(J/ψ,J/ψ), and (ηc,ηc) production through two intermediate photons in electron-positron annihilation at B-factories

We study the processes, $e^{-}{e}^{+}\to {\gamma }^{⁎}{\gamma }^{⁎}\to J/\psi +J/\psi$, and $e^{-}{e}^{+}\to {\gamma }^{⁎}{\gamma }^{⁎}\to {\eta }_c+{\eta }_c$ at $\sqrt{s}=10.6$ GeV in the framework of $4\times 4$ Bethe-Salpeter equation. For $J/\psi +J/\psi$ 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 $J/\psi +J/\psi$ and $\psi \left(2S\right)+\psi \left(2S\right)$ are compatible with the experimental upper limits set by Belle Collaboration, while in the absence of experimental data for ${\eta }_c\left(1S\right)+{\eta }_c\left(1S\right)$, and ${\eta }_c\left(2S\right)+{\eta }_c\left(2S\right)$ production, we have given theoretical prediction of their cross sections, and compared with NRQCD prediction.