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.

Comments: LaTeX, epsf.tex included, 16 pages, 2 figures, corrected version to appear in Physics Letters B