The spectrum of light isovector mesons with $C=+1$ from the COMPASS experiment

Based on the largest event sample of diffractively produced $\pi^-\pi^-\pi^+$, obtained by a pion beam of $190~\rm{GeV/c}$ momentum, the COMPASS collaboration has performed the most advanced partial wave analysis on multi-body final states, using the isobar model. The large number of waves included in the analysis reduces truncation effects. We have used fourteen out of a total of eighty-eight waves, to extract resonance parameters for eleven light meson candidates, most of them observed previously. The coherence of the analysis and the large variety of systematic studies has allowed to determine mass and width of most $a_{J}$ and $\pi_{J}$ states with six different values of $J^{PC}$ below a mass of $2.1~\rm{GeV/c^2}$, with high confidence. We exploited that the production rates of resonant and non-resonant contributions in these fourteen waves vary differently with the four-momentum transfer squared in the reaction. In addition, we have performed the first isobar-freed analysis in diffraction, from which we have determined the shape of the $[\pi\pi]_{S-wave}$ isobar for different $J^{PC}$ of the $3\pi$ system.