Near-threshold structures in the ${D}_{s}^{+}{D}_{s}^{-}$ mass distribution of the decay ${B}^{+}\to {D}_{s}^{+}{D}_{s}^{-}{K}^{+}$

Two near-threshold peaking structures with spin-parities ${J}^{\mathrm{PC}}={0}^{++}$ were recently discovered by the LHCb Collaboration in the ${D}_{s}^{+}{D}_{s}^{-}$ invariant mass distribution of the decay ${B}^{+}\to {D}_{s}^{+}{D}_{s}^{-}{K}^{+}$. The first of them is the resonance $X\left(3960\right)$, whereas the second one, ${X}_{0}\left(4140\right)$, is a structure with the mass around 4140 MeV. To explore their natures and model them, we study the hadronic molecule $M={D}_{s}^{+}{D}_{s}^{-}$ and calculate its mass, current coupling, and width. The mass and current coupling of the molecule are extracted from the QCD two-point sum rule analyses by taking into account vacuum condensates up to dimension 10. To evaluate its full width, we consider the processes $M\to {D}_{s}^{+}{D}_{s}^{-}$, $M\to {\eta }_{c}{\eta }^{\left(\prime \right)}$, and $M\to J/\psi \varphi $. Partial widths of these decays are determined by the strong couplings ${g}_{i},i=1$, 2, 3, 4 at vertices $M{D}_{s}^{+}{D}_{s}^{-}$, $M{\eta }_{c}{\eta }^{\left(\prime \right)}$, and $MJ/\psi \varphi $. They are computed by means of the three-point sum rule method. Predictions for the mass $m=\left(4117±85\right)\text{}\text{}\mathrm{MeV}$ and width ${\Gamma }_{M}=\left(62±12\right)\text{}\text{}\mathrm{MeV}$ of the molecule $M$ are compared with the corresponding LHCb data, and also with our results for the diquark-antidiquark state $X=\left[cs\right]\left[\overline{c}\overline{s}\right]$. We argue that the structure ${X}_{0}\left(4140\right)$ may be interpreted as the hadronic molecule ${D}_{s}^{+}{D}_{s}^{-}$, whereas the resonance $X\left(3960\right)$ can be identified with the tetraquark $X$.