Study of $CP$ Violation in $B^-\rightarrow K^- π^+π^-$ and $B^-\rightarrow K^- σ(600)$ decays in the QCD factorization approach

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