Model of Low-Energy Pion-Pion Scattering

A model of the low-energy pion-pion scattering amplitudes is constructed which is consistent with the usual partial-wave analyticity, elastic unitarity, and crossing symmetry in the $s$- and $p$- wave channels. The crossing relations are expressed as five Roskies conditions on the $s$- and $p$- wave amplitudes. The subtraction constants are chosen to agree with currentalgebra constraints. The partial-wave amplitudes are constructed by the $\frac{N}{D}$ method after approximating the left-hand singularities by a few poles. The pole structure is then determined self-consistently by requiring the Roskies conditions and the current-algebra conditions on the derivatives of the amplitudes at subtraction points to be satisfied. A solution is found which contains the $\rho$ and $\sigma$ mesons as dynamical bound states of the two-pion system. The resulting amplitudes, which should be a good approximation below 1 GeV, are in qualitative agreement with available experimental data. The $I=0\mathrm{}$, $s$-wave amplitude has a broad resonance at about 700 MeV. The location of the $\rho$ meson is in agreement with experiment, but the width is somewhat larger than the experimental value. The $I=2\mathrm{}$, $s$-wave phase shifts are repulsive and small, in agreement with experiment.