How Perfect a Gluon Plasma Can Be in Perturbative QCD?

The shear viscosity to entropy density ratio, \eta /s, characterizes how perfect a fluid is. We calculate the leading order \eta /s of a gluon plasma in perturbation using the kinetic theory. The leading order contribution only involves the elastic gg -> gg (22) process and the inelastic gg<->ggg (23) process. The Hard-Thermal-Loop (HTL) treatment is used for the 22 matrix element, while the exact matrix element in vacuum is supplemented by the gluon Debye mass insertion for the 23 process. Also, the asymptotic mass is used for the external gluons in the kinetic theory. The errors from not implementing HTL and the Landau-Pomeranchuk-Migdal effect in the 23 process, and from the uncalculated higher order corrections, are estimated. Our result for \eta /s lies between that of Arnold, Moore and Yaffe (AMY) and Xu and Greiner (XG). Our result shows that although the finite angle contributions are important at intermediate \alpha_s (\alpha_s \sim 0.01-0.1), the 22 process is still more important than 23 when \alpha_s < 0.1. This is in qualitative agreement with AMY's result. We find no indication that the proposed perfect fluid limit \eta /s \simeq 1/(4\pi) can be achieved by perturbative QCD alone.