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.