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Pre-equilibrium phenomena in Quark-Gluon Plasma
Aleksas Mazeliauskas
Theoretical Physics Department, CERN
December 6, 2019
AM and J. Berges, PRL [arXiv:1810.10554]A. Kurkela, AM PRD [arXiv:1811.03068], PRL [arXiv:1811.03040]G. Giacalone, AM, S. Schlichting, [arXiv:1908.02866]
Isolated quantum systems and universality in extreme conditions
Non-equilibrium QCD descriptions at weak coupling αs → 0
At high energies mid-rapidity is dominated by small Bjorken-x gluons
p ∼ Qs saturation scale � ΛQCD, strong gluon fields Aµ ∼ 1αs� 1
=⇒ classical-statistical simulationsdecoherence of classical fields at τQs � 1
=⇒ kinetic evolution of gluon phase space distribution f
�
J+ J
−
E⌘
B⌘
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���
��������������������������������
������������ ���������������
f ∼ 1αs� 1 1
αs� f � 1 f ∼ 1
Soren Schlichting, Initial Stages 2016Color-Glass Condensate
2 / 17
Non-equilibrium QCD descriptions at weak coupling αs → 0
At high energies mid-rapidity is dominated by small Bjorken-x gluons
p ∼ Qs saturation scale � ΛQCD, strong gluon fields Aµ ∼ 1αs� 1
=⇒ classical-statistical simulationsdecoherence of classical fields at τQs � 1
=⇒ kinetic evolution of gluon phase space distribution f
�
J+ J
−
E⌘
B⌘
���������������� �����������������������������
�������������������������
�����������
����
������������������������������������������ ������������������� �����
�����������������������������������������������������������������������������������������
���
��������������������������������
������������ ���������������
f ∼ 1αs� 1 1
αs� f � 1 f ∼ 1
Soren Schlichting, Initial Stages 2016Color-Glass Condensate
2 / 17
High temperature gauge kinetic theory
Boltzmann equation for distribution f of quark and gluon quasi-particles.Arnold, Moore, Yaffe (2003)[1]
∂τfg,q −pzτ∂pzfg,q = −C2↔2[f ]− C1↔2[f ]
Leading order processes in the coupling constant λ = 4παsNc:
1 2↔ 2 elastic scatterings: gg ↔ gg, qq ↔ qq, qg ↔ gq, gg ↔ qq
= |Mggqq |2 = λ2 16
dFCFC2A
[CF
(u
t+t
u
)− CA
(t2 + u2
s2
)]Hard Thermal Loop resumed propagators, screening mass mD ∼ gT
Microscopic studies of QGP equilibration with QCD kinetic theory:
Bottom-up thermalization of gluons. Kurkela and Zhu (2015) [2]
Equilibration of perturbations. Keegan, Kurkela, AM and Teaney (2016) [3]
Preflow computer code KøMPøST Kurkela, AM, Paquet, Schlichting and Teaney (2018)[4, 5]
Chemical equilibration of quarks and gluons. Kurkela and AM (2018) [6, 7]
Pre-scaling phenomena. AM and Berges (2018)[8]
3 / 17
High temperature gauge kinetic theory
Boltzmann equation for distribution f of quark and gluon quasi-particles.Arnold, Moore, Yaffe (2003)[1]
∂τfg,q −pzτ∂pzfg,q = −C2↔2[f ]− C1↔2[f ]
Leading order processes in the coupling constant λ = 4παsNc:
2 1↔ 2 medium induced collinear radiation: g ↔ gg, q ↔ qg, g ↔ qq
= |Mgqq|2 =
k′2 + p′2
k′2p′2p3Fq(k′;−p′, p)︸ ︷︷ ︸
splitting rate
Resummed multiple scatterings with the medium (LPM suppression).
Microscopic studies of QGP equilibration with QCD kinetic theory:
Bottom-up thermalization of gluons. Kurkela and Zhu (2015) [2]
Equilibration of perturbations. Keegan, Kurkela, AM and Teaney (2016) [3]
Preflow computer code KøMPøST Kurkela, AM, Paquet, Schlichting and Teaney (2018)[4, 5]
Chemical equilibration of quarks and gluons. Kurkela and AM (2018) [6, 7]
Pre-scaling phenomena. AM and Berges (2018)[8]
3 / 17
“Bottom-up” thermalization scenario Baier, Mueller, Schiff, and Son (2001)[10]
Evolution of initially over-occupied hard gluons p ∼ Qs � ΛQCD
I) over-occupied pz ∼ Qs
(Qsτ)1/31� Qsτ � α−3/2
s
II) under-occupied pz ∼√αsQs α−3/2
s � Qsτ � α−5/2s
III) mini-jet quenching pz ∼ α3sQs(Qsτ) α−5/2
s � Qsτ � α−13/5s
pz pz pz
pxpxpx
1 10 100Rescaled occupancy: <pα
sf>/<p>
1
10
100
1000
Anis
otr
opy:
PT/P
L
αs=0
αs=0.03
αs=0.15
αs=0.3
Classical YM
Bottom-Upα
s=0.015
Realisticcoupling
2↔ 2 broadening collinear cascade mini-jet quench
nonthermal attractor
Berges, Boguslavski, Schlichting, Venugopalan (2014) [9]
Kurkela and Zhu (2015), Keegan, Kurkela, AM and Teaney (2016), Kurkela, AM, Paquet, Schlichting and Teaney (2018)
[2, 3, 5, 4]
4 / 17
“Bottom-up” thermalization scenario Baier, Mueller, Schiff, and Son (2001)[10]
Evolution of initially over-occupied hard gluons p ∼ Qs � ΛQCD
I) over-occupied pz ∼ Qs
(Qsτ)1/31� Qsτ � α−3/2
s
II) under-occupied pz ∼√αsQs α−3/2
s � Qsτ � α−5/2s
III) mini-jet quenching pz ∼ α3sQs(Qsτ) α−5/2
s � Qsτ � α−13/5s
pz pz pz
pxpxpx
1 10 100Rescaled occupancy: <pα
sf>/<p>
1
10
100
1000
Anis
otr
opy:
PT/P
L
αs=0
αs=0.03
αs=0.15
αs=0.3
Classical YM
Bottom-Upα
s=0.015
Realisticcoupling
2↔ 2 broadening collinear cascade mini-jet quench
Berges, Boguslavski, Schlichting, Venugopalan (2014) [9]
Kurkela and Zhu (2015), Keegan, Kurkela, AM and Teaney (2016), Kurkela, AM, Paquet, Schlichting and Teaney (2018)
[2, 3, 5, 4]
4 / 17
“Bottom-up” thermalization scenario Baier, Mueller, Schiff, and Son (2001)[10]
Evolution of initially over-occupied hard gluons p ∼ Qs � ΛQCD
I) over-occupied pz ∼ Qs
(Qsτ)1/31� Qsτ � α−3/2
s
II) under-occupied pz ∼√αsQs α−3/2
s � Qsτ � α−5/2s
III) mini-jet quenching pz ∼ α3sQs(Qsτ) α−5/2
s � Qsτ � α−13/5s
pz pz pz
pxpxpx
1 10 100Rescaled occupancy: <pα
sf>/<p>
1
10
100
1000
Anis
otr
opy:
PT/P
L
αs=0
αs=0.03
αs=0.15
αs=0.3
Classical YM
Bottom-Upα
s=0.015
Realisticcoupling
2↔ 2 broadening collinear cascade mini-jet quenchhydro attractor
Berges, Boguslavski, Schlichting, Venugopalan (2014) [9]
Kurkela and Zhu (2015), Keegan, Kurkela, AM and Teaney (2016), Kurkela, AM, Paquet, Schlichting and Teaney (2018)
[2, 3, 5, 4]
4 / 17
Part I: Self-similar evolution at weak couplings
From classical simulations to kinetic theory
7
BGLMV (const. anisotropy)
BMSS (elastic scattering)
Turbulence exponents:
α = -2/3
, β = 0 , γ = 1/3 BD (plasma instabilities)
KM (plasma instabilities)
lattice data
1
δS
αS1/7
αS1/3
αS1/2 1 α
S-1
Mom
entu
m s
pace
ani
sotro
py: Λ
L/Λ
T
Occupancy nHard
Hig
her a
niso
tropy
Smaller occupancy
ξ0=1
ξ0=3/2
ξ0=2
ξ0=4
ξ0=6
n0=1n0=1/4
FIG. 5. Evolution in the occupancy–anisotropy plane. In-dicated are the attractor solutions proposed in (BMSS) [1],(BD) [23], (KM) [25] and (BGLMV) [26], along with the sim-ulations results for different initial conditions shown in blue.
While the original work by Baier, Mueller, Schiff andSon [1] (BMSS) determines the basic properties of thekinetic evolution from self-consistency arguments, theself-similar behavior observed from numerical simula-tions indicates that the framework of turbulent ther-malization [36] can be applied. We continue this anal-ysis by plugging the self-similar distribution (10) intoC(elast)[pT , pz; f ] to extract the scaling behavior µ =3α− 2β + γ. The scaling relation in eq. (15), then reads2α−2β+γ+1 = 0. Since elastic scattering processes areparticle number conserving, a further scaling relation isobtained from integrating the distribution function overpT and rapidity wave numbers ν = pzτ . By use of thescaling form (10), particle number conservation leads tothe scaling relation α−2β−γ+1 = 0. Similarly, approxi-mating the mode energy of hard excitations as ωp � pT inthe anisotropic scaling limit, energy conservation yieldsthe final scaling condition α− 3β − γ + 1 = 0.
Remarkably, the above scaling relations are indepen-dent of many of the details of the underlying field the-ory such as the number of colors, the coupling constantas well as the initial conditions. Instead, they only de-pend on the dominant type of kinetic interactions (suchas 2 ↔ 2 or 2 ↔ 3 scattering processes), the con-served quantities of the system and the number of dimen-sions. More specifically, the dynamics of small-angle elas-tic scattering, along with the conservation laws of quasi-particle number and energy provide the three equationsto determine the scaling exponents. These are straight-forwardly extracted to be
α = −2/3 , β = 0 , γ = 1/3 , (18)
in good agreement with those extracted from our latticesimulations of the temporal evolution of gauge invariantobservables.
The close agreement of the lattice simulations with the
bottom-up scenario appears surprising at first. While inthe latter, it is the Debye scale that provides the scalefor multiple incoherent elastic scatterings and the con-sequent broadening of the longitudinal momentum, theone loop self-energy for anisotropic momentum distri-butions could lead to plasma instabilities even at timesτ � Q−1 log2(α−1
S ). The impact of plasma instabilitieson the first stage of the bottom-up scenario has been con-sidered in [23] (BD). In this scenario, plasma instabili-ties create an overpopulation of the unstable soft modesf(p ∼ mD) ∼ 1/αS , such that the interaction of hard ex-citations with the highly populated soft modes becomesthe dominant process. This process leads to a more effi-cient momentum broadening in the longitudinal directionand changes the evolution of the characteristic momen-tum scales and occupancies. Similar considerations, al-beit including a different range of highly occupied unsta-ble modes8, lead to the detailed weak coupling scenarioin [25] (KM). In this scenario, plasma instabilities playa significant role for the entire thermalization process inthe classical regime and beyond. Yet another scenario ofhow highly occupied expanding non-Abelian fields pro-ceed toward thermalization was proposed in [26]. In thisscenario, it is conjectured that the combination of highoccupancy and elastic scattering can generate a transientBose-Einstein condensate. The evolution of this conden-sate together with elastically scattering quasi-particle ex-citations is argued to generate an attractor with fixedPL/PT anisotropy parameter δs.
While all of these effects can in principle be realizedand have interesting consequences for the subsequentspace-time evolution of the strongly correlated plasma,the infrared physics of momenta around the Debye scaleis crucial in all these scenarios. The properties of thishighly non-linear non-Abelian dynamics can be resolvedconclusively through non-perturbative numerical simula-tions, such as those performed here.
A compact summary of our results in comparison withthe different weak coupling thermalization scenarios isshown in Fig. 5, describing the space-time evolution inthe occupancy–anisotropy plane. The horizontal axisshows the occupancy nHard and the vertical axis themomentum-space anisotropy in terms of the typical lon-gitudinal and transverse momenta ΛT,L. The gray linesindicate the attractor solutions of the different thermal-ization scenarios, while the blue lines show our simula-tion results for different initial conditions. One immedi-ately observes the attractor property, which appears tobe in good agreement with the analytical discussion of theBMSS kinetic equation in the high-occupancy regime [5].
As noted previously, similar attractor solutions werediscovered in relativistic scalar theories that purport
8 The range of highly occupied unstable modes in this scenariois determined within the hard-loop framework in Ref. [24] andparametrically given by modes with momenta pT � mD andpz � mDΛT /ΛL.
1 10 100Rescaled occupancy: <pα
sf>/<p>
1
10
100
1000
An
iso
tro
py
: P
T/P
L
αs=0
αs=0.03
αs=0.15
αs=0.3
Classical YM
Bottom-Upα
s=0.015
Realisticcoupling
Berges, Boguslavski, Schlichting, Venugopalan (2014)[11] Kurkela and Zhu (2015)[2]
classical-statistical Yang-Mills kinetic theory of gluons
Occupancy
Anisotropy
scalingscaling
Self-similar evolution of distribution function
fg(p⊥, pz, τ) = ταfS(τβp⊥, τγpz), α ≈ −2
3, β ≈ 0, γ ≈ 1
3
Universal exponents: α ≈ −23 , β ≈ 0, γ ≈ 1
3
5 / 17
Scaling in leading order QCD kinetic theoryInitial conditions fg = σ0
g2e−(p2⊥+ξ2p2z), σ0 = 0.1, g = 10−3, ξ = 2
Scaling regime is reached at late times
fg(p⊥, pz, τ) = τ−2/3fS(p⊥, τ1/3pz), τ → τ/τref
10−3
10−2
10−1
0.01 0.1 1
PL/P
T
τ
free streamingelastic scatterings
QCD kinetic theory
10−3
10−2
10−1
0.1 1
∆τ/τ = 0.28τ
2/3g2f g
(τ,p
⊥,p
z=
0)
p⊥
elastic scatteringsQCD kinetic theory
0.01
0.1
1
τ
pressure anisotropy τ2/3fg(p⊥, pz = 0, τ)
scaling
Approach to a non-thermal fixed point in full QCD kinetic evolution.scaling phenomena is also seen in cold atoms and scalar kinetic theory: Orioli et al. (2015) [12], Mikheev et al. (2018) [13],
Prufer et al. (2018) [14], Erne et al. (2018) [15]
6 / 17
Pre-scaling regime in QCD kinetic theory
Non-equilibrium dynamics undone by self-similar renormalization
fg(p⊥, p⊥, τ) = τα(τ)fS(τβ(τ)p⊥, τγ(τ)pz)
AM and Berges (2018) [8], cf. Micha and Tkachev (2004) [16]
Scaling exponents α(τ), β(τ), γ(τ) can be time dependent!
−1.5
−1
−0.5
0
0.5
1
1.5
0.01 0.1 1
1/3
1/4
−2/3
−3/4
σ0 = 0.1
expo
nent
s
τ
α(τ)β(τ)γ(τ)
10−3
10−2
10−1
0.1 1
1/p⊥
∆τ/τ = 0.28
τ−
α(τ
)g2f g
(τ,p
⊥,p
z=
0)
p⊥
QCD kinetic theory
0.01
0.1
1
τ
scaling exponents τ−α(τ)fg(p⊥, pz = 0, τ)
scaling
pre-scaling
Time evolution encoded int oa few hydrodynamic degrees of freedomα, β, γ.
7 / 17
The onset of thermalization
Consider intermediate couplings and late times.
−1.5
−1
−0.5
0
0.5
1
1.5
0.1 1 10
1/3
−2/3
expo
nent
s
τ/τref
α(τ)β(τ)γ(τ)
1
10
0.0001 0.001 0.01
anis
otro
pyP
T/P
L
occupancy 〈pλf〉 / 〈p〉
Pre-scaling before isotropization
Thermal scaling seen only at late times.
Early time pre-scaling disconnected from late time hydrodynamics.8 / 17
The onset of thermalization
Consider intermediate couplings and late times.
−1.5
−1
−0.5
0
0.5
1
1.5
0.1 1 10 100 1000 10000
1/3
−2/3
expo
nent
s
τ/τref
α(τ)β(τ)γ(τ)
1
10
0.0001 0.001 0.01
anis
otro
pyP
T/P
L
occupancy 〈pλf〉 / 〈p〉
Pre-scaling before isotropization
Thermal scaling seen only at late times.
Early time pre-scaling disconnected from late time hydrodynamics.8 / 17
Part II: Chemical equilibration
Fermion production in weakly coupled QCD
Initial state is dominated by gluon fields
quark productionQGP
But final state is assumed to be in chemical equilibrium:
QGP expansion described by 3 flavour equation of state (u, d, s):36 fermionic and 16 bosonic degrees of freedom.
hadron production at freeze-out consistent with thermal ensemble.
How can we produce fermions?
Quark production from strong color fields. Tanji, Berges (2017) [17]
Martinez, Sievert, Wertepny (2018) [18]
Leading order kinetics: gluon fusion gg ↔ qq and splitting g ↔ qq.Kurkela, AM (2018) [6, 7]
9 / 17
Chemical equilibration with expansion
Mean free path lmfp ∼ 1λ2T⇒ relaxation time τR ≡ 4πη/s
Tid(τ)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4
τchem
00.20.40.60.8
1
100 101 102 103 104
e q/e q
,eq.
τ/τR
η/s(λ = 20) ≈ 0.4η/s(λ = 10) ≈ 1.0η/s(λ = 5) ≈ 2.8η/s(λ = 1) ≈ 1900
τQs
kinetic theorychem. equilibrium
Chemical equilibration at τchem ∼ 1.2τR(τ) for αs ∼ 0.310 / 17
Physical equilibration time-scales in hadronic collisions
τ = (τ/τR)3/2︸ ︷︷ ︸scaled time variable
× (4πη/s)3/2 × 〈sτ〉−1/2 × (4π2νeff/90)1/2︸ ︷︷ ︸phenomenological input
1
10
100
1000
0 0.5 1 1.5 2 2.5 3 3.5 4
τhydro τchem τtherm
η/s = 0.16
e[G
eV/fm
3]
τ [fm]
totalgluons
fermions Input:η/s ≈ 0.16〈sτ〉 ≈ 4.1 GeV2
νeff ≈ 40
τhydro︸ ︷︷ ︸±10% viscous e(τ)
< τchem︸ ︷︷ ︸±10% fermion eq. e(τ)
< τtherm︸ ︷︷ ︸±10% ideal e(τ)
11 / 17
Part III: Entropy production and hydrodynamicattractors
Particle production in nucleus-nucleus collisions
Most basic question: how many particles will be produced in a collision?
Sorensen, Quark-gluon plasma 4, 2010
⟨dNch
dη
⟩⟨dE⊥dη
⟩0
Final multiplicity is driven by entropy production in pre-equilibrium stages.12 / 17
Boost-invariant equations of motion of 1D expansion at early times
Energy-momentum conservation Tµν = diag (e, PT , PT , PL)
∂τe = −e+ PLτ
,
τ < R
Need microscopic input: constitutive relation PL = PL(e, τ).
Equilibrium: equation of state
PLe≈ 1
3=⇒ e ∝ τ− 4
3 .
Near-equilibrium: viscous constitutive equations
PLe
=1
3− 16
9
η/s
τT+ . . . .
η/s —specific shear-viscosity.
Macroscopic evolution far from equilibrium?
13 / 17
Macroscopic theory of equilibration: hydrodynamic attractors
Apparent emergence of constitutive relations far-from-equilibriumHeller and Spalinski (2015)
PLe
= f
[w =
τTeff
4πη/s
], where Teff ∝ e1/4.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1 1 10
PL/e
w = τTeff/(4πη/s)
QCD kineticsBoltzmann RTA
YM kineticsAdS/CFT
viscous hydro
see reviews by Florkowski, Heller and Spalinski (2017), Romatschke and Romatschke (2017) [?, ?] 14 / 17
Similarities of energy evolution in different theories
Integrate equations of motion to find energy attractor Giacalone, AM, Schlichting (2019)
eτ4/3
(eτ4/3)therm≡ E(w)
0.1 1 10w = τTeff/(4πη/s)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
En
ergy
attr
acto
r:τ
4/3e(τ)/
(τ4/
3e)
hyd
ro
C∞ = 0.87 QCD kinetics
C∞ = 0.92 Boltzmann RTA
C∞ = 0.98 YM kinetics
C∞ = 1.06 AdS/CFT
free streaming
viscous hydro
Universal late/early asymptoticsViscous hydro:E(w � 1) = 1− 2
3πw + . . .Free-streaming (e ∼ τ−1):E(w � 1) = C−1
∞ w4/9
C∞ varies only by 20%!
Final-state entropy: dSdy = A⊥(sτ)τtherm
∝(eτ
43
) 34
τtherm
=(
(eτ43 )0/E(w0)
) 34
15 / 17
Entropy-production from hydrodynamic attractor
One-to-one map of initial energy deposition and final particle multiplicity.⟨dNch
dη
⟩︸ ︷︷ ︸final-state
≈ A⊥Nch
S
4
3C3/4∞
(4πη
s
)1/3(π2
30νeff
)1/3
︸ ︷︷ ︸medium properties
(1
A⊥
⟨dE⊥dη
⟩0
)2/3
︸ ︷︷ ︸initial-state
All relevant-prefactors and powers included!
4πη/s ∼ 1− 3 not very well constrained at higher temperatures.
C∞ = 0.87− 1.06 surprisingly robust among theories.
νeff ≈ 40 for equilibrated QGP.
Nch/S = 6.7± 0.8 Hanus, AM and Reygers (2019) (c.f. 7.7-8.5 in HRG)
A⊥ – transverse area of nuclei overlap.
Can now describe equilibration of initial-state energy density (eτ)0 andrelated it to experimental data!
16 / 17
Summary and Outlook
Scaling and pre-scaling present in full QCD kinetic theory evolution.
α(τ), β(τ), γ(τ)—new hydrodynamic-like degrees of freedom forevolution not around equilibrium.
Ordering of equilibration time-scales at moderate couplings
τhydro < τchem < τtherm
Hydrodynamic attractors as a direct link between initial and finalstates: simple formula for final state entropy.
Outlook:
Can “bottom-up” thermalization be understood as (pre)-scaling +hydrodynamic attractor?
Can chemical equilibration be observed, e.g. pre-equilibrium photons?
Equilibration of event-by-event fluctuation, e.g. with KøMPøST
17 / 17
Initial state energy density
Bjorken formula for initial state energy density
eBjorken0 ≈ 1
τ0A⊥
dEfinal⊥dy
.
Does not include work done during expansion!
dE initial⊥dy
= A⊥(τe)0 >dEfinal⊥dy
.
Including the longitudinal work during expansion in central Pb-Pb get
e0 ≈ 270 GeV/fm3
(τ0
0.1fm/c
)−1( C∞0.87
)−9/8( η/s
2/4π
)−1/2
(A⊥
138fm2
)−3/2(dNch/dη
1600
)3/2 (νeff
40
)−1/2(S/Nch
7.5
)3/2
,
c.f. e ≈ 0.3 GeV/fm3 near QCD cross-over.
18 / 17
Extracting scaling exponents from integral moments
Pre-scaling evolution fg(p⊥, pz, τ) = τα(τ)fS(τβ(τ)p⊥, τγ(τ)pz)
imposes relations between integral moments
nm,n(τ) ≡∫ppm⊥ |pz|nfg(p⊥, pz, τ) ∼ τα(τ)−(m+2)β(τ)−(n+1)γ(τ)
Momentum range of scaling ⇔ number of moments obeying scaling.
Consider 5 triples of moments: {1, p⊥, |pz|}, {1, p2⊥, p
2z},
{p⊥, p2⊥, p⊥|pz|}, {p2
⊥, p3⊥, p⊥|pz|}, {1, p3
⊥, |pz|3}Integrals of Boltzmann equation ⇒ equations of motion for moments
τππµν + πµν = 2ησµν / τ log τ α+ α = α∞τ
µ(τ)−α(τ)+1.
Relaxation to hydrodynamic solution / relaxation to scaling solution
19 / 17
Time dependent exponents
fg(p⊥, p⊥, τ) = τα(τ)fS(τβ(τ)p⊥, τγ(τ)pz)
−1.5
−1
−0.5
0
0.5
1
1.5
0.01 0.1 1
1/3
1/4
−2/3
−3/4
σ0 = 0.1
expo
nent
s
τ
α(τ)β(τ)γ(τ)
Time evolution encoded into a few hydrodynamic degrees of freedom.20 / 17
Dependence on initial conditions
Vary initial gluon occupation σ0 = 0.1, 0.6: fg = σ0g2e−(p2⊥+ξ2p2z)
−1.5
−1
−0.5
0
0.5
1
1.5
0.01 0.1 1
1/3
1/4
−2/3
−3/4
σ0 = 0.1
expo
nent
s
τ
α(τ)β(τ)γ(τ)
−1.5
−1
−0.5
0
0.5
1
1.5
0.01 0.1 1
1/3
1/4
−2/3
−3/4
σ0 = 0.6
expo
nent
sτ
α(τ)β(τ)γ(τ)
scaling
pre-scaling
scaling
pre-scaling
Non-universal pre-scaling evolution of α(τ), β(τ), γ(τ)
21 / 17
Weighted momentum distribution
f(t, p) = tαfS(tβp)
d3p 1pf(p) d3p f(p) d3p pf(p)
energynumberDebye
p/T
Moments of distribution function probe different momentum scales.22 / 17
Energy deposition in high energy nucleus-nucleus collisions
Collisions of glasma sheets in color-glass condensate effective theory
b
y
x
QAs QB
sQAs QB
s
Local saturation scale is proportional to nuclear thickness
Q2s(x⊥) ∝ T (x⊥).
Gluon liberation (up to log–corrections)
gluon number (nτ)0(x⊥) ∝ T<(x⊥),
gluon energy (eτ)0(x⊥) ∝ T<(x⊥)√T>(x⊥).
Can now determine centrality dependence of dNch/dη
23 / 17
Universal centrality dependence of particle multiplicity
Collapse of rescaled multiplicity =⇒ compare with theory models⟨dNch
dη
⟩∝ dStherm
dη︸ ︷︷ ︸equilibration
,dNgluons
dη︸ ︷︷ ︸no equilibration
,
⟨dStherm
dη
⟩︸ ︷︷ ︸
e-by-e fluctuations
.
0
500
1000
1500
〈dN
ch/dη〉
w/ pre-equilibrium
w/o pre-equilibrium
w/ pre-eq.+fluctuations
Xe-Xe 5.44 TeV (×1.37)
Pb-Pb 2.76 TeV
U-U 193 GeV (×1.88)
Au-Au 130 GeV (×2.68)
Cu-Cu 62 GeV (×11.43)
0 20 40 60
centrality [%]
0.91.01.1
rati
o
b
y
x
TA TB
centrality = πb2/σAA
Entropy production and e-by-e fluctuations improve agreement with data.24 / 17
Equilibration of perturbations
Non-linearities change the perturbation spectra
stherm ∝ e230 =⇒ δstherm
stherm=
2
3
δe
e0.
k = 0 perturbation evolution in kinetic theory: δe/(e+ T xx) = const.Keegan, Kurkela, AM and Teaney (2016) [3]
0.85
0.90
0.95
1.00
1.05
1 10 100 1000
89
τQs
δe/(e + T xx) / init. val.δe/e / init. val.
δetherm
etherm=
8
9
δe0
e0
25 / 17
Equilibration of perturbations
Non-linearities change the perturbation spectra
stherm ∝ e230 =⇒ δstherm
stherm=
2
3
δe
e0=
3
4× 8
9
δe0
e0.
k = 0 perturbation evolution in kinetic theory: δe/(e+ T xx) = const.Keegan, Kurkela, AM and Teaney (2016) [3]
0.85
0.90
0.95
1.00
1.05
1 10 100 1000
89
τQs
δe/(e + T xx) / init. val.δe/e / init. val.
δetherm
etherm=
8
9
δe0
e0
25 / 17
Non-thermal fixed point (NTFP) for gauge theories
For f ∼ A2 � 1 classical-statistical Yang-Mills describes gluon evolutionAarts, Berges (2002), Mueller, Son (2004), Jeon (2005)
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0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Glu
on
dis
trib
uti
on
: g2 f(
p T=
Q,p
z,τ)
Longitudinal momentum: pz / Q-4 -3 -2 -1 0 1 2 3 4
0
2
4
6
8
10
Res
cale
d d
istr
ibu
tio
n: (
Qτ)
-α g
2 f(p T
=Q
,pz,
τ)
Rescaled momentum: (Qτ)γ pz / Q
Berges, Schenke, Schlichting, Venugopalan (2014) [19] Berges, Boguslavski, Schlichting, Venugopalan (2014) [9]
initial plasma instabilities scaling rescaledlater evolution
Self-similar scaling =⇒ simplification of non-equilibrium physics
fg(p⊥, pz, τ) = ταfS(τβp⊥, τγpz), τ =
√t2 − z2
Universal exponents: α ≈ −23 , β ≈ 0, γ ≈ 1
3scaling in other systems: Orioli et al. (2015) [12], Mikheev et al. (2018) [13], Prufer et al. (2018) [14], Erne et al. (2018) [15]
26 / 17
Comparison between constant and time dependent exponents
10−3
10−2
10−1
0.1 1
∆τ/τ = 0.28
τ2/3g2f g
(τ,p
⊥,p
z=
0)
p⊥
elastic scatteringsQCD kinetic theory
0.01
0.1
1
τ
10−3
10−2
10−1
0.1 1
1/p⊥
∆τ/τ = 0.28
τ−
α(τ
)g2f g
(τ,p
⊥,p
z=
0)
p⊥
QCD kinetic theory
0.01
0.1
1
τ
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 0.05 0.1 0.15 0.2
τ2/3g2f g
(τ,p
⊥=
1,p
z)
τ1/3pz
elastic scatteringsQCD kinetic theory
0.01
0.1
1
τ
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0 0.05 0.1 0.15 0.2
τ−
α(τ
)g2f g
(τ,p
⊥=
1,p
z)
τγ(τ)pz
QCD kinetic theory
0.01
0.1
1
τ
scaling pre-scaling
.
27 / 17
Estimates of entropy production in central Au-Au collisions at RHIC
Particle multiplicity is directly proportional to entropy at thermalization⟨dS
dy
⟩τtherm
= 〈sτA⊥〉τtherm≈ S
Nch
⟨dNch
dη
⟩.
Muller and Schafer (2011)
Most of entropy production occurs at early times during equilibration.28 / 17
Bibliography I
[1] Peter Brockway Arnold, Guy D. Moore, and Laurence G. Yaffe.Effective kinetic theory for high temperature gauge theories.JHEP, 01:030, 2003, hep-ph/0209353.
[2] Aleksi Kurkela and Yan Zhu.Isotropization and hydrodynamization in weakly coupled heavy-ion collisions.Phys. Rev. Lett., 115(18):182301, 2015, 1506.06647.
[3] Liam Keegan, Aleksi Kurkela, Aleksas Mazeliauskas, and Derek Teaney.Initial conditions for hydrodynamics from weakly coupled pre-equilibrium evolution.JHEP, 08:171, 2016, 1605.04287.
[4] Aleksi Kurkela, Aleksas Mazeliauskas, Jean-Francois Paquet, Soren Schlichting, and DerekTeaney.Effective kinetic description of event-by-event pre-equilibrium dynamics in high-energyheavy-ion collisions.Phys. Rev., C99(3):034910, 2019, 1805.00961.
[5] Aleksi Kurkela, Aleksas Mazeliauskas, Jean-Francois Paquet, Soren Schlichting, and DerekTeaney.Matching the Nonequilibrium Initial Stage of Heavy Ion Collisions to Hydrodynamics withQCD Kinetic Theory.Phys. Rev. Lett., 122(12):122302, 2019, 1805.01604.
28 / 17
Bibliography II
[6] Aleksi Kurkela and Aleksas Mazeliauskas.Chemical equilibration in hadronic collisions.Phys. Rev. Lett., 122:142301, 2019, 1811.03040.
[7] Aleksi Kurkela and Aleksas Mazeliauskas.Chemical equilibration in weakly coupled QCD.Phys. Rev., D99(5):054018, 2019, 1811.03068.
[8] Aleksas Mazeliauskas and Jurgen Berges.Prescaling and far-from-equilibrium hydrodynamics in the quark-gluon plasma.Phys. Rev. Lett., 122(12):122301, 2019, 1810.10554.
[9] J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan.Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies.Phys. Rev., D89(7):074011, 2014, 1303.5650.
[10] R. Baier, Alfred H. Mueller, D. Schiff, and D. T. Son.’Bottom up’ thermalization in heavy ion collisions.Phys. Lett., B502:51–58, 2001, hep-ph/0009237.
[11] Juergen Berges, Kirill Boguslavski, Soeren Schlichting, and Raju Venugopalan.Universal attractor in a highly occupied non-Abelian plasma.Phys. Rev., D89(11):114007, 2014, 1311.3005.
28 / 17
Bibliography III
[12] A. Pineiro Orioli, K. Boguslavski, and J. Berges.Universal self-similar dynamics of relativistic and nonrelativistic field theories nearnonthermal fixed points.Phys. Rev., D92(2):025041, 2015, 1503.02498.
[13] Aleksandr N. Mikheev, Christian-Marcel Schmied, and Thomas Gasenzer.Low-energy effective theory of non-thermal fixed points in a multicomponent Bose gas.2018, 1807.10228.
[14] Maximilian Prufer, Philipp Kunkel, Helmut Strobel, Stefan Lannig, Daniel Linnemann,Christian-Marcel Schmied, Jurgen Berges, Thomas Gasenzer, and Markus K. Oberthaler.Observation of universal dynamics in a spinor Bose gas far from equilibrium.Nature, 563(7730):217–220, 2018, 1805.11881.
[15] Sebastian Erne, Robert Bucker, Thomas Gasenzer, Jurgen Berges, and JorgSchmiedmayer.Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium.Nature, 563(7730):225–229, 2018, 1805.12310.
[16] Raphael Micha and Igor I. Tkachev.Turbulent thermalization.Phys. Rev., D70:043538, 2004, hep-ph/0403101.
28 / 17
Bibliography IV
[17] Naoto Tanji and Juergen Berges.Nonequilibrium quark production in the expanding QCD plasma.Phys. Rev., D97(3):034013, 2018, 1711.03445.
[18] Mauricio Martinez, Matthew D. Sievert, and Douglas E. Wertepny.Multiparticle Production at Mid-Rapidity in the Color-Glass Condensate.JHEP, 02:024, 2019, 1808.04896.
[19] Jurgen Berges, Bjorn Schenke, Soren Schlichting, and Raju Venugopalan.Turbulent thermalization process in high-energy heavy-ion collisions.Nucl. Phys., A931:348–353, 2014, 1409.1638.
28 / 17