Determination of the pion generalized parton distributions at zero skewness

TL;DR

This work addresses the extraction of valence pion GPDs at zero skewness ($\xi=0$) by performing a global QCD analysis of spacelike pion electromagnetic form factor data from electroproduction and elastic scattering. The authors adopt a MMGPDs-inspired ansatz $H_v^q(x,t,Q^2)= q_v(x,Q^2)\, \exp[ t f_v^q(x) ]$ with $f_v^q(x)= \alpha'(1-x)^3 \ln(1/x) + B(1-x)^3 + A x (1-x)^2$, testing three pion PDF inputs—xFitter, JAM21, and MAP23—via $\chi^2$ minimization and Hessian uncertainty analysis. They find that two parameters ($\alpha'$, $A$) suffice to describe the data, with only mild dependence on the PDF choice; the MAP23-based extraction provides the best fit and is adopted as the final GPD set. The resulting predictions for the pion form factor $F_\pi(t)$ and its squared magnitude $|F_\pi(t)|^2$ agree well with experimental data and align with a broad range of theoretical approaches, supporting the reliability of the GPD-based picture and enabling future explorations of pion radius, tomography, and mechanical properties via GPD tomography.

Abstract

We perform a global QCD analysis of the pion electromagnetic form factor (FF) data from pion electroproduction and elastic pion scattering to extract the valence pion generalized parton distributions (GPDs) at zero skewness. The analysis uses three different sets of pion parton distribution functions (PDFs), namely xFitter, JAM21, and MAP23, to construct the GPD ansatz. Through a $χ^2$ minimization and a careful parametrization scan, we determine the profile function parameters and find that only two parameters are sufficient to describe the data. The extracted valence pion GPDs from different analyses have similar $x$-dependence, with minor differences at small momentum transfer. The resulting theoretical predictions for the pion electromagnetic FF and its squared magnitude show good agreement with experimental measurements. Among the three analyses, the one using the MAP23 PDFs provides the best overall fit and is adopted as the final GPD set. Our results offer a consistent determination of the valence pion GPDs, indicating a minor impact of the choice of pion PDFs. The present study provides a solid foundation for future investigations of pion structure, including its charge radius, tomography, and mechanical properties.

Figures (6)

• Figure 1: Comparison of the valence pion PDF $q_v(x)$ at $Q^2 = 4~\mathrm{GeV}^2$ from the xFitter Novikov:2020snp, JAM21 Barry:2021osv, MAP23 Pasquini:2023aaf, and GRVPI1 Gluck:1991ey determinations. For xFitter, the EIG grid is used. The lower panel displays the ratio of each PDF to the JAM21 result.
• Figure 2: Comparison of the extracted valence pion GPDs, $xH_v^q(x,t,Q^2)$, corresponding to Set 1, Set 2, and Set 3, obtained using the xFitter, JAM21, and MAP23 PDFs, respectively, shown at $Q^2 = 4~\text{GeV}^2$ for representative values of $t$.
• Figure 3: Comparison of the experimental data for the pion electromagnetic form factor, $F_\pi(t)$, from pion electroproduction with the theoretical predictions obtained from the three fits.
• Figure 4: Comparison of the experimental data for the squared pion electromagnetic form factor, $|F_\pi(t)|^2$, from elastic pion scattering with the theoretical predictions obtained from the three fits.
• Figure 5: Comparison of the pion electromagnetic form factor, $F_\pi(t)$, obtained from the present analysis (Set 3), with representative theoretical predictions from the DSE approach Chang:2013nia, LFH Brodsky:2007hb, LCQM Puhan:2025pfs, and TSWM of holographic QCD Nasibova:2025wnw, together with recent lattice QCD results Wang:2020nbfDing:2024lfjGao:2021xsm.