Relativistic corrections to prompt $J/\psi$ photo- and hadroproduction

We systematically calculate the relativistic corrections to prompt $J/\psi$ photoproduction and hadroproduction using the factorization formalism of nonrelativistic QCD. Specifically, we include the ${}^3\!S_1^{[1]}$ and ${}^3\!P_{J}^{[1]}$ color-singlet and the ${}^3\!S_1^{[8]}$, ${}^1\!S_{0}^{[8]}$, and ${}^3\!P_{J}^{[8]}$ color-octet channels as well as the effects due to the mixing between the ${}^3\!S_1^{[8]}$ and ${}^3\!D_1^{[8]}$ channels. We provide all the squared hard-scattering amplitudes in analytic form. Assuming the nonrelativistic-QCD long-distance matrix elements to satisfy the velocity scaling rules, we find the relativistic corrections to be appreciable, except in the ${}^3\!S_1^{[1]}$ color-singlet channel of hadroproduction. We also observe significant differences in the line shapes of the relativistic corrections between photoproduction and hadroproduction.