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

Abstract

The decay amplitude analyses of $B^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^{\pm }$ decays in the Dalitz plot performed by LHCb indicate that CP asymmetry for the dominant quasi-two-body decay $B^{\pm }\to \rho (770{)}^0{\pi }^{\pm }$ 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 $\rho (770{)}^0$ resonance and the $S$-wave were clearly observed. We show that the nearly vanishing CP violation in $B^{\pm }\to {\rho }^0{\pi }^{\pm }$ 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 $A_{CP}(B^{\pm }\to {\rho }^0{\pi }^{\pm })$ 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 $\rho$ and $S$-wave contributions in the low $m_{{\pi }^{+}{\pi }^{-}}$ region separated by the sign of value of $\cos \theta$ with $\theta$ being the angle between the two same charged pions measured in the rest frame of the $\rho$ resonance can be explained in terms of the smallness of $A_{CP}(B^{+}\to {\rho }^0{\pi }^{+})$ and the interference between $\rho (770)$ and $\sigma /f_0(500)$. If CP asymmetry in $B^{\pm }\to {\rho }^0{\pi }^{\pm }$ 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-$m_{{\pi }^{+}{\pi }^{-}}$ region between 0.5 and 1.0 GeV is consistent with a nearly vanishing CP violation in $B^{\pm }\to {\rho }^0{\pi }^{\pm }$.

Comments: 18 pages, 5 figures, accepted for publication in PRD. arXiv admin note: text overlap with arXiv:2005.06080, arXiv:2007.02558