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Investigation of $J/ψ\to γ\, π^0 η(π^+π^-, π^0π^0)$ 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 $J/ψ\to γηπ^0$, $J/ψ\to γπ^+π^-$ and $J/ψ\to γπ^0π^0$ radiative decays through the final-state interactions. We obtain the corresponding branching fractions $Br(J/ψ\to γa_0(980) \to γηπ^0) = (0.47\pm0.05) \times 10^{-7}$, $Br(J/ψ\to γf_0(980) \to γπ^+π^-) = 0.37 \times 10^{-7} - 1.98 \times 10^{-6}$, $Br(J/ψ\to γf_0(980) \to γπ^0π^0) = 0.18 \times 10^{-7} - 9.92 \times 10^{-7}$, and predict $Br(J/ψ\to γa_0(980)) = 1.72 \times 10^{-8} - 3.07\times 10^{-7}$ and $Br(J/ψ\to γf_0(980)) = 1.86 \times 10^{-8} - 1.89\times 10^{-5}$. These fractions are within the upper limits of the experimental measurements.