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 $B^\pm \to K^*_i \pi^\pm$ and $B^\pm \to \rho K^\pm$, where $K^*_i$ 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 $K^*_i$ 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, $\phi_3$= arg$(-V_{ud}V_{ub}^*/(V_{cd}V_{cb}^*))$, of the Kobayashi-Maskawa matrix.