Towards nucleon structure function moments and parton momentum fractions from lattice QCD

We calculate the lowest even isovector moment of the $F_2$ structure function in $2+1$-flavour lattice QCD with varying quark masses corresponding to $m_{\pi }\approx [410,360,300]\mathrm{M}\mathrm{e}\mathrm{V}$, at a fixed volume of $V={48}^3\times 96$ and coupling $\beta =5.65$ ($a=0.068(3)\mathrm{f}\mathrm{m}$). We directly compute the physical Compton amplitude using the Feynman-Hellmann approach and extract moments of the physical structure function. We report on the quark-mass dependence of the lowest isovector moment and estimate its value at the physical quark-mass point with $\sim 10\mathrm{\%}$ uncertainty at fixed $Q^2$. By analysing the $Q^2$ dependence of the moments at the $SU(3)$ symmetric point ($m_{\pi }\approx 410\mathrm{M}\mathrm{e}\mathrm{V}$), we separate the leading- and higher-twist contributions and estimate the parton momentum fraction, $⟨x{⟩}_{u-d}$, which agrees with existing results.