The FSI contribution to the observed $B_s$ decays into $K^+K^-$ and $\pi^+K^-$

Abstract

Because at the tree level $B_s\rightarrow K^+K^-$ is Cabibbo-triple suppressed, so its branching ratio should be smaller than that of $B_s\to \pi^+ K^-$. The measurements present a reversed ratio as $R=\mathcal B(B_s\rightarrow \pi^+K^-)/\mathcal B(B_s\rightarrow K^+ K^-)\sim{4.9/33}$. Therefore, It has been suggested that the transition $B_s\to K^+K^-$ 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 (FSI) to $B_s\to K^+K^-$ via sequential processes $B_s\to D^{(*)}_{(s)} \bar D_s^{(*)}\to K^+K^-$ is also substantial and should be superposed on the penguin contribution. Indeed, taking into account of the FSI effects, the theoretical prediction on $R$ is well consistent with the data.