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

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 $B^{+}\to D_s^{+}{D}_s^{-}{K}^{+}$. 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 $\mathcal{M}\to J/\psi \varphi$. 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 $\mathcal{M}J/\psi \varphi$. 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 $X$.

Comments: 11 Pages, 4 Figures and 1 Table