Infrared divergences in three-dimensional gauge theories

The loop expansion in massless three-dimensional gauge theories develops infrared divergences starting at the two-loop level. However, we show that the divergences in QC${\mathrm{D}}_3$ which arise through the three-loop level ($g^6$) can be removed by using a residual gauge freedom in covariant $\alpha$ gauges. Divergences in gauge-invariant amplitudes arise at $g^8$ order, as shown explicitly for the Wilson loop. Massless scalar QED develops a gauge-invariant singularity at order $g^4$, which can be canceled by changing the perturbative vacuum.