Observation of several sources of $CP$ violation in $B^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ decays

Observations are reported of different sources of $CP$ violation from an amplitude analysis of $B^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ decays, based on a data sample corresponding to an integrated luminosity of $3{\mathrm{f}\mathrm{b}}^{-1}$ 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 ${\pi }^{+}{\pi }^{-}$ S-wave at low invariant mass. The presence of $CP$ violation related to interference between the ${\pi }^{+}{\pi }^{-}$ S-wave and the P-wave $B^{+}\to \rho (770{)}^0{\pi }^{+}$ 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.