Holographic hydrodynamization Micha P . Heller m.p.heller@uva.nl - PowerPoint PPT Presentation

Holographic hydrodynamization Micha ł P . Heller m.p.heller@uva.nl University of Amsterdam, The Netherlands & National Centre for Nuclear Research, Poland (on leave) based on 1302.0697 [hep-th] MPH, R. A. Janik & P . Witaszczyk ( PRL 110 (2013) 211602 )

Introduction

Modern relativistic (uncharged) hydrodynamics an EFT of the slow evolution of conserved hydrodynamics is currents in collective media „close to equilibrium” As any EFT it is based on the idea of the gradient expansion u ν u ν = − 1 DOFs : always local energy density and local flow velocity ( ) u µ ✏ EOMs: conservation eqns for systematically expanded in gradients r µ T µ ν = 0 T µ ν terms carrying 2 and more gradients T µ ν = ✏ u µ u ν + P ( ✏ ) { g µ ν + u µ u ν } � ⌘ ( ✏ ) � µ ν � ⇣ ( ✏ ) { g µ ν + u µ u ν } ( r · u ) + . . . perfect fluid stress tensor microscopic bulk viscosity (famous) shear viscosity EoS input : (vanishes for CFTs) 1/16

Applicability of hydrodynamics terms carrying 2 and more gradients T µ ν = ✏ u µ u ν + P ( ✏ ) { g µ ν + u µ u ν } � ⌘ ( ✏ ) � µ ν � ⇣ ( ✏ ) { g µ ν + u µ u ν } ( r · u ) + . . . perfect fluid stress tensor microscopic bulk viscosity (famous) shear viscosity EoS input : (vanishes for CFTs) Naively one might be inclined to associate hydrodynamic regime with small gradients. But this is not how we should think about effective field theories! The correct way is to understand hydrodynamic modes as low energy DOFs. Of course, there are also other DOFs in fluid. The topic of my talk is to use holography to elucidate their imprint on hydro. 2/16

Holographic plasmas and their degrees of freedom

Holography From applicational perspective AdS/CFT is a tool for computing correlation functions in certain strongly coupled gauge theories, such as SYM at large and . N = 4 λ N c For simplicity I will consider AdS 1+4 / CFT 1+3 and focus on pure gravity sector. R ab − 1 2 Rg ab − 6 L 2 g ab = 0 Different solutions correspond to states in a dual CFT with different . h T µ ν i Minkowski spacetime at the boundary UV ds 2 = L 2 n dz 2 + η µ ν dx µ dx ν 0 z 2 +2 π 2 o h T µ ν i z 4 + . . . bulk of AdS N 2 ??? c IR 3/16

Kovtun & Starinets Excitations of strongly coupled plasmas [hep-th/0506184] Consider small amplitude perturbations ( ) on top of a holographic plasma 2 ⌧ T 4 δ T µ ν /N c T µ ν = 1 c T 4 diag (3 , 1 , 1 , 1) µ ν + δ T µ ⌫ ( ∼ e − i ! ( k ) t + i ~ k · ~ x ) 8 π 2 N 2 Dissipation leads to modes with complex , which in the sound channel look like ω ( k ) 3rd Re ω / 2 π T k/ 2 π T Re 3 0.5 1 1.5 2 1st 2nd -0.5 2.5 -1 1st 2 2nd -1.5 1.5 -2 1 3rd 0.5 -2.5 k/ 2 π T ∂ω � k → 0 = c sound � -3 Im 0.5 1 1.5 2 ∂ k � Im ω / 2 π T There are two different kinds of modes: as : slowly evolving and dissipating modes (hydrodynamic sound waves) ω ( k ) → 0 k → 0 all the rest: far from equilibrium (QNM) modes dampened over t therm = O (1) /T 4/16

Lesson 1: for hydrodynamics to work all the other DOFs need to relax. 3rd Re ω / 2 π T k/ 2 π T Re 3 0.5 1 1.5 2 1st 2nd -0.5 2.5 -1 1st 2 2nd -1.5 1.5 -2 1 3rd 0.5 -2.5 k/ 2 π T ∂ω � k → 0 = c sound � -3 Im 0.5 1 1.5 2 ∂ k � Im ω / 2 π T 5/16

3rd Re ω / 2 π T k/ 2 π T Re 3 0.5 1 1.5 2 1st 2nd -0.5 2.5 -1 1st 2 2nd -1.5 1.5 -2 1 3rd 0.5 -2.5 k/ 2 π T ∂ω � k → 0 = c sound � -3 Im 0.5 1 1.5 2 ∂ k � Im ω / 2 π T Observation: No matter how long one waits, there will be always remnants of n-eq DOFs Lesson 2: Hydrodynamic gradient expansion cannot converge 6/16

Dynamical model

Fantastic toy-model [Bjorken 1982] The simplest model in which one can test these ideas is the boost-invariant flow with no transverse expansion. x 1 q In Bjorken scenario dynamics depends only on proper time ( x 0 ) 2 − ( x 1 ) 2 τ = ds 2 = − d τ 2 + τ 2 dy 2 + dx 2 2 + dx 2 3 and stress tensor (for a CFT) is entirely expressed in terms of local energy density with h T µ ν i = diag { � ✏ ( ⌧ ) , p L ( ⌧ ) , p T ( ⌧ ) , p T ( ⌧ ) } x 0 and p T ( ⇥ ) = � ( ⇥ ) + 1 p L ( ⇥ ) = − � ( ⇥ ) − ⇥� 0 ( ⇥ ) 2 ⇥� 0 ( ⇥ ) hadronic gas described mixed phase by hydrodynamics QGP described by τ = 0 AdS/CFT in this scenario pre-equilibrium stage x 1 7/16

1103.3452 [hep-th] PRL 108 (2012) 201602: Hydrodynamization MPH, R. A. Janik & P . Witaszczyk For hydrodynamics to work all the other DOFs need to relax. 3rd Re ω / 2 π T k/ 2 π T Re 3 0.5 1 1.5 2 1st 2nd -0.5 2.5 -1 1st 2 2nd -1.5 1.5 -2 1 3rd 0.5 -2.5 k/ 2 π T ∂ω � k → 0 = c sound � -3 Im 0.5 1 1.5 2 ∂ k � Im ω / 2 π T Surprising consequence e - 3 p L e grey: full evolution 1.4 Large anisotropy at the onset of hydrodynamics 1.2 red: 1st order hydrodynamics 1.0 ✏ − 3 p L ≈ 0 . 6 ✏ to 1 . 0 ✏ 0.8 Thus 0.6 isotropization 0.4 hydrodynamization 6 = thermalization 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t T H t L 8/16

Boost-invariant hydrodynamics In hydrodynamics, the stress tensor is expressed in terms of , and their r ν T u µ Key observation: in Bjorken flow is fixed by the symmetries and takes the form u µ u µ ∂ µ = ∂ τ Its gradients will come thus from Christoffel symbols ( ) ds 2 = − d τ 2 + τ 2 dy 2 + dx 2 2 + dx 2 3 τ y = 1 Γ y τ Lesson: in Bjorken flow hydrodynamic gradient expansion = late time power series At very late times p L = − ✏ − ⌧✏ 0 = p T = ✏ + 1 1 1 2 ⌧✏ 0 T ∼ ✏ ∼ ⌧ 4 / 3 τ 1 / 3 1 In holographic hydrodynamics gradient expansion parameter is T r µ u ν T µ ν = ✏ u µ u µ + P ( ✏ ) { ⌘ µ ν + u µ u ν } − ⌘ ( ✏ ) ✏ + P ( ✏ ) � µ ν + . . . s ( ✏ ) T 1 1 1 1 For Bjorken flow is . T r µ u ν τ = 1 τ 2 / 3 τ 1 / 3 9/16

High order hydrodynamics

Hydrodynamic series at high orders 1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, R. A. Janik & P . Witaszczyk ∞ ✓ ◆ ✏ = 3 1 1 1 T 00 = ✏ ( ⌧ ) ⇠ ( T − 1 r µ u ν ⇠ ⌧ − 2 / 3 ) X ✏ n ( ⌧ − 2 / 3 ) n 8 N 2 c ⇡ 2 ✏ 2 + ✏ 3 ⌧ 2 / 3 + ✏ 4 ⌧ 4 / 3 + . . . ⌧ 4 / 3 n =2 at large orders at low orders factorial growth of gradient behavior is different contributions with order ( n !) 1 /n ∼ (2 π n ) 1 / 2 n · n e First evidence that hydrodynamic expansion has zero radius of convergence! 10/16

Singularities in the Borel plane 1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, R. A. Janik & P . Witaszczyk A standard method for asymptotic series is Borel transform and Borel summation Z ∞ ∞ ∞ 1 1 X ✏ n u n ( u = ⌧ − 2 / 3 ) , X u n , ✏ ( u ) ∼ B ✏ (˜ u ) ∼ n ! ✏ n ˜ Borel sum : ✏ Bs ( u ) = uB ✏ ( t ) exp ( − t/u ) dt 0 n =2 n =2 This makes a difference only if we can find analytic continuation of . B ✏ (˜ u ) P 120 m =0 c m ˜ u m Idea: use Pade approximant to reveal singularities of . B ✏ (˜ u ) B ✏ (˜ u ) = P 120 n =0 d n ˜ u n Im z 0 green dots: zeros numerator those are real 30 singularities gray dots: zeros denominator 20 10 Re z 0 - 10 10 20 30 - 10 those zeros cancel almost perfectly - 20 (up to 10 -150 ) - 30 11/16

Hydrodynamic instantons and hydrodynamic gradient expansion

Singularities of Borel transform and QNMs 1302.0697 [hep-th] PRL 110 (2013) 211602: MPH, R. A. Janik & P . Witaszczyk In Borel summation the outcome depends on the contour connect 0 with . ∞ Here there are two inequivalent contours (blue and orange). �✏ ∼ e − 3 / 2 i ω Borel τ 2 / 3 ⇣ ⌘ ⌧ α Borel + . . . uB ✏ ( t ) exp ( − t/u ) dt 3rd Re ω / 2 π T Re 3 Im u é 0 20 2nd -0.5 2.5 -1 1st 2 1 Z ∞ -1.5 1.5 0 ✏ Bs ( u ) = 10 -2 1 0.5 -2.5 k/ 2 π T Z ∞ 1 ✏ Bs ( u ) = uB ✏ ( t ) exp ( − t/u ) dt -3 0.5 1 1.5 2 0 20 Re u é Im ω / 2 π T 0 - 5 5 10 15 Re ω / 2 π T k/ 2 π T Re 3 0.5 1 1.5 2 1st 2nd -0.5 2.5 -1 1st 2 - 10 2nd -1.5 1.5 -2 1 3rd 0.5 -2.5 - 20 k/ 2 π T ω Borel = 3 . 1193 − 2 . 7471 i Im ω / 2 π T -3 Im 0.5 1 1.5 2 α Borel = − 1 . 5426 + 0 . 5192 i is the frequency of the lowest non-hydrodynamic metric QNM at ! k = 0 ω Borel 12/16