The Free Energy of Hot Gauge Theories with Fermions Through g^5

TL;DR

This work addresses the high-temperature free energy $F$ of gauge theories with fermions by pushing the perturbative expansion to order $g^5$ using hard-thermal-loop resummation to account for Debye screening. The authors perform two- and three-loop vacuum diagram calculations, employing a systematic soft-hard separation between scales $T$ and $gT$ to extract the $g^5$ contributions, including the Debye mass consistently. A key result is that the $g^5$ term has no $\ln g$ piece, i.e., $c'_5=0$, and all overlapping double-frequency sums cancel, with the final expressions given in terms of color and fermion factors; they provide explicit QCD and QED special cases. The findings illuminate the perturbative structure of hot gauge theories, inform lattice tests, and have implications for early-universe thermodynamics, with an independent confirmation by Braaten and Nieto.

Abstract

We compute the free energy density $F$ for gauge theories, with fermions, at high temperature and zero chemical potential. In the expansion $F=T^4 [c_0+c_2 g^2+c_3 g^3+(c'_4\ln g+c_4)g^4+ (c'_5\ln g+c_5)g^5+O(g^6)]$, we determine $c'_5$ and $c_5$ analytically by calculating two- and three-loop diagrams. The $g^5$ term constitutes the first correction to the $g^3$ term and is for the non-Abelian case the last power of $g$ that can be computed within perturbation theory. We find that the $g^5$ term receives no contributions from overlapping double-frequency sums and that $c'_5$ vanishes.

Figures (6)

• Figure 1: The one-loop gluon self-energy.
• Figure 2: Diagrams contributing to the free energy of gauge theories with fermions. The crosses are the "thermal counterterms" arising from the last term of (\ref{['gauge resummation']}).
• Figure 3: The dependence of the free energy density $F$ on the choice of renormalization scale $\bar{\mu}$ for six-flavor QCD with $\alpha_s(T) = 0.1$. The free energy density is normalized in units of the ideal gas result $-({1\over45}d_{\rm A} + {7\over180}d_{\rm F}) \pi^2 T^4$. The dotted, dot-dashed, dashed, and solid lines are the results for $F$ including terms through orders $g^2$, $g^3$, $g^4$, and $g^5$, respectively.
• Figure 4: The same as Fig. \ref{['fig53a']} but for $\alpha_s(T) = 0.02$ without fermions.
• Figure 5: The same as Fig. \ref{['fig53a']} but for $\alpha_s(T) = 0.001$.