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Thermalization and elliptic flow at RHIC

We discuss signatures of thermalisation in heavy ion collisions based on elliptic flow. We then propose a new method to analyse elliptic flow, based on multiparticle azimuthal correlations. This method allows one to test quantitatively the collective behaviour of the interacting system.


Introduction
A highly debated issue in ultrarelativistic heavy ion collisions is whether or not the colliding system reaches local thermal equilibrium at some stage of its evolution. Equilibrium with respect to inelastic collisions strongly constrains the ratios of particle abundances 1 ("chemical" equilibrium), as well as phase-space densities, which are obtained by combining informations from momentum spectra and two-particle HBT correlations; 2 on the other hand, equilibrium with respect to elastic collisions constrains momentum distributions, and implies in particular that they are isotropic in the local rest frame. This is the "kinetic" equilibrium, on which we concentrate here.
Kinetic equilibrium itself has (at least) two facets. One is the equilibration between longitudinal and transverse degrees of freedom, i.e., the implication that in the local rest frame, longitudinal and transverse momenta are of the same order of magnitude. This aspect of thermalisation can be discussed from first principles at the partonic level, 3 and there is now a vast literature on this subject. 4 But experimental signatures deal in fact rather with equilibration among the two transverse degrees of freedom. Due to the high Lorentz contraction at ultrarelativistic energies, the typical transverse scale is much larger than the longitudinal scale, so that this "transverse equilibrium" is probably easier to achieve than "longitudinal-transverse equilibrium".
In Sec. 2, we discuss some experimental signatures of transverse equilibrium, especially elliptic flow, and the relevance of hydrodynamical models in this context. In Sec. 3, we present a new method recently developed to obtain more reliable measurements of elliptic flow, which are required in order to draw definite conclusions on the issue of thermalisation.

When are hydrodynamical models useful?
If the colliding system thermalises, its evolution follows the laws of hydrodynamics; it is therefore natural to use hydrodynamical models in order to define signatures of thermalisation. There are two ingredients in these models: i) an equation of state; ii) initial conditions, i.e., energy density, baryon density and fluid velocity on a space-like hypersurface (typically, at some initial time).
Which parameters are under control? The initial time t 0 , at which the system thermalises, is hopefully short but to a large extent unknown, so that hydrodynamics has little predictive power concerning observables which depend strongly on t 0 , such as thermal photon production. 5 Similarly, there is no serious motivation for studying the very first stages of the collision within the framework of hydrodynamics, either within a one-fluid model 6,7 or a multifluid model, 8 and the relevance of signatures based on such parametrisations is questionable. Finally, there is no well-defined prescription concerning the initial longitudinal fluid velocity and density, for which a variety of parametrisations exist, either "Landau" 9 or "Bjorken" 10 type.
The situation is much clearer concerning transverse degrees of freedom: the transverse collective velocity must be initially zero since each nucleonnucleon collision populates the transverse momentum space randomly; the initial density profile in the transverse plane is strongly constrained by observed multiplicity distributions. 11 Therefore, reliable signatures of thermalisation should rather be sought for in observables associated with transverse degrees of freedom: p T spectra, 12 transverse radii 13 and azimuthal anisotropies, in particular elliptic flow, on which we concentrate here.

Why elliptic flow?
Elliptic flow is defined as a correlation between the azimuth φ of an outgoing particle and the azimuth Φ R of impact parameter: 14 where brackets denote a statistical average. At ultrarelativistic energies, v 2 is positive for noncentral collisions. 15,16 In a hydrodynamical picture, it results from anisotropic pressure gradients in the transverse plane, due to the almondshaped region of the overlap region between the two nuclei. 17 Microscopically, v 2 is created by rescattering among the produced particles, which makes it a sensitive probe of final state interactions: if there are none, it vanishes, while other observables such as p T spectra may still look "thermal". 18 Furthermore, predictions of hydrodynamical models for v 2 are very stable: since v 2 is created by transverse pressure gradients, it strongly depends on the initial density transverse profile, which is well controlled as discussed above; it also depends significantly on the equation of state, on which it may thus provide valuable information; on the other hand, v 2 depends only weakly on arbitrary parameters, such as initial time and longitudinal velocity. 17 Quite remarkably, simple hydrodynamical parametrisations are able to reproduce simultaneously the measured p T spectra, HBT radii and elliptic flow at RHIC. 19

Predictions for v 2
The momentum anisotropy v 2 calculated in hydro models is roughly equal to the anisotropy of the almond-shaped overlap region. 17 This purely geometrical effect dominates the centrality dependence of v 2 , which decreases linearly with the number of participants. Deviations from this behaviour can be used to signal a phase transition 20 or a departure from thermal equilibrium. 21 The latter is expected to occur for the most peripheral collisions, where v 2 should be smaller than the hydro prediction. This is indeed observed in several specific transport models like UrQMD 22 (which however predicts a much too small value of v 2 ), QGSM, 23 and AMPT. 24 With a more systematic study, 25 one could relate the observed centrality dependence of v 2 to the degree of thermalisation of the system. Experimental results vary significantly depending on the method used to analyse elliptic flow. 26,27 We come back to this issue in Sec. 3. Hydrodynamical calculations 28 were also able to predict the p T dependence of v 2 for identified hadrons, in remarkable agreement with experimental results: 29 v 2 is almost linear in p T for pions and significantly smaller for protons. However, these non-trivial features are also reproduced by transport models. 22,23,24 In addition, the latter predict a saturation 25 at high p T which is not seen in hydro, suggesting that many elastic collisions are necessary to build the flow at high p T . This saturation, which is seen in the data, could also be related to hard physics. 30 3 Analysing elliptic flow with multiparticle correlations

Flow from azimuthal correlations
Since the reaction plane Φ R is unknown, v 2 cannot be derived directly from (1). It must be inferred from azimuthal correlations. The standard flow analysis 31 relies on the key assumption that all azimuthal correlations are due to flow, i.e., that angles relative to the reaction plane φ − Φ R are statistically independent. This allows one to write the two-particle correlation as 32 where brackets denote an average over pairs of particles belonging to the same event. One could also use higher order correlations, such as the four-particle correlation, which would give However, such equations are not quite correct, since they neglect various other sources of azimuthal correlations ("nonflow correlations"), which is no longer justified at ultrarelativistic energies. 33

Simple illustration of nonflow correlations
In order to illustrate nonflow correlations, we consider the following example: assume that in each event, M/2 pairs of particles are emitted, where both particles in a pair have collinear momenta, but pairs are emitted with random orientations. Since pairs are emitted randomly, there is no flow (v 2 = 0), but there are azimuthal correlations. In each event, there is a total of M (M − 1)/2 particle pairs, among which M/2 are correlated, hence the two-particle correlation A similar reasoning yields the four-particle correlation: .
Applying Eqs. (2) and (4), or (3) and (5), one would obtain v 2 ∼ 1/ √ M , although there is no flow: this is the typical order at which nonflow correlations spoil the standard flow analysis.

Subtraction of nonflow correlations
The contribution of nonflow correlations can be greatly reduced by combining the informations from two-and four-particle correlations. In Eq. (5), a 4-uplet of particles gives a nonvanishing contribution if it consists of two correlated pairs, either (1,3) and (2,4) or (1,4) and (2,3). Subtracting the corresponding contributions, one obtains .
The l.-h. s. of this equation defines the cumulant of four-particle correlations, c 2 {4}. The r.-h. s. is the contribution of nonflow correlations, of order 1/M 3 , i.e., much smaller than the corresponding contribution to the four-particle correlation (5), of order 1/M 2 . On the other hand, the contribution of flow remains of the same magnitude: from Eqs. (2), (3) and (6), one obtains The subtraction therefore reduces the relative contribution of nonflow effects, and yields a more accurate estimate of the flow. This method was recently applied to STAR data. 27 The resulting value of v 2 are smaller than those obtained with the standard analysis, 26 in particular for the most peripheral collisions: this is precisely where nonflow effects are expected to give the largest contribution since the multiplicity M is smaller. The centrality dependence obtained with this method suggests that departures from thermalisation at RHIC may be larger than was previously thought.
This cumulant expansion can be worked out to arbitrary orders, and allows one to extract the genuine 4-, 6-particle correlations and beyond. 34 The practical implementation of the method is described in detail elsewhere. 35 Flow, which is essentially a collective phenomenon, contributes to all orders, while the relative contribution of nonflow correlations decreases as the order increases. Higher order cumulants therefore provide a unique possibility to check quantitatively that azimuthal correlations are indeed of collective origin.