Pion–nucleon amplitude near threshold: the sigma-term and scattering lengths beyond few loops

The pion–nucleon amplitude is considered in the vicinity of the elastic scattering threshold within a relativistic dynamical model dressing the πNN and πNΔ vertices self-consistently with an infinite number of meson loops. The dressing is formulated as solution of a system of coupled integral equations incorporating unitarity, crossing symmetry and analyticity constraints. The calculated scattering lengths and the sigma-term agree with recent data analyses. The dressing is important in this model both below and at threshold. The contribution of the Δ resonance is discussed, including effects of the consistent dressing of the πNΔ vertex. A comparison with the approaches of chiral perturbation theory and the Bethe–Salpeter equation is outlined.