Proton isovector helicity PDF at NNLO and the twist-3 moment $\tilde{d}_2$ from lattice QCD at physical quark masses

TL;DR

This paper reports a lattice QCD calculation of the proton's isovector helicity PDF at NNLO and the twist-3 moment $ ilde{d}_2$, providing insights into the proton's internal spin structure using advanced theoretical methods.

Contribution

It introduces a first-time lattice QCD determination of the $ ilde{d}_2$ moment in the $ar{ m MS}$ scheme and refines the proton helicity PDF extraction at NNLO accuracy.

Findings

The $ ilde{d}_2^{u-d}(2 ext{ GeV})=0.0024(46)$ was obtained.

Helicity PDFs were extracted in the region $x ext{ in}[0.25,0.75]$ with controlled uncertainties.

The study demonstrates the feasibility of combining lattice QCD with LaMET for detailed proton structure analysis.

Abstract

We present a lattice quantum chromodynamics calculation of the $x$-dependent isovector quark helicity parton distribution function (PDF) of the proton in the large momentum effective theory (LaMET) framework. Through operator product expansion (OPE) we also extract the $\tilde{d}_2$ moment of the twist-3 PDF $g_T(x)$ for the first time in the $\overline{\rm MS}$ scheme, which is proportional to the average color Lorentz force experienced by the quark in the proton. This calculation is performed on a lattice of spacing $a$ = 0.076 fm at physical quark masses. The quasi-PDF matrix elements are measured in proton states boosted to momenta $P_z=\{0, 0.25, 1.02, 1.53\}$ GeV. We first extract the lowest few helicity PDF moments from the renormalization-group (RG) invariant ratios of the matrix elements with OPE. Combined with the matrix elements relevant for $g_T(x)$, we obtain $\tilde{d}_2^{u-d}(2\ {\rm GeV})=0.0024(46)$ at next-to-leading order in $\overline{\rm MS}$. Then, the helicity quasi-PDF matrix elements are renormalized in the hybrid scheme with linear renormalon resummation and Fourier transformed to the $x$-space after an asymptotic extrapolation. The quasi-PDF is perturbatively matched to the $\overline{\rm MS}$ PDF with RG and threshold resummations at next-to-leading power and next-to-next-to-leading logarithmic accuracies. After resummations, we determine the PDF in the region $x\in[0.25,0.75]$ with controlled systematic uncertainties. The end-point regions are then parameterized, combined with the LaMET prediction at moderate $x$, and fitted to the short-distance matrix elements in coordinate space.