Problems of Double Charm Production in $e^{+}{e}^{-}$ Annihilation at $\sqrt{s}=10.6$ GeV

Using the nonrelativistic QCD(NRQCD) factorization formalism, we calculate the color-singlet cross sections for exclusive production processes $e^{+}+e^{-}\to J\mathrm{/}\psi +{\eta }_c$~ and ~$e^{+}+e^{-}\to J\mathrm{/}\psi +{\chi }_{cJ}$~$(J=0,1,2)$ at the center-of-mass energy $\sqrt{s}$=10.6 GeV. The cross sections are estimated to be 5.5fb, 6.7fb, 1.1fb, and 1.6fb for $\eta_c, \chi_{c0}, \chi_{c1}$and$\chi_{c2}$,respectively\mathrm{.}Thecalculated$J/\psi+\eta_c$ production rate is smaller than the recent Belle data by about an order of magnitude, which might indicate the failure of perturbative QCD calculation to explain the double-charmonium production data. The complete $\mathcal{O}$$({\alpha }_s^2)$ color-singlet cross section for ${e^++e^-\to \chi_{c0}+ c\bar {c}}$ is calculated. In addition, we also evaluate the ratio of exclusive to inclusive production cross sections. The ratio of $J\mathrm{/}\psi {\eta }_c$ production to $J\mathrm{/}\psi c\bar{c}$ production could be consistent with the experimental data.