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CLEO and E791 data: A smoking gun for the pion distribution amplitude?

Alexander P. Bakulev, S. V. Mikhailov, N. G. Stefanis

TL;DR

This paper uses improved NLO light-cone sum-rule analysis of the CLEOπγ transition data to constrain the pion distribution amplitude, extracting Gegenbauer coefficients a2 and a4 and the inverse moment ⟨x^{-1}⟩_π. The results favor a non-endpoint-suppressed DA with end-point suppression controlled by λ_q^2≈0.4 GeV^2 (the BMS 'bunch'), while ruling out the asymptotic and CZ DAs at high confidence. An independent inverse-moment sum-rule comparison shows consistent values, and the analysis is cross-checked against E791 di-jet data, which broadly supports the BMS-like DA within uncertainties. Collectively, the work strengthens the case for a nontrivial, end-point-suppressed pion DA consistent with nonlocal QCD sum rules and lattice inputs, with robust cross-validation across multiple observables.

Abstract

The CLEO experimental data on the $πγ$ transition are analyzed to next-to-leading order accuracy in QCD perturbation theory using light-cone QCD sum rules. By processing these data along the lines proposed by Schmedding and Yakovlev, and recently revised by us, we obtain new constraints for the Gegenbauer coefficients $a_{2}$ and $a_{4}$, as well as for the inverse moment $x^{-1}$ of the pion distribution amplitude (DA). The former determine the pion DA at low momentum scale, the latter is crucial in calculating pion form factors. From the results of our analysis we conclude that the data confirm the end-point suppressed shape of the pion DA we previously obtained with QCD sum rules and nonlocal condensates, while the exclusion of both the asymptotic and the Chernyak--Zhitnitsky DAs is reinforced at the $3σ$- and $4σ$-level, respectively. The reliability of the main results of our updated CLEO data analysis is demonstrated. Our pion DA is checked against the di-jets data from the E791 experiment, providing credible evidence for our results far more broadly.

CLEO and E791 data: A smoking gun for the pion distribution amplitude?

TL;DR

This paper uses improved NLO light-cone sum-rule analysis of the CLEOπγ transition data to constrain the pion distribution amplitude, extracting Gegenbauer coefficients a2 and a4 and the inverse moment ⟨x^{-1}⟩_π. The results favor a non-endpoint-suppressed DA with end-point suppression controlled by λ_q^2≈0.4 GeV^2 (the BMS 'bunch'), while ruling out the asymptotic and CZ DAs at high confidence. An independent inverse-moment sum-rule comparison shows consistent values, and the analysis is cross-checked against E791 di-jet data, which broadly supports the BMS-like DA within uncertainties. Collectively, the work strengthens the case for a nontrivial, end-point-suppressed pion DA consistent with nonlocal QCD sum rules and lattice inputs, with robust cross-validation across multiple observables.

Abstract

The CLEO experimental data on the transition are analyzed to next-to-leading order accuracy in QCD perturbation theory using light-cone QCD sum rules. By processing these data along the lines proposed by Schmedding and Yakovlev, and recently revised by us, we obtain new constraints for the Gegenbauer coefficients and , as well as for the inverse moment of the pion distribution amplitude (DA). The former determine the pion DA at low momentum scale, the latter is crucial in calculating pion form factors. From the results of our analysis we conclude that the data confirm the end-point suppressed shape of the pion DA we previously obtained with QCD sum rules and nonlocal condensates, while the exclusion of both the asymptotic and the Chernyak--Zhitnitsky DAs is reinforced at the - and -level, respectively. The reliability of the main results of our updated CLEO data analysis is demonstrated. Our pion DA is checked against the di-jets data from the E791 experiment, providing credible evidence for our results far more broadly.

Paper Structure

This paper contains 6 sections, 3 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Light cone sum rules
  3. Confrontation with the CLEO data
  4. The inverse moment $\langle x^{-1}\rangle_{\pi}$ vs the CLEO data
  5. Comparison with the E791 data
  6. Conclusions

Figures (5)

  • Figure 1: Analysis of the CLEO data on $F_{\pi\gamma^{*}\gamma}(Q^2)$ in terms of error regions around the best-fit point (✚) (broken line: $1\sigma$; solid line: $2\sigma$; dashed-dotted line: $3\sigma$) in the ($a_2$,$a_4$) plane contrasted with various theoretical models explained in the text. The slanted shaded rectangle represents the constraints on ($a_2,~a_4$) posed by the nonlocal QCD sum rules BMS01 for the value $\lambda^2_{\rm q}=0.4$ GeV$^{2}$. All constraints are evaluated at $\mu^2_\text{SY}=5.76~\text{GeV}^{2}{GeV$^2${ }}$ after NLO ERBL evolution.
  • Figure 2: Analysis of the CLEO data: (a) Assuming a twist-4 uncertainty of 30% or equivalently at $\delta^2 = 0.13~\text{GeV}^{2}{GeV$^2${ }}$. (b) Excluding the lowest 6 experimental points up to $Q^2=3~\text{GeV}^{2}{GeV$^2${ }}$. The designations here are the same as in Fig. \ref{['fig1']} and the reference scale is $\mu^2_\text{SY}=5.76~\text{GeV}^{2}{GeV$^2${ }}$.
  • Figure 3: Estimation of the influence of higher-order corrections by varying the reference scale in the range $[Q^2/2, 2Q^2]$. The designations are as in Fig. \ref{['fig1']}.
  • Figure 4: (a) The result of the CLEO data processing for the quantity $\langle x^{-1} \rangle^\text{exp}_{\pi}/3-1$ at the scale $\mu^2_0 \approx 1~\text{GeV}^{2}{GeV$^2${ }}$ in comparison with the theoretical predictions from QCD sum rules, denoted SR. The thick solid-line contour corresponds to the union of $2\sigma$-contours, while the thin dashed-line contour denotes the union of $1\sigma$-contours. The light solid line with the hatched band indicates the mean value of $\langle x^{-1} \rangle^\text{SR}_{\pi}/3-1$ and its error bars in the second part of the Figure. (b) The inverse moment $\langle x^{-1} \rangle^\text{SR}_{\pi}$ shown as a function of the Borel parameter $M^2$ from the nonlocal QCD sum rules at the same scale $\mu^2_0$BMS01; the light solid line is the estimate for $\langle x^{-1} \rangle^\text{SR}_{\pi}$; finally, the dashed lines correspond to its error-bars.
  • Figure 5: Comparison of $\varphi^\text{as}$ (solid line), $\varphi^\text{CZ}$ (dashed line), and the BMS "bunch" of pion DAs (strip, BMS02) with the E791 data E79102. The corresponding $\chi^2$ values are As: 12.56; CZ: 14.15; BMS: 10.96.