CP violation in the interference between $ρ(770)^0$ and $S$-wave in $B^\pm\to π^+ π^-π^\pm$ decays and its implication for CP asymmetry in $B^\pm\to ρ^0π^\pm$

Abstract

The decay amplitude analyses of $B^\pm\to π^+ π^-π^\pm$ decays in the Dalitz plot performed by LHCb indicate that CP asymmetry for the dominant quasi-two-body decay $B^\pm\toρ(770)^0π^\pm$ was found to be consistent with zero in all three approaches for the $S$-wave component and that CP-violation effects related to the interference between the $ρ(770)^0$ resonance and the $S$-wave were clearly observed. We show that the nearly vanishing CP violation in $B^\pm\toρ^0π^\pm$ is understandable within the framework of QCD factorization. There are two $1/m_b$ power corrections, one from penguin annihilation and the other from hard spectator interactions. They contribute destructively to $A_{CP}(B^\pm\toρ^0π^\pm)$ to render it compatible with zero, in contrast to the sizable negative CP asymmetry predicted in most of the existing models. We next show that the measured interference pattern between the $ρ$ and $S$-wave contributions in the low $m_{π^+π^-}$ region separated by the sign of value of $\cosθ$ with $θ$ being the angle between the two same charged pions measured in the rest frame of the $ρ$ resonance can be explained in terms of the smallness of $A_{CP}(B^+\toρ^0π^+)$ and the interference between $ρ(770)$ and $σ/f_0(500)$. If CP asymmetry in $B^\pm\toρ^0π^\pm$ is not negligible as predicted in many existing models, the observed interference pattern will be destroyed. We conclude that the experimental observation of the interference pattern between $P$- and $S$-waves in the low-$m_{π^+π^-}$ region between 0.5 and 1.0 GeV is consistent with a nearly vanishing CP violation in $B^\pm\toρ^0π^\pm$.