Moments of Isovector Quark Distributions from Lattice QCD

TL;DR

This work addresses how to reliably extrapolate lattice QCD moments of twist-2 isovector parton distributions to physical quark masses by incorporating chiral nonanalytic corrections from Nπ and Δπ loops. The authors develop parameterizations that blend leading nonanalytic behavior with heavy-quark constraints, and they show that Δ intermediate states drastically reduce curvature in spin-dependent (helicity and transversity) moments, yielding near-linear chiral behavior, while unpolarized moments still exhibit curvature requiring LNA terms to match experimental moments. Their analysis demonstrates that including Δ contributions is crucial for polarized PDFs and provides extrapolation formulas that bring lattice results into agreement with experimental data, with g_A close to the measured value within uncertainties. The results have meaningful implications for nonperturbative hadron structure and guide future lattice studies toward lighter quark masses, larger volumes, and refined renormalization schemes.

Abstract

We present a complete analysis of the chiral extrapolation of lattice moments of all twist-2 isovector quark distributions, including corrections from N pi and Delta pi loops. Even though the Delta resonance formally gives rise to higher order non-analytic structure, the coefficients of the higher order terms for the helicity and transversity moments are large and cancel much of the curvature generated by the wave function renormalization. The net effect is that, whereas the unpolarized moments exhibit considerable curvature, the polarized moments show little deviation from linearity as the chiral limit is approached.

Figures (10)

• Figure 1: Connected (a) and disconnected (b) contributions to the matrix elements of an operator (indicated by the cross). Such diagrams occur in quenched QCD as well as in full QCD.
• Figure 2: The goodness of fit of the extrapolated values of the first three non-trivial moments to the phenomenological values as a function of $\mu$ calculated using Eq. (\ref{['E:goodfit']}).
• Figure 3: Contributions to the wave function and vertex renormalization of the nucleon matrix elements of the operators ${\cal O}_i^{\mu_1\ldots\mu_n}$, $i=q, \Delta q, \delta q$, in Eq. (\ref{['E:operators']}). Solid, double and dashed lines denote nucleon, $\Delta$ and pion propagators and the crossed circle and box indicate the insertion of the relevant operators. Diagrams $Z_2^N$ and $Z_2^\Delta$ denote the contributions to wave function renormalization (a derivative with respect to the external momentum is implied).
• Figure 4: Contributions to the pion loop renormalization of the matrix elements of the twist--2 operators required to evaluate the moments of the PDFs. The upper panel shows nucleon wave function renormalizations ($Z_2^N$, $Z_2^\Delta$) and spin-independent operator renormalizations. The lower panel shows the contributions to the renormalization of spin-dependent operators, and the shaded region is an estimate of the uncertainty in the Weinberg--Tomozawa term, $Z^{\rm WT}_{1,P} \equiv Z^{\rm N WT}_{1,P} + Z^{\rm \Delta WT}_{1,P}$. The $g_{\pi N\Delta}/g_{\pi NN}$ coupling constant ratio is set to the SU(6) symmetric value of $\sqrt{72/25}$.
• Figure 5: Pion dressing of the matrix elements of the spin-independent (upper panel) and spin-dependent (lower panel) operators in Eq. (\ref{['E:operators']}) for various values of the ratio of coupling constants, $g_{\pi N\Delta}/g_{\pi NN}$. The shading in the lower panel indicates the variation about the dashed curve ($g_{\pi N\Delta}/g_{\pi NN}=\sqrt{72/25}$) caused by the uncertainty in the Weinberg--Tomozawa term. The behavior of $Z_2/Z_{\delta q}$ is similar to that of $Z_2/Z_{\Delta q}$.