Hadronic loop effects on the radiative decays of the first radial excitations of $\eta$ and $\eta'$

Based on the one state assumption of $\eta(1405)$ and $\eta(1475)$, thus, $\eta(1295)$ and $\eta(1405/1475)$ are organized as the first radial excitations of $\eta$ and $\eta'$, respectively, we investigate the productions and radiative decays of these two states in $J/\psi\to \gamma\eta_X\to \gamma\gamma V$, where $\eta_X$ stands for $\eta(1295)$ and $\eta(1405/1475)$ and $V$ for vector mesons $\rho^0, \ \omega, \ \phi$. As we have learned from previous studies that the hadronic decays of these two states receive important contributions from the intermediate $\bar{K}K^*+c.c.$ meson loops due to the triangle singularity mechanism, we show that some measurable effects can also arise from the $\bar{K}K^*+c.c.$ meson loops in their radiative decays. Our calculation shows that the impact of the $\bar{K}K^*+c.c.$ meson loops on the $\eta(1405/1475)$ radiative decays is relatively smaller than on $\eta(1295)$ since the latter has a much larger coupling to $\bar{K}K^*+c.c.$ However, the production of $\eta(1295)$ in the $J/\psi$ radiative decays will be strongly suppressed. As a consequence of the $\bar{K}K^*+c.c.$ meson loop contributions, we find that the mixing angle extracted in the radiative decays of $\eta(1295)$ and $\eta(1405/1475)$ will be different from each other, and both are different from the one determined in other processes.