Free Energy of QCD at High Temperature

TL;DR

Braaten and Nieto develop a three-scale effective-field-theory framework for high-temperature QCD, using dimensional reduction to separate contributions from the scales $T$, $gT$, and $g^2T$ via EQCD and MQCD. They compute the QCD free energy to order $g^5$ by matching full QCD to EQCD for the $T$-scale and evaluating the $gT$-scale within EQCD, while treating the $g^2T$-scale nonperturbatively through MQCD, with a parametrization $f_G$ that involves lattice-determined constants. The paper shows how RG evolution of $f_E$ and the perturbative structure at each scale control logarithmic terms, and it discusses the convergence of perturbation theory, finding that reliable results at realistic temperatures require nonperturbative input for the magnetic sector. The outlined $g^6$ program indicates which higher-loop MQCD and full-QCD calculations are needed, underscoring the EFT approach as a practical roadmap for combining perturbative and lattice methods in hot QCD.

Abstract

Effective-field-theory methods are used to separate the free energy for a nonabelian gauge theory at high temperature $T$ into the contributions from the momentum scales $T$, $gT$, and $g^2T$, where $g$ is the coupling constant at the scale $2 πT$. The effects of the scale $T$ enter through the coefficients in the effective lagrangian for the 3-dimensional effective theory obtained by dimensional reduction. These coefficients can be calculated as power series in $g^2$. The contribution to the free energy from the scale $gT$ can be calculated using perturbative methods in the effective theory. It can be expressed as an expansion in $g$ starting at order $g^3$. The contribution from the scale $g^2T$ must be calculated using nonperturbative methods, but nevertheless it can be expanded in powers of $g$ beginning at order $g^6$. We calculate the free energy explicitly to order $g^5$. We also outline the calculations necessary to obtain the free energy to order $g^6$.