Combined analysis on nature of $X(3960)$, $\chi_{c0}(3930)$, and $X_0(4140)$

In this work, a study of the $D_s\bar{D}_s$ interaction and its couplings to the channels $D\bar{D}$ and $J/\psi \phi$ is performed in a quasipotential Bethe-Salpeter equation approach. The $D_s^+{D}_s^-$ and $D^+{D}^-$ invariant mass spectra in three-body $B$ decays are investigated in order to understand the origin of $X(3960)$, $\chi_{c0}(3930)$, and $X_0(4140)$ structures reported at LHCb. With the help of effective Lagrangians, the potential kernel can be constructed with meson exchanges, from which the scattering amplitudes can be obtained. By inserting it into the three-body decay processes, the invariant mass spectra can be calculated with an additional Breit-Wigner resonance introduced. The $D_s^+{D}_s^-$ invariant mass spectrum in the decay process $B^+\to D^+_sD^-_sK^+$ is well reproduced, and the $X(3960)$ structure can be explained as a molecular state from the $D_s\bar{D}_s$ interaction. The state also exhibits as a peak in the $D^+{D}^-$ invariant mass spectrum in the decay $B^+\to D^+D^-K^+$, however, it is too narrow to reproduce the experimental structure around 3930~MeV. The dip around 4140~MeV in the $D_s^+{D}_s^-$ invariant mass spectrum can be reproduced by the additional resonance with interference effect. However, a small bump instead of a dip is produced from the $D_s\bar{D}_s-J/\psi \phi$ coupled-channel effect, which suggests that the $X_0(4140)$ cannot be interpreted as the $D_s\bar{D}_s-J/\psi \phi$ coupling in the current model.