On gauge-invariant Green function in 2+1 dimensional QED

TL;DR

This paper investigates the gauge-invariant fermion Green function in 2+1 dimensional QED, analyzing its properties and divergences across different gauges, and finds consistent anomalous dimensions in various gauge choices.

Contribution

It provides a detailed analysis of the gauge-invariant Green function in 2+1D QED, demonstrating gauge independence of the anomalous dimension and addressing infra-red divergences.

Findings

Infra-red divergence in temporal gauge is regularizable with a specific anomalous dimension.

Infra-red divergence in Coulomb gauge is un-regularizable, but cancels in physical quantities.

Gauge-invariant Green function has the same anomalous dimension across gauges.

Abstract

Both the gauge-invariant fermion Green function and gauge-dependent conventional Green function in $ 2+1 $ dimensional QED are studied in the large $ N $ limit. In temporal gauge, the infra-red divergence of gauge-dependent Green function is found to be regulariable, the anomalous dimension is found to be $ \eta= \frac{64}{3 \pi^{2} N} $. This anomalous dimension was argued to be the same as that of gauge-invariant Green function. However, in Coulomb gauge, the infra-red divergence of the gauge-dependent Green function is found to be un-regulariable, anomalous dimension is even not defined, but the infra-red divergence is shown to be cancelled in any gauge-invariant physical quantities. The gauge-invariant Green function is also studied directly in Lorentz covariant gauge and the anomalous dimension is found to be the same as that calculated in temporal gauge.