QCD leading order study of the $J/\psi$ leptoproduction at HERA within the nonrelativistic QCD framework

As indicated in our previous paper (Phys.Rev.D96.034002), the existing literatures studying the $J/\psi$ production in deeply inelastic scattering (DIS) in collinear factorization are on the basis of a formalism that will lead to wrong results when the ranges of the transverse momentum or the rapidity of the $J/\psi$ in the laboratory frame do not cover all values possible for them. In this paper, we present the renewed results for the $J/\psi$ production in DIS at HERA within the nonrelativistic QCD framework at QCD leading order (LO). Three different sets of the long-distance matrix elements are employed for comparison. The predictions via the colour-singlet (CS) model at QCD LO are generally below the experimental data especially in the regions where perturbation theory are expected to work well, while the colour-octet contributions are of the same order of magnitude as the CS ones, however, in general make the agreement between theory and experiment better.