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

Abstract

In this work, we provide estimates of the branching ratios, direct CP asymmetries and triple product asymmetries in $$B_{(s)} \rightarrow (\pi \pi )(K\pi )$$ B ( s ) → ( π π ) ( K π ) decays in the perturbative QCD approach, where the $$\pi \pi $$ π π and $$K\pi $$ K π invariant mass spectra are dominated by the vector resonances $$\rho (770)$$ ρ ( 770 ) and $$K^*(892)$$ K ∗ ( 892 ) , respectively. Some scalar backgrounds, such as $$f_0(500,980) \rightarrow \pi \pi $$ f 0 ( 500 , 980 ) → π π and $$K^*_0(1430) \rightarrow K\pi $$ K 0 ∗ ( 1430 ) → K π are also accounted for. The $$\rho (700)$$ ρ ( 700 ) is parametrized by the Gounaris-Sakurai function. The relativistic Breit-Wigner formula for the $$f_0(500)$$ f 0 ( 500 ) and Flatté model for the $$f_0(980)$$ f 0 ( 980 ) are adopted to parameterize the time-like scalar form factors $$F_S(\omega ^2)$$ F S ( ω 2 ) . We also use the D.V. Bugg model to parameterize the $$f_0(500)$$ f 0 ( 500 ) and compare the relevant theoretical predictions from different models. While in the region of $$K\pi $$ K π invariant mass, the $$K^*_0(1430)$$ K 0 ∗ ( 1430 ) is described with the LASS lineshape and the $$K^*(892)$$ K ∗ ( 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 $$B_{(s)} \rightarrow \rho (770)K^*(892)$$ B ( s ) → ρ ( 770 ) K ∗ ( 892 ) from the corresponding four-body decay modes and calculate the relevant polarization fractions. Our prediction of longitudinal polarization fraction for $$B^0\rightarrow \rho (770)^0 K^*(892)^0$$ B 0 → ρ ( 770 ) 0 K ∗ ( 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.

Study of the four-body decays $B^{0}_{S} \rightarrow ππππ$ in the perturbative QCD approach

“…At the same time, for the two-meson DAs, we will discuss the S-wave and P-wave via intermediate resonances f 0 (980), f 0 (500), ρ(770) and ω(782). The corresponding time-like form factors of them are collected below [48]:…”

“…Here φ 0 P (x, ω) and φ T P (x, ω) are twist-2 DAs, φ s P (x, ω), φ t P (x, ω), φ a P (x, ω) and φ v P (x, ω) are twist-3 DAs, which are associating to the longitudinal and transverse polarization. a 0,s,t V and a T,a,v V are Gegenbauer moments, which are determined in Ref [48]. For the time-like factors F P and F ⊥ P , we postulate the approximation F ⊥ P = (f T V /f V )F P .…”

“…The six helicity amplitudes are involved in four body decay of B meson and the relation between total amplitude and other components can be found in Refs [47,48,55]. Then branching ratio from Eq.…”

“…However, the part above the 1/b energy scale depends on differential decay channels, and the Feynman diagram numerical calculation is carried out by using the perturbation theory. For the four-body decay B 0 S → N 1 N 2 → ππππ, the amplitude can be written as [47,48].…”

“…For the resonances ρ 0 , we adopt the Gounaris-Sakurai model, the BW model is used in resonances ω, the f 0 (980) is parameterized by F latté model and f 0 (500) is modeled by the Breit-Wigner function [49][50][51]. The models of the pion pair vector or scalar resonances have been borrowed for the study of quasi-two-body B meson decays and the range of invariant mass in ππ pair varies from 300 MeV to 1100 MeV [9,48]. Besides, the four-body decays mainly cover six helicity amplitudes A h with h = V V (3), V S, SV and SS.…”