Transport coefficients in high temperature gauge theories: (II) Beyond leading log

TL;DR

This work delivers a full leading-order computation of transport coefficients in high-temperature QCD and QED using an effective kinetic theory that includes both $2\leftrightarrow2$ scatterings and $1\leftrightarrow2$ splittings with HTL self-energies and LPM effects. It demonstrates that the dominant Coulomb logarithms induce an expansion in inverse powers of $\ln(1/g)$, and constructs a variational solution in a finite basis to extract $\eta$, $D_q$, and $\sigma$ with controlled accuracy. The study shows that a next-to-leading-log (NLL) approximation closely tracks the full LO result for moderate Debye mass ($m_D/T \lesssim 1$), and provides quantitative insights into the sensitivity to higher-order corrections and the asymptotic, non-convergent character of the inverse-log expansion. In addition, the paper extends the analysis to electrical conductivity in QED plasmas and discusses the practical implications for transport in the early Universe and heavy-ion phenomenology, highlighting a robust LO framework and the limited utility of purely leading-log estimates.

Abstract

Results are presented of a full leading-order evaluation of the shear viscosity, flavor diffusion constants, and electrical conductivity in high temperature QCD and QED. The presence of Coulomb logarithms associated with gauge interactions imply that the leading-order results for transport coefficients may themselves be expanded in an infinite series in powers of 1/log(1/g); the utility of this expansion is also examined. A next-to-leading-log approximation is found to approximate the full leading-order result quite well as long as the Debye mass is less than the temperature.

Figures (7)

• Figure 1: Leading-order value of the quark flavor diffusion constant $D_{\rm q}$, multiplied by $g^4 T$, plotted as a function of $m_{\rm D}/T$. The different curves show the result for SU(3) gauge theory with 0 to 6 flavors of quarks. The $N_{\rm f}\, = 0$ curve is the result when one artificially neglects scattering of quarks off other quarks, and only includes scattering of quarks off gluons (in which case the result is independent of $N_{\rm f}\,$).
• Figure 2: Leading-order value of the shear viscosity $\eta$, multiplied by $g^4/T^3$, as a function of $m_{\rm D}/T$. The different curves show the result for SU(3) gauge theory with 0 to 6 flavors of fermions.
• Figure 3: Sensitivity of the quark diffusion constant in three flavor QCD to various simplifying approximations. Each curve shows the ratio of the answer with the indicated approximations made, to the full leading order answer.
• Figure 4: Leading-order results for the flavor diffusion constant $D_{\rm q}$, multiplied by $g^4 T$, as a function of $m_{\rm D}/T$ in three flavor SU(3) gauge theory. Each curve is computed using a different, but equally valid, leading-order definition of the effective kinetic theory. The solid line shows the result of the implementation discussed in detail in Appendix \ref{['app:22']}, in which $t$ or $u$ channel matrix elements are written as an HTL-corrected scalar quark contribution plus an IR-safe spin-dependent remainder. The dotted line is the result of the analogous procedure using fermionic, rather than scalar, scattering as the template for $t$ channel gauge boson exchange. The dashed lines show what happens if the hard thermal loop self-energy is only included in gauge boson exchange lines when the exchange momentum $Q$ satisfies $Q^2 < T^2$, or alternatively $q^2 < T^2$.
• Figure 5: Leading-order results for the flavor diffusion constant (multiplied by $g^4 T$) in three flavor QCD compared to the expansion in inverse powers of $\ln (\mu_*/m_{\rm D})$ truncated at second and third order. Right panel: zoom-in on the small $m_D/T$ region, showing that the third order truncation of the inverse-log expansion can be an improvement over the second order result, but only for $m_{\rm D}/T \le 0.2$.