Chapman-Enskog calculation of the shear viscosity of quark-gluon plasma including all $2\leftrightarrow 2$ scatterings at finite temperature

Abstract

We use the Chapman-Enskog method to investigate the shear viscosity of the quark-gluon plasma with a focus on its relation to parton cross sections. We use the recently obtained analytical expression for the shear viscosity $\eta$ of a massless quark-gluon gas at chemical equilibrium with Boltzmann statistics and all $2\leftrightarrow 2$ scatterings with arbitrary cross sections. Here we apply this general expression to cross sections at finite temperature that are based on perturbative-QCD and screened with scaled thermal masses $\sqrt{\kappa}\,m_D$ and $\sqrt{\kappa}\,m_F$. We find that the Chapman-Enskog results on $\eta \, g^4/T^3$ versus $m_D/T$ at $\kappa=1$ are qualitatively similar to but higher than the corresponding leading-order results from the AMY framework. We then find that using $\kappa=0.4$ allows the Chapman-Enskog results to match well the corresponding AMY results as it includes the effect of using thermal masses (instead of self-energies) to screen the cross sections. In addition, we show that the shear viscosity-to-entropy density ratio $\eta/s$ is very sensitive to the choice of momentum scale $Q$ in the strong coupling, where the choice of $Q=3T$ leads to $\eta/s \sim 0.15$ for $N_f=0$ or 3 at the QCD phase transition temperature $T_c$. These results lay the foundation for mapping parton cross sections to given shear viscosity in parton transport models and QCD effective kinetic theory.