1. Elliptic flow and shear viscosity in a parton cascade approach G. Ferini INFN-LNS, Catania P. Castorina, M. Colonna, M. Di Toro, V. Greco

2. Outline Momentum anisotropy as a measure of plasma properties  Reminder of v 2 & v 4 First results at RHIC  Transport approach to study finite  /s effects  v 2 scaling with eccentricity and system size dependence on  /s effects of freeze-out  v 2 (p T ) & v 4 (p T )  1/4  <  / s < 1/2   Conclusions

3. A measure of the Interaction: Elliptic Flow x y z pxpx pypy v 2 is the 2nd harmonic Fourier coeff. of the distribution of particles. Perform a Fourier expansion of the momentum space particle distributions Free streaming v 2 =0 The analysis can be extended ! Good probe of early pressure c 2 s =dP/d   0 CASCADE  =10 mb Similar trend in hydro

4. If Elliptic Flow is very large To balance the minimum a v 4 > (10 v 2 -1)/34 is required v 4 > 4.4% if v 2 =25% STAR, J. Phys. G34 (2007) v 2 and v 4 contain rich information on  /s

5. Hydrodynamics No microscopic details (mean free path -> 0) + EoS Parton cascade v 2 saturation pattern reproduced Good description of hadron spectra and v 2 (p T ) Mass ordering of v 2 versus p T D. Molnar & M. Gyulassy, NPA 697 (02) First stage of RHIC Parton elastic 2  2 interactions (finite mean free path)

6. It’s not that perfect …  Is it really zero shear viscosity ? But finite mean free path calls for a transport approach! B. I. Abelev et al., (STAR) Nucl-ex 0801.3466 STAR, J. Phys. G34 (2007) Not too peripheral Not too high p T Not too high harmonics

7.  & v 2 Ideal Hydrodynamics Ideal Hydrodynamics: Indipendent of - impact parameter - system size Bhalerao et al., PLB627(2005) 2v   time B. I. Abelev et al., (STAR) Nucl-ex 0801.3466 This calls for a transport approach! Data show evidence for deviation from hydro scaling v2/  Hydrodynamics

8. Transport approach Collision integral not solved with the geometrical interpretation, but with a local stochastic sampling Solved discretizing the space in  x, y   cells Several checks in ultra- relativistic conditions m=0 Z. Xhu, C. Greiner, PRC71(04) The approach provides a good framework for multiparticle collisions Collisions in a box

9. Small viscosity  Large cross sections  Strong couplings  beyond pQCD Shear Viscosity 1)Hydrodynamics means  =0 2)Quantum mechanism  s > 1/15 : R. Lacey et al., PRL99(2006) 3) 4 SYM + Gauge theory g  ∞: Smaller than any other known fluid! Can we constrain  /s with v n ?

10. v 2 /  and the Shear Viscosity v 2 /  and the Shear Viscosity We can simulate a constant shear viscosity during the HIC Hydrodynamics Relativistic Kinetic theory Cascade code In agreement with data v 2 /  is not constant -> finite viscosity We have used pQCD-like cross section with screening mass The viscosity is kept constant varying  s Au+Au @200 AGeV  = cell index in the r-space

11. Elliptic flow sensitive to the Shear Viscosity Au+Au @ 200 AGeV b=9 fm b=7 fm b=5 fm b=3 fm Sensitivity increasing at larger p T Intermediate p T can say more about  /s

12. Finite minimal viscosity is consistent also with the v 4 v 4 more sensitive to the viscosity Going to higher momentum anisotropies … v4 DATA b=9 fm

13. v 2 and v 4 sensitivity to  /s

14. v 2 /  and v 2 / as a function of p T  Scaling for both v2/ and v2/   Agreement with PHENIX data for v 2 / v 2 /  (p T ) sensitive to  /s Larger violation of the scaling at higher viscosity  /s  1/4  closer to data Similar results for Cu+Cu  /s=1/2   /s=1/  Au+Au @ 200 AGeV Reduced sensitivity to  /s

15. PHENIX PRL (2007) Of course it is more complex… STAR (arXiv 2008)  v 2 /  does not scale! Can a cascade approach account for this? Freeze-out is crucial! v 2 / scales!

16. Elliptic flow sensitive to freeze out Sensitivity increasing with b Au+Au @ 200 AGeV Coalescence plays a role For  <  c =0.7 GeV/fm 3 collisions are switched off  /s=1/4  b=5 fm Indeed at that p T QNS is observed! V 2 (p T ) of partons not directly comparable with data PHENIX b=3 fm

17. v 2 /  and v 2 / with freeze-out V2/   V2/  broken in a way similar to STAR data  Agreement with STAR measurements of v 2 / (about 40% in b=3-9 fm) No freeze-out  /s=1/4  v 2 /  scaling broken v 2 / scaling kept! Cascade can get both features:

18. v 2 (p T ) as a measure of  /s v 2 / scaling reproduced, what about v 2 absolute value?   /s<1/2   too low v 2 (p T ) at p T  1.5 GeV/c  comparison with baryon and meson v 2 (p T ) can constrain  /s better (need for coalescence)  v 4 …. (work in progress)  /s=1/4   /s=1/   /s=1/2  PHENIX

19. Summary  v 2 / (  ) scaling holds at finite  /s  v 2 /  breaking of the scaling  v 2 / scaling  v 2 (p T ) hints at 1/4  <  /s<1/2  (+ coalescence!?)  v 4 (p T ) appears more sensitive to  /s (work in progress) Conclusions Future Developments  Dynamical implementation of Greco-Ko-Levai coalescence model  what does go on for LHC conditions? reproduced by a cascade approach (+ freeze-out)

21. First estimate of Shear Viscosity Larger violation of the scaling at lower viscosity  study of Cu+Cu The v 2 /  scaling point to a viscosity of about 1/4  i.e. around the bound limit Au+Au @ 200 A GeV DATA

23. Romatschke Greiner Averaged elliptic flow

24. V2(pT) in viscous hydrodynamics H. Song & U. Heinz P. Romatschke How sensitive is elliptic flow to finite  /s?

25. Au+Au @ 200 AGeV Scaling of time evolution with the system size HydrodynamicsCascade As in hydro in the early evolution v 2 /  scales with system size At the end a significant breaking is observed