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$CP$ violation for $B^{0}\rightarrow ρ^{0}(ω)ρ^{0}(ω) \rightarrow π^+π^-π^+π^-$ in QCD factorization

In the QCD factorization (QCDF) approach we study the direct $CP$ violation in $\bar{B}^{0}\rightarrowρ^0(ω)ρ^0(ω)\rightarrowπ^+π^-π^+π^-$ via the $ρ-ω$ mixing mechanism. We find that the $CP$ violation can be enhanced by double $ρ-ω$ mixing when the masses of the $π^+π^-$ pairs are in the vicinity of the $ω$ resonance, and the maximum $CP$ violation can reach 28{\%}. We also compare the results from the naive factorization and the QCD factorization.