Parametrizations of the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors and Siegert's theorem

TL;DR

This paper improves the parametrization of the $ o ext{Delta}(1232)$ quadrupole form factors using large $N_c$ relations, correcting violations of Siegert's theorem, and combines pion cloud and valence quark models for better data agreement.

Contribution

It introduces corrected parametrizations that respect Siegert's theorem and combines pion cloud and valence quark models for accurate form factor extrapolations.

Findings

Corrected electric quadrupole form factor parametrization reduces Siegert's theorem violation to order $1/N_c^4$.

The combined model achieves good agreement with experimental data across low, intermediate, and timelike $Q^2$ regions.

Extrapolation to the timelike region is consistent with theoretical constraints and experimental observations.

Abstract

The large $N_c$ limit provides relations that can be used to calculate the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors at low and intermediate $Q^2$ under the assumption of the pion cloud dominance. There are two limitations in those parametrizations. First, the parametrization of the Coulomb quadrupole form factor underestimate the low $Q^2$ data. Second, when extrapolated for the timelike region, the form factors violate Siegert's theorem by terms of the order $1/N_c^2$. We propose here corrections to the parametrization of the electric quadrupole form factor, which violate Siegert's theorem only by terms of the order $1/N_c^4$. Combining the improved large $N_c$ pion cloud parametrizations with the valence quark contributions based on a covariant quark model for the quadrupole transition form factors, we obtain an extrapolation to the timelike region consistent with Siegert's theorem, and accomplish also a very good description of the data.