Double heavy-quarkonium production from electron-positron annihilation in the Bethe-Salpeter formalism

In this work we evaluate the cross section of the process $e^+e^-\to J/\psi \eta_c$ at energy $\sqrt{s}\approx 10.6$ GeV in the Bethe-Salpeter formalism. To simplify our calculation, the heavy quark limit is employed. Without taking the beyond-leading-order contribution(s) into account, the cross section calculated in this scenario is comparable with the experimental data. We also present our prediction for the cross section of double bottomonium production $e^+e^-\to \Upsilon(1S)\eta_b$ for the energy range of $\sqrt{s}\approx (25 \hbox{-} 30)$ GeV which may be experimentally tested, even though there is no facility of this range available at present yet.