Understanding
the Quark-Gluon Plasma
via
String Theory
Hong Liu
Massachusetts Institute of Technology
HL, Krishna Rajagopal, Urs A. Wiedemann
hep-ph/0605178, hep-ph/0607062, hep-ph/0612168
Qudsia Ejaz, Thomas Faulkner, HL, Krishna Rajagopal, Urs Wiedemann
to appear
Plan
• Heavy ion collisions
• Shear viscosity (a quick overview)
• Jet quenching
• J/ψ suppression: a prediction
• N=4 SYM v.s. QCD
Quark-Gluon Plasma
At room temperature, quarks and gluons
are always confined inside colorless
objects (hadrons):
protons, neutrons, pions, …..
Very high temperature (asymptotic freedom):
 Interactions become weak
 quarks and gluons deconfined
 Quark-gluon plasma (QGP)
Infinitely high temperature:
QGP behaves like an ideal gas.
Is there a deconfinement phase transition
separating the hadronic and QGP phases?
Can one create quark-gluon plasma in the lab?
QCD Phase diagram (2006)
Smooth crossover at
TC
170 MeV
Relativistic Heavy ion collisions
Relativistic Heavy Ion Collider
(RHIC)
RHIC: Au+Au
sNN
sNN  200 GeV
: center of mass energy per pair
of nucleons
Au: 197 nucleons; Total: 39.4 TeV
• Energy density (peak)
> 5 GeV/fm3
• Temperature (peak)
~ 300 MeV
LHC: Pb + Pb (2009)
sNN  5,500 GeV
Creating a little Big Bang
Experimental probes of the QGP ?
Some basic questions:
Has the created hot matter reached thermal equilibrium?
If yes, when?
Has the QGP been formed? What are its signatures?
Properties: weakly or strongly coupled?
equation of state? Viscosity? opacity?
QGP at RHIC exists for about 10-23 sec (5 fm), making it
impossible to study it using any external probes.
Quark-gluon fluid of RHIC
• Collective behavior of the observed final-state hadrons
(elliptic flow)
• Interaction of produced hard probes with the medium
(jet quenching, J/Ψ suppression)
Nearly ideal, strongly coupled fluid (sQGP)
Main theoretical tool for strong coupling: Lattice calculation
But information on dynamical quantities: scarce and indirect
New theoretical tools are needed!
String theory to the rescue!
Collective motion and
shear viscosity of sQGP
Collective motion

y
If lots of p+p collisions plus free streaming:
final state momenta uniformly distributed
in azimuth angle  .

x
If interaction  equilibration  pressure  pressure gradients
collective motion
 anisotropy of momenta distribution in
.
Near-perfect fluid discovered
• Elliptic flow
dN
( pT )  1  2 v2 ( pT ) cos  2  
d
Strong signal !
• Rough agreement with
hydrodynamic models based
on perfect liquid.
Created hot matter equilibrates very early: before 1fm.
likely strongly interacting !
Shear viscosity should be small!
Universality of Shear viscosity
• RHIC:

s
• Water

s

• N=4 SYM:

 0.1  0.2
s

Teaney (2003)
~ 10

1
 0.08
4
Policastro, Son, and Starinets (2001)
1
4
• The value s
turned out to be universal for all
strongly coupled QGPs with a gravity description.
Kovton, Son and Starinets (2003) Buchel, J. Liu
• Lattice:

s
 0.13  0.03, at T=1.65 TC
Meyer (2007)
AdS/CFT and Jet quenching
Hard probes
Hard scatterings in p+p collisions produce:
back-to-back high energy quarks ("jets“).
The presence of hot matter modifies the properties of jets.
Jet Quenching
They lose energy!
QGP
1. The number of high energy
particles observed should be
much smaller than expected
from p+p collisions:
Only 20% !
2. monojets: sometimes they
never make out.
QGP
Parton energy loss in QGP
The dominant effect of
the medium on a high
energy parton is
medium-induced
Bremsstrahlung.
S
E  
N C qˆ L2
2
Baier, Dokshitzer, Mueller,
Peigne, Schiff (1996):
q̂ : reflects the ability of the medium to “quench” jets.
Toward understanding Opacity
Experimental estimate:
q̂ :
5-15 GeV2/fm
q̂ :
< 0.1
Hadronic gas:
GeV2/fm
Perturbation theory:
q̂ :
<1
GeV2/fm
Strongly coupled QGP?
Theoretical challenge: non-perturbative calculation of
QCD QGP slightly above TC .
q̂ for
Strategy:
• Need a non-perturbative definition of
• Compute
q̂
q̂
in SYM theory using AdS/CFT
q̂ : a non-perturbative formulation
Hard: weakly coupled
Soft: likely strongly coupled
Assume:
E >> ω >> k┴ >>T
q̂ : multiple rescatterings of hard particles
with the medium
Soft scatterings
Zakharov (1997); Wiedemann (2000)
• Amplitude for a particle propagating in the medium
• High energy limit (eikonal approximation):
Soft scatterings are captured by Light like Wilson lines.
A non-perturbative definition of q̂
Wiedemann
HL, Rajagopal, Wiedemann
Light-like Wilson loop:
L
L: conjugate to the pT
L: length of the medium
Assuming: L  1 / T  L
Thermal average
(Hard to calculate using lattice)
Nonperturbative definition of
q̂
Wilson loop from AdS/CFT
Maldacena (1998); Rey and Yee (1998)
Recipe:
Our (3+1)-dim world,
Wilson loop C
in our world
r 
S(C): area of string
worldsheet with
boundary C

horizon
r0
Black hole in AdS spacetime:
• radial coordinate r,
• horizon: r=r0

• constant r surface: (3+1)-dim Minkowski
spacetime
Extremal configuration
r=r0
extremal string configuration:
string touches the horizon.
Interactions
between the
quark and the
medium
r=r0
two disjoint strings
Interaction of the
string with the
horizon of a black
hole.
Wilson loop
With
S I  S  S0 
 2
8 2a

L L2T 3  O L L4T 5

The corresponding BDMPS transport coefficient reads
qˆ SYM 
 3 / 2 34 
 54 
T 3  26.69  SYM N c T 3
q̂ of N=4 SYM theory
BDMPS transport coefficient reads:
qˆ SYM 
 3 / 2 34 

5
4

T  26.69  SYM N c T
3
3
• It is not proportional to number of scattering centers
1
• Take: N  3,  s  , T  300 MeV
C
2
qˆ SYM  4.5 GeV 2 /fm.
• Experimental estimates: 5-15 GeV2/fm
q̂ and number of degrees of freedom
General conformal field theories (CFT) with a gravity
dual: (large N and strong coupling)
HL,Rajagopal
Wiedemann,
qˆCFT
sCFT

qˆ N 4
sN 4
sCFT : entropy density
For non-conformal theories, it may decrease with RG flow.
an estimate for QCD:
qˆQCD
sQCD
47.5


 0.63
qˆ N 4
sN 4
120
Summary
• In QGP of QCD, the energy loss of a high energy parton
can be described perturbatively up to a non-perturbative
jet-quenching parameter.
• We calculate the parameter in N=4 SYM (not
necessarily full energy loss of SYM)
• It appears to be close to the experimental value.
Quarkonium suppression:
a prediction for LHC or RHIC II
Heavy quarkonium in a QGP
Above TC, light-quark mesons no longer exist due to
deconfinement.
Heavy quarkonia are bound by short-distance Coulomb
interaction: may still exist above TC .
In a QGP, interactions between a quark and an anti-quark are
screened by the plasma. A heavy quark meson will
dissociate when the screening length becomes of order the
bound state size.
J / (c c) : Tdiss = 2.1 TC
 (b b) : Tdiss = 3.6 TC
while their excited states already dissociate above 1.2 TC.
Quarkonium suppression
J/ψ
Quarkonium suppression is a sensitive probe of QGP.
Matsui and Satz (1987)
Connecting lattice QCD directly to heavy ion phenomenology
is difficult:
Heavy quark mesons produced in heavy ion collisions
could move very fast relative to the hot medium:
How does the screening effect depend on the velocity?
Velocity dependence of the Tdiss ?
(not known in QCD)
Static quark potential in N=4 SYM
Maldacena;
Rey, Yee;
Rey, Theisen
Yee;
Brandhuber,
Itzhaki,
Sonnenschein
Yankielowicz
……..
probe brane
Ls
Finding string shape
of minimal energy
event horizon
quarks are screened
2
NC limit:
In the large NC and large   gYM
quark potential
V ( L) 
= energy of open string connecting the pair
 f (TL) ,
LS  0.277 / T
Quark potential at finite velocity
HL,
Rajagopal
Wiedemann
Moving at a finite velocity v
Finding string shape
of minimal energy
Event horizon
LS (v)  LS (0) (1  v )
2 1/ 4
(1  v )
2 1/ 4
1
T
1

1
4
(1  v2 ) T
In a rest frame of quark pair, the medium is boosted:
2
2
 1 
 1  4
 (v )  
 (0) 
T


2
2
1

v
1

v






 (1  v ) T 


1

2
4
4
Velocity dependence of
dissociation temperature
Dissociation temperature Td :
d
Given:
this suggests:
LS (Td )
d: size of a meson
1
LS (v)  (1  v 2 )1/ 4
T
Td (v)
(1  v ) Td (0)
2 1/ 4
What would happen if QCD also has similar velocity scaling?
Has RHIC reached Td for J/ψ ?
Lattice: J/ψ may survive up to 2TC
Similarity of the magnitude of J/ψ suppression
at RHIC and SPS
Karsch, Kharzeev, Satz,
RHIC has not reached Tdiss for J/ψ.
Quarkonium suppression:
a prediction via string theory
HL,Rajagopal,Wiedemann
Heavy quark mesons with larger velocity dissociate at a
lower temperature.
Td (v)
(1  v 2 )1/ 4 Td (0)

Expect significant suppression
at large PT.
J/psi
This effect may be significant
and tested at RHIC II or LHC
Data to come
RHIC: low statistics on J/ψ with 2 < PT < 5 GeV,
no data for PT> 5GeV
Reach in PT will extend to
10 GeV in coming years at
RHIC.
LHC will reach even
wider range.
N=4 SYM versus QCD
N=4 SYM versus QCD
N=4 SYM theory
• Conformal
• no asymptotic freedom,
no confinement
• supersymmetric
• no chiral condensate
• no dynamical quarks, 6 scalar and 4
Weyl fermionic fields in the adjoint
representation.
Physics near
vacuum and
at very high
energy is very
different from
that of QCD
N=4 SYM versus QCD (continued)
N=4 SYM at finite T
QCD at T ~TC -3 TC
• conformal
• near conformal (lattice)
• no asymptotic freedom,
no confinement
• not intrinsic properties of
sQGP
• supersymmetric (badly broken ) • not present
• no chiral condensate
• not present
• no dynamical quarks, 6 scalars • may be taken care of by
proper normalization
and 4 fermions in the adjoint
representation.
N=4 SYM versus QCD
Ideal gas (T= infinity QCD)
Strongly coupled
N=4 SYM at finite T
T=0 QCD confinement
N=4 SYM versus QCD
• It is likely that QCD has a string dual in the large N limit.
Finite-T QCD in a strongly coupled regime could be described
by a black hole in this string theory.
• Universality of black hole (horizon physics)
Universality between QCD and N=4 SYM for observables
probing intrinsic properties of the medium.
Summary
• String theory techniques provide qualitative, and
semi-quantitative insights and predictions regarding
properties of strongly interacting quark-gluon plasma:
• Shear viscosity
• Jet quenching parameter
• Quarkonium suppression (a prediction)
• Expect many more chapters to be written for the marriage
between string theory and physics of QCD in extreme
conditions.
Energy and entropy density
Karsch:hep-lat/0106019
QCD:
N=4 SYM:
Gubser, Klebanov,Peet (1998)
2
3
 (   ) 
N T   free
6
4
2
4
Speed of sound
Karsch, hep-ph/0610024
Jet quenching: monojet phenomenon
STAR collaboration: nucl-ex/0501009
Jet quenching: data (II)
# of particles measured
RAA 
# of particles expected from pp collisions