Transport Theory for the Transport Theory for the Quark-Gluon PlasmaQuark-Gluon Plasma

V. GrecoV. Greco UNIVERSITY of CATANIAUNIVERSITY of CATANIAINFN-LNSINFN-LNS

Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011 Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011

x

yz

0)(

0)(

xj

xT

B

HydrodynamicsHydrodynamicsNo microscopic descriptions (mean free path -> 0, =0)

implying f=feq

+ EoS P()

All the observables are in a way or the otherrelated with the evolution of the phase spacedensity :

What happens if we drop such assumptions?There is a more “general” transport theoryvalid also in non-equilibrium?Is there any motivation to look for it?

pdxd

dNtpxf

33),,(

Picking-up four main results at RHIC Picking-up four main results at RHIC

Nearly Perfect FluidNearly Perfect Fluid,, Large Collective FlowsLarge Collective Flows:: Hydrodynamics good describes dN/dpT + v2(pT) with mass

ordering

BUT VISCOSITY EFFECTS SIGNIFICANT (finite and f ≠feq) High OpacityHigh Opacity, Strong, Strong Jet-quenchingJet-quenching:: RAA(pT) <<1 flat in pT - Angular correlation triggered by jets pt<4 GeV

STRONG BULK-JET TALK: Hydro+Jet model non applicable at pt<8-10 GeV

Hadronization modifiedHadronization modified, Coalescence, Coalescence: B/M anomalous ratio + v2(pT) quark number scaling (QNS) MICROSCOPIC MECHANISM RELEVANT

Heavy quarks strongly interactingHeavy quarks strongly interacting:: small RAA large v2 (hard to get both) pQCD fails: large scattering

rates NO FULL THERMALIZATION ->Transport Regime

BULK BULK (p(pTT~T)~T)

MINIJETS MINIJETS (p(pTT>>T,>>T,QCDQCD))

CGC (x<<1)Gluon saturation

Heavy Quarks Heavy Quarks (m(mqq>>T,>>T,QCDQCD))

Microscopic Microscopic MechanismMechanism

Matters!Matters!

Initial Conditions Quark-Gluon Plasma Hadronization

ppTT>> T , intermediate p>> T , intermediate pTT

m >> T , heavy quarksm >> T , heavy quarks /s >>0 , high viscosity/s >>0 , high viscosity Initial time studies of thermalizationsInitial time studies of thermalizations Microscopic mechanism for HadronizationHadronization can modify QGP observable

Non-equilibrium + microscopic scale are relevant in all the subfields

Plan for the LecturesPlan for the Lectures Classical and Quantum Transport Theory - Relation to Hydrodynamics and dissipative

effects - density matrix and Wigner Function Relativistic Quantum Transport Theory - Derivation for NJL dynamics - Application to HIC at RHIC and LHC

Transport Theory for Heavy Quarks - Specific features of Heavy Quarks - Fokker-Planck Equation - Application to c,b dynamics

),()(

),(),(

0

pxfp

xFxm

p

pxfpd

dp

xd

dx

d

pxdf

pdxd

dPptpxtxtprf

N

iii 33

1

33 ))(())((),,(

For a classical relativistic system of N particles

Gives the probability to find a particle in phase-space

pdxd

dPppxxpxf

N

iii 44

1

44 ))(())((),(

f(x,p) is a Lorentz scalar &P0=(p2+m2)1/2

Classical Transport TheoryClassical Transport Theory

If one is interested to the collective behavior or to the behavior of a typical particleknowledge of f(x,p) is equivalent to the full solution … to study the correlations among particles one should go to f(x1,x2,p1,p2) and so on…

Liouville Theorem: if there are only conservative forces -> phase-space density is a constant o motion

Force

),()(1

0),()( pxfxFpm

pxfp

xFxm

ppx

0

fFfvt

fpr

The non-relativistic reduction

Relativistic Vlasov Equation

Liouville -> Vlasov -> No dissipation + Collision= Boltzmann-Vlasov

DissipationEntropy production

Allowing for scatteringsparticles go in and out phase space

(d/dt) f(x,p)≠0

),]([),()(1

pxfCpxfxFpm px

Collision term

The Collision TermIt can be derived formally from the reduction of the 2-body distribution Function in the N-body BBGKY hierarchy.The usual assumption in the most simple and used case:1) Only two-body collisions2) f(x1,x2,p1.p2)=f(x1,p1) f(x2,p2)

The collision term describe the change in f (x,p) because:a) particle of momentum p scatter with p2 populating the phase space in (p’1,p’2)

),,(),,,( 21]2[

212123

13

23 ppxfppppwpdpdpdCloss

probability finding 2 particles in p e p2

and space x

Probability to makethe transition

Sum over all themomenta the kick-out

The particle in (x,p)

dvpdpdw 1223

13)2112(

),,(),,,( 21]2[

212123

13

23 ppxfppppwpdpdpdCgain

Collision Rate

In a more explicit form and covariant version:

gain

loss

At equilibrium in each phase-space region Cgain =Closs 0)],([ 0 pxfC

0][

fffC

vv

1

Relaxation timetime between 2 collisions

When one is close to equilibrium or when the mfp is very small One can linearize the collision integral in f=f-f0 <<f

What is the f0(x,p)=0?

Local Equilibrium SolutionLocal Equilibrium Solution

The necessary and sufficient condition to have C[f]=0 is

),(),(),(),( 2121 pxfpxfpxfpxf

),(),(),(),(),,,(][ 2121212123

13

23 pxfpxfpxfpxfppppwpdpdpdfC

Noticing that p1+p2=p’1+p’2 such a condition is satisfied by the relativistic extension of the Boltzmann distribution:

It is an equilibrium solution also with LOCAL VALUES of T(x), u(x), m(x)

=1/T temperatureu collective four velocity chemical potential

The Vlasov part gives the constraint and the relation among T,u, locally

Main points:• Boltzmann-Vlasov equation gives the right equilibrium distributions• Close to equilibrium there can be many collisions with vanishing net effect

Relation to HydrodynamicsRelation to Hydrodynamics

0)(

0)(

xj

xT

Ideal Hydro

),(

),(

4

4

pxfpppdT

pxfppdj

)()( 04044 ffpd

mffmfxmFpdfppdj p

InsertingVlasov Eq.

Integral of a divergency

We can see that ideal Hydro can be satisfied only if f=feq , on the other hand the underlying hypothesis of Hydro is that the mean free path is so small that the f(x,p)is always at equilibrium during the evolution.

Similarly ∂T, for f≠feq and one can do the expansion in terms of transportcoefficients: shear and bulk viscosity , heat conductivity [P. Romatschke]

At the same time f≠feq is associated to the entropy production ->

General definitions

Notice in Hydro appear only p-integrated quantities

),(ln1),()( 4 pxfpxfppdxS

0

0444 ln)(...ln)()ln1(

f

fffpd

mffppdffppdS xx

0,0ln)1( xxx

Approach to thermal equlibrium is always associated to entropy production

Entropy Production <-> Thermal EquilibriumEntropy Production <-> Thermal Equilibrium

All these results are always valid and do not rely on the relaxation time approx. more generally:

fffffCpdst

sS ln'2'12

S=0 <-> C[f]=0Collision integral is associated to entropy production but if a local equilibrium

is reached there are many collisions without dissipations!

Boltzmann-Vlasov Eq.

Does such an approach can make sense for a quantum system?

One can account also for the quantum effect of Pauli-Blocking in the collision integral

)1)(1()1)(1( 212121212121 ffffffffffff

does not allow scattering if the final momenta have occupation number =1-> Boltzmann-Nordheim Collision integral

This can appear quite simplistic, but notice that C[f]=0 now is

So one gets the correct quantum equilbrium distribution, but what isF(x,p) for a quantum system?

Quantum Transport TheoryQuantum Transport TheoryIn quantum theory the evolution of a system can be described in terms ofthe density matrix operator:

2/ˆ2

),(~

yxxxAxedy

pxAipy

For any operator one can define the Weyl transform of any operator:

which has the property

The Weyl transform of the density operator is called Wigner function

)()(2

ˆ2

),(

xxe

dyxxe

dypxf

ipyipy

W

fW plays in many respectsthe same role of the distribution function in statistical mechanics

and any expectation value can be calculated as

and by (*)

(*)

Properties of the Wigner Function

)()()(),(

)()()(),(

ppppxfdx

xxxpxfdp

W

W

However for pure state fW can be negative so it cannot be a probabilityOn the other hand if we interpret its absolute value as a probabilty it doesnot violate the uncertainty principle because one can show:

1

),( pxfW 1211

2

1 pxN

dxdp

dN

So if we go in a phase space smaller thanxp<h/2 one can never locate a particleIn agreement with the uncertainty principle

Quantum Transport EquationQuantum Transport Equation

Ht

ˆ,ˆˆ

0ˆ,ˆ

ˆ

xHt

x 0ˆ

2

ˆ,ˆ

ˆ

2

2

xUm

p

txe

dy pyi

One can Wigner transform this or the Schr. Equation

After some calculations one gets the following equation

0),())((2)!12(

1

0

122

kw

kpx

k

WW pxfxUkx

f

m

p

t

f I

This exactly equivalent to the Equation for the denity matrix or the Schr. Eq.NO APPROXIMATION but allows an approximation where h does not appear explicitly and still accounting for quantum evolution when the gradient of the potential are not too strong :

0...),()(12

),()( 332

pxfxUpxfxU WpxWpx

0

wpxWW fUx

f

m

p

t

f

This has the same form of the classicaltransport equation, but it is for exampleexact for an harmonic potential

See : W.B. Case, Am. J. Phys. 76 (2008) 937

Transport Theory in Field TheoryTransport Theory in Field TheoryOne can extend the Wigner function (4x4 matrix):

2/,:)()(:)2(

),(4

4

yxxxxeyd

pxFpy

i

It can be decomposed in 16 indipendent components (Clifford Algebra)

TPVS FFiFFFF

2

1555

For example the vector current

),(4),( 44 pxFpdpxFTrpdj V

In a similar way to what done in Quantum mechanicsone can start from the Dirac equation for the fermionic field

See : Vasak-Gyulassy- Elze, Ann. Phys. 173(1987) 462 Elze and Heinz, Phys. Rep. 183 (1989) 81 Blaizot and Iancu, Phys. Rep. 359 (2002) 355

Just for simplicity lets consider the case with only a scalar field

0:)(::)()(:)2(

),(2 4

4

xxxeRd

pxFmi

p Rip

For the NJL G

...)(2

)()( xR

xx x

This is the semiclassical approximation. If one include higher order derivatives getsan expansion in terms of higher order derivatives of the field and of the Wigner function

1 WF PX

The validity of such an expansion is based on the assumption ħ∂x∂pFW >>1Again the point is to have not too large gradients:

XF typical length scale of the fieldPW typical momentum scale of the system

A very rough estimate for the QGPXF ~ RN ~ 4-5 fm , PW ~ T ~ 1-3 fm-1 -> XF·PW ~ 5-15 >> 1

better for larger and hotter systems

0),()(2

)]([2

pxFx

ixmp

iW

px

Substituting the semiclassical approximation one gets:

There is a real and an imaginary part

0),(ˆ* pxFMp W Which contains the in medium mass-shell

Including more terms in the gradient expansion would have brougth termsbreaking the mass-shell constraint

VSW FFF

Decomposing, using both real and imaginary part and taking the trace

This substitute the force term mF(x) of classical transport

Vlasov Transport Equation in QFTVlasov Transport Equation in QFT

Quantum effects encoded in the fields while f(x,p) evolution appears as the classical one.

Transport solved on latticeTransport solved on lattice

...22 Cfp

Solved discretizing the space in x, ycells

See: Z. Xhu, C. Greiner, PRC71(04)

t0

3x0exact solutio

n3x

Putting massless partons at equilibrium in a boxthan the collision rate is

Rate of collisionsper unit of phase

space

Approaching equilibrium in a box

where the temperature is

Highly non-equilibrated distributions

F.Scardina

anisotropy in p-space

Transport vs Viscous Hydrodynamics in 0+1D

LK

s

TK

/5

1

Knudsen number-1

000 2.14

2 T

sK

Huovinen and Molnar, PRC79(2009)

Transport TheoryTransport Theory

valid also at intermediate pvalid also at intermediate pTT out of equilibrium: out of equilibrium: region of modified hadronization at RHICregion of modified hadronization at RHIC

valid also at high valid also at high /s/sLHC and/or hadronic phaseLHC and/or hadronic phase

Relevant at LHC due to large amount of minijet production Relevant at LHC due to large amount of minijet production

Appropriate for heavy quark dynamics Appropriate for heavy quark dynamics can follow exotic can follow exotic non-equilibriumnon-equilibrium phase CGC: phase CGC:

A unified framework against a separate modelling with a wider range of validity in pT + microscopic level.

Applications of transport approach to the QGP Physics- Collective flows & shear viscosity- dynamics of Heavy Quarks & Quarkonia

0)(

0)(

xj

xT

B

HydrodynamicsHydrodynamicsNo microscopic details

(mean free path -> 0, =0)

+ EoS P()

Parton cascadeParton cascade

v2 saturation pattern reproduced

First stage of RHICFirst stage of RHIC

22 Cfp

Parton elastic 22 interactions

- P=/3)

Information from non-equilibrium: Elliptic FlowInformation from non-equilibrium: Elliptic Flow

xy z

px

py

22

22

xy

xyx

cc22ss=dP/d=dP/d

v2/ measures the efficiencyof the convertion of the anisotropy

from CoordinateCoordinate to Momentum spaceMomentum space

...)2cos(v21 2 TT dp

dN

ddp

dNFourier expansion in p-space

||viscosityviscosity

EEoSoS

Massless gas =3P -> c2s=1/3

Bhalerao et al., PLB627(2005)

More generally one can distinguish:

-Short range: collisions -> viscosityShort range: collisions -> viscosity-Long range: field interactionLong range: field interaction -> -> ≠ ≠ 3P3P

D. Molnar & M. Gyulassy, NPA 697 (02)

2v

time

c2s= 0.6

c2s= 0.1

Measure of Measure of P gradientsP gradients

Hydrodynamics

=0

c2s= 1/3

Parton Cascade

If v2 is very large

To balance the minimum vv44 >0 require >0 require

v4 ~ 4% if v2= 20%

222

4224

4 )(

6)4cos(

yx

yyxx

pp

ppppv

At RHIC a finite vAt RHIC a finite v44 observed observed

for the first time !for the first time !

More harmonics needed to describe an elliptic deformation -> v4

P. Kolb

Viscosity cannot be neglectedViscosity cannot be neglected

but it violates causality, it violates causality, IIII00 order expansion needed -> Israel- order expansion needed -> Israel-Stewart tensor based on entropy Stewart tensor based on entropy increase ∂increase ∂ss

P. Romatschke, PRL99 (07)

y

v

A

F x

yz

x

dissipidealTT

Relativistic Navier-Stokes

two parameters appears +

f ~ feq reduce the pT validity range

Transport approachTransport approach

Collisions -> Collisions -> ≠0≠0Field Interaction -> ≠3PFree streaming

C23 better not to show…

Discriminate short and long range interaction:Collisions (≠0) + Medium Interaction ( Ex. Chiral symmetry breaking)

decrease

We simulate a constant shear viscosityWe simulate a constant shear viscosity

sTn

pTr trtr /

1

415

4)),(( ,

=cell index in the r-space

Neglecting and inserting in (*)

4

1

s3

2

45

24 T

g

T

Pns

2

1

Ttr At T=200 MeVAt T=200 MeV

trtr10 mb10 mb

Time-Space dependent cross Time-Space dependent cross

section evaluated locallysection evaluated locally

V. Greco at al., PPNP 62 (09)G. Ferini et al., PLB670 (09)

(*)cost.)4(15

4

Tn

p

s tr Relativistic Kinetic theory Cascade code

The viscosity is kept constant varying

A rough estimate of A rough estimate of (T) (T)

=cell index in the r-space

a)collisions switched off

for <c=0.7 GeV/fm3

b) b) /s increases in the cross-over /s increases in the cross-over region, faking the smooth region, faking the smooth transition between the QGP and transition between the QGP and the hadronic phasethe hadronic phase

Two kinetic freeze-out schemeTwo kinetic freeze-out scheme

Finite lifetime for the QGP small /s fluid!

At 4/s ~ 8 viscous hydrodynamics is not applicable!

No f.o.

sn

ptr /

1

15

1

This gives also automatically a kind of core-corona effect

4/s >3 too low v2(pT) at pT1.5 GeV/c even with coalescence

4/s =1 not small enough to get the large v2(pT) at pT2 GeV/c

without coalescence

Agreement with Hydro at low pT

Parton Cascade at fixed shear viscosity

Role of ReCo for /s estimate

Hadronic Hadronic /s included /s included

shape for vshape for v22(p(pTT) )

consistent with that consistent with that

needed needed

by coalescenceby coalescenceA quantitative estimate needs an EoS with ≠ 3P : cs

2(T) < 1/3 -> v2 suppression ~~ 30%

-> /s ~ 0.1 may be in ~ 0.1 may be in

agreement agreement with coalescencewith coalescence

Short Reminder from coalescence…Short Reminder from coalescence…

Quark Number ScalingQuark Number Scaling

n

p

nT

2V1

Molnar and Voloshin, PRL91 (03)Greco-Ko-Levai, PRC68 (03)Fries-Nonaka-Muller-Bass, PRC68(03)

2

22)2()(

T

T

q

T

T

M ppd

dNαp

pd

dN

3

22)3()(

T

T

q

T

T

B ppd

dNp

pd

dN

)2cos(v21φ 2q

TT

q

TT

q

dpp

dN

ddpp

dN

Is it reasonable the vIs it reasonable the v2q 2q ~0.08~0.08 needed by needed by Coalescence scaling ?Coalescence scaling ?

Is it compatible with a Is it compatible with a fluid fluid /s /s ~ 0.1-0.2~ 0.1-0.2 ? ?

I° Hot Quark

Effect of Effect of /s of the hadronic phase/s of the hadronic phase

Hydro evolution at /s(QGP) down to thermal f.o. ~50%Error in the evaluation of h/s

Uncertain hadronic /s is less relevant

Effect of Effect of /s of the hadronic phase at LHC/s of the hadronic phase at LHC

RHIC – 4/s=1 + f.o.

RHIC – 4/s=2 +No f.o.

Suppression of v2 respect the ideal 4/s=1

LHC – 4/s=1 + f.o.

At LHC the contamination of mixed and hadronic phase becomes negligibleLonger lifetime of QGP -> v2 completely developed in the QGP phase

S. Plumari, Scardina, Greco in preparation

Impact of the Mean Field and/or Impact of the Mean Field and/or

of the Chiral phase transitionof the Chiral phase transition

- Cascade Boltzmann-Vlasov Transport

- Impact of an NJL mean field dynamics

- Toward a transport calculation with a lQCD-EoS

NJL Mean FieldNJL Mean Field

Two effects:Two effects:

≠ ≠ 3p no longer a massless free gas, c3p no longer a massless free gas, css <1/3 <1/3

Chiral phase transitionChiral phase transition

)()(1)2(

)(4)(3

3

TfTfE

pdTMNgNmTM

pcf

Associated Gap Equation

free gas scalar field interaction

Fodo

r, JE

TP(2

006)NJL

gas

Boltzmann-Vlasov equation for the NJLBoltzmann-Vlasov equation for the NJL

Contribution of the NJL Contribution of the NJL

mean fieldmean field

Numerical solution with Numerical solution with -function test particles-function test particles

Test in a Box with equilibrium Test in a Box with equilibrium ff distribution distribution

Simulating a constant Simulating a constant /s with a NJL mean field/s with a NJL mean field

np15

4

Massive gas in relaxation time approximation

The viscosity is kept modifying locally the cross-section

=cell index in the r-spaceM=0

TheoryCode

=10 mb

Au+Au @ 200 AGeV for central collision, b=0 fm.Au+Au @ 200 AGeV for central collision, b=0 fm.

Dynamical evolution with NJLDynamical evolution with NJL

Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed /s?

S. Plumari et al., PLB689(2010)

- NJL mean field reduce the vNJL mean field reduce the v22 : attractive field : attractive field

- If If /s is fixed effect of NJL compensated by cross section increase/s is fixed effect of NJL compensated by cross section increase

- vv22 /s not modified by NJL mean field dynamics!/s not modified by NJL mean field dynamics!

Next stepNext step - use a quasiparticle model - use a quasiparticle model

with a realistic EoS [vwith a realistic EoS [vss(T)](T)]

for a quantitative estimate of for a quantitative estimate of /s /s

to compare with Hydro…to compare with Hydro…

WB=0 guarantees Thermodynamicaly consistency

Using the QP-model of Heinz-Levai Using the QP-model of Heinz-Levai U.Heinz and P. Levai, PRC (1998)

M(T), B(T) fitted to lQCD [A. Bazavov et al. 0903.4379 ]data on and P

NJL

QP

lQC

D-F

odor

° A. Bazavov et al. 0903.4379 hep-lat

P

Transport approach can pave the way for a Transport approach can pave the way for a

consistency among known vconsistency among known v2,42,4 properties: properties:

breaking of v2(pT)/ & persistence of

v2(pT)/<v2> scaling

vv22(p(pTT), v), v44(p(pTT) at ) at /s~0.1-0.2 can agree with /s~0.1-0.2 can agree with

what needed what needed

by coalescence by coalescence (QNS)(QNS) NJL chiral phase transition do not modify NJL chiral phase transition do not modify

vv22 /s/s

Signature of /s(T): large v4/(v2)2

Summary for ligth QGPSummary for ligth QGP

Next Steps for a quantitative estimate:Next Steps for a quantitative estimate:

Include the effect of an EoS fitted to lQCD Implement a Coalescence + Fragmentation mechanism

2Tf

* vm2

1TT

0)(

0)(

xj

xT

B

A Nearly Perfect FluidA Nearly Perfect Fluid*

),( T

m

T

upE

eq

T

egegpxf

TTf f ~ 120 MeV~ 120 MeV

<<TT> ~ 0.5 > ~ 0.5

For the first time very closeFor the first time very close to ideal Hydrodynamicsto ideal Hydrodynamics

Finite viscosity is not negligible

No microscopic description (->0) Blue shift of dN/dpT hadron spectra Large v2/ Mass ordering of v2(pT)

Jet QuenchingJet Quenching

Nuclear Modification Nuclear Modification FactorFactor

How much modification respect to pp?

Jet gluon radiation observedJet gluon radiation observed:

all hadrons RAA <<1 and flat in pT

photons not quenched -> suppression due to QCD

away

near

Medium

Jet triggered angular Jet triggered angular correl.correl.

Surprises…

In vacuum p/ ~ 0.3 due to Jet fragmentation

Hadronization has been modifiedHadronization has been modified ppTT < 4-6GeV !? < 4-6GeV !?

PHENIX, PRL89(2003)

Baryon/MesonsBaryon/Mesons

Protons not suppressed

QuenchingQuenching

Au+Au

p+p

Jet quenching should affect both

suppression: evidence of jet quenching before fragmentation

Hadronization in Heavy-Ion CollisionsInitial state: no partons in the vacuum but a thermal ensemble of partons -> Use in mediumUse in medium quarksNo direct QCD factorization scale for the bulk: dynamics much less violent (t ~ 4 fm/c)

Parton spectrum

H

Baryon

Meson

Coal.

Fragmentation

V. Greco et al./ R.J. Fries et al., PRL 90(2003)

Fragmentation: energy needed to create quarks from vacuum hadrons from higher pT

partons are already there $ to be close in phase space $

ph= n pT ,, n = 2 , 3 baryons from lower momenta (denser)

Coalescence:

ReCo pushes out soft physics by factors x2 and x3 !ReCo pushes out soft physics by factors x2 and x3 !

More easy to More easy to produce baryons!produce baryons!

HqqMqqH Dfff

Pd

Nd3

3

Hadronization ModifiedHadronization Modified

Baryon/MesonsBaryon/Mesons

Au+Au

p+p

PHENIX, PRL89(2003)

Quark number scalingQuark number scaling

n

TT

qT

T

H nppd

dNp

pd

dN

)()(

22

n

p

nT

2V1

Dynamical quarks are visibleDynamical quarks are visibleCollective flowsCollective flows

)2cos(v21φ 2

TT

q

TT

q

dpp

dN

ddpp

dN

/3)(p3v)(pv

/2)(p2v)(pv

Tq2,TB2,

Tq2,TM2,

Enhancement of vEnhancement of v22

v2q fitted from v2

GKL

Coalescence scalingCoalescence scaling

Heavy QuarksHeavy Quarks

mmc,bc,b >> >> QCDQCD produced by pQCD processes (out of

equilibrium)

eqeq > > QGPQGP with standard pQCD cross section (and also

with

non standard pQCD)non standard pQCD)

Hydrodynamics does not apply to heavy quark dynamics

(f≠feq)

pQCD

“D”QGP- RHIC

Equilibration time

npQCD

v4 more sensitive to both /s and f.o.

v4(pT) at 4s could also be consistent with

coalescence

vv44 generated later than v generated later than v22 : more sensitive to properties at : more sensitive to properties at

TTTTcc

What about v4 ?

Relevance of time scale !Relevance of time scale !

Effect of EOS on vEffect of EOS on v22

Decrease in v2 of about 40%H. Song and U.Heinz

Very Large v4/(v2)2 ratio

Ratio v4/v22 not very much depending on not very much depending on /s/s

and not on the initial eccentricity and not on the initial eccentricity

and not on particle species and not on particle species ……

see also M. Luzum, C. Gombeaud, O. Ollitrault, arxiv:1004.2024

Same Hydro with

the good dN/dpT and v2

/

s

1

1

T/Tc

QGP

2

2

V2 develops earlier at higher /s

V4 develops later at lower /s

-> v-> v44/(v/(v22))2 2 larger larger

Effect of Effect of /s(T) on the anisotropies/s(T) on the anisotropies

Hydrodynamics Effect of finite /s+f.o.

Effect of/s(T) + f.o.

Au+Au@200AGeV-b=8fm |y|<1

vv44/(v/(v22))2 2 ~~ 0.8 signature of 0.8 signature of //ss

close to phaseclose to phase transition!transition!

If the system if very dense one can derive and add the three-body collision that make the transition from the dilute to the dense system:

See: Zhu and Greiner PRC71 (2004)