Low-Energy Pion-Pion Interaction

Low-energy pion-pion scattering is assumed to be dominated by the $\rho$ resonance in the $I=1\mathrm{}$, $J=1\mathrm{}$ channel, the $f^0$ resonance in the $I=0\mathrm{}$, $J=2\mathrm{}$ channel, and a broad resonance of mass about 700 MeV in the $I=0\mathrm{}$, $J=0\mathrm{}$ channel. This simple model is found to be in substantial agreement with: (a) Adler's sum rule for pion-pion scattering, (b) the ${K_1}^0-{K_2}^0$ mass difference, (c) $t$-channel contributions to backward pion-nucleon dispersion relation, (d) Malamud and Schlein's "experimental" determination of the pion-pion phase shifts, (e) constraints on the $s$-wave pion-pion phase shifts at the $K$ mass deduced from the two pion decays of the neutral $K$ mesons, (f) characteristics of the $K_{e4\mathrm{}}$ decay, and (g) odd-pion spectral shape for $K$ decay into three pions.