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

Abstract

Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay ${\rm Y}\to {\eta }_c({\chi }_{cJ})+\gamma$. 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 Buchmüller-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 ${\rm Y}\to {\eta }_c+\gamma$ 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 ${\rm Y}\to {\chi }_{c1}+\gamma$ 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 ${\rm Y}\to {\eta }_c+\gamma$. The discrepancies between theory and experiment deserve further research efforts.

arXiv comment24 pages, 2 figures, 6 tables. Significantly improved in writing; Matches version published in JHEP

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