Shear viscosity of classical Yang-Mills field

TL;DR

This paper calculates the shear viscosity of the classical Yang-Mills field on a lattice using the Green-Kubo formula, revealing its dependence on coupling and temperature, and comparing it with perturbative and anomalous viscosity estimates.

Contribution

It provides the first explicit functional form of shear viscosity dependence on coupling and temperature for the classical Yang-Mills field, bridging weak and strong coupling regimes.

Findings

Shear viscosity scales as approximately 1/g^{1.10-1.88} at weak coupling.

The dependence on coupling is weaker than leading-order perturbation theory.

The shear viscosity estimate aligns with anisotropy analysis in expanding CYM dynamics.

Abstract

We investigate the shear viscosity $\eta$ of the classical Yang-Mills (CYM) field on a lattice by using the Green-Kubo formula, where the shear viscosity is calculated from the time-correlation function of the energy-momentum tensor in equilibrium. Dependence of the shear viscosity $\eta(g,T)$ on the coupling $g$ and temperature $T$ is represented by a scaling function $f_\eta(g^2T)$ as $\eta(g,T)=Tf_\eta(g^2T)$ due to the scaling-invariant property of the CYM. The explicit functional form of $f_\eta(g^2T)$ is successfully determined from the calculated shear viscosity: It turns out that $\eta(g,T)$ of the CYM field is proportional to $1/g^{1.10-1.88}$ at weak coupling, which is a weaker dependence on $g$ than that in the leading-order perturbation theory but consistent with that of the "anomalous viscosity" $\eta\propto 1/g^{1.5}$ under the strong disordered field. The obtained shear viscosity is also found to be roughly consistent with that estimated through the analysis of the anisotropy of the pressure of the CYM dynamics in the expanding geometry with recourse to a hydrodynamic equation.