Direct $CP$ violation of three bodies decay process from the resonance effect

Abstract

The physical state of $\rho$-$\omega$-$\varphi$ mesons can be mixed by the unitary matrix. The decay processes of $\omega \to {\pi }^{+}{\pi }^{-}$ and $\varphi \to {\pi }^{+}{\pi }^{-}$ are from the isospin symmetry breaking. The $\rho -\omega$, $\rho -\varphi$ and $\omega -\varphi$ interferences lead to resonance contribution to produce the strong phases. The $CP$ asymmetry is considered from above isospin symmetry breaking due to the new strong phase for the first order. It has been found the $CP$ asymmetry can be enhanced greatly for the decay process of $B^0\to {\pi }^{+}{\pi }^{-}{\eta }^{{(}^{\prime })}$ when the invariant masses of the ${\pi }^{+}{\pi }^{-}$ pairs are in the area around the $\omega$ resonance range and the $\varphi$ resonance range in perturbative QCD. We also discuss the possibility to search the predicted $CP$ violation at the LHC.