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Evaluation of QED cross sections in strong magnetic fields

Olavi Kiuru

Abstract

Quantum electrodynamics (QED) becomes nonlinear when the magnetic field strength surpasses the critical Schwinger limit $B_Q \approx 4.41\cdot 10^{13}$ G. This limit is surpassed, for example, in the magnetospheres of a specific class of neutron stars known as magnetars, which has important consequences for magnetospheric plasma dynamics due to modifications in scattering cross sections. Using a formalism previously applied to the study of magnetic catalysis, I calculate the cross sections of all tree-level 1-to-2, 2-to-1, and 2-to-2 particle QED scattering processes that do not include a photon propagator. The calculations are done in a strong background magnetic field and the results are implemented into an open-source Python package. This article focuses on presenting the formalism and computational techniques required for the calculations, while the impact of the results on, e.g., magnetospheric plasma dynamics is discussed in a companion letter (Kiuru et al. 2026).

Evaluation of QED cross sections in strong magnetic fields

Abstract

Quantum electrodynamics (QED) becomes nonlinear when the magnetic field strength surpasses the critical Schwinger limit G. This limit is surpassed, for example, in the magnetospheres of a specific class of neutron stars known as magnetars, which has important consequences for magnetospheric plasma dynamics due to modifications in scattering cross sections. Using a formalism previously applied to the study of magnetic catalysis, I calculate the cross sections of all tree-level 1-to-2, 2-to-1, and 2-to-2 particle QED scattering processes that do not include a photon propagator. The calculations are done in a strong background magnetic field and the results are implemented into an open-source Python package. This article focuses on presenting the formalism and computational techniques required for the calculations, while the impact of the results on, e.g., magnetospheric plasma dynamics is discussed in a companion letter (Kiuru et al. 2026).

Paper Structure

This paper contains 31 sections, 188 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Feynman rules
  4. External electron legs
  5. External positron legs
  6. External photon legs
  7. Fermion--photon vertex
  8. Fermion propagator
  9. Photon propagator
  10. Calculating cross sections in a magnetic field
  11. $\mathcal{O}(\alpha_e)$ cross sections
  12. Synchrotron radiation
  13. One-photon pair creation
  14. One-photon pair annihilation
  15. Synchrotron self-absorption
  16. ...and 16 more sections

Figures (7)

  • Figure 1: The fermion propagator in the Furry picture, denoted by a double line with an arrow in the middle, includes all possible interactions with the background magnetic field, denoted here by squiggly lines ending with a $\times$fedotov_advances_2023.
  • Figure 2: Feynman diagram of Compton scattering. Electrons are denoted by $e^-$ and are drawn as double lines with an arrow in the middle. Photons are denoted by $\gamma$ and are drawn as squiggly lines. The Lorentz index of their polarization vectors is written next to the $\gamma$. The direction of the momentum of the particles is denoted by external arrows.
  • Figure 3: Feynman diagram of synchrotron radiation.
  • Figure 4: Fermion self-energy at 1-loop order.
  • Figure 5: Spin-dependent decay rate of the first excited Landau level calculated from the electron self-energy (dashed lines, ghosh_fermion_2024) and synchrotron radiation (solid lines, herold_cyclotron_1982), respectively.
  • ...and 2 more figures