Strong decays of the isovector-scalar $D^{*}{\overline{D}}^{*}$ hadronic molecule

We adopt the effective Lagrangian approach to study the strong decays of the $1^{-}\left(0^{++}\right)D^{*}{\overline{D}}^{*}$ molecular state [denoted as $T_{\psi 0}^a\left(4010\right)$ according to the LHCb naming convention] through triangle diagrams. The decay channels include the open-charm $D\overline{D}$, and the hidden-charm ${\eta }_c\pi$, $J/\psi \rho$, and ${\chi }_{c1}\pi$. The coupling between the $T_{\psi 0}^a\left(4010\right)$ and its constituents $D^{*}{\overline{D}}^{*}$ is obtained by solving for the residue of the scattering $T$-matrix at the pole. Our calculations yield a total width of few tens MeV for the $T_{\psi 0}^a\left(4010\right)$ state using three different form factors, with its main decay channels being ${\eta }_c\pi$ and ${\chi }_{c1}\pi$. The $X\left(4100\right)$ and $X\left(4050\right)$ have similar masses and widths, with both masses being close to the $D^{*}{\overline{D}}^{*}$ threshold. Additionally, their decay final states are consistent with those of the $T_{\psi 0}^a\left(4010\right)$. Therefore, it is likely that they represent the same state and both potentially correspond to the $T_{\psi 0}^a\left(4010\right)$. We suggest that future experiments focus on searching for the $T_{\psi 0}^a\left(4010\right)$ signal in the final states ${\eta }_c{\pi }^{-}$, ${\chi }_{c1}{\pi }^{-}$, and $D^0{D}^{-}$ of the $B^0\to {\eta }_c{\pi }^{-}{K}^{+}$, ${\chi }_{c1}{\pi }^{-}{K}^{+}$, and $D^0{D}^{-}{K}^{+}$ processes, respectively, as well as further investigating its resonance parameters with Flatté-like formula.