Bulk viscosity of QCD matter near the critical temperature
aa r X i v : . [ h e p - ph ] M a y BNL-NT-07/23RBRC-681
Bulk viscosity of QCD matter near the critical temperature
Dmitri Kharzeev a and Kirill Tuchin b,c a) Department of Physics,Brookhaven National Laboratory,Upton, New York 11973-5000, USAb) Department of Physics and Astronomy,Iowa State University,Ames, Iowa, 50011, USAc) RIKEN BNL Research Center,Upton, New York 11973-5000, USA (Dated: November 7, 2018)Kubo’s formula relates bulk viscosity to the retarded Green’s function of the traceof the energy-momentum tensor. Using low energy theorems of QCD for the latter wederive the formula which relates the bulk viscosity to the energy density and pressureof hot matter. We then employ the available lattice QCD data to extract the bulkviscosity as a function of temperature. We find that close to the deconfinementtemperature bulk viscosity becomes large, with viscosity-to-entropy ratio ζ/s ∼ One of the most striking results coming from RHIC heavy ion program is the observationthat hot QCD matter created in Au − Au collisions behaves like an almost ideal liquidrather than a gas of quarks and gluons [1, 2, 3, 4, 5]. Indeed, hydrodynamical simulations ofnuclear collisions at RHIC (see e.g. [6, 7]) indicate that the shear viscosity of QCD plasmais very low even though a quantitative determination is significantly affected by the initialconditions [8]. This observation does not yet have any theoretical explanation due to anenormous complexity of QCD in the regime of strong coupling. This is why the informationinferred from the studies of gauge theories treatable at strong coupling such as N = 4 SUSYYang-Mills theory is both timely and valuable. The study of shear viscosity in this theoryusing the holographic AdS/CFT correspondence has indicated that the shear viscosity η atstrong coupling is small, with the viscosity–to–entropy ratio not far from the conjecturedbound of η/s = 1 / π [9, 10] .However N = 4 SUSY Yang-Mills theory is quite different from QCD; in particular itpossesses exact conformal invariance whereas the breaking of conformal invariance in QCD isresponsible for the salient features of hadronic world including the asymptotic freedom [11],confinement, and deconfinement phase transition at high temperature . Mathematically,conformal invariance implies the conservation of dilatational current s µ : ∂ µ s µ = 0. Sincethe divergence of dilatational current in field theory is equal to the trace of the energy-momentum tensor ∂ µ s µ = θ µµ , in conformally invariant theories θ µµ = 0. In QCD, in thechiral limit of massless quarks the trace of the energy-momentum tensor is also equal to zeroat the classical level. However quantum effects break conformal invariance [13, 14]: ∂ µ s µ = θ µµ = X q m q ¯ qq + β ( g )2 g Tr G µν G µν , (1)where β ( g ) is the QCD β -function, which governs the behavior of the running coupling: µ dg ( µ ) dµ = β ( g ); (2)note that we have included the coupling g in the definition of the gluon fields and have notwritten down explicitly the anomalous dimension correction to the quark mass term.How would this breaking of conformal invariance manifest itself in the transport propertiesof QCD plasma? How big are the effects arising from it? The transport coefficient of theplasma which is directly related to its conformal properties is the bulk viscosity; indeed, it isrelated by Kubo’s formula to the correlation function of the trace of the energy-momentumtensor: ζ = 19 lim ω → ω Z ∞ dt Z d r e iωt h [ θ µµ ( x ) , θ µµ (0)] i . (3)It is clear from (3) that for any conformally invariant theory with θ µµ ≡ θ = 0 the bulkviscosity should vanish.The perturbative evaluation of the bulk viscosity ζ of QCD plasma has been performedrecently [15], and yielded a very small value, with ζ /s ∼ − . The parametric smallnessof bulk viscosity can be easily understood from eqs (3) and (1) which show that ζ ∼ α s , in The effects of conformal symmetry breaking on bulk viscosity of SUSY Yang-Mills theory have beenstudied in the framework of the AdS/CFT correspondence in ref [12]. accord with the result of ref. [15]. This would seem to suggest that bulk viscosity effects inthe quark-gluon plasma are unimportant. However, perturbative expansions at temperaturesclose to the critical one are not applicable, so at moderate temperatures one has to rely onlattice QCD calculations. Lattice calculations of the equation of state become increasinglyprecise; however, the direct calculations of transport coefficients have been notoriously diffi-cult. Two calculations have been reported for shear viscosity [16, 17], including a recent highstatistics study [17]. Both indicate that η/s is not much higher than the conjectured boundof 1 / π ; no lattice calculations of the bulk viscosity have been reported so far. Fortunately,the correlation function of the trace of the energy-momentum tensor in QCD is constrainedby the low-energy theorems, which do not rely on perturbation theory. They can thus beused to express the bulk viscosity in terms of the “interaction measure” h θ i = E − P where E is the energy density and P is the pressure, which are measured on the lattice with highprecision. Such a study is the subject of this Letter.The calculation of the bulk viscosity starts with the Kubo’s formula (3) (we follow thedefinitions and notations of [28]). Introducing the retarded Green’s function we can re-write(3) as ζ = 19 lim ω → ω Z ∞ dt Z d r e iωt iG R ( x ) = 19 lim ω → ω iG R ( ω, ~
0) = −
19 lim ω → ω Im G R ( ω, ~ . (4)The last equation follows from the fact that due to P-invariance, function Im G R ( ω, ~
0) is oddin ω while Re G R ( ω, ~
0) is even in ω . Let us define the spectral density ρ ( ω, ~p ) = − π Im G R ( ω, ~p ) . (5)Using the Kramers-Kronig relation the retarded Green’s function can be represented as G R ( ω, ~p ) = 1 π Z ∞−∞ Im G R ( u, ~p ) u − ω − iε du = Z ∞−∞ ρ ( u, ~p ) ω − u + iε du (6)The retarded Green’s function G R ( ω, ~p ) of a bosonic excitation is related to the EuclideanGreen’s function G E ( ω, ~p ) by analytic continuation G E ( ω, ~p ) = − G R ( iω, ~p ) , ω > . (7)Using (6) and the fact that ρ ( ω, ~p ) = − ρ ( − ω, ~p ) we recover G E (0 , ~
0) = 2 Z ∞ ρ ( u, ~ u du . (8)As we discussed above, the scale symmetry of QCD lagrangian is broken by quantumvacuum fluctuations. As a result the trace of the energy momentum tensor θ acquires anon-zero vacuum expectation value. The correlation functions constructed out of opera-tors θ ( x ) satisfy a chain of low energy theorems (LET) which are a consequence of therenormalization group invariance of observable quantities [18]. These low-energy theoremsentirely determine the dynamics of the effective low-energy theory. This effective theory hasan elegant geometrical interpretation [19]; in particular, gluodynamics can be representedas 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 − P = 0 clearly observed on the lattice for SU (3) gluodynamics [21]; the presence ofquarks [22] including the physical case of two light and a strange quark [23], or consideringlarge 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): G E (0 , ~
0) = Z d x h T θ ( x ) , θ (0) i = (cid:18) T ∂∂T − (cid:19) h θ i T . (9)To relate the thermal expectation value of h θ i T to the quantity ( E − P ) LAT computed onthe lattice, we should keep in mind that(
E − P ) LAT = h θ i T − h θ i , (10)i.e. the zero-temperature expectation value of the trace of the energy-momentum tensor h θ i = − | ǫ v | (11)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 rule2 Z ∞ ρ ( u, ~ u du = − (cid:18) − T ∂∂T (cid:19) h θ i T = T ∂∂T ( E − P ) LAT T + 16 | ǫ v | , (12)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 thespectral density ρ . At high frequency, the spectral density should be described by perturba-tion theory; however, the perturbative (divergent) contribution has been subtracted in the T (cid:144) T c Ζ (cid:144) s FIG. 1: The ratio of bulk viscosity to the entropy density for SU (3) gluodynamics. We have used | ǫ v | = 0 . T c and T c = 0 .
28 GeV [21]. definition of the quantities on the r.h.s. of the sum rule (12), and so we should not includethe perturbative continuum ρ ( u ) ∼ α s u on the l.h.s. as well. In the small frequency region,we will assume the following ansatz ρ ( ω, ~ ω = 9 ζπ ω ω + ω , (13)which satisfies (4) and (5). Substituting (13) in (12) we arrive at ζ = 19 ω (cid:26) T ∂∂T ( E − P ) LAT T + 16 | ǫ v | (cid:27) . (14)A peculiar feature of this result is that the bulk viscosity is linear in the difference E − P ,rather than quadratic as naively implied by the Kubo’s formula. This is similar to thestrong coupling result obtained for the non-conformal supersymmetric Yang-Mills gaugeplasma [12].The parameter ω = ω ( T ) is a scale at which the perturbation theory becomes valid.On dimensional grounds, we expect it to be proportional to the temperature, ω ∼ T . Weestimate it as the scale at which the lattice calculations of the running coupling [29] coincidewith the perturbative expression at a given temperature. In the region 1 < T /T c < ω ≈ ( T /T c ) 1 . For an explicit perturbative expression and a discussion of the properties of ρ ( u ) at small frequencies seee.g. [25]. The results of the numerical calculation using as an input the high precision lattice data[21] are displayed in Fig. 1. One can see that away from T c the bulk viscosity is small, inaccord with the expectations based on the perturbative results [15]. However, close to T c therapid growth of E − P causes a dramatic increase of bulk viscosity. Basing on the latticeresults [16, 17] which indicate that the shear viscosity remains small close to T c , we expectthat bulk viscosity will be the dominant correction to the ideal hydrodynamical behavior inthe vicinity of the deconfinement phase transition.We thank F. 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