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

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 ($\beta =2.13$). The inverse lattice spacing is $a^{-1}=1.73\mathrm{GeV}$, and two spatial volumes of $(2.7\mathrm{fm}{)}^3$ and $(1.8\mathrm{fm}{)}^3$ 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 the fully nonperturbatively renormalized isovector quark momentum fraction $⟨x{⟩}_{u-d}$, the helicity fraction $⟨x{⟩}_{\Delta u-\Delta d}$, and transversity $⟨1{⟩}_{\delta u-\delta d}$, as well as an unrenormalized twist-3 coefficient $d_1$. The ratio of the momentum to helicity fractions, $⟨x{⟩}_{u-d}/⟨x{⟩}_{\Delta u-\Delta d}$, 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, $0.7<⟨1{⟩}_{\delta u-\delta d}<1.1$, in the $\overline{\mathrm{MS}}$ scheme at 2 GeV is obtained. The twist-3 coefficient, $d_1$, though yet to be renormalized, supports the perturbative Wandzura-Wilczek relation.