Radiative decays of $h_c$ to the light mesons ${\eta }^{(\prime )}$: A perturbative QCD calculation

Abstract

We study the radiative decays $h_c\to \gamma {\eta }^{(\prime )}$ in the framework of perturbative QCD and evaluate analytically the one-loop integrals with the light quark masses kept. Interestingly, the branching ratios $\mathcal{B}(h_c\to \gamma {\eta }^{(\prime )})$ are insensitive to both the light quark masses and the shapes of ${\eta }^{(\prime )}$ distribution amplitudes. And it is noticed that the contribution of the gluonic content of ${\eta }^{(\prime )}$ is almost equal to that of the quark-antiquark content of ${\eta }^{(\prime )}$ in the radiative decays $h_c\to \gamma {\eta }^{(\prime )}$. By employing the ratio $R_{h_c}=\mathcal{B}(h_c\to \gamma \eta )/\mathcal{B}(h_c\to \gamma {\eta }^{\prime })$, we extract the mixing angle $\varphi =33.8^{\circ }\pm 2.5^{\circ }$, which is in clear disagreement with the Feldmann-Kroll-Stech result $\varphi =39.0^{\circ }\pm 1.6^{\circ }$ extracted from the ratio $R_{J/\psi }$ with nonperturbative matrix elements $⟨0\mid G_{\mu \nu }^a{\tilde{G}}^{a,\mu \nu }\mid {\eta }^{(\prime )}⟩$, but in consistent with $\varphi =33.5^{\circ }\pm 0.9^{\circ }$ extracted from the asymptotic limit of the ${\gamma }^{\ast }\gamma -{\eta }^{\prime }$ transition form factor and $\varphi =33.9^{\circ }\pm 0.6^{\circ }$ extracted from $R_{J/\psi }$ in perturbative QCD. We also briefly discuss possible reasons for the difference in the determinations of the mixing angle.

Comments: 14 pages, 6 figures and 5 tables