Pre-equilibrium phenomena in Quark-Gluon Plasma · 2020. 1. 6. · Non-equilibrium QCD descriptions at weak coupling s!0 At high energies mid-rapidity is dominated by small Bjorken-xgluons

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⌘

��

��

��

��

��

��

��

��

��

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




(Qsτ)1/31� Qsτ � α−3/2

s


s � Qsτ � α−5/2s


s � Qsτ � α−13/5s

pz pz pz

pxpxpx


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



[2, 3, 5, 4]

4 / 17




(Qsτ)1/31� Qsτ � α−3/2

s


s � Qsτ � α−5/2s


s � Qsτ � α−13/5s

pz pz pz

pxpxpx


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



[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.


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)

! !

!"#$#%&!'()*'(+*%,&-&.$""/(01#%&20+*"%0)&3045&

!"#$%&'(%)*+,-$,)%&#%&'(%#.("/#00123(4%5*6

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!!""##$

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!

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>*8*+?

@01#%

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)*+,-$

,)%83(+4-

;%,&'<$($&7

'! %"" ")'! %) '()%$!!

#"' '! %"'()%$!!

#"'

:.("/#00123(42+,-$,

=#>(',4'#$%,&'<$($&$1,#(1'+#&*#'%#*612?*))30$1+##@3'29#:(3*%#0(',4'#*%#'#&$41#,)'(1# ########8A121#&A1#,7,&14#$,#,&2*%:(7#$%&12')&$%:#'%+#,&$((#5'2#52*4#1@3$($<2$34

%"'!

#"'()

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1<,+:,-=!20,)>,=!!=*?,)$:(5&%&)*/(+>*")*5+(:+,--9*

#

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⊥


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


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

Pre-equilibrium phenomena in Quark-Gluon Plasma · 2020. 1. 6. · Non-equilibrium QCD descriptions at weak coupling s!0 At high energies mid-rapidity is dominated by small Bjorken-xgluons

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