Final state interaction contribution to the observedBsdecays intoK+K−andπ+K−

Abstract

Because at the tree level ${B}_{s}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ is Cabibbo triple suppressed, so its branching ratio should be smaller than that of ${B}_{s}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{K}^{\ensuremath{-}}$. The measurements present a reversed ratio as $R=\mathcal{B}({B}_{s}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{K}^{\ensuremath{-}})/\phantom{\rule{0ex}{0ex}}\mathcal{B}({B}_{s}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}})\ensuremath{\sim}4.9/33$. Therefore, it has been suggested that the transition ${B}_{s}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ is dominated by the penguin mechanism, which is proportional to ${V}_{cb}{V}_{cs}^{*}$. In this work, we show that an extra contribution from the final state interaction to ${B}_{s}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ via sequential processes ${B}_{s}\ensuremath{\rightarrow}{D}_{(s)}^{(*)}{\overline{D}}_{s}^{(*)}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ is also substantial and should be superposed on the penguin contribution. Indeed, taking into account the final state interaction effects, the theoretical prediction on $R$ is well consistent with the data.