Computing the viscosity of the QGP on the lattice

Models treating the system produced in heavy ion collisions at RHIC as an ideal fluid have had significant success in describing the observed large elliptic flow.1), 2) Subsequently the leading corrections due to a finite shear viscosity η were estimated3) and led to a remarkably small upper bound on η. Recent relativistic viscous hydrodynamics simulations4)–6) have started to achieve a tighter control over the systematic sources of uncertainty on the extraction of the shear viscosity to entropy density ratio, η/s. On the theory side, it is therefore important to compute the QCD shear viscosity from first principles to complete the picture. Furthermore, since the heavy ion collision program at LHC will probe the quark-gluon plasma at temperatures about a factor two higher, it is crucial to predict the shear viscosity at ≈ 3Tc, and from there to predict the size of elliptic flow, before experimental data is available. A small transport coefficient is a signature of strong interactions; strong interactions in turn require non-perturbative computational techniques. In this talk I present a lattice calculation of the thermal correlators of the energy-momentum tensor in the Euclidean SU(3) pure gauge theory, and discuss methods to extract the shear and bulk viscosity from them. Computationally, the calculation is challenging enough without the inclusion of dynamical quarks, and physically, the thermodynamic properties of the QGP not too close to Tc do not depend sensitively on the flavor content.7) In perturbation theory,8) the ratio of shear viscosity to entropy density of the pure gauge theory only differs by about 30% from full QCD9) at a fixed value of αs (η/s is smaller in the pure gauge theory). It should be appreciated that this is not much for a quantity which is infinite in the Stefan-Boltzmann limit, and the difference is actually reduced when comparing the pure gauge theory and full QCD at a common value of T/Tc. Lattice calculations of the shear viscosity10)–12) are based on Kubo formulas (see e.g.13) and14)), which relate each transport coefficient to the small frequency behavior of the retarded correlator. By analytic continuation (see Appendix A), the latter is related to the Euclidean two-point function of the conserved current. The significant progress made in12) was to obtain the energy-momentum correlators with an accuracy of about 3% on (isotropic) lattices with temporal resolution up to