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

The dipionic transition of the $X(3872)$ to the $\eta_c$ was investigated using an effective Lagrangian approach. In this study, the $X(3872)$ was assumed to be a $D\bar{D}^\ast + \text{c.c.}$ bound state with the quantum numbers $J^{PC}=1^{++}$ and to decay via triangle and box loops. It is found that the partial decay widths arising from the box loops are one order of magnitude greater than those from the triangle ones. The decay widths are model-dependent, as characterized by the cutoff parameter introduced in the form factor. Moreover, the charged and neutral $D\bar{D}^\ast$ configurations of the $X(3872)$ and its mass also influence the decay widths. With our model parameters, the decay widths for the $X(3872)\to\pi^+\pi^-\eta_c$ can reach up to several tens of keV. We hope that the current calculations within the molecular interpretation will be helpful for the future experiments.