This chapter deals with some infrared problems which arise in gauge theories at finite temperature. Most of these problems are still open, and represent interesting – and difficult – challenges for future investigations.

At zero temperature, the existence of singularities in field theories with massless particles has been known for a long time. It is convenient to classify these singularities into infrared and mass singularities. In order to be definite, let us take QCD as an example.

1. (i) A quark can emit a soft gluon, whose momentum k → 0: because the gluon is massless, this leads to infrared singularities, even when the quark is massive.

2. (ii) A massless quark can emit a gluon whose momentum makes a small angle θ with the initial quark momentum. The region θ → 0 leads to mass (or collinear) singularities, even if the gluon momentum k remains finite.

In both cases, the singularities arise from the fact that the final state quark + gluon is degenerate with the initial one when k → 0 in case (i) and when θ → 0 in case (ii).

Much is known about these infrared and mass singularities at T = 0; the most important information is contained in the Bloch–Nordsieck and Kinoshita–Lee–Nauenberg (KLN) theorems, which allow, in well-defined situations, control of these singularities.