CP asymmetry from the effect of the isospin symmetry breaking during B-meson decay

The direct CP asymmetry in quasi-two-body decays of $B\to (V\to {\pi }^{+}{\pi }^{-})P$ is investigated in the perturbative QCD method, where P represents a pseudoscalar meson and V refers to $\rho$, $\omega$ and $\phi$ mesons, respectively. We present the amplitude of the quasi-two-body decay process and investigate the effects of mixed resonances involving ${\rho }^0-\omega$, ${\rho }^0-\phi$ and $\omega -\phi$, while considering the impact of isospin symmetry breaking. We observe a significant CP asymmetry when the invariant mass of the ${\pi }^{+}{\pi }^{-}$ pair is within the resonance ranges of $\rho$, $\omega$ and $\phi$ mesons. Consequently, we proceed to quantify the regional CP asymmetry in these resonance regions. A significant difference is observed when comparing results obtained with and without interferences of the three vector mesons and isospin conservation. The CP asymmetry results obtained from the three-body decay process, without interference due to isospin conservation by the perturbative QCD method, are in agreement with the newly updated data acquired by the LHCb experiment.