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

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Reminder of v 2 at RHIC Evidences for non-ideal hydrodynamics Transport approach with only 2 2 scatterings /s time dependent cross section renormalized to fix /s v 2 (p T )/ v 2 (p T )/ Relation between v 2 (p T )/ and v 2 (p T )/ scaling ? effects of freeze-out /s Relation of /s with coalescence (QNS) /s hints at /s < 3/4 + coalescence at intermediate p T Outline Momentum anisotropy measure of /s Ferini et al., [nucl-th]

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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 CASCADE =10 mb Similar trend in hydro

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Hydrodynamics No microscopic details (mean free path -> 0, =0) + EoS Parton cascade v 2 saturation pattern reproduced Large =10-15mb (coalesc.includ.) 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

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

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Relation between x and v 2 Ideal Hydrodynamics Ideal Hydrodynamics: Independent of - impact parameter - system size Bhalerao et al., PLB627(2005) 2v time Effect of finite s ?! Data show evidence for deviation from hydro scaling v 2 / Hydrodynamics GeV STAR, PRC77(08)

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

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Study of dissipative effects on Study of dissipative effects on How sensitive is elliptic flow to finite /s? Z. Xu & C. Greiner, PRL 101(08) Agreement for s =0.3 – 0.6 /s=0.15 – 0.08 Dependence on freeze-out Viscous HydroCascade ( 2 2,2 3 ) P. Romatschke, PRL99 (07) Dependence on relaxation time II 0 order expansion with green terms ( D. Rischke )

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Transport approach Solved discretizing the space in x, y cells Collision integral not solved with the geometrical interpretation, but with a local stochastic sampling Z. Xhu, C. Greiner, PRC71(04) In the limit t 0 and 3 x 0 exact solutions of the Boltzmann equation Convergency of v 2 results tested against variable t and 3 x discretization and test particle number. 3x3x

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

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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: ) (*)

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Elliptic flow sensitive to the Shear Viscosity 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 50% increase

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“strong evidence for hydrodynamic scaling of v 2 over a broad selection of the elliptic flow data” Does v 2 / scaling validate ideal hydro? PHENIX PRL 98, (2007) Scaling with Centrality and System Size Such scalings holds also at finite viscosity? Scaling outside the hydro region

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v 2 / and v 2 / as a function of p T bothboth Scaling for both v 2 / and v 2 / for both Au+Au and Cu+Cu Small /s does not break v 2 / the scaling Agreement with PHENIX data for v 2 / Violation of the scaling at higher /s /s 1/4 closer to data, but… Au+Au & AGeV /s=1/2 /s=1/

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PHENIX, PRL98 (2007) Of course it is more complex… STAR, PRC77 (2008) v 2 / does not scale! Can a cascade approach account for this? Freeze-out is crucial! v 2 / scales!

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

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v 2 / and v 2 / with freeze-out V 2 / V 2 / broken in a way similar to STAR data Agreement with PHENIX and STAR scaling of v 2 / The freeze-out lowers the V 2 (p T ) at higher p T … (about 40% in b=3-9 fm) No freeze-out /s=1/4 v 2 / scaling brokenv 2 / scaling kept! Cascade can get both features:

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Short Reminder … Enhancement of v 2 Quark Number Scaling Molnar and Voloshin, PRL91 (03) Fries-Nonaka-Muller-Bass, PRC68(03) Considering only momentum space x - p correlation neglected narrow wave function v 2 for baryon is larger and saturates at higher p T v 2q fitted from v 2 GKL, PRC68(03) v 2q (p T ) fitted. Is it reasonable the v 2 needed by coalescence?

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v 2 (p T ) as a measure of /s v 2 / scaling reproduced, what about v 2 absolute value? /s >3/4 too low v 2 (p T ) at p T 1.5 GeV/c for quantitative estimate an EOS with phase transition ( ≠ 3p) needed! lower the estimate the /s PHENIX /s freeze-out Open the room to need coalescence in the region of QNS Finite /s shape for v 2 (p T ) consistent with that needed by coalescence

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Freeze-out with /s from QGP HG Smooth transition of /s from minimal value (1/4 ) to the value typical of a pion-kaon gas (7/4 ) Previous results with sudden freeze-out confirmed GeV/fm 3 ] /s 1/4 8/4 HG QGP = 0.7GeV/fm 3 Preliminary

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v 2 / ( ) scaling holds at finite /s up to 0.15 Freeze-out at =0.7 GeV/fm 3 or /s change from 1/4 7/4 (in cross-over region): Transport at finite /s pave the way for consistency: breaking of v 2 (p T )/ scaling persistence of v 2 (p T )/ scaling v 2 (p T ) need presence of coalescence at p T > 1.5 GeV with /s<3/4 Summary

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