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(PC) = 1(--), 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-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).
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