Relativistic conformal hydrodynamics and holography M. Stephanov - - PowerPoint PPT Presentation

Relativistic conformal hydrodynamics and holography

• M. Stephanov
• U. of Illinois at Chicago

with R. Baier, P . Romatschke, D. Son and A. Starinets arXiv:0712.2451

Relativistic conformal hydrodynamics and holography – p. 1/2

Motivation

Relativistic Heavy Ion Collisions Traditional path: kinetic description ⇒ hydrodynamics Discovery of sQGP: hydrodynamics but no kinetic description i.e QFT ⇒ hydrodynamics. Strong coupling regime of some SUSY gauge theories can be studied using AdS/CFT (holographic) correspondence. i.e., instead of QFT ⇒ kinetic description (Boltzmann) ⇒ hydrodynamics, QFT ⇒ holographic description ⇒ hydrodynamics This talk: Introduction Hydrodynamics as an effective theory Finding kinetic coeff. by matching to AdS/CFT.

Relativistic conformal hydrodynamics and holography – p. 2/2

Hydrodynamic modeling of R.H.I.C. and v2

Approach: take an equation of state, initial conditions, and solve hydrodynamic equations to get particle yields, spectra, etc. v2 – a measure of elliptic flow is a key observable. Pressure gradient is large in-plane. This translates into momentum anisotropy. To do this the plasma must do work, i.e., pressure×∆V

from Kolb/Heinz review

−10 −5 5 10 −10 −5 5 10 x (fm) y (fm)

v2 is large → 1st conclusion, there is pressure, and it builds very early. I.e., plasma thermalizes early (< 1fm/c). BIG theory question: HOW does it thermalize? and why so fast/early? Need to understand initial conditions Mechanism of thermalization? Plasma instabilities?

Relativistic conformal hydrodynamics and holography – p. 3/2

Small viscosity and sQGP (liquid)

Another surprise: where is the viscosity? Ideal hydro already agrees with data. Adding even a small viscous correction makes the agreement worse (Teaney, Romatschke, . . . ) If the plasma was weakly interacting the viscosity η T 3 ∼ (coupling)−2 would be large. Conclusion: the plasma must be strongly coupled – it is a liquid.

Teaney

(GeV)

T

p 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 )

T

(p

2

v 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

6.8 fm (16-24% Central) ≈ b

= 0.1

• τ

/

s

Γ = 0.2

• τ

/

s

Γ = 0

• τ

/

s

Γ

STAR Data

Can there be an ideal liquid, can η = 0? What if coupling → ∞? Policastro, Kovtun, Son, Starinets found that in an N = 4 super-Yang-Mills theory at ∞ coupling η = s/(4π). And so is in a class of theories with infinite

• coupling. Special to AdS/CFT, or a universal lower bound?

If η s = 1 4π is the lowest bound – data suggests RHIC produced an almost perfect fluid. Need viscous (3D) hydro simulation to confirm. Second-order corrections?

Relativistic conformal hydrodynamics and holography – p. 4/2

Scales and hydrodynamics

Hydrodynamics is an effective macroscopic theory, describing transport of energy, momentum and other conserved quantities. The domain of validity is large distance and time scales (small k and ω). If the underlying kinetic description exists, there is a mean free path, ℓmfp. The scale where hydrodynamics applies is greater than ℓmfp. In a strongly coupled system (e.g., sQGP at RHIC) kinetic description may not

• exist. Then the domain of validity is set by a typical microscopic scale, e.g., T −1.

Hydrodynamics can be described as an expansion in gradients. To lowest order – ideal hydrodynamics. The expansion parameter – kℓmicro.

Relativistic conformal hydrodynamics and holography – p. 5/2

Hydrodynamic degrees of freedom and equations

Densities of conserved quantities. In any field theory at least energy and momentum densities T 00 and T 0i. Convenient covariant variables: ε – T 00 in the local rest frame (where T 0i = 0); and uµ – local 4-velocity (the velocity of the local rest frame). Then, by Lorentz covariance: T µν ≡ ε uµuν + T µν

⊥

where T µν

⊥ – has only spatial components in local rest frame (i.e., uµT µν ⊥

= 0). The components of T µν

⊥ are not independent variables, but (local,

instantaneous) functions of ε and uµ. T µν

⊥

= P(ε)∆µν + terms with gradients where the symmetric, transverse (⊥) tensor with no derivatives is ∆µν ≡ gµν + uµuν , 4 variables and 4 equations: ∇µT µν = 0.

Relativistic conformal hydrodynamics and holography – p. 6/2

First order order hydrodynamics

Without gradient terms – ideal hydrodynamics. To first order in gradients: T µν

⊥

= P(ε)∆µν − η(ε)σµν − ζ(ε)∆µν(∇·u) + higher derivs. | {z } viscous stress Πµν where viscous strain (traceless, or shear part of it): σµν = 2∇µuν

Aµν def

= 1 2∆µα∆νβ(Aαβ + Aβα) − 1 d − 1∆µν∆αβAαβ (∆µν projects on ⊥ uµ). η and ζ – shear and bulk viscosities. T ij – rate of momentum transfer (flow), i.e., force/area ζ(∇·u) – contribution to isotropic pressure due to gradients; ησµν – drag force due to the gradients of velocity ⊥ to the velocity – shear stress.

Relativistic conformal hydrodynamics and holography – p. 7/2

Conformal theories

Why could this be relevant to QCD? QCD at T > 2Tc is almost conformal (but still strongly coupled).

RBC-BI

2 4 6 8 10 12 100 200 300 400 500 600 700 800 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

T [MeV] Tr0 (ε-3p)/T4

p4: Nτ=4 6 8

Meyer

0.1 0.2 1 1.5 2 2.5 3 3.5 4

T/Tc

1/4π RHIC LHC η/s [0704.1801] η/s prelim. ζ/s prelim.

AdS/CFT

Relativistic conformal hydrodynamics and holography – p. 8/2

Scale invariance and Weyl symmetry

Consider a field theory with no scale, self-similar under dilation x → λx (accompanied by appropriate rescaling of fields). λ = const here. Examples: ferromagnet at a critical point, N = 4 SUSY YM. Instead of coordinate rescaling one can formally do gµν → λ−2gµν. One can then promote gµν → λ−2gµν to local symmetry, i.e., generalize the theory to curved space in such a way that the action (as a functional of background metric) is invariant under local Weyl transformations (in addition to GR transforms): gµν → e−2ω(x)gµν. For example, since T µν ≡ δS/δgµν T µ

µ = gµνT µν = −(1/2)δS/δω = 0

Relativistic conformal hydrodynamics and holography – p. 9/2

Conformal hydrodynamics (to 1st order)

Using just tracelessness T µ

µ = 0 constrains these coefficients

(∆µ

µ = d − 1):

P = ε d − 1; ζ = 0. To use Weyl invariance we need transformation properties of hydro variables: By dimensions: T → eωT and ε = # · T d. (We shall use T below.) gµνuµuν = −1 means uµ → eωuµ. Since T µν√−g = δS/δgµν, T µν → e(d+2)ω T µν; More nontrivially, σµν ≡ 2∇µuν transforms homogeneously σµν → e3ωσµν, hence η = # · T d−1.

Relativistic conformal hydrodynamics and holography – p. 10/2

Second-order hydrodynamics

Need to find all possible contributions to T µν

⊥ with 2 derivatives, transforming

homogeneously under Weyl transform. Also: use 0-th order equations: D ln T = − 1 d − 1(∇⊥ · u), Duµ = −∇µ

⊥ ln T,

to convert temporal derivatives (D ≡ uµ∇ν) into spatial (∇µ

⊥ ≡ ∆µα∇α) .

∃ five such terms: Oµν

1

= Rµν − (d − 2) “ ∇µ∇ν ln T − ∇µ ln T ∇ν ln T ” , Oµν

2

= Rµν − (d − 2)uαRαµνβuβ , Oµν

3

= σµ

λσνλ ,

Oµν

4

= σµ

λΩνλ ,

Oµν

5

= Ωµ

λΩνλ .

where Ωµν = ∆µα∆νβ∇[αuβ] – vorticity. Only Oµν

1

and Oµν

2

contribute in linearized hydrodynamics. Oµν

2

= 0 in flat space.

Relativistic conformal hydrodynamics and holography – p. 11/2

Second-order kinetic coefficients

Convenient to use this combination Oµν

1

− Oµν

2

− (1/2)Oµν

3

− 2Oµν

5

equal to

Dσµν +

1 d − 1σµν(∇·u) Stress tensor to 2-nd order: T µν

⊥

= P∆µν − ησµν + ητΠ »

Dσµν +

1 d − 1σµν(∇·u) – + κ h Rµν − (d − 2)uαRαµνβuβ i + λ1σµ

λσνλ + λ2σµ λΩνλ + λ3Ωµ λΩνλ .

The five new coefficients are τΠ, κ, λ1,2,3. Nonlinear term σµν∇·u has until recently been often omitted. We see this term is necessary for conformal invariance.

Relativistic conformal hydrodynamics and holography – p. 12/2

AdS/CFT

The 4d N = 4 SUSY YM theory in strong coupling limit can be represented by a semiclassical gravitational theory in 5d. S = R d5x √−g(R − 2Λ) Recipe for calculating a correlator of, e.g., T µν: Vary boundary value at z = 0 of gµν, then T µν(x) = δS δgµν(x, 0).

Relativistic conformal hydrodynamics and holography – p. 13/2

Kinetic coefficients from AdS/CFT

Example: match the following correlator in hydrodynamics: T xyT xy(ω, k)ret = P − iηω + ητΠω2 − κ 2 [(d − 3)ω2 + k2] . to gravity calculation and find P = π2 8 N 2

c T 4,

η = π 8 N 2

c T 3,

τΠ = 2 − ln 2 2πT , κ = η πT . | {z } new Nontrivial cross-checks in sound and shear channels. Using solution to nonlinear equations found by Heller and Janik (asymptotics at large τ of Bjorken boost-invariant flow): λ1 = η 2πT Bhattacharyya, Hubeny, Minwalla, Rangamani: λ2 = 2η ln 2 πT ; λ3 = 0.

Relativistic conformal hydrodynamics and holography – p. 14/2

In kinetic (weakly coupled) theory: τΠ ∼ η Ts ≫ 1 T . κ = 0(?)

Relativistic conformal hydrodynamics and holography – p. 15/2

Müller-Israel-Stewart

Truncate the gradient expansion at second order. Use Πµν = −ησµν in second-order terms. Resulting equations are hyperbolic (causal) even outside of domain of validity (large gradients) – good for simulations. Transverse momentum modes (shear) obey diffusion equation similar to: ∂tρ = −∇j with j = −D∇ρ Which means ∂tρ = D∇2ρ - parabolic. Disturbance propagates with infinite speed? Problem even for nonrelativistic case? Now use instead: j = −D∇ρ − τ∂tj This system is hyperbolic, with characteristic velocity: vdisc = p D/τ The problem is only in the regime (kℓ 1) where hydrodynamics is

• inapplicable. There are no actual modes which propagate with vdisc.

Relativistic conformal hydrodynamics and holography – p. 16/2

Summary

Hydrodynamics is an expansion in gradients of hydrodynamic variables. In conformal theories (e.g., QCD above 2Tc) the form of the equations (stress tensor) are restricted. To first order: only one viscosity coefficient η. To second order: only 5 (in curved space) coefficients. For N = 4 SUSY YM at strong coupling (and large Nc) the coefficients have been determined using AdS/CFT.

Relativistic conformal hydrodynamics and holography – p. 17/2

Appendix

Relativistic conformal hydrodynamics and holography – p. 18/2

Viscosity on the lattice

Difficult problem: need to get large real-time behavior of a correlation function, from Euclidean (imaginary) time measurements. Numerical noise must be very low. Must assume that extrapolation to large times (low frequencies) is smooth.

Meyer

0.1 0.2 1 1.5 2 2.5 3 3.5 4

T/Tc

1/4π RHIC LHC η/s [0704.1801] η/s prelim. ζ/s prelim.

At T ∼ 1 − 3.5 Tc η/s is close to 1/(4π) The bulk viscosity vanishes quickly above T ∼ 2Tc. The latter is in agreement with trace anomaly calcula- tion by RBC-BI →

2 4 6 8 10 12 100 200 300 400 500 600 700 800 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

T [MeV] Tr0 (ε-3p)/T4

p4: Nτ=4 6 8

Relativistic conformal hydrodynamics and holography – p. 19/2

Entropy and the second law

Relativistic conformal hydrodynamics and holography – p. 20/2