Long-Distance Contributions to Light Vector Mesons and Radiative Decays of $η_{c}$

Abstract: In the present work, we investigate the long-distance contributions to light hadrons and radiative decays of $η<em>c$, where the long-distance contribution is phenomenologically described by the meson loops. Our estimations indicate that the branching fractions of $η_c \to \mathcal{VV},\ \mathcal{V}=(ρ,ω, φ)$ can be roughly reproduced in the range $1.4<α<1.6$. While the parameter range for $η_c \to γ\mathcal{V}$ is determined by the similarity of $η_c \to γ\mathcal{V}/\mathcal{VV}$ and $χ</em>{c1} \to γ\mathcal{V}/\mathcal{VV}$. In this parameter range, our estimations indicate that the branching fractions for the processes $η_c \to γρ<sup>0$ and $η_c \to γω$ are of order $10<sup>{-4}$, while the one for the process $η_c \to γφ$ is of order $10<sup>{-5}$. The estimations in the present work are expected to be accessible by future experiments at BESIII and Belle II.