Edmond Iancu InstitutdePhysiqueThéoriquedeSaclay · New Trends in High-Energy Physics Glasma, Flow...

Nucleus-nucleus collisions: Glasma, Flow & Thermalization

Edmond IancuInstitut de Physique Théorique de Saclay

FLOW FLUCTUATIONS

dNdφ∝ 1 +

∞∑

n=1

2vn cos n(φ− ψn)

=⇒⟨

dNpairs

d∆φ

⟩(flow)∝ 1 +

∞∑

n=1

2⟨

v 2n

⟩cos n(∆φ)

v2e2iψ2 ∝ ε2e2iΦ2 ≡ −{r2e2iφ}{r2}

(Holopainen, Niemi, Eskola, Phys.Rev.C83, 034901 (2011))

ψRP

~pt

φ

ψ2

ψ3

ψ1

MATT LUZUM (IPHT) FLOW FLUCTUATIONS QUARK MATTER 2011 5 / 13

New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 1 / 41

Nucleus-nucleus collisionsIntroduction to AA collisions

Bookkeeping

Inclusive gluon spectrum

Loop corrections

CERN

François Gelis – 2007 Lecture III / III – School on QCD, low-x physics, saturation and diffraction, Copanello, July 2007 - p. 9/65

Initial particle production

■ Dilute regime : one parton in each projectile interact

■ Dense regime : multiparton processes become crucial(+ pileup of many simultaneous scatterings)

“Dense–dense scattering” : much more complicated !

Non–linear effects enter at all levels

in both incoming wavefunctions: gluon saturation

in the scattering process: multiple interactions

in the partonic medium created by the early scattering: final–stateinteractions



Bookkeeping


Loop corrections

CERN







2 CGC weight functions: WY1[ρ1], WY2

[ρ2]

in the scattering process: multiple interactions




Bookkeeping


Loop corrections

CERN








[ρ2]

classical Yang–Mills equations with 2 sources: ρ1, ρ2




Bookkeeping


Loop corrections

CERN








[ρ2]

classical Yang–Mills equations with 2 sources: ρ1, ρ2

kinetic theory, hydrodynamics, quark-gluon plasma, ...


CGC factorization for AA collisions

Numerically solve classical YM equations with 2 sources (2D lattice)

DνFνµ(x) = δµ+ρ1(x) + δµ−ρ2(x)

Decompose the classical field Aµa in Fourier modes

B gluon spectrum for given configurations ρ1 and ρ2 (“event-by-event”)

Average over ρ1 and ρ2 using the CGC distributions of the nuclei⟨

dN

dY d2p⊥

⟩=

∫[Dρ1Dρ2]WYbeam−Y

[ρ1]WYbeam+Y[ρ2]

dN

dY d2p⊥

∣∣∣∣class

B JIMWLK evolution from Ybeam up to the rapidity Y of the produced gluon

Introduction

Bookkeeping

Classical fields

Factorization

● What is the problem ?

● Leading order

● Next to Leading Order

● Initial field perturbation

● JIMWLK Hamiltonian

● Extensions

Summary

CERN

François Gelis – 2007 Lecture III / IV – Hadronic collisions at the LHC and QCD at high density, Les Houches, March-April 2008 - p. 36

What is the problem ?

■ For the single gluon spectrum in AA collisions, one wouldlike to establish a formula such as :

fidN

d3~p

fl

=LLog

ZˆDρ1 Dρ2

˜WYbeam−y [ρ1] Wy+Ybeam

[ρ2]dN

d3~p

˛˛˛˛LO

with∂

∂YWY = HW

Y

p

ρ1ρ2 y + Ybeam- Ybeam

◆ All the leading logs of 1/x1,2 are absorbed in the W ′s

◆ The W ′s obey the JIMWLK evolution equation


The color field of a single nucleusWhat are the chromo-electric and magnetic fields created by aultrarelativistic nucleus ? (ρ(x) ∝ δ(x+) for a left mover)

non-Abelian generalization of the Liénard-Wiechert potentials

Weiszäcker-Williams fields describing quasi-real photons/gluons

Ea ⊥ Ba ⊥ z

E⊥ ·B⊥ = 0 , |E⊥| = |B⊥| ∼1

g

transverse polarizations

chromo-electromagnetic waves

Lorentz contraction: ∝ δ(x+)

Fields vary over a distance ∼ 1/Qs =⇒ gluons typically have k⊥ ∼ QsFields have strength ∼ 1/g =⇒ gluons have occupation numbers ∼ 1/αs


The scattering between two color sheets

Prior to scattering: purely transverse fields, localized near the 2 light-cones

During collision, mutual color rotations induce color charges on the sheets

longitudinal chromo-electric and chromo-magnetic fields

color strings (flux tubes) with typical transverse size 1/Qs


The scattering between two color sheets

Prior to scattering: purely transverse fields, localized near the 2 light-cones

..

...

.

.

.

...

... .

. .

..

. . ...

.

.

During collision, mutual color rotations induce color charges on the sheets

longitudinal chromo-electric and chromo-magnetic fields

color strings (flux tubes) with typical transverse size 1/Qs


Glasma (McLerran and Lappi, 06)

The ‘valence’ charges of the 2 nuclei rapidly separate from each other

The color field between the 2 recessing nuclei: ‘glasma’ (‘glass’ + ‘plasma’)

After a time τ ∼ 1/Qs, the transverse fields are regenerated

By that time, the partonic system becomes dilute (field strengths become ofO(1)), due to longitudinal expansion

Fourier mode decomposition =⇒ gluon productionNew Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 6 / 41

Longitudinal expansion

Gluons liberated by the collision at t = 0 have transverse momenta k⊥ ∼ Qsand generic longitudinal momenta |kz| ∼ x1P

+1 or |kz| ∼ x2P

−2

After a time t ∼ 1/Qs, they separate from each other along the z axis

particles which at time t are located at z have a velocity vz = zt

In any slice of z, the distribution in the transverse plane is roughly isotropic

But the 3-dimensional distribution in momentum is highly anisotropic


Rapidity correlations

Consider pairs of particles in the final state with different rapidities

hadrons which propagate at different angles w.r.t. the collision axis

η =1

2lnp+ pzp− pz

=1

2ln

1 + vz1− vz

= − ln tanθ

2

Construct the 2-hadron correlation in η and p⊥ (cf. talk by Jan Fiete):

C(∆φ,∆η) ≡ dNpair

d2p1⊥dη1d2p2⊥dη2− dN

d2p1⊥dη1

dN

d2p2⊥dη2

interested in the distribution in ∆η = η1 − η2 and in ∆φ = φ1 − φ2

Large ∆η > 1 =⇒ particles with very different longitudinal velocities

they separate from each other after a time τ ∼ 1/Qs ' 0.2 fm

their correlations must have been built at smaller times τ . 1/Qs

Long-range correlations in rapidity teach us about the early stages/glasma


Collective flow in AA collisions (cf. lecture by Jan Fiete)

Left: p+Pb collisions: “trivial” correlations associated with jets

a peak around ∆η = 0 and ∆φ = 0: particles from a same jeta plateau in ∆η narrow in ∆φ around ∆φ = π: recoiling jets

η∆-4

-20

24

φ∆0

2

4

φ∆ dη∆d

pair

N2 d

trig

N1

0.160.180.20

< 35trk

offline = 5.02 TeV, NNNsCMS pPb

< 3 GeV/cT

1 < p

(a)

8 5 Results

passocT < 3 GeV/c, and with the track multiplicity in the range 220 ≤ Noffline

trk < 260. For PbPbcollisions, this Noffline

trk range corresponds to an average centrality of approximately 60%, asshown in Table 1. For both high-multiplicity systems, in addition to the correlation peak near(∆η, ∆φ) = (0, 0) due to jet fragmentation (truncated for better illustration of the full correlationstructure), a pronounced long-range structure is seen at ∆φ ≈ 0 extending at least 4.8 units in|∆η|. This structure was previously observed in high-multiplicity (Noffline

trk ∼ 110) pp collisionsat√

s = 7 TeV [38] and pPb collisions at√

sNN = 5.02 TeV [39–41]. The structure is also prominentin AA collisions over a wide range of energies [2, 12–15, 33, 34, 36, 37]. On the away side(∆φ ≈ π) of the correlation functions, a long-range structure is also seen and found to exhibita magnitude similar to that on the near side for this pT range. In non-central AA collisions,this cos(2∆φ)-like azimuthal correlation structure is believed to arise primarily from ellipticflow [31]. However, the away-side correlations must also contain contributions from back-to-back jets, which need to be accounted for before extracting any other source of correlations.

η∆-4

-2

0

2

4

(radians)

φ∆

0

2

4

φ∆ dη∆d

pair

N2 d

trig

N1

2.4

2.6

2.8

< 260offlinetrk N≤ = 2.76 TeV, 220 NNs(a) CMS PbPb

< 3 GeV/ctrig

T1 < p

< 3 GeV/cassoc

T1 < p

η∆-4

-2

0

2

4

(radians)

φ∆

0

2

4

φ∆ dη∆d

pair

N2 d

trig

N1 3.1

3.2

3.3

3.4

< 260offlinetrk N≤ = 5.02 TeV, 220 NNs(b) CMS pPb

< 3 GeV/ctrig

T1 < p

< 3 GeV/cassoc

T1 < p

Figure 2: The 2D two-particle correlation functions for (a) 2.76 TeV PbPb and (b) 5.02 TeV pPbcollisions for pairs of charged particles with 1 < ptrig

T < 3 GeV/c and 1 < passocT < 3 GeV/c

within the 220 ≤ Nofflinetrk < 260 multiplicity bin. The sharp near-side peak from jet correlations

is truncated to emphasize the structure outside that region.

To investigate the observed correlations in finer detail and to obtain a quantitative comparisonof the structure in the pp, pPb, and PbPb systems, one-dimensional (1D) distributions in ∆φare found by averaging the signal and background 2D distributions over |∆η| < 1 (defined asthe “short-range region”) and |∆η| > 2 (defined as the “long-range region”) respectively, asdone in Refs. [33, 34, 38, 39]. The correlated portion of the associated yield is estimated usingan implementation of the zero-yield-at-minimum (ZYAM) procedure [57]. In this procedure,the 1D ∆φ correlation function is first fitted by a second-order polynomial in the region 0.1 <|∆φ| < 2. The minimum value of the polynomial, CZYAM, is then subtracted from the 1D ∆φcorrelation function as a constant background (containing no information about correlations)such that its minimum is shifted to have zero associated yield. The statistical uncertainty inthe minimum level obtained by the ZYAM procedure, combined with the deviations arisingfrom the choice of fit range in |∆φ|, gives an absolute uncertainty of ±0.003 in the associatedevent-normalized yield that is independent of multiplicity and pT.

Figures 3 and 4 show the 1D ∆φ correlation functions, after applying the ZYAM procedure,for PbPb and pPb data, respectively, in the multiplicity range Noffline

trk < 20 (open) and 220 ≤Noffline

trk < 260 (filled). Various selections of ptrigT are shown for a fixed passoc

T range of 1–2 GeV/cin both the long-range (top) and short-range (bottom) regions, with pT increasing from left to

Right: Pb+Pb collisions: long-range correlations in ∆η narrow in ∆φ

both on the “near side” (∆φ ' 0) and on the “away side” (∆φ ≈ π)New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 9 / 41

Elliptic flow

What is the origin of the double peak structure (∆φ = 0 and π) ?

TWO-PARTICLE CORRELATIONS

Unique long-range correlations in heavy-ion collisions. . .

p+p

0.6

0.65

0.7

0.75

0.8

0 π/2 π 3π/2

∆φ

dN/d∆φ (/Ntrig)

0.6

0.65

0.7

0.75

0.8

0 π/2 π 3π/2

∆φ

dN/d∆φ (/Ntrig)

0

0.01

0.02

0.03

0.04

0 π/2 π 3π/2

∆φ

’Subtracted’ correlationRidge Shoulder

Au+Au

η∆-6 -4 -2 0 2 4 6

η

∆dd

N

trig

gN

1

0

0.2

0.4

0.6

0.8

(rad)φ∆-1 0 1 2 3 4 5

φ∆

ddN

tr

igg

N1

0

0.2

0.4

0.6

0.8Event centrality

0-10%

40-60%

Hydrodynamic calculation

(STAR, arXiv:1010.0690) (PHOBOS, Phys.Rev. C81 (2010) 024904 )

. . . can be generated by purely collective flow.

MATT LUZUM (IPHT) FLOW FLUCTUATIONS QUARK MATTER 2011 4 / 13

R ≡ 〈N1N2〉 − 〈N1〉〈N2〉〈N1〉〈N2〉

∝ v22 cos(2∆φ

)

Maximum at ∆φ = 0 or π; minimum at ∆φ = π/2: elliptic flow


The geometry of a HIC

7

Geometry of a Heavy-Ion Collision

Number of participants (Npart): number of incoming nucleons (participants) in the overlap regionNumber of binary collisions (Nbin or Ncoll): number of equivalent inelastic nucleon-nucleon collisions

Reaction plane

x

z

y

Non-central collision

“peripheral” collision (b ~ bmax)“central” collision (b ~ 0)

Nbin ≥ NpartNew Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 11 / 41

Elliptic flow v2

dN

dφ∝ 1 + 2v2 cos 2φ

v2 : the ‘coefficient of the elliptic flow’

Non-central AA collision: impact parameter b⊥ > 0

The interaction region is (roughly) elliptic

Pressure gradient is larger along the smaller axis (x)

Fluid velocity is proportional to the pressure gradient

Particle emerge predominantly parallel to the fluid velocity

=⇒ the particle distribution is not axially symmetric !


Elliptic flow v2

dN

dφ∝ 1 + 2v2 cos 2φ

v2 : the ‘coefficient of the elliptic flow’

Non-central AA collision: impact parameter b⊥ > 0

The interaction region is (roughly) elliptic

Pressure gradient is larger along the smaller axis (x)

Fluid velocity is proportional to the pressure gradient

Particle emerge predominantly parallel to the fluid velocity

Initial anisotropy in coordinate space yields final anisotropy in momentum


Granularity and fluctuations

ψ2

ψ3

Nucleons are randomly distributed inside a nucleus.

In some events, the shape of the interaction (overlap) region can be quitedifferent from an ellipse

Then one speaks about triangular flow (or higher harmonics)

dN

dφ∝ 1 + 2v2 cos 2(φ−Ψ2) + 2v3 cos 3(φ−Ψ3) + ...

The small disks need not be nucleons: they can also be saturation domains

∆x⊥ ∼ 1/Qs ∼ 0.2 fm � RN ∼ 1 fmNew Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 13 / 41

Centrality dependence for v2 and v3NEW FLOW OBSERVABLES: v3, v4, v5, . . .

ALICE, arXiv:1105.3865

centrality percentile0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.02

0.03

0.04| > 1}!"{2, |2v

| > 1}!"{2, |3v

{2}CGCn# $ 1k

{2}Wn# $ 2k

LESSONS:Glauber may not work either

. MATT LUZUM (IPHT) FLOW FLUCTUATIONS QUARK MATTER 2011 9 / 13

ε2 =〈y2 − x2〉〈y2 + x2〉

Ultra-central collisions 0÷ 5%: the more central, the smaller v2

central collisions: the interaction region is axially symmetricone expects v2 to be proportional to the eccentricityCGC calculations of the initial ε2 compare better than just a randomdistribution of “wounded” nucleons (Glauber)

v3 is much less sensitive to the centrality: controlled by fluctuations


Momentum dependence for v2

[GeV]T

p0 5 10 15 20

2v

0

0.1

0.2

0.3 =2.76 TeV 40-50%NNs Pb+Pb ± hATLAS

=2.76 TeV 40-50%NNs Pb+Pb ± hALICE

=200 GeV 40-60%NNs Au+Au ± hSTAR

=200 GeV 40-50%NNs Au+Au 0π PHENIX

v2 first rises up to 3÷ 4 GeV, then it decreases again

the flow is collective motion which contributes to p⊥sufficiently fast particles cannot be driven by the flow

The high-p⊥ tail: natural when fluctuations occur on smaller scales:1/Qs � RN


p⊥ dependence for vn, n = 2− 6 (ATLAS)vn(n=2-6) vs pT (0.5-12 GeV)

8

Similar pT dependence for all n: rise to 3-4 GeV, then falls

0

0.1

0.2

0-5%

ATLAS Preliminary

EP from full FCal

2v

3v

4v

5v

6v

-1bm Ldt = 8 ò

5-10% 10-20%

[GeV]T

p2 4 6 8 10

0

0.1

0.2

20-30%

[GeV]T

p2 4 6 8 10

30-40%

[GeV]T

p2 4 6 8 10

40-50%

0

0.1

0.2

0-5%

ATLAS Preliminary

EP from full FCal

2v

3v

4v

5v

6v

-1bm Ldt = 8 ò

5-10% 10-20%

[GeV]T

p2 4 6 8 10

0

0.1

0.2

20-30%

[GeV]T

p2 4 6 8 10

30-40%

[GeV]T

p2 4 6 8 10

40-50%Similar p⊥ dependence for all n: rise up to 3-4 GeV, then fall


Hydrodynamics in a nut shell

To describe flow, it is natural to use hydrodynamics

transport of conserved quantities (energy, momentum, electric andbaryonic charges ...) via collective motion with fluid velocity v

This is perhaps best introduced by comparing with thermodynamics

Thermodynamics is about a system in global thermal equilibrium

pressure (P ), temperature (T ), chemical potential (µ) are independentof time and uniform throughout the whole volume V

Hydrodynamics is about local thermal equilibrium

P , T , µ and v can vary with space and time ...

... but they vary so slowly that one can still assume thermal equilibriumto hold locally, in the neighbourhood of any point

equations of motion for P (x), T (x), µ(x), v(x)

This holds when the typical scale for space–time variations (‘system size R’)is much larger than the mean free path ` between two successive collisions


Hydrodynamics in a nut shellTo describe flow, it is natural to use hydrodynamics

transport of conserved quantities (energy, momentum, electric andbaryonic charges ...) via collective motion with fluid velocity v

This is perhaps best introduced by comparing with thermodynamics

Thermodynamics is about a system in global thermal equilibrium

pressure (P ), temperature (T ), chemical potential (µ) are independentof time and uniform throughout the whole volume V

Hydrodynamics is about local thermal equilibrium

P , T , µ and v can vary with space and time ...

... but they vary so slowly that one can still assume thermal equilibriumto hold locally, in the neighbourhood of any point

equations of motion for P (x), T (x), µ(x), v(x)

Gradient expansion in powers of K ≡ `/L� 1 (Knudsen number)

In heavy ion collisions, L ∼ RA: transverse size of the nucleusNew Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 17 / 41

Ideal Hydrodynamics

Hydro equations of motion: the conservation laws

∂µ Tµν = 0 ∂µJ

µB = 0

Tµν (energy-momentum tensor) and JµB (baryonic current) :

fluid 4-velocity: uµ(x) = γ(1,v), γ = 1/√

1− v2

energy density ε(x), pressure P (x), baryon density nB(x)

additional parameters (‘viscosities’) for a non-ideal fluid

Ideal (“inviscid”) fluid ≡ local thermal equilibrium =⇒ isotropy

Tµν(0) = diag (ε, P, P, P ) in the local rest frame at x: uµ(x) = (1, 0)

After a boost to the laboratory frame, this becomes:

Tµν(0) = (ε+ P )uµuν − Pgµν , JµB = nuµB

Tµν : 4 equations, 5 unknown quantities: ε(x), P (x), vi(x) (i = 1, 2, 3)

Use the equation of state ε = ε(P ) to close the system


Ideal Hydrodynamics

Hydro equations of motion: the conservation laws

∂µ Tµν = 0 ∂µJ

µB = 0

Tµν (energy-momentum tensor) and JµB (baryonic current) :

fluid 4-velocity: uµ(x) = γ(1,v), γ = 1/√

1− v2

energy density ε(x), pressure P (x), baryon density nB(x)

additional parameters (‘viscosities’) for a non-ideal fluid

Ideal (“inviscid”) fluid ≡ local thermal equilibrium =⇒ isotropy

Tµν(0) = diag (ε, P, P, P ) in the local rest frame at x: uµ(x) = (1, 0)

After a boost to the laboratory frame, this becomes:

Tµν(0) = (ε+ P )uµuν − Pgµν , JµB = nuµB

Tµν : 4 equations, 5 unknown quantities: ε(x), P (x), vi(x) (i = 1, 2, 3)

E.g.: ideal gas of massless particles: ε = 3P , hence Tµµ = 0


Initial conditions & the thermalization puzzle

One needs the initial conditions ε0(~x), vi0(~x) at some proper time τ0

The isotropy seems inconsistent with the geometry of a heavy ion collision

strong dissymmetry between longitudinal and transverse directions

TµνYM =1

4gµνFαβFαβ − FµαF να

τ = 0+: purely longitudinal fields Ez, Bz

TµνYM =1

2(E2

z +B2z )× diag(1, 1, 1,−1)

negative “longitudinal pressure”

τ ∼ 1/Qs: TµνYM = diag(ε, ε/2, ε/2, 0) : zero “longitudinal pressure”

“Rapid thermalization at some time τ0 . 1 fm” (?)


Viscous hydrodynamics

Deviations from local thermal equilibrium are captured by viscous hydro

Ideal hydro: zeroth order in the gradient expansion =⇒ no dissipation

Dissipative effects are described by higher orders: Tµν = Tµν(0) + Πµν

The first-order terms: ∂µuν or gµν(∂ · u) are weighted by the viscosities

shear viscosity η, bulk viscosity ζ: “transport coefficients”

measure the fluid ability to transfer energy and momentum alongdirections orthogonal to that of the flow

heuristically: friction between the different layers of the fluid

Non-relativistic limit v � 1 =⇒ Navier-Stokes equation (ρ = mass density)

ρ

(∂

∂t+ ~v · ~∇

)~v = −~∇P + η∇2~v + ζ ~∇

(~∇ · ~v

)

First-order relativistic hydro is inconsistent with causality, but this can befixed by adding second-order terms (“Müller-Israel-Stewart theory”)


Shear viscosity

2 layers of a same fluid, one flowing the other one at rest, separated by a wall

Remove the wall: the two layers would not get mixed for ideal flow (η = 0)

η : a measure of a fluid ability to transfer px in the y direction

Uniformflow ux

Stationary

ux

Wall

xy

1

A

dpxdt

= −η duxdy

η ∼ `ρvth ∼ `ε/ρvth

vth average molecularvelocity (� fluid velocity v)

Proportional to the mean free path: ` ∝ 1/(nσ) ∼ 1/α2s

larger for weakly coupled systems ! (Maxwell, 1860)


Initial conditions: from smoothness to granularity

Left: mean field description — a smooth distribution for the nuclear matter

Smooth (or Average) Initial Conditions

Optical Glauber

A

b

B

A B

s

b

Thickness functions: TA,B(s) =∫ ∞

−∞dz ρA,B(z, s)

Overlap function: TAB(b) =∫

d2s TA(s)TB(s− b)

Number of collisions: Ncoll(b) = ABTAB(b)σNNinel

Number of participants:

Npart(b) =

A∫

d2s TA(s)(

1−[1− TB(s− b)σNN

inel]B)

+ B∫

d2s TB(s− b)(

1−[1− TA(s)σNN

inel]A)

Smooth distribution ==> No fluctuations.No εodd nor vodd

Jeon (McGill) MUSIC 9 / 42

Fluctuating Initial Conditions

1 Sample nucleonpositions

Jeon (McGill) MUSIC 10 / 42Right: random distribution of the nucleons inside each nucleus

=⇒ fluctuations (Monte-Carlo Glauber)New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 22 / 41

Initial conditions: from nucleons to Glasma

Left: random superposition of nucleons within the nuclear disks

Monte-Carlo Glauber (nucleons)

size of flucts: Rp ∼ 1 fm

Glasma (color flux tubes)

size of flucts: 1/Qs ∼ 0.2 fm

Right: Glasma simulation (classical Yang–Mills equations with randomlydistributed color charges) (see review by Gelis and Schenke, 2016)


Hydro simulations for v2 (from Luzum and Romatschke, 08)

Early simulations used ideal hydro + Monte-Carlo Glauber (nucleons)

required early equilibration τ0 . 1 fm/c: very puzzling

Glasma generates a larger initial eccentricity ε2

this leaves the place for a larger viscosity, i.e. for deviations from localthermal equilibrium

2

0 100 200 300 400NPart

0

0.02

0.04

0.06

0.08

0.1

v 2

PHOBOS

Glauber

!/s=10-4

!/s=0.08

!/s=0.16

0 1 2 3 4pT [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est.)STAR event-plane

Glauber

!/s=10-4

!/s=0.08

!/s=0.16

0 100 200 300 400NPart

0

0.02

0.04

0.06

0.08

0.1

v 2

PHOBOS

CGC

!/s=10-4

!/s=0.08

!/s=0.16

!/s=0.24

0 1 2 3 4pT [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est).STAR event-plane

CGC!/s=10-4

!/s=0.08

!/s=0.16

!/s=0.24

FIG. 8: (Color online) Comparison of hydrodynamic models to experimental data on charged hadron integrated (left) andminimum bias (right) elliptic flow by PHOBOS [4] and STAR [5], respectively. STAR event plane data has been reduced by20 percent to estimate the removal of non-flow contributions [5, 6]. The line thickness for the hydrodynamic model curvesis an estimate of the accumulated numerical error (due to, e.g., finite grid spacing). The integrated v2 coe!cient from thehydrodynamic models (full lines) is well reproduced by 1

2ep (dots); indeed, the di"erence between the full lines and dots gives

an estimate of the systematic uncertainty of the freeze-out prescription.

Initial condition !/s Ti [GeV] Tf [GeV] "0 [fm/c] a [GeV!1]

Glauber 10!4 0.340 0.14 1 2

Glauber 0.08 0.333 0.14 1 2

Glauber 0.16 0.327 0.14 1 2

CGC 10!4 0.310 0.14 1 2

CGC 0.08 0.304 0.14 1 2

CGC 0.16 0.299 0.14 1 2

CGC 0.24 0.293 0.14 1 2

TABLE I: Summary of parameters used for the viscous hydrodynamics simulations

2

0 100 200 300 400NPart

0

0.02

0.04

0.06

0.08

0.1

v 2

PHOBOS

Glauber

!/s=10-4

!/s=0.08

!/s=0.16

0 1 2 3 4pT [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est.)STAR event-plane

Glauber

!/s=10-4

!/s=0.08

!/s=0.16

0 100 200 300 400NPart

0

0.02

0.04

0.06

0.08

0.1

v 2

PHOBOS

CGC

!/s=10-4

!/s=0.08

!/s=0.16

!/s=0.24

0 1 2 3 4pT [GeV]

0

5

10

15

20

25

v 2 (p

erce

nt)

STAR non-flow corrected (est).STAR event-plane

CGC!/s=10-4

!/s=0.08

!/s=0.16

!/s=0.24

FIG. 8: (Color online) Comparison of hydrodynamic models to experimental data on charged hadron integrated (left) andminimum bias (right) elliptic flow by PHOBOS [4] and STAR [5], respectively. STAR event plane data has been reduced by20 percent to estimate the removal of non-flow contributions [5, 6]. The line thickness for the hydrodynamic model curvesis an estimate of the accumulated numerical error (due to, e.g., finite grid spacing). The integrated v2 coe!cient from thehydrodynamic models (full lines) is well reproduced by 1

2ep (dots); indeed, the di"erence between the full lines and dots gives

an estimate of the systematic uncertainty of the freeze-out prescription.

Initial condition !/s Ti [GeV] Tf [GeV] "0 [fm/c] a [GeV!1]

Glauber 10!4 0.340 0.14 1 2

Glauber 0.08 0.333 0.14 1 2

Glauber 0.16 0.327 0.14 1 2

CGC 10!4 0.310 0.14 1 2

CGC 0.08 0.304 0.14 1 2

CGC 0.16 0.299 0.14 1 2

CGC 0.24 0.293 0.14 1 2

TABLE I: Summary of parameters used for the viscous hydrodynamics simulations

The CGC fits prefer a larger value for the viscosity/entropy density ratio


Hydro simulations for v2 (Gelis and Schenke, 2016)

Early simulations used ideal hydro + Monte-Carlo Glauber (nucleons)

required early equilibration τ0 . 1 fm/c: very puzzling

Glasma generates a larger initial eccentricity ε2

viscous hydro + Glasma yields very good fits to all vn’s with η/s = 0.2

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn

2⟩1

/2

pT [GeV]

ATLAS 30-40%, EP v2 v3 v4 v5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50

⟨vn2⟩1

/2

centrality percentile

η/s = 0.2ALICE data vn{2}, pT>0.2 GeV v2

v3 v4 v5

This value η/s seems still too small to be consistent with weak coupling


Viscosity over entropy density ratioη ∼ `× ε (` : mean free path; ε : energy density)

Thermodynamics: ε+ P = Ts =⇒ ε/s ∼ T ∼ 1/λth

Uncertainty principle =⇒ ideal fluids (η = 0) cannot exist in nature

η

s∼ `

λth∼ mean free path

de Broglie wavelength& ~

Weakly coupled QGP (Arnold, Moore, Yaffe, 2003) :

η

s∼ ~

α2s ln(1/αs)

� ~

Conjectured limit at strong coupling (Kovtun, Son, Starinets, 05)

η

s→ ~

4πwhen λ ≡ g2Nc → ∞ (AdS/CFT)

The value preferred by the fits is at most a few times 1/4π ' 0.08


RHIC serves up the perfect liquid !

‘Strongly–coupled quark–gluon plasma’ or ‘perfect fluid’

Applications of the AdS/CFT correspondence to heavy ion collisions

Two main arguments in favor of strong coupling:

small η/s ratio from the hydro fits to the flow coefficientssmall thermalization time needed for consistency of hydro

These 2 conditions can also be satisfied, or circumvented, at weak coupling


Mean free path at weak coupling (1)

A weakly coupled QGP in thermal equilibrium with temperature T

Thermal (quasi)particles: quarks and gluons with energy and momenta oforder T , whose properties are slightly dressed by the interactions

Number density n ∼ T 3, energy density ε ∼ T 4

Mean free path: typical distance/time between two successive collisions

` ∼ 1

Γ, Γ = collision rate = nσvrel ∼ T 3σ

σ : cross-section for binary collisions

Weak coupling =⇒ the dominant mechanism is 2→ 2 elastic scattering



Coulomb scattering (one gluon exchange), deflection angle θ ' q⊥/pdσ

dq2⊥∼ α2

s

q4⊥

the total cross-section

σ =

∫dq2⊥α2s

q4⊥

... is infrared (q⊥ → 0) divergentIn the plasma, color interactions are screened over a distance λD ∼ 1/gT ,that is, at the momentum scale q2

⊥ ∼ αsT 2 (Debye screening)

`c ∼ 1/(αsT ) = m.f.p. for small-angle scattering: θ ∼ gT/T ∼ galso for color relaxation: even soft collisions can change the color state

Not the relevant m.f.p. for processes which require momentum transfer, likeviscosity: large-angle scattering



Coulomb scattering (one gluon exchange), deflection angle θ ' q⊥/p

dσ

dq2⊥∼ α2

s

q4⊥

the total cross-section

σ =

∫

αsT 2

dq2⊥α2s

q4⊥' αsT 2

In the plasma, color interactions are screened over a distance λD ∼ 1/gT ,that is, at the momentum scale q2

⊥ ∼ αsT 2 (Debye screening)

`c ∼ 1/(αsT ) = m.f.p. for small-angle scattering: θ ∼ gT/T ∼ galso for color relaxation: even soft collisions can change the color state

Not the relevant m.f.p. for processes which require momentum transfer, likeviscosity: large-angle scattering



The transport cross-section: weighted by 1− cos θ ' θ2/2 with θ ' q⊥/p

σtr =

∫dq2⊥q2⊥p2

α2s

q4⊥

σtr =α2s

T 2

∫ T 2

αsT 2

dq2⊥

q2⊥

=α2s

T 2ln

1

αs

Logarithmically sensitive to all “semi-hard” scales gT � q⊥ � T

` ∼ 1

σtrT 3∼ 1

α2sT ln(1/αs)

= m.f.p. for large-angle scattering

`� `c: changing momentum is more difficult than changing the color

Shear viscosity at weak coupling: η ∼ `ε ∼ T 3

α2s ln(1/αs)


η/s in pQCD: from LO to NLO

Left: a compilation of the leading-order result (Arnold, Moore, Yaffe, 2003)by D. Teaney, arXiv:0905.2433

fixed coupling & running coupling with 1-loop and 1-loop beta function

0.1

1

10

0.2 1 10 100 1000η/s

T [GeV]

LO µEQCD

LO µMS

NLO µEQCD

NLO µMS

0.1

1

10

0.2 1 10 100 1000

Right: the recent NLO calculation vs to the LO result (2-loop running)

Ghiglieri, Moore and Teaney, arXiv:1802:09535 (based on kinetic theory)New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 30 / 41

η/s in pQCD: from LO to NLO

Left: a compilation of the leading-order result (Arnold, Moore, Yaffe, 2003)by D. Teaney, arXiv:0905.2433

fixed coupling & running coupling with 1-loop and 1-loop beta function

0.1

1

10

0.2 1 10 100 1000η/s

T [GeV]

LO µEQCD

LO µMS

NLO µEQCD

NLO µMS

0.1

1

10

0.2 1 10 100 1000

Right: the recent NLO calculation vs to the LO result (2-loop running)

at T ∼ 300 MeV, the result is reduced by a factor of 4: η/s ∼ 0.2New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 30 / 41

Kinetic theory in a nut shell

For weakly coupled systems, kinetic theory can be used to describe thenon-equilibrium evolution at intermediate times 1/Qs . τ . τ0

Follow the non-equilibrium evolution of the phase-space occupation numbers

fs(t,x,p) ≡ (2π)3

νs

dNsd3xd3p

s = g, q, q ...; νg = 2(N2c −1), νq = νq = 2Nc

Meaningful when fs � 1/α (which includes f � 1 at weak coupling)

It smoothly matches onto classical Yang-Mills dynamics when f � 1 andonto hydro when off-equilibrium deviations are small: δf � feq

All the observables can be computed from fs(t,x,p)

Tµν(x) =

∫d3p

(2π)3

pµpν

Ep

∑

s

νsfs(t,x,p)


Boltzmann equation

Two types of interactions: external force and collisions among the particles(∂

∂t+ vp ·

∂

∂x+ Fext ·

∂

∂p

)f(t,x,p) = −C[f ]

C[f ]: “collision term”: non-linear in f , non-local in p, but local in x

Variation time scales much larger than the duration of the collision process

e.g. ∆t� 1/(gT ) for 2→ 2 collisions near equilibrium

C[feq]=0: collision term vanishes in thermal equilibrium fs(t,x,p)→ ns(p)

ng(p) =1

eβp − 1, nq,q(p) =

1

eβ(Ep∓µ) + 1

Bose-Einstein for gluons, Fermi-Dirac for quarks

Collisions can be either elastic (e.g. 2→ 2) or inelastic (e.g. 2→ 3)


Elastic collisions

2→ 2 collisions: generally argued to be the LO piece at weak coupling

‘Gain’ and ‘loss’ terms: the gluon p is the one that is measured

Cel[f ] =

∫

p′,k,k′

|M|2(2p)(2p′)(2k)(2k′)

Φ[f ]

Φ[f ] = fpfp′ [1 + fk][1 + fk′ ]− fkfk′ [1 + fp][1 + fp′ ]

5 conserved quantities: particle number and the 4–momentum

It vanishes at (local) thermal equilibrium: Cel[n] = 0 (“detailed balance”)


Elastic collisions

2→ 2 collisions: generally argued to be the LO piece at weak coupling

‘Gain’ and ‘loss’ terms: the gluon p is the one that is measured

Cel[f ] =

∫

p′,k,k′

|M|2(2p)(2p′)(2k)(2k′)

Φ[f ]

Φ[f ] = fpfp′ [1 + fk][1 + fk′ ]− fkfk′ [1 + fp][1 + fp′ ]

It also vanishes in the limit of zero momentum transfer q = |q| → 0

transport cross-section: weighted by a factor q2 at small q


The thermalization puzzle

An off-equilibrium perturbation typically dies away over a time τrel ∼ `/vth

At weak coupling, τrel ∼ ` is parametrically large (in units of λth = 1/T )

τrel

λth∼ η

s∼ 1

α2s ln(1/αs)

� 1

The folklore says: “Fast thermalization is not possible at weak coupling”

Not quite true: recall the NLO calculation of η/s (Ghiglieri et al, arXiv:1802:09535)

at T ∼ 300 MeV, the result is reduced by a factor of 4: η/s ∼ 0.2

The actual problem is the rapid longitudinal expansion

the partonic system becomes more and more dilutecollisions occur less often and have little chances to equilibrate

A solution to that is provided by radiation (“inelastic collisions”)


Anisotropy from the longitudinal expansion

At time t ∼ 1/Qs, gluons are freed from the nuclei with p⊥ ∼ Qs (“hard”)and occupation numbers f0 ∼ 1/α

At later times t� 1/Qs, all the gluons located at z have vz = z/t

the distribution in velocity (or momentum) is highly anisotropic

Focus on midrapidity for definiteness: η ∼ 0 (say, z ∼ 1/Qs)

physics is boost invariant: replace t→ τ ≡√t2 − z2 for any other η

vz 'pzp⊥∼ 1

Qsτ=⇒ pz '

1

τ� p⊥ ∼ Qs when Qsτ � 1


The “Bottom-up” scenario (Baier, Mueller, Schiff, Son ’01)

This would be the distribution in the absence of collisions

Elastic 2→ 2 scattering competes with the longitudinal expansion

pz decreases less fast with τ : pz ∼ Qs

(Qsτ)1/3

numerical calculation by Kurkela et al,arXiv:1805.00961

gluon distribution in the (px, pz) plane as afunction of the scaled time (with η/s = 0.62)

τ

τrel(τ)≡ τT (τ)

η/s

the color scale: the occupation number in therange 0.1 < f < 5

Anisotropy less pronounced than for free streaming, but steadily increasing



The “hard” (p⊥ ∼ Qs) gluons can also suffer “inelastic collisions”

collisions which trigger emissions of softer gluons

The soft gluons are emitted isotropically

they escape the problem of the longitudinal expansion

They can thermalize, over a time ∼ τrel, via elastic 2→ 2 scattering



The “hard” (p⊥ ∼ Qs) gluons can also suffer “inelastic collisions”

collisions which trigger emissions of softer gluons

The hard particles scatter off the soft thermal both

They efficiently lose energy via medium-induced radiation

The distribution approaches local thermal equilibrium


Medium-induced branchings

The prototype: 2↔ 3 inelastic scattering (single scattering)

Both splittings (2→ 3) and recombinations (3→ 2)

The splittings (recombinations) can contribute both to “gain” and to “loss”

Actually, multiple scattering is important (“LPM effect)New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 38 / 41

Hydrodynamisation in the “Bottom-up” scenario(Kurkela and Zhu, arXiv:1506.06647; Kurkela et al, arXiv:1805.00961)

Numerical solutions to the Boltzmann equation with C[f ] = Cel[f ] + Cinel[f ]

Highly anisotropic initial conditions: PL � PT

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tµν/T

µν

id.

τTid./(4πη/s)

λ = 10 (η/s ≈ 0.62)

λ = 15 (η/s ≈ 0.34)

λ = 20 (η/s ≈ 0.22)

λ = 25 (η/s ≈ 0.16)

2nd hydro asympt.

e/3

PT

PL

different values of the coupling (η/s)

λ ≡ g2Nc = 4παsNc

viscous hydro (2nd order in gradients)

hydro: universal curve of the rescaled time

τ


η/s

Kinetic theory ⇒ universal curve quasi-independent of the coupling (or η/s)

Perfect matching with viscous hydro for τ & τrel

... albeit the anisotropy is still pronounced: (PT−PL)/PT ∼ 30%New Trends in High-Energy Physics Glasma, Flow & Thermalization Edmond Iancu 39 / 41




0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tµν/T

µν

id.

τTid./(4πη/s)

λ = 10 (η/s ≈ 0.62)

λ = 15 (η/s ≈ 0.34)

λ = 20 (η/s ≈ 0.22)

λ = 25 (η/s ≈ 0.16)

2nd hydro asympt.

e/3

PT

PL





τ


η/s

The fact that the data for the flow coefficients vn can be well described byviscous hydro does not mean that the QGP has reached local equilibrium !





0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Tµν/T

µν

id.

τTid./(4πη/s)

λ = 10 (η/s ≈ 0.62)

λ = 15 (η/s ≈ 0.34)

λ = 20 (η/s ≈ 0.22)

λ = 25 (η/s ≈ 0.16)

2nd hydro asympt.

e/3

PT

PL





τ


η/s

In proper units, the results are independent of the coupling

In fact, the same conclusion (hydrodynamisation w/o thermalization) hasbeen reached at infinite coupling, within AdS/CFT (Chesler and Yaffe, 2008)


A sequence of effective theories (Kurkela et al, 2018)

A more realistic simulation of a heavy ion collision

initial conditions at τ = 0−: McLerran-Venugopalan modelnumerical solutions to classical Yang-Mills equations (glasma)negative longitudinal pressure at times 0+ < τ < 1/Qs = 0.1 fmmatching onto kinetic theory at τ = 0.2 fmmatching onto viscous hydro at τ = 2 fm

−8 −40

48−8

−4

0

4

8

x (fm)

y (fm)

0.1

1

2

3

4τe3/4 (GeV2)

IP-GlasmaτEKT = 0.2 fm

−1

−0.5

0

0.5

1

0.01 0.1 1

〈PT 〉〈e〉

〈PL〉〈e〉

1/3

IP-Glasma (2+1D Yang-Mills)

Kinetic theory

2+1D hydro.

τ (fm)


THANK YOU FOR YOUR ATTENTION !

... and have a good trip back home!


Edmond Iancu InstitutdePhysiqueThéoriquedeSaclay · New Trends in High-Energy Physics Glasma, Flow...

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