Possible contributions to $e^+e^- \to J/\psi + \eta_c$ due to intermediate meson rescatterings

Inspired by the obvious discrepancies between experiment and non-relativistic QCD (NRQCD) studies of $e^+ e^-\to J/\psi+\eta_c$ at $\sqrt{s}\simeq 10.6$ GeV, we investigate contributions from intermediate meson loops as long-range interaction transitions to this process. The intermediate meson loops include $D\bar D(\bar D^\ast)$, $D\bar D^\ast(D {or} D^\ast)$, $D^\ast \bar D^\ast(D)$ and corresponding $D_s$ intermediate mesons. With the constraints from experimental data on the vertex couplings, we find that the intermediate meson loops account for $2.7\sim 6.7 fb$ of the cross sections within a reasonable range of cut-off energies of the factor parameter. We also investigate contributions from the absorptive part and find that it accounts for approximately $0.58\sim 1.38 fb$. These results imply that contributions from long-range interaction transitions may still play a role in such an energy region.