How perfect can a gluon plasma 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 smoothly connects the two different approximations used by Arnold, Moore, and Yaffe (AMY) and Xu and Greiner (XG). At small alpha(s) (alpha(s) << 1), our result is closer to AMY's collinear result while at larger alpha(s) the finite angle noncollinear configurations become more important and our result is closer to XG's soft bremsstrahlung result. In the region where perturbation is reliable (alpha(s) <= 0.1), we find no indication that the proposed perfect fluid limit eta/s approximate to 1/(4 pi) can be achieved by perturbative QCD alone. Whether this can be achieve for alpha(s) >= 0.1 is still an open question.