Relativistic corrections to prompt $J/ψ$ photo- and hadroproduction

Abstract: 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.