Near-threshold structures in the $D_{s}^{+}D_{s}^{-}$ mass distribution of the decay $B^{+}\rightarrow D_{s}^{+}D_{s}^{-}K^{+}$

Abstract

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^{+}\rightarrow D_{s}^{+}D_{s}^{-}K^{+}$. The first of them is the resonance $X(3960)$, whereas the second one, $X_0(4140)$, is a structure with the mass around $4140~\mathrm{MeV}$. To explore their natures and model them, we study the hadronic molecule $\mathcal{M}=D_s^{+}D_s^{-}$ 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 $\mathcal{M} \to D_{s}^{+}D_{s}^{-}$, $\mathcal{M} \to\eta_{c}\eta^{(\prime)}$, and $\mathcal{ M} \to J/\psi\phi$. Partial widths of these decays are determined by the strong couplings $g_i, \ i=1,2,3,4 $ at vertices $\mathcal{M} D_{s}^{+}D_{s}^{-}$, $\mathcal{M}\eta_{c} \eta^{(\prime)}$, and $\mathcal{M} J/\psi\phi$. They are computed by means of the three-point sum rule method. Predictions for the mass $m=(4117 \pm 85)~\mathrm{MeV}$ and width $\Gamma_{ \mathcal{M}}=(62 \pm 12)~\mathrm{MeV}$ of the molecule $\mathcal{M}$ are compared with the corresponding LHCb data, and also with our results for the diquark-antidiquark state $X=[cs][\overline{c}\overline{s}]$. We argue that the structure $X_0(4140)$ may be interpreted as the hadronic molecule $ D_s^{+}D_s^{-}$, whereas the resonance $X(3960)$ can be identified with the tetraquark $X$.