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

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).

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 v 2, 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 v 2 =v 2 hydro /(1+1.4 K) The transport result smoothly converges to hydro as K goes to 0, as expected

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 10 6 particles. A similar achievement in 3 dimensions would require 10 9 particles.

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

Elliptic flow in ideal hydro v 2 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/ , PHOBOS nucl-ex/ 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, (2006)

Our model for the centrality and system-size dependence of elliptic flow We simply put together eccentricity scaling and deviations from hydro: v 2 /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 σ are free parameters, fit to Phobos Au-Au data for v 2.

Results using Glauber model (data from PHOBOS) The « hydro limit » of v 2 /ε is 0.3, well above the value for central Au-Au collisions. Such a high value would require a very hard EOS (unlikely)

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)

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 section σ: If σ 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.

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 v 2. 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 v 2. Hydro predictions should be done with a smooth crossover, rather than with a first-order phase transition.

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. v 2 /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 v 2 /ε should be higher at LHC due to the longer lifetime of the QGP.

Backup slides

v 2 versus K in a 2D transport model The lines are fits using v 2 =v 2 hydro /(1+K/K 0 ), where K 0 is a fit parameter

v 4 /v 2 2 versus p t Deviations from ideal hydro result in larger values, closer to data (about 1.2) than hydro, but still too low

v 2 versus p t : 2D transport versus hydro

3D transport versus hydro Molnar and Huovinen, Phys. Rev. Lett. 94, (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.