On Perturbative Qcd at Finite Temperature

There is a number of interesting scenarios which are described by quantum field theories at finite temperature (electroweak phase transition, heavy ion collisions, etc.) In this talk I will focus on the deconfinement phase transition of QCD: lattice simulations suggest that hadron matter undergoes a transition into a phase in which quarks are deconfined at a temperature of order 150 MeV. This new state of matter called quark-gluon plasma (QGP) is expected to be studied experimentally at CERN and Brookhaven in the next decade. Here, we will consider the system at high temperature so that it is well inside the QGP phase. The temperature is the typical energy of the system and, when it is high enough, the strong coupling constant is small so that one would expect perturbation theory to describe the QGP accurately. It is specially interesting to consider the limit in which all of the masses scales of the system are smaller than the temperature. However, in this case, particles are massless and one has to deal with infrared divergences. In perturbation theory the infrared problems are fixed by resumming infinite series of diagrams that provide infrared-finite results. The fields that describe a system of particles at finite temperature in the imaginary time formalism have to satisfy periodic (for bosons) or antiperiodic (for fermions) boundary conditions in the Euclidean time variable. This allows us to expand a generic field Φ(x, τ) in terms of its Fourier modes φn(x):