$X_0(2900)$ and $\chi_{c0}(3930)$ in process $B^+\to D^+ D^- K^+$

This study investigates the nature of the $X_0(2900)$ and $\chi_{c0}(3930)$ based on experimental results of the process $B^+$$\to$$D^+ D^- K^+$. We focus on the S-wave $D^{*-}K^{*+}$ and $D_s^+D_s^-$ molecular states, which can be related to the $X_0(2900)$ and $\chi_{c0}(3930)$, respectively. Using effective Lagrangians, we construct the potential kernel of the $D^{*-}K^{*+}$$-$$D^{-}K^{+}$ and $D_s^+D_s^-$$-$$D^+D^-$ interactions with a one-boson-exchange model, and determine the scattering amplitudes and their poles through a quasipotential Bethe-Salpeter equation approach. By incorporating the potential kernel into the three-body decay process $B^+$$\to$$D^+ D^- K^+$, we evaluate the $D^-K^+$ and $D^+D^-$ invariant mass spectra, as well as the Dalitz plot, with Monte Carlo simulation. A satisfactory fit to the $D^-K^+$ and $D^+D^-$ invariant mass spectra is achieved after introducing additional Breit-Wigner resonances, the $X_1(2900)$, $\psi(3770)$, and $\chi_{c2}(3930)$ states. Prominent signals of the $X_0(2900)$ and $\chi_{c0}(3930)$ states appear as peaks in the $D^-K^+$ and $D^+D^-$ invariant mass spectra near 2900 and 3930 MeV, respectively. Clear event concentration of the $X_0(2900)$ and $\chi_0(3930)$ states is evident as strips in the Dalitz plot. The results suggest that both $X_0(2900)$ and $\chi_{c0}(3930)$ can be interpreted as molecular states, with the inclusion of $X_1(2900)$ and $\chi_2(3930)$ necessary to describe structures in the regions near 2900 and 3930 MeV.