Direct CP violation in three-body decays of the $B$ meson by resonance effects

We discuss direct CP violation in three-body decays of $B$ meson such as $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}$, which involves various intermediate resonance states, such as $B^{\pm }\to K_i^{\ast }{\pi }^{\pm }$ and $B^{\pm }\to \rho K^{\pm }$, where $K_i^{\ast }$ represents the $\pi K$ resonance. Due to the large final state interaction phases of the resonances, the CP asymmetry can be as large as the 25% level near the kinematical region where the $K_i^{\ast }$ and $\rho$ resonances overlap. By examining a Dalitz plot of this mode and combining a measurement of the branching ratio for $B^{\pm }\to {\rho }^{\pm }K$, it is possible to extract the weak phase, ${\varphi }_3$= arg$(-V_{ud}{V}_{ub}^{\ast }\mathrm{/}(V_{cd}{V}_{cb}^{\ast }))$, of the Kobayashi-Maskawa matrix.