Radiative decays of $X(3872)$ within $D{\bar D}^*$ molecular framework

Within the framework of nonrelativistic effective field theory, we calculate the radiative decay width of the process $X(3872) \to \bar{D} D \gamma$ while taking into account the $D\bar{D}$ final-state interactions. In this work, the $X(3872)$, with the spin-parity quantum numbers $J^{PC}=1^{++}$, is treated as a $D^*\bar{D} +\rm c.c.$ bound state comprising equal proportions of neutral and charged components. Our numerical calculations predict the tree-level partial decay width of approximately $11.0$ keV for the decay process $X(3872) \to \bar{D}^0 D^0\gamma$, while the partial width for $X(3872) \to D^- D^+\gamma$ is less than $1.0$ keV. It is found that the $D\bar{D}$ rescattering effect enhances the tree-level width of $X(3872) \to \bar{D}^0 D^0\gamma$ by $6\%$. In contrast, the rescattering effect makes a suppression to the charged channel $X(3872)\to D^- D^+\gamma$ by roughly $38\%$. We expect that the present predictions based on the molecular picture of the $X(3872)$ can be tested by future experiments.