A simple plan for the Delta resonance

Abstract

We construct the $\Delta$ resonance as a superposition of a bare $\Delta$ state and the $\pi N$ continuum. It is parametrized by three coupling constants for local $\pi N \Delta$ and $\pi \pi N N$ couplings and the $\Delta$ mass. The latter incorporates the mass renormalization due to the $\pi N \Delta$ interaction, while the results depend only weakly, if at all, on its wave-function renormalization. Three more renormalization constants are needed for the derivative contact interaction. They allow one to generate the $\Delta$ resonance dynamically. A large number of fits test the quality of different model assumptions in the $P_{33}$, $P_{31}$ and $P_{13}$ p-wave $\pi N$ scattering channels.