Eur. Phys. J. C (2021) 81: 806
https://doi.org/10.1140/epjc/s10052-021-09608-5

Regular Article - Theoretical Physics

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

¹ Department of Physics, College of Sciences, Nanjing Agricultural University, 210095, Nanjing, Jiangsu, China
² School of Mathematics and Physics, Changzhou University, 213164, Changzhou, Jiangsu, China
³ College of Sciences, North China University of Science and Technology, 063210, Tangshan, Hebei, China
⁴ Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, 210023, Nanjing, Jiangsu, China

^a liyakelly@163.com
^b yandac@126.com

Received: 25 July 2021
Accepted: 28 August 2021
Published online: 12 September 2021

Abstract

In this work, we provide estimates of the branching ratios, direct CP asymmetries and triple product asymmetries in $B_{(s)}\to (\pi \pi )(K\pi )$ decays in the perturbative QCD approach, where the $\pi \pi$ and $K\pi$ invariant mass spectra are dominated by the vector resonances $\rho (770)$ and $K^{\ast }(892)$, respectively. Some scalar backgrounds, such as $f_0(500,980)\to \pi \pi$ and $K_0^{\ast }(1430)\to K\pi$ are also accounted for. The $\rho (700)$ is parametrized by the Gounaris-Sakurai function. The relativistic Breit-Wigner formula for the $f_0(500)$ and Flatté model for the $f_0(980)$ are adopted to parameterize the time-like scalar form factors $F_S({\omega }^2)$. We also use the D.V. Bugg model to parameterize the $f_0(500)$ and compare the relevant theoretical predictions from different models. While in the region of $K\pi$ invariant mass, the $K_0^{\ast }(1430)$ is described with the LASS lineshape and the $K^{\ast }(892)$ is modeled by the Breit-Wigner function. We find that the decay rates for the considered decay modes agree with currently available data within errors. As a by-product, we extract the branching ratios of two-body decays $B_{(s)}\to \rho (770)K^{\ast }(892)$ from the corresponding four-body decay modes and calculate the relevant polarization fractions. Our prediction of longitudinal polarization fraction for $B^0\to \rho {(770)}^0{K}^{\ast }{(892)}^0$ decay deviates a lot from the recent LHCb measurement, which should be resolved. It is shown that the direct CP asymmetries are large due to the sizable interference between the tree and penguin contributions, but they are small for the tree-dominant or penguin-dominant processes. The PQCD predictions for the “true” triple product asymmetries are small which are expected in the standard model, and consistent with the current data reported by the LHCb Collaboration. Our results can be tested by the future precise data from the LHCb and Belle II experiments.