Exact integral equation for pion-pion scattering involving only physical region partial waves

Abstract

We derive an exact relation for ππ scattering which yields the real parts of the ππ partial wave amplitudes a_l^(I)(s) in the region 4m²_π≤s≤60m²_π, is given i) the I=0 and the I=2 S wave scattering lengths and ii) the 1m a_l^(I)(s) for 4m²_π≤s≤∞, where √s denotes the centre-of-mass energy, l the angular momentum and I the isotopic spin. It also provides i) a system of integral equations to determine the a_l^(I)(s) for 4m²_π≤s≤16m²_π, 16²_π≤s≤∞, and ii) necessary and sufficient conditions for crossing symmetry expressed in terms of the physical region partial wave amplitudes only.