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Transport in strongly coupled QFTs and gauge/gravity duality
Andrei Starinets Rudolf Peierls Centre for Theoretical Physics Oxford University

XIV Lomonosov Conference
Moscow State University Moscow Russia
22 August 2009


Heavy ion collision experiments at RHIC (2000-current) and LHC (2009-??) create hot and dense nuclear matter known as the "quark-gluon plasma" (note: qualitative difference between p-p and Au-Au collisions)
Elliptic flow, jet quenching... - focus on transport in this talk

Evolution of the plasma "fireball" is described by relativistic fluid dynamics (relativistic Navier-Stokes equations): Landau; Bjorken Need to know thermodynamics (equation of state) kinetics (first- and second-order transport coefficients) in the regime of intermediate coupling strength: initial conditions (initial energy density profile) thermalization time (start of hydro evolution) freeze-out conditions (end of hydro evolution)


Quantum field theories at finite temperature/density
Thermodynamics Kinetics

Equilibrium entropy equation of state .......

Near-equilibrium transport coefficients emission rates .........

perturbative non-perturbative Lattice pQCD

perturbative non-perturbative ???? kinetic theory


Energy density vs temperature for various gauge theories
Ideal gas of quarks and gluons

Ideal gas of hadrons
Figure: an artistic impression from Myers and Vazquez, 0804.2423 [hep-th]


AdS/CFT correspondence

conjectured exact equivalence

Latest test: Janik'08

Generating functional for correlation functions of gauge-invariant operators

String partition function

In particular

Classical gravity action serves as a generating functional for the gauge theory correlators


supersymmetric YM theory
Gliozzi,Scherk,Olive'77 Brink,Schwarz,Scherk'77

· Field content:

· Action:

(super)conformal field theory = coupling doesn't run


Dual to QCD? (Polchinski-Strassler)
coupling

energy


Dual to QCD? (Polchinski-Strassler)
coupling

energy


Dual to QCD? (Polchinski-Strassler)
coupling

energy


At zero temperature, N=4 SYM is obviously a very bad approximation to QCD However: At finite temperature it is qualitatively similar to QCD supersymmetry broken non-Abelian plasma (with additional d.o.f.) area law for spatial Wilson loops Debye screening spontaneous breaking of symmetry at high temperature

hydrodynamics


Hydrodynamics: fundamental d.o.f. = densities of conserved charges Need to add constitutive relations!

Example: charge diffusion
Conservation law Constitutive relation [Fick's law (1855)] Diffusion equation Dispersion relation
Expansion parameters:


First-order transport (kinetic) coefficients
Shear viscosity Bulk viscosity Charge diffusion constant

Supercharge diffusion constant Thermal conductivity Electrical conductivity

*

Expect Einstein relations such as

to hold


Second-order transport (kinetic) coefficients
(for theories conformal at T=0)

Relaxation time Second order trasport coefficient Second order trasport coefficient Second order trasport coefficient Second order trasport coefficient
In non-conformal theories such as QCD, the total number of second-order transport coefficients is quite large


10-dim gravity

4-dim gauge theory ­ large N, strong coupling
Holographically dual system in thermal equilibrium M, J, Q
T S

M,J,Q

Gravitational fluctuations

Deviations from equilibrium ????

and B.C. Quasinormal spectrum


Computing transport coefficients from "first principles"
Fluctuation-dissipation theory (Callen, Welton, Green, Kubo) 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


Example: stress-energy tensor correlator in in the limit
Zero temperature, Euclid: Finite temperature, Mink:

(in the limit
The pole (or the lowest quasinormal freq.) Compare with hydro: In CFT: Also,

)

(Gubser, Klebanov, Peet, 1996)


First-order transport coefficients in N = 4 SYM
in the limit
Shear viscosity Bulk viscosity Charge diffusion constant
for non-conformal theories see Buchel et al; G.D.Moore et al Gubser et al.

Supercharge diffusion constant Thermal conductivity Electrical conductivity

(G.Policastro, 2008)


Shear viscosity in

SYM

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

Correction to

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

Buchel, 0805.2683 [hep-th]; Myers, Paulos, Sinha, 0806.2156 [hep-th]


Electrical conductivity in SYM
Weak coupling:

Strong coupling:

*

Charge susceptibility can be computed independently:
D.T.Son, A.S., hep-th/0601157

Einstein relation holds:


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:
Benincasa, Buchel, Naryshkin, hep-th/0610145

· Extended to models with fundamental fermions in the limit
Mateos, Myers, Thomson, hep-th/0610184

· String/Gravity dual to QCD is currently unknown


Universality of shear viscosity in the regime described by gravity duals

Graviton's component obeys equation for a minimally coupled massless scalar. But then . we get Since the entropy (density) is


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., 0806.3797 [hep-th], P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175


Chernai, Kapusta, McLerran, nucl-th/0604032


A viscosity bound conjecture

?
in units of

Minimum of

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


A hand-waving argument

Thus

?

Gravity duals fix the coefficient:


Shear viscosity - (volume) entropy density ratio from gauge-string duality
In ALL theories (in the limit where dual gravity valid) : In particular, in N=4 SYM:

Other higher-derivative gravity actions

Y.Kats and P.Petrov: 0712.0743 [hep-th] M.Brigante, H.Liu, R.C.Myers, S.Shenker and S.Yaida: 0802.3318 [hep-th], 0712.0805 [hep-th]. R.Myers,M.Paulos, A.Sinha: 0903.2834 [hep-th] (and ref. therein ­ many other papers)
for superconformal Sp(N) gauge theory in d=4 Also: The species problem: T.Cohen, hep-th/0702136; A. Dolbado, F.Llanes-Estrada: hep-th/0703132


Shear viscosity - (volume) entropy density ratio in QCD

The value of this ratio strongly affects the elliptic flow in hydro models of QGP


Viscosity-entropy ratio of a trapped Fermi gas
0.7

/s

0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.4 0.6 0.8

T/TF
T.Schafer, cond-mat/0701251 (based on experimental results by Duke U. group, J.E.Thomas et al., 2005-06)


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

25

STAR

v2 (%)

PHENIX

20

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

EOS H

15

10

Perturbative QCD:
h 2 3 4 p (GeV)
T +/-

5
Chernai, Kapusta, McLerran, nucl-th/0604032

0 0

1

SYM:


Elliptic flow with color glass condensate initial conditions
CGC

35
STAR non-flow corrected (est). STAR event-plane /s=10
-4

30

25
/s=0.08

20

v2 (percent)

15
/s=0.16 /s=0.24

10

5 1 2 pT [GeV] 3 4

0 0

Luzum and Romatschke, 0804.4015 [nuc-th]


Elliptic flow with Glauber initial conditions
Glauber

25
STAR non-flow corrected (est.) STAR event-plane /s=10
-4

20
/s=0.08 /s=0.16

15

v2 (percent)

10

5

0 0 1

2 pT [GeV]

3

4

Luzum and Romatschke, 0804.4015 [nuc-th]


Viscosity/entropy ratio in QCD: current status

Theories with gravity duals in the regime where the dual gravity description is valid
(universal limit)

[Kovtun, Son & A.S] [Buchel] [Buchel & Liu, A.S]

QCD: RHIC elliptic flow analysis suggests

QCD: (Indirect) LQCD simulations

H.Meyer, 0805.4567 [hep-th]

Trapped strongly correlated cold alkali atoms

T.Schafer, 0808.0734 [nucl-th]

Liquid Helium-3


Spectral sum rules for the QGP

6

0.1

10 8 6
0.2

5

4

0.05

3

2

0.1

4 2
1 w

1

0.5

0.5

1

1.5

w

In N=4 SYM at ANY coupling
P.Romatschke, D.Son, 0903.3946 [hep-ph]


Photoproduction rate in SYM
SYM , 0.2 SYM , 0.5 SYM ,

0.02

3

1

EM

Nc 2 T

0.015

dk



0.01

0.005

d



0

2

4 kT

6

8

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



Other avenues of (related) research

Bulk viscosity for non-conformal theories (Buchel, Benincasa, Gubser, Moore...)

Non-relativistic gravity duals (Son, McGreevy,... )

Gravity duals of theories with SSB, AdS/CMT (Kovtun, Herzog, Hartnoll, Horowitz...)

Bulk from the boundary, time evolution of QGP (Janik,...)

Navier-Stokes equations and their generalization from gravity (Minwalla,...)

Quarks moving through plasma (Chesler, Yaffe, Gubser,...)


New directions

S. Hartnoll "Lectures on holographic methods for condensed matter physics", 0903.3246 [hep-th]

C. Herzog "Lectures on holographic superfluidity and superconductivity", 0904.1975 [hep-th]

M. Rangamani "Gravity and hydrodynamics: Lectures on the fluid-gravity correspondence", 0905.4352 [hep-th]


THANK YOU


Hydrodynamic properties of strongly interacting hot plasmas in 4 dimensions can be related (for certain models!)

to fluctuations and dynamics of 5-dimensional black holes


AdS/CFT correspondence: the role of J

For a given operator

, identify the source field , e.g.

satisfies linearized supergravity e.o.m. with b.c. The recipe:

To compute correlators of , one needs to solve the bulk supergravity e.o.m. for and compute the on-shell action as a functional of the b.c.

Warning: e.o.m. for different bulk fields may be coupled: need self-consistent solution

Then, taking functional derivatives of

gives


Holography at finite temperature and density

Nonzero expectation values of energy and charge density translate into nontrivial background values of the metric (above extremality)=horizon and electric potential = CHARGED BLACK HOLE (with flat horizon)

temperature of the dual gauge theory

chemical potential of the dual theory


Gauge-string duality and QCD

Approach I: use the gauge-string (gauge-gravity) duality to study N=4 SYM and similar theories, get qualitative insights into relevant aspects of QCD, look for universal quantities (exact solutions but limited set of theories)

Approach II: bottom-up (a.k.a. AdS/QCD) ­ start with QCD, build gravity dual approximation

(unlimited set of theories, approximate solutions, systematic procedure unclear)

(will not consider here but see e.g. GÝrsoy, Kiritsis, Mazzanti, Nitti, 0903.2859 [hep-th])

Approach III: solve QCD

Approach IIIa: pQCD (weak coupling; problems with convergence for thermal quantities)

Approach IIIb: LQCD (usual lattice problems + problems with kinetics)


Over the last several years, holographic (gauge/gravity duality) methods were used to study strongly coupled gauge theories at finite temperature and density

These studies were motivated by the heavy-ion collision programs at RHIC and LHC (ALICE, ATLAS) and the necessity to understand hot and dense nuclear matter in the regime of intermediate coupling

As a result, we now have a better understanding of thermodynamics and especially kinetics (transport) of strongly coupled gauge theories

Of course, these calculations are done for theoretical models such as N=4 SYM and its cousins (including non-conformal theories etc). for QCD

We don't know quantities such as


New transport coefficients in

SYM

Sound dispersion:

Kubo:


Our understanding of gauge theories is limited...

Perturbation theory

Lattice


Conjecture: specific gauge theory in 4 dim = specific string theory in 10 dim

Perturbation theory

Lattice


In practice: gravity (low energy limit of string theory) in 10 dim = 4-dim gauge theory in a region of a parameter space Dual gravity

Perturbation theory

Lattice

Can add fundamental fermions with


Computing real-time correlation functions from gravity

To extract transport coefficients and spectral functions from dual gravity, we need a recipe for computing Minkowski space correlators in AdS/CFT

The recipe of [D.T.Son & A.S., 2001] and [C.Herzog & D.T.Son, 2002] relates

real-time correlators in field theory to Penrose diagram of black hole in dual gravity

Quasinormal spectrum of dual gravity = poles of the retarded correlators in 4d theory
[D.T.Son & A.S., 2001]


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)


Analytic structure of the correlators


p -> 0
- i 4 - i 8 - i 12




-p p

- i 4

- i 8

- i 12

-p

p

T -> 0



-p

p

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

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


Computing transport coefficients from dual gravity

Assuming validity of the gauge/gravity duality, all transport coefficients are completely determined by the lowest frequencies in quasinormal spectra of the dual gravitational background

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

This determines kinetics in the regime of a thermal theory where the dual gravity description is applicable

Transport coefficients and quasiparticle spectra can also be obtained from thermal spectral functions


Spectral function and quasiparticles

6

0.1

5
0.6 0.4
0.2

A
0.8

4

0.05

3
0.2

B
1 2 3 w

2

0.1

1 1 w
-0.2

0.5

10

8

6

C

A: scalar channel B: scalar channel - thermal part C: sound channel
1 1.5 w

4

2

0.5


Is the bound dead?
superconformal Sp(N) gauge theory in d=4

Y.Kats and P.Petrov, 0712.0743 [hep-th] "Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory"

M.~Brigante, H.~Liu, R.~C.~Myers, S.~Shenker and S.~Yaida, ``The Viscosity Bound and Causality Violation,'' 0802.3318 [hep-th], ``Viscosity Bound Violation in Higher Derivative Gravity,'' 0712.0805 [hep-th].

The "species problem" T.Cohen, hep-th/0702136, A.Dobado, F.Llanes-Estrada, hep-th/0703132


A viscosity bound conjecture

Minimum of

in units of

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


Chernai, Kapusta, McLerran, nucl-th/0604032


QCD

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)

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

We still have

J.Mas D.Son, A.S. O.Saremi K.Maeda, M.Natsuume, T.Okamura
1.25 1.2 1.15 1.1 1.05

0.1

0.2

0.3

0.4

0.5


Photon and dilepton emission from supersymmetric Yang-Mills plasma

S. Caron-Huot, P. Kovtun, G. Moore, A.S., L.G. Yaffe, hep-th/0607237


Photon emission from SYM plasma

Photons interacting with matter:

To leading order in

Mimic

by gauging global R-symmetry

Need only to compute correlators of the R-currents


Now consider strongly interacting systems at finite density and LOW temperature


Probing quantum liquids with holography
Low-energy elementary excitations
phonons fermionic quasiparticles + bosonic branch (zero sound)

Quantum liquid in p+1 dim

Specific heat at low T

Quantum Bose liquid

Quantum Fermi liquid (Landau FLT)

Departures from normal Fermi liquid occur in

- 3+1 and 2+1 ­dimensional systems with strongly correlated electrons

- In 1+1 ­dimensional systems for any strength of interaction (Luttinger liquid)

One can apply holography to study strongly coupled Fermi systems at low T


L.D.Landau (1908-1968)


The simplest candidate with a known holographic description is

at finite temperature T and nonzero chemical potential associated with the "baryon number" density of the charge

There are two dimensionless parameters:
is the hypermultiplet mass

is the baryon number density

The holographic dual description in the limit is given by the D3/D7 system, with D3 branes replaced by the AdSSchwarzschild geometry and D7 branes embedded in it as probes.

Karch & Katz, hep-th/0205236


AdS-Schwarzschild black hole (brane) background

D7 probe branes

The worldvolume U(1) field

couples to the flavor current

at the boundary

Nontrivial background value of

corresponds to nontrivial expectation value of

We would like to compute
temperature

- the specific heat at low

- the charge density correlator


The specific heat (in p+1 dimensions):

(note the difference with Fermi

and Bose

systems)

The (retarded) charge density correlator has a pole corresponding to a propagating mode (zero sound) - even at zero temperature

(note that this is NOT a superfluid phonon whose attenuation scales as

)

New type of quantum liquid?


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

At the moment, this method is the only theoretical tool available to study the near-equilibrium regime of strongly coupled thermal field theories

The result for the shear viscosity turns out to be universal for all such theories in the limit of infinitely strong coupling

Influences other fields (heavy ion physics, condmat)


A hand-waving argument

Thus

Gravity duals fix the coefficient:


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


Equations such as limit of string theory

describe the low energy

As long as the dilaton is small, and thus the string interactions are suppressed, this limit corresponds to classical 10-dim Einstein gravity coupled to certain matter fields such as Maxwell field, p-forms, dilaton, fermions

Validity conditions for the classical (super)gravity approximation

- curvature invariants should be small:

- quantum loop effects (string interactions = dilaton) should be small:

In AdS/CFT duality, these two conditions translate into and


The challenge of RHIC (continued
)

Rapid thermalization

??

Large elliptic flow

Jet quenching

Photon/dilepton emission rates


The bulk and the boundary in AdS/CFT correspondence

UV/IR: the AdS metric is invariant under

z plays a role of inverse energy scale in 4D theory

z

5D bulk (+5 internal dimensions)

0 4D boundary