Parton distribution functions beyond leading twist from lattice QCD: The $h_L(x)$ case

TL;DR

This paper presents the first lattice QCD calculation of the twist-3 parton distribution $h_L(x)$ for the proton, providing new insights into its structure beyond leading twist using the quasi-distribution approach.

Contribution

It introduces a novel lattice QCD method to compute the twist-3 distribution $h_L(x)$ and compares it with the Wandzura-Wilczek approximation, expanding understanding of proton structure.

Findings

First lattice QCD calculation of $h_L(x)$

Comparison of $h_L(x)$ with Wandzura-Wilczek approximation

Disentangling quark flavor contributions

Abstract

We report the first-ever calculation of the isovector flavor combination of the chiral-odd twist-3 parton distribution $h_L(x)$ for the proton from lattice QCD. We employ gauge configurations with two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3 fm and lattice spacing of 0.093 fm. The values of the quark masses lead to a pion mass of $260$ MeV. We use a source-sink time separation of 1.12 fm to control contamination from excited states. Our calculation is based on the quasi-distribution approach, with three values for the proton momentum: 0.83 GeV, 1.25 GeV, and 1.67 GeV. The lattice data are renormalized non-perturbatively using the RI$'$ scheme, and the final result for $h_L(x)$ is presented in the $\overline{\rm MS}$ scheme at the scale of 2 GeV. Furthermore, we compute in the same setup the transversity distribution, $h_1(x)$, which allows us, in particular, to compare $h_L(x)$ to its Wandzura-Wilczek approximation. We also combine results for the isovector and isoscalar flavor combinations to disentangle the individual quark contributions for $h_1(x)$ and $h_L(x)$, and address the Wandzura-Wilczek approximation in that case as well.