Prescaling and Far-from-Equilibrium Hydrodynamics in the Quark … · 5=2 s ˝Qs˝˝ 13=5 s 6 / 16....

Prescaling and Far-from-Equilibrium Hydrodynamics inthe Quark-Gluon Plasma

Aleksas Mazeliauskas

Institut fur Theoretische PhysikUniversitat Heidelberg

April 5, 2019

AM and Jurgen Berges, Phys. Rev. Lett. 122, 122301 (2019)

Isolated quantum systems and universality in extreme conditions

Motivation

Relativistic heavy ion collisions

Heavy ion programs at RHIC (since 2000) and LHC (since 2010).

Sorensen, Quark-gluon plasma 4, 2010

Information loss =⇒ macroscopic (hydrodynamic) description of QCD.2 / 16

QCD kinetics

αs → 0, overlap with classical fields, early times

αs ≈ 0.3, overlap with hydrodynamics, late times

3 / 16

Equilibration in heavy ion collisions: weak coupling picture

At high energies and densities — asymptotic freedom αs � 1Gross, Wilczek; Politzer (1973)[1, 2]

�

J+ J

−

E⌘

B⌘

��

��

��

��

��

��

��

��

��

f ∼ 1αs� 1 1� f � 1

αsf ∼ 1

Soren Schlichting, Initial Stages 2016Color-Glass Condensate

QCD kinetic theory — bridge between early and late time dynamics.4 / 16

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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-0.6 -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) [3] Berges, Boguslavski, Schlichting, Venugopalan (2014) [4]

scaling rescaled

Self-similar scaling =⇒ loss of information

fg(p⊥, pz, τ) = ταfS(τβp⊥, τγpz), τ =

√t2 − z2

Universal exponents: α ≈ −23 , β ≈ 0, γ ≈ 1

3Scaling phenomena also seen in scalar theories, cold atom experimentsOrioli et al. (2015) [5], Mikheev et al. (2018) [6], Prufer et al. (2018) [7], Erne et al. (2018) [8] 5 / 16

QCD effective kinetic theory

Weakly coupled quark and gluon quasi-particles in a soft background.Arnold, Moore, Yaffe (2003)[9]

∂τfg,q −pzτ∂pzfg,q︸︷︷︸

expansion

= − C2↔2[f ]︸︷︷︸ − C1↔2[f ]︸︷︷︸Complete leading order description:

elastic 2↔ 2 scatterings: gg ↔ gg, qq ↔ qq, gq ↔ gq, gg ↔ qq

particle number changing 1↔ 2 processes: g ↔ gg, q ↔ qg, g ↔ qq(includes interference effects — LPM suppression)

only parameter — the coupling constant αs.

“Bottom-up” thermalization scenario Baier, Mueller, Schiff, and Son (2001)[10]

I) over-occupied pz ∼ Qs(Qsτ)1/3

1� 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

6 / 16

From classical simulations to kinetic theory

Initial distribution f0 ∼ 1g2θ(Qs −

√p2⊥ + ξ2p2z)

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)[12]

classical-statistical Yang-Mills kinetic theory of gluons

Occupancy

Anisotropy

7 / 16

Far-from-equilibrium dynamics with QCD kinetic theory

Scaling in leading order QCD kinetic theory with fermions

Initial conditions fg = σ0g2e−(p

2⊥+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

Non-thermal fixed point reached in full QCD kinetic evolution.

8 / 16

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 (2019) [13]

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)

Much earlier collapse to scaling solution fS — pre-scaling regime.9 / 16

Pre-scaling regime in QCD kinetic theory

Non-equilibrium dynamics undone by self-similar renormalization


AM and Berges (2019) [13]

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.9 / 16

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

.

10 / 16

Extracting exponents from integral moments

Pre-scaling evolution imposes relations between integral moments

nm,n(τ) ≡ νg∫

d3p

(2π)3pm⊥ |pz|nfg(p⊥, pz, τ),

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

∂ log nm,n(τ)

∂ log τ= α(τ)− (m+ 2)β(τ)− (n+ 1) γ(τ).

where we redefined the exponents to be τα(τ) → exp[∫ τ

1dττ α(τ)

]If all moments nm,n scale with the same α, β, γ ⇒ pre-scaling regime.

Consider 5 triples of moments: {1, p⊥, |pz|}, {1, p2⊥, p2z}, {p⊥, p2⊥, p⊥|pz|},{p2⊥, p3⊥, p⊥|pz|}, {1, p3⊥, |pz|3}

11 / 16

Time dependent exponents


−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 ≤ 312 / 16

Dependence on initial conditions

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

2⊥+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)

Time evolution of exponents =⇒ far-from-equilibrium hydrodynamics

∂µTµν(e, uσ) = 0 ⇐⇒ ∂µT

µν(α(τ), β(τ), γ(τ)) = 0

Hydrodynamics, which is not based on expansion around equilibrium!

13 / 16

Dependence on initial conditions

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

2⊥+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

s

τ

α(τ)β(τ)γ(τ)

scaling

pre-scaling

scaling

pre-scaling

Time evolution of exponents =⇒ far-from-equilibrium hydrodynamics

∂µTµν(e, uσ) = 0 ⇐⇒ ∂µT

µν(α(τ), β(τ), γ(τ)) = 0

Hydrodynamics, which is not based on expansion around equilibrium!13 / 16

Hydrodynamics far-from-equilibrium

Integrals of Boltzmann equation ⇒ equations of motion for moments

∂τf −pzτ∂pzf = −C[f ]

Consider Jµ = νg∫ppµ

p0fp, Iµνσ = νg

∫ppµpνpσ

p0fp,

∂τn+n

τ= −CJ ,

∂τIτxx +

Iτxx

τ= −CxxI ,

∂τIτzz +

3Iτzz

τ= −CzzI ,

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

2α(τ) + 2 log τ∂α(τ)

∂ log τ= −5

τCJn

+ 2τCxxIIτxx

+τCzzIIτzz

Scaling of the collision kernel closes the system.Berges, Mikheev and Mazeliauskas, work in progress

14 / 16

Beyond the first stage of Bottom-up

Dependence on the coupling strength (pure glue simulation)

Vary the coupling constant αs = g2/(4π)

−1.5

−1

−0.5

0

0.5

1

1.5

0.01 0.1 1

1/3

1/4

−2/3

−3/4

expo

nent

s

τ

α(τ)β(τ)γ(τ)

g2 = 10−6

1

10

100

0.01 0.1 1

anis

otro

pyP

T/P

L

occupancy 〈pλf〉 / 〈p〉

g2 = 10−6

g2 = 10−5

g2 = 10−4

g2 = 10−3

g2 = 10−2

g2 = 10−1

g2 = 100



1� Qsτ � α−3/2s



15 / 16



1

10

100

0.01 0.1 1

anis

otro

pyP

T/P

L


g2 = 10−6

g2 = 10−5

g2 = 10−4

g2 = 10−3

g2 = 10−2

g2 = 10−1

g2 = 100

−1.5

−1

−0.5

0

0.5

1

1.5

0.01 0.1 1

1/3

1/4

−2/3

−3/4

expo

nent

s

τ

α(τ)β(τ)γ(τ)

g2 = 10−3



1� Qsτ � α−3/2s



15 / 16



1

10

100

0.01 0.1 1

anis

otro

pyP

T/P

L


g2 = 10−6

g2 = 10−5

g2 = 10−4

g2 = 10−3

g2 = 10−2

g2 = 10−1

g2 = 100

−1.5

−1

−0.5

0

0.5

1

1.5

0.01 0.1 1

1/3

1/4

−2/3

−3/4

expo

nent

s

τ

α(τ)β(τ)γ(τ)

g2 = 10−2



1� Qsτ � α−3/2s



15 / 16



1

10

100

0.01 0.1 1

anis

otro

pyP

T/P

L


g2 = 10−6

g2 = 10−5

g2 = 10−4

g2 = 10−3

g2 = 10−2

g2 = 10−1

g2 = 100

−1.5

−1

−0.5

0

0.5

1

1.5

0.01 0.1 1

1/3

1/4

−2/3

−3/4

expo

nent

s

τ

α(τ)β(τ)γ(τ)

g2 = 10−1



1� Qsτ � α−3/2s



15 / 16



1

10

100

0.01 0.1 1

anis

otro

pyP

T/P

L


g2 = 10−6

g2 = 10−5

g2 = 10−4

g2 = 10−3

g2 = 10−2

g2 = 10−1

g2 = 100

−1.5

−1

−0.5

0

0.5

1

1.5

0.01 0.1 1

1/3

1/4

−2/3

−3/4

expo

nent

s

τ

α(τ)β(τ)γ(τ)

g2 = 100



1� Qsτ � α−3/2s



15 / 16

Outlook on early time dynamics

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

Scaling is present in full QCD kinetic theory evolution.

Found pre-scaling regime — even earlier simplification ofnon-equilibrium QGP evolution. AM and Berges (2019)

Pre-scaling in non-relativistic scalars/cold atoms?

α(τ), β(τ), γ(τ)—new hydrodynamic-like degrees of freedom.

∂µTµν(e, uσ) = 0 ⇐⇒ ∂µT

µν(α, β, γ) = 0

Far-from-equilibrium hydrodynamics?Berges, Mikheev and Mazeliauskas, work in progress

16 / 16

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