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

Ivan Ahumada*, Patrick Copinger†, and James P. Edwards‡

  • *Contact author: ivan.ahumadahernandez@plymouth.ac.uk
  • †Contact author: patrick.copinger@plymouth.ac.uk
  • ‡Contact author: james.p.edwards@plymouth.ac.uk
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Phys. Rev. D 112, 096024 – Published 24 November, 2025

DOI: https://doi.org/10.1103/p2d3-k25t

Abstract

We study low energy photons coupled to scalar and spinor matter in the presence of an arbitrary homogeneous electromagnetic field in a first-quantized (worldline) approach. Utilizing 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 linearization operator. To treat the linearization, several techniques are employed, including a functional expansion based on the proper time formalism and worldline Green’s functions, linearized 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’s function for this path integral satisfying homogeneous Dirichlet boundary conditions and momentum space vertex operators representing the scattering photons already in momentum space.

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Phys. Rev. D 112, 096024– Published 24 November, 2025
  • Received 8 August 2025
  • Accepted 4 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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