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. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Science and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.

An investigation about localized CP asymmetries for the processes of $${\bar{B}}_{s}^{0} \rightarrow \pi ^{+}\pi ^{-}\pi ^{0} (K^{0})$$ B ¯ s 0 → π + π - π 0 ( K 0 ) is presented in this paper. The innovation of this paper is that there is a consideration of three-particle $$\rho ^0(770)$$ ρ 0 ( 770 ) , $$\omega (782)$$ ω ( 782 ) and $$\phi (1020)$$ ϕ ( 1020 ) interferences effect. Generally, $$\omega (782)$$ ω ( 782 ) and $$\phi (1020)$$ ϕ ( 1020 ) both can decay into $$\pi ^+\pi ^-$$ π + π - pair where can cause extremely small contribution from isospin symmetry breaking. Nevertheless, our analysis shows that $${\bar{B}}_{s}^{0} \rightarrow \pi ^+\pi ^-\pi ^{0}(K^{0})$$ B ¯ s 0 → π + π - π 0 ( K 0 ) decay process can bring differential CP asymmetry about 65% (36%) because of isospin symmetry breaking. To be better compared with the data from experimental in the future, we integrate CP asymmetry over the invariant mass and obtain localized CP asymmetry value for the decay $${\bar{B}}_{s}^{0} \rightarrow \pi ^+\pi ^-\pi ^{0} (K^{0})$$ B ¯ s 0 → π + π - π 0 ( K 0 ) . We find that there is an evident signal about CP asymmetry at invariant mass value $$m(\pi ^{+}\pi ^{-})$$ m ( π + π - ) below the mass of $$\rho ^{0}(770)$$ ρ 0 ( 770 ) with the decay $${\bar{B}}_{s}^{0} \rightarrow \pi ^+\pi ^{-}\pi ^{0}$$ B ¯ s 0 → π + π - π 0 .

An investigation about localized CP asymmetries for the processes of $${\bar{B}}_{s}^{0} \rightarrow \pi ^{+}\pi ^{-}\pi ^{0} (K^{0})$$ B ¯ s 0 → π + π - π 0 ( K 0 ) is presented in this paper. The innovation of this paper is that there is a consideration of three-particle $$\rho ^0(770)$$ ρ 0 ( 770 ) , $$\omega (782)$$ ω ( 782 ) and $$\phi (1020)$$ ϕ ( 1020 ) interferences effect. Generally, $$\omega (782)$$ ω ( 782 ) and $$\phi (1020)$$ ϕ ( 1020 ) both can decay into $$\pi ^+\pi ^-$$ π + π - pair where can cause extremely small contribution from isospin symmetry breaking. Nevertheless, our analysis shows that $${\bar{B}}_{s}^{0} \rightarrow \pi ^+\pi ^-\pi ^{0}(K^{0})$$ B ¯ s 0 → π + π - π 0 ( K 0 ) decay process can bring differential CP asymmetry about $$65\%$$ 65 % ($$36\%$$ 36 % ) because of isospin symmetry breaking. To be better compared with the data from experimental in the future, we integrate CP asymmetry over the invariant mass and obtain localized CP asymmetry value for the decay $${\bar{B}}_{s}^{0} \rightarrow \pi ^+\pi ^-\pi ^{0} (K^{0})$$ B ¯ s 0 → π + π - π 0 ( K 0 ) . We find that there is an evident signal about CP asymmetry at invariant mass value $$m(\pi ^{+}\pi ^{-})$$ m ( π + π - ) below the mass of $$\rho ^{0}(770)$$ ρ 0 ( 770 ) with the decay $${\bar{B}}_{s}^{0} \rightarrow \pi ^+\pi ^{-}\pi ^{0}$$ B ¯ s 0 → π + π - π 0 .