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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 $Υ\to η_c(χ_{cJ})+γ$. 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 $\mathcal{O}(α_s)$ corrections are moderate for $η_c$ and $χ_{c2}$ production, and are quite marginal for $χ_{c0}$ and $χ_{c1}$ 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 $χ_{cJ}$, and slightly improve that for $η_c$. With the NRQCD matrix elements evaluated via the Buchmüller-Tye potential model, we find the decay width for $η_c$ production is one-order-of-magnitude larger than $χ_{cJ}$ production, which may provide a good opportunity to search for $Υ\to η_c+γ$ in experiment. In addition, the decay width for $χ_{c1}$ production is several times larger than those for $χ_{c0,2}$. Finally, we find the NLO NRQCD prediction for the branching fraction of $Υ\to χ_{c1}+γ$ 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 $Υ\to η_c+γ$. The discrepancies between theory and experiment deserve further research efforts.