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

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 $J\mathrm{/}\psi \to \gamma \eta {\pi }^0$, $J\mathrm{/}\psi \to \gamma {\pi }^{+}{\pi }^{-}$ and $J\mathrm{/}\psi \to \gamma {\pi }^0{\pi }^0$ radiative decays through the final-state interactions. We obtain the corresponding branching fractions $Br(J/\psi\to \gamma a_0(980) \to \gamma\eta\pi^0) = (0.47\pm0.05) \times 10^{-7}$,$Br(J/\psi\to \gamma f_0(980) \to \gamma\pi^+\pi^-) = 0.37 \times 10^{-7} - 1.98 \times 10^{-6}$,$Br(J/\psi\to \gamma f_0(980) \to \gamma\pi^0\pi^0) = 0.18 \times 10^{-7} - 9.92 \times 10^{-7}$,andpredict$Br(J/\psi\to \gamma a_0(980)) = 1.72 \times 10^{-8} - 3.07\times 10^{-7}$and$Br(J/\psi\to \gamma f_0(980)) = 1.86 \times 10^{-8} - 1.89\times 10^{-5}$. These fractions are within the upper limits of the experimental measurements.