1. Quantifying the properties of Hot QCD Matter – INT Seattle, July 2010
Anisotropies in momentum space
in a Transport Approach
V. Greco
UNIVERSITY of CATANIA
INFN-LNS
Quantifying the properties of Hot QCD Matter – INT Seattle, July 2010

2. Information from non-equilibrium: Elliptic Flowpx py x y z
v2/e measures the efficiency
of the convertion of the anisotropy
from Coordinate to
Momentum space
Fourier expansion in p-space
l=(sr)-1
| viscosity
c2s=dP/de
| EoS
c2s= 1/3
Parton Cascade
2v2/e
time
c2s= 0.6
c2s= 0.1
Measure of
P gradients
Hydrodynamics
l=0
Massless gas e=3P -> c2s=1/3
More generally one can distinguish:
Short range: collisions -> viscosity
Long range: field interaction -> e ≠ 3P
D. Molnar & M. Gyulassy, NPA 697 (02)
Bhalerao et al., PLB627(2005)

3. First stage of RHIC Parton cascade Hydrodynamics + EoS P(e)
Parton elastic 22 interactions
(l=1/sr - P=e/3)
No microscopic details
(mean free path  -> 0, h=0)
+ EoS P(e)
v2 saturation pattern reproduced

4. At RHIC a finite v4 observed
If v2 is very large
More harmonics needed to describe
an elliptic deformation -> v4
To balance the minimum v4 >0 require
v4 ~ 4.4% if v2= 25%
At RHIC a finite v4 observed
for the first time !

5. Outline Results from RHIC Cascade 2<->2 collisions at fixed h/s:
bulk, jets, hadronization, heavy quarks
-> motivation for a transport approach
Cascade 2<->2 collisions at fixed h/s:
Scaling properties of v2(pT)/ex
Link v2(pT) - h/s~ and coalescence
Large v4/(v2)2
Transport Theory with Mean Field at fixed h/s:
NJL chiral phase transition and v2 <-> h/s
Extension to quasiparticle models fitted to lQCD e,P

6. From the State of the Art -> Transport
Initial Conditions
Quark-Gluon Plasma
Hadronization
BULK
(pT~T)
Microscopic
Mechanism
Matters!
CGC (x<<1)
Gluon saturation
MINIJETS
(pT>>T,LQCD)
Heavy Quarks
(mq>>T,LQCD)
From RHIC but more relevant at LHC:
Initial Condition – “exotic” non equilibrium
Bulk – Hydrodynamics BUT large finite viscosities (h,z)
Minijets – perturbative QCD BUT strong Jet-Bulk “talk”
Heavy Quarks – Brownian particle (?) BUT strongly coupled to Bulk
Hadronization – Microscopic mechanism can modify QGP observables
Non-equilibrium + microscopic scale are relevant in all the subfields
A unified framework against a separate modelling can be useful

7. Viscous Hydrodynamics
Relativistic Navier-Stokes (Hooke law like)
but it violates causality,
II0 order expansion needed -> Israel-Stewart tensor based on entropy increase ∂m sm >0
- th,tz two parameters appears
- df (pT) quite arbitrary
- df~ feq reduce the pT validity range
P. Romatschke, PRL99 (07)

8. Transport approach Discriminate short and long range interaction:
Field Interaction -> e≠3P
Free streaming
Collisions -> h≠0
C23 better not to show…
Discriminate short and long range interaction:
Collisions (l≠0) + Medium Interaction ( Ex. Chiral symmetry breaking)
r,T
decrease

9. Motivation for Transport approach
Wider Range of validity in h, z, pT + microscopic level -> hadronization
l->0 Hydrodynamic limit can be derived
It is a 3+1D (viscous hydro 2+1D till now)
No gradient expansion, full calculation
valid also at intermediate pT - out of equilibrium
region of the modified hadronization at RHIC
valid at high h/s -> LHC
include hadronization by coalescence+fragmentation
CGC pT out of equilibrium impact (beyond the difference in ex)
not possibile in hydrodynamics
naturally including Bulk viscosity z

10. Transport ->Cascade approach
Solved discretizing the
space in (h, x, y)a cells
Collision integral not solved with the geometrical interpretation, but with a local stochastic sampling
Z. Xhu, C. Greiner, PRC71(04)
t0
3x0
exact solutions of the Boltzmann equation
D3x
Questions that we want to address:
What scalings survive for a fluid at finite h/s?
Can we constrain /s by v2?
Are both v2(pT) and v4 (pT) consistent with a unique h/s?
Are v2(pT) and v4 (pT) at finite h/s consistent with Quark Number Scaling?

11. We simulate a constant shear viscosity
Relativistic Kinetic theory
Cascade code
(*)
=cell index in the r-space
=cell index in the r-space
Time-Space dependent cross section evaluated locally
The viscosity is kept
constant varying s
(different from D. Molnar arXiV:
P. Huovinen-D. Molnar, PRC79 (2009))
A rough estimate of (T)
Neglecting  and inserting in (*)
At T=200 MeV
tr10 mb
G. Ferini et al., PLB670 (09)
V. Greco at al., PPNP 62 (09)

12. Analizing the scaling of v2(pT)/ex
Is the finite h/s that causes the breaking of v2/e scaling?
The v2 /<v2> scaling validates the ideal hydrodynamics?

13. Relation between ex and v2 in Hydro
Bhalerao et al., PLB627(2005)
STAR, PRC77(08)
Hydrodynamics
2v2/e
time
Ideal Hydrodynamics (no size scale):
v2/e scales with :
- impact parameter
- system size
Does the breaking
come from finite h/s?

14. Parton Cascade – without a freeze-out
v2/ and v2/<v2> as a function of pT
Au+Au & AGeV
4p/s=1
Scaling for both v2/<v2> and v2/ for both Au+Au and Cu+Cu
Agreement with PHENIX data for v2/<v2>
/s1/4 on top to data, but… this is missleading

15. Freeze-out is crucial ! Experimentally… PHENIX PRL 98, 162301 (2007)
v2(pT)/ does not scale!
v2(pT)/<v2> scales!
PHENIX PRL 98, (2007)
Note: Scaling also outside
the pT hydro region
STAR, PRC77 (2008)
Can a cascade approach account for this?
Freeze-out is crucial !

16. At 4ph/s ~ 8 viscous hydrodynamics is not applicable!
Two kinetic freeze-out scheme
Finite lifetime for the QGP small h/s fluid!
collisions switched off
for <c=0.7 GeV/fm3
b) /s increases in the cross-over region, faking the smooth transition between the QGP and the hadronic phase
No f.o.
At 4ph/s ~ 8 viscous hydrodynamics is not applicable!

17. Results with both freeze-out and no freeze-out
No f.o.
No f.o.
AGeV
v2/ scaling broken
v2/<v2> scaling kept!
Cascade at finite h/s + freeze-out :
V2/ broken in a way similar to STAR data
Agreement with PHENIX and STAR scaling of v2/<v2>
Freeze-out + h/s lowers the v2(pT) at higher pT …

18. Short Reminder from coalescence…
Enhancement of v2
Quark Number Scaling
Molnar and Voloshin, PRL91 (03)
Fries-Nonaka-Muller-Bass, PRC68(03)
v2 for baryon is larger
and saturates at higher pT
v2q fitted from v2p
GKL, PRC68(03)
Is it reasonable the v2q ~0.08
needed by
Coalescence scaling ?
Is it compatible with a fluid h/s ~ ?
Greco-Ko-Levai,PRC68(03)

19. Role of Reco for h/s estimate
Parton Cascade at fixed shear viscosity
Hadronic h/s included
-> shape for v2(pT)
consistent with that needed
by coalescence
A quantitative estimate needs
an EoS with e≠ 3P :
vs2(T) < 1/3 -> v2 suppression ~ 30%
-> h/s ~ 0.1 may be in agreement
with coalesccence
Agreement with Hydro at low pT
4/s >3  too low v2(pT) at pT1.5 GeV/c even with coalescence
4/s =1 not small enough to get the large v2(pT) at pT2 GeV/c
without coalescence

20. Uncertain hadronic h/s
Effect of h/s of the hadronic phase
Hydro evolution at h/s(QGP)
down to thermal f.o. -> ~50%
Error in the evaluation of h/s
Uncertain hadronic h/s
is less relevant

21. Effect of h/s of the hadronic phase at LHC
5.5 ATeV , b= 8 fm |y|<1
The mixed phase
becomes irrelevant!

22. What about v4 ? Relevance of time scale !
v4 more sensitive to both h/s and f.o.
v4(pT) at 4ph/s=1-2 could also be consistent with coalescence
v4 generated later than v2 : more sensitive to properties at TTc

23. Very Large v4/(v2)2 ratio Ratio v4/v22 not very much depending on h/s
Same Hydro with
the good dN/dpT and v2
Ratio v4/v22 not very much depending on h/s
and not on the initial eccentricity
and not on particle species
and not on impact parmeter…
See M. Luzum, C. Gombeaud, O. Ollitrault, arxiv:

24. Effect of h/s(T) on the anisotropies
4ph/s 1
T/Tc
QGP 2
V2 develops earlier at higher h/s
V4 develops later at lower h/s
-> v4/(v2)2 larger
v4/(v2)2 ~ 0.8 signature of h/s
close to phase transition!
|y|<1
Hydrodynamics
Effect of finite h/s+f.o.
Effect of h/s(T) + f.o.

25. At LHC v4/(v2)2 large time scale …
5.5 ATeV , b= 8 fm |y|<1
4ph/s=1
4ph/s=1 + f.o.
4ph/s(T) + f.o.
Only RHIC has
the right time
scale to infere the
T dependence of h/s!

26. Impact of the Mean Field and/or of 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

27. NJL Mean Field Two effects:
free gas
scalar field interaction
Associated
Gap Equation
Two effects:
- e ≠ 3p no longer a massless free gas, cs <1/3
- Chiral phase transition
Fodor, JETP(2006)
NJL
gas

28. Boltzmann-Vlasov equation for the NJL
Self-Consistently
derived from NJL
lagrangian
Mass generation affects momenta -> attractive contribution
Contribution of the NJL mean field
Simulating a constant h/s with a NJL mean field
Massive gas in relaxation time approximation
=cell index in the r-space
M=0
The viscosity is kept modifying locally the cross-section

29. Dynamical evolution with NJL
200 AGeV for central collision, b=0 fm.

30. Does the NJL chiral phase transition affect
the elliptic flow of a fluid at fixed h/s?
S. Plumari et al., PLB689(2010)
NJL mean field reduce the v2 : attractive field
If h/s is fixed effect of NJL compensated by cross section increase
v2 <-> h/s not modified by NJL mean field dynamics
Extension to realistic EoS -> quasiparticle model fitted to lQCD

31. Next step - use a quasiparticle model with a realistic EoS [vs(T)]
for a quantitative estimate of h/s
to compare with Hydro…
but still missing the 3-body collisions
and also hadronization…

32. ° A. Bazavov et al. 0903.4379 hep-lat
Using the QP-model of Heinz-Levai
U.Heinz and P. Levai, PRC (1998)
WB=0 guarantees
Thermodynamicaly consistency
M(T), B(T) fitted to lQCD [A. Bazavov et al ]data on e and P e
NJL P QP
lQCD-Fodor
° A. Bazavov et al hep-lat

33. Summary
Transport at finite h/s+ f.o. can pave the way for a consistency among known v2,4 properties:
breaking of v2(pT)/ & persistence of v2(pT)/<v2> scaling
Large v4/(v2)2 ratio signature of h/s(T) (at RHIC)
v2(pT), v4(pT) at h/s~ can agrees with what needed by coalescence (QNS)
NJL chiral phase transition do not modify v2 <-> h/s
Next Steps :
Include the effect of an EoS fitted to lQCD
Implement a Coalescence + Fragmentation mechanism

36. Simulating a constant h/s with a NJL mean field
Massive gas in relaxation time approximation
=cell index in the r-space
M=0
The viscosity is kept modifying locally the cross-section
Theory
Code
s =10 mb

37. Picking-up four main results at RHIC
Nearly Perfect Fluid, Large Collective Flows:
Hydrodynamics good describes dN/dpT + v2(pT) with mass ordering
BUT VISCOSITY EFFECTS SIGNIFICANT
High Opacity, Strong Jet-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 modified, Coalescence:
B/M anomalous ratio + v2(pT) quark number scaling (QNS)
MICROSCOPIC MECHANISM: NO Hydro+Statistical hadronization
Heavy quarks strongly interacting:
small RAA large v2 (hard to get both) pQCD fails: large scattering rates
NO BROWNIAN MOTION, NO FULL THERMALIZATION ->Transport Regime

38. Test in a Box at equilibrium
Calculation for Au+Au running …

39. Boltzmann-Vlasov equation for the NJL
Numerical solution with d-function test particles
Contribution of the NJL mean field
Test in a Box with equilibrium f distribution