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

 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]

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  

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

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

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)

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

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 )

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. 3x3x

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

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: ) (*)

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

“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

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/ 

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!

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

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:

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?

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

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

 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