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

Abstract

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