Localized direct CP violation in $B^\pm\rightarrow \rho^0 (\omega)\pi^\pm\rightarrow \pi^+ \pi^-\pi^\pm$

We study the localized direct CP violation in the hadronic decays $B^\pm\rightarrow \rho^0 (\omega)\pi^\pm\rightarrow\pi^+ \pi^-\pi^\pm$, including the effect caused by an interesting mechanism involving the charge symmetry violating mixing between $\rho^0$ and $\omega$. We calculate the localized integrated direct CP violation when the low invariant mass of $\pi^+\pi^-$ [$m(\pi^+\pi^-)_{low}$] is near $\rho^0(770)$. For five models of form factors investigated, we find that the localized integrated direct CP violation varies from -0.0170 to -0.0860 in the ranges of parameters in our model when $0.750<m(\pi^+\pi^-)_{low}<0.800$\,GeV. This result, especially the sign, agrees with the experimental data and is independent of form factor models. The new experimental data shows that the signs of the localized integrated CP asymmetries in the regions $0.470<m(\pi^+\pi^-)_{low}<0.770$\,GeV and $0.770<m(\pi^+\pi^-)_{low}<0.920$\,GeV are positive and negative, respectively. We find that $\rho$-$\omega$ mixing makes the localized integrated CP asymmetry move towards the negative direction, and therefore contributes to the sign change in those two regions. This behavior is also model independent. We also calculate the localized integrated direct CP violating asymmetries in the regions $0.470<m(\pi^+\pi^-)_{low}<0.770$\,GeV and $0.770<m(\pi^+\pi^-)_{low}<0.920$\,GeV and find that they agree with the experimental data in some models of form factors.