Abstract
The $P$-wave charmonium decays $h_{c}\rightarrow\gamma\eta^{(\prime)}$ 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 $h_{c}\rightarrow\gamma\eta^{(\prime)}$. 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 $\phi=39.3^{\circ}\pm1.0^{\circ}$ as input, we find that the predicted ratio $R_{h_{c}}=\mathcal{B}(h_{c}\rightarrow\gamma\eta)/\mathcal{B}(h_{c}\rightarrow\gamma\eta^{\prime})$ is much smaller than the experiment measurement. While, with $\phi=33.5^{\circ}\pm0.9^{\circ}$ 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 $R_{h_{c}}^{exp}=(30.7\pm11.3\pm8.7)\%$. As a cross-check, the mixing angle $\phi=33.8^{\circ}\pm2.5^{\circ}$ is extracted by employing the ratio $R_{h_{c}}$, and a brief discussion on the difference in the determinations of $\phi$ is given.
Highlights
The hadronic decays of charmonia have played important roles for our understanding of QCD, especially the interplay of perturbative QCD and nonperturbative QCD [1,2,3,4], since the first charmonium-state J=ψ was observed [5,6]
As an illustration, using the Feldmann-Kroll-Stech result about the mixing angle φ 1⁄4 39.3° Æ 1.0° as input, we find that the predicted ratio Rhc 1⁄4 Bðhc → γηÞ=Bðhc → γη0Þ is much smaller than the experiment measurement, while, with φ 1⁄4 33.5° Æ 0.9° extracted from the asymptotic limit of the γÃγ − η0 transition form factor, we obtain Rhc 1⁄4 30.3%, consistent with Rehxcp 1⁄4 ð30.7 Æ 11.3 Æ 8.7Þ%
We have revisited the P-wave charmonium radiative decays hc → γηð0Þ in the B-S formalism, where the internal momentum of hc has been retained in both the soft wave function ψðq Þ and the hard-scattering amplitude Oðq Þ
Summary
The hadronic decays of charmonia have played important roles for our understanding of QCD, especially the interplay of perturbative QCD and nonperturbative QCD [1,2,3,4], since the first charmonium-state J=ψ was observed [5,6]. It is found that the calculations are IR safe and the predicted branching ratios Bðhc → γηð0ÞÞ are much smaller than the experimental measurements This indicates that the relativistic corrections or/and the contributions from the higher Fock-state of hc are highly significant, while from the point of view of NRQCD, the next-to-leading-order Fock-state contributions are suppressed by a relative factor v2ccαs in the decays hc → γηð0Þ [65]. For the contributions of the gluonic content of ηð0Þ in the decays hc → γηð0Þ, the next-to-leading-order effects related to the internal momentum are not substantially suppressed in the major region of the wave function of hc, and the corresponding relativistic corrections are extremely important.
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