Moments of parton distributions functions of the pion from lattice QCD using gradient flow

TL;DR

This paper uses lattice QCD with gradient flow to nonperturbatively compute ratios of pion valence PDF moments up to the sixth order, demonstrating a systematic approach with controlled continuum extrapolation and promising agreement with phenomenology.

Contribution

It introduces a novel lattice QCD method employing gradient flow to accurately determine pion PDF moment ratios, with controlled continuum and flow-time extrapolations.

Findings

Ratios exhibit O(a^2) scaling across ensembles

Results are consistent with phenomenological expectations

Method shows promise for future nucleon structure studies

Abstract

We present a nonperturbative determination of the pion valence parton distribution function (PDF) moment ratios $\left\langle x^{n-1} \right\rangle / \left\langle x \right\rangle$ up to $n=6$, using the gradient flow in lattice QCD. As a testing ground, we employ SU($3$) isosymmetric gauge configurations generated by the OpenLat initiative with a pseudoscalar mass of $m_\pi \simeq 411~\text{MeV}$. Our analysis uses four lattice spacings and a nonperturbatively improved action, enabling full control over the continuum extrapolation, and the limit of vanishing flow time, $t\to0$. The flowed ratios exhibit O($a^2$) scaling across the ensembles, and the continuum-extrapolated results, matched to the $\overline {\text{MS}}$ scheme at $\mu = 2$ GeV using next-to-next-to-leading order matching coefficients, show only mild residual flow-time dependence. The resulting ratios, computed with a relatively small number of configurations, are consistent with phenomenological expectations for the pion's valence distribution, with statistical uncertainties that are competitive with modern global fits. These findings demonstrate that the gradient flow provides an efficient and systematically improvable method to access partonic quantities from first principles. Future extensions of this work will target lighter pion masses toward the physical point, and applications to nucleon structure such as the proton PDFs and the gluon and sea-quark distributions.