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 (2000-2008-2020 ??)

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 densityvs

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 gravity4-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-dimgravity 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/0508092

Strong coupling: A.S., hep-th/0207133

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/0205051, P.Kovtun, A.S., hep-th/0506184)

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 functionsof flavor currents

R.Myers, A.S., R.Thomson, 0706.0162 [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/0406264

perturbative thermal gauge theoryS.Huot,S.Jeon,G.Moore, hep-ph/0608062

Electrical conductivity in SYM

Weak coupling:

Strong coupling:

* Charge susceptibility can be computed independently:

Einstein relation holds:

D.T.Son, A.S., hep-th/0601157

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/0610145

Mateos, Myers, Thomson, hep-th/0610184

A viscosity bound conjecture

P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

Minimum of in units of

Chernai, Kapusta, McLerran, nucl-th/0604032

Chernai, Kapusta, McLerran, nucl-th/0604032

Chernai, Kapusta, McLerran, nucl-th/0604032

Viscosity-entropy ratio of a trapped Fermi gas

T.Schafer, cond-mat/0701251

(based on experimental results by Duke U. group, J.E.Thomas et al., 2005-06)

Chernai, Kapusta, McLerran, nucl-th/0604032

QCD

Viscosity “measurements” at RHIC

Viscosity is ONE of the parameters used in the hydro modelsdescribing the azimuthal anisotropy of particle distribution

-elliptic flow forparticle species “i”

Elliptic flow reproduced for

e.g. Baier, Romatschke, nucl-th/0610108

Perturbative QCD:

SYM:

Chernai, Kapusta, McLerran, nucl-th/0604032

Shear viscosity at non-zero chemical potential

Reissner-Nordstrom-AdS black hole

with three R charges

(Behrnd, Cvetic, Sabra, 1998)

We still have

J.MasD.Son, A.S.O.SaremiK.Maeda, M.Natsuume, T.Okamura

(see e.g. Yaffe, Yamada, hep-th/0602074)

Photon and dilepton emission from supersymmetric Yang-Mills plasmaS. Caron-Huot, P. Kovtun, G. Moore, A.S., L.G. Yaffe, hep-th/0607237

Photons interacting with matter:

Photon emission from SYM plasma

To leading order in

Mimic by 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) vspSYM (solid line)at equal coupling

(and =3)

pQCD (dotted line) vspSYM (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

Equilibrium Near-equilibrium

entropyequation of state

…….

transport coefficientsemission rates

………

perturbative non-perturbative

pQCD Lattice

perturbative non-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/0405231

Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/0408095

“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/0309213, A.S., to appear, P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175

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/0309213, hep-th/0405231

Analytic structure of the correlators

Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092

Strong coupling: A.S., hep-th/0207133

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

CA: scalar channel

B: scalar channel - thermal part

C: sound channel

Pressure in perturbative QCD

Quantum field theories at finite temperature/density

Equilibrium Near-equilibrium

entropyequation of state

…….

transport coefficientsemission rates

………

perturbative non-perturbative

pQCD Lattice

perturbative non-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 modelsdescribing the azimuthal anisotropy of particle distribution

-elliptic flow forparticle species “i”

Elliptic flow reproduced for

e.g. Baier, Romatschke, nucl-th/0610108

Perturbative QCD:

SYM:

Chernai, Kapusta, McLerran, nucl-th/0604032

A hand-waving argument

Gravity duals fix the coefficient:

Thus

Thermal conductivityNon-relativistic theory:

Relativistic theory:

Kubo formula:

In SYM with non-zero chemical potential

One can compare this with the Wiedemann-Franz lawfor 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 channel universality 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/0405231

Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/0408095

“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/0309213, A.S., to appear, P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175

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: