Recent lessons about hydrodynamics from holography

Transcription

1 Recent lessons about hydrodynamics from holography Michał P. Heller University of Amsterdam, The Netherlands & National Centre for Nuclear Research, Poland (on leave) based on [hep-th] MPH, R. A. Janik & P. Witaszczyk (PRL 08 (202) 20602) [hep-th] MPH, R. A. Janik & P. Witaszczyk (PRL 0 (203) 2602) /3

2 Holographic duality Maldacena [hep-th/97200] review: Mc Greevy [hep-th] Z Z D i exp (i d 4 x L) with (operational/ simple ) L [ Tr(d Φi d Φ gym 2 i )+c ijk Tr( Φ i Φj Φk )+d ijkl Tr( Φ i Φj Φk Φl ) ] c ijk Tr( Φ i Φj Φk )+d ijkl Tr( Φ i Φj Φk Φl ) ] holography strongly coupled ( gymn 2 c ) quantum theories of large matrices ( N c ) classical gravity theories in a higher dimensional spacetimes bad news: none of the pheno-relevant QFTs is truly holographic in this sense but : there are significant similarities (especially for QCD and its phases)! but II: we can solve rich strongly coupled systems through solving simple PDEs! 2/3

3 Applied holography holography is thus an interesting th/pheno tool for qualitative insight on otherwise hard-to-calculate ab initio quantities due to (but not only) strong coupling T A 2+ dimensional CFT at T>0 Quantum critical T KT 0 Superfluid g c Insulator strongly coupled fermionic systems g real-time physics of strongly coupled QCD (lattice, pqcd) 3/3

4 Thermalization puzzle at RHIC and LHC Heinz [nucl-th/ ] 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(/4 ) starting on very early (< fm/c) ~ 0 fm hydronized after < fm/c Explaining ab initio this quick applicability of hydro is a major puzzle in QCD@HIC. What can the holography teach us about thermalization in similar models? 4/3

5 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 = ) EOMs: conservation eqns r µ T µ =0 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) 5/3

6 What did we learn from the fluid-gravity duality? We were reminded that gradient expansion needs to be done systematically, e.g. Israel & Stewart, T µν = εu µ u ν + P µν + Π µν + pheno EOM τ Π DΠ µν = Π µν ησ µν We learned something about transport coefficients at strong coupling and we managed to transfer this knowledge to the heavy ion-community (big success!): /s =/ we also know now that /4 is not any fundamental bound Buchel, Myers & Sinha [hep-th] v η/s=0-4 η/s=0.08 standard η/s=0.08 Pade η/s=0.6 standard η/s=0.6 Pade Baier et al [hep-th] p T [GeV] Bhattacharyya et al [hep-th] [ Π µν = ησ µν τ Π DΠ µν + d ] d Πµν ( u) ] + κ [R µν (d 2)u α R α µν β u β + λ η 2 Π µ λπ ν λ λ 2 η Π µ λω ν λ + λ 3 Ω µ λω ν λ Luzum & Romatschke [nucl-th] It opened new perspective to view both the phenomena in fluids and in gravity 6/3

7 Holography, QNMs and hydrodynamics Kovtun & Starinets [hep-th/050684] Consider small amplitude perturbations ( T µ /N 2 c T 4 ) on top of a holographic plasma T µ = 8 2 N 2 c T 4 diag (3,,, ) µ + T µ ( e i!(k) t+i ~ k ~x ) Due to = gymn 2 c! (and N c!?) the temperature T is the only microscopic scale Dissipation leads to modes with complex!(k), which in the sound channel look like Re!/2 T 3rd 2nd k/2 T st 2.5 st - k!0 = c sound!(k)! 0 as k! k/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. behavior of the lowest (hydrodynamic) frequency which is absent for E α and Z 3.ForE z and Z,hydrodynamicfrequenciesarepurelyimaginary(givenbyEqs. (4.6) 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 t RHIC hydro parts (given by Eq. (4.44) for small ω and q), and eventually (for large q)becomesindistinguishableinthetowerofothereigenfrequencies. As an example, dispersion relations for the three lowest 7/3quasinormal 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()/T This is also the meaning in which is fast: 0.5 fm/c x 350 MeV = T ttherm = 0.63!!!

8 Fantastic toy-model [Bjorken 982] x x 0 The simplest, yet phenomenologically interesting field theory dynamics is the boost-invariant flow with 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 8/3 = no elliptic flow (~ central collision) and stress tensor (in conformal case) is entirely expressed in terms of energy density =0 ds 2 = d dy 2 + dx 2 + dx 2 2 ht µ i = diag{ p L ( ) = ( ) 0 ( ) with We are interested both in setting strongly coupled non-equilibrium initial states at =0 and tracking their relaxation towards hydro and in hydro phase as well [sic] = q (x 0 ) 2 (x ) 2 ( ),p L ( ),p T ( ),p T ( )} and p T ( ) = ( )+ 2 0 ( )

9 (Fast) hydrodynamization [hep-th] PRL 08 (202) 20602: General stress tensor here has 3 different components MPH, R. A. Janik & P. Witaszczyk ht µ i = diag{ ( ),p L ( ),p T ( ),p T ( )} Hydro constitutive relations relate them to each other via gradient expansion Obviously we know the hydro form of the stress tensor, but do not know when it applies For this we need to know how non-hydro DOFs relax. We can investigate it numerically! st, 2nd and 3rd order hydro large anisotropy at the onset of hydrodynamics! RHIC fast! T eff ( ) The single most interesting result was that hydrodynamization similar findings in Chesler & Yaffe and /3 = thermalization: Pressure anisotropy is observed to be between 3 p L 0.6 to.0 with hydrodynamics already being a valid description of the stress tensor dynamics.

10 Hydrodynamic series at high orders [hep-th] PRL 0 (203) 2602: MPH, R. A. Janik & P. Witaszczyk So far nothing has been known about the character of hydrodynamic expansion Idea: take a simple flow (here the boost-invariant flow) and using the fluid-gravity duality generate the on-shell form of its hydrodynamic stress tensor at high orders»e n ên T 00 = ( ) X n=2 n ( 2/3 ) n (T r µ u 2/3 ) at low orders behavior is different at large orders factorial growth of gradient contributions with order n First evidence that hydrodynamic expansion has zero radius of convergence! 0/3

11 Why hydro series might be asymptotic? [hep-th] PRL 0 (203) 2602: MPH, R. A. Janik & P. Witaszczyk Famous examples of asymptotic expansions arise in pqfts ] [ There, the number of Feynman graphs grows ~order! at large orders* ] [ We suspect analogous mechanism might work also in the case of hydro series* T µ = u µ u + P ( ){ g µ + u µ u } ( ) µ ( ){ g µ + u µ u }(r u)+... [ [ Π µν = ησ µν τ Π DΠ µν + d ] ] [ d Πµν ( u) [ + κτ [R µν (d 2)u α R α µν β Π DΠ µν + d ] [ ] ] u d Πµν ( u) β + κ [R µν (d 2)u α R α µν β u β + λ η 2 Π µ λπ ν λ λ 2 + λ η Π µ λω ν λ + λ 3 Ω µ λω ν λ. η 2 Π µ λπ ν λ λ 2 η Π µ λω ν λ + λ 3 Ω µ λω ν λ +... /3 st order hydro ( transport coeff) 2nd order hydro (5 transport coeffs)...

12 What controls the fast growth of hydros coeffs? [hep-th] PRL 0 (203) 2602: MPH, R. A. Janik & P. Witaszczyk A standard method for asymptotic series is Borel transform and Borel summation (u) X n u n (u = 2/3 ), B (ũ) n=2 X n=2 n! nũ n, Borel sum : Bs (u) = Z 0 u B (t)exp( t/u)dt B (ũ) reveals singularities leading to 0 radius of convergence 20 Im ué Re!/2 T 5 20 Re ué Z Bs (u) = u 0 B (t)exp( t/u)dt Bs (u) = Z u B (t)exp( t/u)dt e i#u Closer inspection reveals that the closest one to 0 is the lowest non-hydro QNM! 2/ k/2 T Figure 6: Real 0.5and imaginary parts.5 of three2 lowest q 2nd momentum The curves for which 0 as 0 correspo st finite temperature N =4 SYM theory. - st 2nd -.5 behavior of the lowest (hydrodynamic) frequency w Z,hydrodynamicfrequenciesarepurelyimaginary ω-2.5 and q), and presumably move off to infinity 3rdas q k/2 Tfrequency has both real and imaginary parts (give Im Im!/2 T eventually (for large q)becomesindistinguishablein Figure 6: Real and imaginary parts of three lowest example, quasinormal dispersion frequencies relations asfor function the three of spatial lowest qua momentum. The curves for which 0 as 0 correspond (includingto the hydrodynamic one of the soundmode wave) inare theshown dual in 3 2 Re!/2 T Im!/2 T 2nd st k/2 T 3rd

13 Summary Holography allows to do fantastic ab initio calculations! Strong coupling naturally leads to quick applicability of hydrodynamics (RHIC?). At the moment of hydrodynamization, the stress tensor can be very anisotropic. Thus superficially it needs to be distinguished from isotropization/thermalization! Model studies strongly suggest that hydrodynamics is an asymptotic series! Large order behavior knows about the lowest far-from-equilibrium DOF. Open directions Do anisotropies in hydrodynamic regime leave an observational imprint? Is resummed hydrodynamics phenomenologically relevant? Towards holographic heavy ion collisions ( Tuesday ) 3/3