Study of CP violations in $$B^-\rightarrow K^- \pi ^+\pi ^-$$ B-→K-π+π- and $$B^-\rightarrow K^- f_0(500)$$ B-→K-f0(500) decays in the QCD factorization approach

Abstract

Abstract Within the QCD factorization approach, we study the CP violations in $$B^-\rightarrow K^-\pi ^+\pi ^-$$ B-→K-π+π- and $$B^-\rightarrow K^- f_0(500)$$ B-→K-f0(500) decays. We find the experimental data of the localized CP asymmetry in $$B^-\rightarrow K^-\pi ^+\pi ^-$$ B-→K-π+π- decays in the region $$m_{K^-\pi ^+}^2<15$$ mK-π+2<15 $$\mathrm {GeV}^2$$ GeV2 and $$0.08<m_{\pi ^+\pi ^-}^2<0.66$$ 0.08<mπ+π-2<0.66 $$\mathrm {GeV}^2$$ GeV2 can be explained by the interference of two intermediate resonances, $$\rho ^0(770)$$ ρ0(770) and $$f_0(500)$$ f0(500) when the parameters in our interference model are in the allowed ranges, i.e. the relative strong phase $$\delta \in [2.124, 5.976]$$ δ∈[2.124,5.976] and the end-point divergence parameters $$\rho _S\in [5.692, 8]$$ ρS∈[5.692,8] and $$\phi _S \in [0, 2\pi ]$$ ϕS∈[0,2π] . With the obtained allowed ranges for $$\rho _S$$ ρS and $$\phi _S$$ ϕS , we obtain the predictions for the CP asymmetry parameter $$A_{CP} \in [-0.115, -0.151]$$ ACP∈[-0.115,-0.151] and the branching fraction $${\mathcal {B}} \in [3.763, 20.014]\times 10^{-5}$$ B∈[3.763,20.014]×10-5 for $$B^-\rightarrow K^-f_0(500)$$ B-→K-f0(500) decay modes.