Strong decays of the isovector-scalar $D^\ast\bar{D}^\ast$ hadronic molecule

We adopt the effective Lagrangian approach to study the strong decays of the $1^-(0^{++})$ $D^\ast\bar{D}^\ast$ molecular state [denoted as $T_{\psi0}^a(4010)$ according to the LHCb naming convention] through triangle diagrams. The decay channels include the open-charm $D\bar{D}$, and the hidden-charm $\eta_c\pi$, $J/\psi\rho$, and $\chi_{c1}\pi$. The coupling between the $T_{\psi0}^a(4010)$ and its constituents $D^\ast\bar{D}^\ast$ 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 $T_{\psi0}^a(4010)$ state using three different form factors, with its main decay channels being $\eta_c\pi$ and $\chi_{c1}\pi$. The $X(4100)$ and $X(4050)$ have similar masses and widths, with both masses being close to the $D^\ast\bar{D}^\ast$ threshold. Additionally, their decay final states are consistent with those of the $T_{\psi0}^a(4010)$. Therefore, it is likely that they represent the same state and both potentially correspond to the $T_{\psi0}^a(4010)$. We suggest that future experiments focus on searching for the $T_{\psi0}^a(4010)$ signal in the final states $\eta_c\pi^-$, $\chi_{c1}\pi^-$ and $D^0D^-$ of the $B^0\to\eta_c\pi^-K^+$, $\chi_{c1}\pi^- K^+$ and $D^0D^-K^+$ processes, respectively, as well as further investigating its resonance parameters with Flatt\'e-like formula.