Quark models and the structure of theΔ(1232)resonance

Abstract

We investigate the structure of the $\Delta \mathrm{}\left(1232\right)\mathrm{}$ resonance within a model which contains pions ($\pi$) interacting with nucleons ($N$) and $\Delta$'s. With a $\pi \mathrm{NN}$ (or $\pi N\Delta$) vertex form factor obtained from quark models, we demonstrate explicitly that the perturbation series of $\pi -N$ scattering converges rapidly. Thus a crossing-symmetric solution to the $\pi -N$ scattering problem can be obtained reliably by perturbation expansion. We expand the $\pi -N$ scattering series to fourth order in renormalized coupling constants, and then the (3,3) phase shift is calculated using both $T$ and $K$ matrices. The $\Delta$ contribution is found to be dominant in generating the observed resonance. Our calculation and results are compared with those of the cloudy bag model.