Transport Theory for the Transport Theory for the

QuarkQuark--Gluon PlasmaGluon Plasma

V. GrecoV. Greco

UNIVERSITY of CATANIAUNIVERSITY of CATANIA

INFNINFN--LNSLNS

QuarkQuark--GluonGluon Plasma and Plasma and HeavyHeavy--IonIon CollisionsCollisions –– TurinTurin ((ItalyItaly), 7), 7--12 12 MarchMarch 2011 2011

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x

yz

=∂

=∂

0)(

0)(

xj

xT

B

µµ

µνµ

HydrodynamicsHydrodynamicsNo microscopic descriptions

(mean free path λλλλ -> 0, ηηηη=0)

implying f=feq

+ EoS P(εεεε)

All the observables are in a way or the other

related with the evolution of the phase space

density :

What happens if we drop such assumptions?

There is a more “general” transport theory

valid also in non-equilibrium?

Is there any motivation to look for it?

pdxd

dNtpxf

33),,( =

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PickingPickingPickingPickingPickingPickingPickingPicking--------up four main results at RHIC up four main results at RHIC up four main results at RHIC up four main results at RHIC up four main results at RHIC up four main results at RHIC up four main results at RHIC 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 λ and f ≠feq)

�� High OpacityHigh Opacity, , StrongStrong JetJet--quenchingquenching:� 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

� HadronizationHadronization modifiedmodified, , CoalescenceCoalescence::� 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

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BULK BULK

((ppTT~T)~T)

MINIJETS MINIJETS

((ppTT>>T,>>T,ΛΛQCDQCD))

CGC (x<<1)

Gluon saturationHeavyHeavy QuarksQuarks

(m(mqq>>T,>>T,ΛΛQCDQCD))

MicroscopicMicroscopic

MechanismMechanism

MattersMatters!!

Initial Conditions Quark-Gluon Plasma Hadronization

�� ppTT>> T , intermediate >> T , intermediate ppTT

�� m >> T , m >> T , heavyheavy quarksquarks

�� ηη/s >>0 , high /s >>0 , high viscosityviscosity

�� InitialInitial time time studiesstudies of of thermalizationsthermalizations

� Microscopic mechanism for HadronizationHadronization can modify QGP observable

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

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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

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),()(

),(),(

0

pxfp

xFxm

p

pxfpd

dp

xd

dx

d

pxdf

∂∂+

∂∂=

∂∂+

∂∂==

µµ

µ

µ

µ

µ

µ

µ

τττ

pdxd

dPptpxtxtprf

N

iii 33

1

33 ))(())((),,( =−−=∑=

δδrr

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 ))(())((),( =−−=∑=

τδτδ f(x,p) is a Lorentz scalar &

P0=(p2+m2)1/2

ClassicalClassical TransportTransport TheoryTheory

If one is interested to the collective behavior or to the behavior of a typical particle

knowledge 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

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),()(1

0),()( pxfxFpm

pxfp

xFxm

ppx

∂⋅+∂⋅==

∂∂+

∂∂

µµ

µ

µ

0=∇⋅+∇⋅+∂∂

fFfvt

fpr

rrrr

The non-relativistic reduction

Relativistic Vlasov Equation

Liouville -> Vlasov -> No dissipation

+

Collision= Boltzmann-VlasovDissipation

Entropy production

Allowing for scatterings

particles go in and out phase space

(d/dt) f(x,p)≠0

),]([),()(1

pxfCpxfxFpm px =

∂⋅+∂⋅

Collision term

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The Collision Term

It 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 collisions

2) 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 make

the transition

Sum over all the

momenta the kick-out

The particle in (x,p)

σdvpdpdw 1223

13)2112( =′′′′→

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

212123

13

23 ppxfppppwpdpdpdCgain ′′′′′′= ∫

Collision Rate

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In a more explicit form and covariant version:

gain

loss

At equilibrium in each phase-space region Cgain =Closs0)],([ 0 =pxfC

τ0][

fffC

−−≅vv

1 λρσ

τ == Relaxation time

time between 2 collisions

When one is close to equilibrium or when the λmfp is very small

One can linearize the collision integral in δf=f-f0 <<f

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

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LocalLocal EquilibriumEquilibrium SolutionSolution

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:

[ ]µβ +⋅−= )(exp)( uppf juttner

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

β=1/T temperature

u collective four velocity

µ chemical potential

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

Main points:

• Boltzmann-Vlasov equation gives the right equilibrium distributions

• Close to equilibrium there can be many collisions with vanishing net effect

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Relation Relation toto HydrodynamicsHydrodynamics

=∂

=∂

0)(

0)(

xj

xTµ

µ

µνµ

Ideal Hydro

),(

),(

4

4

pxfpppdT

pxfppdj

νµµν

µµ

∫

∫=

=

∫∫∫ −−=

−−∂−=∂=∂ )()( 04044 ffpd

mffmfxmFpdfppdj p ττ

µµ

µµ

µµ

Inserting

Vlasov 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 ∂µTµν , for f≠feq and one can do the expansion in terms of transport

coefficients: 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

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[ ]),(ln1),()( 4 pxfpxfppdxS −= ∫µµ

[ ]0

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

f

fffpd

mffppdffppdS xx −==∂−=−∂−=∂ ∫∫∫ τµ

µµ

µµµ

0,0ln)1( ≥≥− xxx

Approach to thermal equlibrium is always associated to entropy production

EntropyEntropy Production <Production <--> > ThermalThermal EquilibriumEquilibrium

All these results are always valid and do not rely

on the relaxation time approx. more generally:

[ ] fffffCpdst

sS ln'2'12 −−=⋅∇+

∂∂=∂ ∫

rrµ

µ

∆S=0 <-> C[f]=0

Collision integral is associated to entropy production but if a local equilibrium

is reached there are many collisions without dissipations!

Boltzmann-Vlasov Eq.

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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

[ ]µβ +⋅−+=

)(exp1

1)(

uppfFD

So one gets the correct quantum equilbrium distribution, but what is

F(x,p) for a quantum system?

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Quantum Quantum TransportTransport TheoryTheoryIn quantum theory the evolution of a system can be described in terms of

the density matrix operator:

iii

iw ΨΨ=∑ρ̂ [ ]OtTrtO ˆ)()( ρ)=

2/ˆ2

),(~

yxxxAxedy

pxAipy

±== ±−+∫ h

hπ

[ ] dxdypxBpxABATr ∫∫= ),(~

),(~ˆˆ

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

),( −+∗−

−+

−ΨΨ== ∫∫ xxe

dyxxe

dypxf

ipyipy

Whh

hh πρ

π

[ ] ∫== dxdppxOpxfOTrO W ),(~

),(ˆρ̂fW plays in many respects

the same role of the distribution

function in statistical mechanics

and any expectation value

can be calculated as

and by (*)

(*)

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Properties of the Wigner Function

∫

∫=ΨΨ=

=ΨΨ=∗

∗

)()()(),(

)()()(),(

ppppxfdx

xxxpxfdp

W

W

ρ

ρ

[ ] ∫== dxdppxOpxfOTrO W ),(~

),(ˆρ̂

However for pure state fW can be negative so it cannot be a probability

On the other hand if we interpret its absolute value as a probabilty it does

not violate the uncertainty principle because one can show:

hπ1

),( ≤pxfW 1211

2

1 ≤∆∆≤∆⇒≤ pxNdxdp

dN ππππ hh

So if we go in a phase space smaller than

∆x∆p<h/2 one can never locate a particle

In agreement with the uncertainty principle

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Quantum Quantum TransportTransport EquationEquation

[ ]Ht

ˆ,ˆˆ ρρ =

∂∂ [ ] 0ˆ,ˆ

ˆ=−

∂∂

+− xHt

x ρρ0ˆ

2

ˆ,ˆ

ˆ

2

2

=

+−

∂∂

+−∫ xUm

p

txe

dy pyi

ρρπ

h

h

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 rsh

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=∇⋅∇+∂∂+

∂∂

wpxWW fUx

f

m

p

t

f rr

r

r This has the same form of the classical

transport equation, but it is for example

exact for an harmonic potential

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

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Quantum Quantum TransportTransport EquationEquation

[ ]Ht

ˆ,ˆˆ ρρ =

∂∂ [ ] 0ˆ,ˆ

ˆ=−

∂∂

+− xHt

x ρρ0ˆ

2

ˆ,ˆ

ˆ

2

2

=

+−

∂∂

+−∫ xUm

p

txe

dy pyi

ρρπ

h

h

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 rsh

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

h

0=∇⋅∇+∂∂+

∂∂

wpxWW fUx

f

m

p

t

f rr

r

r This has the same form of the classical

transport equation, but it is for example

exact for an harmonic potential

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

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TransportTransport TheoryTheory in in FieldField TheoryTheory

One can extend the Wigner function (4x4 matrix):

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

),(4

4

yxxxxeyd

pxFpy

i±=ΨΨ= ±−+∫ αβαβ π

h

h

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

µνµν

µµ

µµ σγγγγ TPVS FFiFFFF

2

1555 ++++=

For example the vector current

[ ] ),(4),( 44 pxFpdpxFTrpdj Vµµµµ γγ ∫∫ ==ΨΨ=

In a similar way to what done in Quantum mechanics

one can start from the Dirac equation for the fermionic field

[ ] 0)()()( =Ψ−−−∂ xgMVgi sV σγ µµµ[ ] 0)()(())(( =Ψ−−−∂ −−− xxgMxVgi sV αµµµ σγ [ ] 0)()(())(()(

)2( 4

4

=Ψ−−−∂Ψ∫ −−−+⋅ xxgMxVgixe

RdsV

Ripαµµ

µβ σγ

πSee : Vasak-Gyulassy- Elze, Ann. Phys. 173(1987) 462

Elze and Heinz, Phys. Rep. 183 (1989) 81

Blaizot and Iancu, Phys. Rep. 359 (2002) 355

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TransportTransport TheoryTheory in in FieldField TheoryTheory

One can extend the Wigner function (4x4 matrix):

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

),(4

4

yxxxxeyd

pxFpy

i±=ΨΨ= ±−+∫ αβαβ π

h

h

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

µνµν

µµ

µµ σγγγγ TPVS FFiFFFF

2

1555 ++++=

For example the vector current

[ ] ),(4),( 44 pxFpdpxFTrpdj Vµµµµ γγ ∫∫ ==ΨΨ=

In a similar way to what done in Quantum mechanics

one can start from the Dirac equation for the fermionic field

[ ] 0)()()( =Ψ−−−∂ xgMVgi sV σγ µµµ[ ] 0)()(())(( =Ψ−−−∂ −−− xxgMxVgi sV αµµµ σγ

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

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

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

),(2 4

4

=ΨΨ−

−∂⋅−⋅ −−+⋅−

∫ xxxeRd

pxFmi

p Rip σπ

γγ αβραβρ

For the NJL σ=G ΨΨ

...)(2

)()( +∂−≅− xR

xx xσσσ µ

µ This is the semiclassical approximation.

If one include higher order derivatives gets

an 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∂pσ⋅FW >>1

Again the point is to have not too large gradients:

XF typical length scale of the field

PW typical momentum scale of the system

A very rough estimate for the QGP

XF ~ RN ~ 4-5 fm , PW ~ T ~ 1-3 fm-1 -> XF·PW ~ 5-15 >> 1

better for larger and hotter systems

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0),()(2

)]([2

=

∂∂+−−⋅+∂⋅ pxFx

ixmp

iW

px µµσσγγ

Substituting the semiclassical approximation one gets:

There is a real and an imaginary part

{ } 0),(ˆ* =−⋅ pxFMp Wγ Which contains the

in medium mass-shell( ) 0),(

2*2 =− pxFMp S

Including more terms in the gradient expansion would have brougth terms

breaking the mass-shell constraint

[ ]{ } 0),(* =∂∂++∂ ∗∗∗∗∗ pxFmmFpp Spµ

µµννµ

µ

{ } 0),(ˆ)()( ** =∂∂+∂⋅ pxFxMxM Wpxµµγ µ

µγ VSW FFF +=

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

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

VlasovVlasov TransportTransport EquationEquation in QFTin QFT

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

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Transport solved on latticeTransport solved on latticeTransport solved on latticeTransport solved on latticeTransport solved on latticeTransport solved on latticeTransport solved on latticeTransport solved on lattice

...22 +=∂ ↔Cfp µµ

Solved discretizing the space in (η, (η, (η, (η, x, y))))α α α α cells

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

∆t→0

∆3x→0

exact

solution∆3x

Putting massless partons

at equilibrium in a box

than the collision rate is

Rate of collisions

per unit of phase

space

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Approaching equilibrium in a box

where the temperature is

Highly non-equilibrated distributions

F.Scardina

anisotropy in p-space

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Transport vs Viscous Hydrodynamics in 0+1D

λτ

λ→= L

K

s

TK

/5

1

ητ=

Knudsen number-1

000 2.14

2 τπηT

sK <⇒>

Huovinen and Molnar, PRC79(2009)

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Transport TheoryTransport TheoryTransport TheoryTransport TheoryTransport TheoryTransport TheoryTransport TheoryTransport Theory

� validvalid alsoalso at intermediate at intermediate ppTT out of out of equilibriumequilibrium::regionregion of of modifiedmodified hadronizationhadronization at RHICat RHIC

�� validvalid alsoalso at high at high ηη/s/s −−> > LHC and/or LHC and/or hadronichadronic phasephase

�� RelevantRelevant at LHC due at LHC due toto largelarge amountamount of minijet production of minijet production

�� Appropriate Appropriate forfor heavyheavy quark quark dynamicsdynamics

�� can can followfollow exoticexotic nonnon--equilibriumequilibrium phasephase CGC: CGC:

[ ]{ } ...),( 3222* ++=∂∂++∂ ↔↔

∗∗∗∗∗ CCpxfmmFpp pµ

µµννµ

µ

A unified framework against a separate modelling with

a wider range of validity in η, ζ,η, ζ,η, ζ,η, ζ, pT + microscopic level.

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Applications of transport approach

to the QGP Physics

- Collective flows & shear viscosity

- dynamics of Heavy Quarks & Quarkonia

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=∂

=∂

0)(

0)(

xj

xT

B

µµ

µνµ

HydrodynamicsHydrodynamicsNo microscopic details

(mean free path λ -> 0, η=0)

+ EoS P(εεεε)

PartonParton cascadecascade

v2 saturation pattern reproduced

First stage of RHICFirst stage of RHIC

22↔=∂ Cfp µµ

Parton elastic 2→2 interactions

(λ=1/σρ - P=ε/3)

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Information from nonInformation from non--equilibrium: Elliptic Flowequilibrium: Elliptic Flow

xy z

px

py

22

22

xy

xyx +

−=ε

cc22ss==dPdP/d/dεε

v2/ε measures the efficiencyof the convertion of the anisotropy

from CoordinateCoordinate to

MomentumMomentum spacespace

[ ]...)2cos(v21 2 ++= ϕϕ TT dp

dN

ddp

dNFourier expansion in p-space

| | | | | | | | EoSEoS

Massless gas ε=3P -> c2s=1/3

Bhalerao et al., PLB627(2005)

2v2/ε

time

c2s= 0.6

c2s= 0.1

MeasureMeasure of of

P P gradientsgradients

Hydrodynamics

λ=0

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Information from nonInformation from non--equilibrium: Elliptic Flowequilibrium: Elliptic Flow

xy z

px

py

22

22

xy

xyx +

−=ε

cc22ss==dPdP/d/dεε

λ=λ=λ=λ=λ=λ=λ=λ=(σρ)(σρ)(σρ)(σρ)(σρ)(σρ)(σρ)(σρ)−−−−−−−−1 1 1 1 1 1 1 1

v2/ε measures the efficiencyof the convertion of the anisotropy

from CoordinateCoordinate to

MomentumMomentum spacespace

[ ]...)2cos(v21 2 ++= ϕϕ TT dp

dN

ddp

dNFourier expansion in p-space

|| viscosityviscosity

| | | | | | | | EoSEoS

Massless gas ε=3P -> c2s=1/3

Bhalerao et al., PLB627(2005)

More generally one can distinguish:

--Short Short rangerange: : collisionscollisions --> > viscosityviscosity

--Long Long rangerange: : fieldfield interactioninteraction --> > ε ε ≠≠ 3P3P

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

2v2/ε

time

c2s= 0.6

c2s= 0.1

MeasureMeasure of of

P P gradientsgradients

Hydrodynamics

λ=0

c2s= 1/3

Parton Cascade

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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

++−

== ϕ

At RHIC a finite vAt RHIC a finite v44 observedobservedforfor the first time !the first time !

More harmonics needed to describean elliptic deformation -> v4

P. KolbPD

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Viscosity cannot be neglectedViscosity cannot be neglectedViscosity cannot be neglectedViscosity cannot be neglectedViscosity cannot be neglectedViscosity cannot be neglectedViscosity cannot be neglectedViscosity cannot be neglected

but itit violatesviolates causalitycausality, ,

IIII00 orderorder expansionexpansion neededneeded --> > IsraelIsrael--StewartStewart

tensortensor basedbased on on entropyentropy increaseincrease ∂∂µ µ ssµµ >0>0

P. Romatschke, PRL99 (07)

y

v

A

F x

yz

x

∂∂−= η

µνµνµνdissipidealTT Π+=

Relativistic Navier-Stokes

τη,τζ two parameters appears +

δf ~ feq reduce the pT validity range

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PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com

PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com

Transport approachTransport approachTransport approachTransport approachTransport approachTransport approachTransport approachTransport approach

[ ]{ } ...),( 3222* ++=∂∂++∂ ↔↔

∗∗∗∗∗ CCpxfmmFpp pµ

µµννµ

µ

CollisionsCollisions --> > ηη≠≠00Field Interaction -> ε≠3PFree streaming

C23 better not to show…

Discriminate short and long range interaction:

Collisions (λ≠0) + Medium Interaction ( Ex. Chiral symmetry breaking)

ρ,Τ

decrease

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WeWe simulate a simulate a constantconstant shearshear viscosityviscosity

( ) sTn

pTr trtr /

1

415

4)),(( , ηµ

σρσα

αα −

==r

α=cell index in the r-space

Neglecting µ and inserting in (*)

πη

4

1=s

32

45

24 T

g

T

Pns

πε =+=≈

2

1

Ttr ≈σ At T=200 At T=200 MeVMeV

σσtrtr∼∼10 10 mbmb

TimeTime--Space Space dependentdependent cross cross

sectionsection evaluatedevaluated locallylocally

V. Greco at al., PPNP 62 (09)

G. Ferini et al., PLB670 (09)

(*)cost.)4(15

4 =−

=Tn

p

s tr µσηRelativistic Kinetic theory Cascade code

The viscosity is kept

constant varying σ

A A roughrough estimate of estimate of σσ(T) (T)

α=cell index in the r-space

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a) collisions switched off

for ε<εc=0.7 GeV/fm3

b) b) ηη/s /s increasesincreases in the crossin the cross--over over regionregion, , fakingfaking the the smoothsmoothtransitiontransition betweenbetween the QGP and the QGP and the the hadronichadronic phasephase

TwoTwoTwoTwoTwoTwoTwoTwo kinetickinetickinetickinetickinetickinetickinetickinetic freezefreezefreezefreezefreezefreezefreezefreeze--------outoutoutoutoutoutoutout schemeschemeschemeschemeschemeschemeschemescheme

Finite lifetime for the QGP small η/s fluid!

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

No f.o.

sn

ptr /

1

15

1

ησ =

This gives also automatically

a kind of core-corona effect

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� 4πη/s >3 → too low v2(pT) at pT≥1.5 GeV/c even with coalescence

� 4πη/s =1 not small enough to get the large v2(pT) at pT≥2 GeV/c without coalescence

Agreement with Hydro at low pT

Parton Cascade at fixed shear viscosity

Role of ReCo for ηηηη/s estimate

HadronicHadronic ηη/s /s includedincluded

−−>> shapeshape forfor vv22((ppTT) )

consistentconsistent withwith thatthat neededneeded

byby coalescencecoalescence

A quantitative estimate needs

an EoS with ε≠ 3P :

cs2(T) < 1/3 -> v2 suppression ~~ 30%

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

with coalescencewith coalescence

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Short Reminder from coalescenceShort 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 α

[ ])2cos(v21φ

2q ϕ+=TT

q

TT

q

dpp

dN

ddpp

dN

IsIs itit reasonablereasonable the vthe v2q 2q ~0.08~0.08

neededneeded byby

CoalescenceCoalescence scalingscaling ??

IsIs itit compatiblecompatible withwith a a fluidfluid

ηη/s /s ~ 0.1~ 0.1--0.20.2 ? ?

I° Hot Quark

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EffectEffect of of ηηηηηηηη/s of the /s of the hadronichadronic phasephase

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

Uncertain hadronic ηηηη/s

is less relevant

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EffectEffect of of ηηηηηηηη/s of the /s of the hadronichadronic phasephase at LHCat LHC

RHIC – 4πηπηπηπη/s=1 + f.o.

RHIC – 4πηπηπηπη/s=2 +No f.o.

Suppression of v2 respect the ideal 4πηπηπηπη/s=1

LHC – 4πηπηπηπη/s=1 + f.o.

At LHC the contamination of mixed and hadronic phase becomes negligible

Longer lifetime of QGP -> v2 completely developed in the QGP phase

S. Plumari, Scardina, Greco in preparation

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Impact of the Impact of the MeanMean FieldField and/or and/or

of the of the ChiralChiral phasephase transitiontransition

- Cascade −> Boltzmann-Vlasov Transport

- Impact of an NJL mean field dynamics

- Toward a transport calculation with a lQCD-EoS

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NJL NJL MeanMean FieldField

TwoTwo effectseffects::

−− εε ≠≠ 3p no 3p no longerlonger a a masslessmassless freefree gas, gas, ccss <1/3<1/3

−− ChiralChiral phasephase transitiontransition

[ ])()(1)2(

)(4)(3

3

TfTfE

pdTMNgNmTM

pcf

+−

Λ

−−+= ∫ πAssociated

Gap Equation

free gas scalar field interaction

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NJL NJL MeanMean FieldField

TwoTwo effectseffects::

−− εε ≠≠ 3p no 3p no longerlonger a a masslessmassless freefree gas, gas, ccss <1/3<1/3

−− ChiralChiral phasephase transitiontransition

[ ])()(1)2(

)(4)(3

3

TfTfE

pdTMNgNmTM

pcf

+−

Λ

−−+= ∫ πAssociated

Gap Equation

free gas scalar field interaction

Fodo

r, JE

TP(2

006)NJL

gas

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BoltzmannBoltzmann--VlasovVlasov equationequation forfor the NJLthe NJL

Contribution of the NJL mean fieldContribution of the NJL mean field

NumericalNumerical solutionsolution withwith δδ--functionfunction test test particlesparticles

Test in a Box Test in a Box withwith equilibriumequilibrium ff distributiondistribution

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SimulatingSimulating a a constantconstant ηη/s /s withwith a NJL a NJL meanmean fieldfield

npτη15

4=

Massive gas in relaxation time approximation

The viscosity is kept modifying

locally the cross-section

α=cell index in the r-spaceM=0

TheoryCode

σ =10 mb

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

DynamicalDynamical evolutionevolution withwith NJLNJL

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

S. Plumari et al., PLB689(2010)

-- NJL NJL meanmean fieldfield reduce the vreduce the v22 : : attractiveattractive fieldfield

-- IfIf ηη/s /s isis fixedfixed effecteffect of NJL of NJL compensatedcompensated byby cross cross sectionsection increaseincrease

-- vv22 <<−−>> ηη/s /s notnot modifiedmodified byby NJL NJL meanmean fieldfield dynamicsdynamics!!

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NextNext stepstep -- useuse a a quasiparticlequasiparticle modelmodel

withwith a a realisticrealistic EoSEoS [[vvss(T)](T)]

forfor a quantitative estimate of a quantitative estimate of ηηηηηηηη/s /s toto compare compare withwith HydroHydro……

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WB=0 guarantees

Thermodynamicaly consistency

UsingUsing the the QPQP--modelmodel

U.Heinz and P. Levai, PRC (1998)

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

NJL

QP

lQC

D-F

odor

° A. Bazavov et al. 0903.4379 hep-lat

ε

P

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TransportTransport approachapproach can can pavepave the way the way forfor a a consistencyconsistency

amongamong knownknown vv2,42,4 propertiesproperties::

� breaking of v2(pT)/εεεε & persistence of v2(pT)/<v2> scaling

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

by coalescence by coalescence (QNS)(QNS)

�� NJL NJL chiralchiral phase transition do not modify vphase transition do not modify v22 <<<<<<<<−−−−−−−−>>>>>>>> ηηηηηηηη/s/s

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

Summary for Summary for ligthligth QGPQGP

NextNext StepsSteps forfor a quantitative estimate:a quantitative estimate:

� Include the effect of an EoS fitted to lQCD

� Implement a Coalescence + Fragmentation mechanism

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2Tf

* vm2

1TT +≈

=∂

=∂

0)(

0)(

xj

xT

B

µµ

µνµ

A Nearly Perfect FluidA Nearly Perfect FluidA Nearly Perfect FluidA Nearly Perfect FluidA Nearly Perfect FluidA Nearly Perfect FluidA Nearly Perfect FluidA Nearly Perfect Fluid

*),( T

m

T

upE

eq

T

egegpxf−−⋅−−

⋅=⋅=µγ rr

TTff ~ 120 ~ 120 MeVMeV

<<ββTT> ~ 0.5 > ~ 0.5

ForFor the first time the first time veryvery closeclose

toto ideal ideal HydrodynamicsHydrodynamics

Finite viscosity is not negligible

No microscopic description (λ->0)� Blue shift of dN/dpT hadron spectra

� Large v2/ε� Mass ordering of v2(pT)

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Jet QuenchingJet QuenchingJet QuenchingJet QuenchingJet QuenchingJet QuenchingJet QuenchingJet Quenching

Nuclear Modification FactorNuclear Modification Factor

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 correlcorrel..

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Surprises…

In vacuum p/π ~ 0.3 due to Jet fragmentation

HadronizationHadronization has been modifiedhas been modifiedppTT < 4< 4--6GeV !?6GeV !?

PHENIX, PRL89(2003)

Baryon/MesonsBaryon/Mesons

Protons not suppressed

QuenchingQuenching

Au+Au

p+p

� Jet quenching should affect both

π0 suppression: evidence of jetquenching before fragmentation

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Hadronization in Heavy-Ion Collisions

�Initial state: no partons in the vacuum but a

thermal ensemble of partons -> Use in mediumUse in medium quarks

�No 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 , 3baryons from lower momenta (denser)

Coalescence:

ReCoReCo pushespushes out soft out soft physicsphysics byby factorsfactors x2 and x3 !x2 and x3 !

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

∫∫Σ

→Σ

⊗+Φ⊗⊗= HqqMqqH Dfff

Pd

Nd3

3

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HadronizationHadronizationHadronizationHadronizationHadronizationHadronizationHadronizationHadronization ModifiedModifiedModifiedModifiedModifiedModifiedModifiedModified

Baryon/MesonsBaryon/Mesons

Au+Au

p+p

PHENIX, PRL89(2003)

Quark Quark numbernumber scalingscaling

n

TT

qT

T

H nppd

dNp

pd

dN

∝ )()(

22

n

p

nT

2V1

Dynamical quarks are visibleDynamical quarks are visible

Collective flowsCollective flows

[ ])2cos(v21φ

2 ϕ+=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 v2π

GKL

Coalescence scalingCoalescence scaling

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Heavy QuarksHeavy Quarks

−− mmc,bc,b >> >> ΛΛQCDQCD produced by pQCD processes (out of equilibrium)

−− ττeqeq > > ττQGPQGP with standard pQCD cross section (and also with

non standard non standard pQCDpQCD))

Hydrodynamics does not apply to heavy quark dynamics (f≠feq)

pQCD

“D”QGP- RHIC

Equilibration timeEquilibration timeEquilibration timeEquilibration time

npQCD

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