Effective-Lagrangian approach to the theory of pion photoproduction in theΔ(1232)region

Abstract

We investigate theoretical uncertainties and model dependence in the extraction of the nucleon-$\Delta \mathrm{}\left(1232\right)\mathrm{}$ electromagnetic transition amplitudes from the multipole data base. Our starting point is an effective Lagrangian incorporating chiral symmetry, which includes, at the tree level, the pseudovector nucleon Born terms, leading $t$-channel vector-meson exchanges, and $s$- and $u$-channel $\Delta$ exchanges. We express the nucleon-$\Delta$ transition magnetic dipole ($M1\mathrm{}$) and electric quadrupole ($E2\mathrm{}$) amplitudes in terms of two independent gauge couplings at the $\gamma N\Delta$ vertex, and fit these to various multipole data sets. We find a large sensitivity to the method used in unitarizing the amplitude, and extract the $\frac{E2\mathrm{}}{M1\mathrm{}}$ ratio (EMR) to be negative, with a magnitude of around 1.5%. The resonant amplitudes in this work are of interest to the test of topical hadron models inspired by QCD: the sign of the EMR, extracted by us, is in accord with that predicted by most realistic models, and its magnitude lies between the predictions of the quark shell model and the Skyrmion model. Finally, our work provides a phenomenologically satisfying unitary amplitude for pion photoproduction off nucleons, which can be used as a realistic starting point for theoretical studies of this process in complex nuclei.