The centrality dependence of elliptic flow

1. The centrality dependence of elliptic flow Workshop on heavy ion collisions at the LHC: Last call for predictions, May 30, 2007 Jean-Yves Ollitrault, Clément Gombeaud (Saclay), Hans-Joachim Drescher, Adrian Dumitru (Frankfurt) nucl-th/0702075 and arXiv:0704.3553

2. Outline • A model for deviations from ideal hydro. • Centrality and system-size dependence of elliptic flow in ideal hydro: eccentricity scaling. • Eccentricity scaling+deviations from hydro: explaining the centrality and system-size dependence of elliptic flow at RHIC. • Predictions for LHC (in progress).

3. Elliptic flow, hydro, and the Knudsen number • Elliptic flow results from collisions among the produced particles • The relevant dimensionless number is K=λ/R where λ is the mean free path of a parton between two collisions, and R the system size. • K»1: few collisions, little v2, proportional to 1/K. • Ideal hydro is the limit K=0. Does not reproduce all RHIC results. • Viscous hydro is the first-order correction (linear in K) • The Boltzmann transport equation can be used for all values of K. We have solved numerically a 2-dimensional Boltzmann equation (no longitudinal expansion, transverse only) and we find v2=v2hydro/(1+1.4 K) The transport result smoothly converges to hydro as K goes to 0, as expected

4. Why a 2-dimensional transport calculation? • Technical reason: numerical, finite-size computer. • The Boltzmann equation (2 to 2 elastic collisions only) only applies to a dilute gas (particle size « distance between particles). This requires “parton subdivision”. • To check convergence of Boltzmann to hydro, we need both a dilute system and a small mean free path, i.e., a huge number of particles. • In the 2-dimensional case, we were able to reproduce hydro within 1% using 106 particles. A similar achievement in 3 dimensions would require 109 particles.

5. Does v2 care about the longitudinal expansion? Time-dependence of elliptic flow in transport and hydro: Little difference between 2D and 3D ideal hydro. Deviations from hydro should also be similar, but the mean free path λis strongly time-dependent in 3D due to longitudinal expansion. We estimate λ at the time when elliptic flow builds up.

6. Elliptic flow in ideal hydro • v2 in hydro scales like the initial eccentricity ε: requires a thorough knowledge of initial conditions! Recent breakthrough: • ε was underestimated in early hydro calculations: it is increased by fluctuations in the positions of nucleons within the nucleus, which are large for small systems and/or central collisions Miller & Snellings nucl-ex/0312008, PHOBOS nucl-ex/0610037 • The CGC predicts a larger ε than Glauber (binary collisions + participants) scaling. Hirano Heinz Kharzeev Lacey Nara, Phys. Lett. B636, 299 (2006) Adil Drescher Dumitru Hayashigaki Nara, Phys. Rev. C74, 044905 (2006)

7. Our model for the centrality and system-size dependence of elliptic flow We simply put together eccentricity scaling and deviations from hydro: v2/ε= h/(1+1.4 K) Where K-1= σ(1/S)(dN/dy) (S = overlap area between the two nuclei) ε and (1/S)(dN/dy) are computed using a model (Glauber or CGC+fluctuations) as a function of system size and centrality. Both the hydro limit h and the partonic cross section σare free parameters, fit to Phobos Au-Au data for v2.

8. Results using Glauber model(data from PHOBOS) The « hydro limit » of v2/ε is 0.3, well above the value for central Au-Au collisions. Such a high value would require a very hard EOS (unlikely)

9. Results using CGC The fit is exactly as good, but the hydro limit is significantly lower : 0.22 instead of 0.3, close to the values obtained by various groups (Heinz&Kolb, Hirano)

10. LHC: deviations from hydro • How does K evolve from RHIC to LHC ? Recall that • K-1 ~ σ (1/S)(dN/dy) • dN/dy increases by a factor ~ 2 • Two scenarios for the partonic cross sectionσ: • Ifσ is the same, deviations from ideal hydro are smaller by a factor 2 at LHC than at RHIC (12% for central Pb-Pb collisions for CGC initial conditions) • Dimensional analysis suggests σ~T-2~ (dN/dy)-2/3. Then, K decreases only by 20% between RHIC and LHC, and the centrality and system-size dependence are similar at RHIC and LHC.

11. LHC: the hydro limit • Lattice QCD predicts that the density falls by a factor ~ 10 between the QGP and the hadronic phase • If deviations from ideal hydro are large in the QGP, this means that the hadronic phase contributes little to v2. • The density decreases like 1/t : the lifetime of the QGP scales like (dN/dy) : roughly 2x larger at LHC than at RHIC. There is room for significant increase of v2. • Hydro predictions should be done with a smooth crossover, rather than with a first-order phase transition.

12. Summary • The centrality and system-size dependence of elliptic flow measured at RHIC are perfectly reproduced by a simple model based on eccentricity scaling+deviations from hydro • Elliptic flow is at least 25% below the « hydro limit », even for the most central Au-Au collisions • Glauber initial conditions probably underestimate the initial eccentricity. • v2/ε will still increase as a function of system size and/or centrality at LHC, and 12 to 20% below the «hydro limit» for the most central Pb-Pb collisions. • The hydro limit of v2/ε should be higher at LHC due to the longer lifetime of the QGP.

13. Backup slides

14. v2 versus K in a 2D transport model The lines are fits using v2=v2hydro/(1+K/K0), where K0is a fit parameter

15. v4/v22 versus pt Deviations from ideal hydro result in larger values, closer to data (about 1.2) than hydro, but still too low

16. v2 versus pt: 2D transport versus hydro

17. 3D transport versus hydro Molnar and Huovinen, Phys. Rev. Lett. 94, 012302 (2005) For small values of K, i.e., large values of σ, deviations from ideal hydro should scale like 1/σ, which is clearly not the case here.