We demonstrate that for sufficiently high temperature $T$ the behavior of any four-dimensional gauge theory with small coupling constant $\alpha$, at distances beyond the electrical Debye screening length ${\xi }_D\sim \frac{1}{\sqrt{\alpha }T}$, 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 (QC${\mathrm{D}}_3$). We study QC${\mathrm{D}}_3$ in the loop expansion, which is only valid for distances $\ll \frac{1}{\alpha T}$, 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 ${\overline{A}}^2$. 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 QC${\mathrm{D}}_3$ at distances $\gtrsim \frac{1}{\alpha T}$ 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 QC${\mathrm{D}}_3$. The model is massless scalar QE${\mathrm{D}}_3$, which with $N$ charged scalars is soluble in a $\frac{1}{N}$ expansion as $N\to \infty$. Using the $\frac{1}{N}$ expansion, we demonstrate that infrared softening occurs: the long-range behavior of the photon propagator in massless scalar QE${\mathrm{D}}_3$ is less singular than that of free fields. Infrared softening might also occur in QC${\mathrm{D}}_3$, although it cannot be demonstrated to finite order in the loop expansion. The implications of an assumed infrared softening in QC${\mathrm{D}}_3$ 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.

• Received 29 December 1980

DOI:https://doi.org/10.1103/PhysRevD.23.2305