Eur. Phys. J. C (2024) 84: 822
https://doi.org/10.1140/epjc/s10052-024-13214-6

Regular Article – Theoretical Physics

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

Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, 210097, Nanjing, China

^c junhe@njnu.edu.cn

Received: 23 July 2024
Accepted: 6 August 2024
Published online: 17 August 2024

Abstract

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 $D_{}^{\ast -}K_{}^{\ast +}$ 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 $D_{}^{\ast -}K_{}^{\ast +}$–$D^{-}{K}^{+}$ 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, using Monte Carlo simulation. A satisfactory fit to the $D^{-}{K}^{+}$ and $D^{+}{D}^{-}$ invariant mass spectra is achieved after introducing additional Breit–Wigner resonances, $X_1(2900)$, $\psi (3770)$, and ${\chi }_{c2}(3930)$. 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 from the $X_0(2900)$ and ${\chi }_0(3930)$ 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 $D^{-}{K}^{+}$ and $D^{+}{D}^{-}$ invariant mass spectra, respectively.