Amplitude analysis of the $D^+$ → $π^-π^+π^+$ decay and measurement of the $π^-π^+$ S-wave amplitude
An amplitude analysis of the $D^+$ → $π^-π^+π^+$ decay is performed with a sample corresponding to 1.5 fb -1 of integrated luminosity of $pp$ collisions at a centre-of-mass energy $\sqrt{s}$ = 8 TeV collected by the LHCb detector in 2012. The sample contains approximately six hundred thousand candidates with a signal purity of 95%. The resonant structure is studied through a fit to the Dalitz plot where the $π^-π^+$ S-wave amplitude is extracted as a function of $π^-π^+$ mass, and spin-1 and spin-2 resonances are included coherently through an isobar model. The S-wave component is found to be dominant, followed by the $ρ(770)^0π^+$ and $f_2(1270)π^+$ components. A small contribution from the $ω$(782) → $π^-π^+$ decay is seen for the first time in the $D^+$ → $π^-π^+π^+$ decay.