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 $B^\pm \to f_j K^\pm$, where $K^*_i$ and $f_j$ represent the $\pi K$ and $\pi\pi$ resonances, respectively. 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 $f_j$ resonances overlap. By examining a Dalitz plot of this mode, it is possible to extract the weak phase, $\phi_3$= arg$(-V_{ud}V_{ub}^*/(V_{cd}V_{cb}^*))$, of the Kobayashi-Maskawa matrix. This method is applicable to any $B \to PR(R \to PP)$ decay, where $P$ is a pseudo-scalar meson and $R$ is a resonance which decays into two pseudo-scalar mesons.