Exclusive double-charmonium production frome+e−annihilation into two virtual photons

We calculate the contributions from QED processes involving two virtual photons to the cross sections for $e^{+}{e}^{-}$ annihilation into two charmonium states with the same C parity. Generically, the cross sections are three orders of magnitude smaller than those for charmonia with opposite C parity because they are suppressed by a factor of ${\alpha }^2/{\alpha }_s^2.$ However, if both charmonia have quantum numbers $J^{\mathrm{PC}}{=1\mathrm{}}^{--},$ then there is a contribution to the cross section that involves the fragmentation of each photon into a charmonium. The fragmentation contribution is enhanced by powers of $E_{\mathrm{beam}}{/m}_c,$ the ratio of the beam energy to the charm-quark mass, and this enhancement can compensate for the suppression factor that is associated with the coupling constants. In particular, the predicted cross section for $J/\psi +J/\psi$ at the B factories is larger than that for $J/\psi +{\eta }_c.$