Hadronic loop effects on the radiative decays of the first radial excitations of $\eta$ and ${\eta }^{\mathrm{\prime }}$

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