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

摘要

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.