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 ${cal B}(h_cto gammaeta^prime) = (1.94^{+0.70}_{-0.51})times 10^{-3}$ in the nonrelativistic QCD approach, which is in agreement with the BESIII measurement. The prediction of the branching ratio of $h_cto gammaeta$ 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.
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