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arXiv:0712.2451v3 [hep-th] 15 Jul 2008

Preprint typeset in JHEP style - HYPER VERSION

BI-TP 2007/29

INT PUB 07-45

SHEP-07-47

Relativistic viscous hydrodynamics,

conformal invariance, and holography

Rudolf Baier

Fakult¨ at f¨ ur Physik, Universit¨ at Bielefeld, D-33501 Bielefeld, Germany

E-mail: baier@physik.uni-bielefeld.de

Paul Romatschke and Dam Thanh Son

Institute for Nuclear Theory, University of Washington,

Box 351550, Seattle, WA, 98195, USA

E-mail: paulrom@phys.washington.edu, son@phys.washington.edu

Andrei O. Starinets

School of Physics & Astronomy, University of Southampton,

Highfield, Southampton SO17 1BJ, United Kingdom

E-mail: starina@phys.soton.ac.uk

Mikhail A. Stephanov

Department of Physics, University of Illinois, Chicago, IL 60607-7059, USA

E-mail: misha@uic.edu

Abstract: We consider second-order viscous hydrodynamics in conformal field theories at

finite temperature. We show that conformal invariance imposes powerful constraints on the

form of the second-order corrections. By matching to the AdS/CFT calculations of correla-

tors, and to recent results for Bjorken flow obtained by Heller and Janik, we find three (out

of five) second-order transport coefficients in the strongly coupled N = 4 supersymmetric

Yang-Mills theory. We also discuss how these new coefficents can arise within the kinetic

theory of weakly coupled conformal plasmas. We point out that the M¨ uller-Israel-Stewart

theory, often used in numerical simulations, does not contain all allowed second-order terms

and, frequently, terms required by conformal invariance.

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Contents

1. Introduction2

2.Conformal invariance in hydrodynamics

2.1Conformal invariance and Weyl anomalies

2.2 First order hydrodynamics as derivative expansion

2.3Conformal invariance in first-order hydrodynamics

3

3

5

6

3. Second-order hydrodynamics of a conformal fluid

3.1 Second-order terms

3.2 Kubo’s formulas

3.3 Sound Pole

3.4 Shear pole

3.5 Bjorken Flow

7

7

9

9

10

11

4. Second-order hydrodynamics for strongly coupled N = 4 supersymmetric

Yang-Mills plasma

4.1Scalar channel

4.2 Shear channel

4.3 Sound channel

4.4Bjorken flow

13

13

15

15

17

5. Kinetic theory

5.1 Setup

5.2Moment approximation

5.3The structure of the collision integral

17

18

19

21

6. Analysis of the M¨ uller-Israel-Stewart theory

6.1 Causality in first order hydrodynamics

6.2 Hydrodynamic variables and second order hydrodynamics

6.3 Causality and the domain of applicability

6.4 Entropy and the second law of thermodynamics

6.5Additional non-hydrodynamic modes

21

21

22

23

24

25

7. Conclusion26

A. Perturbative solutions of the shear and the sound mode equations27

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1. Introduction

Relativistic hydrodynamics is an important theoretical tool in heavy-ion physics, astrophysics,

and cosmology. Hydrodynamics gives reliable description of the non-equilibrium real-time

macroscopic evolution of a given system. It is an effective description in terms of a few

relevant variables (fields) and it applies to the evolution which is slow, both in space and in

time, relative to a certain microscopic scale [1, 2].

In the most common applications of hydrodynamics the underlying microscopic theory

is a kinetic theory. In this case the microscopic scale which limits the validity of hydrody-

namics is the mean free path ℓmfp. In other words, the parameter controlling the precision of

hydrodynamic approximation is kℓmfp, where k is the characteristic momentum scale of the

process under consideration.

More generally, the underlying microscopic description is a quantum field theory, which

might not necessarily admit a kinetic description. An experimental example of such a system

is the strongly coupled quark-gluon plasma (sQGP) recently discovered at the Relativistic

Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory. The N = 4 supersymmetric

SU(Nc) Yang-Mills theory in the limit of strong coupling provides a theoretical example of

such a system which, in the limit of large number of colors Nc, can be studied analytically

using the AdS/CFT correspondence [3]. In these cases, where kinetic description may be

absent, the role of the parameter ℓmfpis played by some typical microscopic scale. In the

above examples this scale is set by the temperature: ℓmfp∼ T−1.

When the parameter kℓmfpis not too small, one may want to go beyond the first order

in kℓmfp. This is the case, for example, in the early stages of heavy-ion collisions. There are

two sources of corrections beyond the kℓmfporder. First, there are corrections due to thermal

fluctuations of hydrodynamic variables contributing via nonlinearities of the hydrodynamic

equations. The fluctuation corrections lead to nonanalytic low-momentum behavior of certain

correlators [4] (similarly to the chiral logarithms that emerge from loops in chiral perturbation

theory) and are, for example, essential for describing non-trivial dynamical critical behavior

near phase transitions [5]. Such corrections are calculable in the framework of hydrodynamics

with thermal noise.

The second source of corrections are second-order terms (order (kℓmfp)2) in the hydrody-

namic equations, sometimes called the Burnett corrections [6]. These corrections come with

additional transport coefficients. These second-order transport coefficients are not calculable

from hydrodynamics, but have to be determined from underlying microscopic description or

input phenomenologically, similarly to first-order transport coefficients such as viscosity.

In gauge theories with a large number of colors Nc the corrections of the first type

(fluctuation) are suppressed by 1/N2

c

[4] and therefore the corrections of the second type

(Burnett) dominate in the limit of fixed k and Nc → ∞. For this reason, in this paper,

we concentrate on the second type of corrections. Moreover, we shall consider the case of

conformal theories, where the number of second-order transport coefficients is substantially

reduced. In the real-world applications we deal with fluids which are not exactly conformal,

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however, e.g., QCD at sufficiently high temperatures is approximately conformal.

This paper is organized as follows. In Sec. 2 we derive the consequences of conformal

symmetry for hydrodynamics. In Sec. 3 we classify all terms of order k2consistent with

conformal symmetry. In Sec. 4 we compute three of the five new transport coefficients for

the strongly-coupled N = 4 supersymmetric Yang-Mills (SYM) theory using the AdS/CFT

correspondence. In Sec. 5 we show that hydrodynamic equations derived from the kinetic

description (Boltzmann equation) of a weakly coupled conformal theory do not contain all

allowed second-order terms. In Sec. 6, we analyze our findings from the point of view of

the M¨ uller-Israel-Stewart theory [7, 8, 9, 10], which involves only one new parameter at the

second order, and show that this parameter cannot account for all second-order corrections.

Our conclusions are summarized in Sec. 7.

2. Conformal invariance in hydrodynamics

To set the stage, let us emphasize again that hydrodynamics is a controlled expansion scheme

ordered by the power of the parameter kℓmfp, or equivalently, by the number of derivatives

of the hydrodynamic fields. We shall set up this expansion paying particular attention to the

consequences of the conformal invariance on the equations of hydrodynamics.

2.1 Conformal invariance and Weyl anomalies

The hydrodynamic fields are expectation values of certain quantum fields, such as e.g., com-

ponents of the stress-energy tensor, averaged over small but macroscopic volumes and time

intervals. Such averages can, in principle, be calculated in the close-time-path (CTP) formal-

ism [11]. Consider a generic finite-temperature field theory in the CTP formulation. Turning

on external metrics on the upper and lower contours, the partition function is

Z[g1

µν,g2

µν] =

?

Dφ1Dφ2exp?iS[φ1,g1

µν] − iS[φ2,g2

µν]?, (2.1)

where φ1and φ2represent the two sets of all fields living on the upper and lower parts of the

contours, and S[φ,gµν] is the general coordinate invariant action.

The one-point Green’s function of the stress-energy tensor is obtained by differentiating

the partition function (the metric signature here is − + ++):

?T1µν? = −

2i

√−g1

2i

√−g2

δ lnZ

δg1

δ lnZ

δg2

µν

, (2.2)

?T2µν? =

µν

, (2.3)

where ?...? denote the mean value under the path integral and√−g1,2≡

In this paper we consider conformally invariant theories. In such theories the action

S[φ,gµν] evaluated on classical equations of motion δS/δφ = 0 and viewed as a functional of

?

−detg1,2

µν.

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the external metric gµνis invariant under local dilatations, or Weyl transformations:

gµν→ e−2ωgµν, (2.4)

with parameter ω a function of space-time coordinates. As a consequence, classical stress-

energy tensor Tµν

In the conformal quantum theory (2.1) the Weyl anomaly [12, 13] implies

cl≡ δS/δgµνis traceless since gµνTµν

cl= −(1/2)δS/δω = 0.

g1

g2

µν?T1µν? = Wd[g1

µν?T2µν? = Wd[g2

µν], (2.5a)

µν], (2.5b)

where Wdis the Weyl anomaly in d dimensions, which is identically zero for odd d. For d = 4:

a

16π2(RµνλρRµνλρ−4RµνRµν+R2)+

W4[gµν] = −

c

16π2(RµνλρRµνλρ−2RµνRµν+1

3R2), (2.6)

where Rµνλρand Rµν(R) are the Riemann tensor and Ricci tensor (scalar), and for SU(Nc)

N = 4 SYM theory a = c =1

derivatives. In general, for even d = 2k, W2kcontains 2k derivatives of the metric.

Let us now explore the consequences of Weyl anomalies for hydrodynamics. The hydro-

dynamic equations (without noise) do not capture the whole set of CTP Green’s functions,

but only the retarded ones. Hydrodynamics determines the stress-energy tensor Tµν(more

precisely, its slowly varying average over sufficiently long scales) in the presence of an arbi-

trary (also slowly varying) source gµν. The connection to the CTP partition function can be

made explicit by writing

4

?N2

c− 1?[14]. The right-hand side of Eqs. (2.5) contains four

g1

µν= gµν+1

2γµν,g2

µν= gµν−1

2γµν. (2.7)

If γµν= 0 then Z = 1 since the time evolution on the lower contour exactly cancels out the

time evolution on the upper contour. When γµνis small one can expand

lnZ =i

2

?

dx

?

−g(x)γµν(x)Tµν(x) + O(γ2), (2.8)

where Tµν(x) depends on gµν, and is the stress-energy tensor in the presence of the source

gµν. At long distance scales it should be the same as computed from hydrodynamics.

Substituting Eqs. (2.7) and (2.8) into Eq. (2.5), the O(1) and O(γ) terms yield two

equations:

gµνTµν= Wd[gµν], (2.9a)

gµν(x)δ[?−g(x)Tαβ(x)]

δgµν(y)

+

?

−g(x)Tαβ(x)δd(x − y) =

δ

δgαβ(y)(

?

−g(x)Wd[gµν(x)]).

(2.9b)

In odd dimensions, the right hand sides of Eqs. (2.9) are zero. In even dimensions, they

contain d derivatives. In a hydrodynamic theory, where one keeps less than d derivatives,

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