Molecular Nature of X(3872) in B⁰ → K⁰X(3872) and B⁺K⁺X(3872) Decays

We investigate the decays of B⁰ → K⁰X(3872) and B⁺ → K⁺X(3872) based on the picture where the X(3872) resonance is strongly coupled to the $D{\overline{D}}^{*}+c.c.$ channel. In addition to the decay mechanism where the X(3872) resonance is formed from the $c\overline{c}$ pair hadronization with the short-distance interaction, we have also considered the $D{\overline{D}}^{*}$ rescattering diagrams in the long-distance scale, where D and ${\overline{D}}^{*}$ are formed from c and $\overline{c}$ separately. Because of the difference of the mass thresholds of charged and neutral $D{\overline{D}}^{*}$ channels, and the rather narrow width of the X(3872) resonance, at the X(3872) mass, the loop functions of $D^0{\overline{D}}^{*0}$ and $D^{+}{\overline{D}}^{*-}$ are much different. Taking this difference into account, the ratio of $ℬB^0\to K^{0}X(3872)/ℬB^{+}\to K^{+}X(3872)\simeq 0.5$ can be naturally obtained. Based on this result, we also evaluate the decay widths of $B_s^0\to \eta ({\eta }^{\prime })X(3872)$. It is expected that future experimental measurements of these decays can be used to elucidate the nature of the X(3872) resonance.