Treatment of overlapping resonances in 3-body systems and the generalization of the isobar approximation

Abstract

An essential problem when making an isobar approximation in the case of overlapping resonances is the fact that diagrams which contain one particle exchange processes, as e.g. nucleon exchange in the πΔ system, give singularities above the physical threshold and a complicated dependence of the S-matrix on the total energy and the sub-energies. In this paper we treat these singularities explicitly in a manner which is inspired by Faddeev theory. We define resonance-particle scattering amplitudes in a consistent way which fulfils 3-particle unitarity exactly. These amplitudes can be calculated using dispersion relations or the N D method. An important result is that the exchange diagram corresponding to the so called Peierls mechanism should not be regarded as a driving force. It influences the result through a slight change of the resonance particle phase space. An application to the πΔ system will be presented in a subsequent paper.