Towards holographic heavy ion collisions

Transcription

1 Towards holographic heavy ion collisions Michał P. Heller University of Amsterdam, The Netherlands & National Centre for Nuclear Research, Poland (on leave) based on [hep-th] with J. Casalderrey-Solana, D. Mateos & W. van der Schee 1/16

2 Thermalization puzzle Heinz [nucl-th/767] There are overwhelming evidences that relativistic heavy ion collision programs at RHIC and 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(1/ ) starting on very early (< 1 fm/c) ~ 1 fm described by hydro after < 1 fm/c Explaining ab initio this very quick applicability of hydro is a fascinating puzzle What can the holography teach us about equilibration in similar models? 2/16

3 Global equilibrium Thermal = deconfined hqft Bulk black hole AdS-Schwarzschild black hole is described by the metric r = 1 r EH = T event horizon ds 2 BH = 2dtdr r 2 1 T dt 2 + r 2 d~x 2 r down to r = The plasma/black hole thermodynamics is given by T µ = N 2 c T diag (3, 1, 1, 1) µ, s = Area/l 2 P = 1 2 N 2 c 2 VT 3 3/16

4 Going away from equilibrium review: Hubeny & Rangamani [hep-th] 2) QFT: hydrodynamics gravity: fluid/gravity duality amplitude 3) far from equilibrium regime 1) QFT: linear response theory gravity: quasinormal modes QFT: global equilibrium gravity: eternal black hole momentum /16

5 Small amplitude perturbations and dissipation r = 1 r EH = T event horizon g ab (r) e i!(k)t+i~ k ~x down to r = Quasinormal modes are small amplitude perturbations on top of BH that obey - Dirichlet bdry conditions at the bdry - Ingoing bdry conditions at the horizon The latter lead to complex frequencies! and hence dissipation T µ = Nc 2 T diag (3, 1, 1, 1) µ + T µ e i!(k)t+~ k ~x 5/16 exponential decay with time

6 Quasinormal mode spectrum Consider small amplitude perturbations ( [hep-th/5618] Kovtun & Starinets T µ /N 2 c T ) on top of a holographic plasma T µ = N 2 c T diag (3, 1, 1, 1) µ + T µ ( e i!(k) t+i ~ k ~x ) Due to = gymn 2 c!1 (and N c!1?) the temperature T is the only microscopic scale Complex!(k) in the sound channel look like Re!/2 T 3rd 2nd k/2 T 1st st k! = c sound!(k)! as k! 1.5 k/2 T Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatial momentum. The curves for which as correspondtohydrodynamicsoundmodeinthedual finite temperature N = SYM theory. behavior of the lowest (hydrodynamic) frequency which is absent for E α and Z 3.ForE z and Z 1,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (.16) and (.32) for small ω and q), and presumably move off to infinity as q becomes large. For Z 2,thehydrodynamic frequency has both real and imaginary t RHIC hydro parts (given by Eq. (.) for small ω and q), and eventually (for large q)becomesindistinguishableinthetowerofothereigenfrequencies. As an example, dispersion relations for the three lowest 6/16quasinormal frequencies in the soundchannel Im Im!/2 T : slowly evolving and dissipating modes (hydrodynamic sound waves) all the rest: far from equilibrium (QNM) modes dampened over 3rd t therm = O(1)/T This is also the meaning in which is fast:.5 fm/c x 35 MeV = T ttherm =.63!!!

7 Modern relativistic (uncharged) hydrodynamics hydrodynamics is an EFT of the slow evolution of conserved currents in collective media close to equilibrium As any EFT it is based on the idea of the gradient expansion DOFs: always local energy density and local flow velocity u µ ( u u = 1) EOMs: conservation eqns r µ T µ = for T µ systematically expanded in gradients gravity reminded us that all terms allowed by symmetries can enter T µ = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... perfect fluid stress tensor microscopic input: EoS (famous) shear viscosity bulk viscosity (vanishes for CFTs) 7/16

8 Fluid-gravity duality [hep-th] Bhattacharyya Hubeny Minwalla Rangamani T (x) u µ (x) T u µ ds 2 = 2u µ dx µ dr r 2 1 dual spacetime locally looks like a boosted black brane T u µ u dx µ dx + r 2 ( µ + u µ u )dx µ dx r 8/16 + gradient terms.

9 General idea behind the non-equilibrium holography Of interest are geometries which interpolate between far-from-equilibrium states at the boundary at initial time tini and thermalized ones at (some) larger time tiso Minkowski spacetime tini x = t x 1 The stress tensor is read off from nearboundary expansion of dual solution Skenderis et al. (2) bulk of AdS z=1/r The criterium for (local) thermalization is that the stress tensor is to a good accuracy described by hydrodynamics There are two ways of defining n-eq. states: Future horizon AdS Future horizon AdS - shaking equilibrium via QFT sources - defining them without invoking their origin Dynamics Vacuum AdS Source = Source Dynamics Initial state Source = Let s investigate the outcomes of the both! (a) Source = (b) 9/16

10 A typical holographic thermalization process Theory: Boundary (at z=) [hep-th] MPH, D. Mateos, W. van der Schee, D. Trancanelli Numerical experiment: initial profile for the bulk metric absorption by the horizon Curvature (BH subtracted) ht µ i = diag, P (t), P (t), P (t) 1/16

11 The main problem Minkowski spacetime tini x = t x 1 bulk of AdS z=1/r HUGE FREEDOM OF CHOICE Which far from equilibrium initial condition is the closest to the experiment? 11/16

12 Towards a holographic heavy ion collision Operational view: collide holographically two lumps of matter moving at relativistic speeds unfortunately necessarily deconfined, i.e. with ht µ i = O(N 2 c ) x x 1 u State of the art as of June 213: colliding gravitational shock wave solutions 12/16

13 Gravitational shock wave solutions Janik & Peschanski [hep-th/512162] Chesler & Yaffe [hep-th] ds 2 = 1 u 2 (du2 + µ dx µ dx )+u 2 h(x )dx 2 Poincare patch vacuum AdS dual stress tensor: shock wave disturbance moving with the speed of light Solution of Einstein s equations with the negative CC for any longitudinal profile h(x ) u T tt = T zz = T tz = N c 2 h(t ± z) 2 2 z t We will specialize to h(t ± z) =E exp (t ± z) 2 /2! 2. But we re in a CFT, so the only qty that matters is e = E 1/ 13/16

14 v Colliding shocks at ecy '.6 is roughly the value corresponding to Pb nuclei boosted to RHIC energies ecy '.6 µ total energy* E T E/µ µz rgy density E/µ as a function of time v and coordinate z µv Chesler & Yaffe [hep-th].1 2 µz 6 FIG. 2: Energy flux S/µ as a function of time v and longiµv µz tudinal coordinate z. µz = µz = 3 port. Although this is not exactly true for our.2a function of time v and FIG. 1: Energy density E/µ as.75 P? /µ ofiles, the residual error in Einstein s.1 equap /µ ligible µv when the separation of the incoming longitudinal coordinate z..15 Thursday, November 11, 21 hydro ore than a 2few times the shock width..5 he initial data relevant for our metric ansatz disjoint support. Although this.1 is not exactly true for our e (numerically) for the di eomorphism trans.25.1 Gaussian profiles, the residual error in Einstein s equa.5 Wednesday, November 1, 21 e single shock metric (8) from Fe erman 2 6 tions is negligible when the separation of the incoming Eddington-Finkelstein coordinates. In parthursday, November 1 µz is more than a few timesthe shock width. shocks ompute the anisotropy function B± for each significant stopping: To find 2 the initial 2 data 6relevant for 1 our 2 3metric 5ansatz 6 um the result, B = B + B. We of choose + a function. 2: Energy flux S/µ as time the v and longiµv µv 15% slow-down of well Tseparated maxima (1), we solve (numerically) for the di eomorphism transv so the incoming shocks are nal coordinate z. negligibly overlap above the apparent hori- forming again, the single shock metric (8) from Fe ermananisotropy at hydrodynamization FIG. 3: to Longitudinal and transverse pressure as a function nctions µz a and f2 may be found analytically, Graham Eddington-Finkelstein coordinates. In par= is µz = 3 it the full story? of time v, at z = and z = 3/µ. Also shown for compari1/16.2 ticular, we compute the anisotropy function B± for each

15 Dynamical crossover [hep-th] Casalderrey-Solana, MPH, Mateos, van der Schee e left =2e CY e right = 1 8 e CY z T <!!! again, significant stopping: 15% slow-down of T maxima ( v.85 ) hydro kicks in soon after the outer parts of incoming shocks meet more like the old Landau picture 15/16 no stopping: T maxima move with v 1 hydro applicable only at midrapidities and late enough!!! more like what seems to be happening at RHIC and LHC

16 Why is it interesting? Dispels the myth that strong coupling necessarily leads to stopping E/ Another evidence for distinguishing hydrodynamization from thermalization P L E & P T E t z P E equilibrium = 1 3 fast 16/16 t