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
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
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
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
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
Hydrodynamization in weak coupling
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
Hydrodynamization in weak coupling
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
OveroccupiedUnderoccupied
Thermal
Initial
f~1f~α f~α−1
Numerical demonstration of overoccupied part of the diagram
Classical theory never thermalizes or isotropizes
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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, . . .
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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
OveroccupiedUnderoccupied
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
14 / 35
Anisotropy: PT / P
L
Occupancy: f
OveroccupiedUnderoccupied
Thermal
Initial
f~1f~α f~α−1
Isotropic overoccupied: Transmutation of d.o.f’s
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
16 / 35
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
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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.
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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)
19 / 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
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
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
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
OveroccupiedUnderoccupied
Thermal
Initial
f~1f~α f~α−1
Isotropic overoccupied: Transmutation of d.o.f’s
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
OveroccupiedUnderoccupied
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
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.
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
Smooth approach to hydrodynamics AK, Zhu, PRL 115 (2015) 18, 182301
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
Transverse dynamics and preflow
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
Transverse dynamics and preflow
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