Study of the $B^{-}\to K^{-}\eta {\eta }_c$ decay due to the $D\overline{D}$ bound state

We study the $B^{-}\to K^{-}\eta {\eta }_c$ decay by taking into account the $S$-wave contributions from the pseudoscalar meson-pseudoscalar meson interactions within the unitary coupled-channel approach, where the $D\overline{D}$ bound state is dynamically generated. In addition, the contribution from the intermediate resonance $K_0^{*}(1430{)}^{-}$, with $K_0^{*}(1430{)}^{-}\to K^{-}\eta$, is also considered. Our results show that there is a clear peak around 3720 MeV in the $\eta {\eta }_c$ invariant mass distribution, which could be associated with the $D\overline{D}$ bound state. The future precise measurements of the $B^{-}\to K^{-}\eta {\eta }_c$ process at the Belle II and LHCb experiments could be, therefore, used to check the existence of the $D\overline{D}$ bound state, and to deepen our understanding of the hadron-hadron interactions.

Figure 1

The dominant quark-level diagrams for the $B^{-}\to K^{-}\eta {\eta }_c$ decay, where (a)–(d) refer to the internal and external $W^{-}$ emission mechanisms, respectively.

Figure 2

The final-state interactions of the coupled channels $D^0{\overline{D}}^0$, $D^{+}{D}^{-}$, $D_s^{+}{D}_s^{-}$, and ${\eta }^{\prime }{\eta }_c$.

Figure 3

The Dalitz plot for the $B^{-}\to K^{-}\eta {\eta }_c$ decay. The green dash-dotted line and band stand for the mass and width of $X(3700)$, while the blue dashed line and band for the mass and width of the well-established resonance $K_0^{*}(1430)$.

Figure 4

The diagram for the $B^{-}\to K^{-}\eta {\eta }_c$ decay via the intermediate $K_0^{*}(1430)$ resonance.

Figure 5

The $\eta {\eta }_c$ (a), $K^{-}\eta$ (b) and $K^{-}{\eta }_c$ (c) invariant mass distributions of the $B^{-}\to K^{-}\eta {\eta }_c$ decay with $\beta =0.012$, $\phi =0$, and $C=3.0$. The green dash-dotted, the magenta dashed, and the red solid curves represent the contributions from $X(3700)$, $K_0^{*}(1430)$, and the total contributions, respectively.

Figure 6

The doubly differential decay width ${\mathrm{d}}^2\mathrm{\Gamma }/(\mathrm{d}{M}_{\eta {\eta }_c}\mathrm{d}{M}_{K^{-}\eta })$ of the $B^{-}\to K^{-}\eta {\eta }_c$ decay in the $(M_{\eta {\eta }_c}^2,M_{K^{-}\eta }^2)$ plane, where the $X(3700)$ and $K_0^{*}(1430)$ resonances can be clearly seen.

Figure 7

The $\eta {\eta }_c$ (a), $K^{-}\eta$ (b) and $K^{-}{\eta }_c$ (c) invariant mass distributions of the $B^{-}\to K^{-}\eta {\eta }_c$ decay with $\phi =0$, $C=3.0$, as well as four different values of $\beta =0.009$ (red solid), 0.012 (green dash-dotted), 0.015 (magenta long-dashed), and 0.018 (blue dashed).

Figure 8

The $\eta {\eta }_c$ (a), $K^{-}\eta$ (b) and $K^{-}{\eta }_c$ (c) invariant mass distributions of the $B^{-}\to K^{-}\eta {\eta }_c$ decay with $\beta =0.012$, $C=3.0$, as well as six different values of $\phi =0$ (red solid), $\pi /3$ (green dash-dotted), $2\pi /3$ (magenta long-dashed), $\pi$ (blue dashed), $4\pi /3$ (olive dotted), and $5\pi /3$ (yellow dash-dot-dotted).

Figure 9

The $\eta {\eta }_c$ (a), $K^{-}\eta$ (b) and $K^{-}{\eta }_c$ (c) invariant mass distributions of the $B^{-}\to K^{-}\eta {\eta }_c$ decay with $\beta =0.012$, $\phi =0$, as well as three different values of $C=3.0$ (red solid), 2.5 (green dash-dotted), and 2.0 (magenta dashed).