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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 $π^+ π^-$, $π^\pm K^\mp$, and $π^+ K^0$ or $π^- \overline{K}^0$. For $B^0~{\rm or}~ \overline{B}^0 \to π^+ π^-$, 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 $γ\equiv {\rm Arg}~V_{ub}^*$ and $α= π- β- γ$, where $β\equiv {\rm Arg}~V_{td}^*$. Special cases include the vanishing of strong interaction phase differences between amplitudes, the possibility of recovering partial information when $π^+ π^-$ and $π^\pm K^\mp$ decays cannot be distinguished from one another, and the use of a correlation between $γ$ and $α$ in the region of allowed parameters.