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

Abstract

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 ${\text{Bar}B}^0\to {\pi }^{+}{\pi }^{-}{\text{Bar}K}^0$. 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 $\arg [-\frac{V_{ts}{V}_{tb}^{\star }}{V_{us}{V}_{ub}^{\star }}]$.

Comments: 39 pages, 19 figures, 8 tables. Discussion added on Page 14. Misprints also corrected. To be published in Phys. Rev. D