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

Abstract

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 $K^{\ast }(892)$, $K_0^{\ast }(1430)$, $K^{\ast }(1410)$, $K^{\ast }(1680)$ and $K_2^{\ast }(1430)$. By fitting the 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 $0.08<m_{{\pi }^{+}{\pi }^{-}}^2<0.66$ ${\mathrm{GeV}}^2$, we get the end-point divergence parameters in our model, ${\varphi }_S\in [4.75,5.95]$ and ${\rho }_S\in [4.2,8]$. Using these results for ${\rho }_S$ and ${\varphi }_S$, we predict that the $CP$ asymmetry parameter ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}\in [-0.094,-0.034]$ and the branching fraction $\mathcal{B}\in [1.82,20.0]\times 10^{-5}$ for the $B^{-}\to K^{-}\sigma (600)$ decay. In addition, we also analyse 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 ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(B^{-}\to K^{-}[{\pi }^{+}{\pi }^{-}]\to K^{-}{\pi }^{+}{\pi }^{-})=0.585\pm 0.045$, ${\mathcal{A}}_{\mathcal{C}\mathcal{P}}(B^{-}\to [K^{-}{\pi }^{+}]\pi \to K^{-}{\pi }^{+}{\pi }^{-})=0.086\pm 0.021$ and ${{\mathcal{A}}_{\mathcal{C}\mathcal{P}}}^{NR}(B^{-}\to K^{-}{\pi }^{+}{\pi }^{-})=0.061\pm 0.0042$, 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.

Comments: arXiv admin note: text overlap with arXiv:0704.1049 by other authors