Bulk viscosity of QCD matter near the critical temperature

Kubo's formula relates bulk viscosity to the retarded Green's function of the trace of the energy-momentum tensor. Using low energy theorems of QCD for the latter we derive the formula which relates the bulk viscosity to the energy density and pressure of hot matter. We then employ the available lattice QCD data to extract the bulk viscosity as a function of temperature. We find that close to the deconfinement temperature bulk viscosity becomes large, with viscosity-to-entropy ratio zeta/s about 1.

One of the most striking results coming from RHIC heavy ion program is the observation that hot QCD matter created in Au − Au collisions behaves like an almost ideal liquid rather than a gas of quarks and gluons [1,2,3,4,5]. Indeed, hydrodynamical simulations of nuclear collisions at RHIC (see e.g. [6,7]) indicate that the shear viscosity of QCD plasma is very low even though a quantitative determination is significantly affected by the initial conditions [8]. This observation does not yet have any theoretical explanation due to an enormous complexity of QCD in the regime of strong coupling. This is why the information inferred from the studies of gauge theories treatable at strong coupling such as N = 4 SUSY Yang-Mills theory is both timely and valuable. The study of shear viscosity in this theory using the holographic AdS/CFT correspondence has indicated that the shear viscosity η at strong coupling is small, with the viscosity-to-entropy ratio not far from the conjectured bound of η/s = 1/4π [9,10] .
However N = 4 SUSY Yang-Mills theory is quite different from QCD; in particular it possesses exact conformal invariance whereas the breaking of conformal invariance in QCD is responsible for the salient features of hadronic world including the asymptotic freedom [11], confinement, and deconfinement phase transition at high temperature 1 . Mathematically, conformal invariance implies the conservation of dilatational current s µ : ∂ µ s µ = 0. Since the divergence of dilatational current in field theory is equal to the trace of the energymomentum tensor ∂ µ s µ = θ µ µ , in conformally invariant theories θ µ µ = 0. In QCD, in the chiral limit of massless quarks the trace of the energy-momentum tensor is also equal to zero at the classical level. However quantum effects break conformal invariance [13,14]: where β(g) is the QCD β-function, which governs the behavior of the running coupling: note that we have included the coupling g in the definition of the gluon fields and have not written down explicitly the anomalous dimension correction to the quark mass term.
How would this breaking of conformal invariance manifest itself in the transport properties of QCD plasma? How big are the effects arising from it? The transport coefficient of the plasma which is directly related to its conformal properties is the bulk viscosity; indeed, it is related by Kubo's formula to the correlation function of the trace of the energy-momentum tensor: It is clear from (3) that for any conformally invariant theory with θ µ µ ≡ θ = 0 the bulk viscosity should vanish.
The perturbative evaluation of the bulk viscosity ζ of QCD plasma has been performed recently [15], and yielded a very small value, with ζ/s ∼ 10 −3 . The parametric smallness of bulk viscosity can be easily understood from eqs (3) and (1) [16,17], including a recent high statistics study [17]. Both indicate that η/s is not much higher than the conjectured bound The last equation follows from the fact that due to P-invariance, function ImG R (ω, 0) is odd in ω while Re G R (ω, 0) is even in ω. Let us define the spectral density ρ(ω, p) = − 1 π ImG R (ω, p) .
Using the Kramers-Kronig relation the retarded Green's function can be represented as The retarded Green's function G R (ω, p) of a bosonic excitation is related to the Euclidean Green's function G E (ω, p) by analytic continuation Using (6) and the fact that ρ(ω, p) = −ρ(−ω, p) we recover As we discussed above, the scale symmetry of QCD lagrangian is broken by quantum vacuum fluctuations. As a result the trace of the energy momentum tensor θ acquires a non-zero vacuum expectation value. The correlation functions constructed out of operators θ(x) satisfy a chain of low energy theorems (LET) which are a consequence of the renormalization group invariance of observable quantities [18]. These low-energy theorems entirely determine the dynamics of the effective low-energy theory. This effective theory has an elegant geometrical interpretation [19]; in particular, gluodynamics can be represented as a classical theory formulated on a curved (conformally flat) space-time background [20].
At finite temperature, the breaking of scale invariance by quantum fluctuations results in θ = E − 3P = 0 clearly observed on the lattice for SU(3) gluodynamics [21]; the presence of quarks [22] including the physical case of two light and a strange quark [23], or considering large N c [24] does not change this conclusion.
The LET of Ref. [18,19] were generalized to the case of finite temperature in [26,27].
The lowest in the chain of relations reads (at zero baryon chemical potential): To relate the thermal expectation value of θ T to the quantity (E − 3P ) LAT computed on the lattice, we should keep in mind that i.e. the zero-temperature expectation value of the trace of the energy-momentum tensor has to be subtracted; it is related to the vacuum energy density ǫ v < 0. Now, using (8), (9) and (10) we derive the following sum rule This exact relation is the main result of our paper.
In order to extract the bulk viscosity ζ from (12) we need to make an ansatz for the spectral density ρ. At high frequency, the spectral density should be described by perturbation theory; however, the perturbative (divergent) contribution has been subtracted in the definition of the quantities on the r.h.s. of the sum rule (12), and so we should not include the perturbative continuum 2 ρ(u) ∼ α 2 s u 4 on the l.h.s. as well. In the small frequency region, we will assume the following ansatz which satisfies (4) and (5). Substituting (13) in (12) we arrive at A peculiar feature of this result is that the bulk viscosity is linear in the difference E − 3P , rather than quadratic as naively implied by the Kubo's formula. This is similar to the strong coupling result obtained for the non-conformal supersymmetric Yang-Mills gauge plasma [12].
The parameter ω 0 = ω 0 (T ) is a scale at which the perturbation theory becomes valid.
On dimensional grounds, we expect it to be proportional to the temperature, ω 0 ∼ T . We estimate it as the scale at which the lattice calculations of the running coupling [29] coincide with the perturbative expression at a given temperature. In the region 1 < T /T c < 3 we find ω ≈ (T /T c ) 1.4 GeV. Now we are ready to use (14) to extract the bulk viscosity from the lattice data.