Localized directCPviolation forB±→ρ0(ω)π±→π+π−π±in QCD factorization

Abstract

We study the localized direct $CP$ violation in the hadronic decays ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}(\ensuremath{\omega}){\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, including the effect caused by an interesting mechanism involving the charge symmetry violating mixing between ${\ensuremath{\rho}}^{0}$ and $\ensuremath{\omega}$ in the QCD factorization approach. We find that $\ensuremath{\rho}\text{\ensuremath{-}}\ensuremath{\omega}$ mixing makes the localized integrated $CP$ asymmetry move toward the negative direction when the low invariant mass of ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ [$m({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{)}_{\mathrm{low}}$] is near ${\ensuremath{\rho}}^{0}(770)$. The localized integrated direct $CP$ violation obtained in the QCD factorization approach varies from $\ensuremath{-}0.0724$ to $\ensuremath{-}0.0389$ in the ranges of parameters when $0.750lm({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{)}_{\mathrm{low}}l0.800\text{ }\text{ }\mathrm{GeV}$. This result, especially the sign, agrees with the experimental data. We also calculate the localized integrated direct $CP$ violating asymmetries in the QCD factorization approach in the regions $0.470lm({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{)}_{\mathrm{low}}l0.770\text{ }\text{ }\mathrm{GeV}$ and $0.770lm({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{)}_{\mathrm{low}}l0.920\text{ }\text{ }\mathrm{GeV}$. We find that these results agree with the experimental data and are more accurate than the results obtained through the naive factorization approach. It is more clear that $\ensuremath{\rho}\text{\ensuremath{-}}\ensuremath{\omega}$ mixing contributes to the sign change in these two regions.