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 $3{\mathrm{fb}}^{-1}$ of $pp$ collisions recorded with the LHCb detector. The most challenging aspect of the analysis is the description of the behavior 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.

• Received 16 September 2019

DOI:https://doi.org/10.1103/PhysRevD.101.012006

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

© 2020 CERN, for the LHCb Collaboration