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

The analytical expressions of the shear viscosity of both one and two particle species with Boltzmann statistics and $2\to 2$ 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 $2\to 2$ 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 $2\to 2$ 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 $2\to 2$ 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.