Inclusive $J/\psi$ Productions at $e^+ e^-$ Colliders

Inclusive $J/\psi$ productions in $e^+ e^-$ annihilation is studied in the framework of NRQCD. We first review the leading-order calculations of the cross sections for $e^+ e^- \to J/\psi c \bar{c}$ and $e^+ e^- \to J/\psi g g$ and find that their ratio is about 1:1.5 at $\sqrt{s}\simeq 10$\gev. This result is in conflict with the current measurements by the Belle Collaboration, which finds that the process $e^+e^- \to J/\psi c\bar{c}$ accounts for about 2/3 of all the prompt $J/\psi$'s. We show that the discrepancy in the total rate as well as in the $J/\psi$ momentum distributions can be resolved by considering a large renormalization $K$ factor ($K\simeq 4$) for the $J/\psi c \bar {c}$ cross section and by taking into account collinear suppression in the end-point energy region of $J/\psi g g$ production. Detailed studies of the model predictions in terms of the density matrix are performed and various momentum and angular distributions are presented as functions of the $K$ factors. These distributions can be used to determine the normalizations of each subprocess provided that the production and decay angular distributions do not alter much by higher order corrections.