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

Motivated by recent measurements of the radiative decay rates of the $P$-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, because 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 ${\eta }^{\prime }$, which are the major mechanism for $\eta$ and ${\eta }^{\prime }$ production, are employed. $\eta ({\eta }^{\prime })$ 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 ${\mathrm{\Gamma }}_{h_c}$. The applications of these formulas to the radiative decays to $\eta ({\eta }^{\prime })$ of the $P$-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, and the nonperturbative dynamics of charmonium and bottomonium.

Figure 1

Typical Feynman diagrams that contribute to the process $h_c\to \gamma \eta ({\eta }^{\prime })$, where the effective vertex $g^{*}{g}^{*}\eta (\prime )$ denotes two-gluon transitions to $\eta (\prime )$ form factors. Note that the diagrams where the photon is directly emitted from $\eta (\prime )$ and only two gluons link to $h_c$ contribute trivially, due to $CP$ conservation.

Figure 2

Typical Feynman diagrams for two-gluon transitions to ${\eta }^{(\prime )}$ form factors, where (a) denotes the diagrams for two gluons to $|q\overline{q}⟩$ and $|s\overline{s}⟩$, while (b) denotes the diagrams for two gluons to $|gg⟩$.