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

Qi, Jing-Juan (College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China) ; Guo, Xin-Heng (College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China) ; Wang, Zhen-Yang (Physics Department, Ningbo University, Zhejiang 315211, China) ; Zhang, Zhen-Hua (School of Nuclear and Technology, University of South China, Hengyang, Hunan 421001, China) ; Wang, Chao (Center for Ecological and Environmental Sciences, Key Laboratory for Space Bioscience and Biotechnology, Northwestern Polytechnical University, Xi’an 710072, China)

16 April 2019

Abstract: In this work, we study the localized $CP$ violation in ${B}^{-}\to {K}^{-}{\pi }^{+}{\pi }^{-}$ and ${B}^{-}\to {K}^{-}\sigma \left(600\right)$ 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 $\left[\pi \pi \right]$ channels including $\sigma \left(600\right)$, ${\rho }^{0}\left(770\right)$ and $\omega \left(782\right)$ but also from $\left[K\pi \right]$ channels including ${K}_{0}^{*}\left(700\right)\left(\kappa \right)$, ${K}^{*}\left(892\right)$, ${K}_{0}^{*}\left(1430\right)$, ${K}^{*}\left(1410\right)$, ${K}^{*}\left(1680\right)$ and ${K}_{2}^{*}\left(1430\right)$. By fitting the four experimental data ${A}_{CP}\left({K}^{-}{\pi }^{+}{\pi }^{-}\right)=0.678±0.078±0.0323±0.007$ for ${m}_{{K}^{-}{\pi }^{+}}^{2}<15\text{}\text{}{\mathrm{GeV}}^{2}$ and $0.08<{m}_{{\pi }^{+}{\pi }^{-}}^{2}<0.66\text{}\text{}{\mathrm{GeV}}^{2}$, ${A}_{CP}\left({B}^{-}\to {K}_{0}^{*}\left(1430\right){\pi }^{-}\right)=0.061±0.032$, $B\left({B}^{-}\to {K}_{0}^{*}\left(1430\right){\pi }^{-}\right)=\left({39}_{-5}^{+6}\right)×{10}^{-6}$ and $B\left({B}^{-}\to \sigma \left(600\right){\pi }^{-}\to \phantom{\rule{0ex}{0ex}}{\pi }^{-}{\pi }^{+}{\pi }^{-}\right)<4.1×{10}^{-6}$, we get the end-point divergence parameters in our model, ${\varphi }_{S}\in \left[1.77,2.25\right]$ and ${\rho }_{S}\in \left[2.39,4.02\right]$. Using these results for ${\rho }_{S}$ and ${\varphi }_{S}$, we predict that the $CP$ asymmetry parameter ${A}_{CP}\in \left[-0.34,-0.11\right]$ and the branching fraction $B\in \left[6.53,17.52\right]×{10}^{-6}$ for the ${B}^{-}\to {K}^{-}\sigma \left(600\right)$ decay. In addition, we also analyze contributions to the localized $CP$ asymmetry ${A}_{CP}\left({B}^{-}\to {K}^{-}{\pi }^{+}{\pi }^{-}\right)$ from $\left[\pi \pi \right]$, $\left[K\pi \right]$ channel resonances and nonresonance individually, which are found to be ${A}_{CP}\left({B}^{-}\to {K}^{-}\left[{\pi }^{+}{\pi }^{-}\right]\to {K}^{-}{\pi }^{+}{\pi }^{-}\right)=0.509±0.042$, ${A}_{CP}\left({B}^{-}\to \left[{K}^{-}{\pi }^{+}\right]\pi \to {K}^{-}{\pi }^{+}{\pi }^{-}\right)=0.174±0.025$ and ${{A}_{CP}}^{\mathrm{NR}}\left({B}^{-}\to {K}^{-}{\pi }^{+}{\pi }^{-}\right)=0.061±0.0042$, 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 $\left[\pi \pi \right]$ channel resonances while contributions from the $\left[K\pi \right]$ channel resonances and nonresonance are both very small.

Published in: Physical Review D 99 (2019)