Topology and the Infrared Structure of Quantum Electrodynamics

TL;DR

This paper introduces a non-perturbative approach to infrared divergences in quantum electrodynamics using geometric phases, leading to finite, gauge-invariant scattering amplitudes and insights into positronium formation.

Contribution

It reformulates infrared divergence regulation via Berry phases in field space, avoiding perturbative dressing and providing a new perspective on the infrared structure in QED.

Findings

Infrared divergences cancel due to destructive interference among Berry phases.

The approach yields finite, gauge-invariant scattering amplitudes without soft-photon summation.

Identifies a topological flux-induced singularity in the dressed S-matrix at specific energy.

Abstract

We study infrared divergences in quantum electrodynamics using geometric phases and the adiabatic approximation in quantum field theory. In this framework, the asymptotic \textit{in} and \textit{out} states are modified by Berry phases, $e^{i \Delta \alpha_{\text{in}}}$ and $e^{i \Delta \alpha_{\text{out}}}$, which encode the infrared structure non-perturbatively and regulate soft-photon divergences. Unlike the Faddeev--Kulish formalism, which employs perturbative dressing with coherent states, our approach reformulates the effective action in terms of Berry connections in field space. This yields finite, gauge-invariant scattering amplitudes without requiring a sum over soft-photon emissions. We show that infrared divergences cancel to all orders in the bremsstrahlung vertex function $\Gamma^\mu(p_1, p_2)$, due to destructive interference among inequivalent Berry phases. As an application, we study the formation of positronium in the infrared regime and argue that the dressed \(S\)-matrix exhibits a functional singularity at $s = 4 m_e^2$, corresponding to a physical pole generated by topological flux.