Long-Distance Contributions to Light Vector Mesons and Radiative Decays of ${\eta }_c$

In the present work, we investigate the long-distance contributions to light hadrons and radiative decays of ${\eta }_c$, where the long-distance contribution is phenomenologically described by the meson loops. Our estimations indicate that the branching fractions of ${\eta }_c\to \mathcal{V}\mathcal{V},\mathcal{V}=(\rho ,\omega ,\phi )$ can be roughly reproduced in the range $1.4<\alpha <1.6$. While the parameter range for ${\eta }_c\to \gamma \mathcal{V}$ is determined by the similarity of ${\eta }_c\to \gamma \mathcal{V}\mathrm{/}\mathcal{V}\mathcal{V}$ and ${\chi }_{c1}\to \gamma \mathcal{V}\mathrm{/}\mathcal{V}\mathcal{V}$. In this parameter range, our estimations indicate that the branching fractions for the processes ${\eta }_c\to \gamma {\rho }^0$ and ${\eta }_c\to \gamma \omega$ are of order ${10}^{-4}$, while the one for the process ${\eta }_c\to \gamma \phi$ is of order ${10}^{-5}$. The estimations in the present work are expected to be accessible by future experiments at BESIII and Belle II.