Asymptotic Infrared Fractal Structure of the Propagator for a Charged Fermion

TL;DR

This paper presents a gauge-invariant, non-perturbative calculation of the fractional exponent in the electron propagator's branch cut structure caused by infrared effects, revealing a fractal-like, asymptotic infrared structure.

Contribution

It provides the first unambiguous, gauge-invariant determination of the fractional exponent in the charged fermion propagator's infrared structure.

Findings

Derived a gauge-invariant fractional exponent for the propagator.

Confirmed the fractal, branch cut nature of the electron propagator.

Explored analogous effects in soft graviton interactions.

Abstract

It is well known that the long-range nature of the Coulomb interaction makes the definition of asymptotic ``in'' and ``out'' states of charged particles problematic in quantum field theory. In particular, the notion of a simple particle pole in the vacuum charged particle propagator is untenable and should be replaced by a more complicated branch cut structure describing an electron interacting with a possibly infinite number of soft photons. Previous work suggests a Dirac propagator raised to a fractional power dependent upon the fine structure constant, however the exponent has not been calculated in a unique gauge invariant manner. It has even been suggested that the fractal ``anomalous dimension'' can be removed by a gauge transformation. Here, a gauge invariant non-perturbative calculation will be discussed yielding an unambiguous fractional exponent. The closely analogous case of soft graviton exponents is also briefly explored.