Transcript 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

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
Energy density
vs
temperature
QCD deconfinement transition (lattice data)
The challenge of RHIC (continued)
Rapid thermalization
??
Large elliptic flow
Jet quenching
Photon/dilepton emission rates
10-dim gravity
M,J,Q
4-dim gauge theory – large N,
strong coupling
Holographically dual system
in thermal equilibrium
M, J, Q
T
Gravitational fluctuations
S
Deviations from equilibrium
????
+ fluctuations of other fields
and B.C.
Quasinormal spectrum
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!)
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
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
Conservation law
Constitutive relation
[Fick’s law (1855)]
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:
Predictions of hydrodynamics
Hydrodynamics predicts that the retarded correlator
has a “sound wave” pole at
Moreover, in conformal theory
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
Strong coupling: A.S., hep-th/0207133
Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092
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
Spectral functions and quasiparticles in
The slope at zero frequency
determines the kinetic coefficient
Figures show
Peaks correspond to quasiparticles
at different values of
Spectral function and quasiparticles
in finite-temperature “AdS + IR cutoff” model
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]
Transport coefficients in N=4 SYM
in the limit
• Shear viscosity
• Bulk viscosity
• Charge diffusion constant
• Thermal conductivity
• Electrical conductivity
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
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
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
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)
QCD
Chernai, Kapusta, McLerran, nucl-th/0604032
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:
Shear viscosity at non-zero chemical potential
Reissner-Nordstrom-AdS black hole
with three R charges
(see e.g. Yaffe, Yamada, hep-th/0602074)
We still have
(Behrnd, Cvetic, Sabra, 1998)
J.Mas
D.Son, A.S.
O.Saremi
K.Maeda, M.Natsuume, T.Okamura
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
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
 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 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
entropy
equation of state
…….
transport coefficients
emission rates
………
perturbative non-perturbative
Lattice
pQCD
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
 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
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
.
we get
Since the entropy (density) is
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:
)
A viscosity bound conjecture
P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231
Analytic structure of the correlators
Strong coupling: A.S., hep-th/0207133
Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092
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)
Spectral function and quasiparticles
A
B
A: scalar channel
C
B: scalar channel - thermal part
C: sound channel
Pressure in perturbative QCD
Quantum field theories at finite temperature/density
Equilibrium
Near-equilibrium
entropy
equation of state
…….
transport coefficients
emission rates
………
perturbative non-perturbative
Lattice
pQCD
perturbative non-perturbative
????
kinetic theory
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
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:
A hand-waving argument
Thus
Gravity duals fix the coefficient:
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:
Classification of fluctuations and
universality
O(2) symmetry in x-y plane
Shear channel:
Sound channel:
Scalar channel:
Other fluctuations (e.g.
But not the shear channel
) may affect sound 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
.
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., 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”
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
Sound wave pole
Compare:
In CFT: