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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 takingthe beyond-leading-order contribution(s) into account, the cross sectioncalculated in this scenario is comparable with the experimental data. We alsopresent 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 nofacility of this range available at present yet.

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