arXiv2018
In this work, we study the localized $CP$ violation in $B^-\rightarrow K^-π^+π^-$ and $B^-\rightarrow K^- σ(600)$ decays by employing the quasi two-body QCD factorization approach. Both the resonance and the nonresonance contributions are studied for the $B^-\rightarrow K^-π^+π^-$ decay. The resonance contributions include those not only from $[ππ]$ channels including $σ(600)$, $ρ^0(770)$ and $ω(782)$ but also from $[Kπ]$ channels including $K^*(892)$, $K_0^*(1430)$, $K^*(1410)$, $K^*(1680)$ and $K_2^*(1430)$. By fitting the experimental data $\mathcal{A_{CP}}(K^-π^+π^-)=0.678\pm0.078\pm0.0323\pm0.007$ for $m_{K^-π^+}^2<15$ $\mathrm{GeV}^2$ and $0.08<m_{π^+π^-}^2<0.66$ $\mathrm{GeV}^2$, we get the end-point divergence parameters in our model, $ϕ_S \in [4.75, 5.95]$ and $ρ_S\in[4.2, 8]$. Using these results for $ρ_S$ and $ϕ_S$, we predict that the $CP$ asymmetry parameter $\mathcal{A_{CP}} \in [-0.094, -0.034]$ and the branching fraction $\mathcal{B} \in [1.82, 20.0]\times10^{-5}$ for the $B^-\rightarrow K^-σ(600)$ decay. In addition, we also analyse contributions to the localized $CP$ asymmetry $\mathcal{A_{CP}}(B^-\rightarrow K^-π^+π^-)$ from $[ππ]$, $[Kπ]$ channel resonances and nonresonance individually, which are found to be $\mathcal{A_{CP}}(B^-\rightarrow K^-[π^+π^-] \rightarrow K^-π^+π^-)=0.585\pm0.045$, $\mathcal{A_{CP}}(B^-\rightarrow [K^-π^+] π\rightarrow K^-π^+π^-)=0.086\pm0.021$ and $\mathcal{A_{CP}}^{NR}(B^-\rightarrow K^-π^+π^-)=0.061\pm0.0042$, respectively. Comparing these results, we can see that the localized $CP$ asymmetry in the $B^-\rightarrow K^-π^+π^-$ decay is mainly induced by the $[ππ]$ channel resonances while contributions from the $[Kπ]$ channel resonances and nonresonance are both very small.