This study investigates the three-body decay process $B^+ \to D^{ *+} D^- K^+$, aiming to explore the possible origins of $T^*_{ \bar{ c}\bar{ s}0}(2870)^0$ and $\chi_{ c1}(3872)$ as intermediate states. Within the molecular state framework, $T^*_{ \bar{ c}\bar{ s}0}(2870)^0$ and $\chi_{ c1}(3872)$ are considered as possible $\bar{ D}^{ *}K^{ }$ and $D^*\bar{ D}$ molecular states, respectively. Using effective Lagrangians, the interaction kernels of the $\bar{ D}^{ *}K^{ *}$ and $D^*\bar{ D}$ systems are constructed within the one-boson-exchange model. The corresponding rescattering amplitudes and pole positions are obtained by solving the quasipotential Bethe-Salpeter equation. These amplitudes are incorporated into the decay amplitude of the three-body process, and the $D^-K^+$ and $D^{ *+}D^-$ invariant mass spectra are simulated via Monte Carlo methods. To better reproduce the experimental data, additional Breit-Wigner contributions from $T^*_{ \bar{ c}\bar{ s}1}(2900)^0$, $\chi_{ c1}(4010)$, and $h_c(4300)$ are included. The results show a pronounced enhancement near 2900 MeV in the $D^-K^+$ invariant mass spectrum, strongly supporting the interpretation of $T^*_{ \bar{ c}\bar{ s}0}(2870)^0$ as a $\bar{ D}^{ *}K^{ *}$ molecular state. While the $\bar{ D}^{ *}K^{ *}$ molecular state provides a reasonable contribution to the $D^-K^+$ spectrum, the $D^*\bar{ D}$ molecular state yields no significant effect on either the $D^-K^+$ or $D^{ *+}D^-$ distributions. This suggests that the observed $\chi_{ c1}(3872)$ structure around 3872 MeV may not be interpreted as a $D^*\bar{ D}$ molecular state.
Categories: hep-ph