Inelastic Rescattering in B Decays to $ππ$, $πK$, and $K\bar{K}$, and Extraction of $γ$

Abstract

We discuss all contributions from inelastic SU(3)-symmetric rescattering in B decays into a final pair of pseudoscalar mesons $PP = ππ$, $K\bar{K}$, $πK$. FSI-induced modifications of amplitudes obtained from the quark-line approach are described in terms of a few parameters which take care of all possible SU(3)-symmetric forms relevant for final-state interactions. Although in general it appears impossible to uniquely determine FSI effects from the combined set of all $ππ$, $K \bar{K}$, and $πK$ data, drawing some conclusions is feasible. In particular, it is shown that in leading order the amplitudes of strangeness-changing B decays depend on only one additional complex FSI-related parameter apart from those present in the definitions of penguin and tree amplitudes. It is also shown that joint considerations of U-spin-related $ΔS =0$ and $|ΔS|=1$ decay amplitudes are modified when non-negligible SU(3)-symmetric FSI are present. In particular, if rescattering in $B^+ \to K^+\bar{K}^0$ is substantial, determination of the CP-violating weak angle $γ$ from $B^+ \to π^+ K^0$, $B^0_d \to π^-K^+$, $B^0_s \to π^+ K^-$, and their CP counterparts might be susceptible to important FSI-induced corrections.