François Gelis
1
The early stages ofHeavy Ion Collisions
SEWM, Swansea University, July 2012
z
t
François GelisIPhT, Saclay
François Gelis
2
Outline 2
1 Initial state factorization
2 Final state evolution
In collaboration with :
K. Dusling (NCSU)T. Epelbaum (IPhT)T. Lappi (Jyväskylä U.)R. Venugopalan (BNL)
z
t
strong fields classical dynamics
gluons & quarks out of eq. viscous hydro
gluons & quarks in eq. ideal hydro
hadrons kinetic theory
freeze out
François Gelis
3
Stages of a nucleus-nucleus collision 3
z
t
strong fields classical dynamics
gluons & quarks out of eq. viscous hydro
gluons & quarks in eq. ideal hydro
hadrons kinetic theory
freeze out
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3
Stages of a nucleus-nucleus collision 3
Goals :
• evolution from t = −∞ to the start of hydrodynamics• first principles description in QCD• weakly coupled (but strongly interacting...)
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4
What do we need to know about nuclei? 4
Nucleus at rest
• At low energy : valence quarks
• At higher energy :
• Lorenz contraction of longitudinal sizes• Time dilation B slowing down of the internal dynamics• Gluons start becoming important
• At very high energy : gluons dominate
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4
What do we need to know about nuclei? 4
Slightly boosted nucleus
• At low energy : valence quarks
• At higher energy :
• Lorenz contraction of longitudinal sizes• Time dilation B slowing down of the internal dynamics• Gluons start becoming important
• At very high energy : gluons dominate
François Gelis
4
What do we need to know about nuclei? 4
High energy nucleus
• At low energy : valence quarks
• At higher energy :
• Lorenz contraction of longitudinal sizes• Time dilation B slowing down of the internal dynamics• Gluons start becoming important
• At very high energy : gluons dominate
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5
Kinematics 5
• Low typical final state transverse momentum p⊥ . 1 GeV• Incoming partons have low momentum fractions x ∼ p⊥/E
• x ∼ 10−2 at RHIC (E = 200 GeV)• x ∼ 4.10−4 at the LHC (E = 2.76 − 5.5 TeV)
-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
François Gelis
6
Nucleon parton distributions 6
-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
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Nucleon parton distributions 6
Large x : dilute regime
-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
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Nucleon parton distributions 6
Small x : dense regime, gluon saturation
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Saturation domain 7
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7
Saturation domain 7
Saturation criterion [Gribov, Levin, Ryskin (1983)]
αsQ−2︸ ︷︷ ︸
σgg→g
×A−2/3xG(x,Q2)︸ ︷︷ ︸surface density
≥ 1
Q2 ≤ Q2s ≡αsxG(x,Q
2s)
A2/3︸ ︷︷ ︸saturation momentum
∼ A1/3x−0.3
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Color Glass Condensate = effective theory of small x gluons 8
[McLerran, Venugopalan (1994), Jalilian-Marian, Kovner, Leonidov, Weigert(1997), Iancu, Leonidov, McLerran (2001)]
• The fast partons (k+ > Λ+) are frozen by time dilationB described as static color sources on the light-cone :
Jµ = δµ+ρ(x−,~x⊥) (0 < x− < 1/Λ+)
• The color sources ρ are random, and described by a probabilitydistribution WΛ[ρ]
• Slow partons (k+ < Λ+) may evolve during the collisionB treated as standard gauge fieldsB eikonal coupling to the current Jµ : JµAµ
S = −1
4
∫FµνF
µν︸ ︷︷ ︸SYM
+
∫JµAµ︸ ︷︷ ︸
fast partons
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Initial statefactorization
[FG, Lappi, Venugopalan (2008)]
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Renormalization group evolution, JIMWLK equation 10
k-
P-
Λ-
0
fields sources
• The cutoff between the sources and the fields is not physical,and should not enter in observables
• Loop corrections contain logs of the cutoff(physically, due to soft gluon radiation)
• In inclusive Deep Inelastic Scattering: cancelled by letting thedistribution of the sources depend on the cutoff
Λ∂W[ρ]
∂Λ= H
(ρ,δ
δρ
)W[ρ] (JIMWLK equation)
• What about nucleus-nucleus collisions?Are the logs still universal?
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11
Handwaving argument for factorization 11
τcoll ∼ E-1
• The duration of the collision is very short: τcoll ∼ E−1
• Soft radiation takes a long timeB it must happen (long) before the collision
• The projectiles are not in causal contact before the impactB the logarithms are intrinsic properties of the projectiles,independent of the measured observable
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Handwaving argument for factorization 11
τcoll ∼ E-1
• The duration of the collision is very short: τcoll ∼ E−1
• Soft radiation takes a long timeB it must happen (long) before the collision
• The projectiles are not in causal contact before the impactB the logarithms are intrinsic properties of the projectiles,independent of the measured observable
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11
Handwaving argument for factorization 11
τcoll ∼ E-1
space-like interval
• The duration of the collision is very short: τcoll ∼ E−1
• Soft radiation takes a long timeB it must happen (long) before the collision
• The projectiles are not in causal contact before the impactB the logarithms are intrinsic properties of the projectiles,independent of the measured observable
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Power counting 12
In the saturated regime: Jµ ∼ g−1
g−2 g# of external legs g2×(# of loops)
• No dependence on the number of sources Jµ
B infinite number of graphs at each order
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Leading Order 13
• All observables can be expressed in terms of classical solutionsof Yang-Mills equations :
[Dµ,Fµν] = Jν1 + Jν2
• Boundary conditions for inclusive observables :retarded, with A → 0 at x0 = −∞
Example : inclusive spectra at LO
dN1
d3~p
∣∣∣∣LO
∼
∫d4xd4y eip·(x−y) �x�y A(x)A(y)
dNn
d3~p1 · · ·d3~pn
∣∣∣∣LO
=dN1
d3~p1
∣∣∣∣LO
· · · dN1d3~pn
∣∣∣∣LO
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Next to Leading Order 14
Relation between LO and NLO
ONLO =
[1
2
∫u,v
∫k
[akT
]u
[a∗kT
]v+
∫u
[αT
]u
]OLO
• T is the generator of the shifts of the initial field :
exp
[ ∫u
[αT
]u
]O[class. field at τ︷ ︸︸ ︷Aτ( Ainit︸︷︷︸
init. field
)]= O
[Aτ( Ainit + α︸ ︷︷ ︸
shifted init. field
)]
• ak,α are calculable analytically
• Valid for all inclusive observables,e.g. spectra, the energy-momentum tensor, etc...
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Initial state logarithms 15
• In the CGC, upper cutoff on the loop momentum : k± < Λ,to avoid double counting with the sources Jν1,2B logarithms of the cutoff
Central result for factorization
1
2
∫u,v
∫k
[akT
]u
[a∗kT
]v+
∫u
[αT
]u=
= log(Λ+)H1 + log
(Λ−)H2 + terms w/o logs
H1,2 = JIMWLK Hamiltonians of the two nuclei
• No mixing between the logs of the two nuclei
• Since the LO ↔ NLO relationship is the same for all inclusiveobservables, these logs have a universal structure
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Factorization of the logarithms 16
• The JIMWLK Hamiltonian H is self-adjoint :∫[Dρ]W
(HO
)=
∫[Dρ]
(HW
)O
Inclusive observables at Leading Log accuracy
〈O〉Leading Log
=
∫ [Dρ
1Dρ
2
]W1[ρ
1
]W2[ρ
2
]OLO [ρ1, ρ2]︸ ︷︷ ︸
fixed ρ1,2
• Logs absorbed into the scale evolution of W1,2
Λ∂W
∂Λ= HW (JIMWLK equation)
• Universality : the same W’s for all inclusive observables
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Final stateevolution
[Dusling, Epelbaum, FG, Venugopalan (2010)][Dusling, FG, Venugopalan (2011)]
[Epelbaum, FG (2011)]
QS
-1
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Energy momentum tensor at LO 18
QS
-1
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18
Energy momentum tensor at LO 18
Tµν for longitudinal ~E and ~B
TµνLO
(τ = 0+) = diag (ε, ε, ε,−ε)
B far from ideal hydrodynamics
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
2
c0+c
1 Exp(0.427 Sqrt(g
2 µ τ))
c0+c
1 Exp(0.00544 g
2 µ τ)
[Romatschke, Venugopalan (2005)]
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Weibel instabilities for small perturbations 19
[Mrowczynski (1988), Romatschke, Strickland (2003), Arnold, Lenaghan,Moore (2003), Rebhan, Romatschke, Strickland (2005), Arnold, Lenaghan,Moore, Yaffe (2005), Romatschke, Rebhan (2006), Bodeker, Rummukainen(2007),...]
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
2
c0+c
1 Exp(0.427 Sqrt(g
2 µ τ))
c0+c
1 Exp(0.00544 g
2 µ τ)
[Romatschke, Venugopalan (2005)]
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19
Weibel instabilities for small perturbations 19
[Mrowczynski (1988), Romatschke, Strickland (2003), Arnold, Lenaghan,Moore (2003), Rebhan, Romatschke, Strickland (2005), Arnold, Lenaghan,Moore, Yaffe (2005), Romatschke, Rebhan (2006), Bodeker, Rummukainen(2007),...]
• For some k’s, the field fluctuations ak divergelike exp
√µτ when τ→ +∞
• Some components of Tµν have secular divergences whenevaluated at fixed loop order
• When ak ∼ A ∼ g−1, the power counting breaks down andadditional contributions must be resummed :
g e√µτ ∼ 1 at τmax ∼ µ−1 log2(g−1)
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20
Improved power counting 20
Loop ∼ g2 , T ∼ e√µτ
u
Tµν(x)
vΓ2(u,v)
• 1 loop : (ge√µτ)2
• 2 disconnected loops :(ge√µτ)4
• 2 nested loops : g(ge√µτ)3
B subleading
Leading terms at τmax
• All disconnected loops to all ordersB exponentiation of the 1-loop result
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20
Improved power counting 20
Loop ∼ g2 , T ∼ e√µτ
Tµν(x)
• 1 loop : (ge√µτ)2
• 2 disconnected loops :(ge√µτ)4
• 2 nested loops : g(ge√µτ)3
B subleading
Leading terms at τmax
• All disconnected loops to all ordersB exponentiation of the 1-loop result
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20
Improved power counting 20
Loop ∼ g2 , T ∼ e√µτ
Tµν(x)
Γ3(u,v,w)
• 1 loop : (ge√µτ)2
• 2 disconnected loops :(ge√µτ)4
• 2 nested loops : g(ge√µτ)3
B subleading
Leading terms at τmax
• All disconnected loops to all ordersB exponentiation of the 1-loop result
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21
Resummation of the leading secular terms 21
Tµνresummed
= exp
[1
2
∫u,v
∫k
[akT]u[a∗kT]v︸ ︷︷ ︸
G(u,v)
+
∫u
[αT]u
]Tµν
LO[Ainit]
= TµνLO
+ TµνNLO︸ ︷︷ ︸
in full
+ TµνNNLO
+ · · ·︸ ︷︷ ︸partially
• The evolution remains classical, but we must average over aGaussian ensemble of initial conditions
• The 2-point correlation G(u, v) of the fluctuations is known
• Fluctuations ⇔ 1/2 quantum per mode
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21
Resummation of the leading secular terms 21
Tµνresummed
= exp
[1
2
∫u,v
∫k
[akT]u[a∗kT]v︸ ︷︷ ︸
G(u,v)
+
∫u
[αT]u
]Tµν
LO[Ainit]
=
∫[Dχ] exp
[−1
2
∫u,v
χ(u)G−1(u, v)χ(v)
]Tµν
LO[Ainit + χ+ α]
• The evolution remains classical, but we must average over aGaussian ensemble of initial conditions
• The 2-point correlation G(u, v) of the fluctuations is known
• Fluctuations ⇔ 1/2 quantum per mode
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22
Analogous scalar toy model 22
φ4 field theory coupled to a source
L =1
2(∂αφ)
2 −g2
4!φ4 + Jφ
Strong external source: J ∝ Q3
g
• In 3+1-dim, g is dimensionless, and the only scale in the problemis Q
• This theory has unstable modes (parametric resonance)
• Two setups have been studied :
• Fixed volume system• Longitudinally expanding system
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Pathologies in fixed order calculations 23
Tree
-40
-30
-20
-10
0
10
20
30
40
-20 0 20 40 60 80
time
PLO εLO
• Oscillating pressure at LO : no equation of state
• Small NLO correction to the energy density (protected by energyconservation)
• Secular divergence in the NLO correction to the pressure
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23
Pathologies in fixed order calculations 23
Tree + 1-loop
-40
-30
-20
-10
0
10
20
30
40
-20 0 20 40 60 80
time
PLO+NLO εLO+NLO
• Oscillating pressure at LO : no equation of state• Small NLO correction to the energy density (protected by energy
conservation)• Secular divergence in the NLO correction to the pressure
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Spectrum of fluctuations 24
Initial conditions
φ(t0, x) = ϕ(t0, x) +
∫d3k
(2π)32k
[ck e
iλkt0 Vk(x) + c.c.]
• ϕ(t0, x) = classical background field
• ck = complex random Gaussian number:⟨ck⟩= 0 ,
⟨ckc∗l
⟩= (2π)3 k δ(k− l)
• λk, Vk determined by:[−∇2 +
1
2g2ϕ2(t0, x)
]Vk(x) = λ
2kVk(x)
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Resummed energy momentum tensor 25
-1.5
-1
-0.5
0
0.5
1
1.5
2
10 100 1000 10000
time
px+py+pz
ε
• No secular divergence in the resummed pressure
• The pressure relaxes to the equilibrium equation of state
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26
Time evolution of the occupation number 26
0.01
0.1
1
10
100
1000
10000
0 0.5 1 1.5 2 2.5 3 3.5
f k
k
Bose-Einstein with µ=0.54,T=1.31
T/(ωk-µ)-1/2 with µ=0.54,T=1.31
const / k5/3
t = 0
60
200
1000
2000
5000
104
• Resonant peak at early times
• Late times : classical equilibrium with a chemical potential
• µ ≈ m + excess at k = 0 : Bose-Einstein condensation?
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27
Bose-Einstein condensation 27
10-2
10-1
100
101
102
103
104
0 0.5 1 1.5 2 2.5 3 3.5
f k
k
t = 0
50
150
300
2000
5000
104
T/(ωk-µ)-1/2
10-2
10-1
100
101
102
103
104
0 0.5 1 1.5 2 2.5
t = 0
20
40
60
80
100
120
140
• Start with the same energy density, but an empty zero mode
• Very quickly, the zero mode becomes highly occupied
• Same distribution as before at late times
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Volume dependence 28
×10-6
×10-5
×10-4
×10-3
×10-2
×10-1
×100
×101 ×10
2 ×103 ×10
4
f(0)
/ V
time
L = 20 L = 30 L = 40
f(k) =1
eβ(ωk−µ) − 1+ n0δ(k) =⇒ f(0) ∝ V = L3
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Evolution of the condensate 29
10-1
100
101
102
103
104
100
101
102
103
104
105
f 0
time
203 lattice , 256 configurations , V(φ) = g
2φ
4/4!
g2 = 1
2
4
8
• Formation time almost independent of the coupling• Condensate lifetime much longer than its formation time• Smaller amplitude and faster decay at large coupling
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Longitudinal expansion: spectrum of fluctuations 30
Initial conditions
φ(τ0, η, x⊥) = ϕ(τ0, x⊥) +
∫d2k⊥dν
(2π)3
[cνkH
(2)iν (λkτ0)Vk(x⊥) e
−iνη + c.c.]
• ϕ(τ0, x⊥) = classical background field (boost invariant)
• cνk = complex random Gaussian number:⟨cνk⟩= 0 ,
⟨cνkc
∗σl
⟩= (2π)3 δ(k⊥ − l⊥)δ(ν− σ)
• λk, Vk determined by:[−∇2
⊥ +1
2g2ϕ2(τ0, x⊥)
]Vk(x⊥) = λ
2kVk(x⊥)
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Longitudinal expansion: initial time 31
• The EoM is singular when τ→ 0 : one must start at τ0 6= 0
∂2τφ+1
τ∂τφ−
1
τ2∂2ηφ−∇2
⊥φ+g2
6φ3 = 0
• The spectrum of fluctuations of the initial field also depends onτ0, in such a way that the end result is independent of τ0
×10-1
×100
×101
×102
×103
×104
×10-2 ×10
-1 ×100 ×10
1
τ
PT(τ0=0.01)
ε(τ0=0.01)
PT(τ0=0.1)
ε(τ0=0.1)
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Longitudinal expansion : occupation number 32
τ = 0.1 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied
• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalizedsystem)
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32
Longitudinal expansion : occupation number 32
τ = 10 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied
• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalizedsystem)
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32
Longitudinal expansion : occupation number 32
τ = 50 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalized
system)
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32
Longitudinal expansion : occupation number 32
τ = 100 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalized
system)
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32
Longitudinal expansion : occupation number 32
τ = 150 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalized
system)
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32
Longitudinal expansion : occupation number 32
τ = 200 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalized
system)
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32
Longitudinal expansion : occupation number 32
τ = 300 [40 × 40 × 320]
0
0.5
1
1.5
2
2.5
3
k⊥
0
100
200
300
400
500
600
ν
×10-2
×100
×102
×104
×106
100
102
104
• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalized
system)
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33
Longitudinal expansion : equation of state 33
10-2
10-1
100
101
10 100
τ
τ-4/3
τ-1
2PT + P
L
ε
• After a short time, one has 2PT+ P
L≈ ε
• Change of behavior of the energy density: τ−1 → τ−4/3. Sinceone has ∂τε+ (ε+ P
L)/τ = 0, this suggests that P
Lgets close to
ε/3
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34
Longitudinal expansion : isotropization 34
×10-3
×10-2
×10-1
×100
×101
0 50 100 150 200 250 300
τ
2PT + P
L
ε
PT
PL
• At early times, PL
drops much faster than PT
(redshifting of thelongitudinal momenta due to the expansion)
• Drastic change of behavior when the expansion rate becomes smallerthan the growth rate of the unstability
• Eventually, isotropic pressure tensor : PL≈ P
T
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35
Summaryand Outlook
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36
Summary 36
• Factorization of high energy logarithms in AA collisions• universal distributions, also applicable in pA or DIS• controls the rapidity dependence of correlations• limited to inclusive observables
• Resummation of secular terms in the final state evolution• stabilizes the NLO calculation• leads to the equilibrium equation of state• isotropization even with longitudinal expansion• full thermalization on longer time-scales• Bose-Einstein condensation if overoccupied initial state
(so far, all numerical studies done for a toy scalar model)
What’s next? QCD
• generalizable to Yang-Mills theory
• gauge invariant
• seamless integration with the CGC description of AA collisions
• computationally expensive ( ∼ [scalar case] ×3× (N2c − 1))
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37
Extra
ρ1
g 1 g-1
AApA
g-4
g-2
1
g2
g4
p q
p
q
p
q
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38
Dense-dilute collisions 38
ρ1
g 1 g-1
AApA
g-4
g-2
1
g2
g4
p q
p
q
p
q
François Gelis
38
Dense-dilute collisions 38
Expected complications
• More diagrams to consider even at Leading Order• More terms in the evolution Hamiltonian if ρ ∼ g:
g2ρ2(∂
∂ρ
)2∼ g4ρ2
(∂
∂ρ
)4
François Gelis
39
Exclusive processes 39
François Gelis
39
Exclusive processes 39
Example : differential probability to produce 1 particle at LO
dP1
d3~p
∣∣∣∣LO
= F[0]×∫d4xd4y eip·(x−y)�x�yA+(x)A−(y)
∣∣∣z=0
• The vacuum-vacuum graphs do not cancel in exclusivequantities : F[0] 6= 1 (in fact, F[0] = exp(−c/g2)� 1)
• A+ and A− are classical solutions of the Yang-Mills equations,but with non-retarded boundary conditions
François Gelis
40
Thermalization in Yang-Mills theory 40
• Recent analytical work : Kurkela, Moore (2011)
• Going from scalars to gauge fields :
• More fields per site (3 Lorentz components × 8 colors)
• More complicated spectrum of initial conditions
• Expansion : UV overflow on a fixed grid in η
François Gelis
41
BEC and dilepton production 41
François Gelis
41
BEC and dilepton production 41
Two topologies for virtual photons at LO
Connected ω ∼Minv ∼ Qs k⊥ ∼ Qs
Disconnected ω ∼Minv ∼ Qs k⊥ � Qs
B excess of dileptons with k⊥ �Minv
François Gelis
42
Links to Quantum Chaos 42
• Quantum Chaos : how does the chaos at the classical levelmanifests itself in quantum mechanics?
• Berry’s conjecture [M.V. Berry (1977)]High lying eigenstates of such systems have nearly randomwavefunctions. The corresponding Wigner distribution is almostuniform on the energy surface
• Srednicki’s eigenstate thermalization hypothesis[M. Srednicki (1994)]For sufficiently inclusive measurements, these high lyingeigenstates look thermal. If the system starts in a coherent state,decoherence is the main mechanism to thermalization
François Gelis
43
Phenomenology[Dumitru, FG, McLerran, Venugopalan (2008)]
[Dusling, FG, Lappi, Venugopalan (2010)]
François Gelis
44
2-particle correlations in AA collisions 44
[STAR Collaboration, RHIC]
• Long range rapidity correlation
• Narrow correlation in azimuthal angle
• Narrow jet-like correlation near ∆y = ∆ϕ = 0
François Gelis
45
Probing early times with rapidity correlations 45
detection (∼1 m/c)
freeze out (∼10 fm/c)
latest correlation
AB
z
t
• By causality, long range rapidity correlations are sensitive to thedynamics of the system at early times :
τcorrelation ≤ τfreeze out e−|∆y|/2
0 0.5 1 1.5 2
g2µτ
0
0.2
0.4
0.6
0.8
[(g
2µ
)4/g
2]
Bz
2
Ez
2
BT
2
ET
2
[Lappi, McLerran (2006)]
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46
Color field at early time 46
0 0.5 1 1.5 2
g2µτ
0
0.2
0.4
0.6
0.8
[(g
2µ
)4/g
2]
Bz
2
Ez
2
BT
2
ET
2
[Lappi, McLerran (2006)]
François Gelis
46
Color field at early time 46
QS
-1
• The field lines form tubes of transverse size ∼ Q−1s
• Rapidity correlation length : ∆η ∼ α−1s
François Gelis
47
2-hadron correlations from color flux tubes 47
• η-independent fields lead to long range correlations :
• Particles emitted by different flux tubes are not correlatedB (RQs)
−2 sets the strength of the correlation
• At early times, the correlation is flat in ∆ϕ
The collimation in ∆ϕ is produced later by radial flow
François Gelis
47
2-hadron correlations from color flux tubes 47
• η-independent fields lead to long range correlations :
R
QS
-1
• Particles emitted by different flux tubes are not correlatedB (RQs)
−2 sets the strength of the correlation
• At early times, the correlation is flat in ∆ϕ
The collimation in ∆ϕ is produced later by radial flow
François Gelis
47
2-hadron correlations from color flux tubes 47
• η-independent fields lead to long range correlations :
R
QS
-1
• Particles emitted by different flux tubes are not correlatedB (RQs)
−2 sets the strength of the correlation
• At early times, the correlation is flat in ∆ϕ
The collimation in ∆ϕ is produced later by radial flow
François Gelis
47
2-hadron correlations from color flux tubes 47
• η-independent fields lead to long range correlations :
vr
• Particles emitted by different flux tubes are not correlatedB (RQs)
−2 sets the strength of the correlation
• At early times, the correlation is flat in ∆ϕ
The collimation in ∆ϕ is produced later by radial flow
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48
Centrality dependence 48
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 50 100 150 200 250 300 350 400
Am
pli
tud
e
Npart
radial boost model
STAR preliminary
• Main effect : increase of the radial flow velocity with the centralityof the collision
0
0.2
0.4
0.6
0.8
1
1.2
-6 -5 -4 -3 -2 -1 0 1 2
1/N
trig
dN
ch/d
∆η
∆η
pTtrig
= 2.5 GeV
pTassoc
= 350 MeV
Au+Au 0-30% (PHOBOS)
p+p (PYTHIA)ytrig = 0
ytrig=0.75
ytrig=1.5
François Gelis
49
Rapidity dependence 49
0
0.2
0.4
0.6
0.8
1
1.2
-6 -5 -4 -3 -2 -1 0 1 2
1/N
trig
dN
ch/d
∆η
∆η
pTtrig
= 2.5 GeV
pTassoc
= 350 MeV
Au+Au 0-30% (PHOBOS)
p+p (PYTHIA)ytrig = 0
ytrig=0.75
ytrig=1.5
François Gelis
49
Rapidity dependence 49
Prediction at LHC energy
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-4 -2 0 2 4 6 8 10 12
d2N
/(d
2p
T d
2q
T d
yp d
yq)
[GeV
-4]
yq - yp
Pb+Pb at LHC
× 10-2
× 0.5
× 0.65× 0.85
pT = 2 GeV
qT = 2 GeV
yp = 0
yp = -1.5
yp = -3
yp = -4.5
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50
Backup
François Gelis
51
Boundary conditions 51
A+(x)=
∫y
G0++(x, y)[J(y)−V ′(A+(y))
]−G0+−(x, y)
[J(y)−V ′(A−(y))
]A−(x)=
∫y
G0−+(x, y)[J(y)−V ′(A+(y))
]−G0−−(x, y)
[J(y)−V ′(A−(y))
]G̃0+−(p) = 2π z(p) θ(−p0)δ(p
2)
• A+ and A− are solutions of the classical EoM
• Decompose the fields in Fourier modes :
Aε(x) ≡∫
d3~p
(2π)32Ep
[a(+)ε (x0, ~p) e
−ip·x + a(−)ε (x0, ~p) e
+ip·x]
Boundary conditions
x0 = −∞ : a(+)+ (~p) = a
(−)− (~p) = 0
x0 = +∞ : a(+)− (~p) = z(p)a
(+)+ (~p) , a
(−)+ (~p) = z(p)a
(−)− (~p)
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52
Moyal bracket 52
• Wigner transform of the commutator
• Explicit expression :
{{A,B}} =2
h̄A(Q,P) sin
(h̄
2(←∇Q
→∇P
−←∇P
→∇Q
)
)B(Q,P)
• Quantum deformation of the Poisson bracket :
{{A,B}} = {A,B}+ O(h̄2)