Study of $CP$ violation in $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ and $B^{-}\to K^{-}\sigma \mathbf{(}600\mathbf{)}$ decays in the QCD factorization approach

In this work, we study the localized $CP$ violation in $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ and $B^{-}\to K^{-}\sigma (600)$ decays by employing the quasi-two-body QCD factorization approach. Both the resonance and the nonresonance contributions are studied for the $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ 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 $K_0^{*}(700)(\kappa )$, $K^{*}(892)$, $K_0^{*}(1430)$, $K^{*}(1410)$, $K^{*}(1680)$ and $K_2^{*}(1430)$. By fitting the four experimental data ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(K^{-}{\pi }^{+}{\pi }^{-})=0.678\pm 0.078\pm 0.0323\pm 0.007$ for $m_{K^{-}{\pi }^{+}}^2<15{\mathrm{GeV}}^2$ and $0.08<m_{{\pi }^{+}{\pi }^{-}}^2<0.66{\mathrm{GeV}}^2$, ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(B^{-}\to K_0^{*}(1430){\pi }^{-})=0.061\pm 0.032$, $\mathcal{B}(B^{-}\to K_0^{*}(1430){\pi }^{-})=({39}_{-5}^{+6})\times {10}^{-6}$ and $\mathcal{B}(B^{-}\to \sigma (600){\pi }^{-}\to {\pi }^{-}{\pi }^{+}{\pi }^{-})<4.1\times {10}^{-6}$, we get the end-point divergence parameters in our model, ${\varphi }_S\in [1.77,2.25]$ and ${\rho }_S\in [2.39,4.02]$. Using these results for ${\rho }_S$ and ${\varphi }_S$, we predict that the $CP$ asymmetry parameter ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}\in [-0.34,-0.11]$ and the branching fraction $\mathcal{B}\in [6.53,17.52]\times {10}^{-6}$ for the $B^{-}\to K^{-}\sigma (600)$ decay. In addition, we also analyze contributions to the localized $CP$ asymmetry ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(B^{-}\to K^{-}{\pi }^{+}{\pi }^{-})$ from $[\pi \pi ]$, $[K\pi ]$ channel resonances and nonresonance individually, which are found to be ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(B^{-}\to K^{-}[{\pi }^{+}{\pi }^{-}]\to K^{-}{\pi }^{+}{\pi }^{-})=0.509\pm 0.042$, ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(B^{-}\to [K^{-}{\pi }^{+}]\pi \to K^{-}{\pi }^{+}{\pi }^{-})=0.174\pm 0.025$ and ${{\mathcal{A}}_{\mathcal{C}\mathcal{P}}}^{\mathrm{NR}}(B^{-}\to K^{-}{\pi }^{+}{\pi }^{-})=0.061\pm 0.0042$, respectively. Comparing these results, we can see that the localized $CP$ asymmetry in the $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$ decay is mainly induced by the $[\pi \pi ]$ channel resonances while contributions from the $[K\pi ]$ channel resonances and nonresonance are both very small.