Transport coefficients in strongly coupled gauge theories:

insights from string theory

Andrei Starinets

Perimeter Institute for Theoretical Physics

Collaboration: Dam SonGiuseppe PolicastroChris HerzogAlvaro NunezPavel KovtunAlex BuchelJim LiuAndrei ParnachevPaolo Benincasa

References:

hep-th/0205051 hep-th/0205052 hep-th/0302026 hep-th/0309213 hep-th/0405231 hep-th/0406124hep-th/0506144 hep-th/0506184 hep-th/0507026

The goal is to compute transport coefficients (e.g. shear and bulk viscosity) and the speed of sound in thermal QCD

This is of interest for hydrodynamic models applied to RHIC data (elliptic flow)

Transport coefficients are hard to compute from “first principles”, even in perturbation theory. (For example, no perturbative calculation of bulk viscosity in gauge theory is available.)

Lattice approach cannot be used directly.

Transport coefficients and the speed of sound can be computed from string theory for SOME thermal gauge theories in nonperturbative regime

This calculation is based on approach known as “gauge/gravity duality” or “AdS/CFT correspondence”

AdS = Anti de Sitter spaceCFT = Conformal Field Theory

Gauge-gravity duality in string theory

Perturbative string theory: open and closed strings(at low energy, gauge fields and gravity, correspondingly)

Nonperturbative theory: D-branes (“topological defects” in 10d)

Complementary description of D-branes by open (closed) strings:

perturbative gauge theory description OK

perturbative gravity description OK

Computing transport coefficients from “first principles”

Kubo formulae allow one to calculate transport coefficients from microscopic models

In the regime described by a gravity dual the correlator can be computed using AdS/CFT

Fluctuation-dissipation theory(Callen, Welton, Green, Kubo)

Shear viscosity in SYM

Correction to : A.Buchel, J.Liu, A.S., hep-th/0406264

P.Arnold, G.Moore, L.Yaffe, 2001

Universality of

Theorem:

For any thermal gauge theory (with zero chemicalpotential), the ratio of shear viscosity to entropy density is equal to in the regime describedby a corresponding dual gravity theory

Remark:

Gravity dual to QCD (if it exists at all) is currentlyunknown.

A viscosity bound conjecture

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

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

Bulk viscosity and the speed of sound in SYM

is a “mass-deformed” (Pilch-Warner flow)

Finite-temperature version: A.Buchel, J.Liu, hep-th/0305064

The metric is known explicitly for Speed of sound and bulk viscosity:

Epilogue

AdS/CFT gives insights into physics of thermal gauge theories in the nonperturbative regime

General algorithm exists to compute transport coefficients and the speed of sound in any gravity dual

Model-independent statements can presumably be checked experimentally

Hydrodynamics as an effective theory

Thermodynamic equilibrium:

Near-equilibrium:

Eigenmodes of the system of equations

Shear mode (transverse fluctuations of ):

Sound mode:

For CFT we have and

Relation to RHIC IF quark-gluon plasma is indeed formed in heavy ion collisions

IF a hydrodynamic regime is unambiguously proven to exist

THEN hydrodynamic MODELS describe experimental results for e.g. elliptic flows well, provided

Bulk viscosity and speed of sound results are potentially interesting

Example: supersymmetric Yang-Millstheory at finite temperature

Fields:

The theory is conformal (coupling doesn’t run)

In the limit the theory is described by a gravity dual (AdS-Schwarzschild black hole)