Divergences in Real-Time Classical Field Theories at Non-Zero Temperature

TL;DR

This paper investigates divergences in real-time classical field theories at finite temperature, revealing how divergences behave at different loop orders and proposing renormalization strategies for such theories.

Contribution

It provides a perturbative analysis of divergences in hot classical field theories, establishing the divergence structure across loop orders and discussing renormalization methods.

Findings

Linear divergences are determined by classical hard thermal loops.

Higher-loop diagrams are less divergent, with superficial degree decreasing by one per loop.

Two-loop SU(N) self-energy diagrams are verified to match the divergence predictions.

Abstract

The classical approximation provides a non-perturbative approach to time-dependent problems in finite temperature field theory. We study the divergences in hot classical field theory perturbatively. At one-loop, we show that the linear divergences are completely determined by the classical equivalent of the hard thermal loops in hot quantum field theories, and that logarithmic divergences are absent. To deal with higher-loop diagrams, we present a general argument that the superficial degree of divergence of classical vertex functions decreases by one with each additional loop: one-loop contributions are superficially linearly divergent, two-loop contributions are superficially logarithmically divergent, and three- and higher-loop contributions are superficially finite. We verify this for two-loop SU(N) self-energy diagrams in Feynman and Coulomb gauges. We argue that hot, classical scalar field theory may be completely renormalized by local (mass) counterterms, and discuss renormalization of SU(N) gauge theories.