Observation of several sources of $CP$ violation in $B^+ \to π^+ π^+ π^-$ decays
Observations are reported of different sources of $CP$ violation from an amplitude analysis of $B^+ \to π^+ π^+ π^-$ decays, based on a data sample corresponding to an integrated luminosity of $3 \; {\rm fb}^{-1}$ of $pp$ collisions recorded with the LHCb detector. A large $CP$ asymmetry is observed in the decay amplitude involving the tensor $f_2(1270)$ resonance, and in addition significant $CP$ violation is found in the $π^+ π^-$ S-wave at low invariant mass. The presence of $CP$ violation related to interference between the $π^+ π^-$ S-wave and the P-wave $B^+ \to ρ(770)^0 π^+$ amplitude is also established; this causes large local asymmetries but cancels when integrated over the phase space of the decay. The results provide both qualitative and quantitative new insights into $CP$-violation effects in hadronic $B$ decays.