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Scaling of Elliptic Flow for a fluid at Finite Shear Viscosity

Scaling of Elliptic Flow for a fluid at Finite Shear Viscosity. V. Greco M. Colonna M. Di Toro G. Ferini. University of Catania INFN-LNS. From the Coulomb Barrier to the Quark-Gluon Plasma, Erice (Sicily) 22 Sept. 2008. Momentum anisotropy measure of /s. Reminder of v 2 at RHIC

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Scaling of Elliptic Flow for a fluid at Finite Shear Viscosity

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  1. Scaling of Elliptic Flow for a fluid at Finite Shear Viscosity V. Greco M. Colonna M. Di Toro G. Ferini University of Catania INFN-LNS From the Coulomb Barrier to the Quark-Gluon Plasma, Erice (Sicily) 22 Sept. 2008

  2. Momentum anisotropy measure of /s • Reminder of v2 at RHIC • Evidences for non-ideal hydrodynamics • Transport approach with only 22 scatterings • time dependent cross section renormalized to fix /s • Relation between v2(pT)/<v2> and v2(pT)/ scaling ? • effects of freeze-out • Relation of /swith coalescence (QNS) • v2(pT) not only <v2> hints at/s < 3/4 + coalescence at intermediate pT Outline Ferini et al., 0805. 4814 [nucl-th]

  3. A measure of the Interaction: Elliptic Flow z y x py px l=(sr)-1 Free streaming v2=0 c2s=dP/de Perform a Fourier expansion of the momentum space particle distributions CASCADE =10 mb Similar trend in hydro Good probe of early pressure • v2 is the 2nd harmonic Fourier coeff. • of the distribution of particles. The analysis can be extended !

  4. First stage of RHIC Parton cascade Hydrodynamics No microscopic details (mean free path  -> 0, h=0) Parton elastic 22 interactions • Good description of hadron spectra and v2(pT) • Mass ordering of v2 versus pT + EoS v2 saturation pattern reproduced Large s=10-15mb (coalesc.includ.) D. Molnar & M. Gyulassy, NPA 697 (02)

  5. It’s not that perfect … • Is it really zero ideal hydro (zero shear viscosity) ? Not too peripheral B. I. Abelev et al., (STAR), PRC77 (08) Not too high pT Not too high harmonics STAR, J. Phys. G34 (2007)

  6. Relation betweenexand v2 STAR, PRC77(08) Bhalerao et al., PLB627(2005) Hydrodynamics 2v2/e Au+Au@200 GeV time Ideal Hydrodynamics: Independent of - impact parameter - system size Data show evidence for deviation from hydro scaling v2/ Effect of finite h/s ?!

  7. Shear Viscosity Small viscosity Large cross sections Strong couplings  beyond pQCD R. Lacey et al., PRL99(2006) 1)Quantum mechanism h/s > 1/15 : Kinetic Theory Can we constrain /s with v2? 2) 4 SYM + Gauge theory g ->∞: Smaller than any other known fluid!

  8. Study of dissipative effects on <v2> How sensitive is elliptic flow to finite /s? Viscous Hydro Cascade (2<->2,2<->3) P. Romatschke, PRL99 (07) Z. Xu & C. Greiner, PRL 101(08) Agreement for s=0.3 – 0.6 Dependence on tp relaxation time II0 order expansion with green terms (D. Rischke) /s=0.15 – 0.08 Dependence on freeze-out

  9. Collision integral not solved with the geometrical interpretation, but with a localstochastic sampling Z. Xhu, C. Greiner, PRC71(04) Transport approach Solved discretizing the space in (h, x, y)a cells In the limit t0 and 3x0 exact solutions of the Boltzmann equation D3x Convergency of v2 results tested against variable Dt and D3x discretization and test particle number.

  10. Cross section for fixed /s We simulate a constant shear viscosity during the HIC Relativistic Kinetic theory Cascade code =cell index in the r-space We have used pQCD-like cross section with screening mass The viscosity is kept constant varying as

  11. Evolution of cross section with Temperature A rough estimate of (T) can be done using (*) Neglecting and inserting in (*) At T=200 MeVtr10 mb In our code it is evaluated locally(different from D. Molnar arXiV:0806.0026)

  12. Elliptic flow sensitive to the Shear Viscosity Sensitivity increasing at larger pT Intermediate pT can say more about /s Au+Au @ 200 AGeV b=9 fm 50% increase b=7 fm b=5 fm b=3 fm

  13. Does v2/<v2> scaling validate ideal hydro? PHENIX PRL 98, 162301 (2007) “strong evidence for hydrodynamic scaling of v2 over a broad selection of the elliptic flow data” Scaling with Centrality and System Size Such scalings holds also at finite viscosity? Scaling outside the hydro region 

  14. v2/ and v2/<v2> as a function of pT /s=1/2 /s=1/ Au+Au & Cu+Cu@200 AGeV • Scaling for both v2/<v2> and v2/ for both Au+Au and Cu+Cu • Small h/s does not break v2/the scaling • Agreement with PHENIX data for v2/<v2> /s1/4 closer to data, but… Violation of the scaling at higher /s

  15. Of course it is more complex… PHENIX, PRL98 (2007) STAR, PRC77 (2008)  v2/ does not scale! v2/<v2> scales! Can a cascade approach account for this? Freeze-out is crucial!

  16. Elliptic flow sensitive to freeze out For <c=0.7 GeV/fm3 collisions are switched off /s=1/4 Effect of freeze-out b=3 fm Effect of freeze-out increasing with b

  17. v2/ and v2/<v2> with freeze-out /s=1/4 No freeze-out No freeze-out v2/ scaling broken v2/<v2> scaling kept! (about 40% in b=3-9 fm) Cascade can get both features: • V2/broken in a way similar to STAR data • Agreement with PHENIX and STAR scaling of v2/<v2> • The freeze-out lowers the V2(pT) at higher pT …

  18. Quark Number Scaling Enhancement of v2 v2q fitted from v2p GKL, PRC68(03) Short Reminder … Considering only momentum space x - p correlation neglected narrow wave function Molnar and Voloshin, PRL91 (03) Fries-Nonaka-Muller-Bass, PRC68(03) • v2 for baryon is larger and saturates at higher pT v2q(pT) fitted. Is it reasonable the v2 needed by coalescence?

  19. v2(pT) as a measure of /s v2/<v2> scaling reproduced, what about v2 absolute value? h/s 0.1-0.2 + freeze-out Open the room to need coalescence in the region of QNS  PHENIX Finite h/s -> shape for v2(pT) consistent with that needed by coalescence • /s >3/4  too low v2(pT) at pT1.5 GeV/c • for quantitative estimate an EOS with phase transition (e≠ 3p)needed! -> lower the estimate the h/s

  20. Freeze-out with h/s from QGP -> HG h/s HG 7/4p QGP 1/4p e 0.3 1.7 [GeV/fm3] Smooth transition of /s from minimal value (1/4p) to the value typical of a pion-kaon gas (7/4p) Preliminary e0=0.7GeV/fm3 Previous results with sudden freeze-out confirmed

  21. Summary • v2/<v2> () scaling holds at finite /s up to 0.15 • Freeze-out at e0=0.7 GeV/fm3 or • h/s change from 1/4p-> 7/4p (in cross-over region): • Transport at finite h/s pave the way for consistency: • breaking of v2(pT)/ scaling • persistence of v2(pT)/<v2> scaling • v2(pT) need presence of coalescence at pT> 1.5 GeV with 1/4p<h/s<3/4p

  22. Scaling of time evolution with the system size Hydrodynamics Cascade Au+Au @ 200 AGeV As in hydro in the early evolution v2/ scales with system size At the end a significant breaking is observed

  23. If Elliptic Flow is very large To balance the minimum a v4 > (10v2-1)/34 is required v4 > 4.4% if v2=25% STAR, J. Phys. G34 (2007) v2 and v4 contain rich information on /s

  24. Study of dissipative effects at RHIC Transport approach Z. Xu et al., 0711.0961 [nucl-th] /s=0.15 – 0.08 Agreement with data for s=0.3 – 0.6 With only 22 collisions, RHIC v2 is obtained by using a growing cross section 2/3 which yields /s1/4 on average (D. Molnar, 0806.0026 [nucl-th])

  25. h/s HG 7/4p QGP 1/4p e 0.3 1.7 [GeV/fm3]

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