Shear Viscosity of a Gluon Plasma in Perturbative QCD

We calculate the shear viscosity $\eta$ to entropy density $s$ ratio $\eta /s$ of a gluon plasma in kinetic theory including the gg->gg and gg<->ggg processes. Due to the suppressed contribution to $\eta$ in the gg->gg forward scattering, it is known that the gluon bremsstrahlung gg<->ggg process also contributes at the same order $O(\alpha_{s}^{-2})$ in perturbative QCD. Using the Gunion-Bertsch formula for the gg<->ggg matrix element which is valid for the limit of soft bremsstrahlung, we find that the result is sensitive to whether the same limit is taken for the phase space. Using the exact phase space, the gg<->ggg contribution becomes more important to $\eta$ than gg->gg for $\alpha_{s}\gtrsim 2\times 10^{-3}$. Therefore, at $\alpha_{s}=0.1$, $\eta /s\simeq 1.0$, between 2.7 obtained by Arnold, Moore and Yaffe (AMY) and 0.5 obtained by Xu and Greiner. If the soft bremsstrahlung limit is imposed on the phase space such that the recoil effect from the bremsstrahlung gluon is neglected, then the correction from the gg<->ggg process is about 10-30% of the total which is close to AMY's prediction. This shows that the soft bremsstrahlung approximation is not as good as previously expected.