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 $B^{+}\to {\pi }^{+}{K}^0$, $B_d^0\to {\pi }^{-}{K}^{+}$, $B_s^0\to {\pi }^{+}{K}^{-}$, and their CP counterparts might be susceptible to important FSI-induced corrections.

Comments: 31 LaTeX pages, 1 .ps figure