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 ⇒ ⇒ hydrodynamics holographic description Introduction Hydrodynamics as an effective theory This talk: 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. from Kolb/Heinz review 10 y (fm) v2 – a measure of elliptic flow is a key observable. 5 Pressure gradient is large in-plane. This translates 0 into momentum anisotropy. To do this the plasma must do work, i.e., pressure × ∆ V −5 −10 −10 −5 0 5 10 x (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? Teaney 0.2 Ideal hydro already agrees with data. ) T (p 0.18 b ≈ 6.8 fm (16-24% Central) 2 v Adding even a small viscous correction makes the 0.16 STAR Data 0.14 agreement worse (Teaney, Romatschke, . . . ) Γ / τ = 0 s o 0.12 If the plasma was weakly interacting the viscosity 0.1 Γ / τ = 0.1 η s o T 3 ∼ (coupling) − 2 would be large. 0.08 0.06 / = 0.2 Γ τ 0.04 s o Conclusion: the plasma must be strongly coupled 0.02 – it is a liquid. 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 p (GeV) T 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/ (4 π ) . And so is in a class of theories with infinite coupling. Special to AdS/CFT, or a universal lower bound? If η s = 1 4 π 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, ℓ mfp . The scale where hydrodynamics applies is greater than ℓ mfp . 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 – kℓ micro . 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 0 i . Convenient covariant variables: T 00 in the local rest frame (where T 0 i = 0 ); and ε – u µ – local 4-velocity (the velocity of the local rest frame). Then, by Lorentz covariance: T µν ≡ ε u µ u ν + T µν ⊥ where T µν ⊥ – has only spatial components in local rest frame (i.e., u µ T µν = 0 ). ⊥ ⊥ are not independent variables, but (local, The components of T µν instantaneous) functions of ε and u µ . = P ( ε )∆ µν + terms with gradients T µν ⊥ where the symmetric, transverse ( ⊥ ) tensor with no derivatives is ∆ µν ≡ g µν + u µ u ν , 4 variables and 4 equations: ∇ µ T µν = 0 . Relativistic conformal hydrodynamics and holography – p. 6/2

First order order hydrodynamics Without gradient terms – ideal hydrodynamics. To first order in gradients: = P ( ε )∆ µν − η ( ε ) σ µν − ζ ( ε )∆ µν ( ∇· u ) + higher derivs. T µν ⊥ | {z } viscous stress Π µν where viscous strain (traceless, or shear part of it): σ µν = 2 � ∇ µ u ν � = 1 1 � A µν � def 2∆ µα ∆ νβ ( A αβ + A βα ) − d − 1∆ µν ∆ αβ A αβ ( ∆ µν projects on ⊥ u µ ). 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; ησ µν – 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 > 2 T c is almost conformal (but still strongly coupled). Meyer RBC-BI LHC 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 η /s [0704.1801] 12 0.2 RHIC Tr 0 η /s prelim. ( ε -3p)/T 4 10 ζ /s prelim. 8 p4: N τ =4 6 6 0.1 8 1/4 π 4 2 T [MeV] T/T c 0 0 100 200 300 400 500 600 700 800 1 1.5 2 2.5 3 3.5 4 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 g µν → λ − 2 g µν . One can then promote g µν → λ − 2 g µν 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): g µν → e − 2 ω ( x ) g µν . For example, since T µν ≡ δS/δg µν µ = g µν T µν = − (1 / 2) δS/δω = 0 T µ Relativistic conformal hydrodynamics and holography – p. 9/2

Conformal hydrodynamics (to 1st order) Using just tracelessness T µ ( ∆ µ µ = 0 constrains these coefficients µ = d − 1 ): ε P = d − 1; ζ = 0 . To use Weyl invariance we need transformation properties of hydro variables: T → e ω T ε = # · T d . (We shall use T below.) By dimensions: and g µν u µ u ν = − 1 u µ → e ω u µ . means Since T µν √− g = δS/δg µν , T µν → e ( d +2) ω T µν ; More nontrivially, σ µν ≡ 2 � ∇ µ u ν � transforms homogeneously σ µν → e 3 ω σ µν , 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: 1 Du µ = −∇ µ D ln T = − d − 1( ∇ ⊥ · u ) , ⊥ ln T, to convert temporal derivatives ( D ≡ u µ ∇ ν ) into spatial ( ∇ µ ⊥ ≡ ∆ µα ∇ α ) . ∃ five such terms: “ ” = R � µν � − ( d − 2) ∇ � µ ∇ ν � ln T − ∇ � µ ln T ∇ ν � ln T O µν , 1 = R � µν � − ( d − 2) u α R α � µν � β u β , O µν 2 λ σ ν � λ , λ Ω ν � λ , λ Ω ν � λ . O µν O µν O µν = σ � µ = σ � µ = Ω � µ 3 4 5 where Ω µν = ∆ µα ∆ νβ ∇ [ α u β ] – vorticity. contribute in linearized hydrodynamics. Only O µν and O µν 1 2 O µν = 0 in flat space. 2 Relativistic conformal hydrodynamics and holography – p. 11/2

Second-order kinetic coefficients Convenient to use this combination O µν − O µν − (1 / 2) O µν − 2 O µν equal to 1 2 3 5 1 � Dσ µν � + d − 1 σ µν ( ∇· u ) Stress tensor to 2-nd order: = P ∆ µν − ησ µν T µν ⊥ » – h i 1 R � µν � − ( d − 2) u α R α � µν � β u β � Dσ µν � + d − 1 σ µν ( ∇· u ) + ητ Π + κ λ σ ν � λ + λ 2 σ � µ λ Ω ν � λ + λ 3 Ω � µ λ Ω ν � λ . + λ 1 σ � µ The five new coefficients are τ Π , κ , λ 1 , 2 , 3 . Nonlinear term σ µν ∇· 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. d 5 x √− g ( R − 2Λ) R S = Recipe for calculating a correlator of, e.g., T µν : Vary boundary value at z = 0 of g µν , then δS � T µν ( x ) � = δg µν ( x, 0) . Relativistic conformal hydrodynamics and holography – p. 13/2