Revisiting the $P$-wave charmonium radiative decays $h_c\to \gamma {\eta }^{(\prime )}$ with relativistic corrections

Abstract

The $P$-wave charmonium decays $h_c\to \gamma {\eta }^{(\prime )}$ are revisited by taking into account relativistic corrections. The decay amplitudes are derived in the Bethe-Salpeter formalism, in which the involved one-loop integrals are evaluated analytically. Intriguingly, from both the quark-antiquark content and the gluonic content of ${\eta }^{(\prime )}$, the relativistic corrections make significant contributions to the decay rates of $h_c\to \gamma {\eta }^{(\prime )}$. By comparison with the leading-order contributions from the quark-antiquark content (one-loop level), the ones from the gluonic content (tree level) are also important, which is compatible with the conclusion obtained without relativistic corrections. Usually, for $\eta$ production processes, the predicted branching ratios are sensitive to the angle of $\eta -{\eta }^{\prime }$ mixing. As an illustration, using the Feldmann-Kroll-Stech result about the mixing angle $\varphi =39.3^{\circ }\pm 1.0^{\circ }$ as input, we find that the predicted ratio $R_{h_c}=\mathcal{B}(h_c\to \gamma \eta )/\mathcal{B}(h_c\to \gamma {\eta }^{\prime })$ is much smaller than the experiment measurement. While, with $\varphi =33.5^{\circ }\pm 0.9^{\circ }$ extracted from the asymptotic limit of the ${\gamma }^{\ast }\gamma -{\eta }^{\prime }$ transition form factor, we obtain $R_{h_c}=30.3\%$ in consistent with $R_{h_c}^{exp}=(30.7\pm 11.3\pm 8.7)\%$. As a cross-check, the mixing angle $\varphi =33.8^{\circ }\pm 2.5^{\circ }$ is extracted by employing the ratio $R_{h_c}$, and a brief discussion on the difference in the determinations of $\varphi$ is given.

Comments: 34 pages, 4 figures and 7 tables