Nucleon isovector structure functions in (2+1)-flavor QCD with domain wall fermions

TL;DR

This study computes low moments of nucleon structure functions in (2+1)-flavor QCD using domain-wall fermions on two lattice volumes. It nonperturbatively renormalizes isovector moments and analyzes finite-volume and chiral effects, finding a mass-independent ratio of momentum to helicity fractions that agrees with experiment and trends toward experimental values at lighter quark masses. The work reports the quark momentum and helicity fractions, transversity, and twist-3 d1, with renormalization in RI-MOM and conversion to MSbar at 2 GeV, highlighting the role of nonperturbative renormalization and chiral symmetry. It also outlines plans to reach the physical point with lighter pions on larger volumes to reduce systematic uncertainties and sharpen comparisons with experiment.

Abstract

We report on numerical lattice QCD calculations of some of the low moments of the nucleon structure functions. The calculations are carried out with gauge configurations generated by the RBC and UKQCD collaborations with (2+1)-flavors of dynamical domain wall fermions and the Iwasaki gauge action ($β= 2.13$). The inverse lattice spacing is $a^{-1} = 1.73$ GeV, and two spatial volumes of ((2.7{\rm fm})^3) and ((1.8 {\rm fm})^3) are used. The up and down quark masses are varied so the pion mass lies between 0.33 and 0.67 GeV while the strange mass is about 12 % heavier than the physical one. The structure function moments we present include fully non-perturbatively renormalized iso-vector quark momentum fraction, (< x >_{u-d}), helicity fraction, (< x >_{Δu - Δd}), and transversity, (< 1 >_{δu - δd}), as well as an unrenormalized twist-3 coefficient, (d_1). The ratio of the momentum to helicity fractions, (< x >_{u-d}/< x >_{Δu - Δd}), does not show dependence on the light quark mass and agrees well with the value obtained from experiment. Their respective absolute values, fully renormalized, show interesting trends toward their respective experimental values at the lightest quark mass. A prediction for the transversity, (0.7 < < 1 >_{δu -δd} < 1.1), in the (\bar{\rm MS}) scheme at 2 GeV is obtained. The twist-3 coefficient, (d_1), though yet to be renormalized, supports the perturbative Wandzura-Wilczek relation.

Figures (18)

• Figure 1: A nucleon observable, isovector quark momentum fraction, $\langle x\rangle_{u-d}$, from RBC 2-flavor dynamical DWF ensemble with $m_{ud}=0.02$Lin:2008uz, with source-sink separation of 10 and 12: a clear systematic difference is seen. The shorter source-sink separation is not manifestly free of excited-state contamination.
• Figure 2: Effective mass of the nucleon correlation function with Gaussian smearing applied at both source and sink, for quark mass $m_f=0.005$.
• Figure 3: Ratio of the bare, isovector, momentum and helicity fractions, $\langle x\rangle_{u-d}/\langle x\rangle_{\Delta u-\Delta d}$, which is naturally renormalized for DWF. Both volumes are shown, $({\rm 2.7\ fm})^3$ (circles) and $({\rm 1.8\ fm})^3$ (squares). The square symbols have been moved slightly in the plus x-direction. They are in good agreement with experiment which is denoted by the star. No discernible dependence on volume nor pion mass can be detected.
• Figure 4: Signals for the ratio of three- and two-point functions for the bare quark momentum fraction, $\langle x\rangle_{u-d}$. Quark mass $0.005$ (circles) and 0.01 (squares).
• Figure 5: Non-perturbative renormalization factor for the quark momentum fraction, $\langle x \rangle_{u-d}$ . Circles denote the RI-MOM values, squares the $\overline{\rm MS}$ ones. The line denotes a linear fit used to remove the leading $O((ap)^2)$ lattice artifacts.