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

TL;DR

This work addresses low-energy photon scattering in a homogeneous background field within strong-field QED, leveraging a first-quantised worldline formalism. The authors develop three complementary approaches to obtain master formulae for both scalar and spinor matter: (i) a functional-expansion method around a constant background, (ii) a worldline linearisation that re-expresses $N$-photon insertions as couplings to an effective homogeneous field, and (iii) a momentum-space path integral with Dirichlet boundary conditions that yields a momentum-space vertex operator. They derive explicit results for the one-loop effective action and the propagator, including open-line endpoint contributions, and show a replacement rule that maps scalar amplitudes to spinor amplitudes at low energy, extended to open lines. A novel momentum-space representation and recursion in cycle integrals reduce the parameter integrals to tractable forms, enabling high-multiplicity low-energy amplitudes in homogeneous backgrounds. The framework provides a versatile toolkit for analyzing light-by-light scattering and related processes in SFQED, with potential applications to strong-field laser physics and astrophysical settings.

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.