Two-loop QCD corrections to $\hat{I}{\cdot }_b\to J/\ddot{I}ˆ+{\hat{I}}^3$

Abstract

We present a next-to-leading-order (NLO) analysis of the rare radiative decay $\hat{I}{\cdot }_b\to J/\ddot{I}ˆ{\hat{I}}^3$, a flavor-changing transition between bottomonium and charmonium, within the framework of non-relativistic QCD (NRQCD). We systematically compute the complete $\mathcal{O}(\hat{I}{\pm }_s)$ corrections, which include the one-loop QCD corrections to the QED-initiated amplitudes and the two-loop corrections to the QCD-initiated ones. The branching ratio is enhanced from $2.06_{-1.32}^{+2.82}\times 10^{-7}$ at LO to $7.53_{-1.16}^{+5.67}\times 10^{-7}$ at NLO, representing an increase by a factor of about 3.65. The theoretical uncertainties caused by renormalization scale and $m_{b/c}$ masses are also discussed. Furthermore, the renormalization scale dependence is reduced at NLO.