Prescaling and far-from-equilibrium hydrodynamics in the ... · Non-equilibrium QCD descriptions at...

Prescaling and far-from-equilibrium hydrodynamics inthe quark-gluon plasma

Aleksas Mazeliauskas

Institut fur Theoretische PhysikUniversitat Heidelberg

September 23, 2019

AM and Jurgen Berges, Phys. Rev. Lett. 122 (2019) (arXiv:1810.10554)

Isolated quantum systems and universality in extreme conditions

The Universal Law That Aims Time’s Arrow

Article by Natalie Wolchover, Quanta Magazine (2019)

Thermalization—the end stage of non-equilibrium phenomena.

c©Rolando Barry for Quanta Magazine

How do far-from-equilibrium quantum systems begin to thermalize?2 / 18

Heavy ion collision experiments at RHIC and LHC

study new phenomena “by distributing high energy or high nucleon densityover a relatively large volume.” – T.D. Lee, 1974

Sorensen, Quark-gluon plasma 4, 2010

Creation of dense QCD matter at extreme temperatures T ∼ 1012 K.3 / 18

Phase diagram of thermalized QCD matter

neutron star mergers

baryon chemical potential

tem

per

atu

re

heavy ion collisions, early universe

Ding, Karsch, Mukherjee (2015)

Quark-Gluon Plasma—the state with fundamental QCD constituents.4 / 18

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

5 / 18

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↔ 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 resummed propagators, screening mass mD ∼ gTBraaten, Pisarski (1990) [2]

Microscopic studies of QGP equilibration with QCD kinetic theory.Kurkela and Zhu (2015) [3], Keegan, Kurkela, AM and Teaney (2016) [4], Kurkela, AM, Paquet, Schlichting and Teaney

(2018)[5, 6], A. Kurkela, AM (2018) [7, 8], AM, J. Berges (2018)[9]

6 / 18

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 medium induced colinear 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(Landau–Pomeranchuk–Migdal suppression).

Microscopic studies of QGP equilibration with QCD kinetic theory.Kurkela and Zhu (2015) [3], Keegan, Kurkela, AM and Teaney (2016) [4], Kurkela, AM, Paquet, Schlichting and Teaney

(2018)[5, 6], A. Kurkela, AM (2018) [7, 8], AM, J. Berges (2018)[9]

6 / 18

“Bottom-up” thermalization scenario Baier, Mueller, Schiff, and Son (2001)[11]

Evolution of initially over-occupied hard gluons p ∼ Qs � ΛQCD

I) over-occupied pz ∼ Qs

(Qsτ)1/31� Qsτ � α−3/2s

II) under-occupied pz ∼√αsQs α−3/2s � Qsτ � α−5/2s

III) mini-jet quenching pz ∼ α3sQs(Qsτ) α−5/2s � 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) [10]

Kurkela and Zhu (2015), Keegan, Kurkela, AM and Teaney (2016), Kurkela, AM, Paquet, Schlichting and Teaney (2018)

[3, 4, 6, 5]

7 / 18




(Qsτ)1/31� Qsτ � α−3/2s



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



[3, 4, 6, 5]

7 / 18




(Qsτ)1/31� Qsτ � α−3/2s



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



[3, 4, 6, 5]

7 / 18

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)[12] Kurkela and Zhu (2015)[3]

classical-statistical Yang-Mills kinetic theory of gluons

Occupancy

Anisotropy

scalingscaling

8 / 18

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) [13] Berges, Boguslavski, Schlichting, Venugopalan (2014) [10]

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) [14], Mikheev et al. (2018) [15], Prufer et al. (2018) [16], Erne et al. (2018) [17]

9 / 18

Scaling in leading order QCD kinetic theory

Initial conditions fg = σ0g2e−(p

2⊥+ξ

2p2z), σ0 = 0.1, g = 10−3, ξ = 2Scaling 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.

10 / 18

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) [9]

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

Much earlier collapse to scaling solution fS — pre-scaling regime.11 / 18

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

.

12 / 18

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.13 / 18

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 solution ⇔ range of moments obeying scaling.

Integrals of Boltzmann equation ⇒ equations of motion for moments

∂τf −pzτ∂pzf = −C[f ]

Relaxation to hydrodynamic solution / relaxation to scaling solution

τππµν + πµν = 2ησµν / τ log τ α+ α = α∞τ

µ(τ)−α(τ)+1.

Time evolution encoded into few hydrodynamic degrees of freedom

14 / 18

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)

Closely related evolution of moments nm,n with 0 ≤ n,m ≤ 315 / 18

Dependence on initial conditions

Vary initial gluon occupation σ0 = 0.1, 0.6: fg = σ0g2e−(p

2⊥+ξ

2p2z)

−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

New hydrodynamic degrees of freedom α(τ), β(τ), γ(τ) far away fromequilibrium.

16 / 18

The onset of thermalization

Later initialization times =⇒ slower expansionlarger coupling =⇒ stronger interactions

−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)

0.1

0.01 0.1 1

PL/P

T

τT/(4πη/s)

1

10

0.0001 0.001 0.01

anis

otro

pyP

T/P

L

occupancy 〈pλf〉 / 〈p〉

Pre-scaling useful in studying the intermediate regimes of thermalizationscenario.

17 / 18

Summary and Outlook

fg(p⊥, pz, τ) = τα(τ)fS(τβ(τ)p⊥, τγ(τ)pz)

Scaling and pre-scaling present in full QCD kinetic theory evolution.AM and Berges (2019)

Pre-scaling in classical-statistical Yang-Mills/non-relativistic scalars?see work of Schmied, Mikheev, Gasenzer (2018)[18], poster of Chatrchyan, Geier, Berges

α(τ), β(τ), γ(τ)—new hydrodynamic-like degrees of freedom forevolution not around equilibrium.

Extending pre-scaling description towards the onset of thermalization.Berges, AM, Mikheev, work in progress

18 / 18

Bibliography I

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[2] Eric Braaten and Robert D. Pisarski.Soft Amplitudes in Hot Gauge Theories: A General Analysis.Nucl. Phys., B337:569–634, 1990.

[3] Aleksi Kurkela and Yan Zhu.Isotropization and hydrodynamization in weakly coupled heavy-ion collisions.Phys. Rev. Lett., 115(18):182301, 2015, 1506.06647.

[4] 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.

[5] 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.

18 / 18

Bibliography II

[6] 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.

[7] Aleksi Kurkela and Aleksas Mazeliauskas.Chemical equilibration in hadronic collisions.Phys. Rev. Lett., 122:142301, 2019, 1811.03040.

[8] Aleksi Kurkela and Aleksas Mazeliauskas.Chemical equilibration in weakly coupled QCD.Phys. Rev., D99(5):054018, 2019, 1811.03068.

[9] 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.

[10] 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.

[11] 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.

19 / 18

Bibliography III

[12] 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.

[13] 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.

[14] 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.

[15] 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.

[16] 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.

20 / 18

Bibliography IV

[17] 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.

[18] Christian-Marcel Schmied, Aleksandr N. Mikheev, and Thomas Gasenzer.Prescaling in a far-from-equilibrium Bose gas.Phys. Rev. Lett., 122(17):170404, 2019, 1807.07514.

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