Transport coefficients in high temperature gauge theories: (I) Leading-log results

TL;DR

This work derives leading-log transport coefficients for high-temperature gauge theories in the weak-coupling limit, providing analytic and variationally computed expressions for the electrical conductivity $\sigma$, flavor diffusion constants $D_{a}$, and shear viscosity $\eta$ in Abelian and non-Abelian contexts. By formulating a linearized Boltzmann equation with a Hermitian collision operator and recasting the problem as a variational maximization of $Q[\chi]$, the authors extract the leading-log contributions dominated by small-angle scattering, supplemented by hard-thermal-loop screening to regulate IR behavior. They present compact parametric formulas, including explicit group-theory factors and matter-content dependencies, and compare against prior results, clarifying the role of various scattering processes (notably diagrams (A)–(E)) in determining the transport coefficients. The results illuminate how transport coefficients scale with couplings, temperatures, and content of charged species, offering foundational input for hydrodynamic modeling of the quark-gluon plasma and early-universe cosmology, and setting the stage for all-log (beyond leading-log) analyses in a companion work.

Abstract

Leading-log results are derived for the shear viscosity, electrical conductivity, and flavor diffusion constants in both Abelian and non-Abelian high temperature gauge theories with various matter field content.

Figures (4)

• Figure 1: Momentum and species label conventions for $2\to2$ scattering. Time runs from left to right.
• Figure 2: Leading-order diagrams for all $2\leftrightarrow 2$ particle scattering processes in a gauge theory with fermions. Solid lines denote fermions and wiggly lines are gauge bosons. Time may be regarded as running horizontally, either way, and so a diagram such as $(D)$ represents both $f \bar{f} \to gg$ and $gg \to f \bar{f}$. The diagrams of the first row [$(A)$--$(E)$] contribute to the leading log transport coefficients, while the diagrams of the second row [$(F)$--$(J)$], and all interference terms, do not.
• Figure 3: A generic vertex from diagrams ($A$)--($C$) of Fig. \ref{['fig:diagrams']}, to be analyzed in the soft exchange limit. In this figure, the solid line denotes any sort of particle (e.g., a gauge boson or fermion) that is being scattered, and the wavy line represents the exchanged gauge boson.
• Figure 4: Comparison of the exact (leading-log) form for $\chi^e(p)$ to the one and six parameter ansatz results, for the case of an $e^+ e^-$ plasma (left) and a three lepton and five quark plasma (right). The six parameter ansatz curve is essentially indistinguishable from the exact curve except for $p<0.1 T$. The dashed curves look exactly the same for both of our two ansätze.