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On viscosity of Quark Gluon Plasma
Defu Hou
CCNU , Wuhan
RHIC-Star full TOF detector and related physics in China
Hangzhou April 27-29
1
Outlines





Introduction and motivation
Viscosity from Kubo formula
Viscosity from kinetic theory (Boltzmann Eq)
Viscosity from AdS/CFT
Summary
2
QCD under extreme conditions



At very High T or density ( deconfined)
High T (Early universe, heavy-ion collisions)
High density matter ( in the core of neutron stars)
3
Motivations
Experiments aspect:
@ RHIC

Robust collective flows, well described by ideal
hydro with Lattice-based EoS. This indicates very
strong interaction even at early time => sQGP

sQGP seems to be the almost perfect fluid known
/s>= .1-.2<<1
4
Study of dissipative effects on <v2>
How sensitive is elliptic flow to finite /s?
Viscous Hydro
P. Romatschke, PRL99 (07)
Dependence on tp relaxation time
II0 order expansion
with green terms (D. Rischke)
Cascade (2<->2,2<->3)
Z. Xu & C. Greiner, PRL 101(08)
Agreement for s=0.3 – 0.6
/s=0.15 – 0.08
5
Theoretic aspect:
•
To calculate Trsp. Coefs. in FT in highly nontrivial
(nonperturbative ladder resummation) (c around 5)
•
String theory method: AdS/CFT
/s
(D.Son et al 2003)
= 1/4p
. Kinetic theory + uncertainty principle (Gyulassy)
6
Main obstacle for theory




QCD in nonperturbative regime (T~200Mev)
Pertburb. Expansion of QCD is not well behaved for realistic T
For thermodyn.,one can use lattice and resummation techniques
Kinetic coefficients are difficult to extract from lattice
7
Shear Viscosity
Fx
v x
 
A
y
8
9
10
Viscosity from Kubo formula
11
Nonlinear Response
12
13
S. Jeon, PRD 52; Carrington, Hou, Kobes, PRD61
14
Carrington, Hou, Kobes, PRD64 (2001)
15
Hou, hep-ph/0501284
16
17
Viscosity from kinetics theory
18
19
20
Viscosity of hot QCD at finite density
Fluctuation of distribution
(s: species)
Boltzmann Equation
Recast the Boltzmann equation
P.Arnold, G.D.Moore and G.Yaffe,
JHEP 0011(00)001
21
Shear viscosity
With a definition of inner product and expanded distribution functions,
where
22
Collision terms
Performing the integral over dk’ with the help of
Scattering
amplitude
Distribution function term
\chi term
23
Matrix Element
24

Variation method gives
Liu, Hou, Li EPJC 45(2006)
25
Computing transport coefficients from
AdS/CFT
In the regime described by a gravity dual
the correlator can be computed using
AdS/CFT
26
AdS/CFT at finite temperature
Classical Supergravity on AdS-BH×S5
=
conjecture
Witten ‘98
4dim. Large-Nc strongly coupled
SU(Nc) N=4 SYM at finite temperature
(in the deconfinement phase).
27
Field Theory
=
Gauge Theories
QCD
Gravity Theory
Quantum Gravity
String theory
Holography
the large N limit
Supersymmetric Yang Mills
N large
Gravitational theory in 10 dimensions
Calculations
Correlation functions
Quark-antiquark potential
28
AdS/CFT now being applied to
RHIC physics

Viscosity, /s.

EOS

Jet quenching

“Sound” waves

Photon production

Friction …

Heavy quarkonium

Hardron spectrum (ADS/QCD)
29
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
D. Son, P. Kovtun, A.S., hep-th/0405231
30
Shear viscosity in
SYM
P.Arnold, G.Moore, L.Yaffe, 2001
Correction to
: A.Buchel, J.Liu, A.S., hep-th/0406264
31
A viscosity bound conjecture
P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231
32
Universality of
Theorem:
For any thermal gauge theory (with zero chemical
potential), the ratio of shear viscosity to entropy
density is equal to
in the regime described
by a corresponding dual gravity theory
Remark:
Gravity dual to QCD (if it exists at all) is currently
unknown.
33
Possible Mechanisms for Low viscosity

Large cross-section, strong coupling

Anomalous viscosity: turbulence
M. Asakawa, S.A. Bass, B.M., hep-ph/0603092, PRL
See Abe & Niu (1980) for effect in EM plasmas
34
Take moments of


p
 r   p D( p )  p  f (r , p, t )  C  f 
 

t
E


p
with pz2
2
2
4
1
g
B
t
N
1
g
ln
g
1
1
m
2
 O 1 2 c

O
10




3

Nc  1
sT
T3
 A C
M. Asakawa, S.A. Bass, B.M., hep-ph/0603092
See Abe & Niu (1980) for effect in EM plasmas
35
Low viscosity due to Anderson Local.

AL effect renders infinite reduces viscosity
significantly even at weak coupling

Mechanism:coherent backscattering (CBS)
effect
Ginaaki, Hou , Ren PRD 77(2008)
36
Summary
Approches to calculate viscosity




Kubo formula: via correlation functions
of currents
Transport theory: Boltzmann Eqs. (for
weak scattering)
ADS/CFT(strongly coupled)
Lattice calculation (noisy)
37

Thanks
38
Renormalized diffusion
39
Weak Localization (WL)



Anderson proposed (‘58) that electronic diffusion
can vanish in a random potential (AL)
Experiments detected ( Ishimaru 1984,Wolf Maret
1985)
Mechanism:coherent backscattering (CBS) effect
after a wave is multiply scattered many times， its phase
coherence is preserved in the backscattering direction，
the probability of back scattering is enhenced via
constructive interference
40
Viscosity with random medium
System:
quasi-particles
in random potential
Candidate disorder in sQGP ?
1. The islands of heavy state; bound states (Shuryak);
2. The reminiscent of confinement vaccum, say the
domain structure of 't Hooft's monopole
condensation;
3. The disoriented chiral condensate (DCC);
4. CGC
41
42
Response function
43
BS Eq. In Diagrams
44

Localization length

Itinerant states
---- Localized States
45
II Some applications to N=4 SUSY YM Plasma:
Equation of state in strong coupling:
Plasma temperature = Hawking temperature
Near Schwarzschild horizon
z  z h (1   2 )
ds2  d 2 
  1
4 2 2 1
2

dt

d
x
z h2
z h2
Continuating to Euclidean time, t  it
 2 
2d polarcoordinates   , t 
 zh 
To avoid a conic singularity at   0 , the period of t  pz h
4
1
ds  d  2  2 dt 2  2 dx 2
zh
zh
2
2
Recalling the Matsubara formulation
1
T
pz h
46
Free energy = temperature X (the gravity action without metric fluctuations)
E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/9803131.
Consider a 4D Euclidean space of spatial volume V_3 at z  
The EH action of AdS-Schwarzschild:
zh

1
dz V3   1
1

I EH 0  
V3 (20  12)  dt  5 
 4 
4

16pG5
8pG5  
zh 
0
 z
The EH action of plain AdS
I
( 0)
EH
 
1
dz V3   1
0  
V3 (20  12)  dt 5 
4
16pG5
8
p
G
z

5
0

----- To eliminate the conic singularity,   pz n
----- To match the proper length in Euclidean time

4 
( 0)
0
    f     1  4   I GH 0  I GH
2
z
h 

Plasma free energy:
F
1
lim

I 0  I
( 0)
EH
V3
p2 2 4
0  

N c T V3
8
16pG5 z h4
 0

Plasma entropy:
p2 2 3
 F 
S  
N c T V3
 
2
 T V3
EH

47
Bekenstein-Hawking entropy:
Gubser, Klebanov & Pest, PRD54, 3915 (1996)
S BH
1
1 horizonarea
  (horizonarea) measuredin Planckunits  
8
4
4
l
P
1
where l P  10d PlancklengthG 8
------ The metric on the horizon：
ds2 
1
2
2
d
x

d

5
z h2
T hehorizonarea 
V3
 ( thesolid angle of S 5 )  p 6T 3V3
3
zh
------ The gravitational constant of the dual:
 S BH 
1 2 2 3
p N c T V3  S plasma
2
G10  l 
8
P
p4
2 N c2
agree with the entropy extraced from the gravity action.
48
The ratio 3/4:
The plasma entropy density at N c   and   
s  S / V3 
1 2 2 3
Nc p T
2
The free field limit:
the contents of N=4 SUSY YM
number
gauge potential
1
real scalars
6
6
1 2 2 3 1 2 2 3
Nc p T  Nc p T
30
5
Weyl spinors
4
8
7
7 2 2 3
N c2p 2T 3 
Nc p T
240
30
entropy density
1 2 2 3
Nc p T
30
2 2 2 2
Nc p T
3
s
3

 0.75
(0)
s
4
s (0) 
s
 0 .8.
The lattice QCD yields
s0
49
Shear viscosity in strong coupling:
Policastro, Son and Starinets, JHEP09, 043 (2002)
Kubo formula
1
R
   lim 0 Im G xy, xy ( ,0)

G
R
xy, xy
( , q)   dtdxe
i t  i q  x
 (t )  Txy ( x),Txy (0) 
where
50
Gravity dual:
the coefficient of
term of the gravity action
hxy2
pT
1
2
2
2

ds 
 fdt  dx   2 du
u
4u f
2
2
2
2
z
u 2
zh
f  1 u
2
0  u 1
51
The metric fluctuation
pT
1 2
2
2
 

ds 
 fdt  dx   2 du  h (t, z, u)dx dx
u
4u f
2 2
2
Substituting into Einstein equation
R  4g   0
 
 
g  0
 
x 
 g x 
1
and linearize
u
where   h  2 2 hxy
p T
x
y
The Laplace equation of a scalar field
52
Calculation details:
------ Nonzero components of the Christofel (up to symmetris):


ttu  2p 2T 2 u 4  1
utt 
1 2 f 
uuu     
2u f 
iju  2p 2T 2 f ij
1 f  1
  
2  f u 
uji  
1
 ij
2u
i , j  x, y , z
------ Nonzero components of the Ricci tensor:
4p 2T 2
Rtt 
f
u
4p 2T 2
Rij  
 ij
u
Ruu  
1
u2 f
Linear expansion:
     
 xyt 
withnonzerocomponents(up to symmetries) :
1

2f
1
2
 xyz    , z
1
2
1
2
 ytx   xty  
 yzx   xzy   , z
R   R   r 
rxy  
r x y  4h x y  
1
2pT 
1
2pT 
2
2
 xyu  2p 2T 2 f   u , u 
1
2
 yux   xuy   , u
with the only nonzero component
 u

 f

    u , zz   2u 3
  , u   4
u  u

 f

 u

 f
1
 

  
    u , zz   2u 3   , u   

g
g


 
f

u
u

x

x
2

g






53
The solution:
 (t , z, u)  (1  u)
i
 ˆ
2
(1  u )
1
 ˆ
2
 (u)e i ( qz t )
where ˆ 

q
qˆ 
2pT
2pT
 2
d 2
1 i
ˆ 2
2 d
2
u (1  u ) 2   1  (1  i)ˆ u  1  1  i ˆ u
  ˆ  qˆ 
ˆ  i
du 
2
4
du
2



u   0

Heun equation (Fucks equation of 4 canonical singularities)
------trivial when energy and momentum equatl to zero;
------low energy-momentum solution can be obtained perturbatively.

The boundary condition at horizon: u  1
i
 ˆ

i ( qz t )
(1  u ) 2 e
 (t , z, u ) ~ 
i
(1 - u ) 2 ˆ e i ( qz t )

incomingwave  retardedcorrelator
outgoingwave  advancedcorrelator
The incoming solution at low energy and zero momentum:
 (t , z, u)  (1  u)
i
 ˆ
2
ˆ 1  u


1

i
ln
 O(ˆ 2 ) e it

2
2


54
T hequadratic termof I EH ( )  I GH ( )
1
1
f
  N c2p 2T 4  du d 4 x
8
u
0

1
  
 f  
2 2 4


V
N
p
T
lim




4
c
u 0 

u
8
u

u




2
i
1
V4 N c2pT 3  V4 G xyR , xy ( ,0)
16
2
i
 G xyR , xy ( ,0)   N c2pT 3
8
1
8
  N c2pT 3
Viscosity ratio:
V_4 = 4d spacetime volume

s
Elliptic flow of RHIC:

1
 0.08
4p

s
 0.1
Lattice QCD: noisy
55
III. Remarks:
N=4 SYM is not QCD, since
1). It is supersymmetric
2). It is conformal ( no confinement )
3). No fundamental quarks
---- 1) and 2) may not be serious issues since sQGP is in
the deconfined phase at a nonzero temperature. The
supersymmetry of N=4 SYM is broken at a nonzero T.
---- 3) may be improved, since heavy fundamental quarks
may be introduced by adding D7 branes. ( Krach & Katz)
Introducing an infrared cutoff ---- AdS/QCD:
I EH  
1
dz d 4 x g e  R  12

16pG5
where thedilatonfield   cz 2


Karch, Katz, Son & Stephenov
1
dt 2  dx 2  dz 2
2
z
----- Regge behavior of meson spectrum ---- confinement;
----- Rho messon mass gives c  338MeV;
----- Lack of string theory support.
ds2 
56
Deconfinement phase transition:
Hadronic phase:
I EH
hadronic

with ds 2 
Plasma phase:
I EH
plasma

wit h ds 2 
Herzog, PRL98, 091601 (2007)
1
4
 cz 2
R  12
dz
d
x
g
e
16pG5  

1
2
2
2
dt

d
x

dz
z2

1
4
 cz 2
R  12
dz
d
x
g
e
16pG5  



1
4 4 4
2
2
4 4 4
1

p
T
z
dt

d
x

1

p
T z
2
z

1
dz

Hawking-Page transition:
I EH
hadronic
 I EH
plasma
 Tc  0.4917 c  191MeV
2
---- First order transition with entropy jump  N c
---- Consistent with large N_c QCD because of the liberation
of quark-gluon degrees of freedom.
57
58
59
60
Epilogue




AdS/CFT gives insights into physics of thermal
gauge theories in the nonperturbative regime
Generic hydrodynamic predictions can be used to
check validity of AdS/CFT
General algorithm exists to compute transport
coefficients and the speed of sound in any gravity
dual
Model-independent statements can presumably be
checked experimentally
61
62
63
Mechanisms for Low viscosity

Large cross-section, strong coupling

Anomalous viscosity: turbulence
Soft color fields generate anomalous
transport coefficients, which may give the
medium the character of a nearly perfect fluid
even at moderately weak coupling
M. Asakawa, S.A. Bass, B.M., hep-ph/0603092, PRL
See Abe & Niu (1980) for effect in EM plasmas
64