Transport Coefficients in Large $N_f$ Gauge Theory: Testing Hard Thermal Loops

TL;DR

The paper investigates shear viscosity and fermionic diffusion in a gauge theory with a large number of fermion species (large $N_f$) by solving the fermionic kinetic theory at leading order in $1/N_f$ and to all orders in the coupled parameter $g^2N_f$. It benchmarks the Hard Thermal Loop (HTL) approximation against the exact all-orders result, including careful treatment of infrared gauge-boson self-energies and resonance effects. The study finds that HTL performs well at weak to moderate couplings ($g^2N_f\lesssim1$) with renormalization-scale uncertainties, while beyond this regime HTL accuracy deteriorates but remains within a factor of about two due to scale ambiguities. The results provide a controlled test of HTL resummation in a tractable setting and offer insights into transport in QED/QCD-like theories with many fermion flavors.

Abstract

We compute shear viscosity and flavor diffusion coefficients for ultra-relativistic gauge theory with many fermionic species, Nf >> 1, to leading order in 1/Nf. The calculation is performed both at leading order in the effective coupling strength g^2 Nf, using the Hard Thermal Loop (HTL) approximation, and completely to all orders in g^2 Nf. This constitutes a nontrivial test of how well the HTL approximation works. We find that in this context, the HTL approximation works well wherever the renormalization point sensitivity of the leading order HTL result is small.

Figures (8)

• Figure 1: All bubble graphs, left, and the resulting self-energies when they are cut, right.
• Figure 2: Scattering processes which must be considered at leading order .
• Figure 3: Scattering processes which may be important at leading order in the coupling $g$, when we do not expand in $N_{\rm f}$, but which are $N_{\rm f}$ suppressed and can be neglected here. Dashed lines on the bubble graphs show where they are cut to give the diagrams presented. Dotted lines in the scattering processes indicate that gauge bosons are not propagating states in the kinetic theory, and we must consider the process including the states a gauge boson eventually scatters against or decays into.
• Figure 4: Shear viscosity, computed to all orders in $g^2 N_{\rm f}$. The renormalization point $\bar{\mu}$ in the $\overline{\hbox{MS}}$ scheme is $\bar{\mu}=\pi e^{-\gamma_{\rm E}} T$, the "dimensional reduction" answer.
• Figure 5: Diffusion constant, computed to all orders in $g^2 N_{\rm f}$. The renormalization point is the same as in Fig. \ref{['fig:result1']}.