Conformal hydrodynamics beyond supergravity approximation Alex - PowerPoint PPT Presentation

Conformal hydrodynamics beyond supergravity approximation Alex Buchel (Perimeter Institute & University of Western Ontario) Based on: arXiv:0804.3161, 0806.0788, 0808.1601, 0808.1837, and to appear with: Rob Myers, Miguel Paulos, Aninda Sinha 1

Outline of the talk: • Motivation • Conformal hydrodynamics from the gauge theory perspective: ⇒ first order hydrodynamics; = = ⇒ consistency of hydrodynamic description; ⇒ second order (causal) hydrodynamics; = ⇒ ⋆ boost-invariant expansion of a CFT plasma. = • N = 4 SYM gauge theory plasma as a toy model: = ⇒ non-equilibrium AdS/CFT correspondence beyond the supergravity approximation; ⇒ universality of transport of CFT plasma beyond the supergravity approximation. = • Non-universal viscosity bound violation in CFT plasma with c � = a central charges • Conclusions and future directions 2

Motivation ⇒ One of the striking application of the gauge theory/string theory duality to study strongly = coupled gauge theory plasma is the (conjectured) KSS bound: η � s ≥ 4 πk b The bound is saturated at infinitly strong coupling, and in the planar limit. Can this bound be violated? If so, under which conditions? ⇒ Can we test gauge theory/string theory duality in the non-equilibrum setting? = = ⇒ How do we formulate a causal relativistic hydrodynamics, and describe boost-invariant expansion of plasma (which could be of relevance to RHIC/LHC)? 3

First-order 4d conformal hydrodynamics (gauge theory perspctive) ⇒ consider translationary invariant theory in flat space in equilibrium. = In local rest frame   ǫ 0 0 0   0 P 0 0   T µ µ = 0 ⇒ ǫ = 3 P ] T µν = , [for CFT :     0 0 P 0     0 0 0 P Theory is characterized by conserved quantities, in particular the stress-energy tensor T µν : ∂ µ T µν = 0 ⇒ consider slow, macroscopic fluctuations = � � | ¯ q | , ω ≪ T, any other microscopic scale Effective description of such fluctuations is provided by macroscopic hydrodynamics 4

Hydrodynamics is based on two assumptions: a: T µν [fluctuations] are conserved (as in equilibrium) • fluctuations are always on-shell — expect to be a good approximation for b: “Linear response theory is valid” — good approximation from small amplitudes • linear response theory introduces phenomenological parameters into effective description of fluctuations 5

Let u µ = ( u 0 , u i ) — fluid 4-velocity. Introduce a proper (rest) frame for the fluid element [ ∂ µ u ν � = 0 u 0 = 1 , u i = 0 , , off − equilibrium ] � � � � T µν = ( P + ǫ ) u µ u ν + Pη µν + τ µν ⇑ ⇑ equilibrium stress tensor stress tensor due to velocity gradients Definition of the rest frame: τ 00 , τ 0 i = 0 ⇒ T 00 = ǫ ; T 0 i = 0 “Constitutive” relation for remaining components: 6

� � � � ∂ i u j + ∂ j u i − 2 δ ij ∂ k u k 3 δ ij ∂ k u u τ ij = − ζ − η ζ — couples to the trace of the velocity gradients — bulk viscosity [in CFT ζ = 0 ] η — couples to the traceless part of the velocity gradients — shear viscosity = ⇒ stress-energy conservation T 00 + ∂ i T 0 i = 0 ∂ 0 T 0 i + ∂ j ˜ T ij = 0 ∂ 0 ˜ ; T 00 ≡ T 00 − ǫ , and where ˜ � � � � 1 ∂ i T 0 j + ∂ j T 0 i − 2 T ij ≡ T ij − Pδ ij = − ˜ 3 δ ij ∂ k T 0 k + ζ δ ij ∂ k T 0 k η ǫ + P = ⇒ we would like to study on-shell fluctuation, i.e, eigenmodes of the above conservation laws 7

Here we have two types of eigenmodes: the shear mode (transverse fluctuations of the momentum density T 0 i ) a : ǫ + P q 2 = − i η iη Ts q 2 ω = − where we used ǫ + P = Ts sound mode (simultaneous fluctuations of the energy density ˜ T 00 and longitudinal b : component of T 0 i ) ω = c s q − i 4 η � 1 + 3 ζ � q 2 2 3 Ts 4 η c s — the speed of sound η, ζ — shear and bulk viscosities Dispersion relations for the fluctuations are realized (mostly) as poles in equilibrium correlation functions 8

I say ’mostly’ because for the shear mode v = (0 , v y , 0) , ¯ v y = v y ( z ) , xy − is a shear plane < T xy ( z ) T xy (0) > R does not have a pole because it does not couple to energy or momentum fluctuations. Rather, we have Kubo formula (sh.1) 1 � xe iωt < [ T xy ( � η = lim dtd� x ) , T xy (0)] > 2 ω ω → 0 � � 1 G A xy,xy ( ω, 0) − G R = lim xy,xy ( ω, 0) 2 ωi ω → 0 Other correlation functions of T µν will have a diffusive pole (sh.2) 1 D = η G R xz,xz ( ω, q z ) ∼ , iω − Dq 2 Ts z 9

For the sound wave mode ( ζ, η, c s ) : (sw.1) can be extracted from equilibrium 1-point correlation function < T µν > s = ∂P c 2 ∂ǫ v CF T 1 Recall, for conformal theories: ǫ = 3 P , so = s √ 3 (sw.2) 1 < T 00 T 00 > R ∝ ω 2 − c 2 s q 2 + i Γ ωq 2 2 q 2 + O ( q 3 ) there is a pole at ω = c s q − i Γ Recall, for conformal theories: ζ = 0 10

Consistency of hydrodynamic description hydro mode computation produces shear (sh.1) < T xy,xy > R,A +Kubo formula η η shear (sh.2) < T xz,xz > R +pole D = T s sound (sw.1) < T 00 >, < T ii > c s sound (sw.2) < T 00 , 00 > R +pole c s , Γ ⇒ (sh.1) and (sh.2) produces η — must be consistent = � � η 1 + 3 ζ ⇒ (sw.1) and (sw.2) produces c s — must be consistent, also Γ = 4 = is 3 T s 4 η sensitive to D, η 11

= ⇒ First order hydrodynamics is acausal: the linearized equation for a diffusive mode is not hyperbolic (first order in temporal but second order in spatial derivatives) — discontinuity in initial conditions propagates at infinite speed. The acausality is a real problem in numerical simulations. Second order causal hydrodynamics ⇒ Motivated largely by AdS/CFT correspondence (though AdS/CFT strictly speaking was = not needed for this), the effective field theory of conformal hydrodynamics was developed by Braier et.al and Bhattacharyya et.al ⇒ First order hydrodynamics involves first-order gradients of the local 4-velocity ∇ α u β ; = second order hydrodynamics includes 2-order gradients of the local 4-velocity. In principle, one can extend the theory to arbitrary order gravients at the expence of introducing new phenomenological parameters (suplumenting η, ζ at the first order). AdS/CFT provides a first-principle evaluation of ALL phenomenological parameters for a given CFT. = ⇒ The hydrodynamic equations is the familiar one: ∇ µ T µν = 0 12

T µν = ǫu µ u ν + P ∆ µν + Π µν , ∆ µν = g µν + u µ u ν Π µν = − ησ µν + (2nd order terms) , u µ Π µν = 0 , g µν Π µν = 0 where σ µν is symmetric transverse tensor constructed of first derivatives. = ⇒ besides the shear viscosity η , the second-order conformal hydrodynamics is described by 5 additional phenomenological parameters: { τ Π , κ , λ 1 , λ 2 , λ 3 } • τ Π is the relaxation time that ’restores’ causality in first-order hydro • λ 1 is a coupling of a term bilinear in the velocities, which show up in boost-invariant expansion of the plasma • λ 2 , 3 are not needed for irrotational flows 13

Consistency of the second order hydrodynamic description = ⇒ Second-order Kubo formular: � � ( ω, q ) = P − iηω + ητ Π ω 2 − κ ω 2 + q 2 G xy,xy R 2 = ⇒ Dispersion relation for the sound: � � ω = c s q − i Γ q 2 + Γ s τ Π − Γ c 2 q 3 c s 2 where Γ is from the 1st-order hydrodynamics. Notice that looking at q 2 dependence in the second order Kubo formular we can obtain τ Π ; the same phenomenological coefficient can be extracted from the O ( q 3 ) sound wave dispersion relation 14

N = 4 SYM gauge theory plasma as a toy model gauge theory string theory N = 4 SU ( N ) SYM ⇐ ⇒ N-units of 5-form flux in type IIB string theory g 2 ⇐ ⇒ g s Y M ⇒ Consider the theory in the ’t Hooft (planar limit), N → ∞ , g 2 Y M → 0 with Ng 2 = Y M kept fixed. SUGRA is valid Ng s → ∞ . In which case the background geometry is AdS 5 × S 5 = ⇒ Beyong the SUGRA approximation 1 ⇐ ⇒ g s -corrections N -corrections 1 α ′ -corrections ⇐ ⇒ Y M -corrections Ng 2 15

In the planar limit, but for a finite (large) ’t Hooft coupling Ng 2 Y M : d 10 x √− g 1 � R − 1 1 � � 4 · 5!( F 5 ) 2 + · · · + γe − 3 2( ∂φ ) 2 − 2 φ W + · · · S IIB = 16 πG 10 where φ is a dilaton, γ = 1 8 ζ (3)( α ′ ) 3 , and W is constructed from the Weyl tensor C mnpq rsk + 1 W ≡ C hmnk C pmnq C rsp C q 2 C hkmn C rqmn C rsp C q h h rsk and · · · denote other SUGRA modes and higher order α ′ corrections Some features of the α ′ corrected geometry at T � = 0 α ′ � = 0 α ′ = 0 φ = 0 φ � = 0 , depends on r size of S 5 is constant size of S 5 depends on r S = A horizon S � = A horizon use Wald formula 4 G 10 4 G 10 T H ≡ T 0 T H ≡ T 0 (1 + 15 γ ) 16

Non-equilibrium AdS/CFT correspondence beyong the spergravity approximation To obtain retarded correlation function of the boundary stress energy tensor, we study scalar perturbations of the background geometry : g 5 µν → g 5 µν + h xy ( u, x ) It will be convenient to introduce a field ϕ ( u, x ) , ϕ ( u, x ) = u h xy ( u, x ) r 2 0 and use the Fourier decomposition d 4 k � (2 π ) 4 e − iωt + i k · x ϕ k ( u ) ϕ ( u, x ) = Finally, we introduce ω k w ≡ k ≡ , 2 πT 0 2 πT 0 17