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In this work, a study of the $D_s\bar{D}_s$ interaction and its couplings tothe channels $D\bar{D}$ and $J/\psi \phi$ is performed in a quasipotentialBethe-Salpeter equation approach. The $D_s^+{D}_s^-$ and $D^+{D}^-$ invariantmass spectra in three-body $B$ decays are investigated in order to understandthe origin of $X(3960)$, $\chi_{c0}(3930)$, and $X_0(4140)$ structures reportedat LHCb. With the help of effective Lagrangians, the potential kernel can beconstructed with meson exchanges, from which the scattering amplitudes can beobtained. By inserting it into the three-body decay processes, the invariantmass spectra can be calculated with an additional Breit-Wigner resonanceintroduced. 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 beexplained as a molecular state from the $D_s\bar{D}_s$ interaction. The statealso 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 experimentalstructure around 3930~MeV. The dip around 4140~MeV in the $D_s^+{D}_s^-$invariant mass spectrum can be reproduced by the additional resonance withinterference effect. However, a small bump instead of a dip is produced fromthe $D_s\bar{D}_s-J/\psi \phi$ coupled-channel effect, which suggests that the$X_0(4140)$ cannot be interpreted as the $D_s\bar{D}_s-J/\psi \phi$ coupling inthe current model.

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