Production of $J/\psi +c\bar{c}$ through two photons in $e^+e^-$ annihilation

We study the production of $J/\psi+c\bar{c}$ in $e^+e^-$ annihilation through two virtual photons. The cross section is estimated to be 23 fb at $\sqrt{s}=10.6$ GeV, which is smaller by a factor of six than the calculated cross section for the same process but through one virtual photon. As a result, while the annihilation into two photons may be important for certain exclusive production processes, the big gap between the inclusive production cross section $\sigma(e^+e^- \to J/\psi + c\bar{c})\simeq$ 0.9pb observed by Belle and the current nonrelativistic QCD prediction of $\simeq$0.15pb still remains very puzzling. We find, however, as the center-of-mass energy increases ($\sqrt{s}>20$ GeV) the production through two virtual photons $e^+e^- \to 2\gamma^* \to J/\psi + c\bar{c}$ will prevail over that through one virtual photon, because in the former process the photon fragmentation into $J/\psi$ and into the charmed quark pair becomes more important at higher energies.