Quasi-two-body decays $B_{(s)}\to Pρ\to Pππ$ in perturbative QCD approach

Abstract

In this work, we calculate the $CP$-averaged branching ratios and the direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)} \to P (ρ\to) ππ$ by employing the perturbative QCD (PQCD) approach (here $P$ stands for a light pseudoscalar meson $π, K, η$ or $η^{\prime}$). The vector current time-like form factor $F_π$, which contains the final state interactions between the pion pair in the resonant region associated with the $P$-wave states $ρ(770)$ along with the two-pion distribution amplitudes, are employed to describe the interactions between the $ρ$ and the pion pair under the hypothesis of the conserved vector current. We found that (a) the PQCD predictions for the branching ratios and the direct $CP$-violating asymmetries for most considered $B_{(s)} \to P (ρ\to) ππ$ decays agree with currently available data within errors; (b) for ${\cal B}(B \to π^0 ρ^0 \to π^0(π^+ π^-)$, the PQCD prediction is much smaller than the measured one; and (c) for $B^+ \to π^+(ρ^0\to)π^+ π^-$ decay mode, we found a negative $CP$ asymmetry $(-27.5^{+3.0}_{-3.7})\%$, which agrees with other theoretical predictions but different in sign from those as reported by BaBar and LHCb Collaboration.

13 pages, 1 figure