Eur. Phys. J. C (2024) 84: 754
https://doi.org/10.1140/epjc/s10052-024-13087-9

Regular Article - Theoretical Physics

CP-violating observables of four-body $B_{(s)}\to (\pi \pi )(K\overline{K})$ decays in perturbative QCD

¹ School of Mathematics and Physics, Changzhou University, 213164, Changzhou, Jiangsu, China
² College of Sciences, North China University of Science and Technology, 063210, Tangshan, Hebei, China

^c jindui1127@126.com

Received: 7 May 2024
Accepted: 1 July 2024
Published online: 28 July 2024

Abstract

In this work, we investigate six helicity amplitudes of the four-body $B_{(s)}\to (\pi \pi )(K\overline{K})$ decays via an angular analysis in the perturbative QCD (PQCD) approach. The $\pi \pi$ invariant mass spectrum is dominated by the vector resonance $\rho (770)$ together with scalar resonance $f_0(980)$, while the vector resonance $\varphi (1020)$ and scalar resonance $f_0(980)$ are expected to contribute in the $K\overline{K}$ invariant mass range. We extract the two-body branching ratios $\mathcal{B}(B_{(s)}\to \rho \varphi )$ from the corresponding four-body decays $B_{(s)}\to \rho \varphi \to (\pi \pi )(K\overline{K})$ based on the narrow width approximation. The predicted $\mathcal{B}(B_s^0\to \rho \varphi )$ agrees well with the current experimental data within errors. The longitudinal polarization fractions of the $B_{(s)}\to \rho \varphi$ decays are found to be as large as $90\%$, basically consistent with the previous two-body predictions within uncertainties. In addition to the direct CP asymmetries, the triple-product asymmetries (TPAs) originating from the interference among various helicity amplitudes are also presented for the first time. Since the $B_s^0\to {\rho }^0\varphi \to ({\pi }^{+}{\pi }^{-})(K^{+}{K}^{-})$ decay is induced by both tree and penguin operators, the values of the ${\mathcal{A}}_{\text{dir}}^{\text{CP}}$ and ${\mathcal{A}}_{\text{T-true}}^1$ are calculated to be $(21.8_{-3.3}^{+2.7})\%$ and $(-10.{23}_{-1.56}^{+1.73})\%$ respectively. While for pure penguin decays $B^0\to {\rho }^0\varphi \to ({\pi }^{+}{\pi }^{-})(K^{+}{K}^{-})$ and $B^{+}\to {\rho }^{+}\varphi \to ({\pi }^{+}{\pi }^0)(K^{+}{K}^{-})$, both the direct CP asymmetries and “true” TPAs are naturally expected to be zero in the standard model (SM) due to the absence of the weak phase difference. The “fake” TPAs requiring no weak phase difference are usually none zero for all considered decay channels. The sizable “fake” ${\mathcal{A}}_{\text{T-fake}}^1=(-20.{92}_{-2.80}^{+6.26})\%$ of the $B^0\to {\rho }^0\varphi \to ({\pi }^{+}{\pi }^{-})(K^{+}{K}^{-})$ decay is predicted in the PQCD approach, which provides valuable information on the final-state interactions. The above predictions can be tested by the future LHCb and Belle-II experiments.