an indication of nonperturbative behavior and C3 /Q2 contribution at low virtuality. The determination of the crossing point is a challenge for any theoretical model and therefore, the precise experimental determination of it is of relevance. Many models fail to achieve this crossing because their pion DA is defined close to its asymptotic form. The pion DA can be expressed as a series in the Gegenbauer polynomials,   ∞ X

an Q2 Cn3/2 (2x − 1) (5) φπ x, Q2 = 6x (1 − x) 1 + n(even)=2
We can compare different models by looking at the values of the coefficients of the expansion an Q2 . In our case, at Q2 = 1 GeV2 many an coefficients are significant, but we focus our attention in a few terms: a2 = 0.389, a4 = 0.244 and a6 = 0.179. At Q2 = 4 GeV2 we obtain the values a2 = 0.307, a4 = 0.173 and a6 = 0.118, which are close to those obtained by Polyakov [6]. Consistently, our result for the πT F F is similar to that obtained in ref. [6]. At Q2 = 5.76 GeV2 we obtain a2 = 0.292, a4 = 0.161 and a6 = 0.108, which are very different from those of ref. [14]. These author use for their fit BABAR data for the ηT F F [15], together with the pion data. It is therefore not a surprise that these authors come to a different conclusion, namely, that the Belle and the BABAR data cannot be reproduced to the same level of accuracy within the Light Cone Sum Rules approach [16]. However, in an extension of the ideas
developed in the present paper to the η case studied in ref. [17] looking at the state |qi = 12 |u ūi + d d¯ a very different structure of the an coefficients to that of the pion arises. At Q2 = 1 GeV2 , the values of the coefficients are a2 = 0.134 and a4 = 0.352 or, equivalently, at Q2 = 5.76 GeV2 we have a2 = 0.101 and a4 = 0.232. Therefore, that study does not supports the combined use of both data sets We have developed a formalism to describe the πT F F on all experimentally accessible range, and hopefully beyond. The formalism is based on a two energy scale description. The formulation in the low energy scale is nonperturbative, while that of the high energy scale is based on perturbative QCD. The two descriptions are matched at an energy scale Q0 called hadronic scale [18, 19]. We stress the crucial role played by the nonperturbative input at the level of the low energy description. It is an important 6 outcome of this calculation the role played by the 1/Q2 power correction term in determining the slope of the data at high Q2 , despite the fact that they do almost not contribute to the value of the πT F F . We have used a flat π DA, i.e. a constant value for all x [6, 7], which with our normalization becomes φ(x) = 1. Our choice has been motivated by chiral symmetry [5]. Model calculations, Nambu-JonaLasinio (NJL) [20–23] and the ”spectral” quark model [24], give a constant π DA. The πT F F calculated in these models, however, overshoots the data [25], emphasizing the importance of QCD evolution. The calculation shown proves that the BABAR and Belle results can be accommodated in our scheme, which only uses standard QCD ingredients and low energy data. Moreover, at the light of our results, we confirm that at 40 GeV2 we have not yet reached the asymptotic regime which will happen at higher energies. We would like to thank A. V. Pimikov and M. V. Polyakov for useful comments.This work has been partially funded by the Ministerio de Economa y Competitividad and EU FEDER under contract FPA201021750-C02-01, by Consolider Ingenio 2010 CPAN (CSD2007-00042), by Generalitat Valenciana: Prometeo/2009/129, by the European Integrated Infrastructure Initiative HadronPhysics3 (Grant number 283286). [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] S. Uehara et al. [The Belle Collaboration], arXiv:1205.3249 [hep-ex]. B. Aubert et al. [The BABAR Collaboration], Phys. Rev. D 80, 052002 (2009) [arXiv:0905.4778 [hep-ex]]. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22 (1980) 2157. V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. 112 (1984) 173. S. Noguera and V. Vento, Eur. Phys. J. A 46, 197 (2010) [arXiv:1001.3075 [hep-ph]]. M. V. Polyakov, JETP Lett. 90, 228 (2009) [arXiv:0906.0538 [hep-ph]]. A. V. Radyushkin, Phys. Rev. D 80 (2009) 094009 [arXiv:0906.0323 [hep-ph]]. K. Nakamura et al. [Particle Data Group Collaboration], J. Phys. G G 37 (2010) 075021. C. Amsler et al. [Particle Data Group], Phys. Lett. B 667 (2008) 1. H. J. Behrend et al. [CELLO Collaboration], Z. Phys. C 49 (1991) 401. J. Gronberg et al. [CLEO Collaboration], Phys. Rev. D 57 (1998) 33 [arXiv:hep-ex/9707031]. S. V. Mikhailov and N. G. Stefanis, Mod. Phys. Lett. A 24 (2009) 2858 [arXiv:0910.3498 [hep-ph]]. A. E. Dorokhov, arXiv:0905.4577 [hep-ph], arXiv:0909.5111 [hep-ph]. A. P. Bakulev, S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, arXiv:1205.3770 [hep-ph]. P. del Amo Sanchez et al. [BABAR Collaboration], Phys. Rev. D 84, 052001 (2011) [arXiv:1101.1142 [hep-ex]]. A. P. Bakulev, S. V. Mikhailov, A. V. Pimikov and N. G. Stefanis, Phys. Rev. D 84, 034014 (2011) [arXiv:1105.2753 [hep-ph]]. S. Noguera and S. Scopetta, Phys. Rev. D 85 (2012) 054004 [arXiv:1110.6402 [hep-ph]]. M. Traini, A. Mair, A. Zambarda and V. Vento, Nucl. Phys. A 614 (1997) 472. S. Noguera and V. Vento, Eur. Phys. J. A 28 (2006) 227 [arXiv:hep-ph/0505102]. I. V. Anikin, A. E. Dorokhov and L. Tomio, Phys. Lett. B 475 (2000) 361 [hep-ph/9909368]. M. Praszalowicz and A. Rostworowski, Phys. Rev. D 64 (2001) 074003 [arXiv:hep-ph/0105188]. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 66 (2002) 094016 [arXiv:hep-ph/0207266]. A. Courtoy and S. Noguera, Phys. Rev. D 76 (2007) 094026 [arXiv:0707.3366 [hep-ph]]. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 67 (2003) 074021 [arXiv:hep-ph/0301202]. W. Broniowski and E. R. Arriola, arXiv:0910.0869 [Unknown].