[Paper Review] Emphasizing the different trends of the existing data for the $\gamma^*\gamma o \pi^0$ transition form factor

A. P. Bakulev, С. В. Михайлов+2 more|arXiv (Cornell University)|Dec 4, 2012

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper analyzes the $γ^*\gamma \to \pi^0$ transition form factor using light-cone sum rules (LCSRs) with NLO perturbative corrections and twist-4 contributions. It finds that Belle data are consistent with QCD predictions and the BMS pion distribution amplitude bunch, while BaBar data conflict with both, showing no 1σ overlap and a significantly worse fit quality, indicating a fundamental tension between the two experimental datasets at high $Q^2$. The results suggest the BaBar data may be inconsistent with QCD expectations and lattice constraints.

ABSTRACT

The new data on the $\gamma^*\gamma o \pi^0$ transition form factor of the Belle Collaboration are analyzed in comparison with those of BaBar (including the older data of CELLO and CLEO) using an approach based on light-cone sum rules. Performing a 2-, and a 3-parametric fit to these data, we found that the Belle and the BaBar data have no overlap at the $1\sigma$ level. While the Belle data agree with our predictions, the Babar data are in conflict with them.

Motivation & Objective

• To resolve the long-standing tension between BaBar and Belle experimental data on the $γ^*\gamma \to \pi^0$ transition form factor at high $Q^2$.
• To test the consistency of existing data (CELLO, CLEO, BaBar, Belle) with QCD predictions based on light-cone sum rules (LCSRs) and the BMS pion distribution amplitude (DA) bunch.
• To determine whether the BaBar data, which previously showed a violation of QCD scaling, are compatible with theoretical expectations and lattice constraints.
• To quantify the impact of including Belle versus BaBar data on the extracted pion DA parameters and their endpoint behavior.
• To assess the compatibility of the fitted pion DAs with lattice QCD results and theoretical constraints from QCD sum rules with nonlocal condensates (NLC).

Proposed method

• Uses light-cone sum rules (LCSRs) at NLO in QCD perturbation theory, including twist-4 and twist-6 contributions to the form factor.
• Applies a dispersion relation in $q^2$ and quark-hadron duality in the vector channel to model long-distance photon interactions.
• Performs 2-parameter (a2, a4) and 3-parameter (a2, a4, a6) fits to the data using the Gegenbauer expansion of the pion DA.
• Employs the BMS pion DA bunch from QCD sum rules with nonlocal condensates as a theoretical reference point.
• Quantifies data compatibility via $\chi^2_{\text{ndf}}$ and 1$\sigma$ confidence regions in the $(a_2, a_4)$ and $(a_2, a_4, a_6)$ parameter spaces.
• Uses the endpoint derivative $D^{(2)}_{\phi}(\Delta)$ to characterize the slope of the pion DA near $x=0$ and $x=1$.

Experimental results

Research questions

• RQ1 Do the Belle data on the $\gamma^*\gamma \to \pi^0$ form factor agree with QCD predictions based on LCSRs and the BMS pion DA bunch?
• RQ2 Is there a statistically significant discrepancy between the BaBar and Belle data sets at high $Q^2$?
• RQ3 How do the fitted pion distribution amplitudes from the Belle and BaBar data sets compare to each other and to theoretical constraints?
• RQ4 What is the endpoint behavior of the pion DA extracted from Belle versus BaBar data, and how does it compare to lattice QCD results?
• RQ5 Does the inclusion of BaBar data lead to a pion DA that is incompatible with the BMS bunch and lattice constraints?

Key findings

• The 2D fit to the combined CELLO, CLEO, and Belle (CCBe) data yields a $\chi^2_{\text{ndf}} = 0.6$, which is consistent with the theoretical prediction and the BMS pion DA bunch.
• The 2D fit to the CELLO, CLEO, and BaBar (CCBB) data yields a $\chi^2_{\text{ndf}} = 2.0$, indicating a significantly worse fit quality compared to the CCBe set.
• The confidence region for the CCBe fit shows no 1$\sigma$ overlap with the CCBB fit, indicating a statistically significant tension between the two data sets.
• The pion DA extracted from CCBe data has a less pronounced slope near the endpoints ($D^{(2)}_{\phi}(0.05) = 17.2 \pm 8.5$) and is compatible with the BMS bunch, while the CCBB DA has a steeper slope ($D^{(2)}_{\phi}(0.05) = 25.6 \pm 5.25$) and lies outside the BMS region.
• The 3D fits to CCBe and CCBB data do not intersect at the 1$\sigma$ level, confirming the incompatibility of the two data sets at high $Q^2$.
• The CCBe-based pion DA is compatible with lattice QCD constraints on $a_2$, while the CCBB-based DA is not, further indicating inconsistency with lattice results.