Radiative decays of Upsilon(nS) into S-wave and P-wave charmonium

Motivated by very recent measurement of the radiative decays of $\Upsilon(1S)$ to $\chi_{c1}$ at Belle, we use the nonrelativistic QCD factorization theory and calculate the branching fractions of the radiative decays of bottomonium into S-wave and P-wave charmonium, i.e. $\Upsilon(nS) \to \eta_c(nS)+\gamma$ and $\Upsilon(nS)\to \chi_{cJ}+\gamma$. We systematically studied the branching fractions of the radiative decays of bottomonium into charmonium. Compared to the previous calculation, we obtained the analytical expression for the decay widths and considered the color-octet contributions. For $\Upsilon(nS) \to \eta_c(nS)+\gamma$, the relativistic corrections are also obtained. Through the calculation, the theoretical prediction for $\Upsilon(1S)\to\chi_{c1}+\gamma$ is still smaller than the recent Belle measurement. Further theoretical work and experimental analysis are necessary to understand the $\chi_{c1}$ production mechanism in upsilon decays.