Limitations to dimensional reduction at high temperature

The claim that renormalizable four-dimensional field theories in the infinite-temperature limit undergo a reduction to effective three-dimensional ones is analyzed in perturbation theory. An essential ingredient is the finite-temperature renormalization group in the imaginary-time formalism. A precise criterion for the occurrence of complete dimensional reduction is given. This is satisfied only in exceptional cases, and is violated e.g. by φ4 and QCD. These theories dimensionally reduce only up to a given order in perturbation theory. Illustrative one-loop calculations are given on the basis of a novel summation technique. The perturbative structure of QED and QCD at high temperature is examined in detail. Ward identities as well as explicit computations are used to explain why QED dimensionally reduces to all orders, but QCD does not. In addition, a potential instability deriving from an anomalous diagram is identified and cured. (orig.)