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We revisit the coupled channel $K\bar{K}$ interactions and dynamicallygenerate the resonances $f_0(980)$ and $a_0(980)$ within both the isospin andthe physical bases. The $f_0(980)-a_0(980)$ mixing effects are generated in thescattering amplitudes of the coupled channels with the physical basis, whichexploits the important role of the $K\bar{K}$ channel in the dynamical natureof these resonances. With the scattering amplitudes obtained, we investigatethe $f_0(980)$ and $a_0(980)$ contributions to the $J/\psi\to \gamma\eta\pi^0$,$J/\psi\to \gamma\pi^+\pi^-$ and $J/\psi\to \gamma\pi^0\pi^0$ radiative decaysthrough the final-state interactions. We obtain the corresponding branchingfractions $Br(J/\psi\to \gamma a_0(980) \to \gamma\eta\pi^0) = (0.47\pm0.05)\times 10^{-7}$, $Br(J/\psi\to \gamma f_0(980) \to \gamma\pi^+\pi^-) = 0.37\times 10^{-7} - 1.98 \times 10^{-6}$, $Br(J/\psi\to \gamma f_0(980) \to\gamma\pi^0\pi^0) = 0.18 \times 10^{-7} - 9.92 \times 10^{-7}$, and predict$Br(J/\psi\to \gamma a_0(980)) = 1.72 \times 10^{-8} - 3.07\times 10^{-7}$ 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.

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