# Electromagnetic currents for dressed hadrons

## TL;DR

This paper develops an extended method for coupling electromagnetic fields to dressed hadrons, ensuring gauge invariance and revealing that dressing effects cancel in on-shell processes, emphasizing the importance of contact interactions.

## Contribution

It introduces an extended substitution scheme for electromagnetic couplings that accounts for hadronic internal structure and form factors, improving upon previous minimal coupling approaches.

## Key findings

- Dressed currents are gauge invariant for pions and nucleons.
- Dressing effects cancel in on-shell Compton scattering, leaving only bare contributions.
- Contact two-photon processes are necessary for nontrivial dressing effects.

## Abstract

We propose an extension of the minimal-substitution prescription for coupling the electromagnetic field to hadronic systems with internal structure. The resulting rules of extended substitution necessarily distinguish between couplings to scalar and Dirac particles. Moreover, they allow for the incorporation of electromagnetic form factors for virtual photons in an effective phenomenological framework. Applied to pions and nucleons, assumed to be fully dressed to all orders, the resulting dressed currents are shown to be locally gauge invariant. Moreover, half-on-shell expressions of (hadron propagator)$\times$(electromagnetic current) needed in all descriptions of physical processes will lose \textit{all} information about hadronic dressing for real photons. The Ball-Chiu ansatz for the spin-1/2 current is seen to suffer from an incomplete coupling procedure where some essential aspects of the Dirac particle are effectively treated as those of a scalar particle. Applied to real Compton scattering on pions and nucleons, we find that \emph{all} dressing information cancels exactly when external hadrons are on shell, leaving only gauge-invariant bare Born-type contributions with physical masses. Hence, nontrivial descriptions necessarily require contact-type two-photon processes obtained by hadrons looping around two photon insertion points.