Weak phases from B decays to kaons and charged pions

Phases of elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can be obtained using decays of B mesons to \pi^+ \pi^-, \pi^\pm K^\mp, and \pi^+ K^0 or \pi^- \overline{K}^0. For B^0~{\rm or}~ \overline{B}^0 \to \pi^+ \pi^-, one identifies the flavor of the neutral B meson at time of production and studies the time-dependence of the decay rate. The other processes are self-tagging and only their rates need be measured. By assuming flavor SU(3) symmetry and first-order SU(3) breaking, one can separately determine the phases \gamma \equiv {\rm Arg}~V_{ub}^* and \alpha = \pi - \beta - \gamma, where \beta \equiv {\rm Arg}~V_{td}^*. Special cases include the vanishing of strong interaction phase differences between amplitudes, the possibility of recovering partial information when \pi^+ \pi^- and \pi^\pm K^\mp decays cannot be distinguished from one another, and the use of a correlation between \gamma and \alpha in the region of allowed parameters.