Phenomenological $\pi N \to \pi \pi N$ amplitude and analysis of low energy data on total cross sections and 1–D distributions

We develop the phenomenological amplitude of the $\pi N \to \pi \pi N$ reaction taking into account the exchanges of $\Delta$ and $N^{(*)}$ along with the OPE mechanism and the polynomial background derived with the account of isotopic, crossing, $ C $, $ P $ and $ T $ symmetries of strong interactions. The data base consisting of total cross sections in the energy region $ 0.300 \le P_{\rm Lab} \le 500 $ MeV/c and 1D distributions from the bubble–chamber experiments was used to determine free parameters of the amplitude. For the first time there was found an amplitude equally well describing the broad variety of different data in all reaction channels. The best solutions are characterized by $ \chi^{2}_{\rm DF} = $ 1.16, $ N = 411 $. The isobar exchanges are found to be more important than OPE at the considered energies. Despite this the parameters of OPE are statistically significant; however, the $ \pi \pi $–scattering lengths appear different in different solutions