$B_{s,d} \to \pi\pi,\pi K, KK$: Status and Prospects

Several years ago, it was pointed out that the U-spin-related decays $B_d\to\pi^+\pi^-$, $B_s\to K^+K^-$ and $B_d\to\pi^\mp K^\pm$, $B_s\to \pi^\pm K^\mp$ offer interesting strategies for the extraction of the angle gamma of the unitarity triangle. Using the first results from the Tevatron on the $B_s$ decays and the B-factory data on $B_{u,d}$ modes, we compare the determinations of gamma from both strategies, study the sensitivity on U-spin-breaking effects, discuss the resolution of discrete ambiguities, predict observables that were not yet measured but will be accessible at LHCb, explore the extraction of the width difference $\Delta\Gamma_s$ from untagged $B_s\to K^+K^-$ rates, and address the impact of new physics. The data for the $B_d\to\pi^+\pi^-$, $B_s\to K^+K^-$ system favour the BaBar measurement of direct CP violation in $B_d\to\pi^+\pi^-$, which will be used in the numerical analysis, and result in a fortunate situation, yielding $\gamma=(66.6^{+4.3+4.0}_{-5.0-3.0})^\circ$, where the latter errors correspond to a generous estimate of U-spin-breaking effects. On the other hand, the $B_d\to\pi^\mp K^\pm$, $B_s\to \pi^\pm K^\mp$ analysis leaves us with $26^\circ\leq\gamma\leq70^\circ$, and points to a value of the $B_s\to \pi^\pm K^\mp$ branching ratio that is larger than the current Tevatron result. An important further step will be the measurement of mixing-induced CP violation in $B_s\to K^+K^-$, which will also allow us to extract the $B^0_s$--$\bar B^0_s$ mixing phase unambiguously with the help of $B_s\to J/\psi \phi$ at the LHC. Finally, the measurement of direct CP violation in $B_s\to K^+K^-$ will make the full exploitation of the physics potential of the $B_{s,d}\to \pi\pi, \pi K, KK$ modes possible