We investigate in a phenomenological way direct $\mathrm{CP}$ violation in the hadronic decays $B^{\pm ,0\mathrm{}}\to {\pi }^{+}{\pi }^{-}{K}^{\pm ,0\mathrm{}}$ where the effect of $\rho -\omega$ mixing is included. If $N_c^{\mathrm{eff}}$ (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 $\mathrm{CP}$ violating asymmetry $a_{\max }$ in the range $-25\%$ to $+49\%$ for $B^{-}\to {\pi }^{+}{\pi }^{-}{K}^{-}$ and $-24\%$ to $+55\%$ for $B^0\to {\pi }^{+}{\pi }^{-}{K}^0.$ We also find that $\mathrm{CP}$ 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 $N_c^{\mathrm{eff}}$ and hence, a measurement of direct $\mathrm{CP}$ violation in $B^{\pm ,0\mathrm{}}\to {\pi }^{+}{\pi }^{-}{K}^{\pm ,0\mathrm{}}$ would remove the $\mathrm{mod}\left(\pi \right)$ ambiguity in $\arg \left[-V_{\mathrm{ts}}{V}_{\mathrm{tb}}^{\star }{/V}_{\mathrm{us}}{V}_{\mathrm{ub}}^{\star }\right].$

• Received 22 August 2002

DOI:https://doi.org/10.1103/PhysRevD.66.096008