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

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 $[pipi]$ channels including $sigma(600)$, $rho^0(770)$ and $omega(782)$ but also from $[Kpi]$ channels including $K^*(892)$, $K_0^*(1430)$, $K^*(1410)$, $K^*(1680)$ and $K_2^*(1430)$. By fitting the experimental data $mathcal{A_{CP}}(K^-pi^+pi^-)=0.678pm0.078pm0.0323pm0.007$ 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 $rho_Sin[4.2, 8]$. 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 $[pipi]$, $[Kpi]$ channel resonances and nonresonance individually, which are found to be $mathcal{A_{CP}}(B^-rightarrow K^-[pi^+pi^-] rightarrow K^-pi^+pi^-)=0.585pm0.045$, $mathcal{A_{CP}}(B^-rightarrow [K^-pi^+] pi rightarrow K^-pi^+pi^-)=0.086pm0.021$ and $mathcal{A_{CP}}^{NR}(B^-rightarrow K^-pi^+pi^-)=0.061pm0.0042$, 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 $[pipi]$ channel resonances while contributions from the $[Kpi]$ channel resonances and nonresonance are both very small.