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Home > Physical Review D (APS) > Study of <math><mi>C</mi><mi>P</mi></math> violation in <math><msup><mi>B</mi><mo>−</mo></msup><mo>→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup></math> and <math><msup><mi>B</mi><mo>−</mo></msup><mo>→</mo><msup><mi>K</mi><mo>−</mo></msup><mi>σ</mi><mo>(</mo><mn>600</mn><mo>)</mo></math> 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 ${A}_{CP}\left({K}^{-}{\pi}^{+}{\pi}^{-}\right)=0.678\pm 0.078\pm 0.0323\pm 0.007$ for ${m}_{{K}^{-}{\pi}^{+}}^{2}<15\text{}\text{}{\mathrm{GeV}}^{2}$ and $0.08<{m}_{{\pi}^{+}{\pi}^{-}}^{2}<0.66\text{}\text{}{\mathrm{GeV}}^{2}$, ${A}_{CP}({B}^{-}\to {K}_{0}^{*}(1430\left){\pi}^{-}\right)=0.061\pm 0.032$, $B({B}^{-}\to {K}_{0}^{*}(1430\left){\pi}^{-}\right)=\left({39}_{-5}^{+6}\right)\times {10}^{-6}$ and $B({B}^{-}\to \sigma (600){\pi}^{-}\to \phantom{\rule{0ex}{0ex}}{\pi}^{-}{\pi}^{+}{\pi}^{-})<4.1\times {10}^{-6}$, 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 ${A}_{CP}\in [-0.34,-0.11]$ and the branching fraction $B\in [6.53,17.52]\times {10}^{-6}$ for the ${B}^{-}\to {K}^{-}\sigma \left(600\right)$ decay. In addition, we also analyze contributions to the localized $CP$ asymmetry ${A}_{CP}({B}^{-}\to {K}^{-}{\pi}^{+}{\pi}^{-})$ from $\left[\pi \pi \right]$, $\left[K\pi \right]$ channel resonances and nonresonance individually, which are found to be ${A}_{CP}({B}^{-}\to {K}^{-}[{\pi}^{+}{\pi}^{-}]\to {K}^{-}{\pi}^{+}{\pi}^{-})=0.509\pm 0.042$, ${A}_{CP}({B}^{-}\to [{K}^{-}{\pi}^{+}]\pi \to {K}^{-}{\pi}^{+}{\pi}^{-})=0.174\pm 0.025$ and ${{A}_{CP}}^{\mathrm{NR}}({B}^{-}\to {K}^{-}{\pi}^{+}{\pi}^{-})=0.061\pm 0.0042$, 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 $\left[\pi \pi \right]$ 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)**
**Published by: **APS

**DOI: **10.1103/PhysRevD.99.076010

**License: **CC-BY-4.0