Enhanced direct CP violation in $B^{\pm,0} \to \pi^{+} \pi^{-} K^{\pm,0}$

We investigate in a phenomenological way, direct CP violation in the hadronic decays $B^{\pm,0} \to \pi^{+} \pi^{-} K^{\pm,0}$ where the effect of $\rho - \omega$ mixing is included. If $N_{c}^{eff}$ (the effective parameter associated with factorization) is constrained using the most recent experimental branching ratios (to $\rho^{0}K^{0}, \rho^{\pm}K^{\mp}, \rho^{\pm}K^{0}, \rho^{0} K^{\pm}$ and $\omega K^{\pm}$) from the BABAR, BELLE and CLEO Collaborations, we get a maximum CP violating asymmetry, $a_{max}$, in the range -25% to $+49$ for $B^{-} \to \pi^{+}\pi^{-} K^{-}$ and -24% to $+55$ for ${\Bar B}^{0} \to \pi^{+}\pi^{-} {\Bar K}^{0}$. We also find that CP violation is strongly dependent on the Cabibbo-Kobayashi-Maskawa matrix elements. Finally, we show that the sign of $\sin \delta$ is always positive in the allowed range of $N_{c}^{eff}$ and hence, a measurement of direct $CP$ violation in $B^{\pm,0} \to \pi^{+} \pi^{-} K^{\pm,0}$ would remove the mod$(\pi)$ ambiguity in ${\rm arg}[ - \frac{V_{ts}V_{tb}^{\star}}{V_{us}V_{ub}^{\star}}]$.