CP asymmetry in $B^{-}\to {\pi }^{+}{\pi }^{-}{K}^{-}$ and $B^{-}\to K^{+}{K}^{-}{K}^{-}$ decays

The near equality between the measured value for $\mathrm{\mid }{V}_{ub}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{V}_{cb}\mathrm{\mid }$ and its unitarity triangle lower bound from $\sin 2\beta$ measurements suggests that $\alpha \approx {90}^{\circ }$ and $\sin \gamma \approx \cos \beta =0.91$ and a large direct CP violation in B decays would be possible. In this talk, I would like to discuss a recent analysis of the non resonant and resonant CP asymmetry for the decays $B^{-}\to {\pi }^{+}{\pi }^{-}{K}^{-}$ and $B^{-}\to K^{+}{K}^{-}{K}^{-}$ where the CP asymmetry at the ${\chi }_{c0}$ resonance comes from the interference of the non resonant amplitude with the resonant amplitude $B^{\pm }\to {\chi }_{c0}{K}^{\pm }$ $\to {\pi }^{+}{\pi }^{-}{K}^{\pm }$ and $B^{\pm }\to {\chi }_{c0}{K}^{\pm }$ $\to K^{+}{K}^{-}{K}^{\pm }$. Using the the Babar and Belle measurements of the $B^{-}\to {\chi }_{c0}{K}^{-}$ branching ratios, we predict that partial width asymmetry at the ${\chi }_{c0}$ resonance could reach 10% for the $B^{-}\to {\pi }^{+}{\pi }^{-}{K}^{-}$ decay and 16% for the $B^{-}\to K^{+}{K}^{-}{K}^{-}$decay.