New Strategies to Extract $\beta$ and $\gamma$ from $B_d\to \pi^+ \pi^-$ and $B_s\to K^+ K^-$

The decays $B_d\to \pi^+ \pi^-$ and $B_s\to K^+ K^-$ are related to each other by interchanging all down and strange quarks, i.e. through the $U$-spin flavour symmetry of strong interactions. A completely general parametrization of the CP-violating observables of these modes is presented within the framework of the Standard Model, allowing the determination of the angles $\beta$ and $\gamma$ of the unitarity triangle. This strategy is affected neither by penguin contributions nor by any final-state-interaction effects, and its theoretical accuracy is only limited by $U$-spin-breaking corrections. If the $B^0_d - \bar{B^0_d}$ mixing phase $2\beta$ is determined separately, for example with the help of $B_d\to J/\psi K_S$, $\gamma$ can be extracted with a reduced $U$-spin flavour symmetry input. A variant of this strategy to determine $\gamma$, which uses $B_d\to\pi^\mp K^\pm$ instead of $B_s\to K^+ K^-$ and relies -- in addition to the SU(3) flavour symmetry -- on a certain dynamical assumption, is also briefly discussed.