Short-flow-time expansion of non-singlet twist-two operators at next-to-next-to-leading order QCD

TL;DR

This work addresses the challenge of obtaining higher Mellin moments $\langle x^{n-1}\rangle^{h}_q(\mu)$ of non-singlet parton distributions from lattice QCD by leveraging gradient flow to remove power-divergent operator mixing. It derives and presents the NNLO matching coefficients $\zeta_n(t,\mu)$ for $n=1$ to $6$, relating flowed twist-two operators at flow time $t>0$ to the $\overline{\text{MS}}$ scheme. The analytic results, validated by gauge and RG checks and consistent with prior literature, have been applied to extract higher moments of the valence pion on OpenLat ensembles. The methodology establishes a path to extend to the singlet sector and to broader hadron structure studies, including non-forward distributions, using gradient-flow–based determinations of PDFs.

Abstract

The gradient-flow formalism provides a framework for the direct determination of moments of parton distribution functions (PDFs) from lattice QCD calculations. Their conversion from the gradient-flow scheme to $\overline{\text{MS}}$ requires the matching coefficients of the short-flow-time expansion, which can be computed perturbatively. We determine these coefficients for the first six non-singlet PDF moments up to next-to-next-to-leading order in the strong coupling.

Figures (3)

• Figure 1: Generic form of the diagrams contributing to the r.h.s. of \ref{['eq:calcmix']}. All Feynman diagrams in this paper were created using FeynGameHarlander:2020cyhBundgen:2025utt.
• Figure 2: Examples for diagrams contributing to the r.h.s. of \ref{['eq:calcmix']}. (a): tree-level diagram contributing for all $n\geq 1$; (b) and (c): diagrams contributing at for all $n\geq 1$ and $n\geq 2$, respectively; (d), (e), (f): diagrams contributing at for all $n\geq 1,2,3$, respectively. Straight lines denote quarks, curly lines gluons; flow lines are marked by an arrow next to them, which denotes the flow direction; filled/hollow vertex symbols denote regular/flowed vertices; the symbol $\otimes$ denotes one of the operators of \ref{['eq:ops:flon']}.
• Figure 3: Examples for diagrams contributing to the r.h.s. of \ref{['eq:calcmix']}. (a): diagram contributing at for all $n\geq 3$; (b), (c): diagram contributing at for all $n\geq 4,5$, respectively. The notation is the same as in \ref{['fig:dias']}. Note that due to the flow time associated with the operator vertex, these diagrams do not involve scaleless integrals and are therefore non-zero.