From
Quark-Gluon Plasma
to the
Perfect Liquid (II)
Berndt Mueller – Duke University
Krakow School of Theoretical
Physics
Zakopane, 14-22 June 2007
1
Part II
R esults from R H IC
2
The RHIC Facility
…or what a good deal of
M oney and P lanning C an B uy!
STAR
3
Main RHIC results
Important results from RHIC:







Chemical (flavor) and thermal equilibration
Elliptic flow = early thermalization, low viscosity
Collective flow pattern related to valence quarks
Jet quenching = parton energy loss, high opacity
Strong energy loss of c and b quarks
Charmonium suppression not strongly increased
compared with lower (CERN) energies
Photons unaffected by medium at high pT
4
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: ε(x,τi), P(ε), (η,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
5
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 time scale of ~ 5 fm/c if a QGP EoS is assumed.
6
v2(pT) vs. hydrodynamics
Failure of ideal
hydrodynamics
- tells us how
hadrons form
Mass splitting characteristic
property of hydrodynamics
7
Quark number scaling of v2
In the recombination regime, meson and baryon v2 can be obtained
from the quark v2 :
ć pt ö
v ( pt ) = 2 v ç ÷
č2ř
M
2
q
2
ć pt ö
v ( pt ) = 3v ç ÷
č3ř
B
2
q
2
S orensen plot
T,µ,v
qq
Em itting m edium is com posed of
unconfined,flow ing quarks.
8
Photons versus hadrons
No suppression for photons
Suppression of hadrons
9
Radiative energy loss
Radiative energy loss:
q
q
dE / dx : ρΛ κΤ
2
L
Scattering centers = color charges
q
q
Density of
scattering centers
g
δσ
µ
2
qö = ρ ňθ δθ
ş
ρσ
κ
=
Τ
2
λφ
δθ
2
Scattering power of
the QCD medium:
2
2
Range of color force
10
How large is q-hat?

Data are described by a large loss parameter for central collisions:
RHIC data
Loizides,
hep-ph/0608133
2
ö
q ≈ 5 − 20 Γ ες /φµ
sQGP
pQGP
R. Baier
Zhang, Owens,
Wang & Wang,
nucl-th/0701045
2
ö
q ≈ 1 − 2 Γ ες /φµ
Pion gas
Cold nuclear matter
Larger than expected from
perturbation theory ?
11
Part III
Tow ard the
“P erfect” Liquid
12
Ideal gas vs. perfect liquid

An ideal gas is characterized by interactions strong enough
to reach thermal equilibrium (on a reasonable time scale),
but weak enough to neglect their effect on P(n,T).


This ideal can be approached arbitrarily well by diluting the gas
and waiting very patiently ( limit t → ∞ first, then n → 0 ).
A perfect fluid is one that obeys the Euler equations, i.e. a
fluid that has negligible viscosity and infinite thermal
conductivity (relative to gradients).

There is no presumption with regard to the equation of state.
13
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 (d ik + ui uk ) - h (Ńi uk + Ń k ui - 23 d ik Ń ×u )+ Vd ik Ń ×u
Microscopically, η is given by the rate of momentum transport:
p
1
η ≈ 3 npλ f =
3σ tr
Unitarity limit on cross sections suggests that η has a lower bound:
4π
σ tr Ł 2
p
Ţ
p3
ηł
12π
14
Viscosity of materials
Tem perature dependence ofthe shearviscosity oftypical
fluids:
4π ×
QuickTimeŞ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
15
Lower bound on η/s ?
A heuristic argum entfor(η/s)m in is obtained using s ≈ 4 n :
ćl f ö
h » n ( p v )ç ÷ »
č v ř
1
3
1
12
će ö
s ç ÷t
čnř
f
The uncertainty relation dictates that τf (ε/n ) ≥ ,and thus:
h
h
ηł
s≈
s
12
4π
A llknow n m aterials obey this condition!
For N =4 S U (N c)S Y M theory the bound is saturated atstrong coupling:
é
ů
s ę
135 V(3)
ú
h =
1+
+
L
ú
4p ę (8 g 2 N )3/ 2
c
ë
ű
16
Bounds on η from v2
R elativistic viscous
hydrodynam ics:
∂ µ T µν = 0 with
T
µν
µ ν
= (ε + P)u u − Pg
µν
µ ν
ν
µ
+ η (∂ u + ∂ u − trace)
Boost invar’t hydro requires η/s ≈ 0.1.
Γs / τ 0 ≈ η / s
D. Teaney
η/s < 0.3 confirmed by 2-D viscous
hydro calculation (Baier & Romatschke).
N=4 SUSY YM theory (g2Nc  1):
η/s = 1/4π (Policastro, Son, Starinets).
Absolute lower bound on η/s ?
17
QGP viscosity – collisions
B aym ,H eiselberg,… .
Classical expression for shear viscosity:
h » npl
1
3
D anielew icz & G yulassy,
P hys.R ev.D 31,53 (85)
f
Collisional mean free path in medium:
l
(C )
f
= (ns
A rnold,M oore & Y affe
)
-1
tr
Transport cross section in QCD medium:
s
tr
5p 2
» 2 a s I (a s )
p
ć
1 ö
I (a s ) = (1 + 2a s )ln ç1 + ÷- 2
č as ř
Collisional shear viscosity of QGP:
T
9s
hC »
»
s tr 100pa s2 ln a s- 1
18
QCD matter
Low T (Prakash et al.)
using experimental
data for 2-body
interactions.
High T (Yaffe et al.)
using perturbative
QCD
R H IC
data
1/4π
19
Part IV
A nom alous V iscosity
20
Spontaneous color fields
C olorcorrelation
length
M .Strickland,hepph/0511212
Tim e
N onabelian
Q uasiabelian
N oise
Length (z)
21
QGP viscosity – anomalous
Classical expression for shear viscosity:
h » 13 npl
f
Momentum change in one coherent domain:
Dp » gQ a B a rm
Anomalous mean free path in medium:
l
( A)
f
» rm
p2
(Dp )
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
3 g Q B rm g Q B rm
22
Expansion ⇔ Anisotropy
Perturbed equilibrium distribution:
QGP
X -s p a c e
f ( p ) = f 0 ( p ) éë1 + f1 ( p ) (1 ± f 0 ( p ) )ůű
f 0 ( p ) = exp[- um p m / T ]
For shear flow of ultrarelat. fluid:
QGP
P -s p a c e
5h / s i j 1
f1 ( p ) =
p p - 3 d ij )(Ńu )ij
2 (
E pT
(Ńu )ij = 12 (Ńiu j + Ń j ui )-
1
3
d ij Ń ×u
A nisotropic m om entum distributions generate instabilities ofsoftfield
m odes.G row th rate Γ ~ f1(p).
 Shear flow alw ays results in the form ation ofsoftcolor fields;
 Size controlled by f1(p), i.e.(u)and η/s.
23
Turbulence ⇒ Diffusion
Vlasov-Boltzmann transport of thermal partons:
é¶
ů
p
×Ń r + F ×Ń p ú f (r , p, t ) = C [ f ]
ę +
ęë ¶ t E p
ú
ű
with Lorentz force
F = gQ
a
(E
a
+ v´ B
a
)
a
E ,B
a
r (t ')
r (t ) = r
Assuming E , B random Ţ Fokker-Planck eq:
é¶
ů
p
×Ń r - Ń p ×D( p ) ×Ń p ú f (r , p, t ) = C éë f ů
ę +
ű
¶
t
E
ęë
ú
p
ű
Diffusion is dominated by
chromo-magnetic fields:
with diffusion coefficient
t
Dij ( p ) =
ňdt ' Fi (r (t '), t ') Fj (r , t ) .
2
dt
'
B
(
t
')
B
(
t
)
ş
B
τm
ň
-Ą
24
Shear viscosity
Take m om ents of
∂
ů
p
⋅Ń r − Ń p ⋅D( p ) ⋅Ń p  f (r , p, t ) = C  f ů
ę +
ű
∂
t
E
ę
ű
p

Dij ( p) =
w ith p z2
τ
α
β
δτ
∋
Φ
ρ
(
τ
∋
),
τ
∋
Υ
(
ρ
,
ρ
)
Φ
ρ, τ)
) αβ
ϕ (
ň ι(
−Ą
ρα
ρα ρ ρα
Φ = γ ( Ε + ϖ× Β ) = χολορφορχε
2
4
-1
F tm
Nc
1
g
ln
g
1
1
-2
= O (1) 2
+ O (10 )
ş
+
3
3
h
N c - 1 sT
T
h A hC
+
+i
2
dt
'
F
(
t
')
F
(
t
)
=
F
τ m ş qö
ň i
M .A sakaw a,S .A .B ass,B .M .,
P R L 96:252301 (2006)
= jet quenching parameter !!!
P rog Theo P hys 116:725
(2007)
25
Who wins?
Smallest viscosity dominates in system with several sources of viscosity
Anomalous viscosity
ηΑ ć 1 
:  2

σ č γ Ńυ ř
Ńu : t
-1
3/5
Collisional viscosity
ηΧ
36π
≈
σ 50 γ 4 λν γ −1
® Anomalous viscosity wins out at small g and t
Estimate for turbulent color field intensity:
F
2
: t
- 3.1
26
Part IV
E xploring the
“P erfect” Liquid
27
Connecting jets with the medium
H ard partons probe the m edium via the density ofcolored scattering
centers:
qö = ρ ňθ2 δθ2 ( δσ / δθ2 )
Ifkinetic theory applies,therm algluons are quasi-particles that
experience the sam e m edium .Then the shearviscosity is:
p
η ≈ Cr p l f ( p) = C
In Q C D ,sm allangle scattering
dom inates:
W ith ⟨p ⟩ ~ T and s ≈ 4 ρ one
finds:
3
From R H IC
data:
3
s tr ( p)
4 qö
σ tr ( p) »
sör
η 5 T3
»
s 4 qö
M ajum der,B M ,
W ang,hepph/0703085
Τ
T0 ≈ 335 Μες , θ0 ≈ 1 − 2 Γ ες /φµ →
≈ 0.12 − 0.24
θ0
2
3
0
28
Strong vs. weak coupling
T3
A tstrong coupling,
qö
“blackness”.
(fπ/Τ)
is a m ore faithfulm easure ofm edium
N =4 S Y M
QCD
4
(ln T)
λ −2
-4
( 4π )
−1
5T 3
4 qö
η/s
xGπ ( ξ)ξ→0
λ −1/2
1
ln (T / Tc )
~1
ln (1 / λ )
29
Fate of the “lost” energy (I)
p+p
Away-side jet
Au+Au
Trigger jet
Lost energy of away-side jet is redistributed to angles away from
180° and low transverse momenta pT < 2 GeV/c (Mach cone?).
30
Fate of the “lost” energy (II)

Near side jet phenomenon:

Longitudinal broadening of jet cones.
“Ridge” contains all additional energy:
In-medium fragmentation
P ossible explanation:Longitudinal
diffusion ofradiated gluons in random ,
transverse color-m agnetic fields (A .
M ajum der,… )
Jet
Jet
B
η
φ
Finaldistribution ofinm edium radiated
gluons
x
y
z
31
Summary
The m attercreated in heavy ion collisions form s a highly color
opaque plasm a,w hich has an extrem ely sm allshearviscosity.This
(nearly) “perfectliquid ” behaves like a strongly coupled plasm a of
quarks and gluons,possibly due to the presence ofstrong turbulent
colorfields,especially atearly tim es w hen the expansion is m ost
rapid.
P enetrating probes,i.e.jets,heavy quarks,photons,are the best
w ay to furtherprobe this m edium .The extended dynam ic range of
R H IC IIand LH C ,togetherw ith detailed theoreticalm odeling and
sim ulations,w illallow us to quantitatively determ ine its essential
transportproperties.
32