Investigation of $J/\psi \rightarrow \gamma \, \pi ^0 \eta (\pi ^+\pi ^-, \pi ^0\pi ^0)$ radiative decays including final-state interactions

Abstract

We revisit the coupled-channel $K\bar{K}$ interactions and dynamically generate the resonances $f_0(980)$ and $a_0(980)$ within both the isospin and the physical bases. The $f_0(980)$–$a_0(980)$ mixing effects are generated in the scattering amplitudes of the coupled channels with the physical basis, which exploits the important role of the $K\bar{K}$ channel in the dynamical nature of these resonances. With the scattering amplitudes obtained, we investigate the $f_0(980)$ and $a_0(980)$ contributions to the $J/\psi \rightarrow \gamma \eta \pi ^0$, $J/\psi \rightarrow \gamma \pi ^+\pi ^-$ and $J/\psi \rightarrow \gamma \pi ^0\pi ^0$ radiative decays through the final-state interactions. We obtain the corresponding branching fractions $\mathrm{Br}(J/\psi \rightarrow \gamma a_0(980) \rightarrow \gamma \eta \pi ^0) = (0.48\pm 0.03) \times 10^{-7}$, $\mathrm{Br}(J/\psi \rightarrow \gamma f_0(980) \rightarrow \gamma \pi ^+\pi ^-) = (0.52-2.08) \times 10^{-7}$, $\mathrm{Br}(J/\psi \rightarrow \gamma f_0(980) \rightarrow \gamma \pi ^0\pi ^0) = (0.26-1.04) \times 10^{-7}$, and predict $\mathrm{Br}(J/\psi \rightarrow \gamma a_0(980)) = (1.24 - 1.61) \times 10^{-7}$ and $\mathrm{Br}(J/\psi \rightarrow \gamma f_0(980)) = (0.69 - 4.00) \times 10^{-7}$. These fractions are within the upper limits of the experimental measurements.