Radiative $h_{c/b}$ decays to $\eta$ or $\eta^\prime$

Motivated by recent measurements of the radiative decay rates of the \emph{P}-wave spin singlet charmonium $h_c$ to the light meson $\eta$ or $\eta^\prime$ by the BESIII Collaboration, we investigate the decay rates of these channels at order $\alpha \alpha_s^4$. The photon is radiated mainly from charm quark pairs in the lowest order Feynman diagrams, since the diagrams where a photon radiated from light quarks are suppressed by $\alpha_s$ or the relative charm quark velocity $v$, due to Charge parity conservation. The form factors of two gluons to $\eta$ or $\eta^\prime$ are employed, which are the major mechanism for $\eta$ and $\eta^\prime$ productions. $\eta(\eta^\prime)$ is treated as a light cone object when we consider that the parent charmonium mass is much heavier than that of the final light meson. We obtain the branching ratio ${\cal B}(h_c\to \gamma\eta^\prime) = (1.94^{+0.70}_{-0.51})\times 10^{-3}$ in the nonrelativistic QCD approach, which is in agreement with the BESIII measurement. The prediction of the branching ratio of $h_c\to \gamma\eta$ is also within the range of experimental error after including the larger uncertainty of the total decay width $\Gamma_{h_c}$. The applications of these formulae to the radiative decays to $\eta(\eta^\prime)$ of the \emph{P}-wave spin singlet bottomonium $h_b(nP)$ are presented. These studies will shed some light on the $\eta - \eta^\prime$ mixing effects, the flavor SU(3) symmetry breaking, as well as the nonperturbative dynamics of charmonium and bottomonium.