Next-to-leading-order QCD corrections to the radiative decays ${\chi }_{bJ}\to J\mathrm{/}\psi +\gamma$

Within nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) QCD corrections to ${\chi }_{bJ}\to J\mathrm{/}\psi +\gamma$ with $J=0,1,2$. We determine all six independent helicity amplitudes, including the one-loop corrections to the QED amplitude and the two-loop corrections to the QCD amplitude, and retain the interference between the QED and QCD amplitudes in the decay widths. The NLO corrections substantially enhance the ${\chi }_{b0}$ and ${\chi }_{b1}$ decay widths, while reducing the ${\chi }_{b2}$ decay width by approximately $36\mathrm{\%}$. Including the QED amplitude leads to sizable destructive interference for ${\chi }_{b0}$ and an enhancement for ${\chi }_{b2}$, but has only a small effect on the ${\chi }_{b1}$ decay width. Two treatments of photon fragmentation, based on the NRQCD matrix element and the experimentally determined $J\mathrm{/}\psi$ decay constant, yield NLO unpolarized decay widths that agree at the few percent level or better. Using the decay constant scheme at ${\mu }_R=m_b$, we obtain NLO central branching fractions of $1.03\times {10}^{-6}$, $1.45\times {10}^{-5}$, and $1.99\times {10}^{-6}$ for the three channels, respectively. The NLO corrections reduce the renormalization scale dependence, particularly for ${\chi }_{b2}$, although a sizable dependence remains for ${\chi }_{b0}$. At fixed $m_b$, the branching fractions decrease with increasing $r=m_c\mathrm{/}{m}_b$, more strongly for ${\chi }_{b1}$ and ${\chi }_{b2}$ than for ${\chi }_{b0}$. For $pp$ collisions at $\sqrt{s}=14\mathrm{T}\mathrm{e}\mathrm{V}$ and an integrated luminosity of $100{\mathrm{f}\mathrm{b}}^{-1}$, we estimate about $1.9\times {10}^4$ and $4.8\times {10}^4$ ${\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}\gamma$ events from ${\chi }_{b0}$ and ${\chi }_{b2}$ decays, respectively. Measurements with the existing Belle $\mathrm{{\rm Y}}(2S)$ sample remain challenging because of the small expected yields. A larger $\mathrm{{\rm Y}}(2S)$ sample at Belle II would improve the prospects for observing these radiative decays.