1. Heavy-ion collisions at RHIC Jean-Yves Ollitrault, IPhT Saclay IPN Orsay, December 16, 2009

2. Experiments: The relativistic heavy-ion programme RHIC at Brookhaven : Two beams of atomic Au or Cu nuclei, accelerated at energies up to 100 GeV per nucleon (since 2000) LHC at CERN: Two beams of atomic Pb nuclei will be accelerated at energies up to 2.7 TeV per nucleon (2010?)

3. Theory: The phase diagram of strong interactions Can we learn anything about hot gauge theories by smashing heavy ions together? Do heavy-ion collisions have anything to do with temperature and thermodynamics?

4. Phases of a heavy-ion collision at RHIC Lorentz contraction: colliding pancakes Strongly-interacting matter is formed: this is the nonperturbative regime of QCD. If interactions are strong enough, it may reach local thermal equilibrium. Then, it expands like a fluid. Eventually, one measures yields, single-particle spectra and correlations between identified hadrons. Can we make sense out of all this?

5. Outline Heavy-ion chemistry and the phase diagram Microscopic modeling of heavy-ion collisions: how much nuclear structure do we need? Elliptic flow and the « perfect fluid » estimating the viscosity of hot QCD from RHIC data Higher harmonics and flow fluctuations open issues

6. Heavy-ion chemistry Hadron yields in heavy-ion collisions are well described by statistical models : grand-canonical ensemble for an ideal gas of hadrons at temperature T, baryon chemical potential μ, volume V: 3 parameters only Andronic et al nucl-th/0511071 Technical subtleties: particle identification is over a limited range in momentum: must be extrapolated using a reasonable spectrum only stable particles (with respect to strong interactions) are detected : include « feeddown » from resonance decays

7. Heavy-ion chemistry Using data at different energies, the statistical model gives a line in the (T, μ) plane. The temperature at RHIC is compatible with the deconfinement temperature predicted by lattice QCD calculations (can be done at μ=0 only) The « hadronization curve » might tell us about the phases of QCD Andronic at al 0911.4806

8. Modeling heavy-ion collisions Inputs: nuclear density profile (from electron scattering experiments) nucleon-nucleon inelastic cross section σ (pp data + smooth interpolation) Miller et al nucl-ex/0701025

9. Monte-Carlo Glauber calculations Miller et al nucl-ex/0701025 Assume independent nucleons (interactions and Pauli blocking neglected except for possible hard-core interactions) Draw randomly positions of nucleons according to the nuclear density Treat nucleons as black disks of surface σ. A collision occurs between projectile and target nucleons if their disks overlap in transverse space. Nucleons which collide at least once (filled disks) are participants

10. Why we need Glauber calculations Miller et al nucl-ex/0701025 They are used to estimate experimentally the centrality of the collision from the multiplicity N ch of particles seen in detectors, assuming N ch =a * (number of participants) +b * (number of nucleon-nucleon collisions) And also less trivial information about the size and shape of the fireball…

11. RHIC matter as an expanding fluid Fluid dynamics = only well-controlled description of an expanding, strongly-interacting system. Macroscopic description: systematic gradient expansion ∂ μ ((є+P)u μ u ν -Pg μν ) + η ∂ μ ∂ ρ …+ ∂ μ ∂ ρ ∂ σ … = 0 ideal viscous 2 nd order terms ~1/R ~1/R 2 where R = system size A large system expands as an ideal fluid Is a Au (or Cu) nucleus large enough ? Not sure… Or, equivalently, is the viscosity η small enough ?

12. Viscosity and strong coupling The stronger the interactions, the smaller the viscosity η: in transport theory, η scales like the mean free path λ So η may well be small in strongly-coupled QCD… but lattice QCD is not yet able to give a reliable estimate of η An exact result for N=4 supersymmetric gauge theories at strong coupling, obtained using the gauge/string duality: η/s=1/4π (s=entropy density) Kovtun Son Starinets hep-th/0405231 How close is the QCD η to the string theory prediction?

13. A specific observable: 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 result in more particles near φ=0 or π (elliptic flow)

14. Anisotropic flow Fourier series expansion of the azimuthal distribution: Using the φ→-φ and φ→φ+π symmetries of overlap area: dN/dφ=1+2v 2 cos(2φ)+2v 4 cos(4φ)+… v 2 = ( means average value) is elliptic flow v 4 = is a (much smaller) « higher harmonic » higher harmonics v 6, etc are 0 within experimental errors.

15. Elliptic flow at RHIC One of the first observables measured at RHIC (2 nd physics paper): elliptic flow is as large as predicted by ideal hydro

16. Eccentricity scaling v 2 scales by construction like the eccentricity ε of the initial density profile, defined as : yy xx Ideal fluid dynamics ∂ μ T μν =0 is scale invariant : x μ →λx μ This implies: v 2 /ε independent of system size R Viscous corrections are expected to scale like -η/R. ε depends on the collision centrality

17. Viscous hydro results for elliptic flow ~ system size R Lines: fits using (1+α/R) -1, α = fit parameter Deviations to v 2 /ε=constant clearly scale like η/R A direct way of estimating the viscosity η experimentally ! Masui et al arxiv:0908.0403

18. Testing the scaling on RHIC data η/s~0.15-0.3 at RHIC Caveat: v 2 is measured, ε comes from a model! Masui et al arxiv:0908.0403

19. 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.

20. Eccentricity fluctuations Depending on where the participant nucleons are located within the nucleus at the time of the collision, the actual shape of the overlap area may vary: the orientation and eccentricity of the ellipse defined by participants fluctuates. Assuming that v 2 scales like the eccentricity, eccentricity fluctuations translate into v 2 (elliptic flow) fluctuations

21. ε in Monte-Carlo Glauber calculations Miller et al nucl-ex/0701025 The eccentricity of the ellipse defined by the participants, ε part, is much larger than the standard eccentricity ε std computed from a smooth density profile (esp. for smaller systems)

22. Anisotropic flow in hydrodynamics Ideal gas (weakly-coupled particles) in global thermal equilibrium. The phase-space distribution is (Boltzmann) dN/d 3 pd 3 x = exp(-E/T) Isotropic! A fluid moving with velocity v is in (local) thermal equilibrium in its rest frame: dN/d 3 pd 3 x = exp(-(E-p.v)/T) Not isotropic: Momenta parallel to v preferred At RHIC, the fluid velocity depends on φ: typically v(φ)=v 0 +2ε cos(2φ)

23. The simplicity of v 4 Within the approximation that particle momentum p and fluid velocity v are parallel (good for large momenta) dN/dφ=exp(2ε p cos(2φ)/T) Expanding to order ε, the cos(2φ) term is v 2 =ε p/T Expanding to order ε 2, the cos(4φ) term is v 4 =½ (v 2 ) 2 Hydrodynamics has a universal prediction for v 4 /(v 2 ) 2 ! Should be independent of equation of state, initial conditions, centrality, particle momentum and rapidity, particle type

24. PHENIX results for v 4 PHENIX data for charged pions Au-Au collisions at 100+100 GeV 20-60% most central 5 The ratio is independent of p T, as predicted by hydro. But… the value is significantly larger than 0.5

25. More data on v 4 : centrality dependence Data > hydro Small discrepancy between STAR and PHENIX data Au-Au collision per nucleon 100+100 GeV

26. Why fluctuations change v 4 /(v 2 ) 2 The reference directions x and y are not known experimentally: thus v 2 = and v 4 = are not measured directly v 2 from 2-particle correlations: = v 4 from 3-particle correlations: = If v 2 and v 4 fluctuate, the measured v 4 /(v 2 ) 2 is really / 2. Inserting the prediction from hydrodynamics, [v 4 /(v 2 ) 2 ] exp =½ / 2

27. Data versus eccentricity fluctuations Fluctuations explain most of the discrepancy between data and hydro, except for central collisions which suggest / 2 =3 By symmetry, v 2 =0 for central collisions, except for fluctuations! Eccentricity fluctuations can be modelled using a Monte-Carlo program provided by the PHOBOS collaboration: Throw the dice for the positions of nucleons, with probability given by the nuclear density: No free parameter! Gombeaud JYO arxiv:0907.4664

28. Summary Fluid dynamics is the most promising approach to describe the bulk of particle production in heavy-ion collisions. In 2005, the « perfect liquid » picture was popular. Now, we have learned that viscous corrections are important : elliptic flow is 25 % below the «ideal hydro », even for central Au-Au collisions ! The phenomenology of heavy-ion collisions has made a lot of progress in the last few years: we now have quantitative relations between observables and properties of hot QCD. It is crucial to have a better control on the initial stage of the collision: although thermalization loses memory of initial conditions, so that the detailed structure of the initial state is lost, but nontrivial information remains through fluctuations. Fluctuations tell us about the (partonic or nucleonic) structure of incoming nuclei, and are not yet understood.

29. Backup slides

30. 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.

31. A simple (non-standard) test of ideal-fluid behaviour Ideal hydrodynamics ∂ μ T μν =0 is scale invariant : x μ →λx μ is still a solution. Scale invariance can be tested by varying the size of the colliding nuclei (Cu-Cu versus Au-Au) or the centrality of the collision (peripheral<central) Pt spectra are centrality independent to a good approximation, and this is also true in ideal hydro.

32. How well do we know the initial density profile? Drescher Dumitru Hayashigaki Nara, nucl-th/0605012 This is the initial eccentricity obtained assuming a smooth density profile. It is strongly model- dependent! We need a good control of initial conditions

33. PHOBOS data for v 2 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!