Small-$x$ gluon GPD constrained from deeply virtual $J/\psi$ production and gluon PDF through universal-moment parameterization

We phenomenologically constrain the small-$x$ and small-$\xi$ gluon generalized parton distributions (GPDs) with the deeply virtual $J/\psi$ production (DV$J/\psi$P) in the framework of GPDs through universal moment parameterization (GUMP). We use a hybrid cross-section formula combining collinear factorization to the next-to-leading order (NLO) accuracy of the strong coupling $\alpha_s$, with corrections from non-relativistic QCD to account for the power corrections due to the heavy $J/\psi$ mass. We reach reasonable fit to the measured differential cross-sections of DV$J/\psi$P by H1 at Hadron-Electron Ring Accelerator (HERA) as well as forward gluon PDFs from JAM22 global analysis. We find that both NLO and non-relativistic corrections are significant for heavy vector meson productions. Of course, the gluon GPD we obtain still contain considerable freedom in need of inputs from other constraints, particularly in the distribution-amplitude-like region.