Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium

Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay $\Upsilon\to \eta_c(\chi_{cJ})+\gamma$. Both the helicity amplitudes and the helicity decay widths are obtained. It is the first computation for the processes involving both bottomonium and charmonium at two-loop accuracy. By employing the Cheng-Wu theorem, we are able to convert most of complex-valued master integrals (MIs) into real-valued MIs, which makes the numerical integration much efficient. Our results indicate the $\mathcal{O}(\alpha_s)$ corrections are moderate for $\eta_c$ and $\chi_{c2}$ production, and are quite marginal for $\chi_{c0}$ and $\chi_{c1}$ production. It is impressive to note the NLO corrections considerably reduce the renormalization scale dependence in both the decay widths and the branching fractions for $\chi_{cJ}$, and slightly improve that for $\eta_c$. With the NRQCD matrix elements evaluated via the Buchm\"uller-Tye potential model, we find the decay width for $\eta_c$ production is one-order-of-magnitude larger than $\chi_{cJ}$ production, which may provide a good opportunity to search for $\Upsilon\to \eta_c+\gamma$ in experiment. In addition, the decay width for $\chi_{c1}$ production is several times larger than those for $\chi_{c0,2}$. Finally, we find the NLO NRQCD prediction for the branching fraction of $\Upsilon\to \chi_{c1}+\gamma$ is only half of the lower bound of the experimental data measured recently by {\tt Belle}. Moreover, there exists serious contradiction between theory and experiment for $\Upsilon\to \eta_c+\gamma$. The discrepancies between theory and experiment deserve further research efforts.