$\gamma, J/\psi$ production through ISR process in electron-positron annihilation at B-factories

We study the Initial state radiation (ISR) process, $e^- e^+ \rightarrow^*\gamma\rightarrow \gamma +J/\psi$ observed at high luminosity $e^- e^+$ colliders such as B-factories, with the ISR photon being emitted through the initial electron or positron, prior to their annihilation. The emitted $J/\Psi$ is produced through the photon fragmentation process, with the produced $J/\psi$ in turn decaying into a pair of leptons, $\mu^-+\mu^+$ through the leptonic decay process, $J/\Psi\rightarrow \gamma^*\rightarrow \gamma \mu^- \mu^+$. The cross section for the $J/\Psi$ production, and the branching ratio for the leptonic decay of $J/\Psi\rightarrow \mu^- \mu^+$, both calculated in the framework of Bethe-Salpeter equation are used to work out the cross section for the complete process, $e^- e^+ \rightarrow \gamma +J/\psi\rightarrow \gamma+\mu^- +\mu^+$. Our results are compared with the BaBar data, and other models. This study is then extended to production and decays of $\Upsilon(1S)$ through the ISR process.