Early Isotropization of the Quark Gluon Plasmaepelbaum/Santiago_29_10.pdf · Santiago de Compostela, 29th October 2013 Thomas EPELBAUM IPhT THOMAS EPELBAUM Early Isotropization of

Early Isotropization ofthe Quark Gluon Plasma

Santiago de Compostela, 29th October 2013Thomas EPELBAUM

IPhT

THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 0 / 28

OUTLINE

1 MOTIVATION

2 THEORETICAL FRAMEWORK

3 A PROOF OF CONCEPT: SCALAR FIELD THEORY

4 YANG-MILLS THEORY

5 CONCLUSION


HEAVY ION COLLISIONS: THE GENERAL PICTURE

QGP

Hydrodynamics

viscousideal

Hadron gas



0 100 200 300 400

NPart

0

0.02

0.04

0.06

0.08

0.1

v2

PHOBOS

CGC

η/s=10-4

η/s=0.08

η/s=0.16

η/s=0.24

[LUZUM, ROMATSCHKE (2008)]THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 1 / 28


Viscous Hydrodynamics

I) Macroscopic theoryII) Few parameters: PL, PT , ε, ~uIII) Need input:

1) Equation of state f(PL, PT ) = ε2) Small anisotropy3) Initialization: ε(τ0), PL(τ0)? ...4) viscous coefficients: shear viscosity η,...5) Short isotropization time





1) Equation of state f(PL, PT ) = ε2) Small anisotropy3) Initialization: ε(τ0), PL(τ0)? ...4) viscous coefficients: shear viscosity η,...5) Short isotropization timeNon

e ofthisiseasy

toget f

rom

QCD



Early transition: the problem

CGC

Glasma Isotropization?Time scale?

Hydrodynamics

QGP

Huge anisotropy Small anisotropy(negative PL)

Long time puzzle: Does (fast) isotropization occur?


2 THEORETICAL FRAMEWORKHow to deal with a Heavy Ion CollisionThe Color Glass CondensateThe Classical statistical approximation


HOW TO STUDY THE TRANSITION?

Strongly coupled method: AdS/QCD?

SupersymmetricSU(N) gauge theory

Black hole

String theory



Weakly coupled method: QCD

SupersymmetricSU(N) gauge theory

Black hole

String theory

QCD



Weakly coupled QCD with only gluons

-310

-210

-110

1

10

-410

-310

-210

-110 1

-310

-210

-110

1

10

HERAPDF1.0

exp. uncert.

model uncert.

parametrization uncert.

x

xf

2 = 10 GeV2Q

vxu

vxd

xS

xg

H1 and ZEUS

-310

-210

-110

1

10

u

du



Weakly coupled method at dense regime:αs � 1 but fgluon ∼ 1

αs


TWO ADDITIONAL FEATURES: SATURATION AND TIME DILATION











Gluon saturation when emission = recombination⇒ Saturation scale Qs



u

du












THE COLOR GLASS CONDENSATE [MCLERRAN, VENUGOPALAN (1993)]

THE MAIN ASSUMPTIONS

• Fast gluons are "frozen" by time dilation.

• Described as static color sources J located on the light cone axis

• Small x→ Gluon saturation→ J ∼ Q3s α

−1/2s .

• Slow gluons are the standard gauge field Aµ ∼ Qs α−1/2s .

• System boost-invariant→ Aµ rapidity independant.

Langrangian of theory reads

L = −14FµνF

µν + JµAµ



Theoretical framework (Weakly coupled but strongly interacting)

x+x−

CGCJ− J+

E ,B

LO: ε =12

(~E

2+ ~B

2)

︸︷︷︸Classicalcolor fields

DµFµν = Jν︸︷︷︸

Color sourceson the light cone

[KRASNITZ, VENUGOPALAN (1998)]



ε = E2⊥ +B2

⊥ + E2L +B2

L

PT = E2L +B2

L

PL = E2⊥ +B2

⊥ − E2L −B2

L



0 0.5 1 1.5 2

g2µτ

0

0.2

0.4

0.6

0.8

[(g2µ

)4/g

2]

Bz

2

Ez

2

BT

2

ET

2

[LAPPI, MCLERRAN (2006)]



ε = E2⊥︸︷︷︸0

+ B2⊥︸︷︷︸

0

+E2L +B2

L

PT = E2L +B2

L

PL = E2⊥︸︷︷︸0

+ B2⊥︸︷︷︸

0

−E2L −B2

L

Initial Tµν is (ε, ε, ε,−ε)!



Strong anisotropy at early time

-0.05

0

0.05

0.1

0 0.5 1 1.5 2 2.5 3

g2τ P

L /

(g

2µ

)3,

g

2τ P

T /

(g

2µ

)3

g2µτ

τ PLτ PT

[GELIS, FUKUSHIMA (2012)]



Strong anisotropy at early time

-0.05

0

0.05

0.1

0 0.5 1 1.5 2 2.5 3

g2τ P

L /

(g

2µ

)3,

g

2τ P

T /

(g

2µ

)3

g2µτ

τ PLτ PT

{∂τε+

ε+PLτ

= 0limτ→0+

ε = cst ⇒ PL = −ε

ε = 2PT + PL ⇒ PT = ε

[GELIS, FUKUSHIMA (2012)]


THE COLOR GLASS CONDENSATE AT NLO

E2(x) = E2(x)︸︷︷︸LO

+12

∫~k

∣∣e~k(x)∣∣2︸︷︷︸NLO

+ · · ·

e~k(x) perturbation to E(x) created by a plane waveof momentum ~k in the remote past.



0 500 1000 1500 2000 2500 3000 3500

g2 µ τ

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

max

τ2 T

ηη /

g4

µ3 L

η

c0+c

1 Exp(0.427 Sqrt(g

2 µ τ))

c0+c

1 Exp(0.00544 g

2 µ τ)

[ROMATSCHKE, VENUGOPALAN (2006)]

Small Fluctuations grow exponentially (Weibel instability)



• Because of instabilities, the NLO correction eventually becomes as largeas the LO⇒ Important effect, should be included

• NLO alone will grow forever⇒ unphysical effect, should be taken care of

-40

-30

-20

-10

0

10

20

30

40

-20 0 20 40 60 80

time

PLO εLO

-40

-30

-20

-10

0

10

20

30

40

-20 0 20 40 60 80

time

PNLO εNLO

• Such growing contributions are present at all orders of the perturbativeexpansion

How to deal with them?




THE CLASSICAL-STATISTICAL METHOD

• At the initial time τ = τ0, take:

~E0(τ0,~x) = ~E0(τ0,~x) +∫~k

c~k~e~k(τ0,~x)

where c~k are random coefficients:⟨c~kc~k ′

⟩∼ δ~k~k ′

• Solve the Classical equation of motion DµFµν = Jν

• Compute⟨~E

2(τ,~x)

⟩, where 〈〉 is the average on the c~k (Monte-Carlo)

• One can show that this resums all the fastest growing terms at eachorder, leading to a result that remain bounded when τ→∞[GELIS, LAPPI, VENUGOPALAN (2008)]

This gives: LO+NLO+Subset of higer orders


IMPLICATIONS OF THE CLASSICAL-STATISTICAL METHOD

LO: Classical solution



LO: Classical solution

NLO:Parabolic

approximationaround LO



NLO:Parabolic

approximationaround LO



Resummed:keep thecompletepotential







-200

-150

-100

-50

0

50

100

150

200

0 50 100 150 200 250

time

g=0.5 g=1 g=2 g=4 g=8 ε/3


3 A PROOF OF CONCEPT: SCALAR FIELD THEORYThe TheoryNumerical results


SCALAR FIELD THEORY

Adapted coordinate system to describe a Heavy Ion Collision?

x⊥

z

System boost invariant in z direction


SCALAR FIELD THEORY

Proper time/rapidity coordinate system

η = cstτ = cst

τ0

x+x−


SCALAR FIELD THEORY

The model

Initial conditions: classical statistical method

φ(τ0, x⊥,η) = ϕ0(τ0, x⊥) +∑k⊥,ν

cνk⊥eiνη aν,k⊥(τ0, x⊥)

Time evolution: Klein Gordon equation

[∂2

∂τ2 +1τ

∂

∂τ−∇2

⊥ −1τ2

∂2

∂η2

]︸︷︷︸

�

φ+g2

6φ3 = 0


HOW INTERACTIONS COMPETE WITH EXPANSION?

Initial anisotropy

PT

PL



Interactions isotropize the system

PT

PL



Expansion dilutes the system

PT

PL



Expansion ≶ Interactions for realistic αs?

PT

PL


3 A PROOF OF CONCEPT: SCALAR FIELD THEORYThe TheoryNumerical results


TµνRESUM [DUSLING, TE, GELIS, VENUGOPALAN (2012]

×10-1

×100

×101

×102

×103

×104

×10-2 ×10-1 ×100 ×101

τ

PT(τ0=0.01)

ε(τ0=0.01)

PT(τ0=0.1)

ε(τ0=0.1)


TµνRESUM [DUSLING, TE, GELIS, VENUGOPALAN (2012]

×10-3

×10-2

×10-1

×100

×101

0 50 100 150 200 250 300

τ

2PT + P

L

ε

PT

PL


ε BEHAVIOUR

10-2

10-1

100

101

10 100

τ

τ-4/3

τ-1

2PT + P

L

ε

Bjorken Law: ∂τε+ ε+PLτ

= 0THOMAS EPELBAUM Early Isotropization of the Quark Gluon Plasma 15 / 28

HYDRODYNAMICS: IDEAL AND VISCOUS

Equation of state: ε = 2PL + PT

IDEAL HYDRO

Isotropic system

Tµνideal = εuµuν − P(gµν − uµuν)

VISCOUS HYDRO

Anisotropic system

Tµν = Tµνideal + ηπµν

In our case

PT =ε

3+

2η3τ

PL =ε

3−

4η3τ


HYDRODYNAMICS: IDEAL AND VISCOUS

Equation of state: ε = 2PL + PT

IDEAL HYDRO

Isotropic system

Tµνideal = εuµuν − P(gµν − uµuν)

VISCOUS HYDRO

Anisotropic system

Tµν = Tµνideal + ηπµν

In our case

PT =ε

3+

2η3τ

PL =ε

3−

4η3τ


FIRST ORDER VISCOUS HYDRODYNAMICS

BJORKEN’s Law (coming from ∂µTµν = 0):

∂τε+ε+ PL

τ= 0 → ∂τε+

43ε

τ=

43η

τ2

assuming η = η0τ

and STEFAN-BOLTZMANN entropy s ≈ ε 34

∂τε+43ε

τ=

43η

s︸︷︷︸cte

ε34

τ2

At a given time, knowing ε, PT , PL and assuming• an EOS• STEFAN-BOLTZMANN entropy• η = η0

τ

• ηs = cte

gives a very simple hydro model.




∂τε+ε+ PL

τ= 0 → ∂τε+

43ε

τ=

43η

τ2

assuming η = η0τ


∂τε+43ε

τ=

43η

s︸︷︷︸cte

ε34

τ2


τ

• ηs = cte





∂τε+ε+ PL

τ= 0 → ∂τε+

43ε

τ=

43η

τ2

assuming η = η0τ


∂τε+43ε

τ=

43η

s︸︷︷︸cte

ε34

τ2


τ

• ηs = cte



COMPARISON WITH HYDRO: ISOTROPIZATION

×10-3

×10-2

×10-1

×100

×101

0 50 100 150 200 250 300

τ

2PT + P

L

ε

PT

PL



×10-3

×10-2

×10-1

50 100 150 200 250 300

τ

τ0=70 Field theory



×10-3

×10-2

×10-1

50 100 150 200 250 300

τ

τ0=150 Field theory



×10-3

×10-2

×10-1

50 100 150 200 250 300

τ

τ0=200 Field theory


COMPARISON WITH HYDRO: VISCOSITY

PT − PL =2ητ

0.1

1

10

100

0 50 100 150 200 250 300

η /

s

τ

effective η / s

perturbation theory

AdS/CFT bound1

4π

∼ 104

g4


COMPARISON WITH HYDRO: VISCOSITY

see also [ASAKAWA, BASS, MULLER (2006-07)]

0.1

1

10

100

0 50 100 150 200 250 300

η /

s

τ

effective η / s

perturbation theory

AdS/CFT bound1

4π

∼ 104

g4


4 YANG-MILLS THEORYThe theoryNumerical results


THE NLO SPECTRUM

• Need to know ~e~k(τ0,~x) at the time τ0 we start the numerical simulation

• For practical reasons, we must start in the forward light cone (τ0 > 0)

x− x+

?

~e~k(x) ∼t 7→−∞

eik.x

This can be done analytically [TE,GELIS 1307:1765]


THE NLO SPECTRUM

Result of [TE,GELIS 1307:1765]

eiν~k⊥

= iν(Fi,− − Fi,+) eη

ν~k⊥(x) = Di (Fi,− − Fi,+)

with

Fi,+k (x) ∼ eiνη U

†1(~x⊥)

∫~p⊥

ei~p⊥·~x⊥ U1(~p⊥ +~k⊥)(

p2⊥τ

2k⊥

)iν[δij −

2pi⊥pj⊥

p2⊥

]ε

jkλ .

• U†1 depends on the color source J+ of the first nucleus

• Analogous formula for Fi,−.


4 YANG-MILLS THEORYThe theoryNumerical results


YM ON A LATTICE

Gauge potential Aµ → link variables (exact gauge invariance on the lattice)

aL aT

N

L

Lη

x

y

Numerical parameters• Transverse lattice size L = 64, transverse lattice spacing QsaT = 1• Longitudinal lattice size N = 128, longitudinal lattice spacing aL = 0.016• Number of configurations for the Monte-Carlo Nconf = 200 to 2000• Initial time Qsτ0 = 0.01


EOM ON A LATTICE

Writing

Eµ(x) =xµ

Uµ(x) = x µ U†µ(x) =x + µµ

and

Uµν(x) =x µ

ν U†µν(x) = x

µ

ν

Uµ−ν(x) =

x µ

ν U†µ−ν(x) =

x

µ

ν .

We can therefore rewrite the EOM as

∂τxI=

−i2gaIa2

J

∑J

x I

J−

x

I

J+

x I

J−

x

I

J

∂τ

x I = − i g aIxI

I


NUMERICAL RESULTS [TE,GELIS 1307:2214]

αs = 8 10−4 (g = 0.1)

+ 10-3

+ 10-2

+ 10-1

+ 100

+ 101

1

Tµν /

Qs4

τ [fm/c]

0.01 0.1 2 3 4

- 10-3

- 10-2

- 10-1

- 100

- 101

0.1 1.0

Qs τ

10 20 30 40

ε

PT

PL



αs = 8 10−4 (g = 0.1)

-1

0

1/3

1/2

+1

0.1 1.0 10.0

1

Qs τ

τ [fm/c]

0.01 0.1

10.0 20.0 30.0 40.0

2 3 4

PT / ε

PL / ε

LO


RENORMALIZATION PROCEDURE

⟨E2

L,div⟩

∼ Q2sk

2⊥,max + k4

⊥,max + k4⊥,max ln2 νmax

τ+ ...

∼

x

+

x

︸︷︷︸Q2sk2

⊥,max

+

x

︸︷︷︸k4

⊥,max

+

x

+

x

︸︷︷︸k4

⊥,max ln2 νmaxτ

+...

3 last diagrams can be substracted with a simulation where

Aaµ(x) = 0 + aa

µ(x)

E2L "fine" (B2

L too)



⟨E2

T,div⟩

∼ Q2s

ν2maxτ2 + k2

⊥,maxν2

maxτ2 + k4

⊥,max ln2 νmax

τ+ ...

∼

x

+

x

︸︷︷︸Q2s

ν2maxτ2

+

x

︸︷︷︸k2

⊥,maxν2

maxτ2

+

x

+

x

︸︷︷︸k4

⊥,max ln2 νmaxτ

+...

3 last diagrams can be substracted with a simulation where

Aaµ(x) = 0 + aa

µ(x)

How to deal with the first 2? → ad-hoc fit for the time being.

Otherwise E2L and B2

L behaves as τ−2 at early time.



ε = E2T + B2

T + E2L︸︷︷︸

fine

+ B2L︸︷︷︸

fine

PT = E2L︸︷︷︸

fine

+ B2L︸︷︷︸

fine

PL = E2T + B2

T − E2L︸︷︷︸

fine

− B2L︸︷︷︸

fine



〈PT〉phys. = 〈PT〉 backgd.+ fluct.

− 〈PT〉 fluct.only

〈ε, PL〉phys. = 〈ε, PL〉 backgd.+ fluct.︸︷︷︸

computed

− 〈ε, PL〉 fluct.only︸︷︷︸

computed

+ A τ−2︸︷︷︸fitted

.

Ad-hoc term only one to satisfy Bjorken law and EOS:

∂ττ−α + 2τ−α−1 = 0



How come that problematic divergent diagrams behaves as mass terms?

In the continuum limit, they don’t exist local gauge invariant operators ofdimension two.

On the lattice though, they coud be terms like

g2 ν2max

k2⊥,maxτ

2 TrF2,

where

Fµν(x) ∼x µ

ν−

x

µ

ν



αs = 2 10−2 (g = 0.5)

+ 10-3

+ 10-2

+ 10-1

+ 100

1

Tµν /

Qs4

τ [fm/c]

0.01 0.1 2 3 4

- 10-3

- 10-2

- 10-1

- 100

0.1 1.0

Qs τ

10 20 30 40

ε

PT

PL



αs = 2 10−2 (g = 0.5)

+ 10-3

+ 10-2

+ 10-1

+ 100

1

Tµν /

Qs4

τ [fm/c]

0.01 0.1 2 3 4

- 10-3

- 10-2

- 10-1

- 100

0.1 1.0

Qs τ

10 20 30 40

ε

PT

PL



αs = 2 10−2 (g = 0.5)

-1

0

1/3

1/2

+1

0.1 1.0 10.0

1

Qs τ

τ [fm/c]

0.01 0.1

10.0 20.0 30.0 40.0

2 3 4

PT / ε

PL / ε

LO



αs = 2 10−2 (g = 0.5)

-1

0

1/3

1/2

+1

0.1 1.0 10.0

1

Qs τ

τ [fm/c]

0.01 0.1

10.0 20.0 30.0 40.0

2 3 4

PT / ε

PL / ε

LO


ANOMALOUSLY SMALL VISCOSITY

Assuming simple first order viscous hydrodynamics

ε ≈ ε0τ− 4

3︸︷︷︸Ideal hydro

− 2η0τ−2︸︷︷︸

first order correction

we can compute the dimensionless ratio (η = η0τ−1)

ηε−34 . 1

In contrast, perturbation theory at LO gives ηε−34 ∼ 300.

If the system is closed from being thermal

ε−34 ∼ s =⇒ η

sNot far from

14π


CONCLUSION

• Correct NLO spectrum from first principles

• Fixed anisotropy for g = 0.5 at τ ∼ 1fm/c

• No need for strong coupling to get isotropization

• Compatible with viscous hydrodynamical expansion

• Assuming simple first order viscous hydrodynamics

ηε−34 . 1


CONCLUSION



1) Equation of state f(PL, PT ) = ε2) Small anisotropy3) Initialization: ε(τ0), PL(τ0)? ...4) viscous coefficients: shear viscosity η,...5) Short isotropization time


Documents

Early Isotropization of the Quark Gluon Plasmaepelbaum/Santiago_29_10.pdf · Santiago de Compostela, 29th October 2013 Thomas EPELBAUM IPhT THOMAS EPELBAUM Early Isotropization of