Holographic thermalization - an update Michał P Heller University of Amsterdam, The Netherlands & National Centre for Nuclear Research, Poland (on leave) partly based on arxiv:202098 [hep-th] (+D Mateos+D Trancanelli+W van der Schee), arxiv:033452 [hep-th] & arxiv:2030755 [hep-th] (+R Janik,+P Witaszczyk) /7
Motivation: fast thermalization at RHIC The goal of heavy ion collisions programs is to explore the phase diagram of QCD There are significant evidences that relativistic heavy ion collision program at RHIC (no also at the LHC) created strongly coupled quark-gluon plasma (sqgp) Successful description of experimental data is based on hydrodynamic simulations of an almost perfect fluid of /s =O(/4 ) starting on very early (< fm/c) Heinz (2004) ~ 0 fm thermalized after < fm/c This very fast thermalization or rather hydronization is a puzzle!!! 2/7
Ho does the holography enter the game? QCD in the regime of interest -- T = O(200 MeV) -- is not eakly coupled* None of the available first principle methods is suited for describing time-dependent phenomena involving QCD matter in this regime: - pqcd: convergence issues hen perturbative results are extrapolated to realistic coupling*; - Lattice: euclidean setup does not allo to calculate real-time quantities ithout strong assumptions The idea of AdS/CFT applied to QCD is to calculate things of phenomenological interest for QCD community in holographic models of QCD and treat results as: - ballpark estimates of QCD quantities: /s =O(/4 ) is the most famous example of such qty; - suggestions of ne phenomena in QCD matter: later in the presentation In the folloing I ill overvie hat holography can say about thermalization puzzle, ie I ill assume that strong coupling dominates pre-equilibrium RHIC dynamics 3/7
Questions to be asked Hydrodynamics is an effective theory in hich stress tensor is expressed in terms of temperature, velocity* and their gradients T µ = (T )u µ u + p(t )( µ + u µ u µ ) (T ) µ (u, ru)+ and its conservation gives EOMs for temperature and velocity Main question: ho long does it take for the stress tensor in holographic gauge theories to achieve hydro form starting from some far-from-equilibrium initial state? Preliminary studies gave phenomenologically interesting ansers The question is of course hether this is a generic statement Holography: What is the mechanism behind thermalization at strong coupling? Eventually e need to make (ballpark!!!) predictions Phenomenology: Ho does the hydronization process proceed at strong coupling? 4/7
Tool: AdS/CFT correspondence From applicational perspective AdS/CFT is a tool for computing correlation functions in certain strongly coupled gauge theories, such as N =4SYM at large N c and The idea is to replace a QFT problem (eg collision of to lumps of matter) by its gravity counterpart, solve Einstein s equations, translate back to QFT and learn sth To make the story complete, e need a dictionary beteen QFT and gravity The first entry is that the vacuum state of holographic CFT is dual to AdS spacetime Minkoski spacetime at the boundary 0 bulk of AdS z=/r IR UV 5/7 ds 2 = L2 z 2 (dz2 + µ dx µ dx ) being a solution of R ab 2 Rg ab 6 L 2 g ab =0 There are also other solutions and to interpret them in CFT e need to understand the dictionary better!
Holographic dictionary made simple bdry 0 bulk of AdS z=/r ds 2 = L2 z 2 (dz2 + g µ (z,x)dx µ dx ) Other solutions of R ab 2 Rg 6 ab L 2 g ab =0 are deformations of AdS space Asymptotic ( z 0) behavior of deformation carries information about dual <operator> Solving Einstein s equations gives g µ = µ + g µ (4) (x)z 4 + ith @ µ g (4) and µ g (4) z 0 µ =0 µ =0 g (4) µ smells like <stress tensor> of 4-dimensional theory We can make it very precise by µ! g µ (0) (x) When e evaluate the bulk action on-shell, e ill schematically see the folloing familiar structure Z q Z Z q S on shell = d 4 x det g (0) g(0) µ g (4) µ + = d 4 xj O + = d 4 x det g (0) g(0) µ T µ + This nails it don! Z hqf T [g (0) µ ] e is gravity ht µ i = q 2 det g (0) 6/7 g (0) µ log Z hqf T [g (0) ] 2 Nc µ g(4) 2 2 µ = µ g (0)
Black holes in AdS Thermal hqft = Bulk black hole AdS-Scharzschild black hole is described by the metric boundary @ r = r EH = T event horizon ds 2 BH = 2dtdr r 2 4 T 4 r 4 dt 2 + r 2 d~x 2 don to singularity @ r = 0 The plasma/black hole thermodynamics is given by T µ = 8 2 N 2 c T 4 diag (3,,, ) µ, s = Area/4l 2 P = 2 N 2 c 2 VT 3 7/7
Quasinormal modes and holography boundary @ r = Consider small amplitude perturbations; r EH = T event horizon g ab (r) e i!(k)t+i~ k ~x They obey: - Dirichlet bdry conditions at the bdry - Ingoing bdry conditions at the horizon Because of the latter! is complex (QNM)! don to singularity @ r = 0 Its imaginary part encodes dissipation: T µ = 8 2 N 2 c T 4 diag (3,,, ) µ + T µ e i!(k)+~ k ~x exponential decay Holographic thermalization is as efficient as the one at RHIC in the folloing sense RHIC 05 fm/c x 350 MeV = T tiso = 063 vs loest QNM of AdS BH T / Im(frequency) = O() Fast thermalization in holography persists also if e go beyond small perturbations 8/7
Creating non-equilibrium states Of interest are geometries hich interpolate beteen far-from-equilibrium states at the boundary at initial time tini and thermalized ones at (some) larger time tiso Minkoski spacetime 0 tini x 0 = t x The stress tensor is read off from nearboundary expansion of dual solution Skenderis et al (2000) bulk of AdS z=/r The criterium for (local) thermalization is that the stress tensor is to a good accuracy described by hydrodynamics There are to ays of defining n-eq states: Future horizon AdS Future horizon AdS - shaking equilibrium via QFT sources - defining them ithout invoking their origin Dynamics Vacuum AdS Source = 0 Source 0 Dynamics Initial state Source = 0 We ll go for the latter as it seems more generic! (a) Source = 0 (b) 9/7
Example : isotropization arxiv:202098 [hep-th] The simplest model of thermalization is probably isotropization of the stress tensor ht µ i = diag, 3 2 3 P (t), 3 + 3 P (t), 3 + 3 P (t) Symmetries dictate* an ansatz for the dual metric ds 2 =2dtdr Adt 2 + 2 e 2B dx 2 + 2 e B (dx 2 2 + dx 2 3) Solving Einstein s equations close to the boundary reads B = r 4 b 4 (t)+ r b0 4(t)+ 2 2r 6 b00 4(t)+ 4r 3 b(3) 4 (t)+, =r A = r 2 r 4 a 4 here and, = 3 8 2 N c 2 a 4 P (t) = 3 8 2 N c 2 b 4 (t) 2 7r 8 b 4(t) 2 3 7r 9 b 4(t)b 0 4(t)+ 7r 8 b 4(t) 2 + and Crucial thing to notice: At any constant bulk time, bulk arps contain infinitely many scales set by its derivatives This is a bulk notion far-from-equilibrium state Anisotropy goes to zero for large enough time and e recover a thermal state 0/7 P (t) and
What can be done ith it? arxiv:202098 [hep-th] st thing to do is to verify hether fast thermalization is generic in this model Hence, e considered a large set of bulk non-equilibrium states given by B(r) and at some initial bulk time hypersurface and plasma s energy density (constant in time) 0 examples of initial states encoded geometrically including once supported mostly in the UV, mostly in the IR, in the middle and spread evenly beteen UV and IR (z = /r) We considered 200 of such guys and solved the initial value problem As you remember I said that loest QNM s thermalization time is similar to the one obtained by solving GR equations in the nonlinear regime Is it a coincidence? 2nd thing to do is to compare P (t) obtained from L and NL Einstein s equations /7
Time evolution of pressure anisotropy (L/NL) 69% 32% 5% 69% 34% 0% 22% 53% 67% 47% 25% 83% 22% 27% 32% 54% 2/7
Histogram of thermalization times (> 2000 states) 05 fm/c x 350 MeV = T tiso = 063 3/7
Example I: boost-invariant flo x x 0 The simplest, yet phenomenologically interesting field theory dynamics is the boost-invariant flo ith no transverse expansion x = relevant for central rapidity region In Bjorken scenario dynamics depends only on proper time hadronic gas mixed phase described by hydrodynamics QGP described by pre-equilibrium stage AdS/CFT in this scenario 4/7 = no elliptic flo (~ central collision) and stress tensor (in conformal case) is entirely expressed in terms of energy density =0 ds 2 = d 2 + 2 dy 2 + dx 2 + dx 2 2 ht µ i = diag{ p L ( ) = ( ) 0 ( ) ith We are interested in setting strongly coupled non-equilibrium initial states at =0 (and also at >0) and tracking their unforced relaxation toards hydrodynamics = q (x 0 ) 2 (x ) 2 ( ),p L ( ),p T ( ),p T ( )} and p T ( ) = ( )+ 2 0 ( )
All order viscous hydrodynamics amounts ization and transition to hydrodynamics ould occu pressed in terms of flo velocities uµ and their deriva- to presentfor su ciently larg the energy-momentum tensor a series of terms In exfigure hich a e ould presentmerge this plot for 20 traject tives ithing coe cients being proportional to as appropriate µ ization transition hydrodynam pressed in terms of flo velocities u andthetheir corresponding derivato 20and different initialtostates It is c poers of Tef f, the proportionality constants being Figure a e presentmodes this plot ith coe cients beingofproportional to appropriate from the plot In that nonhydrodynamic are transporttives coe cients Forµ the case N = 4 plasma, µ µ µ to(t 20) different ofr Tef being htconstants i = { (T ) the + P (T )} uinitial u + P +evolution, initial ht = is0 notand f, the important incorresponding the stage of plasma Hydrodynamics: the abovepoers mentioned form of itproportionality an assumption µ µ from plot that transport coe cients For the case N = 4 plasma, for all the sets of the initial data, for nonhydrodynam > 065 the cu but a result of a derivation from AdS/CFT [7] of Hydrothe initial stage of pla the above are mentioned of Tµ isequations not an assumption merge intouaµimportant single curveincharacteristic of hydrodyn dynamic equations just theform conservation In the conformal hydrodynamics have gradients of only µ forballethe setsa of data, anisot for In Figure sho plotinitial of pressure a result derivation from AdS/CFT Hydro= but 0, hich are of by aconstruction first-order differ- [7] ics µt F () into a single curve characterist 3pL merge dynamic are just the conservation equations 2 8 for a selected profile and com ential equations forequations Tef f µ µ uconstruction @µ = plasma @, so But here to boost-invariant symmetries its gradients are trivial (Christoffels) ics In Figure b e sho a,plot this ith the corresponding curves for st 2nd of andp 0, hich are by first-order differin thedue case conformal this µ T of= () order hydrodynamics example leads to aential universal form of dynamical equa 3p L We 2 Fobserve 8 on for this a selected p equations forfirst Tef forder µ of boost-invariant the one perfect agreement tions forof thethis ith the corresponding curves fo Inscale therinvariant case conformal plasma this hand, ( ) T =quantity 0 in the boost-invariant Because hydro is this a ast order ODEith for hydrodyna µ forequa > 063order and, hydrodynamics on the other hand, quite size Weaobserve o leads to a universal form of first order dynamical = Tef (3) f3 pressure anisotropy that regime hich is neverth the oneinhand, a perfect agreement tions for the scale invariant quantity 2 2 4 = Tef dimensionless qty f We define Tef f by ( ) = Nc Tef f ( ) and usecompletely explained by dissipative hydrodynamics for > 063 and, on the other han 8 = T namely In (3) order topressure study the transitionintothat hydrodynamic ef f anisotropy regime h F HL more detail, completely e ill adopt a numerical criterion forht d Fhydro () explained by dissipative d =, (4) namely malization hich is the deviation of from the d In order to study the transition to d 35 order hydro more expression (5)e ill adopt a numerica detail, d F () hydro here Fhydro () is completely determined in terms, of the = (4) Equations of hydro: 25 malization d hich is the deviation of d For N = 4 plasma d transport coe cients of the theory < 0005 order hydro (5) rd order expression 3 at strong coupling Fhydro ()/ is knon explicitly up to 5 Fhydro () perfect here Fhydro ()rdis completely determined in terms of the terms corresponding to 3 order hydrodynamics [3] d transport coe cients of the theory For N = 4 Despite plasma the beildering variety of fluid d the nonequilibr 05 <0 rd order 2 3 2 at strong coupling F ()/ is knon explicitly up to hydro ()exist, 2 log 2 5 2 45 log 2 + 24 log 2 evolution, e ill sho belofhydro that there hoe rd + + + + to972 3 3 order hydrodynamics [3] surprising 02 regularities 04 06in the 08 3-05 some dynamics 3 9 terms 27 2corresponding 2 Despite the beildering variety of t (5) st 3rd order 2hydro 2nd 2 Initial and final entropy from the 2 log 2 5 2 45 log 2 + 24 log 2 evolution, e ill Apart sho belo thatene th -5 + + + + momentum tensor components, a very important ph some surprising regularities in the dy 3 9 27 2 2 972 3 3 cal property of the evolving plasma system is its ent (5) -25 This is quite reminiscent of [2] here all-order hydrodynamics andarea finaland entropy Apart density S (per Initial transverse unit (spacetime as postulated in terms of linearized AdS dynamics momentum tensor components, a ver pidity) In the general time-dependent case, the pr 5/7 cal property of the evolving plasma sy = Approaching hydrodynamics
Holographic prediction hydro Reriting equations of hydrodynamics in a form d d = F hydro(), allos to explicitly see hether non-hydro modes already relaxed hen curves coincide! st, 2nd and 3rd order hydro large anisotropy at the onset of hydrodynamics Note that hydronization time is not given by the convergence radius of the hydro expansion! Chesler & Yaffe 09064426 Lublinsky & Shuryak 0704647 and 09054069 The single most interesting result is that hydronization occurs ell before isotropization! Pressure anisotropy is observed to be beteen 3p L 06 to0 ith hydrodynamics already being a valid description of the stress tensor dynamics similar findings in Chesler & Yaffe 09064426 and 03562 6/7
Summary Holographic prediction : strong coupling gives pheno-ok hydronization times 05 fm/c x 350 MeV = T tiso = 063 Our input: e verified this on a large set on initial conditions (more than 2000!!!) arxiv:202098 [hep-th] (+D Mateos +D Trancanelli +W van der Schee) Holographic prediction II: large anisotropy at the onset of hydrodynamics arxiv:033452 [hep-th] arxiv:2030755 [hep-th] (+R Janik,+P Witaszczyk) Our input: e understood it is generic in the boost-invariant flo 7/7