We study direct CP violation in the hadronic decay $B^{ \pm} \to \rho^{ 0}\pi^{ \pm}$, including the effect of $\rho - \omega$ mixing. We find that the CP violating asymmetry is strongly dependent on the CKM matrix elements, especially the Wolfenstein parameter $\eta$. For fixed $N_{ c}$ (the effective parameter associated with factorization), the CP violating asymmetry, $a$, has a maximum of order $30%-50%$ when the invariant mass of the $\pi^{ +}\pi^{ -}$ pair is in the vicinity of the $\omega$ resonance. The sensitivity of the asymmetry, $a$, to $N_{ c}$ is small. Moreover, if $N_{ c}$ is constrained using the latest experimental branching ratios from the CLEO collaboration, we find that the sign of $\sin \delta$ is always positive. Thus, a measurement of direct CP violation in $B^{ \pm} \to \rho^{ 0}\pi^{ \pm}$ would remove the mod$(\pi)$ ambiguity in ${ \rm arg}[ - \frac{ V_{ td}V_{ tb}^{ \star}}{ V_{ ud}V_{ ub}^{ \star}}]$.
Categories: hep-ph