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Shear viscosity of the Quark-Gluon Plasma from a virial expansion

We calculate the shear viscosity $η$ in the quark-gluon plasma (QGP) phase within a virial expansion approach with particular interest in the ratio of $η$ to the entropy density $s$, i.e. $η/s$. The virial expansion approach allows us to include the interactions between the partons in the deconfined phase and to evaluate the corrections to a single-particle partition function. In the latter approach we start with an effective interaction with parameters fixed to reproduce thermodynamical quantities of QCD such as energy and/or entropy density. We also directly extract the effective coupling $\ga_{\rm V}$ for the determination of $η$. Our numerical results give a ratio $η/s\approx 0.097$ at the critical temperature $T_{\rm c}$, which is very close to the theoretical bound of $1/(4π)$. Furthermore, for temperatures $T\leq 1.8 T_{\rm c}$ the ratio $η/s$ is in the range of the present experimental estimates $0.1-0.3$ at RHIC. When combining our results for $η/s$ in the deconfined phase with those from chiral perturbation theory or the resonance gas model in the confined phase we observe a pronounced minimum of $η/s$ close to the critical temperature $T_{\rm c}$.