Dressed Dirac Propagator from a Locally Supersymmetric ${\cal N}=1$ Spinning Particle

TL;DR

This paper develops a worldline supersymmetric approach to compute the dressed Dirac propagator with multiple photons, providing a compact formula and novel fermionic state representation.

Contribution

It introduces a new worldline supersymmetric method using a coherent state basis and gravitino integration to derive the dressed Dirac propagator.

Findings

Derived a compact off-shell master formula for scattering amplitudes.

Implemented a coherent state basis for fermionic degrees of freedom.

Reproduced the numerator of the dressed Dirac propagator via gravitino path integral.

Abstract

We study the Dirac propagator dressed by an arbitrary number $N$ of photons by means of a worldline approach, which makes use of a supersymmetric ${\cal N} = 1$ spinning particle model on the line, coupled to an external Abelian vector field. We obtain a compact off-shell master formula for the tree level scattering amplitudes associated to the dressed Dirac propagator. In particular, unlike in other approaches, we express the particle fermionic degrees of freedom using a coherent state basis, and consider the gauging of the supersymmetry, which ultimately amounts to integrating over a worldline gravitino modulus, other than the usual worldline einbein modulus which corresponds to the Schwinger time integral. The path integral over the gravitino reproduces the numerator of the dressed Dirac propagator.