Final state interactions in three-meson systems: Analysis of data on p¯p→${\mathrm{\pi }}^0$${\mathrm{\pi }}^0$${\mathrm{\pi }}^0$ and ηη${\mathrm{\pi }}^0$ at rest

We provide a theoretical and computational formalism for the study of three-body annihilation of pp¯. It includes (i) an analysis of three-body equations with the purpose of extracting the leading singularities and (ii) an analysis of low-energy ππ, KK¯, ηη, and πη amplitudes and the reconstruction of them from available experimental data. Based on this formalism, we perform an analysis of data on pp¯→${\mathrm{\pi }}^0$ ${\mathrm{\pi }}^0$ ${\mathrm{\pi }}^0$ and ηη${\mathrm{\pi }}^0$ at rest. The data are fitted well by pure ${}_{}{}^1{}_{}{}^{}$ ${\mathit{S}}_0$ annihilation and require two I=0,${\mathit{J}}^{\mathit{P}\mathit{C}}$=$0^{++}$ resonances with ${\mathit{M}}_1$=1335±40 MeV, ${\mathrm{\Gamma }}_1$=${255}_{\mathrm{-}40}^{+60\mathrm{}}$ MeV, and ${\mathit{M}}_2$=1505±20 MeV, ${\mathrm{\Gamma }}_2$=150±20 MeV. The latter explains features of the data previously interpreted in an alternative way as due to ${\mathit{A}}_{\mathit{X}}$(1515) with ${\mathit{J}}^{\mathit{P}\mathit{C}}$=$2^{++}$. As a second step, we evaluate the magnitude of the amplitudes due to triangle singularities which result from rescatterings of outgoing mesons. Taking into account these rescattering processes enhances the magnitudes of low-lying resonances, notably ${\mathit{f}}_0$(975).