Holography, thermalization and heavy-ion collisions I Michał P. Heller Perimeter Institute for Theoretical Physics, Canada National Centre for Nuclear Research, Poland 1202.0981 [PRL 108 191601 (2012)] with Mateos, van der Schee & Tancanelli 1304.5172 [JHEP 1309 026 (2013)] with Mateos, van der Schee & Triana
Introduction
New kind of string pheno see also lectures by Umut Gursoy and Karl Landsteiner The natural domain of string pheno is the realm of BSM & early Universe cosmology 1 st decade of the 21 st century: holography and quark-gluon plasma at RHIC String theory making impact in a brand new way: s = 1 4 1203.1810 Bożek & Broniowski Why exciting? Geometrizes certain QFTs. New ab initio tool w/r weak coupling lattice 1/15
Heavy ion collisions primer (RHIC *2000, LHC *2010) 1-2 fm see also Umut Gursoy s lectures 10 fm Successful pheno for soft observables: use hydro, ht µ i = F [T,u ], as early as then: Initial n-eq state is intrinsically anisotropic (expansion axis L vs. transversal plane?): These lectures: ab initio ht L L i ht?? i =0 ht µ i time F [T,u ] 2/15 = 1 in strongly-coupled QFTs from gravity
Some of the key questions motivating these lectures Lecture 1: how long does it take ht µ i to equilibrate in strongly-coupled QFTs? Lecture II: what is ht µ i(t, ~x) after a collision of 2 strongly-interacting objects? Lecture III: what is relativistic hydrodynamics? 3/15
AdS gravity (< 2009)
Key notions in holography Ab initio studies of a large class non-abelian QFTd s = understanding geometriesd+1 Works also for certain non-conformal QFTs, but simplest for (appropriate) CFTs Geometries4+1 (QCD lives in 4D) are governed by the EOMs (+ bdry conditions) of: S = 1 2 l 3 P Z d 5 x p g R + 12 + matter + O(R)2 L2 relying on EOMs: N 2 c L3 l 3 P 1 ~ neglecting those: = g 2 YMN c 1 1 These lectures: R ab ~ strongly-coupled SYM (CFT) 2 Rg 6 ab L 2 g ab =0 N =4 see also Jan Plefka s lectures 4/15
Properties of Anti-de Sitter (AdS) spacetime CFT4 vacuum = AdS5: SO(2, 4) are its isometries We want N =4 SYM to live in Minkowski space Poincare patch: µ ds 2 = g ab dx a dx b = L2 u 2 du 2 + µ dx µ dx x i x 0 UV laser x 0 u IR u = 1 extremal horizon laser u =0 EOMs require boundary conditions at u =0: for g ab, µ plays this role here 5/15
Einstein s equations in AdS and dual Of course, we are interested in excited states: ds 2 = L2 u 2 ht µ i du 2 + g µ (u, x)dx µ dx 1 Solving R ab for g µ (u, x) around gives*: 2 Rg 6 ab L 2 g ab =0 u =0 g µ (u, x) = µ + t µ u 4 +... with µ t µ =0 & @ µ t µ =0 Indeed, one can show that t µ = C ht µ i with, for N =4SYM, C = 2 2 Nc 2 Points of departure: 2) fix it by avoiding naked singularities 1) assume ht µ i 1) static Lecture 3 next 2 slides Lectures 1 & 2 6/15 3) get ht µ i 2) perturb bdry metric 3) getht µ i 2) solve EOMs 1) set g ab t=0
Strongly-coupled QGP = black brane Equilibrium strongly-coupled QGP: ht µ i = diag(e,p,p,p) µ and E = O(Nc 2 ): ds 2 = L2 u 2 du 2 (1 u 4 /u 4 0) 2 S = A hor 4G N 1+u 4 /u 4 0 dt 2 +(1+u 4 /u 4 0)d~x 2 TdS= de Simplest n-eq states: linear response theory at finite temperature: Z Z 3) get ht µ i = d 3 k d! e i!t+i~ k ~x G R (!,k) g u = u 0 horizon absorbtion black brane 2) perturb bdry metric a bit µ u =0 1) static plasma Holographic thermalization = horizon formation* and subsequent equilibration 7/15
Quasinormal modes: dofs of strongly-coupled QGP Z Z ht µ i = d 3 k d! e i!t+i~ k ~x G R (!,k) g µ Z d! :??? Singularities in the lower-half!-plane are single poles (QNMs) for each value of k see hep-th/0506184 by Kovtun & Starinets 3 2.5 Re! n /2 T Re 3rd 2nd -0.5 0.5 1 1.5 2 k/2 T 1st 2 1.5 1st -1-1.5 2nd Thus 1 0.5 k/2 T -2-2.5 0.5 1 1.5 2-3 Im! n /2 T Figure 6: Real and imaginary parts of three lowest quasinormal frequencies as function of spatial momentum. The curves for which 0 as 0 correspondtohydrodynamicsoundmodeinthedual finite temperature N =4 SYM theory. ht µ i = X n Z d 3 kc µ n e! n(k)t+i ~ k ~x with behavior of the lowest (hydrodynamic) frequency which is absent for E α and Z 3.ForE z and Z 1,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.16) and (4.32) 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 parts (given by Eq. (4.44) for small ω and q), and eventually (for large q)becomesindistinguishableinthetowerofothereigenfrequencies. 8/15 As an These lectures: ~ nonlinear interactions between QNMs studied using AdS gravity 3rd exponential decay in 1/T slow decay (hydro)
Going non-equilibrium (2009 ++): homogeneous isotropization 1202.0981 [PRL 108 191601 (2012)] with Mateos, van der Schee & Tancanelli 1304.5172 [JHEP 1309 026 (2013)] with Mateos, van der Schee & Triana
Homogeneous isotropization What is the simplest n-eq setup available? It is: ht µ i = diag E, E 2 3 3 P(t), E 3 + 1 3 P(t), E 3 + 1 3 P(t) µ 3 8 N c 2 2 T 4 see also 0812.2053 with Chesler & Yaffe E = const =, no ~x -dependence no hydro, but sensitive to nonlinearities functions of t and r Dual metric ansatz: t = const 3) get P(t) 2) solve EOMs 1) set g ab t=0 ds 2 /L 2 = 2 dt dr r 2 numerical GR see 1309.1439 by Chesler & Yaffe & 1407.1849 by van der Schee 9/15 A dt 2 + 2 e 2B dx 2 L + 2 e B dx 2 P E P(t) 3 dynamical eqs. to ~get @ t A,@ t & 2 constraints (1 on initial data) @ t B
Initial conditions Let us solve all Einstein s eqs. near the boundary, i.e. for r =0 and we look at B : B = b(t) 2r 4 b 0 (t) 2r 5 7b 00 (t) 24r 6 b (3) (t) 8r 7 +... with b(t) = 2 2 3N 2 c P(t) Solving ht µ i(t, ~x) for t>0 for requires knowing (@ t ) n ht µ i for all n 0 t=0 c µ ~ you need to specify occupation numbers for all QNMs see also 0906.4423 with Beuf, Janik & Peschanski Not all B t=0 will do: some lead to naked singularities. Nontrivial conditions on ht µ i see also 0806.2141 by Janik & Witaszczyk 10/15
Holographic thermalization Theory: out in t boundary absorption by the horizon absorption by the horizon t z absorption by the horizon Numerical experiment: initial profile 3B E r 4 r/r hor ht µ i = diag E, E 3 2 3 P(t), E 3 + 1 3 P(t), E 3 + 1 3 P(t) µ tt We watch genuinely n-eq states relax the way they want to relax (bdry: µ )! 11/15
Sample processes vs. linear response theory 1202.0981, 1304.5172 above: P/E full P/E = 1X n=1 c n e i! n(k)t +cc P/E = X10 n=1 c n e i! n(k)t +cc above: the corresponding Re and Im of cn s Surprising linearity despite seemingly large deviations from equilibrium 12/15
Genericity of 1/T relaxation time at strong coupling Homogeneous isotropization: ht µ i = diag E, 1 3 E 2 3 P(t), 1 3 E + 1 3 P(t), 1 3 E + 1 3 P(t) µ 1000 different excited states: all equilibrate within 1.2 / T pheno: 1 fm 400 MeV = O(1) surprising linearity By now confirmed in many other setups (see also Lecture 2) 13/15
Summary of Lecture 1
Notions certain states of a class of strongly-coupled QFTs = higher dimensional geometries dofs of strongly-coupled QGP = QNMs of dual black branes: ht µ i = X n Z d 3 kc µ n e! n(k)t+i ~ k ~x with exponential decay in 1/T slow decay (hydro) equilibration in strongly coupled QFTs = dual horizon formation and equilibration: absorption by the horizon t 3B E r 4 absorption by the horizon 14/15 tt r/r hor
Lessons Real-time dynamics of QFTs requires 1-many initial conditions: B = b(t) 2r 4 b 0 (t) 2r 5 7b 00 (t) 24r 6 b (3) (t) 8r 7 +... with b(t) = 2 2 3N 2 c Holography makes it manageable by adding r R ab 1 2 Rg ab P(t) 6 L 2 g ab =0 Indications that equilibration in 1/T at strong coupling can be generic: Confirmed in many other setups*. Is t eq T = O(1) becoming new /s =1/4? 15/15