Abstract
The recent results on anisotropic flow in ultrarelativistic nuclear collisions along with recent methodical developments and achievements in the understanding of the phenomena, are reviewed. The emphasis is given to the elliptic flow results.
ANISOTROPIC FLOW FROM AGS TO RHIC
[5mm] S. A. VOLOSHIN
[5mm] Department of Physics and Astronomy, Wayne State University
666 W. Hancock, Detroit, MI 48201, U.S.A.
[8mm]
1 Introduction
In recent years, the subject of anisotropic flow in ultrarelativistic nuclear collisions attracts an increasing attention of heavy ion community. One of the main reasons for that is the sensitivity of anisotropic flow, and in particular elliptic flow [1], to the evolution of the system at the very early times [2].
Anisotropic flow is defined as azimuthal asymmetry in particle distribution with respect to the reaction plane (the plane spanned by the beam direction and the impact parameter). It is called flow for it is a collective phenomena, but it does not necessarily imply hydrodynamic flow. It is convenient to characterize the magnitude of this asymmetry using Fourier decomposition of the azimuthal distributions. Then the first harmonic describes socalled directed flow, and the second harmonic corresponds to elliptic flow; nonzero higher harmonics can be also present in the distribution. The corresponding Fourier coefficients, are used to quantify the effect [3].
The two reasons for anisotropic flow are the original asymmetry in the configuration space (noncentral collisions !) and rescatterings. In a case of elliptic flow the initial “ellipticity” of the overlap zone is usually characterized by the quantity , assuming the reaction plane being plane. With the system expansion the spatial anisotropy decreases. This is the reason for high sensitivity of elliptic flow to the evolution of the system in the very early times [2, 4], of the order of 2–5 fm/c, independent of the model.
Due to the lack of space we will not discuss in detail all recent developments regarding directed flow. Briefly mention a couple. In [5] a very interesting qualitative prediction is given: it is shown that the radial flow (isotropic expansion in the transverse plane) and an incomplete baryon stopping should lead to a “wiggle” in the rapidity dependence of baryon directed flow; should change it’s sign three times with rapidity! This effect was also observed in a hydro model [6]. Once found it would be a strong evidence for the spacemomentum correlations caused by radial flow. In a recent paper [7] the important question of the role of the momentum conservation in directed flow measurements are discussed. Concrete recommendations for the analysis have been worked out.
Recently, many new results regarding elliptic flow have been obtained in all directions: experimental measurements, improving analysis methods, and theoretical understanding of the underlying physics of the phenomena. I will try to mention the most important results in all three directions, but concentrate mostly on the last two questions. A more complete picture of the recent experimental results can be found in [8].
2 Improving the methods
A significant progress in theoretical description of anisotropic flow demands the accuracy in measurements. Thus the corresponding methods are evolving in the directions of improving both, the statistical uncertainties, and understanding systematics in the measurements. For the first, improving on the statistical errors, we mention extensive use of proper weights (the best would be ), which leads to the improvement of the reaction plane resolution by 10%20% and reduction of the statistical errors, and using the scalarproduct approach [9]. In the scalarproduct method the flow is given by:
(1) 
where is a unit vector associated with a particle of a given rapidity and transverse momentum, and are flow vectors for the “full” event and subevents “a” and “b”:
(2) 
where sum is over all particles in an (sub)event. This method (which is also easy to implement and analyze) in addition to the flow angle takes into account the flow effects on the magnitude of the flow vector and as a result gives smaller statistical errors.
In is not possible to determine the reaction plane in the collision directly. Therefore, any measurement of the anisotropy in particle production with respect to the reaction plane is based on the measurements of particle azimuthal correlations among themselves. Those correlations to a different degree (depending on what exactly is analyzed) include the contribution from the correlations that are not related to the orientation of the reaction plane (e.g. resonance decays), and often called nonflow contribution. For a reliable interpretation of the results the nonflow contribution should be estimated or, better, measured.
Fig. 1: STAR [10]. Centrality dependence of the correlation between subevent flow angles.
Fig. 2: Simulations [27]. The results from 2 (triangles), 4 (stars) and 6particle (crosses) cumulants.
As an example, we discuss the estimate of nonflow contribution in STAR elliptic flow measurements [10]. An important observation for that is on the centrality dependence of the nonflow effects. The azimuthal correlation between two particles can be written as
(3) 
where is the harmonic, and the average is taken over all pairs of particles. The represents the contribution to the pair correlation from nonflow effects. Then, the correlation between two subevent flow angles is
(4) 
where is the multiplicity for a subevent. Here, we have taken into account that the strength of the nonflow correlations scale in inverse proportion to the multiplicity: . What is important is that the nonflow contribution to is approximately independent of centrality. The typical shape of , see, for example, Fig. 1, is peaked at midcentral events due to the fact that for peripheral collisions, is small, and for central events, is small. In the estimates [10] of the systematic errors, the authors set the quantity . The justification for this value was the observation of similar correlations for the first and higher harmonics (it has been investigated up to the sixth harmonic). One could expect the nonflow contribution to be of similar order of magnitude for all these harmonics, and model simulations support this conclusion. Given the value , one simply estimates the contribution from nonflow effects to the measurement of from the plot of using Eq.(4) (see circlepoint errorbars in Fig.6). The relative contribution of nonflow effects is largest for very central and very peripheral bins (where, the reaction plane resolution is smallest!).
Anisotropic flow is a genuine multiparticle phenomena (which justifies use of the term collective flow). It means that if one considers manyparticle correlations instead of twoparticle correlations, the relative contribution of nonflow effects (due to few particle clusters) would decrease. Considering manyparticle correlations, one has to subtract the contribution from correlations in the lowerorder multiplets and use cumulants instead of simple correlation functions [11]. For example, correlating four particles, one gets
(5) 
In this expression, two factors of “2” in front of the middle term correspond to the two ways of pairing (1,3)(2,4) and (1,4)(2,3) and account for the possibility to have nonflow effects in the first pair and flow correlations in the second pair and vice versa. The factor “2” in front of the last term is due to the two ways of pairing. The pure fourparticle nonflow correlation is omitted from this expression (see below on the possible magnitude of such a contribution). If one subtracts from the expression (5) twice the square of the expression (3), one is left with only the flow contributions
(6) 
where the notation is used for the cumulant. A very elegant way of calculating cumulants in flow analysis with the help of the generating function is proposed in [11]. The simulations [27], see Fig. 2, confirm that using 4particle cumulants reliably removes nonflow contributions. It also shows that even in the presence of genuine 4particle correlations (due to clusters decaying into 4particles, such were introduced into simulations) those correlations are combinatorially suppressed compared to real flow correlations and there is no real need for use of higher order cumulants in flow analysis.
The high precision results available in modern high statistics and large acceptance experiments become sensitive to another effect usually neglected in flow analysis, namely, eventbyevent flow fluctuations. The latter can have two different origins: “real” flow fluctuations – fluctuations at fixed impact parameter and fixed multiplicity (see, for example, [12, 13]), and impact parameter variations among events from the same centrality bin in a case where flow does not fluctuate at fixed impact parameter. Note that these fluctuations affects any kind of analysis, including the “standard” one based on pair correlations. The reason is that any flow measurements are based on correlations between particles, which are sensitive only to certain moments of the distribution in . In the pair correlation approach with the reaction plane determined from the second harmonic, the correlations are proportional to . Averaging over events gives , which in general is not equal to . The 4particle cumulant method involves the difference between 4particle correlations and (twice) the square of the 2particle correlations. It is usually assumed that this difference comes from nonflow correlations. Note, however, that this difference () could be due to flow fluctuations. Let us consider an example where the distribution in is flat from to . Then, a simple calculation would lead to the ratio of the flow values from the standard 2particle correlation method and 4particle cumulants as large as .
3 The physics of elliptic flow
Many important developments in this area: better understanding of the transverse momentum dependence of anisotropic flow in low region with the help of the “blast wave” model [14, 15], attempts [18] to describe in high region accounting for the parton energy loss – “jet quenching”, calculation in a parton cascade model [19], analysis of the anisotropies in Color Glass Condensate [13], and detailed analysis of the elliptic flow in the hydro models [23, 24, 25].
A very interesting development in this field is an attempt to calculate elliptic flow in Color Glass Condensate – classical field approach to describe ultrarelativistic nuclear collisions. One of the important consequences in this approach [13] is strong eventbyevent fluctuations in . As it has been already discussed, such fluctuations would manifest themselves in the difference of the flow results derived from 2 and 4particle correlations.
3.1 Transverse momentum dependence
I would like to mention here what is usually referred to as “hydro inspired”, “blast wave” or “expanding shell”, models. Such models consider particle production from a thermal source in a form of an expanding shell with the radial expansion velocity having some azimuthal modulation. The case of directed flow was discussed in [16]. That model was used to fit E877 data and gave quite reasonable results [16]. The model was generalized for the elliptic flow case in [14]. Later it was used to fit STAR data and was further generalized for the case of the elliptic shape shell [15]. In this approach:
(7) 
where , , and are modified Bessel functions, and where , and . The assumptions of this model are boostinvariant longitudinal expansion and freezeout at constant temperature on a thin shell, which expands with a transverse rapidity exhibiting a second harmonic azimuthal modulation, . Here, is the azimuthal angle (measured with respect to the reaction plane) of the boost of the source element on the freezeout hypersurface [14], and and are the mean transverse expansion rapidity () and the amplitude of its azimuthal variation, respectively. In Fig. 3, the fit to the minimumbias data with is shown as the dotted lines. The relatively poor fit led the authors to introduce a spatially anisotropic freezeout hypersurface, with one extra parameter, , describing the variation in the azimuthal density of the source elements, . This additional parameter leads to a good description of the data, shown as the solid lines in Fig. 1. A positive value of the parameter would mean that there are more source elements moving in the direction of the reaction plane. The model predicts a specific dependence of the elliptic flow on the particle mass. This massdependent effect is larger for lower temperatures () and larger transverse rapidities ().
The behavior of at large transverse momenta is also very interesting. One of the possibilities is that the anisotropy at such transverse momenta is due to path length dependent nuclear modification of the parton fragmentation function (jet quenching). High parton produced in the direction of long axis of the overlapping region exhibits more inelastic (in addition to elastic) collisions than that emitted along the short axis. It results in smaller probability to fragment into high hadron. The effect depends on the density of the media and thus the observed anisotropy could serve as a measure on this very density [18] (and features of the energy loss itself). The transverse momenta, where saturates could also help in understanding the origin of the particles in the region of 2–5 GeV/c: do they acquire their transverse momentum due to multiple scattering or they come from a fragmentation of even higher parton? The preliminary STAR data [28], Fig. 4, support the idea of flow saturation at high .
3.2 Hydro and low density limits
The values of elliptic flow measured at RHIC are comparable to that in hydrodynamic models. There was a clear disagreement at lower energies. As it has been mentioned in the introduction, for elliptic flow one needs rescatterings. Denser the matter and more rescattering means higher elliptic flow. The current understanding is that in limit of zero mean free path, the hydro limit, one gets the largest possible values of elliptic flow.
Interesting that the flow values obtained in parton cascade calculations [19] at different transport opacities could in principle significantly exceed the flow values from hydro calculations. It raises a question about validity of the assumption that the largest values of flow can be reached in hydro model. Two lines shown in Fig, 6 correspond to the results of this model for two values of transport opacity, corresponding to 35 (upper line) and 13 times higher gluon density as given by HIJING model. Note, however, that this opacity was calculating assuming 1 mb gluon transport cross section and assuming the hadronization picture when the number of gluons equals the number of hadrons. If one would, for example, consider a system as a constituent quark gas, one would have to increase the cross section approximately by 3 times and quark density by approximately 2 to 3 times. That would give the opacities very close to that needed to describe the data.
Fig. 5: Preliminary NA49 results [17] on flow centrality dependence.
Fig. 6: STAR [27]. centrality dependence from 2 and 4particle correlations. Lines are the predictions from a parton cascade model[19] (see text).
In the hydro limit elliptic flow is basically proportional to the original spatial ellipticity of the nuclear overlapping region [1, 23, 24], . In the opposite limit, usually called the low density limit [20, 21], elliptic flow depends also on the particle density in the transverse plane: , where is the area of the overlapping zone. It results in a different centrality dependencies of the elliptic flow in these two limits. The comparison of the results on elliptic flow from this point of view was first done in [21]. In this picture, the transition to the deconfinement would lead to some wiggles in dependence, (“kinks”) [22, 20, 21]).
Fig. 7: PHOBOS results on overlayed with [26].
Fig. 8: as a function of particle rapidity density for three colliding energies. Hydro limits taken from [23].
One indication that at RHIC energies flow is still proportional to the particle density can be seen from Fig. 7 (taken from [26]) which show that closely follows . The 3D hydro calculations [25] also cannot describe the pseudorapidity dependence of elliptic flow, once more indicating that the hydro description could be not correct in spite of the large values of measured at RHIC.
Taking the recent (year 1, GeV) STAR results on elliptic flow from 4particle cumulants [27] (no nonflow effects) and plotting them along preliminary NA49 [8] and E877 results, Fig. 8, also suggests that even at RHIC energies elliptic flow continue to rise with particle density. (Note possible systematic errors in Fig. 8 of the order of 1020% due to uncertainties in centrality measurements. However, this uncertainty does not alter the general trend). Decisive measurements could be already at full RHIC energies of GeV. If would continue to rise with particle density (which increases for about 15% in this energy range), it could give difficulties to hydrodynamic interpretation.
Acknowledgments. The author thanks the organizers of the very productive and exciting workshop for the invitation. Numerous discussions with Art Poskanzer are greatly appreciated.
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