Study of $$B_{(s)} \rightarrow (\pi \pi )(K\pi )$$ decays in the perturbative QCD approach

In this work, we provide estimates of the branching ratios, direct $CP$ asymmetries and triple product in $B_{(s)} \to (\pi\pi)(K\pi)$ decays perturbative QCD approach, where $\pi\pi$ $K\pi$ invariant mass spectra are dominated by vector resonances $\rho(770)$ $K^*(892)$, respectively. Some scalar backgrounds, such as $f_0(500,980) \pi\pi$ $K^*_0(1430) K\pi$ also accounted for. The $\rho(700)$ is parametrized Gounaris-Sakurai function. relativistic Breit-Wigner formula for $f_0(500)$ Flatt\'e model $f_0(980)$ adopted to parameterize time-like form factors $F_S(\omega^2)$. We use D.V. Bugg compare relevant theoretical predictions from different models. While region mass, $K^*_0(1430)$ described with LASS lineshape $K^*(892)$ modeled find that decay rates considered modes agree currently available data within errors. As a by-product, extract ratios two-body \rho(770)K^*(892)$ corresponding four-body calculate polarization fractions. Our prediction longitudinal fraction $B^0\to \rho(770)^0 K^*(892)^0$ deviates lot recent LHCb measurement, which should be resolved. It shown large due sizable interference between tree penguin contributions, but they small tree-dominant or penguin-dominant processes. PQCD expected standard model, consistent current reported Collaboration.Our results can tested future precise Belle II experiments.