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Direct CP violation in $\bar{B}^0 \to ρ^0(ω)ρ^0(ω) \to π^+π^-π^+π^-$

We study the direct CP violation in $\bar{B}^0 \to ρ^0(ω)ρ^0(ω) \to π^+π^-π^+π^-$ (with unpolarized $ρ^0(ω)$) via the $ρ-ω$ mixing mechanism which causes a large strong phase difference and consequently a large CP violating asymmetry when the masses of the $π^+π^-$ pairs are in the vicinity of the $ω$ resonance. Since there are two $ρ(ω)$ mesons in the intermediate state $ρ-ω$ mixing contributes twice to the first order of isospin violation, leading to an even larger CP violating asymmetry (could be 30% -- 50% larger) than in the case where only one $ρ(ω)$ meson is involved. The CP violating asymmetry depends on the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and the hadronic matrix elements. The factorization approach is applied in the calculation of the hadronic matrix elements with the nonfactorizable effects being included effectively in an effective parameter, $N_c$. We give the constraint on the range of $N_c$ from the latest experimental data for the branching ratios for $\bar{B}^0 \toρ^0ρ^0$ and $\bar{B}^0 \toρ^+ρ^-$. We find that the CP violating asymmetry could be very large (even more than 90% for some values of $N_c$). It is shown that the sensitivity of the CP violating asymmetry to $N_c$ is large compared with its smaller sensitivity to the CKM matrix elements. We also discuss the possibility to remove the mod $(π)$ ambiguity in the determination of the CP violating phase angle $α$ through the measurement of the CP violating asymmetry in the decay $\bar{B}^0\to ρ^0(ω)ρ^0(ω) \to π^+π^-π^+π^-$.