# A Hydrodynamic Description of Heavy Ion Collisions at the SPS and RHIC

###### Abstract

A hydrodynamic + cascade model of relativistic heavy ion collisions is presented and compared to available hadronic data from the SPS to RHIC. The model consistently reproduces the radial and elliptic flow data for different particles, collision energies, and impact parameters. Three ingredients are essential to the success: (a) a reasonable EOS exhibiting the hard and soft features of the QCD phase transition, (b) thermal hadronization at the phase boundary, and (c) subsequent hadronic rescattering. Some features of the RHIC data are readily explained: (i) the observed elliptic flow and its dependence on and mass, (ii) the anomalous ratio for , (iii) the difference in the slope parameters measured by the STAR and PHENIX collaborations, and (iv) the respectively strong and weak impact parameter dependence of the and slope parameters. For an EOS without the hard and soft features of the QCD phase transition, the broad consistency with the data is lost.

## I Introduction

### i.1 Reaching the Macroscopic Limit

Excited nuclear matter has been created by colliding Pb and Au ions at the SPS () and RHIC () accelerators QM97 ; QuarkMatter99 ; QM2001 . For infinitely large nuclei, the excited matter can be characterized by macroscopic quantities – pressure, temperature, viscosity, etc. Lattice QCD simulations indicate that for temperatures larger than , confined nuclear matter morphs into a phase of deconfined quarks and gluons – the Quark Gluon Plasma (QGP) ShuryakQGP ; Lattice-Eos . The possibility of observing the QGP in real nuclei has motivated the heavy ion experimental program.

If the system is macroscopic then thermodynamics describes the static properties of the matter and hydrodynamics describes the dynamic properties of the matter. In fact, the observed particle ratios are remarkably close to the particle ratios in an ideal gas of hadrons at a temperature, Stachel-Thermal ; Becattini-Thermal ; Sollfrank-Thermal ; Redlich-Thermal . This suggests that the system evolved from a state close to thermal equilibrium at the phase transition boundary. However, the same thermal description reproduces the hadron ratios in proton-proton and collisions, where the system size is small and equilibration seems impossible. The success of thermodynamics seems to reflect phase space rather than the equilibration of macroscopic system.

Therefore, a thermodynamic (static) description, divorced from a hydrodynamic (dynamic) description, can not unambiguously signal a macroscopic state. It is important that elementary proton-proton and collisions do not exhibit hydrodynamic behavior. An analysis of hadronic spectra SZ-FlowProfile shows little sign of the transverse expansion predicted by hydrodynamics. Thus, the excited systems produced in these elementary collisions are macroscopic.

In contrast, experiments with heavy ions do show evidence for a hydrodynamic expansion. Momentum correlations, colloquially known as , are observed at the SPS and RHIC. In PbPb collisions, the particles emerge from the collision with a collective transverse velocity of approximately RadialFlow-Review . This radial flow is firmly established from a combined analysis of particle spectra, HBT correlations, and deuteron coalescence Uli-Expand . In non-central collisions, the particles emerge with an elliptic flow (see for exampleQuarkMatter99 ; QM2001 ). Elliptic flow is quantified by , the asymmetry of the angular distribution

(1) |

where is measured around the beam axis with respect to the impact parameter. Radial and elliptic flow data are measured as a function of transverse momenta, particle type, impact parameter, and collision energy. This wealth of momentum correlations severely constrains viable models of the heavy ion collision.

Several microscopic models have been used to explain the available heavy-ion data. The first is a dilute parton model, which is quantified with the HIJING event generator HIJING . Dilute parton models are based upon the extrapolation of perturbative QCD from high down to a scale of . For central AuAu collisions at RHIC, HIJING predicted a mini-jet multiplicity of , which is insufficient to generate the strong hydrodynamic response observed at RHIC Molnar-Elliptic . The second is a string model, which is quantified with the UrQMD event generator. In string-models non-interacting strings decay into hadrons which subsequently interact. Due to the small transverse pressure at early times, UrQMD predicted a decrease in elliptic flow from the SPS to RHIC UrQMD-Elliptic . A increase was observed. In contrast to these microscopic models, hydrodynamic calculations at the SPS and RHIC, give a good description of the observed radial and elliptic flows Sollfrank-BigHydro ; Schlei-BigHydro ; Kolb-UU ; Kolb-LowDensity ; Kolb-Radial ; Htoh , but offer no insight into the microscopic mechanism of equilibration.

Accepting the macroscopic approach and its limitations, the phase transition to the QGP influences both the radial and elliptic flows. Lattice simulations indicate Lattice-Eos that over a wide range of energy densities , the temperature and pressure are nearly constant and the speed of sound is approximately zero, . Because the speed of sound is small in this range, the pressure can not effectively accelerate the matter HS-soft ; Rischke-Log . However, when the initial energy density is well above the transition region, the matter enters the hard QGP phase. The speed of sound approaches and the pressure drives collective motion. At a time of , the energy density at the SPS and RHIC are very approximately and NA49-EnergyDensity ; Phobos-Multiplicity . Based on these experimental estimates, the hard QGP phase is expected to live significantly longer at RHIC than at the SPS. The final radial and elliptic flows of the produced particles should reflect this difference VanHove-T ; Ollitrault-MixedPhase ; Kataja-MixedPhase ; Ollitrault-Elliptic ; Kolb-UU ; Rischke-Px .

Certainly, hydrodynamics is not applicable when the particles decouple from the collision and this “freezeout” must be modeled in order to compare the observed hadron spectra to hydrodynamic calculations. Usually, a naive freezeout prescription is taken: A “freezeout Temperature” , is specified; thermal and chemical equilibrium are assumed for ; the spectrum of particles passing through the isotherm is calculated; is finally adjusted to match the single particle spectrum of pions and nucleons. Of course, this prescription is unrealistic and takes away from the predictive power of hydrodynamics. Nevertheless, the approach successfully describes many radial and elliptic observables from the SPS to RHIC Sollfrank-BigHydro ; Schlei-BigHydro ; Kolb-LowDensity ; Kolb-Radial .

However, the naive freezeout prescription fails in a number of respects. First, on the time scale of the collision , hadronic reactions do not alter the hadron composition and chemical equilibrium is not maintained (see for example Uli-Chemical ; HydroUrqmd ; UrQMD-Hagedorn ). Therefore in the late hadronic stages, chemical freezeout must be modeled in order to describe the resonance contribution and the particle ratios. Second, different particle types freezeout at different times and with different transverse velocities. With a universal freezeout temperature, the transverse flow of the strange particles , is never reproduced Sorge-Strange . Third, at the SPS and RHIC, integrated elliptic flow is a strong function of the freezeout temperature. When the universal freezeout temperature is adjusted to match the nucleon spectrum, the integrated pion elliptic flow is too large Kolb-Flow . In reality, pions and nucleons freezeout at different times and temperatures.

To model freezeout, Bass and Dumitru HydroUrqmd replaced the hadronic phase of the hydrodynamics with a hadronic transport model, UrQMD. In this approach, the switch from hydro to cascade is made at a switching temperature, . The spectrum of particles leaving the surface is taken as the input to the cascade and the attendant theoretical problems are ignored. The approach worked. Chemical freezeout was incorporated into a comprehensive dynamical picture. The flow of the multi-strange baryons was reproduced. When a similar hydro+cascade model Htoh was applied to non-central collisions, elliptic flow was also reproduced at the 20% level. Furthermore, with the indeterminacy removed, these “simple” boost invariant hydro+cascade models were rather predictive – only and the ratio have to be specified.

### i.2 Brief Summary

In this work, we compare the hydro+cascade model of Htoh to the flow systematics at the SPS and RHIC. The model uses hydrodynamics to model the initial stage of the collision, and the hadronic cascade, Relativistic Quantum Molecular Dynamics (RQMD v2.4), to model the final stages of the collision Sorge-RQMD .

The Equation of State (EOS) is varied systematically and results are compared to the whole body of flow data from the SPS to RHIC. A family of EOS, labeled by the value of the Latent Heat (LH) is constructed; LH4, LH8, LH16 denote increasingly soft EOS with latent heats respectively. As a limiting case, the latent heat is made very large forming LH. A Resonance Gas (RG) EOS is also studied. For an EOS with both the hard and soft features of the QCD phase transition, the model is broadly consistent with the body of available data. The best overall consistency with the data is found with LH8. For an EOS with only hard (e.g. RG) or only soft features (e.g. LH), the broad consistency with the data is lost.

The model consists of three distinct components. The first component solves the equations of relativistic hydrodynamics in the transverse plane, assuming Bjorken scaling Bjorken-83 . The switching surface, or the isotherm where , is found. The sensitivity of the model results to will be discussed in a separate publication where chemical freezeout is also addressed Teaney-Chemical . The second component converts the macroscopic hydrodynamic variables on the switching surface into hadrons according to the Cooper-Frye prescription augmented with a theta function rejecting backward moving particles CooperFrye . Finally, the third component, the hadronic cascade RQMD sequentially rescatters the generated hadrons and models the hadronic freezeout stage of the collision. Throughout the analysis the role of hadronic rescattering is assessed. In all figures, the “Hydro+RQMD” curves incorporate hadronic rescattering and resonance decays while the “Hydro Only” curves only incorporate resonance decays.

As outlined in the abstract, several features of the first RHIC data are readily explained in the course of this analysis.

## Ii Model Description and the EOS

### ii.1 Hydrodynamics

Relativistic hydrodynamics is a set of conservation laws for the stress tensor () and for the conserved currents (), and . In equilibrium, and are related to the bulk properties of the fluid by the relations, and LL-Hydro . Here is the energy density, is the pressure, is the number density of the corresponding current, and is the proper velocity of the fluid. In strong interactions, the conserved currents are isospin (), strangeness (), and baryon number (). For the hydrodynamic evolution, isospin symmetry is assumed and the net strangeness is set to zero; therefore only the baryon current is considered below.

The equations of motion may be expressed in terms of the variables and

(2) | |||||

Integrating over the transverse plane, one finds that net baryon number per unit spatial rapidity, , and the transverse momentum per unit rapidity, as well as , are conserved. The energy per unit rapidity, , decreases due to the work done per unit time MG84-Work by the pressure in the longitudinal direction, .

For an ideal fluid, entropy conservation can be derived LL-Hydro , . The entropy current is defined as , where is the entropy density and is the fluid 4-velocity. For a Bjorken expansion entropy conservation becomes

(3) |

Integrating over the transverse plane, we find that

(4) |

is a constant of the motion. This relation is monitored to test the accuracy of the solution.

These equations are solved numerically with a Gudunov method Hydro-Leveque . Using second order operator splitting Hydro-Leveque , a single time step separately updates the x-direction, the y-direction, and the loss terms on the r.h.s. of Eq. 2. Different splittings gave only negligibly different results. The simple RHLLE Riemann solver was used for the updates in the x and y directions Rischke-RHLLE ; Rischke-PlasmaTest . A second order (in ) Runge-Kutta stepper was used for the r.h.s. update.

### ii.2 Initial Conditions

To model the initial conditions, the entropy and baryon distributions at a Bjorken time of , are made proportional to the distribution of participating nucleons in the transverse plane. Since entropy and baryon number are conserved per unit rapidity, the final yields of pions and nucleons are then proportional to the number of participants. The initial conditions are similar to sWN (entropy per Wounded Nucleon) initial conditions in Kolb-Centrality and to the initial conditions of Ollitrault-Elliptic .

For all subsequent discussions, we consider two identical (for simplicity) nuclei with atomic number A and B, and nucleon distributions and , collide along the z-axis with impact parameter , pointing in the x-direction from the center of nucleus A, , to the center of nucleus B, . The nucleon distribution is parameterized as a Woods-Saxon distribution, , and is normalized to the atomic number A. The parameters are , . The used is 4% below the value used by the STAR collaboration SN402 . The number of participating nucleons per unit area, , at a position in the transverse plane is given by

Here, is the thickness of a nucleus at position (x,y) and is the inelastic nucleon-nucleon cross section. For the sake of comparison, is taken as both at the SPS and RHIC. For large , , and often Eq. II.2 is re-written in terms of exponents.

With the number of participants specified, the initial entropy and (net) baryon densities at time , are then fixed with two constants and with

(6) | |||||

(7) |

The two dimensionless constants and are the entropy and net baryon number produced per unit spatial rapidity per participant. At the SPS (see Sect. IV.1), and were adjusted to fit the total yield of charged particles and the net yield of protons, respectively. At RHIC, was adjusted to match the PHOBOS multiplicity Phobos-Multiplicity . At the, time the ratio was not known and was estimated from UrQMD simulations to be . This gives the ratio . Later, the STAR and PHOBOS collaborations measured the ratios, and respectively STAR-ppbar ; Phobos-ppbar . Since the measured ratio is close to the ratio initially used, and since a full simulation takes several CPU days, the UrQMD-based estimate was used throughout this work. This makes the model yield approximately 15% too low and the model proton yield approximately 15% too high. This correction will be accounted for in future works. A summary of the parameters is given in Table 1.

Parameter/Value | PbPb SPS | AuAu RHIC |
---|---|---|

8.06 | 14.42 | |

0.191 | 0.096 | |

(fm) | 1.0 | 1.0 |

(mb) | 33 | 33 |

42 | 150 | |

8.2 | 16.7 | |

6.4 | 11.2 | |

5.4 | 11.0 | |

4.5 | 7.9 |

Two quantities, which will be used extensively in the analysis in Sect. III and Sect. V, are defined as

(8) | |||||

(9) |

where the average is taken over the initial entropy distribution of Eq. 6. These quantities are plotted as a function of the number of participants relative to central collisions in Fig. 1.

measures the initial elliptic deformation of the overlap region and grows approximately linearly with .

For the calculation presented, the entropy and therefore the number of charged particles scales as the number of participants. Recently, the experiments have reported that the charged particle multiplicity grows slightly faster than the number of participants PHENIX-Centrality ; Phobos-Centrality . This slight growth can be incorporated into hydrodynamics Kolb-Centrality , but instead the experimental is compared directly to the model . This makes the model impact parameter slightly larger than the impact parameter determined by the experimental collaborations.

### ii.3 Equation of State

To solve the equations of motion, we need an Equation of State (EOS), or a relation between the pressure () and the energy and baryon densities ( and respectively). In many previous hydrodynamic calculations, a bag model EOS is used Sollfrank-BigHydro ; HydroUrqmd ; HydroUrqmdHBT ; Raju-ResonanceGas . This has some advantages, since the degrees of freedom are explicit in both phases. However a typical bag model results in an EOS with a large latent heat, LH=. Furthermore it is difficult to adjust the latent heat independently of in such models of the phase transition.

We have taken a more pragmatic approach and have constructed a thermodynamically consistent EOS with a variable latent heat in the and plane. First, note the following two derivatives which apply along the path where is constant

(10) | |||||

(11) |

The first of these is simply the definition of the speed of sound. The second relation is surprising: it does not contain the chemical potential explicitly. (It follows by noting that

For smooth flows, entropy and baryon number are separately conserved. If at some initial time is constant everywhere in space, the two conservation laws imply that is constant everywhere in space time LL-Hydro . For the initial conditions specified in Sect. II.2, is constant in space and remains constant as the system evolves. Therefore, the pressure is needed only along the path . This may be directly verified by fully differentiating and noting that the derivatives of the pressure only appear as the speed of sound,

Strictly speaking, transverse shock waves develop near the phase transition and invalidate the assumption of entropy conservation. However, numerical and analytical evidence has shown that entropy production in hydrodynamic simulations of nucleus-nucleus collisions is at most a few percent Ollitrault-RiemmanMixed . Below, entropy production is ignored and the pressure is specified along the trajectory .

The EOS consists of three pieces: a hadronic phase, a mixed phase, and a QGP phase. In strong interactions, Baryon number (B), Strangeness (S), and Isospin (I) are conserved and therefore the EOS depends on and ,and . In the hadronic phase, the thermodynamic quantities –the pressure (p), the energy density (), the entropy density (s), and number densities ( where Q=B,S,I)– are taken as ideal gas mixtures of the lowest SU(3) multiplets of mesons and baryons. The mix includes the pseudo-scalar meson octet () and singlet (), the vector meson octet () and singlet (), the baryon and anti-baryon octets and the baryon and the anti-baryon decuplets. Specifically, and are given by

(12) | |||||

(13) | |||||

(14) | |||||

(15) |

Here the sum is over the hadrons species, are the quantum numbers of the i-th hadron, is for bosons but for fermions, and for example, is the pressure of a simple ideal Bose gas. A fast numerical method for evaluating the thermodynamic quantities of simple Bose/Fermi gases has been constructed Thermo-Pons . For a given T, are determined by the requirements that total strangeness () and isospin() be zero and that . The thermodynamic quantities are then taken as functions of along the adiabatic path where . This hadronic EOS is taken up to a temperature of or an energy density (see Fig. 2). The squared speed of sound is approximately 1/5 in this hadronic gas.

Above the hadronic phase, only the speed of sound squared, , is specified. For the mixed phase the speed of sound was made approximately zero, . The width of the mixed phase (see Fig. 2) is the Latent Heat (LH), LH. LH is taken as a parameter and is adjusted to form phase diagrams LH8, LH16,… with latent heats, , . Above the mixed phase, , the degrees of freedom are taken as massless and the speed of sound is accordingly, . We also consider two limiting cases: a Resonance Gas (RG) EOS and LH. For a RG EOS, the speed of sound is constant above . For LH, the mixed phase continues forever () and there is no ideal plasma phase.

With the speed of sound specified in all phases, Eq. 10 and 11 are integrated to find the pressure and entropy along the adiabatic path specified by the initial conditions, . The full phase diagram for SPS initial conditions is shown in Fig. 2. In Sect. IV and Sect. V,

the subset of the EOS consistent with the available radial and elliptic flow data is found.

### ii.4 RQMD and the Cooper Frye Formula

Given the initial conditions and the EOS, the equations of motion are integrated in time. As the system expands and cools, the mean free path becomes much less than the nuclear radius, and the system breaks up into free-streaming particles. Typically, hydro practitioners Sollfrank-BigHydro ; Schlei-BigHydro model the breakup or stage by finding a hypersurface in space and time where the temperature equals some freezeout temperature, .

This simple picture in which all particles are emitted from a single space-time surface is however unrealistic. Different particles have different hadronic cross sections and suffer their last interaction at different times. They are emitted over a space-time region rather than on a sharp surface. Further, particles in the center rescatter for a longer time than particles in the periphery. To model this physics, the spectrum of particles the space-time surface is taken as the input to the hadronic cascade, RQMD Sorge-RQMD . Subsequently, the particles rescatter. Below, the space-time surface is referred to as the surface rather than the surface. The attending problems with this approach are described below after the details of the input distribution to RQMD are described.

For the family of EOS discussed above, for , the fluid is made up of a collection of ideal gases of fermions and bosons. In order to conserve energy and momentum across the surface, the spectrum of crossing the surface is given by CooperFrye ; CooperFrye-HBT ,

(16) | |||||

where is a differential element of the freezeout hypersurface and is the Bose/Fermi distribution function

The differential elements of the hypersurface can be separated into time-like () and space-like () surface elements. For time-like surfaces, the integrand in Eq. 16 is positive and there is a frame (the rest frame of the surface), where = (dV,0,0,0). The spectrum of Eq. 16 is simply a thermal spectrum boosted by the flow velocity in the frame of the surface. (In practice, the surface velocity is small for most time-like surfaces). The yield of leaving a surface element, , is simply , as may be found by integrating the left and right sides with and going to the rest frame of the matter.

For space-like surfaces, the integrand in Eq. 16 is both positive and negative depending on the momentum of the particle. When the integrand is positive, the particle is leaving the surface and when it is negative the particle is entering the surface. We reject particles entering the hydrodynamic surface. The distribution of particles exiting a space-like surface is

(17) | |||||

It is this distribution that we take as the input distribution for RQMD. For a discussion of the problem of space-like surfaces see Freezeout-Theta ; Freezeout-Bugaev . The number of particles leaving a differential surface element is given by a more complicated formula which is again found by integrating both sides of the equation with Teaney-Thesis . For a stationary surface in the rest frame, the formula has the simple interpretation as the number of particles evaporating from the surface per area per unit time. A consequence of the theta-function is that energy, momentum, and particle number are not exactly conserved across the transition surface. However, the error is only for central (peripheral b=8.0 fm) AuAu collisions at RHIC.

## Iii The Space-Time Evolution

### iii.1 The Hydrodynamic Solution

This section reviews the hydrodynamic evolution for different EOS used in this work. The evolution at the SPS and RHIC is summarized in Fig. 3.

The switching isotherm, (shown in the middle), is particularly important since in the hydro+cascade approach, the particles are injected into RQMD with the velocity distribution of this isotherm.

For EOS with a phase transition (LH8) there are three phases and three corresponding stages in the acceleration history. (i) an explosive QGP phase (), in which the matter accelerates rapidly, (ii) a soft mixed phase (), in which the matter free streams with constant velocity and (iii) a hadronic phase (), in which the hadronic pressure produces additional acceleration.

The QGP phase dictates the duration and transverse size of the collision. At RHIC, the QGP pressure drives the matter outward, rapidly increasing the radius, which in turn shortens the overall lifetime. Therefore, approximately doubling the total multiplicity from the SPS to RHIC increases the total lifetime only slightly, from to . All the additional multiplicity is absorbed by a slightly larger transverse radius. Similarly, for a RG EOS the acceleration is robust and continuous and increasing the total multiplicity only slightly increases the radius and lifetime. In bulk, the radii and lifetimes of RG are similar to LH8.

By contrast, for LH, the stiff QGP phase is absent and the mixed phase is dominant at high energy densities. The strong transverse acceleration associated with LH8, is replaced with a slow evaporative process. The radius of the system slowly shrinks as a function of time. Unlike LH8, increasing the total multiplicity increases the lifetime rather than the radius. Between the SPS and RHIC the lifetime increases from from to . Summarizing, the QGP drives a transverse expansion; the transverse expansion increases the radius and shortens the overall lifetime compared to an EOS without the QGP push.

To quantify the input velocity distributions into RQMD, Fig. 4 plots the transverse fluid

rapidity versus the radius along the switching isotherm at the SPS and RHIC. For LH8 and RG, the transverse rapidity shows a linear rise with radius. A linear flow profile is often used in phenomenological fits to the particle spectra SSH-FlowProfile ; this calculation validates this approach.

For an EOS with a phase transition to the QGP (LH8), the acceleration is initially large but subsequently stalls in the mixed phase. By contrast, for an EOS without the phase transition (RG), the acceleration is robust and continuous. Therefore, although the initial transverse acceleration is smaller for a RG than for LH8, the RG velocity at the end of the SPS mixed phase is larger than for LH8. At RHIC, where for LH8 the QGP phase lives longer, the RG velocity is only larger. Although mean lifetimes and radii of the RG EOS are similar to LH8, the change in the velocity distributions from the SPS to RHIC are markedly different. Comparing LH8 to LH at RHIC,the LH8 flow velocity is approximately twice as large as the LH flow velocity.

Nevertheless, it should be noted that if freezeout is taken as , then the differences between the flow velocities of the EOS is smeared out by the hadron phase, as can be seen by examining the mean flow velocities on the curves in Fig. 3. Indeed, the hadronic phase of LH (which lives longer, since it is born with no transverse velocity) can partially compensate for the weak initial acceleration. Along the isotherm, the flow velocities of LH, RG and LH8 are roughly comparable.

To characterize the flow in non-central collisions for the EOS used in this work, we follow Kolb Kolb-UU and calculate a quantity derived from the stress tensor for

(18) |

where the denotes an average over the transverse plane weighted with the entropy per area per unit spatial rapidity, . is related to the final momentum anisotropy of the particle distribution Ollitrault-Elliptic which ultimately is related to Kolb-Flow .

The general trends seen in follow from the discussion above on the hydrodynamic response of each EOS. LH8 shows a strong early response followed by a stall and subsequent flattening as the matter distribution becomes almost spherical. At RHIC the strong early response lives substantially longer as the matter spends a larger fraction of its total lifetime in the QGP phase. For LH the matter is initially stalled but rapidly accelerates as it slowly enters the hadronic phase (see Fig. 3 (e) and (f)).

### iii.2 Qualitative Predictions of the Hydrodynamic Response

We can now make some qualitative predictions from the hydrodynamic solution. Assume momentarily that the final hadron momentum distributions reflect the boundary between the mixed and hadronic phases or in Fig. 3. Then with the curves presented in the last section, LH8 predicts two qualitative changes. First with Fig. 5, between the SPS and RHIC, weighted elliptic flow should increase by almost a factor of two as the QGP replaces the mixed phase and dominates the early evolution. Second with Fig. 4, the total transverse momentum should increase by 30% as the QGP drives additional transverse motion. The flow differentiates LH8 from a RG EOS and from LH. Since RG EOS accelerates continuously and does not stall in the mixed phase, the transverse momentum is larger at both the SPS and RHIC. In addition, for SPS collision energies, the elliptic flow () for a RG EOS is almost a factor of two larger than for LH8. For LH, the transverse momentum is very low until the very end.

To make these qualitative predictions quantitative and to compare the hydrodynamic solution to experimental data, it is essential to model hadronic freezeout. Between the time when (the solid symbols) and (the open symbols) elliptic flow changes dramatically for each EOS. This is especially true for LH8 at the SPS, where the mixed phase abruptly stalls the development of elliptic flow but then the hadronic phase rapidly completes the development. The differences in the early acceleration tend to get washed out by the hadronic stage. Indeed, even LH has a reasonable radial and elliptic flow by . The extent to which signatures of the early QGP acceleration remain in the final spectra depends on whether the freezeout temperature should be taken as or . The breakup of a heavy ion collision can only be addressed with hadronic cross sections and expansion rates.

### iii.3 RQMD – Input and Response

Relativistic Quantum Molecular Dynamics (RQMD) Sorge-RQMD is a hadronic transport computer code which incorporates many known hadronic cross sections. RQMD has been used extensively to model the heavy ion dynamics Sorge-Strange ; Sorge-elliptic ; Sorge-PreEquilibrium ; Sorge-kink ; Sorge-Time . Briefly, when two particles come within , they elastically scatter or form a resonance. Resonance formation and decay dominate the evolution. The principal reactions are , , , and . Only binary collisions are considered in this hadron cascade. Before discussing the response of RQMD, we first consider the input.

The time distributions are found by projecting the entropy distribution on the switching isotherm in Fig. 3 (or ) onto the axis. In Fig. 6

the fraction of pions (or entropy) injected into RQMD per unit is plotted as a function for each EOS. For LH8, very few particles are evaporated from space-like surfaces at early times, and at () particles are emitted in bulk from the time component of the switching surface. For LH, particles are continuously evaporated from the transition surface and the radius slowly decreases. Therefore, the time distribution is relatively uniform. Finally for a RG EOS, the freezeout surface is not box-like and particles are also emitted into RQMD slowly and continuously.

Now consider the dynamic response of the hadronic cascade. For LH8, the hydrodynamic input into RQMD can be characterized as a simple thermal model with a linearly rising flow profile with a uniform radial distribution except at the edge of the distribution where there is a small maximum. Once this input distribution is taken, the hadrons re-scatter within RQMD and different particles decouple from the cascade at different times. Fig. 7 (a) plots the

mean emission time (the time of last interaction) versus the mass of the particle species. Also shown is , when all collisions in RQMD are switched off and only resonance decays are allowed. The mean number of collisions experienced by a particle is shown in Fig. 7 (b). The mesons scatter approximately once after their principal resonances ( etc.) decay and decouple around . In contrast, due to strong meson-baryon resonances , nucleons and hyperons ( and ) scatter approximately twice and decouple around . The and are emitted directly from the phase boundary since they have small hadronic cross sections.

The duration of the hadronic stage dictates the spatial extent of the final source. In Fig 7 (c), the mean radius is shown as a function of particle mass with and without re-scattering in the hadronic cascade. For comparison, we apply the simple formulas: We assume all particles are emitted from the switching surface at a mean radius and a mean time , with a constant radial velocity (see Fig. 7 (a) and (c)). Since is emitted directly from the switching surface, = and . With the formula distance = velocity time, we have

(19) |

where () is the freezeout radius (time) of particle x and is the freezeout drift velocity. This velocity incorporates a thermal drift velocity and the flow velocity of the source. Given a constant velocity as a function of mass , a very simple fit to the freezeout radii is obtained, as shown in Fig. 7(c). Thus, hadronic cross sections dictate the final freezeout radii of the source.

Hadronic cross-sections also dictate the spatial geometry in non-central collisions. In non-central collisions the ellipticity of the source at freezeout is quantified by the spatial anisotropy,

(20) |

Here, the averages are taken over points of last interaction in the cascade. is negative for the initial almond-shaped distribution but positive for a cucumber-shaped distribution. Fig. 7(d) shows without re-scattering but with resonance decays (Hydro Only) and with hadronic re-scattering (Hydro+RQMD). The initial elliptic flow () changes the overall geometry () by the end of the RQMD stage. At the end of the hydrodynamic stage is negative, indicating that the source retains at least some of its initial almond distribution. becomes positive as the system evolves and the momentum asymmetry changes the source geometry. For nucleons, is almost +3% for modest impact parameters; this may have observable consequences STAR-EllipticParticle .

### iii.4 Impact Parameter Dependence of the Space Time Evolution

In the previous section, we discussed how hadronic cross sections control the lifetime and geometry of the final hadronic distributions. Now the impact parameter is varied and the freezeout distributions are modified. In non-central collisions, the number of charged particles scales as the number of participants; therefore the lifetime of the hadronic stage should also scale as the number of participants. However, the hadronic lifetime is also a function of the cross section, the radius, and the expansion rate (). These depend respectively on the particle species, the r.m.s. radius of the initial geometry, and the EOS. In Fig. 8, the different contributions

to the total lifetime are studied. We plot the mean emission time , divided by size , as a function of the number of participants for different particles and EOS. To set the absolute scale, the “free” axes show directly at two impact parameters.

Consider first the LH8 curves (a): The total lifetime for all particle species falls by approximately 30% from central (b=0 fm) to peripheral (b=8 fm) collisions. The order of particle emission remains as the impact parameter is varied: First rare species () are emitted, then mesons () and finally baryons (). For the , which is representative of the hydrodynamic stage, the curves in Fig. 8(a) are flat at the 15% level, indicating that the total lifetime scales roughly with the size of the overlap region. For pions, the total lifetime also scales with . For protons, indicative of baryon emission, the total lifetime does not quite scale as but rather depends on the absolute number of charged particles in addition to the geometry. This is natural since the freezeout of protons is controlled by the formation of resonances.

For LH, does change more rapidly than for LH8 This is especially true for nucleons. However, for and the difference in the dependence of is small and to a reasonable approximation, the lifetimes of and scale with for all EOS. Changing the EOS simply moves the various curves up and down on Fig. 8 (a) and (b). A RG EOS was also studied (not shown) and the lifetime and dependence were quite similar to LH8.

Eq. 19 relates the freezeout radii and geometry of the different particles to a freezeout time and a single freezeout drift velocity () and a single freezeout radius (). This simple formula was found to be applicable to all impact parameters with approximately the same drift velocity as in central collisions. The lifetimes and radii () all scale with the root mean square radius.

Given the rather simple scaling of lifetimes and radii as a function of impact parameter, it is natural to consider the density of pions at freezeout as first done in HydroUrqmd . Since pion number is approximately conserved during the cascading process, we have and the freezeout entropy density is

(21) |

This quantity is shown as a function of the number of participants for RHIC collisions in Fig. 9.

The freezeout entropy density is roughly constant as a function of impact parameter. In addition, the freezeout density is independent of EOS in spite of differences in transverse velocity gradients. It has been argued that central PbPb collisions cool to a lower temperature than peripheral collisions since transverse and longitudinal velocity gradients are larger in peripheral collisions Hung-Freezeout . However at least in the model, freezeout is not driven by the expansion rate; rather, the freezeout condition reflects a density where the mean free path becomes comparable to the radius of the nucleus.

Next we hold the collision geometry fixed and examine the changes in lifetime, radius and freezeout density as the initial entropy density (, the collision multiplicity) is increased. Fig. 10 (a) and (b) shows the lifetime and emission

radius for a PbPb collision at b=6 fm as the multiplicity per participant is increased from the SPS to the RHIC domain. For LH8, the radius increases but the emission time does not, while for LH the situation is exactly reversed.

This behavior is readily understood as entropy conservation in the transverse plane. Entropy conservation (see Eq. 4) relates the entropy density to the Bjorken time (), the total conserved entropy per unit rapidity (), and the effective area () of the source with the schematic relation,

(22) |

As seen with Fig. 10(c), the entropy density at freezeout () is roughly constant as a function of beam energy. The freezeout time may therefore be related to freezeout entropy density with

(23) |

For LH8, the strong transverse acceleration rapidly increases the area and lowers the entropy density to . Consequently, as the multiplicity is doubled, the total lifetime increases by only 20%. For LH, the radius does not increase but the lifetime increases significantly. Thus the transverse expansion, together with entropy conservation, ultimately determine the total lifetime.

## Iv Radial Flow From the SPS to RHIC

### iv.1 The SPS

In a traditional hydrodynamic calculation Sollfrank-BigHydro ; Schlei-BigHydro ; Kolb-Flow , the pion and nucleon yields fix the total entropy and baryon number in the initial conditions. The freezeout temperature is adjusted to fit the pion and proton spectra. In the hydro+cascade approach advocated here, the freezeout temperature is not a parameter since particles decouple from the cascade when their collision rates become small. Therefore, the pion and nucleon yields set the total entropy and baryon number as before, but the slope parameters provide significant information about the EOS. In particular, the latent heat of the phase transition which best matches the pion and nucleon spectrum is .

In the previous section, we discussed a family of EOS, each with a different latent heat. Now we show in Fig. 11,

the calculated net proton spectrum for the resonance gas EOS and for EOS LH4-LH16. The experimental and theoretical spectra are absolutely normalized. The two numbers parameterizing the initial conditions and (see Table 1) are adjusted to match the height of these spectra. The model curves have been multiplied by a factor of 0.93 to account for the fact that the data is 5% central. Once the height of the spectrum is tuned, the shape of the spectrum is determined by the course of the hydrodynamic evolution, or more generally, by pressure gradients and the duration of the collision. Therefore, it is significant that hydrodynamics generates a flow , which is needed to explain the spectra. This flow velocity was extracted from a variety of thermal analyses Uli-Expand .

For EOS with large latent heats (e.g. LH16), the spectrum is too soft. This is because the hydrodynamic system spends a long time in the mixed phase in which pressure gradients do not generate collective motion. Bag-model equations of state, employed in many hydrodynamic calculations Sollfrank-BigHydro ; HydroUrqmd , typically have a latent heat from which makes the EOS rather soft. This large latent heat is usually compensated by adjusting the freezeout temperature Sollfrank-BigHydro . An EOS with a modest first order phase transition (e.g. LH4 and LH8) generally reproduces the shape of the spectra in Fig 11. Unfortunately, a RG EOS can also reproduce the shape of the spectrum and additional experimental information is needed to separate EOS.

The slope systematics of , and provide the necessary information. The presence of a phase transition stalls the acceleration Rischke-Lifetime ; HS-soft ; therefore information about the velocity at the end of the mixed phase can separate a RG EOS from LH8. At the SPS, the spectra of the different particle species are all reasonably exponential and a slope parameter, is extracted. Specifically the data are fit to the following form,

(24) |

where . In this parameterization, the slope parameter , is directly related to the . The model spectra are fit with Eq. 24 over the range corresponding to the WA97 WA97-Slopes experimental acceptance (); the slope parameters are shown versus particle mass with and without RQMD in Fig. 12(a).

First look at the “Hydro Only” curves. The slopes increase approximately linearly with mass as is expected in a thermal expanding source model SZ-FlowProfile ; SSH-FlowProfile ; SR-FlowProfile . The non-monotonous increasing mass in the “Hydro Only” curves is due to resonance decays and the baryon content of the particles. Once RQMD is included, the slopes are modified by hadronic rescattering leading to mass dependence characteristic of differential freezeout Sorge-Strange . Note the following features. First, in the model the gives a good measure of the flow velocity at the end of the mixed phase. Second, note the increase in the nucleon and slope parameters and the small in the pion slope parameter, due to cooling. As the hadron gas expands, the pions excite and resonances and drive additional transverse motion in the nucleon and hyperon sectors. However, the pions increase the nucleon only at the expense of their own kinetic energy. In a traditional hydrodynamic approach, the hydrodynamic evolution would be continued to match the slope of the nucleon spectrum. Judging from Fig. 12(a), this is misguided, as a nucleon receives much of its momentum after pions have decoupled. As shown below, the nucleon receives about 20% of its transverse kinetic energy from the pion “wind”, irrespective of the colliding energy. To incorporate the rich freezeout dynamics of a cascade, different freezeout temperatures and velocities should be taken for different particles Hung-Freezeout .

A comparison to the available data on slope systematics is given in Fig. 12: (b) shows the slopes for different types of EOS while (c) shows the sensitivity to the latent heat. Although RG and LH4 are capable of reproducing the pion and nucleon spectra, they significantly over-predict the slope parameters of , and . This is because LH4 and RG already have developed a substantial flow velocity at the end of the mixed phase. The slope parameter of the is a sensitive measure of the flow velocity at the end of the mixed phase since the flow velocity is amplified by the mass in the approximate formula . LH, by contrast, under-predicts the slope parameters of and , indicating that is too small at the end of the mixed phase. With LH8, the acceleration is modest – but significant – and the slope systematics are generally reproduced. In the model, only an EOS which has both a stiff and soft piece is capable of reproducing general trends seen in the particle spectra.

### iv.2 Qualitative Changes at RHIC

It was argued above that LH8 provides the best description of the radial flow at SPS: Now the same EOS is used to make predictions for RHIC. At RHIC, the initial energy density is well above the phase transition, and the large early pressure is expected to drive collective motion. In Fig. 13, the nucleon

spectrum for the SPS and RHIC are shown with and without the hadronic rescattering in RQMD. Two features are immediately observed: 1. The increases as the collision energy is increased from the SPS to RHIC Kataja-MixedPhase ; HydroUrqmd ; Ollitrault-MixedPhase . 2. The spectra without hadronic rescattering are reasonably well described by a single exponential ( they look linear on the log plot shown). Once rescattering is included the spectra are curved; this curvature grows from the SPS to RHIC. Describing the spectra with a single slope parameter, although useful in summarizing a large variety of data, is only approximate.

### iv.3 The from the SPS to RHIC and Beyond

To summarize the bulk energy transport in the model we show in Fig. 14:

(a) the for nucleons as a function of collision energy () for different EOS, and (b) the for different particle species as a function of mass for different collision energies (particle multiplicities).

Fig. 14(a) demonstrates that the additional entropy gets converted into additional transverse motion for each EOS. For each EOS, the hadronic contribution to the mean remains constant and is approximately 20% for LH8 and RG, but is approximately 30% for LH. LH is a special case, and we may say that the RQMD contribution is approximately 20% and is independent of the underlying EOS.

Fig. 14(b) demonstrates how the increase in the mean influences the spectra of different particles by plotting versus mass Sorge-Strange . At the SPS the flow velocity at the end of the mixed phase is relatively small – . The slopes before the RQMD phase show a linear rise characteristic of hydro, . When the flow velocity is small, hadronic rescattering changes the linear mass dependence significantly, giving the characteristic shape observed at the SPS. As the flow velocity increases from the SPS to RHIC and beyond, the linear rise with mass becomes increasingly steep and hadronic rescattering, while still contributing to 20% of the for nucleons, does not change the overall mass dependence. The qualitative shape of the mass dependence of therefore gives a good measure of the flow velocity at the end of the mixed phase. Since this flow velocity is different for different EOS, the mass dependence of the can therefore separate the different EOS studied.

### iv.4 The Flow Profile from the SPS to RHIC

The curvature in the spectrum is a signature of a radially flowing thermal source. The general features can be understood from a simple thermal model. For a cylindrically symmetric shell, which expands longitudinally in a boost invariant fashion and which freezes out in an instant with constant temperature T, and a radial velocity , the spectrum is given by SZ-FlowProfile ; SSH-FlowProfile ; SR-FlowProfile

For we have

(26) |

Generally, increasing the velocity increases the curvature of the final spectrum for heavy particles. Increasing the mass also increases the curvature of the flow profile. The shape of the spectrum, together with the mass dependence of the observed particle, may provide a good experimental measure of the velocity of the source at hadronization.

Hadronic rescattering changes the curvature seen in the thermal spectra discussed above. First, different particle types have different hadronic cross sections and therefore freezeout at different times and with different velocities. The curvatures in the final spectra measure these different freezeout velocities. Second, the cascade generates additional transverse flow predominantly in the low region of the spectra. To quantify these effects, we divide the spectra into a low region, and a high region, . We then fit an exponential in both domains. Thus, there is a low slope and high slope. We have checked that this parameterization gives a good description of the shape for all the spectra considered. Fig. 15 shows the low

and high slopes as a function of the particle mass, with and without the RQMD after-burner. The curves illustrate the collective acceleration which occurs during hadronic rescattering, and illustrate an interplay between freezeout and hydrodynamic behavior. First look at the “Hydro Only” curves: the curvature (i.e. the difference between the low and the high slopes) increases with mass as expected from Eq. IV.4. When the cascade is included, rescattering changes this mass dependence. The curvature no longer increases but remains approximately constant after the nucleon mass. It is useful to compare the flow of the nucleon and the . The nucleon has a smaller mass which, according to Eq. IV.4, decreases the curvature relative to the . However, the nucleon decouples later than the and through hadronic rescattering develops larger transverse velocity, which increases the curvature. In the end, the and the nucleon have approximately the same spectral shape. In contrast, the , which has approximately the same mass as the nucleon but which decouples early, has very little spectral curvature. To summarize, an interplay between the differential freezeout dynamics and the curved thermal spectra of Eq. IV.4 results in rich features in the final spectra of different particles.

Before discussing their impact parameter dependence of these rich features (see Sect. IV.6), we study the sensitivity of the spectra to the EOS. The mass dependence of the slopes is a feature of an expanding thermal source and differential freezeout. It is not a feature of the underlying EOS. In Fig. 15, the low and high slopes are shown for three different EOS. For this discussion, the direct comparison of model and data nucleon spectra in Fig. 18 may be helpful. For the high slopes for all particles, there is a simple ordering, , which reflects (through Eq. 26) the ordering of the transverse flow, . In the low region, the ordering is more complex and reflects the space-time structure of the freezeout surface for different EOS. LH evaporates particles shrinking radially inward. This causes an enhancement of the particle yield at low and gives LH a significant slope in the low region. Still, with LH8 the shows much more flow at low than it does with LH16 and LH indicating a large velocity at the end of the mixed phase.

The curvature in the spectra at small is a consequence of the mass dependence of Eq. IV.4 and hadronic rescattering, as discussed above. Now the role of hadronic rescattering, or the “pion wind”, is studied in detail with Fig. 16.

At low , where the cascade is most effective, all the nucleons come from the center of the nucleus, as can be seen in Fig. 16(b). Accordingly, the number of collisions is larger and the nucleons are accelerated more. At high , the nucleon spectrum (recall Fig. 13) is simply shifted with 2-3 collisions by a constant amount, approximately , which increases the slope. These collisions happen over a time scale of and the collision rate is therefore . Collecting these observations, nucleons coming from the center of the collision freezeout last, populate the low region, and are kicked the most by the pion wind.

### iv.5 Comparison to Central RHIC Data

Now we compare model predictions to the first RHIC spectra. Keep in mind the two major predictions of hydrodynamics introduced in Sect. IV.2. First, should increase significantly. Since LH8 was found to give the best agreement to SPS flow data, LH8 should give the best agreement at RHIC. Out of all the EOS studied, the flow velocity increases the most for LH8. Second, the spectra should show the flow profile of Eq. IV.4. This profile is sensitive to the particle mass and flow velocity. At RHIC therefore, LH8 predicts a significant change in slope from low to high .

model spectra for three different EOS compared to data for , and . For each particle type, the transverse mass spectrum is strong. Generally, LH under-predicts the flow profile while LH8 reproduces the spectrum. The data indicate a strong macroscopic transverse response, as expected of an EOS with speed of sound . Some caveats must be mentioned. It is known that the transverse mass spectrum is sensitive to the details of the initial profile and longitudinal expansion Sollfrank-BigHydro . In particular, the transverse mass spectrum is modified if the initial entropy density is distributed according to binary collisions Kolb-Centrality . However, even when the entropy is distributed entirely according to binary collisions, the change in the spectrum is small. The strong increase in radial flow from the SPS to RHIC is reproduced by the hydrodynamic response of LH8. Thus, prediction (1) is borne out by the first spectra at RHIC. Next, look at the shape of the spectrum in Fig 18. This flattening at low is characteristic of a flow profile. We expect a smaller flattening in the kaon spectrum. Thus, the rich flow profile of Eq. IV.4 is also borne out in the first RHIC data and prediction (2) is correct.

It is worthwhile to plot the and spectra on the same plot. The spectra almost cross for It was recently pointed out that the measured ratio is several times above the expected ratio from jet fragmentation and from a hydrodynamic calculation that does not incorporate chemical freezeout Vitev-Baryon . The ratio is readily explained in a simple hydro/thermal model with additional hadronic scattering. The thermal input into RQMD is roughly summarized by Eq. IV.4. Above without rescattering, the slope parameters of pions and nucleons approach the universal value . This slope is given by Eq. 26 with the parameters for and . Accounting for hadronic rescattering, the nucleon slope at large approaches and is better described by and . Hadronic rescattering therefore increases the nucleon flow velocity slightly, from to . We then adjust to match the experimental ratio STAR-ppbar , . Then with all the parameters specified, we draw Eq. IV.4 for and in Fig. 19. A source expanding with a collective velocity of and hadronizing according to a thermal prescription at a temperature of , generates the observed ratio once pion nucleon scattering is taken into account.

### iv.6 The Impact Parameter Dependence of Radial Flow

In peripheral collisions, hydrodynamic features should disappear since the mean free path becomes comparable to the root mean square radius . Because ideal hydrodynamics is scale invariant, the hydrodynamic stage of the model does not capture finite size effects which become increasingly important at larger impact parameters. However, this does not mean that the radial velocity is independent of impact parameter. The hydrodynamic lifetime scales approximately as ; the flow velocity reflects this lifetime. Finite size effects make the total lifetime decrease more quickly than . Finite size effects are most important during the freezeout stage which is modeled with RQMD and therefore the hydro+cascade approach can capture some non-trivial features of the impact parameter dependence.

In the low region, pion-nucleon scattering in RQMD is largely responsible for the anti-proton slope measured by the STAR collaboration STAR-Spectra . In the high region measured by the PHENIX collaboration PHENIX-Spectra , the hydrodynamic stage of the model is much more significant than hadronic rescattering. Fig. 20 shows the low and high slope parameters as a function of the number of participants in the collision. For each particle, the open symbols show the slope parameter without RQMD and the closed symbols show the slope parameter with RQMD. For both pions and kaons in the low and high regions, only a small impact parameter dependence is observed. The pion spectrum is cooled at all impact parameters.

Compare the nucleon and the slope parameters. In the low region, shown in Fig. 20(a), the nucleon slope exhibits a very rapid dependence on the number of participants. Approximately 40% of this slope is due to RQMD and the “Hydro Only” curve for the nucleon is relatively flat as a function of impact parameter. The nucleon and the have approximately the same mass but the is devoid of the strong resonance which drives the flow in the nucleon system. Therefore, the “Hydro Only” curve for nucleons is similar to the “Hydro+RQMD” curve for the .

Next, compare the nucleon and the slopes in the high region, shown in Fig. 20(b). Here is less significant and RQMD is responsible for only of the nucleon slope. Consequently, RQMD makes up the small difference between the nucleon and the “Hydro Only” curves, and the final slope parameters of the two particles are similar in shape and magnitude. Experimentally, the difference in slope parameters between the nucleons and the can be used to assess the contribution of the hadronic phase in the low region.

Turning to the experimental data, we first examine the low region as measured by the STAR collaboration. We compare the model b-dependence of the anti-proton and slope parameters to the experimental data in Fig. 21 and 22. Comparing the anti-proton slopes for different EOS, we see that the rapid b-dependence is not a consequence of a change in EOS. Note that LH has a larger slope at small than LH8. This is an artifact of the exponential fit. The spectrum for LH in central collisions is shown in Fig 18 and is not exponential. The measured spectra are well described by an exponential in this region STAR-Spectra . Coarsely, the model reproduces the strong b-dependence of the anti-protons and the weaker b-dependence of the kaons. However, for more peripheral collisions, the measured slope parameters fall somewhat faster than the model predicts. Naively, this indicates that in the most peripheral collisions the hydrodynamic description is only beginning to work and finite size effects (e.g viscosity) should be taken into account in the QGP and mixed phases which are modeled with the hydrodynamics. The “Hydro Only” curves presented in Fig. 21 and 22 definitely do not reproduce the strong b-dependence of the slopes. With RQMD on the other hand, finite size effects during the late hadronic stages are modeled and most of the rapid b-dependence seen in the data is reproduced.

Turning to the high region, the absolutely normalized model spectra are compared directly to the absolutely normalized PHENIX spectra for different centralities and particles in Fig. 23.

For and the spectrum is reproduced in the most central bin (0-5% central with ) and in the semi-peripheral bin (30-60% central with ). For pions, the spectral shape in the most central bin is reproduced. However, in the semi-peripheral bin, the pion spectrum resembles a power-law rather than a thermal spectrum. This change in shape from peripheral to central has been attributed to jet-quenching Wang-JetQuenching ; GDavid-JetQuenching .

## V Elliptic Flow from the SPS to RHIC

In non-central collisions the particles emerge with an elliptic flow. The spectator matter flies down the beam pipe and the excited nuclear matter is formed in the transverse plane with an almond shaped distribution. Subsequently, if pressure develops in the system, the pressure gradients are larger in the impact parameter direction (the x-direction) than in the y-direction. Then, the excited matter expands preferentially in the x-direction. The magnitude of this elliptic response is quantified experimentally by expanding the distributions in a Fourier series

(27) | |||||

where is measured around the z-axis relative to the impact parameter, which points in the x direction. The flow, , gives a measure of the dynamic response of the excited nuclear matter to the initial anisotropy.

The initial spatial anisotropy is quantified using the parameter (see Eq. 9). The hydrodynamic response is linear in Ollitrault-Elliptic and therefore is sometimes divided by to compare different impact parameters and nuclei Sorge-kink ; Voloshin-LowDensity . As the system expands, the eccentricity decreases. Since is the driving force behind the elliptic flow, the elliptic development finishes before the radial development. Therefore, elliptic flow is generated by the early pressure, although this statement must be qualified (see below). The spatial anisotropy that remains after the collision is quantified by (see Eq. 20). measures how much of the initial spatial anisotropy was not used during the collision for the production of elliptic flow.

### v.1 Qualitative Changes from the SPS to RHIC

In Sect. IV.1 radial flow was used to constrain the EOS. The best (though certainly not unique) description of the data was given by LH8. For consistency, we require the same EOS to describe the elliptic flow data.

Fig. 24(a) shows integrated elliptic flow of pions

as a function of the total multiplicity for different EOS. Fig. 24(b) shows the relative contribution of RQMD to the integrated elliptic flow. Note that elliptic flow increases for all EOS and dramatically so for LH8. Assume momentarily that elliptic flow for Hydro+RQMD stops developing at a temperature of . (Note however that the radial flow develops well below this temperature). The dramatic increase of elliptic flow in Fig. 24(a) can be understood as the dynamic response of the QGP pressure. Recall Fig. 5(b), which plots anisotropy of the hydrodynamic stress tensor versus time and pay particular attention to the points (the solid symbols) on the LH8 and RG curves (ignore LH for now). At the SPS, the anisotropy of the stress tensor increases rapidly at first and then stalls. The final stress tensor anisotropy is small at the end of the mixed phase. At RHIC, elliptic flow develops more rapidly and stalls only when the anisotropy is large. Thus, provided the elliptic flow stops developing at , the elliptic flow increases dramatically as the QGP pressure appears. For a RG EOS at the SPS, there is no mixed phase and no stall and consequently the RG elliptic flow is significantly larger than LH8 and the data. However with RHIC collision energies, LH8 begins to behave as an ideal QGP. Consequently, at RHIC the RG elliptic flow is only 20% larger than that of LH8 and of the data.

To understand when elliptic flow stops developing, it is important to track the spatial geometry of the underlying source. When the spatial anisotropy, , is negative, the pressure drives elliptic flow. However, as approaches zero, the pressure gradients drive radial motion rather than elliptic motion. The elliptic development then stops. Fig. 24(c) shows the spatial anisotropy, , for pions as a function of multiplicity from the SPS to RHIC. Compare the LH8 curves (the stars, the solid squares, and the open squares) seen in Fig. 24(c). The open squares (LH8 Hydro Only) show the spatial anisotropy, , at the end of the mixed phase, the closed squares (LH8 Hydro+RQMD) show after the cascade, and the stars show when the hydrodynamic evolution is continued to . After the mixed phase (LH8 Hydro Only), the matter retains some of its initial almond shape. Continuing the hydrodynamics destroys the initial almond shape completely and increases by a factor of (see Fig. 24(c)). Cascading also changes the almond shape but increases