Effective theories for relativistic fields at high temperature

Relativistic thermal field theories, including those describing the behavior of elementary particles of the Standard Model at high temperatures, are known to be affected by infrared divergences. In particular, perturbative calculations show that Bosonic Matsubara zero-modes lead to divergent results in the limit when the particles are massless. Resummation of such divergences yields an effective, temperature-dependent mass which is directly related to Debye screening. In quantum chromodynamics (QCD) at finite temperature, this phenomenon reveals the intrinsically non-perturbative nature of color-magnetic fields, which persists in the deconfined phase, even at very high temperatures at which the quark-gluon plasma is expected to be weakly coupled. The generation of dimensionful scales, which become parametrically well-separated at high temperatures, however, allows one to formulate a hierarchy of effective theories (electro-static QCD and magneto-static QCD) capturing the non-perturbative physics of hot QCD at large distances. The goal of this thesis project is to study the formulation of effective thermal field theories in the continuum and on the lattice, and to work out their application in different physical problems of relevance for collider physics and for the physics of the early Universe.