early isotropization of the quark gluon plasmaepelbaum/santiago_29_10.pdf · santiago de...
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Early Isotropization ofthe Quark Gluon Plasma
Santiago de Compostela, 29th October 2013Thomas EPELBAUM
IPhT
THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 0 / 28
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OUTLINE
1 MOTIVATION
2 THEORETICAL FRAMEWORK
3 A PROOF OF CONCEPT: SCALAR FIELD THEORY
4 YANG-MILLS THEORY
5 CONCLUSION
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HEAVY ION COLLISIONS: THE GENERAL PICTURE
QGP
Hydrodynamics
viscousideal
Hadron gas
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HEAVY ION COLLISIONS: THE GENERAL PICTURE
0 100 200 300 400
NPart
0
0.02
0.04
0.06
0.08
0.1
v2
PHOBOS
CGC
η/s=10-4
η/s=0.08
η/s=0.16
η/s=0.24
[LUZUM, ROMATSCHKE (2008)]THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 1 / 28
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HEAVY ION COLLISIONS: THE GENERAL PICTURE
Viscous Hydrodynamics
I) Macroscopic theoryII) Few parameters: PL, PT , ε, ~uIII) Need input:
1) Equation of state f(PL, PT ) = ε2) Small anisotropy3) Initialization: ε(τ0), PL(τ0)? ...4) viscous coefficients: shear viscosity η,...5) Short isotropization time
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HEAVY ION COLLISIONS: THE GENERAL PICTURE
Viscous Hydrodynamics
I) Macroscopic theoryII) Few parameters: PL, PT , ε, ~uIII) Need input:
1) Equation of state f(PL, PT ) = ε2) Small anisotropy3) Initialization: ε(τ0), PL(τ0)? ...4) viscous coefficients: shear viscosity η,...5) Short isotropization timeNon
e ofthisiseasy
toget f
rom
QCD
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HEAVY ION COLLISIONS: THE GENERAL PICTURE
Early transition: the problem
CGC
Glasma Isotropization?Time scale?
Hydrodynamics
QGP
Huge anisotropy Small anisotropy(negative PL)
Long time puzzle: Does (fast) isotropization occur?
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2 THEORETICAL FRAMEWORKHow to deal with a Heavy Ion CollisionThe Color Glass CondensateThe Classical statistical approximation
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HOW TO STUDY THE TRANSITION?
Strongly coupled method: AdS/QCD?
SupersymmetricSU(N) gauge theory
Black hole
String theory
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HOW TO STUDY THE TRANSITION?
Weakly coupled method: QCD
SupersymmetricSU(N) gauge theory
Black hole
String theory
QCD
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HOW TO STUDY THE TRANSITION?
Weakly coupled QCD with only gluons
-310
-210
-110
1
10
-410
-310
-210
-110 1
-310
-210
-110
1
10
HERAPDF1.0
exp. uncert.
model uncert.
parametrization uncert.
x
xf
2 = 10 GeV2Q
vxu
vxd
xS
xg
H1 and ZEUS
-310
-210
-110
1
10
u
du
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HOW TO STUDY THE TRANSITION?
Weakly coupled method at dense regime:αs � 1 but fgluon ∼ 1
αs
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
Gluon saturation when emission = recombination⇒ Saturation scale Qs
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
u
du
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
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TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION
THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 4 / 28
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2 THEORETICAL FRAMEWORKHow to deal with a Heavy Ion CollisionThe Color Glass CondensateThe Classical statistical approximation
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
THE MAIN ASSUMPTIONS
• Fast gluons are "frozen" by time dilation.
• Described as static color sources J located on the light cone axis
• Small x→ Gluon saturation→ J ∼ Q3s α
−1/2s .
• Slow gluons are the standard gauge field Aµ ∼ Qs α−1/2s .
• System boost-invariant→ Aµ rapidity independant.
Langrangian of theory reads
L = −14FµνF
µν + JµAµ
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
Theoretical framework (Weakly coupled but strongly interacting)
x+x−
CGCJ− J+
E ,B
LO: ε =12
(~E
2+ ~B
2)
︸ ︷︷ ︸Classicalcolor fields
DµFµν = Jν︸︷︷︸
Color sourceson the light cone
[KRASNITZ, VENUGOPALAN (1998)]
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
ε = E2⊥ +B2
⊥ + E2L +B2
L
PT = E2L +B2
L
PL = E2⊥ +B2
⊥ − E2L −B2
L
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
0 0.5 1 1.5 2
g2µτ
0
0.2
0.4
0.6
0.8
[(g2µ
)4/g
2]
Bz
2
Ez
2
BT
2
ET
2
[LAPPI, MCLERRAN (2006)]
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
ε = E2⊥︸︷︷︸0
+ B2⊥︸︷︷︸
0
+E2L +B2
L
PT = E2L +B2
L
PL = E2⊥︸︷︷︸0
+ B2⊥︸︷︷︸
0
−E2L −B2
L
Initial Tµν is (ε, ε, ε,−ε)!
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
Strong anisotropy at early time
-0.05
0
0.05
0.1
0 0.5 1 1.5 2 2.5 3
g2τ P
L /
(g
2µ
)3,
g
2τ P
T /
(g
2µ
)3
g2µτ
τ PLτ PT
[GELIS, FUKUSHIMA (2012)]
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THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]
Strong anisotropy at early time
-0.05
0
0.05
0.1
0 0.5 1 1.5 2 2.5 3
g2τ P
L /
(g
2µ
)3,
g
2τ P
T /
(g
2µ
)3
g2µτ
τ PLτ PT
{∂τε+
ε+PLτ
= 0limτ→0+
ε = cst ⇒ PL = −ε
ε = 2PT + PL ⇒ PT = ε
[GELIS, FUKUSHIMA (2012)]
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THE COLOR GLASS CONDENSATE AT NLO
E2(x) = E2(x)︸ ︷︷ ︸LO
+12
∫~k
∣∣e~k(x)∣∣2︸ ︷︷ ︸NLO
+ · · ·
e~k(x) perturbation to E(x) created by a plane waveof momentum ~k in the remote past.
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THE COLOR GLASS CONDENSATE AT NLO
0 500 1000 1500 2000 2500 3000 3500
g2 µ τ
1e-13
1e-12
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
max
τ2 T
ηη /
g4
µ3 L
η
c0+c
1 Exp(0.427 Sqrt(g
2 µ τ))
c0+c
1 Exp(0.00544 g
2 µ τ)
[ROMATSCHKE, VENUGOPALAN (2006)]
Small Fluctuations grow exponentially (Weibel instability)
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THE COLOR GLASS CONDENSATE AT NLO
• Because of instabilities, the NLO correction eventually becomes as largeas the LO⇒ Important effect, should be included
• NLO alone will grow forever⇒ unphysical effect, should be taken care of
-40
-30
-20
-10
0
10
20
30
40
-20 0 20 40 60 80
time
PLO εLO
-40
-30
-20
-10
0
10
20
30
40
-20 0 20 40 60 80
time
PNLO εNLO
• Such growing contributions are present at all orders of the perturbativeexpansion
How to deal with them?
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2 THEORETICAL FRAMEWORKHow to deal with a Heavy Ion CollisionThe Color Glass CondensateThe Classical statistical approximation
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THE CLASSICAL-STATISTICAL METHOD
• At the initial time τ = τ0, take:
~E0(τ0,~x) = ~E0(τ0,~x) +∫~k
c~k~e~k(τ0,~x)
where c~k are random coefficients:⟨c~kc~k ′
⟩∼ δ~k~k ′
• Solve the Classical equation of motion DµFµν = Jν
• Compute⟨~E
2(τ,~x)
⟩, where 〈〉 is the average on the c~k (Monte-Carlo)
• One can show that this resums all the fastest growing terms at eachorder, leading to a result that remain bounded when τ→∞[GELIS, LAPPI, VENUGOPALAN (2008)]
This gives: LO+NLO+Subset of higer orders
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IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD
LO: Classical solution
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IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD
LO: Classical solution
NLO:Parabolic
approximationaround LO
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IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD
NLO:Parabolic
approximationaround LO
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IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD
Resummed:keep thecompletepotential
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IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD
Resummed:keep thecompletepotential
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IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD
Resummed:keep thecompletepotential
-200
-150
-100
-50
0
50
100
150
200
0 50 100 150 200 250
time
g=0.5 g=1 g=2 g=4 g=8 ε/3
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3 A PROOF OF CONCEPT: SCALAR FIELD THEORYThe TheoryNumerical results
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SCALAR FIELD THEORY
Adapted coordinate system to describe a Heavy Ion Collision?
x⊥
z
System boost invariant in z direction
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SCALAR FIELD THEORY
Proper time/rapidity coordinate system
η = cstτ = cst
τ0
x+x−
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SCALAR FIELD THEORY
The model
Initial conditions: classical statistical method
φ(τ0, x⊥,η) = ϕ0(τ0, x⊥) +∑k⊥,ν
cνk⊥eiνη aν,k⊥(τ0, x⊥)
Time evolution: Klein Gordon equation
[∂2
∂τ2 +1τ
∂
∂τ−∇2
⊥ −1τ2
∂2
∂η2
]︸ ︷︷ ︸
�
φ+g2
6φ3 = 0
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HOW INTERACTIONS COMPETE WITH EXPANSION?
Initial anisotropy
PT
PL
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HOW INTERACTIONS COMPETE WITH EXPANSION?
Interactions isotropize the system
PT
PL
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HOW INTERACTIONS COMPETE WITH EXPANSION?
Expansion dilutes the system
PT
PL
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HOW INTERACTIONS COMPETE WITH EXPANSION?
Expansion ≶ Interactions for realistic αs?
PT
PL
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3 A PROOF OF CONCEPT: SCALAR FIELD THEORYThe TheoryNumerical results
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TµνRESUM [DUSLING, TE, GELIS, VENUGOPALAN (2012]
×10-1
×100
×101
×102
×103
×104
×10-2 ×10-1 ×100 ×101
τ
PT(τ0=0.01)
ε(τ0=0.01)
PT(τ0=0.1)
ε(τ0=0.1)
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TµνRESUM [DUSLING, TE, GELIS, VENUGOPALAN (2012]
×10-3
×10-2
×10-1
×100
×101
0 50 100 150 200 250 300
τ
2PT + P
L
ε
PT
PL
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ε BEHAVIOUR
10-2
10-1
100
101
10 100
τ
τ-4/3
τ-1
2PT + P
L
ε
Bjorken Law: ∂τε+ ε+PLτ
= 0THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 15 / 28
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HYDRODYNAMICS: IDEAL AND VISCOUS
Equation of state: ε = 2PL + PT
IDEAL HYDRO
Isotropic system
Tµνideal = εuµuν − P(gµν − uµuν)
VISCOUS HYDRO
Anisotropic system
Tµν = Tµνideal + ηπµν
In our case
PT =ε
3+
2η3τ
PL =ε
3−
4η3τ
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HYDRODYNAMICS: IDEAL AND VISCOUS
Equation of state: ε = 2PL + PT
IDEAL HYDRO
Isotropic system
Tµνideal = εuµuν − P(gµν − uµuν)
VISCOUS HYDRO
Anisotropic system
Tµν = Tµνideal + ηπµν
In our case
PT =ε
3+
2η3τ
PL =ε
3−
4η3τ
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FIRST ORDER VISCOUS HYDRODYNAMICS
BJORKEN’s Law (coming from ∂µTµν = 0):
∂τε+ε+ PL
τ= 0 → ∂τε+
43ε
τ=
43η
τ2
assuming η = η0τ
and STEFAN-BOLTZMANN entropy s ≈ ε 34
∂τε+43ε
τ=
43η
s︸︷︷︸cte
ε34
τ2
At a given time, knowing ε, PT , PL and assuming• an EOS• STEFAN-BOLTZMANN entropy• η = η0
τ
• ηs = cte
gives a very simple hydro model.
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FIRST ORDER VISCOUS HYDRODYNAMICS
BJORKEN’s Law (coming from ∂µTµν = 0):
∂τε+ε+ PL
τ= 0 → ∂τε+
43ε
τ=
43η
τ2
assuming η = η0τ
and STEFAN-BOLTZMANN entropy s ≈ ε 34
∂τε+43ε
τ=
43η
s︸︷︷︸cte
ε34
τ2
At a given time, knowing ε, PT , PL and assuming• an EOS• STEFAN-BOLTZMANN entropy• η = η0
τ
• ηs = cte
gives a very simple hydro model.
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FIRST ORDER VISCOUS HYDRODYNAMICS
BJORKEN’s Law (coming from ∂µTµν = 0):
∂τε+ε+ PL
τ= 0 → ∂τε+
43ε
τ=
43η
τ2
assuming η = η0τ
and STEFAN-BOLTZMANN entropy s ≈ ε 34
∂τε+43ε
τ=
43η
s︸︷︷︸cte
ε34
τ2
At a given time, knowing ε, PT , PL and assuming• an EOS• STEFAN-BOLTZMANN entropy• η = η0
τ
• ηs = cte
gives a very simple hydro model.
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COMPARISON WITH HYDRO: ISOTROPIZATION
×10-3
×10-2
×10-1
×100
×101
0 50 100 150 200 250 300
τ
2PT + P
L
ε
PT
PL
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COMPARISON WITH HYDRO: ISOTROPIZATION
×10-3
×10-2
×10-1
50 100 150 200 250 300
τ
τ0=70 Field theory
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COMPARISON WITH HYDRO: ISOTROPIZATION
×10-3
×10-2
×10-1
50 100 150 200 250 300
τ
τ0=150 Field theory
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COMPARISON WITH HYDRO: ISOTROPIZATION
×10-3
×10-2
×10-1
50 100 150 200 250 300
τ
τ0=200 Field theory
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COMPARISON WITH HYDRO: VISCOSITY
PT − PL =2ητ
0.1
1
10
100
0 50 100 150 200 250 300
η /
s
τ
effective η / s
perturbation theory
AdS/CFT bound1
4π
∼ 104
g4
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COMPARISON WITH HYDRO: VISCOSITY
see also [ASAKAWA, BASS, MULLER (2006-07)]
0.1
1
10
100
0 50 100 150 200 250 300
η /
s
τ
effective η / s
perturbation theory
AdS/CFT bound1
4π
∼ 104
g4
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4 YANG-MILLS THEORYThe theoryNumerical results
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THE NLO SPECTRUM
• Need to know ~e~k(τ0,~x) at the time τ0 we start the numerical simulation
• For practical reasons, we must start in the forward light cone (τ0 > 0)
x− x+
?
~e~k(x) ∼t 7→−∞
eik.x
This can be done analytically [TE,GELIS 1307:1765]
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THE NLO SPECTRUM
Result of [TE,GELIS 1307:1765]
eiν~k⊥
= iν(Fi,− − Fi,+) eη
ν~k⊥(x) = Di (Fi,− − Fi,+)
with
Fi,+k (x) ∼ eiνη U
†1(~x⊥)
∫~p⊥
ei~p⊥·~x⊥ U1(~p⊥ +~k⊥)(
p2⊥τ
2k⊥
)iν[δij −
2pi⊥pj⊥
p2⊥
]ε
jkλ .
• U†1 depends on the color source J+ of the first nucleus
• Analogous formula for Fi,−.
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4 YANG-MILLS THEORYThe theoryNumerical results
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YM ON A LATTICE
Gauge potential Aµ → link variables (exact gauge invariance on the lattice)
aL aT
N
L
Lη
x
y
Numerical parameters• Transverse lattice size L = 64, transverse lattice spacing QsaT = 1• Longitudinal lattice size N = 128, longitudinal lattice spacing aL = 0.016• Number of configurations for the Monte-Carlo Nconf = 200 to 2000• Initial time Qsτ0 = 0.01
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EOM ON A LATTICE
Writing
Eµ(x) =xµ
Uµ(x) = x µ U†µ(x) =x + µµ
and
Uµν(x) =x µ
ν U†µν(x) = x
µ
ν
Uµ−ν(x) =
x µ
ν U†µ−ν(x) =
x
µ
ν .
We can therefore rewrite the EOM as
∂τxI=
−i2gaIa2
J
∑J
x I
J−
x
I
J+
x I
J−
x
I
J
∂τ
x I = − i g aIxI
I
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NUMERICAL RESULTS [TE,GELIS 1307:2214]
αs = 8 10−4 (g = 0.1)
+ 10-3
+ 10-2
+ 10-1
+ 100
+ 101
1
Tµν /
Qs4
τ [fm/c]
0.01 0.1 2 3 4
- 10-3
- 10-2
- 10-1
- 100
- 101
0.1 1.0
Qs τ
10 20 30 40
ε
PT
PL
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NUMERICAL RESULTS [TE,GELIS 1307:2214]
αs = 8 10−4 (g = 0.1)
-1
0
1/3
1/2
+1
0.1 1.0 10.0
1
Qs τ
τ [fm/c]
0.01 0.1
10.0 20.0 30.0 40.0
2 3 4
PT / ε
PL / ε
LO
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RENORMALIZATION PROCEDURE
⟨E2
L,div⟩
∼ Q2sk
2⊥,max + k4
⊥,max + k4⊥,max ln2 νmax
τ+ ...
∼
x
+
x
︸ ︷︷ ︸Q2sk2
⊥,max
+
x
︸ ︷︷ ︸k4
⊥,max
+
x
+
x
︸ ︷︷ ︸k4
⊥,max ln2 νmaxτ
+...
3 last diagrams can be substracted with a simulation where
Aaµ(x) = 0 + aa
µ(x)
E2L "fine" (B2
L too)
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RENORMALIZATION PROCEDURE
⟨E2
T,div⟩
∼ Q2s
ν2maxτ2 + k2
⊥,maxν2
maxτ2 + k4
⊥,max ln2 νmax
τ+ ...
∼
x
+
x
︸ ︷︷ ︸Q2s
ν2maxτ2
+
x
︸ ︷︷ ︸k2
⊥,maxν2
maxτ2
+
x
+
x
︸ ︷︷ ︸k4
⊥,max ln2 νmaxτ
+...
3 last diagrams can be substracted with a simulation where
Aaµ(x) = 0 + aa
µ(x)
How to deal with the first 2? → ad-hoc fit for the time being.
Otherwise E2L and B2
L behaves as τ−2 at early time.
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RENORMALIZATION PROCEDURE
ε = E2T + B2
T + E2L︸︷︷︸
fine
+ B2L︸︷︷︸
fine
PT = E2L︸︷︷︸
fine
+ B2L︸︷︷︸
fine
PL = E2T + B2
T − E2L︸︷︷︸
fine
− B2L︸︷︷︸
fine
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RENORMALIZATION PROCEDURE
〈PT〉phys. = 〈PT〉 backgd.+ fluct.
− 〈PT〉 fluct.only
〈ε, PL〉phys. = 〈ε, PL〉 backgd.+ fluct.︸ ︷︷ ︸
computed
− 〈ε, PL〉 fluct.only︸ ︷︷ ︸
computed
+ A τ−2︸ ︷︷ ︸fitted
.
Ad-hoc term only one to satisfy Bjorken law and EOS:
∂ττ−α + 2τ−α−1 = 0
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RENORMALIZATION PROCEDURE
How come that problematic divergent diagrams behaves as mass terms?
In the continuum limit, they don’t exist local gauge invariant operators ofdimension two.
On the lattice though, they coud be terms like
g2 ν2max
k2⊥,maxτ
2 TrF2,
where
Fµν(x) ∼x µ
ν−
x
µ
ν
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NUMERICAL RESULTS [TE,GELIS 1307:2214]
αs = 2 10−2 (g = 0.5)
+ 10-3
+ 10-2
+ 10-1
+ 100
1
Tµν /
Qs4
τ [fm/c]
0.01 0.1 2 3 4
- 10-3
- 10-2
- 10-1
- 100
0.1 1.0
Qs τ
10 20 30 40
ε
PT
PL
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NUMERICAL RESULTS [TE,GELIS 1307:2214]
αs = 2 10−2 (g = 0.5)
+ 10-3
+ 10-2
+ 10-1
+ 100
1
Tµν /
Qs4
τ [fm/c]
0.01 0.1 2 3 4
- 10-3
- 10-2
- 10-1
- 100
0.1 1.0
Qs τ
10 20 30 40
ε
PT
PL
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NUMERICAL RESULTS [TE,GELIS 1307:2214]
αs = 2 10−2 (g = 0.5)
-1
0
1/3
1/2
+1
0.1 1.0 10.0
1
Qs τ
τ [fm/c]
0.01 0.1
10.0 20.0 30.0 40.0
2 3 4
PT / ε
PL / ε
LO
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NUMERICAL RESULTS [TE,GELIS 1307:2214]
αs = 2 10−2 (g = 0.5)
-1
0
1/3
1/2
+1
0.1 1.0 10.0
1
Qs τ
τ [fm/c]
0.01 0.1
10.0 20.0 30.0 40.0
2 3 4
PT / ε
PL / ε
LO
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ANOMALOUSLY SMALL VISCOSITY
Assuming simple first order viscous hydrodynamics
ε ≈ ε0τ− 4
3︸ ︷︷ ︸Ideal hydro
− 2η0τ−2︸ ︷︷ ︸
first order correction
we can compute the dimensionless ratio (η = η0τ−1)
ηε−34 . 1
In contrast, perturbation theory at LO gives ηε−34 ∼ 300.
If the system is closed from being thermal
ε−34 ∼ s =⇒ η
sNot far from
14π
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CONCLUSION
• Correct NLO spectrum from first principles
• Fixed anisotropy for g = 0.5 at τ ∼ 1fm/c
• No need for strong coupling to get isotropization
• Compatible with viscous hydrodynamical expansion
• Assuming simple first order viscous hydrodynamics
ηε−34 . 1
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CONCLUSION
Viscous Hydrodynamics
I) Macroscopic theoryII) Few parameters: PL, PT , ε, ~uIII) Need input:
1) Equation of state f(PL, PT ) = ε2) Small anisotropy3) Initialization: ε(τ0), PL(τ0)? ...4) viscous coefficients: shear viscosity η,...5) Short isotropization time
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