Amplitude analysis of the $B^+ \to \pi^+ \pi^+ \pi^-$ decay

The results of an amplitude analysis of the charmless three-body decay $B^+ \to \pi^+ \pi^+ \pi^-$, in which $CP$-violation effects are taken into account, are reported. The analysis is based on a data sample corresponding to an integrated luminosity of $3 \; {\rm fb}^{-1}$ of $pp$ collisions recorded with the LHCb detector. The most challenging aspect of the analysis is the description of the behaviour of the $\pi^+ \pi^-$ S-wave contribution, which is achieved by using three complementary approaches based on the isobar model, the K-matrix formalism, and a quasi-model-independent procedure. Additional resonant contributions for all three methods are described using a common isobar model, and include the $\rho(770)^0$, $\omega(782)$ and $\rho(1450)^0$ resonances in the $\pi^+ \pi^-$ P-wave, the $f_2(1270)$ resonance in the $\pi^+ \pi^-$ D-wave, and the $\rho_3(1690)^0$ resonance in the $\pi^+ \pi^-$ F-wave. Significant $CP$-violation effects are observed in both S- and D-waves, as well as in the interference between the S- and P-waves. The results from all three approaches agree and provide new insight into the dynamics and the origin of $CP$-violation effects in $B^+ \to \pi^+ \pi^+ \pi^-$ decays.