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

Abstract

The physical state of $\rho$-$\omega$-$\phi$ mesons can be mixed by the unitary matrix. The decay processes of $\omega \rightarrow \pi^{+}\pi^{-}$ and $\phi \rightarrow \pi^{+}\pi^{-}$ are from the isospin symmetry breaking. The $\rho-\omega$, $\rho-\phi$ and $\omega-\phi$ 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 $B^{0}\rightarrow\pi^+\pi^{-}\eta^{(')}$ when the invariant masses of the $\pi^+\pi^{-}$ pairs are in the area around the $\omega$ resonance range and the $\phi$ resonance range in perturbative QCD. We also discuss the possibility to search the predicted $CP$ violation at the LHC.