On One-Loop Gap Equations for the Magnetic Mass in d=3 Gauge Theory

TL;DR

This paper critically examines one-loop gap equations for the magnetic mass in three-dimensional gauge theories, highlighting issues with gauge dependence and infrared instability, and suggests that higher-loop calculations are necessary for reliable results.

Contribution

The paper analyzes previous one-loop gap equation approaches, demonstrating their limitations and emphasizing the need for two-loop level calculations in determining the magnetic mass.

Findings

One-loop gap equations yield gauge-dependent and unphysical results.

Pinch technique self-energy can produce inconsistent residues, indicating infrared issues.

Higher-loop calculations are necessary for consistent magnetic mass determination.

Abstract

Recently several workers have attempted determinations of the so-called magnetic mass of d=3 non-Abelian gauge theories through a one-loop gap equation, using a free massive propagator as input. Self-consistency is attained only on-shell, because the usual Feynman-graph construction is gauge-dependent off-shell. We examine two previous studies of the pinch technique proper self-energy, which is gauge-invariant at all momenta, using a free propagator as input, and show that it leads to inconsistent and unphysical result. In one case the residue of the pole has the wrong sign (necessarily implying the presence of a tachyonic pole); in the second case the residue is positive, but two orders of magnitude larger than the input residue, which shows that the residue is on the verge of becoming ghostlike. This happens because of the infrared instability of d=3 gauge theory. A possible alternative one-loop determination via the effective action also fails. The lesson is that gap equations must be considered at least at two-loop level.