One-loop gap equations for the magnetic mass ind=3gauge theory

Recently several workers have attempted determinations of the so-called magnetic mass of $d=3\mathrm{}$ 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 results. 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 ghost-like. This happens because of the infrared instability of $d=3\mathrm{}$ 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 the two-loop level.