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

Abstract

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 $\mathcal{B}(h_c\to \gamma {\eta }^{\prime })=(1.94_{-0.51}^{+0.70})\times 10^{-3}$ 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.

6 pages, 2 figures, and 1 table;To appear in Physical Review D

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