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Hydrodynamization at weak coupling

Aleksi KurkelaAK, Wiedemann in progress

AK, Mazeliauskas, Paquet, Schlichting, Teaney, in progressKeegan, AK, Mazeliauskas, Teaney JHEP 1608 (2016) 171

AK, Zhu PRL 115 (2015) 18, 182301AK, Lu PRL 113 (2014) 18, 182301AK, Moore JHEP 1111 (2011) 120AK, Moore JHEP 1112 (2011) 044

Oxford, July 2017

1 / 35

Motivation?

Pre thermal plasma Locally thermalised plasmaLorentz contracted nuclei

Soft physics of HIC described by relativistic hydrodynamics

∂µTµν = 0

Gradient expansion around local thermal equilibrium

Tµν = Tµνeq. − η2∇<µuν> + . . .

2 / 35

Motivation?

Anisotropy: PL/P

T

Time: τ

+1

0

Hydro

τi

At early times pre-equilibrium evolution

Hydro simulations start at intialization time τi

3 / 35

Motivation:

Anisotropy: PL/P

T

Time: τ

+1

0

Hydro

τi

Pre-eq.

If prethermal evolution converges smoothly to hydro,independence of unphysical τi

In most current pheno: either free streaming, or nothing at all

4 / 35

Motivation:

Anisotropy: PL/P

T

Time: τ

+1

0

Hydro

τi

Pre-eq.

If prethermal evolution converges smoothly to hydro,independence of unphysical τi

In most current pheno: either free streaming, or nothing at all

4 / 35

Motivation:

In AA collisions: pre-equilibrium evolution ∼ 10% of the evolution

Pre-equilibrium evolution major uncertainty affects η/s, etc

In pA collisions: currently no quantitative description

even if the system becomes hydrodynamical, ”pre-equilibrium”evolution O(1) of the evolution

pp collisions: ?????

5 / 35

Motivation:

In AA collisions: pre-equilibrium evolution ∼ 10% of the evolutionPre-equilibrium evolution major uncertainty affects η/s, etc




5 / 35

Motivation:





5 / 35

Motivation:





5 / 35

Hydrodynamization in weak coupling

Anisotropy: PT / P

L

Occupancy: f

OveroccupiedUnderoccupied

Thermal

Initial

f~1f~α f~α−1

Color Glass Condensate: Initial condition overoccupiedMcLerran, Venugopalan PRD49 (1994) , PRD49 (1994); Gelis et. al Int.J.Mod.Phys. E16 (2007),

Ann.Rev.Nucl.Part.Sci. 60 (2010)

f(Qs) ∼ 1/αs, Qs ∼ 2GeV

Expansion makes system underoccupied before thermalizingBaier et al PLB502 (2001)

f(Qs)� 1

6 / 35


Anisotropy: PT / P

L

Occupancy: f

Thermal

Kinetic theory Classical

YM

Initial

f~1f~α f~α−1

Both

Degrees of freedom:

f � 1: Classical Yang-Mills theory (CYM)f � 1/αs: (Semi-)classical particles, Eff. Kinetic Theory (EKT)

6 / 35


Anisotropy: PT / P

L

Occupancy: f

Thermal

Kinetic theory Classical

YM

Initial

f~1f~α f~α−1

Both

Transmutation of fields to particles: Field-particle dualitySon, Mueller PLB582 (2004) 279-287; Jeon PRC72 (2005) 014907; Mathieu et al EPJ. C74 (2014)

2873; AK et al PRD89 (2014) 7, 074036

1� f � 1/αs

”Bottom-up thermalization” of underoccupied system

6 / 35

Strategy at weak coupling

Anisotropy: PL/P

T

Time: τ

+1

0

Hydro

τi~1fm/cCYM

EKT

τEKT

~0.1 fm/c

Strategy: Switch from CYM to EKT at τEKT , 1� f � 1/αs

From EKT to hydro at τi, PL/PT ∼ 1

7 / 35

Early times 0 < Qsτ . 1: classical evolution

Time: Qsτ-1

PT/ε

PL/ε

+1

0

Epelbaum & Gelis, PRL. 111 (2013) 23230

Melting of the coherent boost invariant CGC fieldsInitial condition from CGC: MV-model, JIMWLK

After τ ∼ 1/Qs, fields decohere, PL > 0

8 / 35

Later times Qsτ > 1: classical evolution

Berges et al. Phys.Rev. D89 (2014) 7, 074011

Anisotropy: PT / P

L

Occupancy: f


Thermal

Initial

f~1f~α f~α−1

Numerical demonstration of overoccupied part of the diagram

Classical theory never thermalizes or isotropizes

9 / 35

Effective kinetic theory of Arnold, Moore, YaffeJHEP 0301 (2003) 030

Soft and collinear divergences lead to nontrivial matrix elementssoft: screening, Hard-loop; collinear: LPM, ladder resum

= Re

∗

No free parameters; LO accurate in the αs → 0, αsf → 0 limit, for

∆t ∼ ω−1 > Typical scattering time ∼ 1/(α2T )

Caveat: in anistropic systems screening complicated. Here withisotropic screening. Also no fermions here, yet

plasma instabilities, . . .

10 / 35

Outline

Hydrodynamization and thermalization of homogenous systems

Hydrodynamization with spatial inhomogenities

Hydrodynamization as decay of non-hydrodynamic modes andanalytic structure of Green functions

11 / 35

Anisotropy: PT / P

L

Occupancy: f


Thermal

Initial

f~1f~α f~α−1

Isotropic overoccupied: Transmutation of d.o.f’s

Isotropic underoccupied: Radiative break-up

Effect of longitudinal expansion: Hydrodynamization

12 / 35

Overoccupied cascade AK, Moore JHEP 1112 (2011) 044

What happens if you have too many soft gluons, f ∼ 1/αs.

ln(p)

ln(f)

Thermal

f ~ 1

Initial condition

(eβp-1)-1

1/α

Q

13 / 35

Overoccupied cascade AK, Moore JHEP 1112 (2011) 044

What happens if you have too many soft gluons, f ∼ 1/αs.No longitudinal expansion.

ln(p)

ln(f)

Thermal

f ~ 1

Initial condition

(eβp-1)-1

Self-similar cascade

pmax

~ t1/7

f(pmax

)~ t-4/7

1/α

Q

τinit ∼ [σn(1 + f)]−1 ∼(Q

T

)7 1

α2sT� 1

α2sT∼ τthem.

c.f. Bokuslawski’s talk

13 / 35

Overoccupied cascade AK, Lu, Moore, PRD89 (2014) 7, 074036

0.01 0.1 1 10

Momentum p~ = p/Q (Qt)

-1/7

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

Res

cale

d o

ccupan

cy:

f~=

λ f

(Q

t)4/7

Lattice (continuum extrap.)

Lattice (large-volume)

Lattice and Kinetic Thy. Compared

Form of cascade from classical lattice simulation,

1� f . 1/αs

Large-volume: (Qa)=0.2, (QL)=51.2, Cont. extr.: down to (Qa)=0.1, (QL)=25.6, Qt=2000, m̃ = 0.0813 / 35

Overoccupied cascade AK, Lu, Moore, PRD89 (2014) 7, 074036

0.01 0.1 1 10

Momentum p~ = p/Q (Qt)

-1/7

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

Res

cale

d o

ccupan

cy:

f~=

λ f

(Q

t)4/7

Kinetic thy (discrete-p)

Lattice (continuum extrap.)

Lattice (large-volume)

Lattice and Kinetic Thy. Compared

Same system, very different degrees of freedom

1 . f � 1/αs

Numerical demonstration of field-particle duality13 / 35

Ending of the overoccupied cascade AK, Lu PRL 113 (2014) 18, 182301

1 10 100

Rescaled time: λ2Tt (1+C

2 logλ

-1)

0.5

1

2

Mom

entu

m s

cale

s

<p>/<p>T

T*/T

m/λ1/2

T

λ=0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0

teq

Thermal equilibrium reached within the kinetic theory

teq ≈72.

1 + 0.12 log λ−11

λ2T

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Anisotropy: PT / P

L

Occupancy: f


Thermal

Initial

f~1f~α f~α−1


Isotropic underoccupied: Bottom-up thermalization

Effect of longitudinal expansion: Hydrodynamization

15 / 35

Underoccupied cascade: Formation of thermal bath

1/m

Soft modes quick to emit

Γel

αΓel ∼ α2

s

n

m2D

∼ α2s

∫p f

αs∫p f/p

nsoft ∼ αs Γel t

Low-p: easy to thermalize

Can dominate the dynamicsscattering, screening, . . .

⇒ Few energetic “jets” propagating in thermalbath

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Underoccupied cascade: Radiational breakup

⊥Δp2~qt^

In vacuum: on-shell splitting kin. disallowed

In medium:

frequent soft scatterings with medium, mom. diffusion: ∆p2 ∼ q̂tScatterings lead to virtuality: P 2 ∼ q̂tNow offshell particle may split collinearly: tf ∼ Q/P 2 ∼

√Q/q̂

Splitting time (per particle) tsplit(Q) ∼ 1αstf ∼ 1

αs

√Qq̂

QED: Landau, Pomeranchuk, Migdal 1953.

QCD: Baier Dokshitzer Mueller Peigne Schiff hep-ph/9607355

17 / 35

Underoccupied cascade: Radiational breakup

E

t (E)split

t (E/2)split

t (E/4)

. . .

. . .

. . .

. . .

TE/4E/2

split

Successive splittings happen in faster times scales:

tquench(Q) ∼ tsplit(Q) + tsplit(Q/2) + tsplit(Q/4) + . . . ∼ tsplit(Q)

Once the parton has had time to split it cascades its energy to IR,T increases.

18 / 35

Bottom-up thermalization AK, Lu, PRL 113 (2014) 18, 182301

0.1 1 10 100Momentum: p/T

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

Occ

up

ancy

: f

Initial condition

Q/T = 404.9λ = 0.1

Start with an underoccupied initial condition p ∼ Qafter a very short time, an IR bath is created (1↔ 2–processes)

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1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

Occ

up

ancy

: f

λ2Tt(T/Q)

1/2=100

502512.56.25

Q/T = 404.9λ = 0.1

teq

, 147

More energy flows to the IR, temperature increases, “Bottom-up”When “bottom” reaches final T , “up” is quenched

AK, Moore JHEP 1112 (2011) 044

teq ∼ (Q/T )1/21

α2sT

19 / 35



1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1O

ccu

pan

cy:

f

λ2Tt(T/Q)

1/2= 200

250300

Q/T = 404.9λ = 0.1

teq

, 147

Hardest scales reach equilibrium last.Close resemblance to Blaizot, Iancu, Mehtar-tani for jets PRL 111 (2013) 052001

19 / 35

Anisotropy: PT / P

L

Occupancy: f


Thermal

Initial

f~1f~α f~α−1


Isotropic underoccupied: Radiative break-up

Application to HIC: effect of longitudinal expansion

20 / 35

Route to equilibrium in EKT AK, Zhu, PRL 115 (2015) 18, 182301

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

Anisotropy: PT / P

L

Occupancy: f


Thermal

Initial

f~1f~α f~α−1

Initial condition (f ∼ 1/αs) from classical field thy calculationLappi PLB703 (2011) 325-330

In the classical limit (αs → 0, αsf fixed), no thermalization

At small values of couplings, clear Bottom-Up behaviour

Features become less defined as αs grows

21 / 35

Route to equilibrium in EKT

p2f(p⊥, pz)

pz−pz

p⊥

1

10

100

0.1 1 10

anis

otro

py

occupancy

p2f(p⊥, pz)

pz−pz

p⊥

1

10

100

0.1 1 10

anis

otro

py

occupancy

p2f(p⊥, pz)

pz−pz

p⊥

1

10

100

0.1 1 10

anis

otro

py

occupancy

p2f(p⊥, pz)

pz−pz

p⊥

1

10

100

0.1 1 10

anis

otro

py

occupancy

22 / 35

Smooth approach to hydrodynamics AK, Zhu, PRL 115 (2015) 18, 182301

αs = 0.03

1 10 100 1000 10000Time: Qτ

0.01

0.1

1

Com

ponen

ts o

f τ4

/3Τ

µ ν/Q

8/3

Kinetic thy.

1/3 τ4/3

ε

τ4/3

PL

τ4/3

PT

Kinetic theory converges to hydro smoothly and automatically

Approach to hydro fixed by perturbative η/sArnold et al. JHEP 0305 (2003) 051

∂τ ε = −4

3

ε

τ+

4η

3τ2, PL =

ε

3− 4η

3τ

23 / 35


αs = 0.03

1 10 100 1000 10000Time: Qτ

0.01

0.1

1

Com

ponen

ts o

f τ4

/3Τ

µ ν/Q

8/3

Kinetic thy.

1st order hydro

1/3 τ4/3

ε

τ4/3

PL

τ4/3

PT

Kinetic theory converges to hydro smoothly and automatically

Approach to hydro fixed by perturbative η/sArnold et al. JHEP 0305 (2003) 051

∂τ ε = −4

3

ε

τ+

4η

3τ2, PL =

ε

3− 4η

3τ23 / 35


αs = 0.3

1 10 100Qτ

0.001

0.01

Kinetic thy.

1st order hydro

2nd order hydro

0.1 1 10τ[fm/c]

0.001

0.01

Co

mp

on

ents

of

τ4

/3T

µ ν/Q

8/3

1/3 τ4/3

ε

τ4/3

PL

τ4/3

PT

For realistic couplings, hydrodynamics reached around . 1fm/c.

Hydro gives a good description even when PL/PT ∼ 1/5

24 / 35

Outline

Hydrodynamization and thermalization of homogenous systems

Hydrodynamization with spatial inhomogenities

Hydrodynamization as decay of non-hydrodynamic modes andanalytic structure of Green functions

25 / 35

Transverse dynamics and preflowa

a

a

t = 0 t = 1fm/c

a

a

Pre-equilibrium evolution leads to:

smearing of the nuclear geometry

Generation of preflow due to gradients26 / 35

Transverse dynamics and preflowa

a

a

t = 0 t = 1fm/c

a

a

Nuclear radius R� cτi ∼ Nucleon radius Rp

Transverse structure small perturbation within the causal horizon

Linear response theory for the transverse structures26 / 35

Transverse dynamics and preflow

Non-equilibrium Green functions on top of non-equilibriumbackground E computed in kinetic theory Keegan et al.JHEP 1608 (2016) 171

27 / 35

Linearized perturbations in EKT Keegan et al. 1605.04287

Transverse perturbations characterized by wavenumber k

f(x⊥,p) = f̄(p) + exp(ix · k)δf(p)

(∂τ −

pzτ∂pz

)f = C[f ]

(∂τ −

pzτ∂pz + ik · p

)f = C[f̄ , f ]

For thermal f̄ : large wavelenght pert. described by hydro

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1k/T

0.55

0.6

0.65

0.7

0.75

Re[

ω(k

)/k ]

Dispersion relation λ=10

2nd order hydro

Ideal hydro cs

2=1/3

EKTω(k)

k= c2s +

4

3

η

e+ p

(csτπ −

2

3cs

η

e+ p

)k2

For larger k, c2s → 1, with polynomialdecay

no plot unfortunately...

28 / 35

Hydrodynamization of perturbations Keegan et al.JHEP 1608 (2016)

δT xx = δee

[13e+ 1

3ητπk2 + η

2τ −2(λ1−ητπ)

9τ2

]− ikδT 0x

e

[η − 1

τ

(η2

2e + ητπ2 − 2

3λ1

)]

k ∼ 1/Rnucleus

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1 10 100 1000

k/Teff

k = 0.01Qs

δTxx/

T00

τQs

kinetic theory2nd order hydro1st order hydroideal hydro

k ∼ 1/Rproton

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1 10 100 1000

k = 0.1Qs

δTxx/

T00

τQs

Perturbations hydrodynamize also at Qτ ∼ {10, 20}.

29 / 35

Hydrodynamization of perturbations Keegan et al.JHEP 1608 (2016)

δT xx = δee

[13e+ 1

3ητπk2 + η

2τ −2(λ1−ητπ)

9τ2

]− ikδT 0x

e

[η − 1

τ

(η2

2e + ητπ2 − 2

3λ1

)]

k ∼ 1/0.5Rnucleus

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1 10 100 1000

k = 0.2Qs

δTxx/

T00

τQs

k ∼ 1/0.25Rnucleus

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

1 10 100 1000

k = 0.4Qs

k/Teff > 0.6δT

xx/

T00

τQs

No hydrodynamics for the large-k modes

29 / 35

Green function in coordinate space

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

δe response to δe perturbationQsτ = 10

τ/τπ = 0.99

τ2E(r)

r/(τ − τ0)

free streamingkinetic theory

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

δe response to δe perturbationQsτ = 20

τ/τπ = 1.6τ2E(r)

r/(τ − τ0)

free streamingkinetic theory

Nanscent formation of dip in the origin hall mark of hydro

30 / 35

Green function in coordinate space

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

δe response to δe perturbation

Qsτ = 50

τ/τπ = 3.1

2nd hydro{

τ2E(r)

r/(τ − τ0)

kinetic theoryQsτ = 10Qsτ = 20

−3.0

−2.0

−1.0

0.0

1.0

2.0

3.0

4.0

0 0.2 0.4 0.6 0.8 1 1.2 1.4

δe response to δe perturbation

Qsτ = 500

τ/τπ = 15

2nd hydro{

τ2E(r)

r/(τ − τ0)

kinetic theoryQsτ = 10Qsτ = 20

Evolution after Qτi > {10, 20}, evolution described by hydro

31 / 35


IP-glasma + EKT + Hydro:

0

20

40

60

80

100eτ

4/3/[norm]

First hydro starts Second hydro starts Third hydro starts

0

20

40

60

80

100

-3 -2 -1 0 1 2 3

eτ4/3/[norm]

x fm

-3 -2 -1 0 1 2 3x fm

-3 -2 -1 0 1 2 3x fm

τ = 0.4 fm τ = 0.8 fm τ = 1.2 fm

τinit = 0.4 fmτinit = 0.8 fmτinit = 1.2 fm

τ = 1.5 fm τ = 2.0 fm τ = 3.0 fm

AK, Mazeliauskas, Paquet, Schlichting, Teaney, in progress

Initialization time dependence removed

32 / 35


Without EKT:

0

20

40

60

80

100eτ

4/3/[norm]

First hydro starts Second hydro starts Third hydro starts

0

20

40

60

80

100

-3 -2 -1 0 1 2 3

eτ4/3/[norm]

x fm

-3 -2 -1 0 1 2 3x fm

-3 -2 -1 0 1 2 3x fm

τ = 0.4 fm τ = 0.8 fm τ = 1.2 fm

τinit = 0.4 fmτinit = 0.8 fmτinit = 1.2 fm

τ = 1.5 fm τ = 2.0 fm τ = 3.0 fm

AK, Mazeliauskas, Paquet, Schlichting, Teaney, in progress

Strong dependence on initialization time!

33 / 35

What’s missing?

Public code to put this to use, (KoMPoST?)Soon: aK, Mazeliauskas, Paquet, Schlichting, Teaney

Role of fermions

production of quarks, chemical equilibration Berges et al. PRC 95 (2017)

Role of far-from-equilibrium plasma instabilities

Extensions to small systems

What is the smallest droplet of (weakly coupled) liquid that can be?Strong coupling: Chesler JHEP 1603 (2016) 146

Can flow-like behaviour arise from only few collisions? Flowwithout hydro?

Non-equilibrium correlation functions in strong coupling?Casalderrey-Solana, Meiring, van der Schee

34 / 35

What’s missing?

Jet loses energy to the medium: Jet thermalization

Same physics governs thermalization of the bulk and the medium

Experimental characterization of jet thermalization will teachabout hydrodynamization at intermediate coupling

35 / 35