Quasi-two-body decays \(B_{(s)}\to P f_2(1270)\to P\pi\pi\) in the perturbative QCD approach

Abstract

In this work, we calculate the \(CP\)-averaged branching ratios and direct \(CP\)-violating asymmetries of the quasi-two-body decays \(B_{(s)}\to P f_2(1270)\to P\pi\pi\) with the two-pion distribution amplitude \(\Phi_{\pi\pi}^{\rm D}\) by using the perturbative QCD factorization approach, where \(P\) represents a light pseudoscalar meson \(K, \pi, \eta\) and \(\eta^{\prime}\). The relativistic Breit-Wigner formula for the \(D\)-wave resonance \(f_2(1270)\) is adopted to parameterize the time-like form factor \(F_{\pi}\), which contains the final state interactions between the pions in the resonant regions. The consistency of theoretical results with data can be achieved by determining the Gegenbauer moments of the \(D\)-wave two-pion distribution amplitudes. The decay rates for the considered decay modes are generally in the order of \(10^{-9}\) to \( 10^{-6}\). The integrated direct \(CP\) asymmetries for the charged modes agree with the BABAR and Belle measurements. As a by-product, we extract the branching ratios of \(B_{(s)}\to Pf_2(1270)\) from the corresponding quasi-two-body decay modes, which still need experimental tests at the ongoing and forthcoming experiments.