Moments of nucleon isovector structure functions in $2+1+1$-flavor QCD

Understand

We present results on the isovector momentum fraction, $\langle x \rangle_{u-d}$, helicity moment, $\langle x \rangle_{\Delta u-\Delta d}$, and the transversity moment, $\langle x \rangle_{\delta u-\delta d}$, of the nucleon obtained using nine ensembles of gauge configurations generated by the MILC collaboration using $2+1+1$-flavors of dynamical highly improved staggered quarks (HISQ).

• The correlation functions are calculated using the Wilson-Clover action and the renormalization of the three operators is carried out nonperturbatively on the lattice in the RI${}^\prime$-MOM scheme.
• The data have been collected at lattice spacings $a \approx 0.15,\ 0.12,\ 0.09,$ and 0.06 fm and $M_\pi \approx 310,\ 220$ and 135 MeV, which are used to obtain the physical values using a simultaneous chiral-continuum-finite-volume fit.
• The final results, in the $\overline{MS}$ scheme at 2 GeV, are $\langle x \rangle_{u-d} = 0.173(14)(07)$, $\langle x \rangle_{\Delta u-\Delta d} = 0.213(15)(22)$ and $\langle x \rangle_{\delta u-\delta d} = 0.208(19)(24)$, where the first error is the overall analysis uncertainty and the second is an additional systematic uncertainty due to possible residual excited-state contributions.