Transport properties of strongly coupled gauge theories from string theory

1
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
2
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

3
The challenge of RHIC
Energy density vs temperature
QCD deconfinement transition (lattice data)
4
The challenge of RHIC (continued)
Rapid thermalization
??
Large elliptic flow
Jet quenching
Photon/dilepton emission rates
5
4-dim gauge theory large N, strong coupling
10-dim gravity
M,J,Q
Holographically dual system in thermal
equilibrium
M, J, Q
T S
Deviations from equilibrium
Gravitational fluctuations
????
fluctuations of other fields
and B.C.
Quasinormal spectrum
6
Transport (kinetic) coefficients

• Shear viscosity
• Bulk viscosity
• Charge diffusion constant
• Thermal conductivity
• Electrical conductivity

Expect Einstein relations such as
to hold
7
Gauge/gravity dictionary determines correlators
of gauge-invariant operators from gravity (in the
regime where gravity description is valid!)
Maldacena Gubser, Klebanov, Polyakov Witten
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
8
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)

9
Hydrodynamics fundamental d.o.f. densities of
conserved charges
Need to add constitutive relations!
Example charge diffusion
Conservation law
Constitutive relation
Ficks law (1855)
Diffusion equation
Dispersion relation
Expansion parameters
10
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
11
Predictions of hydrodynamics
Hydrodynamics predicts that the retarded
correlator
has a sound wave pole at
Moreover, in conformal theory
12
Now look at the correlators obtained from gravity
The correlator has poles at
The speed of sound coincides with the hydro
prediction!
13
Analytic structure of the correlators
Strong coupling A.S., hep-th/0207133
Weak coupling S. Hartnoll and P. Kumar,
hep-th/0508092
14
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)
15
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

16
Spectral functions and quasiparticles in
The slope at zero frequency determines
the kinetic coefficient
Peaks correspond to quasiparticles
Figures show at different values of
17
Spectral function and quasiparticles
in finite-temperature AdS IR cutoff
model
18
Holographic models with fundamental fermions
Thermal spectral functions of flavor currents
Additional parameter makes life
more interesting
R.Myers, A.S., R.Thomson, 0706.0162 hep-th
19
Transport coefficients in N4 SYM
in the limit

• Shear viscosity
• Bulk viscosity
• Charge diffusion constant
• Thermal conductivity
• Electrical conductivity

20
Shear viscosity in SYM
perturbative thermal gauge theory S.Huot,S.Jeon,G.
Moore, hep-ph/0608062
Correction to A.Buchel, J.Liu,
A.S., hep-th/0406264
21
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
22
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

23
A viscosity bound conjecture
Minimum of in units of
P.Kovtun, D.Son, A.S., hep-th/0309213,
hep-th/0405231
24
Chernai, Kapusta, McLerran, nucl-th/0604032
25
Chernai, Kapusta, McLerran, nucl-th/0604032
26
Chernai, Kapusta, McLerran, nucl-th/0604032
27
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)
28
QCD
Chernai, Kapusta, McLerran, nucl-th/0604032
29
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/0610108
Perturbative QCD
Chernai, Kapusta, McLerran, nucl-th/0604032
SYM
30
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)
J.Mas D.Son, A.S. O.Saremi K.Maeda, M.Natsuume,
T.Okamura
We still have
31
Photon and dilepton emission from
supersymmetric Yang-Mills plasma
S. Caron-Huot, P. Kovtun, G. Moore, A.S., L.G.
Yaffe, hep-th/0607237
32
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
33
Photoproduction rate in SYM
(Normalized) photon production rate in SYM for
various values of t Hooft coupling
34
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)
35
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

36
THE END
37
Some results

• Shear viscosity/entropy ratio

• in the limit described by gravity duals
  
• universal for a large class of theories

• Bulk viscosity for non-conformal theories

• 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 N4 SYM

38

• 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

39
Quantum field theories at finite
temperature/density
Equilibrium
Near-equilibrium
transport coefficients emission rates
entropy equation of state .
perturbative
non-perturbative
perturbative
non-perturbative
????
Lattice
kinetic theory
pQCD
40
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

• Stimulating for experimental/theoretical
  research in other fields

41
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

42
Universality of shear viscosity in the regime
described by gravity duals
Gravitons component obeys equation for a
minimally coupled massless scalar. But then
.
Since the entropy (density) is
we get
43
Example 2 (continued) 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
44
A viscosity bound conjecture
P.Kovtun, D.Son, A.S., hep-th/0309213,
hep-th/0405231
45
Analytic structure of the correlators
Strong coupling A.S., hep-th/0207133
Weak coupling S. Hartnoll and P. Kumar,
hep-th/0508092
46
Example 2 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)
47
Spectral function and quasiparticles
A
B
A scalar channel
C
B scalar channel - thermal part
C sound channel
48
Pressure in perturbative QCD
49
Quantum field theories at finite
temperature/density
Equilibrium
Near-equilibrium
transport coefficients emission rates
entropy equation of state .
perturbative
non-perturbative
perturbative
non-perturbative
????
Lattice
kinetic theory
pQCD
50
Thermal spectral functions and holography



Andrei Starinets
Perimeter Institute for Theoretical Physics
Strong Fields, Integrability and Strings
program Isaac Newton Institute for Mathematical
Sciences Cambridge
July 31, 2007
51
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/0610108
Perturbative QCD
Chernai, Kapusta, McLerran, nucl-th/0604032
SYM
52
A hand-waving argument
Thus
Gravity duals fix the coefficient
53
(No Transcript)
54
Thermal conductivity
Non-relativistic theory
Relativistic theory
Kubo formula
In
SYM with non-zero chemical potential
One can compare this with the Wiedemann-Franz
law for the ratio of thermal to electric
conductivity
55
Classification of fluctuations and universality
O(2) symmetry in x-y plane
Shear channel
Sound channel
Scalar channel
Other fluctuations (e.g. )
may affect sound channel
But not the shear channel
universality of
56
Universality of shear viscosity in the regime
described by gravity duals
Gravitons component obeys equation for a
minimally coupled massless scalar. But then
.
Since the entropy (density) is
we get
57
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

58
Effect of viscosity on elliptic flow
59
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
60
Sound wave pole
Compare

In CFT