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