Transcript New Directions” - Duke University

Why is the
Quark-Gluon Plasma
a “Perfect” Liquid ?
Berndt Mueller – YITP / Duke
RIKEN Workshop
Special thanks
to M. Asakawa
and S.A. Bass
8-9 July 2006
1
Lecture I

What does Lattice QCD tell us about the QGP ?

What do RHIC experiments tell us about the QGP ?

What is a “perfect” fluid ?

What are the origins of viscosity ?
2
What does Lattice QCD
tell us about the QGP ?
A: So far, a lot about thermodynamic properties and
response to static probes, a little bit about spectral
functions, (almost) nothing about transport properties.
3
Lattice - EOS

2
30
F. Karsch et al.

RHIC
Indication
of weak or
strong
coupling?
# d.o.f .:
45s
 2 3
2 T
4
Phase coexistence
Study of LQCD at fixed
baryon density r = B/V
De Forcrand & Kratochvila
hep-lat/0602024
5
Lattice - susceptibilities
 XY
2

ln Z (T , i )  XY  X Y
 X Y
XS   xi si ni
C XS  #
XS  X
S2  S
S
2
i
R. Gavai & S. Gupta, hep-lat/0510044
pQGP
6
Heavy quark potentials
Important for insight into medium effects on J/Y, 
Effective coupling as(r,T)
Kaczmarek et al.
Color singlet potential
Kaczmarek et al.
7
Lattice – spectral functions
Spectral functions via analytic
continuation using the maximum
entropy method: J/Y, etc.
Asakawa & Hatsuda
(At present only for quenched QCD.)
Karsch et al.
J/Y may survive up to 1.5 – 2 Tc !
8
What do RHIC experiments
tell us about the QGP ?
A: So far, a lot about transport properties, a little bit
about thermodynamic properties, and (almost) nothing
about spectral functions and response to static probes.
9
The real road to the QGP
…is the
Relativistic Heavy Ion Collider
STAR
10
Space-time picture
Bjorken formula
Pre-equil. phase
eq
dN ( ) / dy
s ( )
dV ( ) / dy
(dN / dy )final

2
R 
s( 0  1 fm/c)  30/fm
Liberation of
saturated low-x
glue fields (CGC)
or T ( 0 )  275 MeV
in Au+Au (200 GeV)
11
3
RHIC results
Important results from RHIC:







Chemical (flavor) and thermal equilibration
Jet quenching = parton energy loss, high opacity
Elliptic flow = early thermalization, low viscosity
Collective flow pattern related to valence quarks
Strong energy loss of c and b quarks
Charmonium suppression not significantly increased
compared with lower (CERN) energies
Photons unaffected by medium at high pT, medium
emission at low pT in agreement with models
12
0

vs. g in Au+Au (vs. p+p)
Yield in A+A
No suppression for photons
Suppression of hadrons
2
d N AA / dpT dy
RAA ( pT ) 
2
TAA  d  NN / dpT dy 
Area density
of p+p coll’s
in A+A
Cross section
in p+p coll’s
Without nuclear effects:
RAA = 1.
13
Photons from the medium
D’Enterria, Peressounko nucl-th/0503054
0 = 0.15 fm/c, T = 570 MeV
Experiment uses internal conversion of g
Turbide, Rapp, Gale PRC 69 014903 (2004)
0 = 0.33 fm/c, T = 370 MeV
14
Hard Probes 2006, June 15, 2006 – G. David, BNL
Radiative energy loss
Radiative energy loss:
q
q
dE / dx
r L kT
2
L
Scattering centers = color charges
q
q
Density of
scattering centers
g
d

2
qˆ  r  q dq
 r kT 
2
dq
f
2
2
Scattering power of
the QCD medium:
2
Range of color force
15
q-hat at RHIC
Eskola et al.
RHIC
RHIC data
?
sQGP?
QGP
“Baier plot”
Pion gas
Cold nuclear matter
16
Collision Geometry: Elliptic Flow
Reaction
plane
 Bulk evolution described by
relativistic fluid dynamics,
 F.D assumes that the medium is
in local thermal equilibrium,
 but no details of how equilibrium
was reached.
 Input: e(x,i), P(e), (h,etc.).
z
y
x
Elliptic flow (v2):
• Gradients of almond-shape surface will lead
to preferential expansion in the reaction plane
• Anisotropy of emission is quantified by 2nd
Fourier coefficient of angular distribution: v2
 prediction of fluid dynamics
17
Elliptic flow is created early
time evolution of the energy density:
initial energy density distribution:
spatial
eccentricity
momentum
anisotropy
P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000) 054909
Model calculations suggest that flow anisotropies are generated at the earliest
stages of the expansion, on a timescale of ~ 5 fm/c if a QGP EoS is assumed.
18
v2(pT) vs. hydrodynamics
Failure of ideal
hydrodynamics
- tells us how
hadrons form
Mass splitting characteristic
property of hydrodynamics
19
Quark number scaling of v2
In the recombination regime, meson and baryon v2 can be obtained
from the quark v2 :
 pt 
v  pt   2v  
 2
M
2
q
2
 pt 
v  pt   3v  
 3
B
2
q
2
qqq
T,,v
qq
 Emitting medium is composed of
unconfined, flowing quarks.
20
From QGP to hadrons
Full 3-d Hydrodynamics
UrQMD
Hadronization
hadronic
rescattering
Cooper-Frye
formula
QGP evolution
Monte Carlo
TC
Nonaka & Bass, nucl-th/0510038
Hirano et al. nucl-th/0511046
TSW
Low (no) viscosity
t fm/c
High
viscosity
Agreement
with data for
hQGP = 0
21
Bounds on h from v2
Relativistic viscous
hydrodynamics:
  T   0 with
T

 
 (e  P)u u  Pg

 


 h ( u   u  trace)
Boost invariant hydro with T00 ~ 1
requires h/s ~ 0.1.
D. Teaney
s /  0  h / s
N=4 SUSY YM theory (g2Nc  1):
h/s = 1/4 (Policastro, Son, Starinets).
Absolute lower bound on h/s ?
QGP(T≈Tc) = sQGP ?
h/s = 1/4 implies f ≈ 0.3 d !
22
Ultra-cold Fermi-Gas
• Li-atoms released from an optical trap
(J. Thomas et al./Duke) exhibit elliptic
flow analogous to that observed in
relativistic heavy-ion collisions
23
What is an
“ideal” or “perfect”
liquid ?
24
Ideal gas vs. perfect liquid

An ideal gas is one that has strong enough interactions to
reach thermal equilibrium (on a reasonable time scale), but
weak enough interactions so that their effect on P(n,T) can
be neglected.


This ideal can be approached arbitraily by diluting the gas and
waiting very patiently (limit t   first, then V  ).
A perfect fluid is one that obeys the Euler equations, i.e. a
fluid that has zero viscosity and infinite thermal conductivity.

There is no presumption with regard to the equation of state.
 Even an imperfect fluid obeys the Euler equations in the limit of
negligible velocity, density, and temperature gradients.
25
What is viscosity ?
Shear and bulk viscosity are defined as coefficients in the
expansion of the stress tensor in gradients of the velocity field:
Tik  e ui uk  P  ik  ui uk   h  i uk   k ui  23  ik   u     ik   u
Microscopically, h is given by the rate of momentum transport:
h  np f 
1
3
p
3 tr
Unitarity limit on cross sections suggests that h has a lower bound:
4
 tr  2
p
p3
 h
12
26
Viscosity of plasmas
Shear viscosity of supercooled one-component plasma fluids:
h
h 
mnP a 2
*
Ichimaru
e2
Interaction measure  
aT
27
Lower bound on h/s ?
Argument [Kovtun, Son & Starinets, PRL 94 (2005) 111601] based on
duality between thermal QFT and string theory on curved
background with D-dimensional black-brane metric, e.g.:
2
r
ds  2
R
2
  r  2
 R
2
  1   dt   dxi   2
i 1
  r 
 r
4
0
4
3
2
1
 r 
2
1

dr


 r 
4
0
4
Kubo formula for shear viscosity:
1
3
it
h  lim
dt d x e Txy  t , x  , Txy  0, 0  

 0 2
Dominated by absorption of (thermal) gravitons by the black hole:
 abs   
8 G
3
it
Txy  t , x  , Txy  0, 0   
dt
d
x
e
 a (horizon area)
 0


 abs (0)
a
s
a
Therefore:
h


because s 
16 G 16 G 4
4G
28
Lower bound on h/s – ctd.
A heuristic argument for (h/s)min is obtained using s ~ 4n :
 f  1  e 
h  n  p v     12 s   f
n
 v 
1
3
But the uncertainty relation dictates that f (e/n)  h, and thus:
h
12
s
4
s
(It is unclear whether this relation holds in the nonrelativistic domain, where
s/n can be much larger than 4. But is obeyed by all known substances.)
For N=4 susy SU(Nc) Yang-Mills the bound is saturated at strong coupling:

s 
135  (3)
h
1

3/
2
4  8 g 2 N 
c





29
Exploring strong coupling

Ability to perform analytical strong coupling calculations in
N=4 susy SU(Nc) YM and success with h have motivated
other applications:







Equation of state [Gubser, Klebanov, Tseytlin, hep-th/9805156]
Spectral densities [Teaney, hep-ph/0602044]
Jet quenching parameter [Liu, Rajagopal, Wiedemann, hep-ph/0605178]
Heavy quark energy loss [Herzog, Karch, Kovtun, Kozcaz, Yaffe,hepth/0605158]
Heavy quark diffusion [Casalderrey-Solana, Teaney, hep-ph/0605199]
Drag force on heavy quark [Gubser, hep-th/0605182 ]
…and continuing!
30
Some results from duality
1
r0 
T

3
45  (3)

Equation of state: e  e 0 

3/
2
 4 32  g 2 N 
c



s 
135  (3)

Shear viscosity: h 
1


4   8 g 2 N 3/ 2
c


(3+1)-D world





1.7652...
2
3
Jet quenching parameter: qˆ 
g Nc  T 1 


3/
2
  g2N 
2  54 
c

dp
 2 1/ 2 2 p
Heavy quark drag force:
   g Nc  T
dt
2
m
 3/ 4  34 
r0
1/ 2
horizon




31
Lecture II

Does “perfect” fluidity imply “strong coupling” ?

What is “anomalous” viscosity ?

Derivation of the anomalous viscosity

Manifestations of anomalous QGP transport processes
32
Today…
…we ask the question:
Is strong coupling really
necessary for small h/s ?
33
What is viscosity ?
Shear and bulk viscosity are defined as coefficients in the
expansion of the stress tensor in gradients of the velocity field:
Tik  e ui uk  P  ik  ui uk   h  i uk   k ui  23  ik   u     ik   u
Microscopically, h is given by the rate of momentum transport:
h  13 np f 
 tr
a s2
p
2
p
3 tr
4
  tr  2
p
p3
h
12
a small value of h implies strong coupling !?!
34
Stellar accretion disks
“A complete theory of accretion disks requires a knowledge of the viscosity.
Unfortunately, viscous transport processes are not well understood. Molecular viscosity
is so small that disk evolution due to this mechanism of angular momentum transport
would be far too slow to be of interest. If the only source of viscosity was molecular,
then  ~ h/r ~  vT, where  is the particle mean free path and vT is the thermal
velocity. Values appropriate for a disk around a newly formed star might be r ~ 1014 cm,
n ~ 1015 cm-3,  ~ 10-16 cm2, so that  ~ 10 cm, and vT ~ 105 cm/s . The viscous
accretion time scale would then be r2/(12) > 1013 yr! Longer by a factor of 105 - 106
than the age conventionally ascribed to such disks. Clearly if viscous accretion
explains such objects, there must be an anomalous source of viscosity. The same
conclusion holds for all the other astronomical objects for which the action of accretion
disks have been invoked.”
(From James Graham – Astronomy 202, UC Berkeley)
http://grus.berkeley.edu/~jrg/ay202/lectures.html
The solution is: String theory?
Unfortunately, NO.
35
Anomalous viscosity
B
Magnetic tension T    B 2  /(4 )
Differentially rotating disc
with weak magnetic field B
shows an instability
(Chandrasekhar)
“Anomalous”, i.e.
non-collisional
viscosity
Spontaneous angular momentum
transfer from inner mass to outer mass
is amplified by interaction with the
rotating disk and leads to instability
(Balbus & Hawley – 1991).
36
Anomalous viscosity:
A ubiquitous phenomenon
37
Anomalous viscosity on the WWW
Google search:
Results 1 - 10 of about 322,000 for
anomalous viscosity. (0.22 seconds)
Chaotic Dynamics, abstract chao-dyn/9509002
Anomalous Viscosity, Resistivity, and Thermal Diffusivity of the Solar Wind Plasma
Authors: Mahendra K. Verma (IIT Kanpur, India)
In this paper we have estimated typical anomalous viscosity, resistivity, and thermal difffusivity of the
solar wind plasma. Since the solar wind is collsionless plasma, we have assumed that the dissipation
in the solar wind occurs at proton gyro radius through wave-particle interactions. Using this dissipation
length-scale and the dissipation rates calculated using MHD turbulence phenomenology [Verma et al.,
1995a], we estimate the viscosity and proton thermal diffusivity. The resistivity and electron's thermal
diffusivity have also been estimated. We find that all our transport quantities are several orders of
magnitude higher than those calculated earlier using classical transport theories of Braginskii. In this
paper we have also estimated the eddy turbulent viscosity.
38
Anomalous viscosity - origins
39
Anomalous viscosity - usage

Plasma physics:


Astrophysics - dynamics of accretion disks:


A.V. = large viscosity induced in nearly collisionless plasmas by
long-range fields generated by plasma instabilities.
A.V. = large viscosity induced in weakly magnetized, ionized
stellar accretion disks by orbital instabilities.
Biophysics:

A.V. = The viscous behaviour of nonhomogenous fluids or
suspensions, e.g., blood, in which the apparent viscosity
increases as flow or shear rate decreases toward zero.
(From: http://www.biology-online.org/dictionary)
40
Can the QGP viscosity be anomalous?

Can the extreme opaqueness of the QGP (seen in experiments) be
explained without invoking super-strong coupling ?

Answer may lie in the peculiar properties of turbulent plasmas.

Plasma “turbulence” = random, nonthermal excitation of coherent
field modes with power spectrum similar to the vorticity spectrum in a
turbulent fluid [P(k) ~ 1/k2].

Plasma turbulence arises naturally in plasmas with an anisotropic
momentum distribution (Weibel-type instabilities).

Expanding plasmas (such as the QGP at RHIC) always have
anisotropic momentum distributions.

Soft color fields generate anomalous transport coefficients, which
may give the medium the character of a nearly perfect fluid even at
moderately weak coupling.
41
QGP viscosity – collisions
Baym, Heiselberg, ….
Classical expression for shear viscosity:
Danielewicz & Gyulassy,
Phys. Rev. D31, 53 (85)
h  np f
1
3
Collisional mean free path in medium:

(C )
f
  n tr 
1
Transport cross section in QCD medium:
5 2
 tr  2 a s I a s 
p

1 
I a s   1  2a s  ln 1    2
 as 
Collisional shear viscosity of QGP:
T
9s
hC 

 tr 100a s2 ln a s1
42
QGP viscosity – anomalous
Classical expression for shear viscosity:
B
h  13 np f
a
p
p
Momentum change in one coherent domain:
rm
p  gQ a B a rm
Anomalous mean free path in medium:
 (f A)  rm
p2
 p 
2
p2
 2 2 2
g Q B rm
Anomalous viscosity due to random color fields:
3
9
sT
np 3
hA  2 2 2
 2 42 2
3g Q B rm g Q B rm
43
Color instabilities
Spontaneous generation of color fields requires infrared instabilities.
Unstable modes in plasmas occur generally when the momentum
distribution of a plasma is anisotropic (Weibel instabilities – 1959).
Such conditions are satisfied in HI collisions:
py
Longitudinal expansion locally “red-shifts” the
longitudinal momentum components of released
small-x gluon fields (CGC) from initial state:
px
beam
pz
 px 
2

 p 
y
2
Q
2
s
 pz 
2
for 
1
Qs
In EM case, instabilities saturate due to effect on
charged particles. In YM case, field nonlinearities
lead to saturation (competition with Nielsen-Olesen
instability?)
44
Weibel (two-stream) instability
v
v
rv
rv
B
45
HTL formalism
Nonabelian Vlasov equations
describe interaction of “hard” and
“soft” color field modes and
generate the “hard-thermal loop”
effective theory:
“soft”
“hard”
k ~ gT (gQs)
k ~ T (Qs)
D F
dp 
 gQ a F a u
d
dQ a
 gf abc Ab u Q c
d

J ( x) 

 gJ
  d Q ( )u
i

i

( )  ( x  xi ( ))
i

ˆ light-like
ui  (1, v)
Effective HTL theory permits systematic
study of instabilities of “soft” color fields.
LHTL

p
p
dp
g Nc
a
a
a
b
1
 4 F F  2  f ( p) F
F
p
( p  D)2ab
2
46
HTL instabilities
Find HTL modes for axisymmetric f ( p )  f eq

p 2   ( p  n )2

For any   0 there always exist modes with  ( k )  i( k ) for some k .
For   0 (oblate), there exist two such unstable modes;
for - 1    0 (prolate), only one unstable mode exists.
=10
Wavelength and growth rate of unstable modes
can be calculated perturbatively:
q=/8
g2
3
d
p
2
2
k  mD  2 
f ( p)  (k / mD ;  )
3
 (2 ) p
Mrowczynski, Strickland et al., Arnold et al.
47
From instability to “turbulence”
Non-abelian growth
Kolmogorov-type power spectrum of
coherent field excitations
[Arnold, Moore, Yaffe, hep-ph/0509226]
Exponential growth
saturates when
B2 > g2 T4.
Quasi-abelian growth
Turbulent power
spectrum
48
Space-time picture
Color correlation
length
M. Strickland, hep-ph/0511212
Time
Nonabelian
Quasiabelian
Noise
Length (z)
49
Anomalous viscosity
Formal derivation
(Sorry – using Chapman-Enskog)
50
Expansion  Anisotropy
Perturbed equilibrium distribution:
QGP
X-space
f ( p)  f 0 ( p) 1  f1 ( p) 1  f 0 ( p)  
f 0 ( p)  exp[u p  / T ]
For shear flow of ultrarelat. fluid:
QGP
P-space
5h / s i j 1
f1 ( p) 
p p  3  ij   u ij
2 
E pT
 u ij  12  iu j   j ui   13  ij   u
Anisotropic momentum distributions generate instabilities of soft field
modes. Growth rate  ~ f1(p).
 Shear flow always results in the formation of soft color fields;
 Size controlled by f1(p), i.e. (u) and h/s.
51
Turbulence  Diffusion
Vlasov-Boltzmann transport of thermal partons:


p
 r  F  p  f (r , p, t )  C  f 
 
 t E p

with Lorentz force
F  gQ
a
E
a
 v B
a

a
E ,B
a
r (t ')
r (t )  r
Assuming E , B random  Fokker-Planck eq:


p
 r   p D ( p )  p  f (r , p, t )  C  f 
 
 t E p

Diffusion is dominated by
chromo-magnetic fields:
with diffusion coefficient
t
Dij ( p ) 


dt ' Fi  r (t '), t '  F j  r , t  .
2
dt
'
B
(
t
')
B
(
t
)

B
m

52
Shear viscosity
Take moments of


p
 r   p D( p)  p  f (r , p, t )  C  f 
 
 t E p

with pz2
4
1
Nc g B  m
g
ln
g
1
1
2
 O 1 2
 O 10 


3
3
h
Nc  1
sT
T
h A hC
1
2
2
M. Asakawa, S.A. Bass, B.M., hep-ph/0603092
See Abe & Niu (1980) for effect in EM plasmas
53
ηA - the feedback loop
p
• Longitudinal flow induces momentum anisotropy:
1
2Tzz
Txx  Tyy
 10
h u
pz
s T
 Anisotropy grows with shear viscosity
_η
• Soft color fields are proportional to Δ:
B
2
2
gT
3
h
s
• The anomalous viscosity is inversely proportional
3
to B2:
Self-consistency
sT
1
hA
g 2 B2  m
h
• Shear viscosity η stabilizes due to:
hA
s
1
h

1

 T
 3
 g u
1/ 2



1
h A hC
54
Time evolution of viscosity
Initial
state
CGC ?
QGP and
hydrodynamic
expansion
h hA
viscosity:
??
Hadronization
h hC
Hadronic phase
and freeze-out
h hHG
Smallest viscosity dominates in system with several sources of viscosity
Anomalous viscosity
hA
s
 1
 3
 g u
1/ 2



Collisional viscosity
hC
36

s 25 g 4 ln a s1
55
Manifestations?

Possible effects on QGP probes:
Jet
Jet


Longitudinal broadening of jet cones
(observed – “ridge”)
Anomalous diffusion of charm and
bottom quarks (observed)
B
x
y
z


Synchrotron-style radiation of soft,
nonthermal photons ?
Au+Au 20-30%
Field induced quarkonium
dissociation ?
b
a
c

No unstable modes for quarks:
quasi-particle picture of QGP is
compatible with low viscosity
c
b
56
Summary
Conclusion:
Because the matter created in heavy ion collisions expands rapidly,
it forms a turbulent color plasma, which has an anomalously small
shear viscosity. Transport phenomena involving quarks and gluons
are strongly influenced by the turbulent color fields, especially at
early times, when the expansion is rapid.
57
Additional slides
58
Anisotropic HTL modes
HTL effective action for general parton momentum distribution f ( p ):
LHTL
a 
1 a a g 2
d3 p
p
p
a
b
 F F 
f
p
F
F


a

4
2  (2 )3 E p
p

D

ab
For axisymmetric f ( p )  f eq

p   ( p  n)
2
2

the gluon propagator (with cos q  kˆ  n ) can be written as:
2
2
2

 ij (k ,  , q )   Aij  Cij   A  (k    a  g ) Bij  (    )Cij   Dij   G
Collective modes are found as poles of  ij :
2
2

(
k
,

,
q
)

k


a  0
 A

1
  G (k ,  ,q )
1
 (k 2   2  a  g )(    2 )   2 sin 2 q  0
For small  the functions a , ,g , of k ,  ,q are explicitly known
up to O( ) [Romatschke & Strickland, hep-ph/0304092].
59
Shear perturbation
Shear flow:
Bjorken flow:

i j
1
f1 ( p) 
p
p


3  ij  (u )ij
2
E pT
 u ij
1
 diag  1, 1, 2  
3
with   5h / s
u 
1

 u ij


2
1
Drift term:   v  r  f 0 ( p)  f 0 (1  f 0 )
p
p

p


i j
3
E pT
 t

Vlasov term:  p D( p)  p f ( p) 
3Q 2  g 2 B 2  m
N
2
c
 1 E 3pT 2
f 0 (1  f 0 ) u  pi p j  13 p 2 
Collision term: C  f   f 0 (1  f 0 ) u  pi p j  13 p 2  I  ( p ) 
60
Estimate of hA
Unstable modes are soft: k
m
2
2
D
 measures momentum anisotropy

Characteristic domain size: rm
1/ 2
Saturation condition for unstable modes: g A
Energy density of turbulent fields:
B
k
k4
2
g
2
2
g Q
 gT 
1
2
2
 gT
B rm
2
2
4

3/ 2
 gT 
3
9s
Anomalous viscosity: h A
3 3/ 2
4g 
36 s
Compare with collisional viscosity: hC 
25 g 4 ln a s1
61
Turbulence  Diffusion
E a , Ba
V-B transport of thermal partons:
 v      igQ a Aa   F  p  f (r , p, Q, t )  C  f 


with Lorentz force
F  gQ
a
E
a
 v B
a
r (t ')
r (t )  r

Assuming E, B random, V-B eq.  F-P eq:
Most important for diffusion -


p
 r   p D( p)   p  f (r , p, t )  C  f 
 
 t E p

with past trajectory r (t '), so that r (t )  r
chromo-magnetic fields:
t
Dij ( p) 
 dt ' F  r (t '), t ' F  r , t  .
i

2
dt
'
B
(
t
'
)
B
(
t
)

B
m

 p  D( p)  p
j
2

g Q
2
B m
2
2  N  1 E
2
c
2
p
 p 
p
2

62
Shear viscosity
Take moments of


p
2







D
(
p
)

f
(
r
,
p
,
t
)

C
f
with
p

r
p
p
z
 

t
E


p
d3 p
16
2
 (2 )3 E pz v r f (r , p)  45 e u
2
m
d3 p
4

2
2
D
p

D
(
p
)

f
(
r
,
p
)

B
 m u
p
 (2 )3 E z p
15 T
  5h / s
hA
4
eT

s 15 mD2 B2  m
hep-ph/0603092
PRL (June30)
63
Diffusion  Viscosity
Diffusion corresponds to “anomalous” viscosity:
hA
4
eT

s 15 mD2 B2  m
But recall that B2 itself depends on anisotropy f1(p) ~ viscosity h !
Take
B
2
 b0 g e
2
h/s
T

4T
 
s  15b0 g 3 u
hA
u  h  h A and  m
1/ 2




h A u
1
:
mD
T u
g 3/ 2
64
ηB - the feedback loop
• Longitudinal flow induces momentum anisotropy:
p
2Tzz
8
8 h
1


Txx  Tyy 5T
T s
pz
 Anisotropy grows with shear viscosity
_η
• Soft color fields are proportional to Δ:
B
2
The Münchhausen
effect
gT 
2
3
• The anomalous viscosity is inversely proportional
3
to B2:
sT
hA
g 2 B2  m
• Shear viscosity η stabilizes due to:
1
h

1

1
h A hC
65
hB vs. hC
Compare anomalous viscosity
with HTL (weak coupling) result
for collisional viscosity
1/ 2

( N  1) T 
  O(1)
3 
s 
10 b0 N c g 
hB
2
c
1
g 3/ 2
hC
5
 4
s
g ln g 1
 hB indeed dominates at weak coupling !
hC
g
hB
4
g
3
g2T3h
T3
g2
2
g2
g2
 m ( gT )1
66