Calculation of the shear viscosity in SU(3) gluodynamics

Abstract

We perform a lattice Monte-Carlo calculation of the two-point functions of the energy-momentum tensor at finite temperature in the SU(3) gauge theory. Unprecedented precision is obtained thanks to a multi-level algorithm. The lattice operators are renormalized non-perturbatively and the classical discretization errors affecting the correlators are corrected for. A robust upper bound for the shear viscosity to entropy density ratio is derived, η/s < 1.0, and our best estimate is η/s = 0.134(33) at T = 1.65Tc under the assumption of smoothness of the spectral function in the low-frequency region.PACS numbers: 12.38. Gc, 12.38.Mh, Introduction.-Models treating the system produced in heavy ion collisions at RHIC as an ideal fluid have had significant success in describing the observed flow phenomena [1,2]. Subsequently the leading corrections due to a finite shear viscosity were computed [3], in particular the flattening of the elliptic flow coefficient v 2 (p T ) above 1GeV. It is therefore important to compute the QCD shear and bulk viscosities from first principles to establish this description more firmly. Small transport coefficients are a signature of strong interactions, which lead to efficient transmission of momentum in the system. Strong interactions in turn require non-perturbative computational techniques. Several attempts have been made to compute these observables on the lattice in the SU(3) gauge theory [4,5]. The underlying basis of these calculations are the Kubo formulas, which relate each transport coefficient to a spectral function ρ(ω) at vanishing frequency. Even on current computers, these calculations are highly non-trivial, due to the fall-off of the relevant correlators in Euclidean time (as x −5 0 at short distances), implying a poor signal-to-noise ratio in a standard Monte-Carlo calculation. The second difficulty is to solve the ill-posed inverse problem for ρ(ω) given the Euclidean correlator at a finite set of points. Mathematically speaking, the uncertainty on a transport coefficient χ is infinite for any finite statistical accuracy, because adding ǫωδ(ω) to ρ(ω) merely corresponds to adding a constant to the Euclidean correlator of order ǫ, while rendering χ infinite. Therefore smoothness assumptions on ρ(ω) have to be made, which are reasonable far from the one-particle energy eigenstates, and can be proved in the hard-thermal-loop framework [6].

Viscosities in the Gluon-Plasma within a Quasiparticle Model

“…Contrary to the popular view, where a large quasiparticle mean free path implies large η/s [24], our results suggest that a quasiparticle description is still admitted for the strongly coupled gluon-plasma. [7], diamonds and triangles from [8]). In addition, the unitarity limit [5] η/s = 1/(4π) (dotted curve) is depicted.…”

“…Apart from the first-principle calculations of bulk (ζ) and shear (η) viscosities by means of lattice QCD [7,8] a variety of other approaches was proposed. Among these the rigorous perturbative calculations starting either from the Boltzmann equation [9,10] or from the Kubo formalism [11] have to be mentioned.…”

The shear channel spectral function in hot Yang-Mills theory

“…It would be very interesting to quantitatively compare this behavior to existing lattice and AdS predictions (see e.g. [7,29]), which in fact is a question that will be addressed in the near future [32].…”

“…[5,6]). At the same time, a reliable nonperturbative determination of transport coefficients, in principle available from the zero frequency limit of the corresponding (Minkowskian) spectral functions, has proven extremely challenging, as the lattice approach is restricted to the Euclidean formulation of the theory [7][8][9]. As a result, a considerable amount of attention has shifted towards the gauge/gravity duality, which has in particular led to the famous claim of the QGP being a nearly 'ideal' fluid, with a shear viscosity to entropy ratio close to the conjectured limit η/s ≥ 1/(4π) [10] (see also [11,12] for some interesting recent developments).…”

Viscous Properties of Strongly Interacting Matter at High Temperature

“…This bold effort obtained η/s ≈ 0.5 in the temperature range 1.6 < T /T c < 2.2, albeit with uncertainties of order 100%. An improved calculation by Meyer [34,35] Another interesting approach has been the computation of η/s in N = 4 supersymmetric SU(N c ) Yang-Mills theory (SYM). In that theory it is possible to do calculations in the large coupling limit [36] and in the weak coupling limit [37], and without too much imagination it is possible to find a paramtrization that interpolates smoothly between the two limits.…”