A Covariant model for the nucleon and the $\Delta$

TL;DR

This paper develops a covariant quark model for the nucleon and Delta baryon, describing their electromagnetic properties and transition form factors, highlighting the importance of wave function components with orbital angular momentum.

Contribution

It introduces a covariant spectator formalism with zero orbital angular momentum wave functions to model nucleon and Delta electromagnetic form factors, including the gamma N to Delta transition.

Findings

All four nucleon electromagnetic form factors are described.

The magnetic transition form factor G_M* is modeled but requires pion cloud effects near Q^2=0.

Pure S-wave models yield zero electric and Coulomb transition form factors.

Abstract

The covariant spectator formalism is used to model the nucleon and the $\Delta$(1232) as a system of three constituent quarks with their own electromagnetic structure. The definition of the ``fixed-axis'' polarization states for the diquark emitted from the initial state vertex and absorbed into the final state vertex is discussed. The helicity sum over those states is evaluated and seen to be covariant. Using this approach, all four electromagnetic form factors of the nucleon, together with the {\it magnetic} form factor, $G_M^*$, for the $\gamma N \to \Delta$ transition, can be described using manifestly covariant nucleon and $\Delta$ wave functions with {\it zero} orbital angular momentum $L$, but a successful description of $G_M^*$ near $Q^2=0$ requires the addition of a pion cloud term not included in the class of valence quark models considered here. We also show that the pure $S$-wave model gives electric, $G_E^*$, and coulomb, $G^*_C$, transition form factors that are identically zero, showing that these form factors are sensitive to wave function components with $L>0$.