Mellin Moments of Proton Unpolarized GPDs at Nonzero Skewness from Lattice QCD with a Neural-Network Parametrization
We use lattice QCD to determine the generalized form factors (GFFs) entering the first four Mellin moments of the unpolarized isovector $u-d$ proton GPDs at nonzero skewness. The framework combines the polynomiality of the moments in the skewness $ξ$, with the short-distance expansion formalism, allowing the lattice-extracted reduced Ioffe-time distributions (ITDs) to be expressed in terms of GFFs. In order to extract the GFFs by fitting these relations to the lattice data, we parameterize each GFF with an artificial neural network (ANN) as a continuous function of the invariant momentum transfer squared $t$, which accommodates the kinematic coverage of the data in a flexible way. With this fit, we determine the fourteen GFFs that enter the first four Mellin moments, including the skewness-dependent GFFs. From the extracted GFFs, we obtain the quark total angular momentum of the isovector combination, $J^{u-d}=0.296(11)(7)$ in the $\overline{\mathrm{MS}}$ scheme at $μ=2~\mathrm{GeV}$, in agreement with previous lattice determinations based on nonlocal operators and with a smaller total uncertainty. We also determine the impact-parameter distributions of the first three Mellin moments in an unpolarized and in a transversely polarized proton. We finally identify and discuss the principal systematic uncertainties which enter this extraction, and the challenges specific to the nonzero-skewness regime.