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

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 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^-$. The predicted asymmetry is rather small, being $0.17 \sin \gamma$ for the $B^- \to \pi^+ \pi^- K^-$ decay and $0.14 \sin \gamma$ for the $B^- \to K^+ K^- K^-$decay.