Thermal spectral functions and holography Andrei Starinets (Perimeter Institute) “Strong Fields, Integrability and Strings” program Isaac Newton Institute for Mathematical Sciences Cambridge, 31.VII.2007

Experimental and theoretical motivation  Heavy ion collision program at RHIC, LHC ( ??)  Studies of hot and dense nuclear matter  Abundance of experimental results, poor theoretical understanding: - the collision apparently creates a fireball of “quark-gluon fluid” - need to understand both thermodynamics and kinetics -in particular, need theoretical predictions for parameters entering equations of relativistic hydrodynamics – viscosity etc – computed from the underlying microscopic theory (thermal QCD) -this is difficult since the fireball is a strongly interacting nuclear fluid, not a dilute gas

The challenge of RHIC QCD deconfinement transition (lattice data) Energy density vs temperature

The challenge of RHIC (continued ) Rapid thermalization Large elliptic flow Jet quenching Photon/dilepton emission rates ??

M,J,Q Holographically dual system in thermal equilibrium M, J, Q T S Gravitational fluctuations Deviations from equilibrium ???? and B.C. Quasinormal spectrum 10-dim gravity 4-dim gauge theory – large N, strong coupling + fluctuations of other fields

Transport (kinetic) coefficients Shear viscosity Bulk viscosity Charge diffusion constant Thermal conductivity Electrical conductivity * Expect Einstein relations such as to hold

Gauge/gravity dictionary determines correlators of gauge-invariant operators from gravity (in the regime where gravity description is valid!) For example, one can compute the correlators such as by solving the equations describing fluctuations of the 10-dim gravity background involving AdS-Schwarzschild black hole Maldacena; Gubser, Klebanov, Polyakov; Witten

Computing finite-temperature correlation functions from gravity  Need to solve 5d e.o.m. of the dual fields propagating in asymptotically AdS space  Can compute Minkowski-space 4d correlators  Gravity maps into real-time finite-temperature formalism (Son and A.S., 2001; Herzog and Son, 2002)

Hydrodynamics: fundamental d.o.f. = densities of conserved charges Need to add constitutive relations! Example: charge diffusion [Fick’s law (1855)] Conservation law Constitutive relation Diffusion equation Dispersion relation Expansion parameters:

Similarly, one can analyze another conserved quantity – energy-momentum tensor: This is equivalent to analyzing fluctuations of energy and pressure We obtain a dispersion relation for the sound wave:

Hydrodynamics predicts that the retarded correlator has a “sound wave” pole at Moreover, in conformal theory Predictions of hydrodynamics

Now look at the correlators obtained from gravity The correlator has poles at The speed of sound coincides with the hydro prediction!

Analytic structure of the correlators Weak coupling: S. Hartnoll and P. Kumar, hep-th/ Strong coupling: A.S., hep-th/

Example: R-current correlator in in the limit Zero temperature: Finite temperature: Poles of = quasinormal spectrum of dual gravity background (D.Son, A.S., hep-th/ , P.Kovtun, A.S., hep-th/ )

Two-point correlation function of stress-energy tensor Field theory Zero temperature: Finite temperature: Dual gravity  Five gauge-invariant combinations of and other fields determine  obey a system of coupled ODEs  Their (quasinormal) spectrum determines singularities of the correlator

The slope at zero frequency determines the kinetic coefficient Peaks correspond to quasiparticles Figures show at different values of Spectral functions and quasiparticles in

Spectral function and quasiparticles in finite-temperature “AdS + IR cutoff” model

Holographic models with fundamental fermions Additional parameter makes life more interesting… Thermal spectral functions of flavor currents R.Myers, A.S., R.Thomson, [hep-th]

Transport coefficients in N=4 SYM Shear viscosity Bulk viscosity Charge diffusion constant Thermal conductivity Electrical conductivity in the limit

Shear viscosity in SYM Correction to : A.Buchel, J.Liu, A.S., hep-th/ perturbative thermal gauge theory S.Huot,S.Jeon,G.Moore, hep-ph/

Electrical conductivity in SYM Weak coupling: Strong coupling: * Charge susceptibility can be computed independently: Einstein relation holds: D.T.Son, A.S., hep-th/

Universality of Theorem: For a thermal gauge theory, the ratio of shear viscosity to entropy density is equal to in the regime described by a dual gravity theory Remarks: Extended to non-zero chemical potential: Extended to models with fundamental fermions in the limit String/Gravity dual to QCD is currently unknown Benincasa, Buchel, Naryshkin, hep-th/ Mateos, Myers, Thomson, hep-th/

A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/ , hep-th/ Minimum of in units of

Chernai, Kapusta, McLerran, nucl-th/

Viscosity-entropy ratio of a trapped Fermi gas T.Schafer, cond-mat/ (based on experimental results by Duke U. group, J.E.Thomas et al., )

Chernai, Kapusta, McLerran, nucl-th/ QCD

Viscosity “measurements” at RHIC Viscosity is ONE of the parameters used in the hydro models describing the azimuthal anisotropy of particle distribution -elliptic flow for particle species “i” Elliptic flow reproduced for e.g. Baier, Romatschke, nucl-th/ Perturbative QCD: SYM: Chernai, Kapusta, McLerran, nucl-th/

Shear viscosity at non-zero chemical potential Reissner-Nordstrom-AdS black hole with three R charges (Behrnd, Cvetic, Sabra, 1998) We still have J.Mas D.Son, A.S. O.Saremi K.Maeda, M.Natsuume, T.Okamura (see e.g. Yaffe, Yamada, hep-th/ )

Photon and dilepton emission from supersymmetric Yang-Mills plasma S. Caron-Huot, P. Kovtun, G. Moore, A.S., L.G. Yaffe, hep-th/

Photons interacting with matter: Photon emission from SYM plasma To leading order in Mimicby gauging global R-symmetry Need only to compute correlators of the R-currents

Photoproduction rate in SYM (Normalized) photon production rate in SYM for various values of ‘t Hooft coupling

How far is SYM from QCD? pQCD (dotted line) vs pSYM (solid line) at equal coupling (and =3) pQCD (dotted line) vs pSYM (solid line) at equal fermion thermal mass (and =3)

Outlook  Gravity dual description of thermalization ?  Gravity duals of theories with fundamental fermions: - phase transitions - heavy quark bound states in plasma - transport properties  Finite ‘t Hooft coupling corrections to photon emission spectrum  Understanding 1/N corrections  Phonino

THE END

Some results in the limit described by gravity duals universal for a large class of theories Bulk viscosity for non-conformal theories Shear viscosity/entropy ratio: in the limit described by gravity duals in the high-T regime (but see Buchel et al, to appear…) model-dependent R-charge diffusion constant for N=4 SYM:

 Non-equilibrium regime of thermal gauge theories is of interest for RHIC and early universe physics  This regime can be studied in perturbation theory, assuming the system is a weakly interacting one. However, this is often NOT the case. Nonperturbative approaches are needed.  Lattice simulations cannot be used directly for real-time processes.  Gauge theory/gravity duality CONJECTURE provides a theoretical tool to probe non-equilibrium, non-perturbative regime of SOME thermal gauge theories

Quantum field theories at finite temperature/density EquilibriumNear-equilibrium entropy equation of state ……. transport coefficients emission rates ……… perturbativenon-perturbative pQCD Lattice perturbativenon-perturbative kinetic theory ????

Epilogue  On the level of theoretical models, there exists a connection between near-equilibrium regime of certain strongly coupled thermal field theories and fluctuations of black holes  This connection allows us to compute transport coefficients for these theories  The result for the shear viscosity turns out to be universal for all such theories in the limit of infinitely strong coupling  At the moment, this method is the only theoretical tool available to study the near-equilibrium regime of strongly coupled thermal field theories  Stimulating for experimental/theoretical research in other fields

Three roads to universality of  The absorption argument D. Son, P. Kovtun, A.S., hep-th/  Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/  “Membrane paradigm” general formula for diffusion coefficient + interpretation as lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theorem P. Kovtun, D.Son, A.S., hep-th/ , A.S., to appear, P.Kovtun, A.S., hep-th/ , A.Buchel, J.Liu, hep-th/

Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then. Since the entropy (density) is we get

Example 2 (continued): stress-energy tensor correlator in in the limit Finite temperature, Mink: Zero temperature, Euclid: (in the limit ) The pole (or the lowest quasinormal freq.) Compare with hydro:

A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/ , hep-th/

Analytic structure of the correlators Weak coupling: S. Hartnoll and P. Kumar, hep-th/ Strong coupling: A.S., hep-th/

Example 2: stress-energy tensor correlator in in the limit Finite temperature, Mink: Zero temperature, Euclid: (in the limit ) The pole (or the lowest quasinormal freq.) Compare with hydro: In CFT: Also,(Gubser, Klebanov, Peet, 1996)

Spectral function and quasiparticles A B C A: scalar channel B: scalar channel - thermal part C: sound channel

Pressure in perturbative QCD

Quantum field theories at finite temperature/density EquilibriumNear-equilibrium entropy equation of state ……. transport coefficients emission rates ……… perturbativenon-perturbative pQCD Lattice perturbativenon-perturbative kinetic theory ????

Thermal spectral functions and holography Andrei Starinets “Strong Fields, Integrability and Strings” program Isaac Newton Institute for Mathematical Sciences Cambridge July 31, 2007 Perimeter Institute for Theoretical Physics

Viscosity “measurements” at RHIC Viscosity is ONE of the parameters used in the hydro models describing the azimuthal anisotropy of particle distribution -elliptic flow for particle species “i” Elliptic flow reproduced for e.g. Baier, Romatschke, nucl-th/ Perturbative QCD: SYM: Chernai, Kapusta, McLerran, nucl-th/

A hand-waving argument Gravity duals fix the coefficient: Thus

Thermal conductivity Non-relativistic theory: Relativistic theory: Kubo formula: InSYM with non-zero chemical potential One can compare this with the Wiedemann-Franz law for the ratio of thermal to electric conductivity:

Classification of fluctuations and universality O(2) symmetry in x-y plane Scalar channel: Shear channel: Sound channel: Other fluctuations (e.g. ) may affect sound channel But not the shear channeluniversality of

Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then. Since the entropy (density) is we get

Three roads to universality of  The absorption argument D. Son, P. Kovtun, A.S., hep-th/  Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/  “Membrane paradigm” general formula for diffusion coefficient + interpretation as lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theorem P. Kovtun, D.Son, A.S., hep-th/ , A.S., to appear, P.Kovtun, A.S., hep-th/ , A.Buchel, J.Liu, hep-th/

Effect of viscosity on elliptic flow

Computing transport coefficients from “first principles” Kubo formulae allows one to calculate transport coefficients from microscopic models In the regime described by a gravity dual the correlator can be computed using the gauge theory/gravity duality Fluctuation-dissipation theory (Callen, Welton, Green, Kubo)

Sound wave pole Compare: In CFT: