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. 
K. Hagiwara, E. Kou, Z.-H. Lin, C.-F. Qiao, G.-H. Zhu