CP violating phase $\gamma$ and the partial widths asymmetry in $B^- \to \pi^+ \pi^- K^-$ and $B^- \to K^+ K^- K^-$ decays

Abstract

Motivated by recent Belle measurement of the branching ratios $B^- \to \pi^+ \pi^- K^-$ and $B^- \to K^+ K^- K^-$ and $B^\pm \to \chi_{c0} K^\pm$, we investigate the CP violating asymmetry in the partial widths for the decays $B^- \to \pi^+ \pi^- K^-$ and $B^- \to K^+ K^- K^-$, which results from the interference of the nonresonant decay amplitude with the resonant amplitude $B^\pm \to \chi_{c0} K^\pm $ $ \to\pi^+ \pi^- K^\pm $ and $B^\pm \to \chi_{c0} K^\pm$ $ \to K^+ K^- K^-$. By taking $\gamma \simeq 58^o$ we predict that the partial widths asymmetry is reaching 10% for the $B^- \to \pi^+ \pi^- K^-$ decay and 16% for the $B^- \to K^+ K^- K^-$decay.