françois gelis the early stages of heavy ion...

François Gelis

1

The early stages ofHeavy Ion Collisions

SEWM, Swansea University, July 2012

z

t

François GelisIPhT, Saclay

François Gelis

2

Outline 2

1 Initial state factorization

2 Final state evolution

In collaboration with :

K. Dusling (NCSU)T. Epelbaum (IPhT)T. Lappi (Jyväskylä U.)R. Venugopalan (BNL)

z

t

strong fields classical dynamics

gluons & quarks out of eq. viscous hydro

gluons & quarks in eq. ideal hydro

hadrons kinetic theory

freeze out

François Gelis

3

Stages of a nucleus-nucleus collision 3

z

t

strong fields classical dynamics

gluons & quarks out of eq. viscous hydro

gluons & quarks in eq. ideal hydro

hadrons kinetic theory

freeze out

François Gelis

3

Stages of a nucleus-nucleus collision 3

Goals :

• evolution from t = −∞ to the start of hydrodynamics• first principles description in QCD• weakly coupled (but strongly interacting...)

François Gelis

4

What do we need to know about nuclei? 4

Nucleus at rest

• At low energy : valence quarks

• At higher energy :

• Lorenz contraction of longitudinal sizes• Time dilation B slowing down of the internal dynamics• Gluons start becoming important

• At very high energy : gluons dominate

François Gelis

4


Slightly boosted nucleus





François Gelis

4


High energy nucleus





François Gelis

5

Kinematics 5

• Low typical final state transverse momentum p⊥ . 1 GeV• Incoming partons have low momentum fractions x ∼ p⊥/E

• x ∼ 10−2 at RHIC (E = 200 GeV)• x ∼ 4.10−4 at the LHC (E = 2.76 − 5.5 TeV)

-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

François Gelis

6

Nucleon parton distributions 6

-310

-210

-110

1

10

-410

-310

-210

-110 1

-310

-210

-110

1

10

HERAPDF1.0

exp. uncert.

model uncert.


x

xf

2 = 10 GeV2Q

vxu

vxd

xS

xg

H1 and ZEUS

-310

-210

-110

1

10

François Gelis

6


Large x : dilute regime

-310

-210

-110

1

10

-410

-310

-210

-110 1

-310

-210

-110

1

10

HERAPDF1.0

exp. uncert.

model uncert.


x

xf

2 = 10 GeV2Q

vxu

vxd

xS

xg

H1 and ZEUS

-310

-210

-110

1

10

François Gelis

6


Small x : dense regime, gluon saturation

François Gelis

7

Saturation domain 7

François Gelis

7

Saturation domain 7

Saturation criterion [Gribov, Levin, Ryskin (1983)]

αsQ−2︸︷︷︸

σgg→g

×A−2/3xG(x,Q2)︸︷︷︸surface density

≥ 1

Q2 ≤ Q2s ≡αsxG(x,Q

2s)

A2/3︸︷︷︸saturation momentum

∼ A1/3x−0.3

François Gelis

8

Color Glass Condensate = effective theory of small x gluons 8

[McLerran, Venugopalan (1994), Jalilian-Marian, Kovner, Leonidov, Weigert(1997), Iancu, Leonidov, McLerran (2001)]

• The fast partons (k+ > Λ+) are frozen by time dilationB described as static color sources on the light-cone :

Jµ = δµ+ρ(x−,~x⊥) (0 < x− < 1/Λ+)

• The color sources ρ are random, and described by a probabilitydistribution WΛ[ρ]

• Slow partons (k+ < Λ+) may evolve during the collisionB treated as standard gauge fieldsB eikonal coupling to the current Jµ : JµAµ

S = −1

4

∫FµνF

µν︸︷︷︸SYM

+

∫JµAµ︸︷︷︸

fast partons

François Gelis

9

Initial statefactorization

[FG, Lappi, Venugopalan (2008)]

François Gelis

10

Renormalization group evolution, JIMWLK equation 10

k-

P-

Λ-

0

fields sources

• The cutoff between the sources and the fields is not physical,and should not enter in observables

• Loop corrections contain logs of the cutoff(physically, due to soft gluon radiation)

• In inclusive Deep Inelastic Scattering: cancelled by letting thedistribution of the sources depend on the cutoff

Λ∂W[ρ]

∂Λ= H

(ρ,δ

δρ

)W[ρ] (JIMWLK equation)

• What about nucleus-nucleus collisions?Are the logs still universal?

François Gelis

11

Handwaving argument for factorization 11

τcoll ∼ E-1

• The duration of the collision is very short: τcoll ∼ E−1

• Soft radiation takes a long timeB it must happen (long) before the collision

• The projectiles are not in causal contact before the impactB the logarithms are intrinsic properties of the projectiles,independent of the measured observable

François Gelis

11

Handwaving argument for factorization 11

τcoll ∼ E-1

space-like interval

• The duration of the collision is very short: τcoll ∼ E−1

• Soft radiation takes a long timeB it must happen (long) before the collision

• The projectiles are not in causal contact before the impactB the logarithms are intrinsic properties of the projectiles,independent of the measured observable

François Gelis

12

Power counting 12

In the saturated regime: Jµ ∼ g−1

g−2 g# of external legs g2×(# of loops)

• No dependence on the number of sources Jµ

B infinite number of graphs at each order

François Gelis

13

Leading Order 13

• All observables can be expressed in terms of classical solutionsof Yang-Mills equations :

[Dµ,Fµν] = Jν1 + Jν2

• Boundary conditions for inclusive observables :retarded, with A → 0 at x0 = −∞

Example : inclusive spectra at LO

dN1

d3~p

∣∣∣∣LO

∼

∫d4xd4y eip·(x−y) �x�y A(x)A(y)

dNn

d3~p1 · · ·d3~pn

∣∣∣∣LO

=dN1

d3~p1

∣∣∣∣LO

· · · dN1d3~pn

∣∣∣∣LO

François Gelis

14

Next to Leading Order 14

Relation between LO and NLO

ONLO =

[1

2

∫u,v

∫k

[akT

]u

[a∗kT

]v+

∫u

[αT

]u

]OLO

• T is the generator of the shifts of the initial field :

exp

[ ∫u

[αT

]u

]O[class. field at τ︷︸︸︷Aτ( Ainit︸︷︷︸

init. field

)]= O

[Aτ( Ainit + α︸︷︷︸

shifted init. field

)]

• ak,α are calculable analytically

• Valid for all inclusive observables,e.g. spectra, the energy-momentum tensor, etc...

François Gelis

15

Initial state logarithms 15

• In the CGC, upper cutoff on the loop momentum : k± < Λ,to avoid double counting with the sources Jν1,2B logarithms of the cutoff

Central result for factorization

1

2

∫u,v

∫k

[akT

]u

[a∗kT

]v+

∫u

[αT

]u=

= log(Λ+)H1 + log

(Λ−)H2 + terms w/o logs

H1,2 = JIMWLK Hamiltonians of the two nuclei

• No mixing between the logs of the two nuclei

• Since the LO ↔ NLO relationship is the same for all inclusiveobservables, these logs have a universal structure

François Gelis

16

Factorization of the logarithms 16

• The JIMWLK Hamiltonian H is self-adjoint :∫[Dρ]W

(HO

)=

∫[Dρ]

(HW

)O

Inclusive observables at Leading Log accuracy

〈O〉Leading Log

=

∫ [Dρ

1Dρ

2

]W1[ρ

1

]W2[ρ

2

]OLO [ρ1, ρ2]︸︷︷︸

fixed ρ1,2

• Logs absorbed into the scale evolution of W1,2

Λ∂W

∂Λ= HW (JIMWLK equation)

• Universality : the same W’s for all inclusive observables

François Gelis

17

Final stateevolution

[Dusling, Epelbaum, FG, Venugopalan (2010)][Dusling, FG, Venugopalan (2011)]

[Epelbaum, FG (2011)]

QS

-1

François Gelis

18

Energy momentum tensor at LO 18

QS

-1

François Gelis

18

Energy momentum tensor at LO 18

Tµν for longitudinal ~E and ~B

TµνLO

(τ = 0+) = diag (ε, ε, ε,−ε)

B far from ideal hydrodynamics

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

2

c0+c

1 Exp(0.427 Sqrt(g

2 µ τ))

c0+c

1 Exp(0.00544 g

2 µ τ)

[Romatschke, Venugopalan (2005)]

François Gelis

19

Weibel instabilities for small perturbations 19

[Mrowczynski (1988), Romatschke, Strickland (2003), Arnold, Lenaghan,Moore (2003), Rebhan, Romatschke, Strickland (2005), Arnold, Lenaghan,Moore, Yaffe (2005), Romatschke, Rebhan (2006), Bodeker, Rummukainen(2007),...]

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

2

c0+c

1 Exp(0.427 Sqrt(g

2 µ τ))

c0+c

1 Exp(0.00544 g

2 µ τ)

[Romatschke, Venugopalan (2005)]

François Gelis

19

Weibel instabilities for small perturbations 19

[Mrowczynski (1988), Romatschke, Strickland (2003), Arnold, Lenaghan,Moore (2003), Rebhan, Romatschke, Strickland (2005), Arnold, Lenaghan,Moore, Yaffe (2005), Romatschke, Rebhan (2006), Bodeker, Rummukainen(2007),...]

• For some k’s, the field fluctuations ak divergelike exp

√µτ when τ→ +∞

• Some components of Tµν have secular divergences whenevaluated at fixed loop order

• When ak ∼ A ∼ g−1, the power counting breaks down andadditional contributions must be resummed :

g e√µτ ∼ 1 at τmax ∼ µ−1 log2(g−1)

François Gelis

20

Improved power counting 20

Loop ∼ g2 , T ∼ e√µτ

u

Tµν(x)

vΓ2(u,v)

• 1 loop : (ge√µτ)2

• 2 disconnected loops :(ge√µτ)4

• 2 nested loops : g(ge√µτ)3

B subleading

Leading terms at τmax

• All disconnected loops to all ordersB exponentiation of the 1-loop result

François Gelis

20



Tµν(x)




B subleading



François Gelis

20



Tµν(x)

Γ3(u,v,w)




B subleading



François Gelis

21

Resummation of the leading secular terms 21

Tµνresummed

= exp

[1

2

∫u,v

∫k

[akT]u[a∗kT]v︸︷︷︸

G(u,v)

+

∫u

[αT]u

]Tµν

LO[Ainit]

= TµνLO

+ TµνNLO︸︷︷︸

in full

+ TµνNNLO

+ · · ·︸︷︷︸partially

• The evolution remains classical, but we must average over aGaussian ensemble of initial conditions

• The 2-point correlation G(u, v) of the fluctuations is known

• Fluctuations ⇔ 1/2 quantum per mode

François Gelis

21

Resummation of the leading secular terms 21

Tµνresummed

= exp

[1

2

∫u,v

∫k

[akT]u[a∗kT]v︸︷︷︸

G(u,v)

+

∫u

[αT]u

]Tµν

LO[Ainit]

=

∫[Dχ] exp

[−1

2

∫u,v

χ(u)G−1(u, v)χ(v)

]Tµν

LO[Ainit + χ+ α]

• The evolution remains classical, but we must average over aGaussian ensemble of initial conditions

• The 2-point correlation G(u, v) of the fluctuations is known

• Fluctuations ⇔ 1/2 quantum per mode

François Gelis

22

Analogous scalar toy model 22

φ4 field theory coupled to a source

L =1

2(∂αφ)

2 −g2

4!φ4 + Jφ

Strong external source: J ∝ Q3

g

• In 3+1-dim, g is dimensionless, and the only scale in the problemis Q

• This theory has unstable modes (parametric resonance)

• Two setups have been studied :

• Fixed volume system• Longitudinally expanding system

François Gelis

23

Pathologies in fixed order calculations 23

Tree

-40

-30

-20

-10

0

10

20

30

40

-20 0 20 40 60 80

time

PLO εLO

• Oscillating pressure at LO : no equation of state

• Small NLO correction to the energy density (protected by energyconservation)

• Secular divergence in the NLO correction to the pressure

François Gelis

23

Pathologies in fixed order calculations 23

Tree + 1-loop

-40

-30

-20

-10

0

10

20

30

40

-20 0 20 40 60 80

time

PLO+NLO εLO+NLO

• Oscillating pressure at LO : no equation of state• Small NLO correction to the energy density (protected by energy

conservation)• Secular divergence in the NLO correction to the pressure

François Gelis

24

Spectrum of fluctuations 24

Initial conditions

φ(t0, x) = ϕ(t0, x) +

∫d3k

(2π)32k

[ck e

iλkt0 Vk(x) + c.c.]

• ϕ(t0, x) = classical background field

• ck = complex random Gaussian number:⟨ck⟩= 0 ,

⟨ckc∗l

⟩= (2π)3 k δ(k− l)

• λk, Vk determined by:[−∇2 +

1

2g2ϕ2(t0, x)

]Vk(x) = λ

2kVk(x)

François Gelis

25

Resummed energy momentum tensor 25

-1.5

-1

-0.5

0

0.5

1

1.5

2

10 100 1000 10000

time

px+py+pz

ε

• No secular divergence in the resummed pressure

• The pressure relaxes to the equilibrium equation of state

François Gelis

26

Time evolution of the occupation number 26

0.01

0.1

1

10

100

1000

10000

0 0.5 1 1.5 2 2.5 3 3.5

f k

k

Bose-Einstein with µ=0.54,T=1.31

T/(ωk-µ)-1/2 with µ=0.54,T=1.31

const / k5/3

t = 0

60

200

1000

2000

5000

104

• Resonant peak at early times

• Late times : classical equilibrium with a chemical potential

• µ ≈ m + excess at k = 0 : Bose-Einstein condensation?

François Gelis

27

Bose-Einstein condensation 27

10-2

10-1

100

101

102

103

104

0 0.5 1 1.5 2 2.5 3 3.5

f k

k

t = 0

50

150

300

2000

5000

104

T/(ωk-µ)-1/2

10-2

10-1

100

101

102

103

104

0 0.5 1 1.5 2 2.5

t = 0

20

40

60

80

100

120

140

• Start with the same energy density, but an empty zero mode

• Very quickly, the zero mode becomes highly occupied

• Same distribution as before at late times

François Gelis

28

Volume dependence 28

×10-6

×10-5

×10-4

×10-3

×10-2

×10-1

×100

×101 ×10

2 ×103 ×10

4

f(0)

/ V

time

L = 20 L = 30 L = 40

f(k) =1

eβ(ωk−µ) − 1+ n0δ(k) =⇒ f(0) ∝ V = L3

François Gelis

29

Evolution of the condensate 29

10-1

100

101

102

103

104

100

101

102

103

104

105

f 0

time

203 lattice , 256 configurations , V(φ) = g

2φ

4/4!

g2 = 1

2

4

8

• Formation time almost independent of the coupling• Condensate lifetime much longer than its formation time• Smaller amplitude and faster decay at large coupling

François Gelis

30

Longitudinal expansion: spectrum of fluctuations 30

Initial conditions

φ(τ0, η, x⊥) = ϕ(τ0, x⊥) +

∫d2k⊥dν

(2π)3

[cνkH

(2)iν (λkτ0)Vk(x⊥) e

−iνη + c.c.]

• ϕ(τ0, x⊥) = classical background field (boost invariant)

• cνk = complex random Gaussian number:⟨cνk⟩= 0 ,

⟨cνkc

∗σl

⟩= (2π)3 δ(k⊥ − l⊥)δ(ν− σ)

• λk, Vk determined by:[−∇2

⊥ +1

2g2ϕ2(τ0, x⊥)

]Vk(x⊥) = λ

2kVk(x⊥)

François Gelis

31

Longitudinal expansion: initial time 31

• The EoM is singular when τ→ 0 : one must start at τ0 6= 0

∂2τφ+1

τ∂τφ−

1

τ2∂2ηφ−∇2

⊥φ+g2

6φ3 = 0

• The spectrum of fluctuations of the initial field also depends onτ0, in such a way that the end result is independent of τ0

×10-1

×100

×101

×102

×103

×104

×10-2 ×10

-1 ×100 ×10

1

τ

PT(τ0=0.01)

ε(τ0=0.01)

PT(τ0=0.1)

ε(τ0=0.1)

François Gelis

32

Longitudinal expansion : occupation number 32

τ = 0.1 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104

• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied

• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalizedsystem)

François Gelis

32


τ = 10 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104

• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied

• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalizedsystem)

François Gelis

32


τ = 50 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104

• Note : ν is the Fourier conjugate of the rapidity η• Initially, only the ν = 0 modes are occupied• Rapid expansion of the distribution in ν (νmax ∼ τ2/3 for a thermalized

system)

François Gelis

32


τ = 100 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104


system)

François Gelis

32


τ = 150 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104


system)

François Gelis

32


τ = 200 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104


system)

François Gelis

32


τ = 300 [40 × 40 × 320]

0

0.5

1

1.5

2

2.5

3

k⊥

0

100

200

300

400

500

600

ν

×10-2

×100

×102

×104

×106

100

102

104


system)

François Gelis

33

Longitudinal expansion : equation of state 33

10-2

10-1

100

101

10 100

τ

τ-4/3

τ-1

2PT + P

L

ε

• After a short time, one has 2PT+ P

L≈ ε

• Change of behavior of the energy density: τ−1 → τ−4/3. Sinceone has ∂τε+ (ε+ P

L)/τ = 0, this suggests that P

Lgets close to

ε/3

François Gelis

34

Longitudinal expansion : isotropization 34

×10-3

×10-2

×10-1

×100

×101

0 50 100 150 200 250 300

τ

2PT + P

L

ε

PT

PL

• At early times, PL

drops much faster than PT

(redshifting of thelongitudinal momenta due to the expansion)

• Drastic change of behavior when the expansion rate becomes smallerthan the growth rate of the unstability

• Eventually, isotropic pressure tensor : PL≈ P

T

François Gelis

35

Summaryand Outlook

François Gelis

36

Summary 36

• Factorization of high energy logarithms in AA collisions• universal distributions, also applicable in pA or DIS• controls the rapidity dependence of correlations• limited to inclusive observables

• Resummation of secular terms in the final state evolution• stabilizes the NLO calculation• leads to the equilibrium equation of state• isotropization even with longitudinal expansion• full thermalization on longer time-scales• Bose-Einstein condensation if overoccupied initial state

(so far, all numerical studies done for a toy scalar model)

What’s next? QCD

• generalizable to Yang-Mills theory

• gauge invariant

• seamless integration with the CGC description of AA collisions

• computationally expensive ( ∼ [scalar case] ×3× (N2c − 1))

François Gelis

37

Extra

ρ1

g 1 g-1

AApA

g-4

g-2

1

g2

g4

p q

p

q

p

q

François Gelis

38

Dense-dilute collisions 38

ρ1

g 1 g-1

AApA

g-4

g-2

1

g2

g4

p q

p

q

p

q

François Gelis

38

Dense-dilute collisions 38

Expected complications

• More diagrams to consider even at Leading Order• More terms in the evolution Hamiltonian if ρ ∼ g:

g2ρ2(∂

∂ρ

)2∼ g4ρ2

(∂

∂ρ

)4

François Gelis

39

Exclusive processes 39

François Gelis

39

Exclusive processes 39

Example : differential probability to produce 1 particle at LO

dP1

d3~p

∣∣∣∣LO

= F[0]×∫d4xd4y eip·(x−y)�x�yA+(x)A−(y)

∣∣∣z=0

• The vacuum-vacuum graphs do not cancel in exclusivequantities : F[0] 6= 1 (in fact, F[0] = exp(−c/g2)� 1)

• A+ and A− are classical solutions of the Yang-Mills equations,but with non-retarded boundary conditions

François Gelis

40

Thermalization in Yang-Mills theory 40

• Recent analytical work : Kurkela, Moore (2011)

• Going from scalars to gauge fields :

• More fields per site (3 Lorentz components × 8 colors)

• More complicated spectrum of initial conditions

• Expansion : UV overflow on a fixed grid in η

François Gelis

41

BEC and dilepton production 41

François Gelis

41

BEC and dilepton production 41

Two topologies for virtual photons at LO

Connected ω ∼Minv ∼ Qs k⊥ ∼ Qs

Disconnected ω ∼Minv ∼ Qs k⊥ � Qs

B excess of dileptons with k⊥ �Minv

François Gelis

42

Links to Quantum Chaos 42

• Quantum Chaos : how does the chaos at the classical levelmanifests itself in quantum mechanics?

• Berry’s conjecture [M.V. Berry (1977)]High lying eigenstates of such systems have nearly randomwavefunctions. The corresponding Wigner distribution is almostuniform on the energy surface

• Srednicki’s eigenstate thermalization hypothesis[M. Srednicki (1994)]For sufficiently inclusive measurements, these high lyingeigenstates look thermal. If the system starts in a coherent state,decoherence is the main mechanism to thermalization

François Gelis

43

Phenomenology[Dumitru, FG, McLerran, Venugopalan (2008)]

[Dusling, FG, Lappi, Venugopalan (2010)]

François Gelis

44

2-particle correlations in AA collisions 44

[STAR Collaboration, RHIC]

• Long range rapidity correlation

• Narrow correlation in azimuthal angle

• Narrow jet-like correlation near ∆y = ∆ϕ = 0

François Gelis

45

Probing early times with rapidity correlations 45

detection (∼1 m/c)

freeze out (∼10 fm/c)

latest correlation

AB

z

t

• By causality, long range rapidity correlations are sensitive to thedynamics of the system at early times :

τcorrelation ≤ τfreeze out e−|∆y|/2

0 0.5 1 1.5 2

g2µτ

0

0.2

0.4

0.6

0.8

[(g

2µ

)4/g

2]

Bz

2

Ez

2

BT

2

ET

2

[Lappi, McLerran (2006)]

François Gelis

46

Color field at early time 46

0 0.5 1 1.5 2

g2µτ

0

0.2

0.4

0.6

0.8

[(g

2µ

)4/g

2]

Bz

2

Ez

2

BT

2

ET

2

[Lappi, McLerran (2006)]

François Gelis

46

Color field at early time 46

QS

-1

• The field lines form tubes of transverse size ∼ Q−1s

• Rapidity correlation length : ∆η ∼ α−1s

François Gelis

47

2-hadron correlations from color flux tubes 47

• η-independent fields lead to long range correlations :

• Particles emitted by different flux tubes are not correlatedB (RQs)

−2 sets the strength of the correlation

• At early times, the correlation is flat in ∆ϕ

The collimation in ∆ϕ is produced later by radial flow

François Gelis

47



R

QS

-1





François Gelis

47



vr





François Gelis

48

Centrality dependence 48

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250 300 350 400

Am

pli

tud

e

Npart

radial boost model

STAR preliminary

• Main effect : increase of the radial flow velocity with the centralityof the collision

0

0.2

0.4

0.6

0.8

1

1.2

-6 -5 -4 -3 -2 -1 0 1 2

1/N

trig

dN

ch/d

∆η

∆η

pTtrig

= 2.5 GeV

pTassoc

= 350 MeV

Au+Au 0-30% (PHOBOS)

p+p (PYTHIA)ytrig = 0

ytrig=0.75

ytrig=1.5

François Gelis

49

Rapidity dependence 49

0

0.2

0.4

0.6

0.8

1

1.2

-6 -5 -4 -3 -2 -1 0 1 2

1/N

trig

dN

ch/d

∆η

∆η

pTtrig

= 2.5 GeV

pTassoc

= 350 MeV

Au+Au 0-30% (PHOBOS)

p+p (PYTHIA)ytrig = 0

ytrig=0.75

ytrig=1.5

François Gelis

49

Rapidity dependence 49

Prediction at LHC energy

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-4 -2 0 2 4 6 8 10 12

d2N

/(d

2p

T d

2q

T d

yp d

yq)

[GeV

-4]

yq - yp

Pb+Pb at LHC

× 10-2

× 0.5

× 0.65× 0.85

pT = 2 GeV

qT = 2 GeV

yp = 0

yp = -1.5

yp = -3

yp = -4.5

François Gelis

50

Backup

François Gelis

51

Boundary conditions 51

A+(x)=

∫y

G0++(x, y)[J(y)−V ′(A+(y))

]−G0+−(x, y)

[J(y)−V ′(A−(y))

]A−(x)=

∫y

G0−+(x, y)[J(y)−V ′(A+(y))

]−G0−−(x, y)

[J(y)−V ′(A−(y))

]G̃0+−(p) = 2π z(p) θ(−p0)δ(p

2)

• A+ and A− are solutions of the classical EoM

• Decompose the fields in Fourier modes :

Aε(x) ≡∫

d3~p

(2π)32Ep

[a(+)ε (x0, ~p) e

−ip·x + a(−)ε (x0, ~p) e

+ip·x]

Boundary conditions

x0 = −∞ : a(+)+ (~p) = a

(−)− (~p) = 0

x0 = +∞ : a(+)− (~p) = z(p)a

(+)+ (~p) , a

(−)+ (~p) = z(p)a

(−)− (~p)

François Gelis

52

Moyal bracket 52

• Wigner transform of the commutator

• Explicit expression :

{{A,B}} =2

h̄A(Q,P) sin

(h̄

2(←∇Q

→∇P

−←∇P

→∇Q

)

)B(Q,P)

• Quantum deformation of the Poisson bracket :

{{A,B}} = {A,B}+ O(h̄2)

françois gelis the early stages of heavy ion...

Documents