1. How perfect is the RHIC liquid? Jean-Yves Ollitrault, IPhT Saclay BNL colloquium, Feb. 24, 2009 based on Drescher Dumitru Gombeaud & JYO, arXiv:0704.3553 Gombeaud, Lappi, JYO, arXiv:0901.4908

2. Two Lorentz-contracted nucleicollide A nucleus-nucleus collision at RHIC « hard » processes, accessible to perturbative calculations High-density, strongly-interacting hadronic matter (quark- gluon plasma?) is created and expands, and eventually reaches the detectors as hadrons.

3. Hot QCD… If interactions are strong enough, the matter produced in a nucleus-nucleus collision at RHIC reaches local thermal equilibrium The thermodynamics of strongly coupled, hot QCD can be computed on the lattice: equation of state correlation functions However, some quantities are very hard to compute on the lattice, such as the viscosity Karsch, hep-lat/0106019

4. …and string theory… Using the AdS/CFT correspondence, one can compute exactly the viscosity to entropy ratio in strongly coupled supersymmetric N=4 gauge theories, and it has been postulated that the result is a universal lower bound. η/s=ħ/4πk B Kovtun Son Starinets hep-th/0405231

5. …and RHIC ? In 2005, a press release claimed that nucleus-nucleus collisions at RHIC had created a perfect liquid, with essentially no viscosity. Since then, many works on viscous hydrodynamics Martinez, Strickland arXiv:0902.3834 Luzum Romatschke arXiv:0901.4588 Bouras Molnar Niemi Xu El Fochler Greiner Rischke arXiv:0902.1927 Song Heinz arXiv:0812.4274 Denicol Kodama Koide Mota arXiv:0807.3120 Molnar Huovinen arXiv:0806.1367 Chaudhuri arXiv:0801.3180 Dusling Teaney arXiv:0710.5932 For a given substance, the minimum of η/s occurs at the liquid-gas critical point : are we seing the QCD critical point? Csernai et al nucl-th/0604032

6. Outline Ideal hydrodynamics, what it predicts Viscous corrections & dimensional analysis The system size dependence of v 2 : a natural probe of viscous effects Comparison with data: how large are viscous corrections? Other observables Conclusions

7. Elliptic flow Non-central collision seen in the transverse plane: the overlap area, where particles are produced, is not a circle. A particle moving at φ=π/2 from the x-axis is more likely to be deflected than a particle moving at φ=0, which escapes more easily. φ Initially, particle momenta are distributed isotropically in φ. Collisions results in positive v 2.

8. Eccentricity scaling v 2 scales like the eccentricity ε of the initial density profile, defined as : yy xx Ideal (nonviscous) hydrodynamics (∂ μ T μν = 0) is scale invariant: v 2 = α ε, where α constant for all colliding systems (Au-Au and Cu-Cu) at all impact parameters at a given energy This eccentricity depends on the collision centrality, which is well known experimentally.

9. Caveat: hydro predictions for v 2 are model dependent The initial eccentricity ε is model dependent The color-glass condensate prediction is 30% larger than the Glauber-model prediction Drescher Dumitru Hayashigaki Nara, nucl-th/0605012 Eccentricity fluctuations are important PHOBOS collaboration, nucl-ex/0510031 The ratio v 2 /ε depends on the equation of state A harder equation of state gives more elliptic flow for a given eccentricity.

10. No scale invariance in data v 2 /ε is not constant at a given energy: non-viscous hydro fails ! But viscosity breaks scale invariance

11. How viscosity breaks scale invariance: dimensional analysis in fluid dynamics η = viscosity, usually scaled by the mass/energy density: η/ε ~ λ v thermal, where λ mean free path of a particle R = typical (transverse) size of the system v fluid = fluid velocity ~ v thermal because expansion into the vacuum The Reynolds number characterizes viscous effects : Re ≡ R v fluid /(η/ε) ~ R/λ Viscous corrections scale like the viscosity: ~ Re -1 ~ λ/R. If viscous effects are not « 1, the hydrodynamic picture breaks down! In this talk, I use instead the Knudsen number K= λ/R 1/K ~ number of collisions per particle

12. A simplified approach to viscous effects Our motivation : study arbitrary values of the Knudsen number K≡λ/R : beyond the validity of hydrodynamics. The theoretical framework = Boltzmann transport theory, which means : particles undergoing 2 → 2 elastic collisions, easily solved numerically by Monte-Carlo simulations. One recovers ideal hydro for K→0 (we check this explicitly). One should recover viscous hydro to first order in K (not checked). The price to pay : dilute system (λ» interparticle distance), which implies, ideal gas equation of state : no phase transition ; connection with real world (data) not straightforward Additional simplifications : 2 dimensional system (transverse only), massless particles. Extensions to 3 d and massive particles under study.

13. Elliptic flow versus time Convergence to ideal hydro clearly seen!

14. Elliptic flow versus K v 2 =α ε/(1+1.4 K) Elliptic flow increases with number of collisions (~1/K) Smooth convergence to ideal hydro as K→0

15. How is K related to RHIC data? The mean free path of a particle in medium is λ=1/σn 1/K= σnR ~ (σ/S)(dN/dy), where σ is a (partonic) cross section S is the overlap area between nuclei (dN/dy) is the particle multiplicity per unit rapidity. K can be tuned by varying the system size and centrality S

16. The centrality dependence of v 2 explained 1.Phobos data for v 2 2.ε obtained using Glauber or CGC initial conditions +fluctuations 3.Fit with v 2 =α ε/(1+1.4 K) assuming 1/K=(σ/S)(dN/dy) with the fit parameters σ and α. K~0.3 for central Au-Au collisions v 2 : 30% below ideal hydro!

17. From cross-section σ to viscosity η Viscosity describes momentum transport, which is achieved by collisions among the produced particles. For a gas of massless particles with isotropic cross sections, transport theory gives η=1.264 k B T/σc (remember, more collisions means lower viscosity) The entropy is proportional to the number of particles S=4Nk B This yields our estimates η/s~0.16-0.19 (ħ/k B ) depending on which initial conditions we use.

18. Other observables : HBT radii ptpt RsRs RoRo For particles with a given momentum p t along x R o measures the dispersion of x last -v t last R s measures the dispersion of y last Where (t last,x last,y last ) are the space-time coordinates of the particle at the last scattering HBT puzzle : R o /R s ~1.5 in hydro, 1 in RHIC data

19. R o and R s versus K Au-Au, b=0 As the number of collisions increases, R o increases and R s decreases, but this is a very slow process: The hydro limit requires a huge number of collisions ! Radii Ro Rs HBT puzzle R/λ=1/K

20. The HBT puzzle revisited R o /R s ptpt A simulation with R/λ=3 (inferred from elliptic flow) gives a value compatible with data, and significantly lower than hydro.

21. Conclusions The centrality and system size dependence of elliptic flow is a specific probe of viscous effects in heavy ion collisions at RHIC. Viscosity is important : elliptic flow is 25 % below the «hydro limit», even for central Au-Au collisions ! Quantitative understanding of RHIC results in the soft momentum sector requires viscosity, probably larger than the lower bound from string theory. Viscous effects are larger on HBT radii than on elliptic flow: the experimental value R o /R s ~1 is consistent with estimates of viscous effects inferred from v 2.

22. Backup slides

23. Estimating the initial eccentricity Nucleus 1 Nucleus 2 Participant Region x y b Until 2005, this was thought to be the easy part. But puzzling results came: 1. v 2 was larger than predicted by hydro in central Au-Au collisions. 2. v 2 was much larger than expected in Cu-Cu collisions. This was interpreted by the PHOBOS collaboration as an effect of fluctuations in initial conditions [Miller & Snellings nucl-ex/0312008] In 2005, it was also shown that the eccentricity depends significantly on the model chosen for initial particle production. We compare two such models, Glauber and Color Glass Condensate.

24. Dimensionless numbers in fluid dynamics They involve intrinsic properties of the fluid (mean free path λ, thermal/sound velocity c s, shear viscosity η, mass density ρ) as well as quantities specific to the flow pattern under study (characteristic size R, flow velocity v) Knudsen number K= λ/R K «1 : local equilibrium (fluid dynamics applies) Mach number Ma= v/c s Ma«1 : incompressible flow Reynolds number R= Rv/(η/ρ) R»1 : non-viscous flow (ideal fluid) They are related ! Transport theory: η/ρ~λc s implies R * K ~ Ma Remember: compressible+viscous = departures from local eq.

25. Dimensionless numbers in the transport calculation Parameters: –Transverse size R –Cross section σ (~length in 2d!) –Number of particles N Other physical quantities –Particle density n=N/R 2 –Mean free path λ=1/σn –Distance between particles d=n -1/2 Relevant dimensionless numbers: –Dilution parameter D=d/λ=(σ/R)N -1/2 –Knudsen number K=λ/R=(R/σ)N -1 The hydrodynamic regime requires both D «1 and K«1. Since N=D -2 K -2, a huge number of particles must be simulated. (even worse in 3d) The Boltzmann equation requires D «1 This is achieved by increasing N (parton subdivision)

26. Test of the Monte-Carlo algorithm: thermalization in a box Initial conditions: monoenergetic particles. Kolmogorov test: Number of particles with energy <E in the system Versus Number of particles with energy <E in thermal equilibrium

27. Elliptic flow versus p t Convergence to ideal hydro clearly seen!

28. Particle densities per unit volume at RHIC (MC Glauber calculation) The density is estimated at the time t=R/c s (i.e., when v 2 appears), assuming 1/t dependence. The effective density that we see through elliptic flow depends little on colliding system & centrality ! H-J Drescher (unpublished)

29. v 4 data (Bai Yuting, STAR)

30. Higher harmonics : v 4 Recall : RHIC data above ideal hydro Boltzmann also above ideal hydro but still below data preferred value from v 2 fits

31. Work in progress Extension to massive particles –Small fraction of massive particles embedded in a massless gas –Study how the mass-ordering of v 2 appears as the mean free path is decreased Extension to 3 dimensions with boost-invariant longitudinal cooling –Repeat the calculation of Molnar and Huovinen –Different method: boost-invariance allows dimensional reduction: Monte-Carlo is 3d in momentum space but 2d in coordinate space, which is much faster numerically.

32. Boltzmann versus hydro RoRo ptpt hydro Boltzmann again converges to ideal hydro for small Kn However, R o is not very sensitive to thermalization pt dependence is already present in (CGC-inspired) initial conditions R/λ~0.1 R/λ~0.5 R/λ~ 3 R/λ~ 10

33. Radii too small RoRo ptpt due to the hard equation of state and 2 dimensional geometry