Phenomenology of low-energy ππ scattering

Exact representations of the low-energy ππ amplitudes by (for the S-waves) generalized effective range formulas and by (for the P-wave) a resonance plus background expression are derived from inverse-amplitude partial-wave dispersion relations, and applied to the phenomenological analysis of a typical set of phase shifts deduced from peripheral pion production. With the aid of rigorous crossing conditions it is found that the simplest satisfactory account of the S-waves is provided by a current algebra type solution with (Adler) zeros and small scattering lengths. Solutions without zeros (S-wave dominant) are completely ruled out. Inelasticity is important in describing the rise of the isoscalar phase shift above the σ resonance, but can be neglected below 1 GeV in other channels. The P-wave is ?-dominated, but its width is sensitive to background terms. The phenomenological importance of rigorous results is stressed.