Low-energy multi-photon scattering at tree-level and one-loop order in a homogeneous electromagnetic field

TL;DR

This paper develops a worldline approach to analyze low-energy multi-photon scattering in a homogeneous electromagnetic field, providing compact expressions and revealing new relationships between scalar and spinor amplitudes at tree and one-loop levels.

Contribution

It introduces a novel worldline formalism for low-energy photon scattering in electromagnetic fields, extending scalar-to-spinor amplitude relations to tree level.

Findings

Derived compact expressions for photon and background-dressed effective actions.

Extended scalar-to-spinor amplitude conversion rules to tree-level amplitudes.

Developed a new worldline Green function for momentum space propagators.

Abstract

We study low energy photons coupled to scalar and spinor matter in the presence of an arbitrary homogeneous electromagnetic field in a first-quantised (worldline) approach. Utilising a Fock-Schwinger gauge for both the scattering photons and homogeneous background, simple compact expressions are found for both the photon- and background-dressed effective action and propagator in scalar and spinor quantum electrodynamics. The low-energy limit allows identification of the coupling of the scattering photons as one of an effective homogeneous superposition of their field strengths, with amplitudes following from application of a suitable linearisation operator. To treat the linearisation, several techniques are employed, including a functional expansion based on the proper time formalism and worldline Green functions, linearised vertex operators under a worldline path integral, and a matrix expansion in the field strengths. We find, in particular, that a replacement rule converting scalar amplitudes to spinor amplitudes at one-loop order can, surprisingly, be extended to tree level amplitudes in the low energy limit. Finally, we discuss a novel worldline representation of the momentum space matter propagators, obtaining a suitable worldline Green function for this path integral satisfying homogeneous Dirichlet boundary conditions and momentum space vertex operators representing the scattering photons already in momentum space.