Effective classical theory for real-time SU($N$) gauge theories at high temperature

We derive an effective classical theory for real-time SU($N$) gauge theories at high temperature. By separating off and integrating out quantum fluctuations we obtain a 3D classical path integral over the initial fields and conjugate momenta. The leading hard mode contribution is incorporated in the equations of motion for the classical fields. This yields the gauge invariant hard thermal loop (HTL) effective equation of motion. No gauge-variant terms are generated as in treatments with an intermediate momentum cut-off. Quantum corrections to classical correlation functions can be calculated perturbatively. The 4D renormalizability of the theory ensures that the 4D counterterms are sufficient to render the theory finite. The HTL contributions of the quantum fluctuations provide the counterterms for the linear divergences in the classical theory.