Viscosity in strongly coupled gauge theories Lessons from string theory

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1 Viscosity in strongly coupled gauge theories Lessons from string theory Pavel Kovtun KITP, University of California, Santa Barbara A.Buchel, (University of Western Ontario) C.Herzog, (University of Washington, Seattle) J.Liu, (University of Michigan) J.Mas, (University Santiago de Compostella) A.Nunez, (New York University) A.Parnachev, (Rutgers University) G.Policastro, (Munich University) D.Son, (University of Washington, Seattle) A.Starinets, (Perimeter Institute, Waterloo) D.Teaney, (SUNY, Stony Brook) A review of many people s work

2 Basic setup Relativistic field theories: In d = 3+1 dimensions Translation + rotation invariant Have stable thermal equilibrium state For simplicity µ = 0, not near critical point Small fluctuations on top of thermodynamic equilibrium state: Slowly relaxing d.o.f.: ǫ = T 00, π i = T 0i = ǫ+p v i, ρ = J 0 Obey classical equations hydrodynamics Does not depend on microscopic details of the theory

3 Conservation laws: µ T µν =0 Hydrodynamic fluctuations { t ǫ = π t π i = j T ij Constitutive relations: { T ij = δ ij [ P + v 2 sδǫ γ ζ π ] γ η ( i π j + j π i 2 3 δij π ) +... γ η η ǫ+p, γ ζ ζ ǫ+p, v2 s = P/ ǫ Viscosities η, ζ input from microscopic physics Two eigenmodes: Shear mode: π (t, k) = e γ ηk 2t π (0, k) Sound mode: π (t, k) = e 1 2 (γ ζ+ 4 3 γ η)k 2 t [ ] π (0, k) cos(kv s t) iˆkv s sin(kv s t)δǫ(0, k) Long-wavelength response is controlled by a small number of kinetic coefficients

4 Correlation functions in the hydrodynamic limit Hydrodynamic modes hydrodynamic singularities at small ω, k. Example: S tx,tx (ω, k) = Kubo formulas 1 η = lim ω 0 2ω 2γ η k 2 (ǫ+p)t ω 2 + (γ η k 2 relaxation of transverse momentum ) 2 dt e iωt d 3 x [T xy (t, x), T xy (0)] Connection to microscopic physics: Viscosities can be extracted from (small-frequency limits of) real-time correlation functions

5 Physical meaning of viscosity z Example: flow of gas Internal friction momentum exchange between layers of fluid x T zx = η v z flux of x-momentum in z-direction Particle number per unit surface per unit time: (nv th ) Momentum per unit surface per unit time: (nv th )(mv) Flux up: nv th m(v 0 l mfp v z ), flux down: nv thm(v 0 + l mfp v z ), T zx = flux up flux down η nv th ml mfp mv th σ (mt)1/2 σ Ideal gas has infinite viscosity

6 How viscosity is measured WEAK COUPLING STRONG COUPLING Prof. J.Maxwell, Mrs.Maxwell, (attic, 1860) RHIC, many people (Brookhaven, 2000) Strong coupling is more difficult

7 How viscosity is computed Gases: Kinetic theory η (mt) 1/2 1 Density-independent, grows with T σ Plasmas: Boltzmann-Vlasov equation η (mt) 1/2 ( T2 e 4 ln Λ ) Coulomb log Λ l dt e 2 T3/2 e 3 n 1/2 due to small-angle scatt. Liquids: Complicated: use molecular dynamics η exp( E a T ) Typical: decreases with T Weakly coupled QFT: Boltzmann-Vlasov equation, or resum Feynman diagrams η T3 g 4 ln g 1 (Log absent if no gauge fields) Non-abelian gauge theories: at T T c need lattice Evaluate Euclidean correlators, invert to find real-time spectral functions... η =???

8 Q: Are there any toy models where viscosity can be analytically computed at strong coupling? A: N=4 supersymmetric Yang-Mills theory, SU(N c ), λ = g 2 N c Why N=4 SYM? Dual string description (AdS/CFT correspondence) allows to compute correlation functions at strong coupling What is N=4 SYM? Gauge fields + 4 fermions + 6 scalars in adjoint of SU(N c ) Conformal theory, λ is a tunable parameter (does not run) Supersymmetric, but SUSY not essential at finite temperature ǫ = 3P, v s = 1 3, ζ = 0, at any non-zero temperature ǫ, P, η are finite in the limit λ

9 AdS/CFT correspondence (J.Maldacena hep-th/ , review: hep-th/ ) large N c, d=4, N=4 SYM = IIB strings on AdS 5 S 5 λ ( R 2 α ) 2 λ 4πN c g s string corrections to SUGRA string loops horizon z=z h AdS 5 strings boundary z=0 R e h(x)t(x) field = Z string [g(x, z 0) = h(x)] R 3,1 T µν (x) h µν (x, z 0) field theory J µ (x) A µ (x, z 0) trf 2 (x) ϕ(x, z 0). T µν T αβ δ2 ln Z string [h] δh µν δh αβ δ 2 δh µν δh αβ S cl [h] Duality unproven, but many consistency checks performed

10 How to compute viscosity from AdS/CFT (G.Policastro, D.Son, A.Starinets hep-th/ , hep-th/ , PK, D.Son, A.Starinets hep-th/ , PK, A.Starinets hep-th/ ) S cl = d 4 xdz g g µν µ ϕ ν ϕ = d 4 x g g zz ϕ(x, z) z ϕ(x, z) z 0 G ret (k) 1 z 3 f(k, z) z f( k, z) z 0 f(k, z) satisfies e.o.m. f(k, z) 1 as z 0 f(k, z) outgoing as z z h (breaks time-reversal invariance) For G ret µν,αβ (k), need linearized metric perturbations h ρλ coupled ODEs Shear viscosity η T xy T xy, need h xy (x, z) h y x = ϕ (minimal massless scalar), and decouples (true more generally) In the hydro limit ω/t 1, k /T 1, can solve analytically Computing real-time correlation functions in strongly coupled SYM amounts to solving a wave equation on a background with BH

11 Spectral function for stress (PK, A.Starinets hep-th/ , D.Teaney hep-ph/ ) 1 ( ω χ( ω) χ T=0 ( ω) ) [ ] 1 π 2 Nc 2T4 0.8 χ(ω, k) = 2 Im G ret xy,xy(ω, k) χ(ω) ω, ω 2πT χ(ω) χ T=0 (ω) e γω, ω 2πT η = π 8 N2 c T ω ω 2πT T 3 by conformal invariance, N 2 c counts d.o.f. Spectral function for conserved energy-momentum 1.2 χ tx,tx ( ω) 10 χ tt,tt ( ω) ω Hydrodynamic peaks clearly visible in dual classical gravity ω

12 Singularities of G ret (ω, k) (A.Nunez, A.Starinets hep-th/ , PK, A.Starinets hep-th/ ) G ret xy,xy(ω) Im( ω) Re( ω) Infinite series of poles ω n = 2πnT(±1 i) as n For conserved densities, ω 0 0 as k 0 Hydro poles agree with Kubo formula G ret tx,tx(ω) G ret tt,tt(ω) Re( ω) Re( ω) Im( ω) -5 Im( ω) Singularities of G ret (ω, k) are (quasi)normal modes of the dual gravity background

13 Universality of η/s (PK, D.Son, A.Starinets hep-th/ , A.Buchel hep-th/ ) Hydro damping rate is set by η/s Shear viscosity η N 2 c, but η/s is finite in the N c limit π s/s 0 η/s At strong coupling: s = 3 4 s(λ=0) [1 + O(λ 3/2 )] η/s = 1 4π [1 + O(λ 3/2 )] 0 λ η s = 1 4π does not depend on the dual gravity background!!! Conformal invariance, SUSY, 3+1 dimensions not essential for the universality η/s = 1 4π for a large class of strongly coupled field theories

14 Speculation: Lower bound on shear viscosity?!! η/s 1 at small coupling η/s is finite at large coupling Natural to assume η/s 1 4π in SYM Is η s 1 4π universal? Prove from first principles?

15 Correction η/s = 1 4π Generalizations (A.Buchel, J.Liu, A.Starinets hep-th/ ) [1 + O( 1 λ 3/2 )] known explicitly in N=4 SYM Add mass terms to N=4 SYM (destroys scale invariance, reduces SUSY): η/s = 1 4π, ζ/η 4.56(v2 s 1 3 ) (P.Benincasa, A.Buchel, A.Starinets hep-th/ ) Non-zero chemical potential: η/s = 1 4π in N=4 SYM, does not depend on µ T (D.Son, A.Starinets hep-th/ , J.Mas hep-th/ , K.Maeda, M.Natsuume, T.Okamura hep-th/ )

16 Photon production from SYM (PK, A.Starinets, to appear) k dγ dk k2 η µν e k/t 1 Im Πret µν(ω, k) ω2 =k x k/t Thick line: x 3 e x 1 Thin line: x 3 e x 1 f SYM(x) At small x: f SYM (x) const, Im Π µ µ(ω=k) k, consistent with hydrodynamics

17 Summary Viscosity can be relatively easily computed for some strongly coupled theories such as N=4 SYM Ratio of viscosity to entropy density is 1/4π for a large class of strongly coupled gauge theories. Lower bound η/s 1 4π? Limitations of the AdS/CFT approach: SUSY theories, contain scalar fields Conformal in the UV rather than AF Fundamental matter N f N c 1/λ corrections doable, 1/N c corrections hard Search for universal properties of strongly interacting thermal gauge theories Other questions can be addressed in AdS/CFT: energy loss by a heavy quark, existence of resonances, photon production, thermalization... Can AdS/CFT be useful for heavy-ion physics?