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We present results on the isovector momentum fraction, $\langle x\rangle_{u-d}$, helicity moment, $\langle x \rangle_{\Delta u-\Delta d}$, andthe transversity moment, $\langle x \rangle_{\delta u-\delta d}$, of thenucleon obtained using nine ensembles of gauge configurations generated by theMILC collaboration using $2+1+1$-flavors of dynamical highly improved staggeredquarks (HISQ). The correlation functions are calculated using the Wilson-Cloveraction and the renormalization of the three operators is carried outnonperturbatively on the lattice in the RI${}^\prime$-MOM scheme. The data havebeen collected at lattice spacings $a \approx 0.15,\ 0.12,\ 0.09,$ and 0.06 fmand $M_\pi \approx 310,\ 220$ and 135 MeV, which are used to obtain thephysical values using a simultaneous chiral-continuum-finite-volume fit. Thefinal 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 anadditional systematic uncertainty due to possible residual excited-statecontributions. These results are consistent with other recent latticecalculations and phenomenological global fit values.

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