The resonance effect for the CP asymmetry associated with the process $ω\rightarrow π^{+}π^{-}π^{0}$

Abstract: The direct CP asymmetry in the weak decay process of hadrons is commonly attributed to the weak phase of the CKM matrix and the indeterminate strong phase. We propose a way of creating a strong phase difference between two decay paths involving vector mesons $V= \omega,\rho$ decaying to $\pi^{+}\pi^{-}\pi^{0}$ considering the G-parity suppressed decay process $\rho^{0} \rightarrow \pi^{+}\pi^{-}\pi^{0} $. It can lead to a significant CP asymmetry in the interference region. We consider the effects of $\omega$ and $\rho$ mixing on their decays to $\pi^{+}\pi^{-}\pi^{0}$. Meanwhile, we calculate the CP asymmetry for the process $B\rightarrow \pi^{+}\pi^{-}\pi^{0}P$ (P refers to pseudoscalar meson) through $\omega(\rho) \rightarrow \pi^{+}\pi^{-}\pi^{0}$. Besides, we also calculate the integral results of different phase space regions. We hope that our work can help provide some theoretical guidance for future experimental studies of CP asymmetry in these decays.