Looking for the exotic $X_0(2866)$ and its $J^P=1^{+}$ partner in the ${\overline{B}}^0\to D^{(*)+}{K}^{-}{K}^{(*)0}$ reactions

We propose two reactions, ${\overline{B}}^0\to K^0{D}^{+}{K}^{-}$ and ${\overline{B}}^0\to K^{*0}{D}^{*+}{K}^{-}$, which have been already measured at Belle, to look into the $J^P=0^{+}$, $X_0(2866)$ state and a $1^{+}$ partner of molecular $D^{*}{\overline{K}}^{*}$ nature by looking at the $D^{+}{K}^{-}$ and $D^{*+}{K}^{-}$ invariant mass distributions, respectively. Very clear peaks over the background are predicted and the branching ratios for the production of these states are evaluated to facilitate the task of determining the needed statistics for their observation. We conclude that with the upgrade of Belle II clear peaks should be seen in both reactions for the two resonances discussed.

Figure 1

Diagrammatic decay of ${\overline{B}}^0\to D^{*+}{K}^{*-}{K}^{*0}$ at the quark level.

Figure 2

(a) Rescattering of $D^{*+}{K}^{*-}$ to give the resonance $R_1$; (b) further decay of $R_1$ into $D^{*+}{K}^{-}$.

Figure 3

$\frac{d\mathrm{\Gamma }}{d{M}_{\mathrm{inv}}}$ for $R_1$ production and $\frac{d{\mathrm{\Gamma }}_{\mathrm{bac}}}{d{M}_{\mathrm{inv}}}$ for background in the ${\overline{B}}^0\to K^{*0}{D}^{*+}{K}^{-}$ reaction in global arbitrary units. $M_{\mathrm{inv}}$ is the $D^{*+}{K}^{-}$ invariant mass.

Figure 4

$\frac{d{\mathrm{\Gamma }}^{\prime }}{d{M}_{\mathrm{inv}}}$ for $R_0$ production and $\frac{d{\mathrm{\Gamma }}_{\mathrm{bac}}^{\prime }}{d{M}_{\mathrm{inv}}}$ for background in the ${\overline{B}}^0\to K^0{D}^{+}{K}^{-}$ reaction in global arbitrary units. $M_{\mathrm{inv}}$ is the $D^{+}{K}^{-}$ invariant mass.