Weak phases fromBdecays to kaons and charged pions

Phases of elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix can be obtained using decays of $B$ mesons to ${\pi }^{+}{\pi }^{-}$, ${\pi }^{\pm }{K}^{+}$, and ${\pi }^{+}{K}^0$ or ${\pi }^{-}{\overline{K}}^0$. For $B^0$ 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 to be measured. By assuming flavor SU(3) symmetry and first-order SU(3) breaking, one can separately determine the phases $\gamma \equiv \mathrm{Arg} V_{\mathrm{ub}}^{*}$ and $\alpha =\pi -\beta -\gamma$, where $\beta \equiv \mathrm{Arg} V_{\mathrm{td}}^{*}$. 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.