Inclusive Charmonium Production via Double $c \bar c$ in $e^+e^-$ Annihilation
Motivated by the recent observation of double charm quark pair production by the Belle Collaboration, we calculate the complete ${\cal O}(α_{s}^{2})$ inclusive production cross sections for $η_{c}$, $J/ψ$, and $χ_{cJ}$(J=0, 1, 2) plus $c\bar{c}$ in $e^+ e^-$ annihilation through a virtual photon. We consider both color-singlet and color-octet contributions, and give the analytical expressions for these cross sections. The complete color-singlet calculations are compared with the approximate fragmentation calculations as functions of the center-of-mass energy $\sqrt{s}$. We find that most of the fragmentation results substantially overestimate the cross sections (e.g. by a factor of $\sim$4 for $χ_{c1}$ and $χ_{c2}$) at the Belle and BaBar energy $\sqrt{s}=10.6$GeV. The fragmentation results become a good approximation only when $\sqrt{s}$ is higher than about 100GeV. We further calculate the color-octet contributions to these cross sections with analytical expressions. We find that while the color-octet contribution to $J/ψ$ inclusive production via double charm is negligible (only about 3%), the color-octet contributions to $χ_{c1}$ and $χ_{c2}$ can be significant.