françois gelis the early stages of heavy ion...

François Gelis

The early stages ofHeavy Ion Collisions

SEWM, Swansea University, July 2012

François GelisIPhT, Saclay

François Gelis

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)

strong fields classical dynamics

gluons & quarks out of eq. viscous hydro

gluons & quarks in eq. ideal hydro

hadrons kinetic theory

freeze out

François Gelis

Stages of a nucleus-nucleus collision 3

strong fields classical dynamics

gluons & quarks out of eq. viscous hydro

gluons & quarks in eq. ideal hydro

hadrons kinetic theory

freeze out

François Gelis

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

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

Slightly boosted nucleus

François Gelis

High energy nucleus

François Gelis

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)

-110 1

HERAPDF1.0

exp. uncert.

model uncert.

parametrization uncert.

2 = 10 GeV2Q

H1 and ZEUS

François Gelis

Nucleon parton distributions 6

-110 1

HERAPDF1.0

exp. uncert.

model uncert.

2 = 10 GeV2Q

H1 and ZEUS

François Gelis

Large x : dilute regime

-110 1

HERAPDF1.0

exp. uncert.

model uncert.

2 = 10 GeV2Q

H1 and ZEUS

François Gelis

Small x : dense regime, gluon saturation

François Gelis

Saturation domain 7

François Gelis

Saturation domain 7

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

αsQ−2︸︷︷︸

σgg→g

×A−2/3xG(x,Q2)︸︷︷︸surface density

Q2 ≤ Q2s ≡αsxG(x,Q

A2/3︸︷︷︸saturation momentum

∼ A1/3x−0.3

François Gelis

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

∫FµνF

µν︸︷︷︸SYM

∫JµAµ︸︷︷︸

fast partons

François Gelis

Initial statefactorization

[FG, Lappi, Venugopalan (2008)]

François Gelis

Renormalization group evolution, JIMWLK equation 10

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

)W[ρ] (JIMWLK equation)

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

François Gelis

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

τcoll ∼ E-1

François Gelis

τcoll ∼ E-1

space-like interval

François Gelis

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

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

∣∣∣∣LO

∫d4xd4y eip·(x−y) �x�y A(x)A(y)

d3~p1 · · ·d3~pn

∣∣∣∣LO

· · · dN1d3~pn

∣∣∣∣LO

François Gelis

Next to Leading Order 14

Relation between LO and NLO

ONLO =

∫u,v

[a∗kT

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

[ ∫u

]O[class. field at τ︷︸︸︷Aτ( Ainit︸︷︷︸

init. field

[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

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

∫u,v

[a∗kT

= 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

Factorization of the logarithms 16

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

∫[Dρ]

Inclusive observables at Leading Log accuracy

〈O〉Leading Log

∫ [Dρ

]W1[ρ

]W2[ρ

]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

Final stateevolution

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

[Epelbaum, FG (2011)]

François Gelis

Energy momentum tensor at LO 18

François Gelis

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

0.0001

ηη /

1 Exp(0.427 Sqrt(g

2 µ τ))

1 Exp(0.00544 g

2 µ τ)

[Romatschke, Venugopalan (2005)]

François Gelis

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

0.0001

ηη /

1 Exp(0.427 Sqrt(g

2 µ τ))

1 Exp(0.00544 g

2 µ τ)

[Romatschke, Venugopalan (2005)]

François Gelis

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

Improved power counting 20

Loop ∼ g2 , T ∼ e√µτ

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

Tµν(x)

B subleading

François Gelis

Tµν(x)

Γ3(u,v,w)

B subleading

François Gelis

Resummation of the leading secular terms 21

Tµνresummed

∫u,v

[akT]u[a∗kT]v︸︷︷︸

G(u,v)

[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

Resummation of the leading secular terms 21

Tµνresummed

∫u,v

[akT]u[a∗kT]v︸︷︷︸

G(u,v)

[αT]u

]Tµν

LO[Ainit]

∫[Dχ] exp

∫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

Analogous scalar toy model 22

φ4 field theory coupled to a source

2(∂αφ)

2 −g2

4!φ4 + Jφ

Strong external source: J ∝ Q3

• 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

Pathologies in fixed order calculations 23

-20 0 20 40 60 80

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

Pathologies in fixed order calculations 23

Tree + 1-loop

-20 0 20 40 60 80

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

Spectrum of fluctuations 24

Initial conditions

φ(t0, x) = ϕ(t0, x) +

∫d3k

(2π)32k

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 +

2g2ϕ2(t0, x)

]Vk(x) = λ

2kVk(x)

François Gelis

Resummed energy momentum tensor 25

10 100 1000 10000

px+py+pz

• No secular divergence in the resummed pressure

• The pressure relaxes to the equilibrium equation of state

François Gelis

Time evolution of the occupation number 26

0 0.5 1 1.5 2 2.5 3 3.5

Bose-Einstein with µ=0.54,T=1.31

T/(ωk-µ)-1/2 with µ=0.54,T=1.31

const / k5/3

• Resonant peak at early times

• Late times : classical equilibrium with a chemical potential

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

François Gelis

Bose-Einstein condensation 27

0 0.5 1 1.5 2 2.5 3 3.5

T/(ωk-µ)-1/2

0 0.5 1 1.5 2 2.5

• 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

Volume dependence 28

×10-6

×10-5

×10-4

×10-3

×10-2

×10-1

×101 ×10

2 ×103 ×10

L = 20 L = 30 L = 40

f(k) =1

eβ(ωk−µ) − 1+ n0δ(k) =⇒ f(0) ∝ V = L3

François Gelis

Evolution of the condensate 29

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

g2 = 1

• 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

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

Longitudinal expansion: initial time 31

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

∂2τφ+1

τ∂τφ−

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

×10-2 ×10

-1 ×100 ×10

PT(τ0=0.01)

ε(τ0=0.01)

PT(τ0=0.1)

ε(τ0=0.1)

François Gelis

Longitudinal expansion : occupation number 32

τ = 0.1 [40 × 40 × 320]

×10-2

• 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

τ = 10 [40 × 40 × 320]

×10-2

• 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

τ = 50 [40 × 40 × 320]

×10-2

• 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

τ = 100 [40 × 40 × 320]

×10-2

system)

François Gelis

τ = 150 [40 × 40 × 320]

×10-2

system)

François Gelis

τ = 200 [40 × 40 × 320]

×10-2

system)

François Gelis

τ = 300 [40 × 40 × 320]

×10-2

system)

François Gelis

Longitudinal expansion : equation of state 33

10 100

τ-4/3

2PT + P

• 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

François Gelis

Longitudinal expansion : isotropization 34

×10-3

×10-2

×10-1

0 50 100 150 200 250 300

2PT + P

• 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

François Gelis

Summaryand Outlook

François Gelis

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

g 1 g-1

François Gelis

Dense-dilute collisions 38

g 1 g-1

François Gelis

Dense-dilute collisions 38

Expected complications

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

g2ρ2(∂

)2∼ g4ρ2

François Gelis

Exclusive processes 39

François Gelis

Exclusive processes 39

Example : differential probability to produce 1 particle at LO

∣∣∣∣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

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

BEC and dilepton production 41

François Gelis

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

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

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

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

François Gelis

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

Probing early times with rapidity correlations 45

detection (∼1 m/c)

freeze out (∼10 fm/c)

latest correlation

• 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

[Lappi, McLerran (2006)]

François Gelis

Color field at early time 46

0 0.5 1 1.5 2

g2µτ

[Lappi, McLerran (2006)]

François Gelis

Color field at early time 46

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

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

François Gelis

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

Centrality dependence 48

0 50 100 150 200 250 300 350 400

radial boost model

STAR preliminary

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

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

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

Rapidity dependence 49

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

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

Rapidity dependence 49

Prediction at LHC energy

-4 -2 0 2 4 6 8 10 12

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

Backup

François Gelis

Boundary conditions 51

A+(x)=

G0++(x, y)[J(y)−V ′(A+(y))

]−G0+−(x, y)

[J(y)−V ′(A−(y))

]A−(x)=

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

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

• Decompose the fields in Fourier modes :

Aε(x) ≡∫

(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

Moyal bracket 52

• Wigner transform of the commutator

• Explicit expression :

{{A,B}} =2

h̄A(Q,P) sin

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

fs740 gps time and frequency system - srs …fs740 gps time...

franc¸ois gelis , raju venugopalan september 18, 2018 ·...

nucleosynthesis in classical novae & x-ray bursts jordi...

installation instructions single package …e).… ·...

1e channel

3 dk l s i 12 ail2010 rd kaplan symposium, april ·...

contracts 1e

chapter 17 performance development. 17- 2 management 1e 17-...

chapter 12 managing team performance. 12- 2 management 1e...

workshop april 4th, 2007 lithography a uc discovery...

overview - microsoft · energy required: mtf vs. mf and icf...

ph - hanna instruments · what is ph? 0 2 4 6 8 10 12 14...

chapter 13 managers as leaders. 13- 2 management 1e 13- 2...

uspas course on photocathode...

technimatter: a new direction for sewm? · introduction t =...

an overview of the jupiter icy moons orbiter (jimo)€¦ ·...

workshop on detection limits without noise … · loveland...

1e learning

diapositiva 1e

sewm'14 keynote: mining events from multimedia streams