$CP$ violation for $B^0\to {\rho }^0(\omega ){\rho }^0(\omega )\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ in QCD factorization

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