Strong decays of the isovector-scalar D^̄^ hadronic molecule

We adopt the effective Lagrangian approach to study the strong decays of the 1⁻(0⁺⁺) D^̄^ molecular state [denoted as Tψ0ᵃ(4010) according to the LHCb naming convention] through triangle diagrams. The decay channels include the open-charm DD̄, and the hidden-charm ηcπ, J/ψρ, and χc1π. The coupling between the Tψ0ᵃ(4010) and its constituents D^̄^ is obtained by solving for the residue of the scattering T-matrix at the pole. Our calculation yields a total width of $(12.0$$-$$35.4)$ MeV for the Tψ0ᵃ(4010) state, with its main decay channels being ηcπ and χc1π. The $X(4100)$ and $X(4050)$ have similar masses and widths, with both masses being close to the D^̄^ threshold. Additionally, their decay final states are consistent with those of the Tψ0ᵃ(4010). Therefore, it is likely that they represent the same state and both potentially correspond to the Tψ0ᵃ(4010). We suggest that future experiments focus on searching for the Tψ0ᵃ(4010) signal in the final states ηcπ⁻, χc1π⁻ and D⁰D⁻ of the B⁰→ηcπ⁻K⁺, χc1π⁻ K⁺ and D⁰D⁻K⁺ processes, respectively, as well as further investigating its resonance parameters with Flatt\'e-like formula.