Shear viscosity of a massless quark-gluon gas in chemical equilibrium including all $2\leftrightarrow 2$ cross sections

TL;DR

The paper tackles the problem of determining the shear viscosity $\eta$ of a massless quark-gluon gas in chemical equilibrium subject to all $2\leftrightarrow 2$ parton scatterings, including inelastic channels. Using the first-order Chapman-Enskog method, it derives a compact expression $\eta = \frac{T}{10}\boldsymbol{\gamma}^{\mathrm{T}} \mathbf{C}^{-1} \boldsymbol{\gamma}$ for the multi-species system, with the collision matrix $\mathbf{C}$ built from collision brackets and cross sections. It provides explicit results for a massless QGP with $N_f$ flavors, including the decomposition of $\mathbf{C}$ into seven cross-section channels and an isotropic-cross-section formula $\eta^{\rm iso} = \frac{12T}{5} \frac{A}{B}$; it also verifies the correct single-species limits, including equal-rate and equal-cross-section scenarios, and discusses the isotropic, energy-independent case via an effective cross section. The work offers analytical tools for implementing parton transport models and connecting to finite-temperature QCD cross sections to study $\eta/s$ in the quark-gluon plasma, with broad applicability to transport theory in high-energy nuclear physics.

Abstract

The analytical expressions of the shear viscosity of both one and two particle species with Boltzmann statistics and $2 \rightarrow 2$ elastic scatterings are known from the Chapman-Enskog method and have been shown to be quite accurate. The expression for a multi-species hadronic gas under $2 \rightarrow 2$ elastic scatterings is also known. Here we use the Chapman-Enskog method to derive the shear viscosity of a massless quark-gluon gas of $N_f$ quark flavors in chemical equilibrium subjected to all $2 \rightarrow 2$ parton scatterings including for the first time inelastic scatterings. We then verify the relation in a general single-species limit, where the shear viscosity of the quark-gluon gas should reduce to the result for a single particle species. In addition, we show the explicit analytical result in terms of the seven independent cross sections for the special case of isotropic and energy-independent cross sections. The analytical relations derived here can be useful for determining the shear viscosity of parton transport models with any $2 \rightarrow 2$ scattering cross sections. They can also be coupled with finite temperature QCD cross sections to help study the shear viscosity of the quark gluon plasma.