High-temperature Yang-Mills theories and three-dimensional quantum chromodynamics

We demonstrate that for sufficiently high temperature T the behavior of any four-dimensional gauge theory with small coupling constant, at distances beyond the electrical Debye screening length ₃1{T}, is determined precisely by the corresponding three-dimensional theory. This is the magnetic sector of the original theory, and in the non-Abelian case it is a Yang-Mills theory like three-dimensional quantum chromodynamics (QCD₃). We study QCD₃ in the loop expansion, which is only valid for distances 1, in both covariant and Coulomb gauges. At a finite order in the loop expansion, the presence of logarithmic infrared divergences signals the appearance of new operators in the operator-product expansion. For example, in a covariant gauge, the gauge self-energy develops infrared divergences at two-loop order associated with the operator A^2. Infrared divergences in the Wilson loop are also considered and shown to cancel below the order at which gauge-invariant local operators can appear in the operator-product expansion. The infrared structure of QCD₃ at distances 1 cannot be directly probed in the loop expansion, however. We present a simpler model which is calculable in this infrared limit, and which might serve as a prototype for QCD₃. The model is massless scalar QED₃, which with N charged scalars is soluble in a 1N expansion as N. Using the 1N expansion, we demonstrate that infrared softening occurs: the long-range behavior of the photon propagator in massless scalar QED₃ is less singular than that of free fields. Infrared softening might also occur in QCD₃, although it cannot be demonstrated to finite order in the loop expansion. The implications of an assumed infrared softening in QCD₃ for the magnetic sector of Yang-Mills theories at high temperatures are also discussed. In particular, we consider the possibility that, if the softening is sufficiently great, there is screening of hot non-Abelian magnetic fields and possible confinement of primordial magnetic monopoles.