CP violation induced by the real part of the interference term in $\rho^0 - \omega$ mixing
To circumvent the severe numerical cancellations in the standard integrated CP asymmetry ($A_{CP}$) near the $\omega$ mass $m_\omega$, we propose a modified CP-violating observable, $A_{CP}^{\Re}$, which explicitly highlights the contribution of the real part of the interference term. Subsequently, we employ the three-body hadronic decay $B^{-} \rightarrow \pi^{+} \pi^{-} \pi^{-}$ within the Perturbative QCD (PQCD) approach as a primary case study to validate this theoretical framework. Furthermore, this method naturally eliminates the smooth, sign-preserving continuum background originating from broad scalar resonances like the $f_0(500)$. This generalized framework provides clean and robust theoretical guidance for recovering localized CP-violation signals that might otherwise be masked by coarse experimental binning at future high-luminosity colliders.