$B_{s,d}\to \pi \pi ,\pi K,KK$: 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 $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ from untagged $B_s\to K^+K^-$rates,andaddresstheimpactofnewphysics\mathrm{.}Thedataforthe$B_d\to\pi^+\pi^-$,$B_s\to K^+K^-$ 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 $\gamma =(66.6_{-5.0-3.0}^{+4.3+4.0}{)}^{\circ }$, 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 ${26}^{\circ }\le \gamma \le {70}^{\circ }$, 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 $B_s^0$–$\bar B^0_s$mixingphaseunambiguouslywiththehelpof$B_s\to J/\psi \phi$ 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.