$(J\mathrm{/}\psi ,J\mathrm{/}\psi )$, and $({\eta }_c,{\eta }_c)$ production through two intermediate photons in electron-positron annihilation at B-factories

We study the processes, $e^{-}{e}^{+}\to {\gamma }^{\ast }{\gamma }^{\ast }\to J\mathrm{/}\psi +J\mathrm{/}\psi$, and $e^{-}{e}^{+}\to {\gamma }^{\ast }{\gamma }^{\ast }\to {\eta }_c+{\eta }_c$ at $\sqrt{s}=10.6$ GeV in the framework of $4\times 4$ Bethe-Salpeter equation. For $J\mathrm{/}\psi +J\mathrm{/}\psi$ production, the dominant contribution is through fragmentation process, while for ${\eta }_c+{\eta }_c$ production, the quark rearrangement diagrams contribute. Our results of cross section for $J\mathrm{/}\psi +J\mathrm{/}\psi$ and $\psi (2S)+\psi (2S)$ are compatible with the experimental upper limits set by Belle Collaboration, while in the absence of experimental data for ${\eta }_c(1S)+{\eta }_c(1S)$, and ${\eta }_c(2S)+{\eta }_c(2S)$ production, we have given theoretical prediction of their cross sections, and compared with NRQCD prediction.