# Revisiting the radiative decays $J/\psi \rightarrow \gamma\eta^{(\prime)}$ in perturbative QCD

## TL;DR

This paper reevaluates the radiative decays of J/ψ to η and η′ mesons within perturbative QCD, analyzing loop integrals and mixing parameters to reconcile theoretical predictions with experimental data.

## Contribution

The study provides a detailed analytical evaluation of loop integrals in J/ψ radiative decays, examining the impact of η-η′ mixing angles and nonperturbative matrix elements on decay ratios.

## Key findings

- Loop integrals are insensitive to light quark masses.
- Branching ratios are unaffected by η(′) distribution amplitude shapes.
- Using specific mixing angles, theoretical ratios agree with experimental results.

## Abstract

In the framework of perturbative QCD, the radiative decays $J/\psi\rightarrow\gamma\eta^{(\prime)}$ are revisited in detail, where the involved one-loop integrals are evaluated analytically with the light quark masses kept. We have found that the sum of loop integrals is insensitive to the light quark masses and the branching ratios $\mathcal{B}(J/\psi\rightarrow\gamma\eta^{(\prime)})$ barely depend on the shapes of $\eta^{(\prime)}$ distribution amplitudes. With the parameters of $\eta-\eta^{\prime}$ mixing extracted from low energy processes and $J/\psi\rightarrow\gamma\eta^{(\prime)}$ by means of nonperturbative matrix elements $\langle0|G_{\mu\nu}^a\tilde{G}^{a,\mu\nu}|\eta^{(\prime)}\rangle$ based on $U_{A}(1)$ anomaly dominance argument, we could not give the ratio $R_{J/\psi}$ in agreement with experimental result. However, using the parameters, especially the mixing angle $\phi=33.5^{\circ}\pm0.9^{\circ}$, extracted from $\gamma^{\ast}\gamma-\eta^{\prime}$ transition form factor measured at $q^{2}=112~\mathrm{GeV}^{2}$ by BaBar collaboration, we obtain $R_{J/\psi}=4.70$ in good agreement with $R_{J/\psi}^{exp}=4.65\pm0.21$. As a crossing check, with $\Gamma^{exp}(\eta^{(\prime)}\rightarrow\gamma\gamma)$ and our results for $J/\psi\rightarrow\gamma\eta^{(\prime)}$, we get $\phi=33.9^{\circ}\pm0.6^{\circ}$. The difference between the determinations of $\phi$ is briefly discussed.