Inelastic Rescattering in B Decays to $\pi \pi$, $\pi K$, and $K\bar{K}$, and Extraction of $\gamma $

Abstract

We discuss all contributions from inelastic SU(3)-symmetric rescattering in B decays into a final pair of pseudoscalar mesons $PP = \pi \pi$, $K\bar{K}$, $\pi K$. 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 $\pi \pi$, $K \bar{K}$, and $\pi K$ 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 $\Delta S =0$ and $|\Delta S|=1$ 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 $\gamma$ from $B^+ \to \pi ^+ K^0$, $B^0_d \to \pi^-K^+$, $B^0_s \to \pi ^+ K^-$, and their CP counterparts might be susceptible to important FSI-induced corrections.