A General Analysis of $\gamma$ Determinations from $B\to\pi K$ Decays

We present a general parametrization of $B^\pm\to\pi^\pm K, \pi^0K^\pm$ and $B_d\to\pi^0 K, \pi^\mp K^\pm$ decays, taking into account both electroweak penguin and rescattering effects. This formalism allows -- among other things -- an improved implementation of the strategies that were recently proposed by Neubert and Rosner to probe the CKM angle $\gamma$ with the help of $B^\pm\to\pi^\pm K, \pi^0K^\pm$ decays. In particular, it allows us to investigate the sensitivity of the extracted value of $\gamma$ to the basic assumptions of their approach. We find that certain SU(3)-breaking effects may have an important impact and emphasize that additional hadronic uncertainties are due to rescattering processes. The latter can be controlled by using SU(3) flavour symmetry arguments and additional experimental information provided by $B^\pm\to K^\pm K$ modes. We propose a new strategy to probe the angle $\gamma$ with the help of the neutral decays $B_d\to\pi^0 K, \pi^\mp K^\pm$, which is theoretically cleaner than the $B^\pm\to\pi^\pm K, \pi^0K^\pm$ approach. Here rescattering processes can be taken into account by just measuring the CP-violating observables of the decay $B_d\to\pi^0K_{\rm S}$. Finally, we point out that an experimental analysis of $B_s\to K^+K^-$ modes would also be very useful to probe the CKM angle $\gamma$, as well as electroweak penguins, and we critically compare the virtues and weaknesses of the various approaches discussed in this paper. As a by-product, we point out a strategy to include the electroweak penguins in the determination of the CKM angle $\alpha$ from $B\to\pi\pi$ decays.