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

In this work, we study the localized CP violation in B−→K−π+π− and B−→K−σ(600) decays by employing the quasi-two-body QCD factorization approach. Both the resonance and the nonresonance contributions are studied for the B−→K−π+π− decay. The resonance contributions include those not only from [ππ] channels including σ(600), ρ0(770) and ω(782) but also from [Kπ] channels including K0*(700)(κ), K*(892), K0*(1430), K*(1410), K*(1680) and K2*(1430). By fitting the four experimental data ACP(K−π+π−)=0.678±0.078±0.0323±0.007 for mK−π+2<15 GeV2 and 0.08<mπ+π−2<0.66 GeV2, ACP(B−→K0*(1430)π−)=0.061±0.032, B(B−→K0*(1430)π−)=(39−5+6)×10−6 and B(B−→σ(600)π−→π−π+π−)<4.1×10−6, we get the end-point divergence parameters in our model, ϕS∈[1.77,2.25] and ρS∈[2.39,4.02]. Using these results for ρS and ϕS, we predict that the CP asymmetry parameter ACP∈[−0.34,−0.11] and the branching fraction B∈[6.53,17.52]×10−6 for the B−→K−σ(600) decay. In addition, we also analyze contributions to the localized CP asymmetry ACP(B−→K−π+π−) from [ππ], [Kπ] channel resonances and nonresonance individually, which are found to be ACP(B−→K−[π+π−]→K−π+π−)=0.509±0.042, ACP(B−→[K−π+]π→K−π+π−)=0.174±0.025 and ACPNR(B−→K−π+π−)=0.061±0.0042, respectively. Comparing these results, we can see that the localized CP asymmetry in the B−→K−π+π− decay is mainly induced by the [ππ] channel resonances while contributions from the [Kπ] channel resonances and nonresonance are both very small.