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 ${e^++e^-\to J/\psi+\eta_c}$ and $e^++e^-\to J/\psi + \chi_{cJ}$ $(J=0,1,2)$ 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. The calculated $J/\psi\eta_c$ production rate is smaller than the recent Belle data by about an order of magnitude, indicating the failure of perturbative QCD calculation to explain the double charmonium production data. In addition, we also calculate the $\chi_{c0}$ inclusive double charm production cross section of ${e^++e^-\to \chi_{c0}+ c\bar {c}}$ in perturbative QCD and find that its value, which should be much larger than, is however saturated by the calculated cross sections of ${e^++e^-\to J/\psi + \chi_{c0}}$ plus ${e^++e^-\to \psi(2S) + \chi_{c0}}$. This again indicates that the perturbative QCD calculation for the charmonium (e.g. $\chi_{c0}$) inclusive double charm production seriously underestimates the production rates, and nonperturbative QCD effects have to be considered at low energies such as $\sqrt{s}$=10.6 GeV.