Anisotropies in momentum space Anisotropies in momentum space in a Transport Approachin a Transport Approach

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

Quantifying the properties of Hot QCD Matter – INT Seattle, July 2010 Quantifying the properties of Hot QCD Matter – INT Seattle, July 2010

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

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)

If v2 is very large

To balance the minimum vv44 >0 require >0 require

v4 ~ 4.4% if v2= 25%

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

OutlineOutline Results from RHICResults from RHIC

bulk, jets, hadronization, heavy quarksbulk, jets, hadronization, heavy quarks

motivation for a transport approachmotivation for a transport approach

Cascade 2<->2 collisions at fixed Cascade 2<->2 collisions at fixed /s/s: Scaling properties of v2(pT)/x

Link v2(pT) - /s~0.1-0.2 and coalescence

Large vLarge v44/(v/(v22))22

Transport Theory with Mean Field at fixed Transport Theory with Mean Field at fixed /s/s:

NJL chiral phase transition and vNJL chiral phase transition and v2 2 /s/s

Extension to quasiparticle models fitted to lQCD Extension to quasiparticle models fitted to lQCD ,P,P

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

From RHIC but more relevant at LHCFrom RHIC but more relevant at LHC: :

Initial ConditionInitial Condition – “exotic” non equilibrium Bulk Bulk – Hydrodynamics BUTBUT large finite viscosities () MinijetsMinijets – perturbative QCD BUTBUT strong Jet-Bulk “talk” Heavy QuarksHeavy Quarks – Brownian particle (?) BUT strongly coupled to Bulk HadronizationHadronization – Microscopic mechanism can modify QGP observables

From the State of the Art From the State of the Art Transport Transport

Non-equilibrium + microscopic scale are relevant in all the subfields A unified framework against a separate modelling can be useful

Viscous HydrodynamicsViscous Hydrodynamics

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 (Hooke law like)

two parameters appears

f (pT) quite arbitrary

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

Motivation for Transport approachMotivation for Transport approach

It is a 3+1D (viscous hydro 2+1D till now)

No gradient expansion, full calculation

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

valid at high valid at high /s /s LHCLHC

include hadronization by coalescence+fragmentationinclude hadronization by coalescence+fragmentation

CGC pCGC pTT out of equilibrium impact ( out of equilibrium impact (beyond the difference in beyond the difference in xx)) not possibile in hydrodynamics

naturally including Bulk viscosity naturally including Bulk viscosity

Wider Range of validity in Wider Range of validity in p pTT + microscopic level -> hadronization + microscopic level -> hadronization

0 Hydrodynamic limit can be derived0 Hydrodynamic limit can be derived

TransportTransportCascade approachCascade approach

...22 Ifp

Solved discretizing the space in x, ycells

Collision integral not solved with the geometrical interpretation, but with a local stochastic sampling

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

exact solutions of the Boltzmann equation

Questions that we want to addressQuestions that we want to address:

What scalings survive for a fluid at finite What scalings survive for a fluid at finite s?s? Can we constrain Can we constrain /s by v/s by v22??

Are both vAre both v22(p(pTT) and v) and v44 (p (pTT) consistent with a unique ) consistent with a unique /s?/s?

Are vAre v22(p(pTT) and v) and v44 (p (pTT) at finite ) at finite /s consistent with Quark Number /s consistent with Quark Number

Scaling?Scaling?

t0

3x03x

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)

(*)

(different from D. Molnar arXiV:0806.0026P. Huovinen-D. Molnar, PRC79 (2009))

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

Analizing the Analizing the

scaling of vscaling of v22(p(pTT)/)/xx

Is the finite Is the finite /s that causes the breaking of v/s that causes the breaking of v22// scaling? scaling?

The vThe v22 /<v /<v22> scaling validates the ideal hydrodynamics?> scaling validates the ideal hydrodynamics?

Relation betweenRelation betweenxxand vand v2 2 in Hydroin Hydro

Ideal Hydrodynamics Ideal Hydrodynamics (no size (no size scale)scale):

v2/ scales with :- impact parameter - system size

Bhalerao et al., PLB627(2005)

2v

time

Hydrodynamics

STAR, PRC77(08)

Does the breaking Does the breaking

come from finite come from finite /s?/s?

vv22// and v and v22/<v/<v22> as a function of p> as a function of pTT

Scaling for bothboth v2/<v2> and v2/ for bothboth Au+Au and Cu+Cu

Agreement with PHENIX data for v2/<v2>/s1/4 on top to data, but… this is but… this is missleadingmissleading

Parton Cascade – without a freeze-outwithout a freeze-out

Au+Au & Cu+Cu@200 AGeV

4/s=1

PHENIX PRL 98, 162301 (2007)

Note: Scaling also outside the pT hydro region

STAR, PRC77 (2008)

vv22(p(pTT)/)/ does not does not scale!scale!

Can a cascade approach account for this?

vv22(p(pTT)/<v)/<v22> scales!> scales!

Experimentally…Experimentally…

Freeze-out is crucial !Freeze-out is crucial !

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

Au+Au@200 AGeV

Cascade at finite Cascade at finite /s + freeze-out /s + freeze-out ::

VV22// broken in a way similar to STAR data Agreement with PHENIX and STAR scaling of v2/<v2> Freeze-out + Freeze-out + /s lowers the v/s lowers the v22(p(pTT) at higher p) at higher pTT … …

vv22// scaling broken scaling broken vv22/<v/<v22> scaling kept!> scaling kept!

Results with both freeze-out Results with both freeze-out and no freeze-outand no freeze-out

No f.o.No f.o.

Short Reminder from coalescence…Short Reminder from coalescence…

/3)(p3v)(pv

/2)(p2v)(pv

Tq2,TB2,

Tq2,TM2,

Enhancement of vEnhancement of v22Quark Number ScalingQuark Number Scaling

n

p

nT

2V1

Molnar and Voloshin, PRL91 (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

v2 for baryon is larger and saturates at higher pT

v2q fitted from v2

GKL, PRC68(03)Greco-Ko-Levai,PRC68(03)

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 ? ?

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 : vs

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

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

agreement agreement with coalesccencewith coalesccence

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 /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

The mixed phase The mixed phase becomes irrelevant!becomes irrelevant!

Pb+Pb @ 5.5 ATeV , b= 8 fm |y|<1

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 !

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

and not on impact parmeterand not on impact parmeter……See 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!

At LHC v4/(v2)2 large time scale …

Pb+Pb @ 5.5 ATeV , b= 8 fm |y|<1

44/s=1/s=1

44/s=1 + f.o./s=1 + f.o.

44/s(T) + f.o/s(T) + f.o..

44/s=1/s=1

44/s=1 + f.o./s=1 + f.o.

44/s(T) + f.o/s(T) + f.o..

Only RHIC has Only RHIC has the right timethe right time

scale to infere the scale to infere the T dependence of T dependence of /s!/s!

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

of the Chiral phase transitionof the Chiral phase transition

- From Cascade to 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

Mass generation affects momenta Mass generation affects momenta attractive contribution attractive contribution

np15

4

Massive gas in relaxation time approximation

The viscosity is kept modifying locally the cross-section

=cell index in the r-spaceM=0

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

Self-Consistentlyderived from NJLlagrangian

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)

Extension to realistic EoS quasiparticle model fitted to lQCD

- 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…

but still missing the 3-body collisions

and also hadronization…

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

° A. Bazavov et al. 0903.4379 hep-lat

NJL

QP

lQC

D-F

odor

P

Transport at finite Transport at finite /s+ f.o. can pave the way for a /s+ f.o. 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

Large v4/(v2)2 ratio signature of /s(T) (at

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

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

vv22 /s/s

SummarySummary

Next Steps :Next Steps :

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

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

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 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: NO Hydro+Statistical hadronization

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

rates NO BROWNIAN MOTION, NO FULL THERMALIZATION ->Transport Regime

Test in a Box at equilibrium Test in a Box at equilibrium

Calculation for Au+Au running …

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