Dispersion relations and low – energy meson - meson interactions

This thesis deals with some general work on the use of inverse amplitude dispersion relations to describe low-energy ππ and πK scattering, and how the sub-threshold amplitudes may then be used to describe noa-leptonic decays. In chapter one we introduce the ideas which form the background to the structure of meson-iueson scattering. In chapter two we investigate a four parameter family of solutions to the ππ partial-wave dispersion relations using the inverse amplitude method assuming elastic unitarity. The S-waves have sub-threshold zeros consistent with the Adler condition and inelastic effects are estimated and found to be small below the rho-meson mass. In chapter three we analytically continue the sub-threshold ππ amplitude found previously to fit the structure of the Dalitz plot in the non-leptonic decays K→3π and ∫→3π. In chapter four we review the unitary effective-range expansions which have been used to describe ππ scattering, and we examine a new unitary effective-range expansion which we use to describe the S-waves of πK scattering giving some estimate of the left-hand cut contributions to the amplitude. In chapter five we extend these amplitudes by making a careful analysis of the left-hand cut and circle cut contributions to the π K partial-wave dispersion relations using the inverse amplitude method. Finally in chapter six we investigate how the ∫ and its associated SU(3) generalization, the Ϭ (962), fit into the overall picture we are able to conclude from our calculations.