Pion-Pion Scattering and Low-Energy Pion Production

A model for the reaction $\pi +N\to 2\pi +N$ at low energies, with $\pi -\pi$ scattering lengths as parameters, discussed in a previous analysis is extended by removing the assumption that $\frac{a_0}{a_2}=\frac{5}{2}$. The parameters are determined by using the ${\pi }^{+}$ angular distribution from ${\pi }^{-}+p\to {\pi }^{-}+{\pi }^{+}+n$ at 430 Mev and the total cross section for ${\pi }^{+}+p\to {\pi }^{+}+{\pi }^{+}+n$ at 470 Mev. We find two acceptable sets for the $\pi -\pi$ scattering lengths: $a_0=0.50\mathrm{}$ ${\mathrm{\mu }}^{-1\mathrm{}}$, $a_1=0.07\mathrm{}$ ${\mathrm{\mu }}^{-1\mathrm{}}$, $a_2=0.16\mathrm{}$ ${\mathrm{\mu }}^{-1\mathrm{}}$ and $a_0=0.65\mathrm{}$ ${\mathrm{\mu }}^{-1\mathrm{}}$, $a_1=0.07\mathrm{}$ ${\mathrm{\mu }}^{-1\mathrm{}}$, $a_2=-0.14\mathrm{}$ ${\mathrm{\mu }}^{-1\mathrm{}}$. The total cross sections for the various charge channel and for single partial waves are computed. The predictions of this model are in qualitative agreement with experiment.