The role of the short-distance interaction in $e^+e^-\to \gamma X(3872)$

In this study, we analyze the cross-section data from the $e^+e^-\to \gamma J/\psi\omega$ process to explore both short-distance and long-distance interactions for the radiative transition $Y(4200)\to \gamma X(3872)$. We investigate the short-distance effects through the E1 transition among the $c\bar{c}$ components, and the long-distance effects via hadronic loop diagrams. Our numerical analysis reveals that short-distance interactions play a significantly larger role in the radiative transition than that of the long-distance interactions. This finding underscores the importance of the compact $c\bar{c}$ components in both the initial and final states for accurately understanding the cross-section $\sigma[e^+e^-\to \gamma J/\psi\omega]$. Furthermore, with the help of relative branch ratio $\mathcal{R}$ we estimate the $\Gamma[Y(4200)\to\gamma X(3872)]$ implied in the the experimental study. Finally, we also discuss the possible existence of $\psi(4040)$ signal within the cross-section data.